Applications of optimal portfolio management by Dimitrios Bisias
Submitted to the Sloan School of Management in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2015 c Massachusetts Institute of Technology 2015. All rights reserved.
Author...... Sloan School of Management June 22, 2015
Certified by...... Andrew W. Lo Charles E. and Susan T. Harris Professor of Finance Thesis Supervisor
Accepted by...... Patrick Jaillet Dugald C. Jackson Professor, Department of Electrical Engineering and Computer Science Co-director, Operations Research Center 2 Applications of optimal portfolio management by Dimitrios Bisias
Submitted to the Sloan School of Management on June 22, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research
Abstract
This thesis revolves around applications of optimal portfolio theory. In the first essay, we study the optimal portfolio allocation among convergence trades and mean reversion trading strategies for a risk averse investor who faces Value- at-Risk and collateral constraints with and without fear of model misspecification. We investigate the properties of the optimal trading strategy, when the investor fully trusts his model dynamics. Subsequently, we investigate how the optimal trading strategy of the investor changes when he mistrusts the model. In particular, we assume that the investor believes that the data will come from an unknown member of a set of unspecified alternative models near his approximating model. The investor believes that his model is a pretty good approximation in the sense that the relative entropy of the alternative models with respect to his nominal model is small. Concern about model misspecification leads the investor to choose a robust optimal portfolio allocation that works well over that set of alternative models. In the second essay, we study how portfolio theory can be used as a framework for making biomedical funding allocation decisions focusing on the National Institutes of Health (NIH). Prioritizing research efforts is analogous to managing an invest- ment portfolio. In both cases, there are competing opportunities to invest limited resources, and expected returns, risk, correlations, and the cost of lost opportunities are important factors in determining the return of those investments. Can we apply portfolio theory as a systematic framework of making biomedical funding allocation decisions? Does NIH manage its research risk in an efficient way? What are the challenges and limitations of portfolio theory as a way of making biomedical funding allocation decisions? Finally in the third essay, we investigate how risk constraints in portfolio opti- mization and fear of model misspecification affect the statistical properties of the market returns. Risk sensitive regulation has become the cornerstone of international financial regulations. How does this kind of regulation affect the statistical properties of the financial market? Does it affect the risk premium of the market? What about the volatility or the liquidity of the market?
3 Thesis Supervisor: Andrew W. Lo Title: Charles E. and Susan T. Harris Professor of Finance
4 Acknowledgments
I would like to express my gratitude to my advisor and mentor, Professor Andrew W. Lo, for his continuing support and advice over all the years I spent at MIT. His immense knowledge in diverse research areas, enthusiasm, hard work, outstanding leadership and motivation have been a source of inspiration. Working with him has been an honor and privilege and I could not have imagined having a better advisor and mentor for my Ph.D study. I would also like to thank the rest of my thesis committee: Professor Dimitri P. Bertsekas for comments that greatly improved this thesis and for his great books that made me love the field of optimization in the first place and Professor Leonid Kogan who provided his insight and expertise that greaty assisted this research. In addition I would like to thank Dr. James F. Watkins, MD for his invaluable help, insights and contribution to the second part of this research. Moreover, I would like to thank Dr. Paul Mende, Dr. Saman Majd and Dr. Eric Rosenfeld whom I had the fortune of being their teaching assistant in finance classes. Paul’s experience in quantitative trading made me realize what career I would like to follow and I am grateful for this. Being part of MIT and in particular the ORC and LFE communities has been a blessing and I consider myself very fortunate to be among very interesting and smart people. I will always remember my years at MIT with nostalgia and joy and I hope that I ’ll be able to express my gratitude in the future several times. My life at MIT would not be so complete and joyful if I didn’t have good lifelong friends to spend time and have productive discussions with. In particular, I would like to thank Nick Trichakis and his wife Lena, Christos and Elli Nicolaides, Markos and Sophia Trichas, Thomas and Anastasia Trikalinos, the golden coach George Pa- pachristoudis and Gerry Tsoukalas. Last but not least I would like to thank my parents Giorgo and Roula and my sister Katerina for their unconditional love and support. I owe to them everything and this thesis is dedicated to them.
5 6 Contents
1 Optimal trading of arbitrage opportunities under constraints 29 1.1 Literaturereview ...... 31 1.2 Analysis ...... 32 1.2.1 Models...... 33 1.2.2 Constraints ...... 34 1.2.3 Solution ...... 36 1.2.4 Connection with Ridge and Lasso regression ...... 46 1.3 Results...... 47 1.3.1 Convergencetrades ...... 47 1.3.2 Mean reversion trading opportunities ...... 56 1.4 Conclusions ...... 56
2 Optimal trading of arbitrage opportunities under model misspecifi- cation 57 2.1 Literaturereview ...... 59 2.2 Analysis ...... 60 2.2.1 Alternative models representation ...... 61 2.2.2 Modelsetup...... 63 2.3 Solution ...... 65 2.3.1 Nofearofmodelmisspecification ...... 65 2.3.2 Fear of model misspecification no constraints ...... 67 2.3.3 Fear of model misspecification with VaR and margin constraints 70 2.4 Results...... 72
7 2.4.1 Convergencetradeswithoutconstraints ...... 73 2.4.2 Mean reversion trading strategies without constraints . . . . . 78 2.4.3 Convergencetradeswithconstraints...... 92 2.4.4 Mean reversion trading strategies with constraints ...... 111 2.5 Conclusions ...... 129
3 Estimating the NIH Efficient Frontier 131 3.1 NIHBackgroundandLiteratureReview ...... 132 3.2 Methods...... 136 3.2.1 FundingData ...... 136 3.2.2 BurdenofDiseaseData ...... 139 3.2.3 Applying Portfolio Theory ...... 142 3.3 Results...... 147 3.3.1 SummaryStatistics...... 147 3.3.2 EfficientFrontiers...... 148 3.4 Discussion...... 151
4 Impact of model misspecification and risk constraints on market 157 4.1 Literaturereview ...... 158 4.2 Analysis ...... 159 4.2.1 Modelsetup...... 159 4.2.2 Varyingconstraints ...... 161 4.2.3 Varying risk aversions ...... 165 4.2.4 Varying constraints and risk aversions ...... 168 4.2.5 Varying fear of model misspecification ...... 168 4.3 Conclusions ...... 170
A Technical Notes 173
8 List of Figures
1-1 Ellipsoids. Ellipsoids of poor investment opportunities for N=2 con- vergence trades at times t =0.3, 0.6, 0.9...... 42
1-2 Weights for the case of uncorrelated spreads and collateral constraint. Weights for the case of uncorrelated spreads...... 45
1-3 VaR constraints, positive correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing VaR constraints (K=1). Initialwealthis$100...... 48
1-4 VaR constraints, negative correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively cor- related (ρ = −0.5) convergence trades, while facing VaR constraints (K=1).Initialwealthis$100...... 48
1-5 VaR constraints, positive correlations, tight constraints. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100...... 49
1-6 VaR constraints, negative correlations, tight constraints. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100...... 49
9 1-7 Wealth evolution under VaR constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under theVaRconstraint. Initialwealthis$100...... 50
1-8 Relation between final wealth and frequency the VaR con- straint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence tradesarelow...... 50
1-9 Margin constraints, positive correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively cor- related (ρ = 0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100...... 52
1-10 Margin constraints, negative correlations. Wealth distribution at t =0.25, 0.5, 0.75, 1 for an investor who invests in two negatively cor- related (ρ = −0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100...... 52
1-11 Margin constraints, positive correlations, more collateral needed. Wealth distribution at t =0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ =0.5) convergence trades, while facing margin constraints (Collateral = 2). Initial wealth is $100...... 53
1-12 Margin constraints, negative correlations, more collateral needed. Wealth distribution at t =0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while fac- ing margin constraints (Collateral = 2). Initial wealth is $100. . . . . 53
1-13 Wealth evolution under margin constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under the margin constraint. Initial wealth is $100...... 54
10 1-14 Relation between final wealth and frequency the margin con- straint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence tradesarelow...... 54
1-15 Positions evolution under VaR constraints. Typical path of the positions in two convergence trading opportunities under VaR con- straints...... 55
1-16 Positions evolution under margin constraints. Typical path of the positions in two convergence trading opportunities under margin constraints...... 55
2-1 Partial derivative of the value function with respect to S for
a single convergence trade. VS as a function of time at S = 1 for different values of the robustness multiplier for a single convergence trade...... 76
2-2 Distortion drift for a single convergence trade. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier for a single convergence trade...... 76
2-3 Distortion drift terms for a single convergence trade. Distor- tion drift terms as a function of time at S = 1 for ν = 1 for a single convergence trade. The first term corresponds to a positive distor- tion drift that reduces the wealth of the investor since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. . . .. 77
2-4 Optimal weight of a single convergence trade. Weight of the convergence trading strategy as a function of time at S = 1 for different valuesoftherobustnessmultiplier...... 77
2-5 Partial derivative of the value function with respect to S for
a single mean reversion trading strategy. VS as a function of time at S = 1 for different values of the robustness multiplier. . . . . 80
11 2-6 Distortion drift for a single mean reversion trading strategy. Distortion drift as a function of time at S = 1 for different values of therobustnessmultiplier...... 81
2-7 Distortion drift terms for a single mean reversion trading strategy. Distortion drift terms as a function of time at S = 1 for ν = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor, since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities...... 81
2-8 Optimal weight of a single mean reversion trading strategy. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier...... 82
2-9 Optimal weights of two uncorrelated mean reversion trading
strategies for S1 = 1 and S2 = 2. Weights of the mean reversion
trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0...... 84
2-10 Ratio of the optimal weights. Ratio of the optimal weights of the
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0...... 84
2-11 Partial derivative of the value function with respect to S1 and
S2 at S1 =1 and S2 =2 when ρ =0. Partial derivative of the value
function with respect to S1 and S2 as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0...... 85
12 2-12 Optimal weights of two positively correlated mean reversion
trading strategies for S1 = 1 and S2 = 2. Weights of the mean
reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0.5...... 85
2-13 Optimal weights of two negatively correlated mean reversion
trading strategies for S1 = 1 and S2 = 2. Weights of the mean
reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5...... 86
2-14 Optimal weights of two uncorrelated mean reversion trading
strategies for S1 = 1 and S2 = 1. Weights of the mean reversion
trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0...... 88
2-15 Ratio of the optimal weights at S1 =1 and S2 =1 when ρ =0. Ratio of the optimal weights of the mean reversion trading strategies
as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0...... 88
2-16 Partial derivative of the value function with respect to S1 and
S2 at S1 =1 and S2 =1 when ρ =0. Partial derivative of the value
function with respect to S1 and S2 as a function of time at S1 = 1 and
S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0...... 89
2-17 Optimal weights of two positively correlated mean reversion
trading strategies for S1 = 1 and S2 = 1. Weights of the mean
reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9...... 89
13 2-18 Partial derivative of the value function with respect to S1 and
S2 at S1 = 1 and S2 = 1 when ρ = 0.9. Partial derivative of
the value function with respect to S1 and S2 as a function of time at
S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9...... 90
2-19 Ratio of the optimal weights at S1 =1 and S2 =1 when ρ =0.9. Ratio of the magnitude of the optimal weights of the mean reversion
trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9...... 90
2-20 Optimal weights of two negatively correlated mean reversion
trading strategies for S1 = 1 and S2 = 1. Weights of the mean
reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.8...... 91
2-21 Partial derivative of the value function with respect to S for
a single convergence trade when L = 0.1 and L = 100. VS as a function of time at S = 1 for different values of the robustness multiplier. The solid line is when L = 100 and the dotted line is for L =0.1...... 94
2-22 Partial derivative of the value function with respect to S for
a single convergence trade when L = 0.1. VS as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 0.1...... 95
2-23 Optimal weight of a single convergence trade when L = 0.1. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 0.1...... 95
14 2-24 Optimal weight of a single convergence trade when L = 1. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 1...... 96
2-25 Distortion drift for a single convergence trade when L = 0.1. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 0.1. ... 96
2-26 Distortion drift for a single convergence trade when L = 1. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 1. .... 97
2-27 Distortion drift terms for a single convergence trade when L = 0.1. Distortion drift terms as a function of time at S = 1 for ν = 1 and L =0.1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor and it is bounded above due to the collateral constraint, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. 97
2-28 Distortion drift terms for a single convergence trade when L =1. Distortion drift terms as a function of time at S = 1 for ν =1 and L = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor and it is bounded above due to the collateral constraint, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. 98
2-29 Optimal weight of a single convergence trade when L =0.1 and L = 100. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The solid line is when L = 100 and the dotted line is for L =0.1...... 98
15 2-30 Optimal weights of two uncorrelated convergence trades for
S1 =1 and S2 =2 when L =0.5. Weights of the convergence trades
as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.5...... 100
2-31 Value of the normalized wealth variance for two uncorrelated
convergence trades at S1 = 1 and S2 = 2 when L = 0.5. Value of the normalized wealth variance for two uncorrelated convergence
trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.5...... 101
2-32 Optimal weights of two uncorrelated convergence trades for
S1 =1 and S2 =2 when L =0.05. Weights of the convergence trades
as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05...... 101
2-33 Value of the normalized wealth variance for two uncorrelated
convergence trades at S1 = 1 and S2 = 2 when L = 0.05. Value of the normalized wealth variance for two uncorrelated convergence
trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05...... 102
2-34 Optimal weights of two positively correlated convergence trades
for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence
trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0.5 and the rhs of the VaR constraint is L =0.05...... 103
16 2-35 Value of the normalized wealth variance for two positively cor-
related convergence trades at S1 =1 and S2 =2 when L =0.05. Value of the normalized wealth variance for two positively correlated
convergence trades as a function of time at S1 = 1 and S2 = 2 for dif- ferent values of the robustness multiplier. The correlation coefficient is ρ =0.5 and the rhs of the VaR constraint is L =0.05...... 104
2-36 Optimal weights of two negatively correlated convergence trades
for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence
trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =0.05...... 104
2-37 Value of the normalized wealth variance for two negatively
correlated convergence trades at S1 = 1 and S2 = 2 when L = 0.05. Value of the normalized wealth variance for two nega-
tively correlated convergence trades as a function of time at S1 = 1
and S2 = 2 for different values of the robustness multiplier. The cor- relation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =0.05...... 105
2-38 Optimal weights of two uncorrelated convergence trades for
S1 =1 and S2 =1 when L =0.05. Weights of the convergence trades
as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05...... 107
2-39 Value of the normalized wealth variance for two uncorrelated
convergence trades at S1 = 1 and S2 = 1 when L = 0.05. Value of the normalized wealth variance for two uncorrelated convergence
trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05...... 107
17 2-40 Optimal weights of two positively correlated convergence trades
for S1 = 1 and S2 = 1 when L = 0.05. Weights of the convergence
trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.8 and the rhs of the VaR constraint is L =0.05...... 108
2-41 Value of the normalized wealth variance for two positively cor-
related convergence trades at S1 =1 and S2 =1 when L =0.05. Value of the normalized wealth variance for two positively correlated
convergence trades as a function of time at S1 = 1 and S2 = 1 for dif- ferent values of the robustness multiplier. The correlation coefficient is ρ =0.8 and the rhs of the VaR constraint is L =0.05...... 109
2-42 Optimal weights of two negatively correlated convergence trades
for S1 = 1 and S2 = 1 when L = 8. Weights of the convergence
trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.8 and the rhs of the VaR constraint is L =0.05...... 109
2-43 Value of the normalized wealth variance for two negatively
correlated convergence trades at S1 = 1 and S2 = 1 when L = 0.05. Value of the normalized wealth variance for two nega-
tively correlated convergence trades as a function of time at S1 = 1
and S2 = 1 for different values of the robustness multiplier. The cor- relation coefficient is ρ = −0.8 and the rhs of the VaR constraint is L =0.05...... 110
2-44 Partial derivative of the value function with respect to S for a single mean reversion trading strategy and a collateral con-
straint with L =0.7. VS as a function of time at S = 1 for different values of the robustness multiplier for L =0.7...... 113
18 2-45 Distortion drift terms for a single mean reversion trading strategy and a collateral constraint with L = 0.7. Distortion drift terms as a function of time at S = 1 for ν = 2 and for L = 0.7. The first term corresponds to a positive distortion drift that reduces the wealth of the investor, since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. The first term is bounded aboveduetothecollateralconstraint...... 113
2-46 Optimal weight of a single mean reversion trading strategy with a collateral constraint with L = 0.7. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier and for L =0.7...... 114
2-47 Partial derivative of the value function with respect to S for a single mean reversion trading strategy with different collat-
eral constraints. VS as a function of time at S = 1 for different values of the robustness multiplier and different collateral constraints. The solid line is for L = 70 and the dotted line for L =0.7...... 114
2-48 Optimal weight of a single mean reversion trading strategy with different collateral constraints. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier and different collateral constraints. The solid line is for L = 70 and the dotted line for L =0.7...... 115
2-49 Optimal weights of two uncorrelated mean reversion trading
strategies for S1 = 1 and S2 = 2 when L = 3. Weights of the
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =3. .... 117
19 2-50 Value of the normalized wealth variance for two uncorrelated
mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 3. Value of the normalized wealth variance for two uncorrelated
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =3. .... 118
2-51 Optimal weights of two uncorrelated mean reversion trading
strategies for S1 = 1 and S2 = 2 when L = 2. Weights of the
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =2. .... 118
2-52 Value of the normalized wealth variance for two uncorrelated
mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 2. Value of the normalized wealth variance for two uncorrelated
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =2. .... 119
2-53 Optimal weights of two uncorrelated mean reversion trading
strategies for S1 = 1 and S2 = 2 when L = 7. Weights of the
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =7. .... 119
2-54 Value of the normalized wealth variance for two uncorrelated
mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 7. Value of the normalized wealth variance for two uncorrelated
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =7. .... 120
20 2-55 Optimal weights of two positively correlated mean reversion
trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights
of the mean reversion trading strategies as a function of time at S1 =
1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.5 and the rhs of the VaR constraint is L =7...... 121
2-56 Value of the normalized wealth variance for two positively
correlated mean reversion trading strategies at S1 = 1 and
S2 =2 when L =7. Value of the normalized wealth variance for two positively correlated mean reversion trading strategies as a function
of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.5 and the rhs of the VaR constraint is L =7...... 122
2-57 Optimal weights of two negatively correlated mean reversion
trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights
of the mean reversion trading strategies as a function of time at S1 =
1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =7...... 122
2-58 Value of the normalized wealth variance for two negatively
correlated mean reversion trading strategies at S1 = 1 and
S2 =2 when L =7. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function
of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =7...... 123
21 2-59 Optimal weights of two uncorrelated mean reversion trading
strategies for S1 = 1 and S2 = 1 when L = 2. Weights of the
mean reversion trading strategies as a function of time at S1 = 1 and
S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =2. .... 125
2-60 Value of the normalized wealth variance for two uncorrelated
mean reversion trading strategies at S1 = 1 and S2 = 1 when L = 2. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function of time at
S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =2...... 125
2-61 Optimal weights of two positively correlated mean reversion
trading strategies for S1 = 1 and S2 = 1 when L = 2. Weights
of the mean reversion trading strategies as a function of time at S1 =
1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.9 and the rhs of the VaR constraint is L =2...... 126
2-62 Value of the normalized wealth variance for two positively
correlated mean reversion trading strategies at S1 = 1 and
S2 =1 when L =2. Value of the normalized wealth variance for two positively correlated mean reversion trading strategies as a function
of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.9 and the rhs of the VaR constraint is L =2...... 127
22 2-63 Optimal weights of two negatively correlated mean reversion
trading strategies for S1 = 1 and S2 = 1 when L = 8. Weights
of the mean reversion trading strategies as a function of time at S1 =
1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =8...... 127
2-64 Value of the normalized wealth variance for two negatively
correlated mean reversion trading strategies at S1 = 1 and
S2 =1 when L =8. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function
of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =8...... 128
3-1 NIH time series flowchart. Flowchart for the construction of NIH appropriations time series. “NIH Approp.” denotes NIH appropria- tions; “PHS Gaps” denotes Institute funding by the U.S. Public Health Service; “Complete Approp.” denotes the union of these two series; “FY Change” allows for the change in government fiscal years; “4Q FY” time series refers to the resulting series in which all years are treated as having four quarters of three months each...... 138
3-2 Appropriations data. NIH appropriations in real (2005) dollars, categorizedbydiseasegroup...... 138
3-3 YLL time series flowchart. Flowchart for the construction of years of life lost (YLL) time series. “WONDER Chapter Age Group” refers to a query to the CDC WONDER database at the chapter level, strati- fied by age group at death; “US Pop.” is the United States population from census data as expressed in the WONDER dataset; and “US GDP”denotesU.S.grossdomesticproduct...... 140
23 3-4 YLL data. Panel (a): Raw YLL categorized by disease group. Panel (b): Population-normalized YLL (with base year of 2005), categorized by disease group. Both panels are based on data from 1979 to 2007. 141
3-5 Efficient frontiers. Efficient frontiers for (a) all groups except HIV and AMS, γ = 0; (b) all groups except HIV and AMS, γ = 5; (c) all groups except HIV and AMS without the dementia effect, γ = 0; and (d) all groups except HIV and AMS without the dementia effect, γ =5; basedonhistoricalROIfrom1980to2003...... 148
4-1 Price of the risky asset as a function of the aggregate market supply under varying constraints. We assume that we have 5 agents with the same risk aversion coefficients. The red plot assumes the same L = 30 for all the agents, while the blue assumes L to be
different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 =50. 163
4-2 Price of the risky asset as a function of the aggregate market supply under tightening constraints. We assume that we have 5 agents with the same risk aversion coefficients. The blue plot assumes
L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 =
40, L5 = 50 and the red assumes that each Li is reduced by 20%. . . . 164
4-3 Price of the risky asset as a function of the aggregate market supply with less variable constraints. We assume that we have 5 agents with the same risk aversion coefficients. The blue plot assumes
L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 =
40, L5 = 50 and the red assumes that L1 = 20, L2 = 25, L3 = 30, L4 =
35, L5 =40...... 164
4-4 Price of the risky asset as a function of the aggregate market supply with constraints and varying risk aversions. We assume that we have 5 agents with same constraints but different risk aversion coefficients. The blue plot assumes L = 30 for each agent, while the red line assumes that the agents are unconstrained...... 166
24 4-5 Price of the risky asset as a function of the aggregate market supply with tightening constraints and varying risk aversions. We assume that we have 5 agents with same constraints but different risk aversion coefficients. The blue plot assumes L = 30 for each agent, while the red line assumes that L =20foreachagent...... 167
25 26 List of Tables
3.1 IoM recommendations. 12 major recommendations of the 1998 Institute of Medicine panel in four large areas for improving the process ofallocatingresearchfunds...... 133 3.2 ICD mapping. Classification of ICD-9 (1978–1998) and ICD-10 (1999– 2007) Chapters and NIH appropriations by Institute and Center to 7 disease groups: oncology (ONC); heart lung and blood (HLB); diges- tive, renal and endocrine (DDK); central nervous system and sensory (CNS) into which we placed dementia and unspecified psychoses to create comparable series as there was a clear, ongoing migration noted from NMH to CNS after the change to ICD-10 in 1999; psychiatric and substance abuse (NMH); infectious disease, subdivided into estimated HIV (HIV) and other (AID); maternal, fetal, congenital and pediatric (CHD). The categories LAB and EXT are omitted from our analysis. 137 3.3 Return summary statistics. Summary statistics for the ROI of disease groups, in units of years (for the lag length) and per-capita- GDP-denominated reductions in YLL between years t and t +4 per dollar of research funding in year t−q, based on historical ROI from 1980to2003...... 147 3.4 ROI example. An example of the ROI calculation for HLB from 1986. 147 3.5 Portfolio weights. Benchmark, single- and dual-objective optimal portfolio weights (in percent), based on historical ROI from 1980 to 2003...... 150
27 28 Chapter 1
Optimal trading of arbitrage opportunities under constraints
In financial economics, an arbitrage is an investment opportunity that is too good to be true when there are no market frictions. In actual financial markets however, there are frictions and even if there are arbitrage opportunities the investors may not be able to fully exploit them due to the constraints they face. We will explore two kinds of risky arbitrage opportunities when there are market frictions. The first one is a case of a textbook arbitrage, a convergence trade strategy. The second is a case of a statistical arbitrage, a mean reversion trading strategy. These two strategies are two of the most popular trading strategies that hedge funds follow, so studying them in detail when there are market frictions is a valuable exercise. A convergence trade is a trading strategy consisting of long/short positions in two similar assets, where we buy the cheap asset, we short the expensive asset and we wait until the prices of the two assets to converge which we know it will happen for sure some particular time in the future. An example of this trade involves the difference in price between the on-the-run and the most recent off-the-run security. An on- the-run security is the most recently issued, and hence most liquid, of a periodically issued security. Since an on-the-run security is more liquid it trades at a premium to off-the-run securities [29]. A convergence trade involves taking a long position in the most recent off-the-run security and shorting the on-the-run security. The on-
29 the-run will become off-the-run upon the issue of a newer security and then there will be almost no difference between the two securities in our trade so their prices will converge. Another example involves investing in Treasury STRIPS with identical maturity dates but different prices. A mean reversion trading strategy involves investing in an asset or a portfolio of assets whose value is a mean reverting process. Since most price series in the equity space follow random walk, this strategy most commonly involves investing in a port- folio of non-mean-reverting assets whose value is a stationary mean reverting series. These price series that can be combined in such a way are called cointegrating. A classic statistical arbitrage example is the pairs trading, which is the first type of algorithmic mean reversion trading strategy invented by institutional traders, report- edly by the trading desk of Nunzio Tartaglia at Morgan Stanley [64]. The statistical arbitrage pairs trading strategy bets on the convergence of the prices of two similar assets whose prices have diverged without a fundamental reason for this. These arbitrage opportunities are risky under market frictions. In particular the first case is exposed to the “divergence risk”, i.e. the fact that the pricing differential between the two similar assets can diverge arbitrarily far from 0 prior to its conver- gence at some particular time in the future. The second case is exposed both to the “divergence risk” and to the “horizon risk”, in other words the fact that the times at which the spread will converge to its long run mean are uncertain. We will explore the optimal portfolio allocation of a risk averse investor who invests in N convergence trades or mean reverting trading strategies, while facing constraints. In particular, we will study the optimal trading strategy when he faces VaR constraints or collateral constraints. Risk sensitive regulation, such as the VaR constraint, has lately become a central component of international financial regu- lations. Collateral or margin constraints, where the investor has to have sufficient wealth to secure the liabilities taken by short positions, have been ubiquitous in the financial transactions for centuries and margin calls have been behind several crises including the LTCM debacle[54]. In the rest of this chapter, we will discuss the relevant literature review. Then we
30 will discuss about the setup of the model and the constraints, we will find the optimal trading strategy of the investor and finally we will explore the characteristics of this optimal strategy.
1.1 Literature review
Merton studied the problem of optimal portfolio allocation in a continuous time set- ting without any market frictions [59]. The optimal portfolio involves two terms: a market timing term and a hedging demand term. The first term is a myopic term that represents the optimal allocation if you were interested at each time instant t only for an horizon dt ahead. The second term represents the investor’s additional demand due to the covariance of the wealth process with the attractiveness of the available investment opportunities. Although Merton gives an analytical general solution this is expressed in terms of the partial derivatives of the value function and additional work is needed to derive the solution in terms of the model parameters. Additionally it assumes that there are no market frictions. Optimal trading of mean reversion trading strategies have been studied by both Boguslavsky and Boguslavskaya [15] and Jurek and Yang [48]. They have found analytical solutions for the optimal weight of a single mean reverting trading strategy for risk averse CRRA investors. Their analysis is similar with the one in Kim and Omberg [49], where they assume that there is a risk free asset with a constant risk- free rate and a single risky asset with a mean reverting risk premium, which implies a mean reverting instantaneous Sharpe ratio. In all the cases they have assumed that there are no market frictions whatsoever. Longstaff and Liu [53] have studied the problem of optimal trading of a single convergence trade under a margin constraint. For the single convergence trade case both VaR constraints and margin constraints collapse in the same constraint and the problem is significantly easier. In addition by studying only convergence trades they have taken out one important dimension of risk, the horizon risk, keeping only the divergence risk. Brennan and Schwarz [17] have also studied the problem of optimal
31 trading of a single convergence trade including transaction costs when the arbitrage potential is restricted by position limits. The literature is rich with papers that study the existence of an equilibrium where there exists mispricings. This persistence of mispricings is typically attributed to agency problems, frictions or some kind of risk. Unlike textbook arbitrages, which generate riskless profits and require no capital commitments, exploiting real-world mispricings requires the assumption of some kind of risk. Shleifer and Vishny [74] emphasized that risks such as the uncertainty about when the pricing differential will converge to 0 and the possibility of a divergence of the mispricing prior to its elimination may play a role in limiting the size of positions that arbitrageurs are willing to take, contributing to the persistence of the arbitrage in equilibrium. Basak and Choitoru [6] also showed that arbitrage can persist in equilibrium when there are frictions. They study dynamic models with log utility and heterogeneous beliefs in the presence of margin requirements and other portfolio constraints. With respect to the constraints, Basak and Shapiro [7] study the problem of opti- mal trading strategy of a risk averse investor who faces finite horizon VaR constraints in a complete markets setting using the martingale representation approach [4]. Here again there are no constraints in the optimal portfolio allocation at each time t but there is only one constraint in the wealth at some finite horizon. Finally, Geanakoplos [33] studies the collateral constraints, how these determine an equilibrium leverage and how this leverage changes over time, the so-called leverage cycles. Let us now discuss about the setup of the model and the constraints and find the optimal trading strategy of the investor.
1.2 Analysis
We assume we have a risk averse investor maximizing the expected continuously compounded rate of return or equivalently the expected logarithm of his final wealth
E(lnWT ). There are two cases to consider. In the first case, the investor can invest in a risk-free asset and N non-redundant convergence trades, modeled as correlated
32 Brownian bridges. In the second case, the investor can invest in a risk-free asset and N non-redundant mean reversion trading strategies, modeled as a multivariate Ornstein-Uhlenbeck (OU) process. The investor faces two kinds of constraints: VaR constraints or collateral constraints. We determine the optimal trading strategy and its characteristics in both cases.
1.2.1 Models
As we mentioned already, a convergence trade is a trading strategy consisting of long/short positions in two similar assets, where we buy the cheap asset, we short the expensive asset and we wait until the prices of the two assets to converge which we know it will happen for sure some time in the future. The spread of the convergence trade can be modeled as a Brownian bridge driven by K Brownian motions, which has the property that the spread will converge to 0 almost surely at some determined time in the future. The stochastic differential equation governing the spread of the trade is given by: aS K dS = − t dt + σ dZ (1.1) t T − t k kt Xk=1 where St is the spread of the trade, a is a parameter controlling the rate of the mean reversion to 0, T is the horizon of the investor which is also the time at which the
K spread goes to 0 with probability 1 and Zt is a Brownian motion in R . We can see that the reversion to 0 grows stronger as t → T . Therefore, the investment opportunities get better as the spread gets larger and t → T , since then the drift term pushing the spread towards 0 gets larger. A mean reversion trading strategy involves investing in a stationary portfolio of non-mean reverting assets, whose value is a mean reverting process. The value of the portfolio can be modeled as an Ornstein-Uhlenbeck (OU) process. The stochastic differential equation governing it is given by:
K
dSt = −φ(St − S¯)dt + σkdZkt Xk=1 33 In our case we have N of these mean reverting processes and we assume that they are modeled as a multivariate Ornstein-Uhlenbeck process, which is defined by the following stochastic differential equation:
¯ dSt = −Φ(St − S)dt + σdZt (1.2)
Above Φ is a N-by-N square transition matrix that characterizes the deterministic portion of the evolution of the process, S¯ is the vector representing the unconditional mean of the process, σ is a N-by-K matrix that drives the dispersion of the process
K and Zt is a Brownian motion in R . The Ornstein-Uhlenbeck process has the nice property that its conditional distri- bution is normal at all times, with mean equal to
−Φτ Et[St+τ ]= S¯ + e (St − S¯)
and covariance matrix independent of St [60]. We assume that Φ has eigenvalues with positive real part, so that the conditional expectation approaches to S¯ as t →∞. The Ornstein-Uhlenbeck process captures the two important dimensions of risk in all relative value trades: the “horizon risk”, in other words the fact that the times at which the spread will converge to its long run mean are uncertain and the “divergence risk”, i.e. the fact that the pricing differential can diverge arbitrarily far from its long run mean prior to its convergence. The Brownian bridge captures only the “divergence” risk, since by its definition we assume that the investor has perfect information about the magnitude of the mispricing at some future date T , i.e. we assume that the date T on which the mispricing will be eliminated is known ahead with certainty.
1.2.2 Constraints
We consider two kinds of constraints: VaR and collateral constraints. The VaR constraint is a widely used statistical risk measure, adopted both by the regulators
34 and the private sector. It is the cornerstone of the capital regulations adopted by Basel regulations. Both the 1996 market risk amendment of the original 1988 Basel accord and the Basel II regulations have been built on the notion of Value-at-Risk [47]. The Value at risk (VaR) at α-level is defined as the threshold value such that the probability of losses greater than the threshold is less than α. In our case we consider instantaneous VaR constraints which amount for determining an upper bound in the wealth volatility, since locally the diffusion processes have normal distributions. Therefore, the instantaneous VaR constraints are given by:
θT Σθ ≤ LW 2 where θ is a N by 1 vector of positions, Σ is the instantaneous covariance matrix of the spreads, L is some proportionality constant that determines the tightness of the constraint and W is the investor’s wealth. Collateral or margin constraints have been ubiquitous in the financial transactions for centuries. Even Shakespeare in the “Merchant of Venice” points out the importance of the collateral, as Shylock charged Antonio no interest rate but he asked for a pound of flesh as a collateral. The collateral constraints provide protection against mark-to-market losses whenever an investor generates a liability by shorting an asset. Therefore, they require that the investor’s wealth is bounded below by the collateral necessary to secure the liabilities. They are given by:
N
λi|θi| ≤ W i=1 X where λi is the collateral necessary to secure the liability in spread i. In our work, each unit of arbitrage should be understood as being relative to a fixed face or notional amount and therefore each λi is a percentage of this fixed face value or notional amount.
35 1.2.3 Solution
Let us now find the optimal trading strategy of a risk averse investor who maximizes
the expected logarithm of his final wealth E(lnWT ). We consider two cases:
• The investor invests in the risk free asset and in N correlated convergence trades.
• The investor invests in the risk free asset and in N correlated mean reversion trading strategies.
For both cases our analysis is similar. For both cases we have:
N
Wt = θitSit + θ0tB0t ∀t ∈ [0, T ] (1.3) i=1 X
where θit is the investor’s position in opportunity i for i = 1, ··· , N, θ0t is the in-
vestor’s position in the risk free asset, Sit is the spread of the convergence trade or
the value of the mean reverting portfolio and B0t is the price of the risk free asset.
The process θt is adapted to the filtration generated by the Brownian motion Zt.
The investor solves the following problem:
maximizeθ∈Θ E(lnWT ) subject to (1.4) N dWt = i=1 θitdSit + θ0tdB0t
dSt = µP(S, t)dt + σ(S, t)dZt
where Θ is the set of admissible trading strategies. Let us first define ∀t ∈ [0, T ] Ft = N θt/Wt ∈ R .
36 For the convergence trades case, investor’s wealth satisfies the following stochastic differential equation:
N a dW = W rdt + θ S (− i − r)dt + θT σdZ t t it it T − t t i=1 X dW N a t = rdt + F S (− i − r)dt + F T σdZ W it it T − t t t i=1 X
By applying Ito’s Lemma we have that:
N a d(ln(W )) = rdt + F S (− i − r)dt − 1/2F T ΣF dt + F T σdZ t it it T − t t t t t i=1 X Therefore it is:
T ln(WT ) = ln(Wt)+ rs ds Zt T N a 1 + F S (− i − r ) − F T ΣF ds is is T − t s 2 s s Zt i=1 ! T X T + Fs σdZs (1.5) Zt Assuming constant interest rate, we have:
Et(ln(WT )) = ln(Wt)+ r(T − t) T N a 1 + E F S (− i − r ) − F T ΣF ds t is is T − t s 2 s s Zt i=1 ! ! T X T + Et( Fs σdZs) (1.6) Zt
37 For the mean reversion trading strategies case, investor’s wealth satisfies the following stochastic differential equation:
N T ¯ T dWt = Wtrdt + θit(−Φi (St − S) − rSit)dt + θ σdZt i=1 X dW N t = rdt + F (−ΦT (S − S¯) − rS )dt + F T σdZ W it i t it t t t i=1 X
where Φi is the i’th row of the transition matrix Φ. By applying Ito’s Lemma we have that:
N T ¯ T T d(ln(Wt)) = rdt + Fit(−Φi (St − S) − rSit)dt − 1/2Ft ΣFtdt + Ft σdZt i=1 X Therefore it is:
T ln(WT ) = ln(Wt)+ rs ds Zt T N 1 + F (−ΦT (S − S¯) − rS ) − F T ΣF ds is i s is 2 s s Zt i=1 ! T X T + Fs σdZs (1.7) Zt Assuming constant interest rate we have:
Et(ln(WT )) = ln(Wt)+ r(T − t) T N 1 + E F (−ΦT (S − S¯) − rS ) − F T ΣF ds t is i s is 2 s s Zt i=1 ! ! T X T + Et( Fs σdZs) (1.8) Zt Under VaR constaints it is: T Ft ΣFt ≤ L< ∞ ∀t
38 Under the margin constraints it is:
N
λi|Fit| ≤ 1 ∀t i=1 X N N T Ft ΣFt = FitFjtσij i=1 j=1 X X N N σ ≤ λ λ |F ||F | ij T T a martingale. Consequently, Et( t Fs σdZs) is equal to 0. Maximizing Et(ln(WT )) is equivalentR to maximizing the third term is equations (1.6), (1.8) for both the cases respectively. Let’s now stydy in detail the solution for both cases for both the constraints. VaR constraint Maximizing Et(ln(WT )) under the VaR constraint is equivalent to solving ∀t the following QCQP: T 1 T minimize Ft µt + Ft ΣFt 2 (1.9) T subject to Ft ΣFt ≤ L where a1 S1t( T −t + r) . µt = . (1.10) aN SNt( + r) T −t for the convergence trades case and T ¯ Φ1 (St − S)+ rS1t . µt = . (1.11) T ¯ ΦN (St − S)+ rSNt for the mean reversion trading strategies case. 39 We can easily solve the problem 1.9 by applying the KKT conditions or by ge- opt opt ometry (see Appendix). Ft ,λt are optimal iff they satisfy the following KKT conditions ([10]): T opt opt • Primal feasibility: Ft ΣFt ≤ L opt • Dual feasibility: λt ≥ 0 opt T opt opt • Complementary slackness: λt (Ft ΣFt − L)=0 opt opt • Minimization of the Lagrangean: Ft = argmin L(Ft,λt ) By solving the KKT conditions (see Appendix for details) we find that: −1 T −1 −Σ µtWt if µt Σ µt ≤ L opt θt = −1 Σ µtWt T −1 − if µ Σ µt ≥ L T Σ−1 t µt µt r L This is equivalently written as: −1 T −1 opt Σ µtWt opt µt Σ µt θt = − opt where 1 + λt = max 1, 1+ λt r L ! Let’s now discuss more the properties of the solution. The investor has logarithmic preferences. Therefore, he is a myopic optimizer - there is no hedging demand [59]. At each time t he looks dt ahead and decides how to trade in an optimal way. There are two cases to consider: T −1 • Case 1: At time t: µt Σ µt ≤ L In this case, the optimal solution is the unconstrained myopic optimal solution, since it satisfies the VaR constraint. For the convergence trades case, this is equivalent to the spread St being in the T −1 a1 aN ellipsoid Et = {S | S (AtΣ At)S ≤ L} where At = diag( T −t + r, ··· , T −t + r). 40 N a1 The volume of the ellipsoid Et is shrinking as t → T , since vol(E)= i=1( T −t + r)−1 det(Σ)vol(B(0, 1)) where B(0, 1) is the unit sphere. Figure 1-1Q shows this shrinkingp ellipsoid at three time instants. For the mean reversion trading strategies case, this is equivalent to the spread or value of the trade being inside the convex set C = {S | (S−S¯)T ((Φ+rI)T Σ−1(Φ+ rI))(S − S¯)+2rS¯T Σ−1(Φ + rI))(S − S¯) ≤ L − r2S¯T Σ−1S¯}, which in the case of r = 0 is the ellipsoid C = {S | (S − S¯)T (ΦT Σ−1Φ)(S − S¯) ≤ L. If S¯ = 0 this convex set is also an ellipsoid. These ellipsoids characterize poor opportunities where the constraints are not active. What constitutes poor investment opportunities changes over time for the case of convergence trades, while it remains invariant for the mean reversion trades case. For the case of convergence trades, the same spreads initially can be considered poor investment opportunities, where the investor does not bind the constraint, he is more conservative, but after some time they can be considered good opportunities and the investor becomes more aggressive and binds the constraint. Informally, when the investment opportunities are poor, the spreads are more likely to widen which then would lead to mark-to-market losses and the investor would not have sufficient wealth to take advantage the better investment oppor- tunities and simultaneously satisfy the VaR constraints. Therefore, the investor is more conservative. T −1 • Case 2: At time t: µt Σ µt > L Now the unconstrained myopic optimal solu- tion does not satisfy the VaR constraint. This case is equivalent to the spread St being outside the shrinking ellipsoid Et for the convergence trades case or the set C for the mean reversion trades case. Now the investment opportunities are good. The investor wants to invest the unconstrained optimal trading strategy, but due to the VaR constraint invests in the proportion of this optimal trading strategy necessary to satisfy the VaR constraint. 41 Ellipsoids of poor investment opportunities for t=0.3, 0.6, 0.9. 0.15 0.1 0.05 0 −0.05 Spread 2 −0.1 −0.15 −0.2 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 Spread 1 Figure 1-1: Ellipsoids. Ellipsoids of poor investment opportunities for N=2 conver- gence trades at times t =0.3, 0.6, 0.9. Margin constraint Maximizing Et(ln(WT )) under the margin constraint is equivalent to solving ∀t the following convex program: T 1 T minimize Ft µt + Ft ΣFt 2 (1.12) N subject to i=1 λi|Fit| ≤ 1 P a1 S1t( T −t + r) . µt = . aN SNt( + r) T −t for the convergence trades case and T ¯ Φ1 (St − S)+ rS1t . µt = . T ¯ ΦN (St − S)+ rSNt 42 for the mean reversion trading strategies case. opt opt Let’s apply the KKT conditions. Ft , νt are optimal iff they satisfy the KKT conditions: N • Primal feasibility: i=1 λi|Fit| ≤ 1 P opt • Dual feasibility: νt ≥ 0 opt N • Complementary slackness: νt ( i=1 λi|Fit| − 1)=0 P opt opt • Minimization of the Lagrangean Ft = argmin L(Ft,λt ) This program cannot be solved analytically in general. Again there are two cases to consider. −1 • Case 1: At time t: kΛΣ µtk1 ≤ 1 where Λ = diag(λ1, ··· ,λN ) In this case, the optimal solution is the unconstrained myopic optimal solution, since it satisfies the margin constraint. −1 For the convergence trades, this is equivalent to having at time t: kΛΣ AtSk1 ≤ a1 aN 1 where Λ = diag(λ1, ··· ,λN ) and At = diag( T −t + r, ··· , T −t + r). In this case we have that St is inside a “diamond” in N dimensional space, which shrinks as t → T . −1 For the mean reversion trades, this is equivalent to having at time t: kΛΣ (Φ(St− S¯)+ rSt)k1 ≤ 1 where Λ = diag(λ1, ··· ,λN ). Informally again, when the investment opportunities are poor, the spreads are more likely to widen which then would lead to mark-to-market losses and the in- vestor would not have sufficient wealth to take advantage the better investment opportunities and have enough wealth for the collateral necessary to secure the liabilities. 43 −1 • Case 2: At time t: kΛΣ µtk1 ≥ 1 where Λ = diag(λ1, ··· ,λN ). Now the investment opportunities are good, the unconstrained myopic optimal solution does not satisfy the collateral constraint and the constraint binds at the optimal solution. Uncorrelated opportunities There is a special case when the trading opportunities are uncorrelated, where we can solve analytically the KKT conditions (see Appendix for details). In that case the optimal positions are given by: µit opt + sign(−µit)(| | − ν ) opt λi t θit = 2 Wt (1.13) σi λi We observe the following: • First of all for the convergence trades, in case the spread is positive we short the spread as we would expect and in case it is negative we are long the spread. For T ¯ the mean reversion trades, the sign is the opposite of the sign of Φi (St−S)+rSit. • Second, if µt is high relative to the collateral then the magnitude of the position is higher. • Third, if the variability of the opportunity is high the magnitude of the corre- sponding position is low. • Finally the more interesting property of the solution is that it has a cutoff value, the dual variable, and if the absolute value of µt over the collateral is greater than the dual variable the position is different from zero otherwise the position is 0. It is: N |λ µ | νopt = 0 if i it ≤ 1 t σ2 i=1 i X 44 and N |λ µ | νopt > 0 if i it > 1 t σ2 i=1 i X The dual variable is 0 when the investment opportunities are poor. It is easy to see that when the margin constraint binds we have: µit opt + sign(−µit)(| | − νt ) ˜ opt λi ˜ Fit = 2 andkF k1 = 1 (1.14) σi 2 λi Weights in different arbitrage opportunities. 1 0.9 0.8 0.7 Weights 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 Figure 1-2: Weights for the case of uncorrelated spreads and collateral con- straint. Weights for the case of uncorrelated spreads. In Figure 1-2 we can see an example of how we invest in different convergence trades when there is no correlation among them, with λ = 1 and volatilities equal to 1 for all the opportunities. The height of each bar is the absolute value of µit and we invest only in those spreads where the µit is larger than the dual variable. N If i=1 |µit| < 1, then the dual variable is 0, we invest in all the opportunities and N theP collateral constraint does not bind. If i=1 |µit| > 1 as is in the figure then the margin constraint binds, the dual variableP is positive and we can find it as follows. We start from the maximum µit and then reduce it until the sum of the weights is equal to 1 where each weight is the distance between the absolute value of µit and ν. 45 1.2.4 Connection with Ridge and Lasso regression Before we explore further the properties and the results of the optimal trading strate- gies, it would be interesting to digress for a while and see what connection there is between our problems and the regularized regressions. In the basic form of regularized regression, the goal is not only to have a good fit, but also regression coefficients that are “small”. Two of the most common forms of regularized regressions are the Ridge and Lasso regression. Ridge regression shrinks the regression coefficients by imposing a penalty on their size [42]. Equation 1.15 is one of the ways to write the Ridge problem. N p 2 minimize (yi − β0 − xijβj) i=1 j=1 (1.15) p 2 subject to Pj=1 βj ≤ t P P The Ridge regression coefficients solution is similar to the optimal trading strategy followed by a risk averse investor with logarithmic preferences, who can choose among N diffusion processes and faces VaR constraints. In both cases we have this propor- tional shrinkage where we reduce all the weights by a constant. Lasso regression is another common form of a regularized regression. It can be used as a heuristic for finding a sparse solution. It does a kind of continuous subset selection [16]. Equation 1.16 is one of the ways to write the Lasso problem. N p 2 minimize (yi − β0 − xijβj) i=1 j=1 (1.16) p subject to Pj=1 kβjk ≤ t P P The Lasso regression coefficients solution is similar to the optimal trading strategy followed by a risk averse investor with logarithmic preferences, who can choose among N diffusion processes and faces margin constraints. Therefore, we can expect that in this case we will have a sparse solution where the weights of several of the opportu- nities will be 0. 46 1.3 Results Let us move on now to the results first for the convergence trades and then for the mean reversion trading strategies. 1.3.1 Convergence trades VaR constraints. We have simulated the optimal trading strategy for N = 2 correlated convergence trading opportunities under VaR constraints. We find the following: • It is often optimal for an investor to underinvest i.e. not to bind the constraint. • The investor typically experiences losses early before locking at a profit as we can see in Figures 1-3, 1-4, 1-5, 1-6. • Tighter constraints lead to less variability and less skewness in the distribution of wealth. They also lead to less final wealth as we can see in Figures 1-5, 1-6. • The wealth is higher when the opportunities hedge each other, as we can see in Figures 1-4, 1-6. This makes sense because when the constraints are binding we care more about losing money which would then lead surely to liquidation when the investment opportunities are better and therefore we prefer the op- portunities to hedge each other. Figure 1-7 shows a typical path for the wealth evolution using the same noise process for positive and negative correlation under the VaR constraint. We see clearly this hedging effect where negative correlation leads to more wealth. • When the initial values of the convergence trades are low, the constraints bind for a small percentage of time and final wealth is negatively correlated to the percentage of time the constraints bind. Figure 1-8 shows this effect. • The final portfolio wealth is highly positively skewed as it is obvious in Figures 1-3, 1-4, 1-5, 1-6 47 For all the simulations we used: σ1 = σ2 = 1, a1 = a2 = 1,S[0] = [1; 1],rf = 0.06, number of steps = 1000. Distribution of wealth Time 0.25 rho 0.5 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Distribution of wealth Time 0.5 rho 0.5 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Distribution of wealth Time 0.75 rho 0.5 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Distribution of wealth Time 1 rho 0.5 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-3: VaR constraints, positive correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing VaR constraints (K=1). Initial wealth is $100. Time 0.25 rho −0.5 K 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.5 rho −0.5 K 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.75 rho −0.5 K 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 1 rho −0.5 K 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-4: VaR constraints, negative correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing VaR constraints (K=1). Initial wealth is $100. 48 Time 0.25 rho 0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.5 rho 0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.75 rho 0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 1 rho 0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-5: VaR constraints, positive correlations, tight constraints. Wealth distribution at t =0.25, 0.5, 0.75, 1 for an investor who invests in two positively cor- related (ρ =0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100. Time 0.25 rho −0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.5 rho −0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.75 rho −0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 1 rho −0.5 K 0.25 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-6: VaR constraints, negative correlations, tight constraints. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100. 49 1400 rho 0.5 rho −0.5 1200 1000 800 600 Final wealth 400 200 0 0 200 400 600 800 1000 1200 Simulation step Figure 1-7: Wealth evolution under VaR constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under the VaR constraint. Initial wealth is $100. 1600 1400 1200 1000 800 Final wealth 600 400 200 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency the constraint binds Figure 1-8: Relation between final wealth and frequency the VaR constraint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence trades are low. 50 Margin constraints. We have also simulated the optimal trading strategy for N = 2 correlated convergence trading opportunities under margin constraints using the same noise process as with the VaR constraints. We have similar results with the case of VaR constraints as we see in Figures 1-9, 1-10, 1-11, 1-12, 1-13 with the following important differences: • When the constraints bind, it is often the case that the position in one of the convergence trades is 0, i.e. we have less diversification, sparse solution. Figure 1-15 shows a typical path of the positions in two convergence trading opportu- nities under VaR constraints, where we see that they tend to be different than 0. Figure 1-16 shows the evolutions of the positions in two convergence trading opportunities under margin constraints for the same exactly asset processes as before. We clearly see that often we invest only in one position, as we expected due to the similarity of the positions with the Lasso regression coefficients. • The final wealth is less skewed and smaller with respect to the case of VaR constraints. 51 Time 0.25 rho 0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.5 rho 0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.75 rho 0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 1 rho 0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-9: Margin constraints, positive correlations. Wealth distribution at t =0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ =0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100. Time 0.25 rho −0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.5 rho −0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.75 rho −0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 1 rho −0.5 Collateral 1 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-10: Margin constraints, negative correlations. Wealth distribution at t =0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100. 52 Time 0.25 rho 0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.5 rho 0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.75 rho 0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 1 rho 0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-11: Margin constraints, positive correlations, more collateral needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing margin con- straints (Collateral = 2). Initial wealth is $100. Time 0.25 rho −0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.5 rho −0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 0.75 rho −0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time 1 rho −0.5 Collateral 2 100 50 0 0 500 1000 1500 2000 2500 3000 3500 4000 Figure 1-12: Margin constraints, negative correlations, more collateral needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing margin con- straints (Collateral = 2). Initial wealth is $100. 53 700 rho 0.5 rho −0.5 600 500 400 300 Final wealth 200 100 0 0 200 400 600 800 1000 1200 Simulation step Figure 1-13: Wealth evolution under margin constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under the margin constraint. Initial wealth is $100. 1600 1400 1200 1000 800 Final wealth 600 400 200 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency the constraint binds Figure 1-14: Relation between final wealth and frequency the margin con- straint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence trades are low. 54 800 Convergence trade 1 Convergence trade 2 600 400 200 0 Position −200 −400 −600 −800 0 100 200 300 400 500 600 700 800 900 1000 Simulation step Figure 1-15: Positions evolution under VaR constraints. Typical path of the positions in two convergence trading opportunities under VaR constraints. 400 Convergence trade 1 Convergence trade 2 300 200 100 0 Position −100 −200 −300 −400 0 100 200 300 400 500 600 700 800 900 1000 Simulation step Figure 1-16: Positions evolution under margin constraints. Typical path of the positions in two convergence trading opportunities under margin constraints. 55 1.3.2 Mean reversion trading opportunities Similar results we get also by simulating the optimal trading strategy for N = 2 correlated mean reversion trading opportunities under VaR and margin constraints. This makes sense since without loss of generality, we have assumed that S¯ = 0 and that Φ is a diagonal matrix, which makes the mean reversion trades case similar to the convergence trades case. They are different in that the drift term of the convergence trades for the same spreads S gets better and better as t → T , while for the mean reversion trades it remains constant. 1.4 Conclusions We explored the optimal portfolio allocation of a risk averse investor who invests in N convergence trades or mean reverting trading strategies, while facing constraints. In particular, we studied the optimal trading strategy when he faces VaR constraints or collateral constraints. The optimal trading strategy is found by solving at each time t a convex program for both cases, we characterized the solution of the convex program and we found the properties of the optimal trading strategy. In all the chapter, we have assumed that the investor completely trusts his model and he is certain about the dynamics of the opportunities he faces. What happens if the model is just an approximation? What happens if the investor believes that opportunities’ dynamics come from an unknown member of a set of unspecified models near an approximating model? Concern about model misspecification will change the optimal trading strategy of the investor and this is the topic of the next chapter. 56 Chapter 2 Optimal trading of arbitrage opportunities under model misspecification A decision maker maximizes a utility function subject to a model. Standard control theory helps a decision maker to make optimal decisions when his model is correct. Robust control theory helps him to make good decisions when his model is approxi- mately correct. In this chapter we will use methods of robust control theory to find the optimal portfolio allocation of a risk averse investor, who invests in convergence trades or mean reverting trading strategies, and is not completely confident about the dynamics of his models. In particular, we assume that the investor believes that the data comes from an unknown member of a set of alternative models near his nominal model. These alternative models are statistically difficult to distinguish from the nominal model. The investor believes that his model is a pretty good approximation in the sense that the discrepancies between the alternative models and his nominal model are small. We will use the relative entropy to characterize the discrepancies between different models. Concern about model misspecification leads the investor to choose a trading strategy that is robust over the alternative models. Three questions come naturally at this point: 57 • What does it mean to have a robust trading strategy? A robust trading strategy is a strategy that works well over the set of alternative probability models. We evaluate the worst performance of a given strategy over that set of alternative probability models and we pick the one that maximizes this worst case performance. It is essentially a “max-min” problem, a two-player game in which a maximizing player chooses the best response to a malevolent player who can disturb the stochastic model within limits. • Why would we be interested in a robust decision rule over alternative models? Why don’t we take a Bayesian approach, where we put a prior distribution over the set of alternative probability models? This could be another approach, but this set of alternative models may be too large or too difficult for the investor to come up with a well behaved, plausible prior distribution. In addition, we might want our solution to work well over any kind of prior distribution [41]. • Why do we use the relative entropy to measure the discrepancy between an alternative and the nominal model? There are other ways to measure discrepancies between alternative probability models, like Prokhorov distance [9] but the relative entropy with respect to a measure P has nice properties and it is more tractable. It is given by: dQ D(Q)= log( )dQ dP Z and it is a convex function of the measure Q. In the rest of this chapter, we will review the relevant literature. Then we will discuss about the set of the alternative models, the relative entropy and equivalent ways to formulate our problem of the optimal robust portfolio allocation of the in- vestor. Subsequently, we will find the optimal robust trading strategy of the investor and finally we will explore the characteristics of this robust strategy. 58 2.1 Literature review Whittle [79], [80] has studied mathematical methods for answering the question of how to make decisions when you don’t fully trust your model. Lars Hansen and Thomas Sargent in [41] have studied how to make economic decisions in the face of model misspecification by modifying and extending aspects of robust control theory. Their work revolves mostly around the linear-quadratic regulator framework, where there is a certainty equivalence principle that allows a deterministic presentation of the control theory. Gilboa and Schmeidler in [34] have studied the max-min expected utility problem where the decision maker has multiple priors and maximizes his expected utility as- suming that nature chooses a probability measure to minimize his expected utility. The minimization is over a closed and convex set of finitely additive probability mea- sures. Their axiomatic treatment views this set of non-unique priors as an expression of the agent’s preferences and the priors are not cast as distortions to a nominal model. Lars Hansen et al. [40] have studied robust decision rules when the agent fears that the data are generated by a statistical perturbation of an approximating model that is either a controlled diffusion process or a control measure over continuous functions of time. They describe how stochastic formulations of robust control “constraint problems” can be viewed in terms of Gilboa and Schmeidler’s max-min expected utility model. They show the connection between the penalty robust control problem and the constraint robust control problem, two closely related problems and formulate the Hamilton Jacobi Bellman equations for various two-player zero sum continuous time games that are defined in terms of a Markov diffusion process. We extend their framework to the problem of optimal robust trading rules for a risk averse investor who does not trust his model dynamics, believes that his nominal model is a good approximation to the real model and invests in arbitrage opportunities. Fleming and Souganidis [77] present how the Bellman-Isaacs condition defines a Bellman equation for a two-player zero-sum game in which both players decide at time 59 0 or recursively. In other words, they show that the freedom to exchange orders of maximization and minimization guarantees that equilibria of games where the choices are done under mutual commitment at time 0 and of games where the choices are done sequentially by both agents coincide. Anderson et al. [3] show how the set of perturbed models in our formulations is difficult to distinguish statistically from the approximating model given a finite sample of timeseries observations. Jacobson [44] and Whittle [78] studied risk sensitive optimal control in the context of discrete-time linear quadratic regulator decision problems. They showed how the risk-sensitive control law can be computed by equivalently solving a robust penalty problem. We will now discuss first how to represent the alternative probability models over which we want our decision rules to be robust and how relative entropy can be used to describe their discrepancies from the nominal model. We will formulate two closely related nonsequential problems and the corresponding recursive HJB equations and finally we will find the optimal robust portfolio allocation for a risk averse investor who is not confident about the dynamics of his models and wants to invest in convergence trades or mean reversion trading strategies. 2.2 Analysis In Chapter 1 we saw that in the case when there is no model misspecification, the investor wants to find the optimal portfolio allocation that solves the following prob- lem: maximizeθ∈Θ E(lnWT ) subject to dWt = θtdSt + θ0tdBt (2.1) dSt = µ(S, t)dt + σ(S, t)dZt dBt = rBdt 60 N Here we have St ∈ R and we have studied the following special cases: a S K dS = − i it dt + σ dZ it T − t ik kt Xk=1 dSt = −Φ(St − S¯)dt + σdZt and Θ is the set of admissible trading strategies: Θ= θ |θT Σθ ≤ LW 2 for the case of VaR constraints and N Θ= θ | λi|θi| ≤ W ( i=1 ) X for the case of the margin constraints. In this Chapter, the investor doubts his model dSt = µ(S, t)dt + σ(S, t)dZt. To capture this doubt of the investor, we surround the approximating model with a cloud of models that are statistically difficult to distinguish and we add a malevolent agent who picks the worst possible model. The investor wants to find the optimal trading strategy that solves the following problem: maxθ∈Θ minQ∈Q EQ(lnWT ) (2.2) where Θ is the set of admissible trading strategies and Q is the set of alternative probability models. Problem 2.2 fits the max-min expected utility model of Gilboa and Schmeidler [34], where Q is a set of multiple different priors. Let’s now discuss how we represent the set of alternative probability models. 2.2.1 Alternative models representation We use martingales to represent perturbations to the probability models and relative entropy to measure the discrepancy between our nominal model and the alternative 61 models. To understand better our continuous time formulations, we digress for a while by borrowing an example from [41]. Let’s consider a discrete time approximating model and its innovations ǫt which are i.i.d Gaussian shocks. An alternative model alters the distribution of these shocks. We use martingales to represent distortions to the probabilities. Letπ ˆt(ǫ) be the alternative density of the shock ǫt+1 based on date t information. Then the random t πˆj−1(ǫ) variable Mt = j=1 mj, where mj = π(ǫ) and M0 = 1, is a martingale and is a ratio of the jointQ alternative density over the joint nominal density. We define the entropy of the alternative distribution associated with Mt as the expected likelihood ratio with respect to the distorted distribution E(Mtlog(Mt)). It has the property that it is always non-negative and it is equal to 0, only when there is no distortion to the nominal distribution. Similarly, in our continuous time formulations we will use martingales to represent distortions to the nominal probability model. We will construct an alternative model ˆ t ˆ by replacing Zt in our model by Zt + 0 hsds, where Zt is a Brownian motion under the alternative measure Q and ht is an adaptedR process that models the distortion, such t 1 t T 0 hsdZs− 0 h hsds that the process ξt = e 2 s is a martingale. Therefore, the nominal model R R is misspecified by allowing the conditional mean of the shock vector in the alternative models to feed back arbitrarily on the history up to date t. Since ξ0 = 1 we have that E(ξt) = 1. Since in addition, ξt > 0, we can define a probability measure Q such dQ that Q(A) = E[1AξT ], in other words ξT = dP is the Radon-Nikodym derivative of Q with respect to P, where the measures Q and P are equivalent. In fact one can always define a process ht so that for any measure Q the Radon-Nikodym derivative dQ of Q with respect to P, dP , is given by the exponential martingale ξT . In this way, our distorted models are: • For the convergence trades, a S K dS = − i it dt + σ (dZˆ + h dt) (2.3) t T − t ik kt kt Xk=1 62 • For the mean reversion trades, dSt = −Φ(St − S¯)dt + σ(dZˆt + htdt) (2.4) where Zˆt is a Brownian motion under Q. Why is it a Brownian motion under Q? The answer lies with the Girsanov theorem [30] that states that if a process ht is such that dQ ξt is a martingale and ξT = dP is the Radon-Nikodym derivative of Q with respect ˆ t to P, then the process Z(t) = Zt − 0 hsds is a Brownian motion under measure Q. Therefore, we parameterize Q by theR choice of the drift distortion adapted process ht. Similarly with the discrete time case, we measure the discrepancy between mea- sures Q and P as the relative entropy D(Q) (see Appendix for derivation), T 1 D(Q)= E [hT h ]dt (2.5) 2 Q t t Z0 . This is to be expected, since the relative entropy between a multivariate Gaussian distribution N(µ,I) and the multivariate standard normal distribution is D(Q) = 1 T 2 µ µ (See Appendix for derivation) and htdt is the conditional mean of the process dZt under the alternative probability measure Q. To express the notion that the nominal model is a good approximation to the real model that generate the spread dynamics, we either restrain the alternative models by D(Q) ≤ η or we penalize them with the magnitude of the entropy. 2.2.2 Model setup Having described the set of alternative distributions, we are ready to formulate the problem a risk averse investor faces who distrusts his model dynamics. As in [40] we define two closely related problems: 63 • A multiplier robust control problem. maxθ∈Θ minQ EQ(lnWT )+ νD(Q) subject to dWt = θtdSt + θ0tdBt dSt = µ(S, t)dt + σ(S, t)(dZˆt + htdt) (2.6) dBt = rBdt dξt = ξthtdZt dQ where ξT = dP and D(Q) is given by equation 2.5. Here in essence there is an implicit restriction manifested by the nonnegative penalty parameter ν. • A constrained robust control problem. maxθ∈Θ minQ EQ(lnWT ) subject to dWt = θtdSt + θ0tdBt dSt = µ(S, t)dt + σ(S, t)(dZˆt + htdt) (2.7) dBt = rBdt dξt = ξthtdZt D(Q) ≤ η dQ where ξT = dP and D(Q) is given by equation 2.5. In both cases the minimizing malevolent agent chooses the distortion process ht taken θ as given and the maximizing investor chooses the optimal strategy taken ht as given. We index the family of multiplier robust control problems by ν and the family of constrained robust control problems by η. Obviously the two problems are related, since the robustness parameter ν can be interpreted as the Lagrange multiplier on the constraint D(Q) ≤ η. Actually we can show that if V (ν) is the optimal value of the multiplier robust problem and K(η) is the optimal value of the constrained robust problem then we have: K(η) = maxν≥0V (ν) − νη [40]. Therefore we will be only interested in finding V (ν). 64 2.3 Solution We will solve 2.6 by solving the corresponding Hamilton Jacobi Bellman (HJB) equa- tion. We will solve the HJB equation for the case that there are no constraints in the admissible trading strategies and when the trading strategies are constrained by VaR or collateral considerations. But first let’s digress for a while and solve the HJB equation for the case when there is no fear of model misspecification. 2.3.1 No fear of model misspecification For now we assume that the investor completely trusts the dynamics of his models dSt = µ(S, t)dt + σ(S, t)dZt. He chooses the trading strategy θ ∈ Θ that solves the problem 2.1 where Θ is the set of admissible trading strategies: Θ= θ|θT Σθ ≤ LW 2 for the case of VaR constraints and N Θ= θ| λi|θi| ≤ W ( i=1 ) X for the case of the margin constraints. In this case the HJB equation is: T T max Vt + VW (Wr + θ (µ(S, t) − rSt)) + VS µ(S, t) θ∈Θ T +1/2VWW θ Σθ + VWSΣθ +1/2trace(ΣVSS)=0 where Σ = σσT and V (W, S, t) is the value function of the investor subject to the terminal condition V (W, S, T ) = ln(W ). Due to the logarithmic preferences of the investor it is: V (W, S, t) = ln(W )+ H(S, t), 1 1 therefore VWS = 0, VW = W and VWW = − W 2 . We also define ∀t ∈ [0, T ] Ft = θt/Wt ∈ RN . 65 The HJB equation becomes: T T max Vt +(r + F (µ(S, t) − rSt)) + VS µ(S, t) F ∈F T − 1/2F ΣF +1/2trace(ΣVSS)=0 where F is the set of admissible trading strategies: F = F |F T ΣF ≤ L for the case of VaR constraints and N F = F | λi|Fi| ≤ 1 ( i=1 ) X for the case of the margin constraints. The optimal trading strategy is the solution to the following convex problem T T min F (−µ(S, t)+ rSt)+1/2F ΣF (2.8) F ∈F as we also proved with a different method in Chapter 1, where µt = −µ(S, t)+ rSt and in particular it is: a1 S1t( T −t + r) . µt = . aN SNt( + r) T −t for the convergence trades case and T ¯ Φ1 (St − S)+ rS1t . µt = . T ¯ ΦN (St − S)+ rSNt for the mean reversion trading strategies case. 66 2.3.2 Fear of model misspecification no constraints In this section we assume that there are no constraints in the trading strategies fol- lowed by the risk averse investor and the investor is not confident about the dynamics of his models. The Hamilton Jacobi Bellman equation for the problem 2.6 is given by: T T max min Vt + VW (Wr + θ (µ(S, t) − rSt)) + VS µ(S, t) θ h ν +1/2V θT Σθ + V Σθ +1/2trace(ΣV )+ V θT σh + V T σh + hT h =0 WW WS SS W S 2 where Σ = σσT and V (W, S, t) is the value function of the investor subject to the terminal condition V (W, S, T ) = ln(W ). The malevolent agent picks the worst case distortion drift process ht and the investor maximizes against the worst case scenario. N After defining ∀t ∈ [0, T ] Ft = θt/Wt ∈ R the HJB equation becomes: T T max min Vt + WVW (r + F (µ(S, t) − rSt)) + VS µ(S, t) F h ν +1/2W 2V F T ΣF +WV ΣF +1/2trace(ΣV )+WV F T σh+V T σh+ hT h =0 WW WS SS W S 2 The inner minimization problem is a convex quadratic problem. The first order conditions are: T T WVW σ F + σ VS + νh =0 σT (WV F + V ) h = − W S ν The optimal value of the inner minimization problem is: W 2V 2 F tΣF + V T ΣV +2WV F T ΣV g(F )= − W S S W S 2ν 67 Plugging this back into the HJB equation we have: T T max Vt + WVW (r + F (µ(S, t) − rSt)) + VS µ(S, t) F 2 T +1/2W VWW F ΣF + WVWSΣF +1/2trace(ΣVSS) W 2V 2 F tΣF + V T ΣV +2WV F T ΣV − W S S W S =0 2ν Due to the logarithmic preferences of the investor it is: V (W, S, t) = lnW + H(S, t) 1 1 and in that case VW = W VWW = − W 2 VWS = 0,VS(W, S, t) = HS(S, t) and the T σ (F +HS) minimizing drift distortion h = − ν independent of the wealth. The HJB equation now becomes: T T max Vt + r + F (µ(S, t) − rSt)+ VS µ(S, t) F F T ΣF + V T ΣV +2F T ΣV − 1/2F T ΣF +1/2trace(ΣV ) − S S S =0 SS 2ν The optimal trading strategy is the solution to the following convex quadratic prob- lem: ΣV 1 1 maximize F T (µ(S, t) − rS − S ) − (1 + )F T ΣF (2.9) t ν 2 ν t t The first order conditions are: ΣV (S , t) 1 µ(S, t) − rS − S t =(1+ )ΣF opt t ν ν t opt 1 −1 ΣVS(St, t) Ft = 1 Σ (µ(S, t) − rSt − ) 1+ ν ν ν V (S , t) F opt = Σ−1(µ(S, t) − rS ) − S t t ν +1 t ν +1 We clearly see that as ν →∞ the optimal trading strategy converges to the one where we have no fear of model misspecification. This is to be expected since at this case the problems 2.1 and 2.6 are equivalent. It is interesting to find the conditions under which these weights are equal to the weights when there is no fear of model misspecification. When there is a fear of model misspecification, the optimal weights are a convex 68 −1 combination of Σ (µ(S, t)−rSt), i.e.the weights without model misspecification and −VS. Therefore these weights are equal to the weights when there is no fear of model opt misspecification, when Vs + F = 0, which is equivalent to hmin = 0. Of course this is expected since in that case there would be no distortion drift and the HJB equation would be the same as the benchmark case of no model misspecification. After plugging in the optimal trading strategy to the HJB equation, it becomes: 1 T −1 Vt + r +1/2 trace(ΣVSS)+1/2 1 (µ(S, t) − rSt) Σ (µ(S, t) − rSt) 1+ ν µ(S, t) − rS 1 + V T µ(S, t) − − 1/2 V T ΣV =0 S ν +1 ν +1 S S T σ (W VW F +VS) We can plug in the optimal trading strategy to h = − ν to find that: (σT F opt + σT V ) h = − S min ν T 1 −1 ΣVS T 1 (σ 1+ Σ (µ(S, t) − rSt − ν )+ σ VS) h = − ν min ν σT (Σ−1(µ(S, t) − rS )+ V ) h = − t S min ν +1 We consider two cases: • Convergence trades. The optimal trading strategy is given by: ν V θopt = − Σ−1A S + S W (2.10) t ν +1 t t ν +1 t a1 aN where At = diag( T −t + r, ··· , T −t + r). The optimal trading strategy is a convex combination of the strategy without fear of model misspecification and ν 1 −VS with weights ν+1 and ν+1 . From the symmetry of the problem we have: H(S, t)= H(−S, t) from which we get VS(St, t)= −VS (−St, t). • Mean reversion trades. The optimal trading strategy is given by: ν V θopt = − Σ−1(Φ(S − S¯)+ rS )+ S W (2.11) t ν +1 t t ν +1 t 69 The optimal trading strategy is again a convex combination of the strategy ν 1 without fear of model misspecification and −VS with weights ν+1 and ν+1 . For the special case where S¯ = 0 we have: ν V θopt = − Σ−1((Φ + rI)S )+ S W (2.12) t ν +1 t ν +1 t 2.3.3 Fear of model misspecification with VaR and margin constraints In this section we assume that the investor is not confident about the dynamics of his models and he faces either VaR or margin constraints. The Hamilton Jacobi Bellman equation for the problem 2.6 is given by: T T max min Vt + VW (Wr + θ (µ(S, t) − rSt)) + VS µ(S, t) θ∈Θ h ν +1/2V θT Σθ + V Σθ +1/2trace(ΣV )+ V θT σh + V T σh + hT h =0 WW WS SS W S 2 where Σ = σσT and V (W, S, t) is the value function of the investor subject to the terminal condition V (W, S, T ) = ln(W ). As previously Θ is the set of admissible trading strategies: Θ= θ|θT Σθ ≤ LW 2 for the case of VaR constraints and N Θ= θ| λi|θi| ≤ W ( i=1 ) X for the case of the margin constraints. We can proceed like the previous case where we had no constraints and we will get 70 the following HJB equation: T T max Vt +(r + F (µ(S, t) − rSt)) + VS µ(S, t) F ∈F F T ΣF + V T ΣV +2F T ΣV − 1/2F T ΣF +1/2trace(ΣV ) − S S S =0 SS 2ν where F is the set of admissible trading strategies: F = F |F T ΣF ≤ L for the case of VaR constraints and N F = F | λi|Fi| ≤ 1 ( i=1 ) X for the case of the margin constraints. The optimal trading strategy is the solution to the following convex problem: T ΣVS 1 1 T maximize F (µ(S, t) − rSt − ) − (1 + )F ΣFt ν 2 ν t (2.13) subject to F ∈ F We clearly see again that as ν →∞ the optimal trading strategy converges to the one where we have no fear of model misspecification as expected. Using a similar proof as in Chapter 1, we can show (see Appendix) that in the case the investor faces VaR constraints the optimal portfolio is given by: 1 −1 T −1 1 2 1 1+ Σ µt if µt Σ µt ≤ L(1 + ν ) opt ν Ft = −1 Σ µt T −1 1 2 if µ Σ µt ≥ L(1 + ) T Σ−1 t ν µt µt r L ΣVS (St,t) where µt = µ(S, t) − rSt − ν . 71 This is also equivalently written as: opt 1 −1 Ft = 1 Σ µt 1+ ν + λ T −1 1 1 µt Σ µt where 1 + ν + λ = max(1 + ν , L ). q We consider two cases: • Convergence trades. The optimal trading strategy is the solution to the follow- ing convex problem: T ΣVS 1 1 T minimize F (AtSt + )+ (1 + )F ΣFt ν 2 ν t (2.14) subject to F ∈ F a1 aN where At = diag( T −t + r, ··· , T −t + r). • Mean reversion trades. The optimal trading strategy is the solution to the following convex problem: T ΣVS 1 1 T minimize F (Φ(St − S¯)+ rSt + )+ (1 + )F ΣFt ν 2 ν t (2.15) subject to F ∈ F For the special case where S¯ = 0, we have the following problem: T ΣVS 1 1 T minimize F ((Φ + rI)St + )+ (1 + )F ΣFt ν 2 ν t (2.16) subject to F ∈ F 2.4 Results We will investigate how the optimal trading strategy changes as a result of mistrust of the model dynamics. We will first study the case where we have no constraints and then the case where we have VaR constraints. 72 2.4.1 Convergence trades without constraints We consider the case where we have N = 1 arbitrage opportunity and there are no constraints. We will study the case where N = 2 when we have constraints. Due to the symmetry of S around 0 it suffices to study only what happens when S ≥ 0, since the symmetry implies that the value function is an even function of the spread S and its partial derivative with respect to the spread is an odd function of S for each t. The optimal weight in the arbitrage opportunity is given by: 1 a σ2V (S , t) F opt = − (( + r)S + S t ) (2.17) t 2 1 t σ (1 + ν ) T − t ν opt σ(F +VS ) and the minimizing distortion drift is given by: hmin = − ν . Comparing to the case where there is no fear of model misspecification we see that now the variance 1 increases by multiplying by a factor of (1 + ν ) and the drift increases by adding 2 σ VS (St,t) ν . When S is positive, one would think that there are three cases to consider: • If VS > 0 then F < 0. In this case there is a tradeoff between the two terms σF opt in hmin. The first term − ν corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is shorting the spread, while σVS the second term − ν corresponds to a negative distortion drift that points to worse investment opportunities. AtStν • If − σ2 < VS < 0 then F < 0 and both the terms in hmin correspond to positive distortion drifts with the first one reducing the wealth of the investor and the second term pointing to worse investment opportunities. AtStν • If VS < − σ2 then F > 0. There is now again a tradeoff between the two terms in hmin. Now the first term corresponds to a negative distortion drift reducing the wealth of the investor since in this case the investor is long the spread, and the second term corresponds to a positive distortion pointing to worse investment opportunities. 73 A little more thought though will exclude the last two cases, since we expect the value function V to be a non-decreasing function of S for nonnegative values of the spread, since higher values of the spread S correspond to better investment opportunities. That will lead to non-negative values of VS for S ≥ 0. The HJB equation is given by: 2 2 1 a 2 S Vt + r +1/2 σ VSS +1/2 1 ( + r) 2 1+ ν T − t σ a ( a + r)S 1 + V (− S + T −t ) − 1/2 V 2σ2 =0 S T − t ν +1 ν +1 S We solve the HJB equation numerically using the method of finite differences [75]. In the following figures we have assumed that rf = 0, σ = 1, a = 0.01 and T = 1. We observe the following: • VS becomes larger and larger as t → T for each value of ν as we see in Figure 2-1 until some value close to the horizon where it starts going down. In addition, VS is higher for higher values of ν. • For very low values of ν the drift distortion hmin is positive and becomes larger as t → T as we see in Figure 2-2. For higher values of ν the drift distortion starts negative and after some point increases as t → T to positive values. As we showed in the previous section, when hmin = 0, the optimal weight is equal to the optimal weight in the case where there is no fear of model misspecification. We can see this in Figure 2-4, where very close at the time where hmin crosses 0, the optimal weight graph crosses the one when ν = 100. • Figure 2-3 shows the two terms of the distortion drift for ν = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. In this tradeoff the first term is losing at the beginning which makes the drift distortion negative but as t → T it increases in a fast rate making the drift distortion positive. 74 • The fact that VS becomes larger as t → T for each value of ν until a point close a to the horizon in combination with the fact that the drift term T −t increases in a hyperbolic way leads to F becoming larger (in absolute value) as t → T (Figure 2-4). The risk averse investor becomes more aggressive as the time to T becomes smaller despite the fact that as t → T the malevolent agent picks a more adverse distortion drift (Figure 2-2). This is due to the fact that the improvement in the investment opportunities is so substantial that dominates the fact that the distortion drift gets also larger. • For very low values of ν the investor is more conservative than the case without model misspecification for all t. For higher values on ν at the beginning the investor is more aggressive and as t → T becomes more conservative comparing to the case without model misspecification (Figure 2-4). This is because at the a beginning the drift term T −t is very low comparing to VS making the total drift term lower for large values of ν than smaller values of ν, which leads to lower magnitude of weight. This situation changes as time to horizon T gets smaller. • As ν →∞ the optimal weight in the strategy converges to the optimal weight when there is no fear of model misspecification as we have argued before. 75 Vs as a function of time for S = 1 0.16 nu is: 0.01 nu is: 0.1 0.14 nu is: 1 nu is: 10 nu is: 100 0.12 0.1 0.08 Vs 0.06 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-1: Partial derivative of the value function with respect to S for a single convergence trade. VS as a function of time at S = 1 for different values of the robustness multiplier for a single convergence trade. hmin as a function of time for S = 1 0.1 nu is: 0.01 nu is: 0.1 0.08 nu is: 1 nu is: 10 nu is: 100 0.06 0.04 hmin 0.02 0 -0.02 -0.04 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-2: Distortion drift for a single convergence trade. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier for a single convergence trade. 76 Drift distortion and its two terms as a function of time for S = 1 0.1 h h1 0.08 h2 0.06 0.04 0.02 0 Distortion -0.02 -0.04 -0.06 -0.08 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-3: Distortion drift terms for a single convergence trade. Distortion drift terms as a function of time at S = 1 for ν = 1 for a single convergence trade. The first term corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. Optimal weights as a function of time for S = 1 0 nu is: 0.01 nu is: 0.1 nu is: 1 nu is: 10 -0.02 nu is: 100 -0.04 -0.06 Weights -0.08 -0.1 -0.12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-4: Optimal weight of a single convergence trade. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. 77 2.4.2 Mean reversion trading strategies without constraints We consider first the case where we have N = 1 mean reversion trading strategy and S¯ = 0 i.e. dSt = −φStdt + σdZt Due to the symmetry of S around 0 it suffices to study only what happens when S ≥ 0, since the symmetry implies that the value function is an even function of the spread S and its partial derivative with respect to the spread is an odd function of S for each t. The optimal weight in the trading strategy is given by: 1 σ2V (S , t) F opt = − ((φ + r)S + S t ) t 2 1 t σ (1 + ν ) ν ν V (S , t) = − (φ + r)S − S t σ2(ν + 1) t ν +1 and the minimizing distortion drift is given by: σ(F opt + V ) h = − S min ν (φ + r)S σV = t − S σ(ν + 1) ν +1 Comparing to the case where there is no fear of model misspecification we see that when the investor does not trust the model dynamics, the variance increases by mul- 2 1 σ VS(St,t) tiplying with a factor of (1 + ν ) and the drift increases by adding ν . If VS is non-negative and decreases as a function of time t, then the increase in the drift gets smaller and smaller and the investor gets more and more conservative as time passes by. In this case the distortion drift also gets larger and larger as time passes by, which explains why the investor gets more and more conservative. When S is positive, we have VS ≥ 0, since higher values of S correspond to better opt investment opportunities. If VS ≥ 0 then F ≤ 0 for S ≥ 0. Therefore we see that σF opt there is a tradeoff between the two terms in hmin. The first term − ν corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is 78 σVS shorting the spread, while the second term − ν corresponds to a negative distortion drift that points to worse investment opportunities. The HJB equation is given by: 2 2 1 2 S (φ + r)S 1 2 2 Vt +r+1/2 σ VSS +1/2 1 (φ+r) 2 +VS(−φS+ )−1/2 VS σ =0 1+ ν σ ν +1 ν +1 We solve the HJB equation numerically using the method of finite differences. In the following figures we have assumed that rf = 0, σ = 1, φ = 1 and T = 1. We observe the following: • VS becomes smaller and smaller as t → T for each value of ν as we see in Figure 2-5. This makes sense, since as t → T there is less time to take advantage of the mean reversion trading strategy. In addition, VS is higher for higher values of ν. • The drift distortion hmin is positive and becomes larger and larger as t → T for each value of the robustness multiplier as we see in Figure 2-6. Figure 2-7 shows the two terms of the distortion drift for ν = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. In this tradeoff the first term wins making the drift distortion positive. • The fact that VS becomes smaller and smaller as t → T for each value of ν leads to F becoming smaller and smaller (in absolute value) as t → T ( Figure 2-8). In other words the risk averse investor becomes more conservative as the time to T becomes smaller. This is to be expected, since as t → T the malevolent agent picks a more adverse distortion drift (Figure 2-6) causing the investor to be more cautious. • The lower the value of ν the more conservative the investor is as we see in Figure 2-8. This is because ν is the robustness multiplier and lower values of it puts 79 less penalty in the distorting alternative distribution, leading to higher positive drift distortions (Figure 2-6). • As ν →∞ the optimal weight in the strategy converges to the optimal weight when there is no fear of model misspecification as we have argued before. Vs as a function of time for S = 1 0.45 nu is: 1 nu is: 10 0.4 nu is: 100 0.35 0.3 0.25 Vs 0.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-5: Partial derivative of the value function with respect to S for a single mean reversion trading strategy. VS as a function of time at S = 1 for different values of the robustness multiplier. 80 hmin as a function of time for S = 1 0.5 nu is: 1 0.45 nu is: 10 nu is: 100 0.4 0.35 0.3 0.25 hmin 0.2 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-6: Distortion drift for a single mean reversion trading strategy. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. Drift distortion and its two terms as a function of time for S = 1 0.7 h h1 0.6 h2 0.5 0.4 0.3 0.2 Distortion 0.1 0 -0.1 -0.2 -0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-7: Distortion drift terms for a single mean reversion trading strat- egy. Distortion drift terms as a function of time at S = 1 for ν = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor, since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. 81 Optimal weights as a function of time for S = 1 -0.5 nu is: 1 nu is: 10 -0.55 nu is: 100 -0.6 -0.65 -0.7 -0.75 Weights -0.8 -0.85 -0.9 -0.95 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-8: Optimal weight of a single mean reversion trading strategy. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier. 82 Let us now consider the case where we have N = 2 mean reversion trading strate- gies and again S¯ = 0. We solve numerically the HJB equation using the method of finite differences. We have assumed that rf = 0, T = 1, 2 0 Φ= 0 1 and 1 ρ Σ= ρ 1 In Figure 2-9 we plot the weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. We have chosen S1 = 1 and S2 = 2, since for these values the drift is the same for both the strategies. We have assumed that there is no correlation between the two trading strategies. We observe the following: • First of all when there is no fear of model misspecification the weights of the two strategies are the same and they do not change over time. • When there is a fear of model misspecification the investor becomes more and more conservative over time just like in the N = 1 case. It is interesting to note that the weight is higher for the first trading strategy where the φ coefficient is higher. That makes sense since “ceteris parebus” we would expect the Vs to be higher for the strategy with the stronger rate of mean reversion (φ coefficient). This is shown is Figure 2-11. • Figure 2-10 shows the ratio of the weights of the two trading strategies. We observe that this is higher for smaller values of the robustness multiplier and it is reduced to 1 as t → T . In Figure 2-12 we have assumed that there is a correlation ρ = 0.5 between the two trading strategies. Now the weights are smaller than before due to the positive correlations but they have the same properties as before. In the case when there is a 83 Optimal weights as a function of time for S1 = 1 and S2 = 2 -1 Spread 1 nu is: 1 Spread 2 nu is: 1 -1.1 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 -1.2 Spread 2 nu is: 100 -1.3 -1.4 -1.5 Weights -1.6 -1.7 -1.8 -1.9 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-9: Optimal weights of two uncorrelated mean reversion trading strategies for S1 =1 and S2 =2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. Ratio of optimal weights as a function of time for S1 = 1 and S2 = 2 1.3 nu is: 1 nu is: 10 nu is: 100 1.25 1.2 1.15 Weights 1.1 1.05 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-10: Ratio of the optimal weights. Ratio of the optimal weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. negative correlation (Figure 2-13 ) the weights are larger, since now the opportunities hedge each other, otherwise the properties remain the same. 84 Vs1 Vs2 as a function of time for S1 = 1 and S2 = 2 1.4 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 1.2 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 1 0.8 Vs 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-11: Partial derivative of the value function with respect to S1 and S2 at S1 =1 and S2 =2 when ρ =0. Partial derivative of the value function with respect to S1 and S2 as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. Optimal weights as a function of time for S1 = 1 and S2 = 2 -0.6 -0.7 -0.8 Spread 1 nu is: 1 Spread 2 nu is: 1 -0.9 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -1 Weights -1.1 -1.2 -1.3 -1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-12: Optimal weights of two positively correlated mean reversion trading strategies for S1 =1 and S2 =2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0.5. 85 Optimal weights as a function of time for S1 = 1 and S2 = 2 -2 Spread 1 nu is: 1 Spread 2 nu is: 1 -2.2 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 -2.4 Spread 2 nu is: 100 -2.6 -2.8 -3 Weights -3.2 -3.4 -3.6 -3.8 -4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-13: Optimal weights of two negatively correlated mean reversion trading strategies for S1 =1 and S2 =2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5. 86 So far we have examined the two trading strategies at values where the drifts are the same. We now choose S1 = 1 and S2 = 1, since for these values the drifts differ for the two strategies. Figure 2-14 shows the weights of the strategies over time for different values of the robustness multiplier when there is no correlation between the two trading strategies. We now observe the following: • First of all when there is no fear of model misspecification the ratio of the weights of the two strategies is the same as the ratio of their drifts normalized by their variances and it does not change over time. • When there is a fear of model misspecification the investor becomes more and more conservative over time just like in the N = 1 case as we see in 2-14. • VS1 is higher than VS2. That makes sense since “ceteris parebus” we would expect the Vs to be higher for the strategy with the stronger rate of mean reversion (φ coefficient). This is shown is Figure 2-16. • Figure 2-15 shows the ratio of the weights of the two trading strategies. We observe that this is higher for smaller values of the robustness multiplier and after initially increasing it finally converges to 1 as t → T . The reason that initially the ratio is less than the one for the case where there is no fear of model misspecification is that initially the ratio of VS1 is less than the ratio of VS2 the drifts. A very interesting case arises when there is a high positive correlation like ρ =0.9 between the two trading strategies. In this case the investor uses the second trading strategy with the lower drift as a hedge for the first strategy as it is shown in Figure 2- 17 where the investor is shorting the first strategy while he is long the second strategy. Now VS2 is negative (Figure 2-18) which is to be expected since the investor is long the asset and higher values of S2 lead to worse investment opportunities for a long investor. Figure 2-19 shows the ratio of the magnitudes of the optimal weights as a function of time. 87 Optimal weights as a function of time for S1 = 1 and S2 = 1 -0.5 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -1 Weights -1.5 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-14: Optimal weights of two uncorrelated mean reversion trading strategies for S1 =1 and S2 =1. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. Ratio of optimal weights as a function of time for S1 = 1 and S2 = 1 2.12 nu is: 1 nu is: 10 2.1 nu is: 100 2.08 2.06 2.04 2.02 Weights 2 1.98 1.96 1.94 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-15: Ratio of the optimal weights at S1 = 1 and S2 = 1 when ρ = 0. Ratio of the optimal weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. Finally Figure 2-20 shows the weights when there is negative correlation ρ = −0.8. The results are similar with the case of no correlation, although now the weights are higher due to the negative correlation, which makes the two strategies good hedges. 88 Vs1 Vs2 as a function of time for S1 = 1 and S2 = 1 1.2 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 1 Spread 1 nu is: 100 Spread 2 nu is: 100 0.8 0.6 Vs 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-16: Partial derivative of the value function with respect to S1 and S2 at S1 =1 and S2 =1 when ρ =0. Partial derivative of the value function with respect to S1 and S2 as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. Optimal weights as a function of time for S1 = 1 and S2 = 1 4 3 2 1 0 -1 Spread 1 nu is: 0.1 Spread 2 nu is: 0.1 Weights Spread 1 nu is: 1 -2 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 -3 -4 -5 -6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-17: Optimal weights of two positively correlated mean reversion trading strategies for S1 =1 and S2 =1. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9. 89 Vs1 Vs2 as a function of time for S1 = 1 and S2 = 1 2.5 V nu is: 0.1 S1 V nu is: 0.1 S2 V nu is: 1 2 S1 V nu is: 1 S2 V nu is: 10 S1 V nu is: 10 S2 1.5 1 Vs 0.5 0 -0.5 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-18: Partial derivative of the value function with respect to S1 and S2 at S1 = 1 and S2 = 1 when ρ = 0.9. Partial derivative of the value function with respect to S1 and S2 as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9. Ratio of optimal weights as a function of time for S1 = 1 and S2 = 1 1.65 nu is: 1 nu is: 10 nu is: 100 1.6 1.55 1.5 Weights 1.45 1.4 1.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-19: Ratio of the optimal weights at S1 =1 and S2 =1 when ρ =0.9. Ratio of the magnitude of the optimal weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9. 90 Optimal weights as a function of time for S1 = 1 and S2 = 1 -3.5 -4 -4.5 -5 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 -5.5 Spread 1 nu is: 100 Spread 2 nu is: 100 -6 Weights -6.5 -7 -7.5 -8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-20: Optimal weights of two negatively correlated mean reversion trading strategies for S1 =1 and S2 =1. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.8. 91 2.4.3 Convergence trades with constraints We consider first the case where we have N = 1 arbitrage opportunity and there are collateral constraints i.e. |F | ≤ L. Due to the symmetry of the constraint and the spread S around 0 it suffices to study only what happens when S ≥ 0, since the symmetry implies that the value function is an even function of the spread S and its partial derivative with respect to the spread is an odd function of S for each t. The optimal weight in the arbitrage opportunity is given by: 2 2 1 a σ VS(St,t) a σ VS (St,t) 2 1 − 2 1 (( T −t + r)St + ν ) if|( T −t + r)St + ν | ≤ Lσ (1 + ν ) σ (1+ ν ) opt 2 a σ VS (St,t) 2 1 Ft = −L if( + r)St + ≥ Lσ (1 + ) T −t ν ν 2 a σ VS (St,t) 2 1 L if( T −t + r)St + ν ≤ −Lσ (1 + ν ) opt σ(F +VS) and the minimizing distortion drift is given by: hmin = − ν . The HJB equation is given by: 2 ( a +r)S 2 1 a 2 S a T −t 1 2 2 Vt + r +1/2 σ VSS +1/2 1 ( T −t + r) σ2 + VS(− T −t S + ν+1 ) − 1/2 ν+1 VS σ =0 1+ ν 2 if |( a + r)S + σ VS(St,t) | ≤ Lσ2(1 + 1 ) T −t t ν ν 2 2 1 2 a σ 1 1 2 2 a Vt + r +1/2 σ VSS − 2ν VS − VS( T −t S − L ν ) − 2 (1 + ν )σ L + L( T −t + r)S =0 2 a σ VS (St,t) 2 1 if ( T −t + r)St + ν ≥ Lσ (1 + ν ) 2 2 1 2 a σ 1 1 2 2 a Vt + r +1/2 σ VSS − VS − VS( S + L ) − (1 + )σ L − L( + r)S =0 2ν T −t ν 2 ν T −t 2 a σ VS (St,t) 2 1 if ( + r)St + ≤ −Lσ (1 + ) T −t ν ν We solve the HJB equation numerically using the method of finite differences. In the following figures we have assumed that rf = 0, σ = 1, a = 0.01 and T = 1. We observe the following: • VS is lower when the constraint is tighter (Figure 2-21). Figure 2-22 shows a typical behaviour of VS over time at S = 1 and L = 0.1 for different values of the robustness multiplier. 92 • The lower the value of the robustness multiplier, the more time it takes to bind the constraint. For very low values of ν the investor is more conservative than the case without model misspecification for all t. When the constraint is relatively tight it is the case that the investor is more conservative when the robustness multiplier is lower (Figure 2-23). When the constraint is relatively loose (L is higher) it might be the case that for not very low values of ν the investor is initially more aggressive and as t → T becomes more conservative comparing to the case without model misspecification (Figure 2-24). This is a because at the beginning the drift term T −t is very low comparing to VS making the total drift term lower for large values of ν than smaller values of ν, which leads to lower magnitude of weight. This situation changes as time to horizon T gets smaller. • The fact that typically VS becomes larger as t → T for each value of ν until a a point close to the horizon in combination with the fact that the drift term T −t increases in a hyperbolic way leads to F becoming larger (in absolute value) as t → T (Figure 2-23, 2-24) till it binds the collateral constraint. The risk averse investor becomes more aggressive as the time to T becomes smaller despite the fact that as t → T the malevolent agent picks a more adverse distortion drift. This is due to the fact that the improvement in the investment opportunities is so substantial that dominates the fact that the distortion drift gets also larger. • For very low values of ν the drift distortion hmin is always positive. When the constraint is tight enough the drift distortion is positive for all values of the robustness multiplier (see Figure 2-25). When the constraint is not very tight for not very low values of ν the drift distortion starts negative and after some point increases as t → T to positive values (see Figure 2-26). • Figure 2-27 shows the two terms of the distortion drift for ν = 1 when L =0.1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is shorting the spread and it is bounded above due to the collateral constraint, while the second term corresponds to a negative 93 distortion drift that points to worse investment opportunities. In this tradeoff the first term wins and the final distortion drift is positive. For higher values of L (see Figure 2-28) we see that the first term is losing at the beginning which makes the drift distortion negative, explaining why initially the investor is more aggressive than the case when there is no fear of model misspecification, but as t → T it increases at a fast rate making the drift distortion finally positive, which explains the fact that after a while the investor becomes more conservative comparing to the case without fear of model misspecification. It is interesting to note that after the collateral constraint binds the distortion drift evolution is determined by the evolution of VS and therefore it might also undergo some initial reduction before increasing to its upper bound dictated by the constraint. • The tighter the collateral constraint the more conservative the investor is even when the constrains does not bind (Figure 2-29). • As ν →∞ the optimal weight in the strategy converges to the optimal weight when there is no fear of model misspecification as we have argued before. Vs as a function of time for S = 1 0.16 0.14 0.12 0.1 0.08 Vs 0.06 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-21: Partial derivative of the value function with respect to S for a single convergence trade when L = 0.1 and L = 100. VS as a function of time at S = 1 for different values of the robustness multiplier. The solid line is when L = 100 and the dotted line is for L =0.1. 94 ×10-3 Vs as a function of time for S = 1 5.5 nu is: 0.01 L is: 0.1 nu is: 0.1 L is: 0.1 5 nu is: 1 L is: 0.1 nu is: 10 L is: 0.1 nu is: 100 L is: 0.1 4.5 4 3.5 3 Vs 2.5 2 1.5 1 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-22: Partial derivative of the value function with respect to S for a single convergence trade when L = 0.1. VS as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 0.1. Optimal weights as a function of time for S = 1 0 -0.02 nu is: 0.01 L is: 0.1 nu is: 0.1 L is: 0.1 nu is: 1 L is: 0.1 nu is: 10 L is: 0.1 -0.04 nu is: 100 L is: 0.1 -0.06 Weights -0.08 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-23: Optimal weight of a single convergence trade when L = 0.1. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 0.1. 95 Optimal weights as a function of time for S = 1 0 nu is: 0.01 L is: 1 nu is: 0.1 L is: 1 -0.01 nu is: 1 L is: 1 nu is: 10 L is: 1 nu is: 100 L is: 1 -0.02 -0.03 -0.04 -0.05 Weights -0.06 -0.07 -0.08 -0.09 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-24: Optimal weight of a single convergence trade when L =1. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 1. hmin as a function of time for S = 1 0.1 0.09 nu is: 0.01 L is: 0.1 nu is: 0.1 L is: 0.1 0.08 nu is: 1 L is: 0.1 nu is: 10 L is: 0.1 0.07 nu is: 100 L is: 0.1 0.06 0.05 hmin 0.04 0.03 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-25: Distortion drift for a single convergence trade when L = 0.1. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 0.1. 96 hmin as a function of time for S = 1 0.1 nu is: 0.01 L is: 1 nu is: 0.1 L is: 1 0.08 nu is: 1 L is: 1 nu is: 10 L is: 1 nu is: 100 L is: 1 0.06 0.04 hmin 0.02 0 -0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-26: Distortion drift for a single convergence trade when L = 1. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F | ≤ 1. Drift distortion and its two terms as a function of time for S = 1 0.1 h h1 h2 0.08 0.06 0.04 Distortion 0.02 0 -0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-27: Distortion drift terms for a single convergence trade when L = 0.1. Distortion drift terms as a function of time at S = 1 for ν = 1 and L = 0.1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor and it is bounded above due to the collateral constraint, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. 97 Drift distortion and its two terms as a function of time for S = 1 0.08 h h1 h2 0.06 0.04 0.02 Distortion 0 -0.02 -0.04 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-28: Distortion drift terms for a single convergence trade when L = 1. Distortion drift terms as a function of time at S = 1 for ν = 1 and L = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor and it is bounded above due to the collateral constraint, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. Optimal weights as a function of time for S = 1 0 -0.02 -0.04 -0.06 Weights -0.08 -0.1 -0.12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time Figure 2-29: Optimal weight of a single convergence trade when L =0.1 and L = 100. Weight of the convergence trading strategy as a function of time at S =1 for different values of the robustness multiplier. The solid line is when L = 100 and the dotted line is for L =0.1. 98 Let us now consider the case where we have N = 2 convergence trades. We solve numerically the HJB equation using the method of finite differences. We have assumed that rf = 0, T = 1, 0.04 a = 0.02 and 1 ρ Σ= ρ 1 Additionally we assume that we have a VaR constraint F T ΣF ≤ L. In Figures 2-30 and 2-32 we plot the weights of the convergence trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of L and different values of the robustness multiplier. At these values of S1 and S2 the drift is the same for both the strategies. We have assumed that there is no correlation between the two trading strategies. Figures 2-31 and 2-33 show F T ΣF , the normalized wealth variance, as a function of time for the different values of the robustness multiplier. We observe the following: • When the VaR constraint binds, and there is a fear of model misspecification we invest more on the spread with the higher rate of mean reversion, due to higher VS. The asymmetry between the two convergence trades goes down as t → T and that is why when the VaR constraint binds, the difference in the weights of the two strategies have to go down, as we see in Figures 2-30 and 2-32 where L =0.5 and L =0.05 respectively. Moreover, the investment in the spread with the higher rate of mean reversion is higher than the corresponding investment when there is no fear of model misspecification. This is due to the 2 2 fact that in both cases we have the same VaR constraint F1 + F2 = L, but in the fist case there is an asymmetry causing F1 to be higher and F2 to be lower than the corresponding weights in the second case. • The lower the value of the robustness multiplier, the more time it takes to bind the constraint, just like in the N = 1 case. 99 • When the VaR constraint does not bind the weights of both of the strategies in- crease at t → T due to the improvement of the investment opportunities. When the constraint is relatively loose (L is higher) it might be the case that for not very low values of ν the investor is initially more aggressive and as t → T be- comes more conservative comparing to the case without model misspecification a (Figures 2-30 and 2-32). This is because at the beginning the drift term T −t is very low comparing to VS making the total drift term lower for large values of ν than smaller values of ν, which leads to lower magnitude of weight. Optimal weights as a function of time for S1 = 1 and S2 = 2 0 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 -0.1 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.2 -0.3 Weights -0.4 -0.5 -0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-30: Optimal weights of two uncorrelated convergence trades for S1 =1 and S2 =2 when L =0.5. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.5. 100 VaR Constraint as a function of time for S1 = 1 and S2 = 2 0.6 nu is: 1 nu is: 10 nu is: 100 0.5 0.4 0.3 Weights 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-31: Value of the normalized wealth variance for two uncorrelated convergence trades at S1 =1 and S2 =2 when L =0.5. Value of the normalized wealth variance for two uncorrelated convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.5. Optimal weights as a function of time for S1 = 1 and S2 = 2 -0.02 Spread 1 nu is: 1 Spread 2 nu is: 1 -0.04 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.06 -0.08 -0.1 Weights -0.12 -0.14 -0.16 -0.18 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-32: Optimal weights of two uncorrelated convergence trades for S1 =1 and S2 =2 when L =0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05. 101 VaR Constraint as a function of time for S1 = 1 and S2 = 2 0.06 nu is: 1 nu is: 10 nu is: 100 0.05 0.04 0.03 Weights 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-33: Value of the normalized wealth variance for two uncorrelated convergence trades at S1 =1 and S2 =2 when L =0.05. Value of the normalized wealth variance for two uncorrelated convergence trades as a function of time at S1 =1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05. 102 Figures 2-34 and 2-35 show the weights and the lhs of the constraint for the case of positive correlation, while Figures 2-36 and 2-37 cover the case for negative correlations. The properties are similar with the ones for the uncorrelated case. Optimal weights as a function of time for S1 = 1 and S2 = 2 0 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 -0.02 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.04 -0.06 Weights -0.08 -0.1 -0.12 -0.14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-34: Optimal weights of two positively correlated convergence trades for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0.5 and the rhs of the VaR constraint is L =0.05. 103 VaR Constraint as a function of time for S1 = 1 and S2 = 2 0.06 nu is: 1 nu is: 10 nu is: 100 0.05 0.04 0.03 Weights 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-35: Value of the normalized wealth variance for two positively correlated convergence trades at S1 =1 and S2 =2 when L =0.05. Value of the normalized wealth variance for two positively correlated convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0.5 and the rhs of the VaR constraint is L =0.05. Optimal weights as a function of time for S1 = 1 and S2 = 2 -0.04 Spread 1 nu is: 1 Spread 2 nu is: 1 -0.06 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 -0.08 Spread 2 nu is: 100 -0.1 -0.12 -0.14 Weights -0.16 -0.18 -0.2 -0.22 -0.24 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-36: Optimal weights of two negatively correlated convergence trades for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =0.05. 104 VaR Constraint as a function of time for S1 = 1 and S2 = 2 0.06 nu is: 1 nu is: 10 nu is: 100 0.05 0.04 0.03 Weights 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-37: Value of the normalized wealth variance for two negatively correlated convergence trades at S1 =1 and S2 =2 when L =0.05. Value of the normalized wealth variance for two negatively correlated convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L =0.05. 105 So far we have examined the two trading strategies at values where the drifts are the same. We now choose S1 = 1 and S2 = 1, since for these values the drifts differ for the two strategies. Figure 2-38 shows the weights of the strategies over time for different values of the robustness multiplier when there is no correlation between the two trading strategies and when L =0.05, while Figure 2-39 shows the lhs of the VaR constraint, the normalized wealth variance, over time. We now observe the following: • First of all when there is no fear of model misspecification the ratio of the weights of the two strategies is the same as the ratio of their drifts normalized by their variances and it does not change over time independent on whether the VaR constraint binds or not. • When the VaR constraint binds and there is a fear of model misspecification the investor invests more on the spread with the higher drift. Since the constraint is 2 2 F1 + F2 = L if F1 increases (decreases) over time, then F2 decreases (increases) over time. In Figure 2-38 F1 decreases over time (in magnitude), because the asymmetry between the two strategies goes down as t → T . The existence of this asymmetry is also the cause that when the robustness multiplier is lower the difference in the weights of the two strategies is larger than when the robustness multiplier is higher. • The lower the value of the robustness multiplier, the more time it takes to bind the VaR constraint. • When the VaR constraint does not bind then the investor becomes more and more aggressive, due to the improvement of the investment opportunities. 106 Optimal weights as a function of time for S1 = 1 and S2 = 1 -0.02 -0.04 -0.06 -0.08 -0.1 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 -0.12 Spread 2 nu is: 10 Spread 1 nu is: 100 Weights Spread 2 nu is: 100 -0.14 -0.16 -0.18 -0.2 -0.22 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-38: Optimal weights of two uncorrelated convergence trades for S1 =1 and S2 =1 when L =0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05. VaR Constraint as a function of time for S1 = 1 and S2 = 1 0.06 nu is: 1 nu is: 10 nu is: 100 0.05 0.04 0.03 Weights 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-39: Value of the normalized wealth variance for two uncorrelated convergence trades at S1 =1 and S2 =1 when L =0.05. Value of the normalized wealth variance for two uncorrelated convergence trades as a function of time at S1 =1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L =0.05. 107 It is interesting to note the weights when there is a positive correlation high enough to be long the second spread and use it as a hedge to the first one. In Figure 2-40 we plot the weights of the two trading strategies and we observe that they have different signs. Figure 2-41 shows the normalized wealth variance as a function of time. When the VaR constraint does not bind both the weights increase in magnitude due to the improvement in the investment opportunities. When the VaR constraint binds, both of the weights become larger in magnitude. This is due to the fact that the asymmetry between the two strategies is reduced towards the asymmetry in the case without fear of model misspecification. Finally Figures 2-42 and 2-43 show what happens when there is a negative corre- lation. The results are similar with the case of no correlation. Optimal weights as a function of time for S1 = 1 and S2 = 1 0.2 0.1 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 0 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.1 Weights -0.2 -0.3 -0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-40: Optimal weights of two positively correlated convergence trades for S1 = 1 and S2 = 1 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.8 and the rhs of the VaR constraint is L =0.05. 108 VaR Constraint as a function of time for S1 = 1 and S2 = 1 0.06 nu is: 1 nu is: 10 nu is: 100 0.05 0.04 0.03 Weights 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-41: Value of the normalized wealth variance for two positively correlated convergence trades at S1 =1 and S2 =1 when L =0.05. Value of the normalized wealth variance for two positively correlated convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.8 and the rhs of the VaR constraint is L =0.05. Optimal weights as a function of time for S1 = 1 and S2 = 1 -0.1 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 -0.15 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.2 -0.25 Weights -0.3 -0.35 -0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-42: Optimal weights of two negatively correlated convergence trades for S1 =1 and S2 =1 when L =8. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.8 and the rhs of the VaR constraint is L =0.05. 109 VaR Constraint as a function of time for S1 = 1 and S2 = 1 0.06 nu is: 1 nu is: 10 nu is: 100 0.05 0.04 0.03 Weights 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-43: Value of the normalized wealth variance for two negatively correlated convergence trades at S1 =1 and S2 =1 when L =0.05. Value of the normalized wealth variance for two negatively correlated convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.8 and the rhs of the VaR constraint is L =0.05. 110 2.4.4 Mean reversion trading strategies with constraints We consider first the case where we have N = 1 mean reversion trading strategy with S¯ = 0 and we have a collateral constraint |F | < L. Again due to the symmetry of S around 0 it suffices to study only what happens when S ≥ 0, since the symmetric behaviour of the spread dynamics combined with the symmetry of the constraint imply that the value function is an even function of the spread S and its partial derivative with respect to the spread is an odd function of S for each t. The optimal weight in the trading strategy is given by: 2 2 1 σ VS(St,t) σ VS (St,t) 2 1 − 2 1 ((φ + r)St + ν ) if |(φ + r)St + ν | ≤ Lσ (1 + ν ) σ (1+ ν ) opt 2 σ VS (St,t) 2 1 Ft = −L if (φ + r)St + ≥ Lσ (1 + ) ν ν 2 σ VS (St,t) 2 1 L if (φ + r)St + ν ≤ −Lσ (1 + ν ) opt σ(F +VS) and the minimizing distortion drift is given by: hmin = − ν . When S is positive, we have VS ≥ 0, since higher values of S correspond to better investment opt opportunities. If VS ≥ 0 then −L ≤ F ≤ 0 for S ≥ 0. Therefore we see that again σF opt there is a tradeoff between the two terms in hmin. The first term − ν corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is σVS shorting the spread and it is bounded above, while the second term − ν corresponds to a negative distortion drift that points to worse investment opportunities. The HJB equation is given by: 2 2 1 2 S (φ+r)S 1 2 2 Vt + r +1/2 σ VSS +1/2 1 (φ + r) σ2 + VS(−φS + ν+1 ) − 1/2 ν+1 VS σ =0 1+ ν 2 if |(φ + r)S + σ VS (St,t) | ≤ Lσ2(1 + 1 ) t ν ν 2 2 1 2 σ 1 1 2 2 Vt + r +1/2 σ VSS − 2ν VS − VS(φS − L ν ) − 2 (1 + ν )σ L + L(φ + r)S =0 2 σ VS(St,t) 2 1 if (φ + r)St + ν ≥ Lσ (1 + ν ) 2 2 1 2 σ 1 1 2 2 Vt + r +1/2 σ VSS − VS − VS(φS + L ) − (1 + )σ L − L(φ + r)S =0 2ν ν 2 ν 2 σ VS(St,t) 2 1 if (φ + r)St + ≤ −Lσ (1 + ) ν ν 111 We solve the HJB equation numerically using the method of finite differences. In the following figures we have assumed that rf = 0, σ = 1, φ = 1 and T = 1 for two different constraints, one tight and one very loose. We have similar results with the case where there are no constraints. In addition we observe the following: • VS becomes higher the less tight the constraint is for each value of the robustness multiplier (Figure 2-47). This is to be expected since the tighter the constraints the less we can take advantage the better investment opportunities when S gets higher. • Figure 2-45 shows the two terms of the distortion drift for ν = 2 an L = 0.7. The first term corresponds to a positive distortion drift that reduces the wealth of the investor since the investor is shorting the spread and it is bounded above due to the constraint, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. In this tradeoff the first term wins making the drift distortion positive. • Figures 2-46, 2-48 shows the optimal weight when S = 1 over time for different values of ν. For L =0.7 we see that for high values of ν the constraint binds for all the time period, for lower values of ν the constraint binds initially and then F is reduced after some time t. Finally for even lower values of ν the constraint does not bind at all. • Figure 2-48 shows that when the constraint is tighter the investor is more con- servative even when the constraint does not bind. This is expected since the tighter the constraint the smaller is the VS. This difference in the weights becomes smaller as t → T . 112 Vs as a function of time for S = 1 0.3 nu is: 1 L is: 0.7 nu is: 2 L is: 0.7 nu is: 3 L is: 0.7 0.25 0.2 0.15 Vs 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-44: Partial derivative of the value function with respect to S for a single mean reversion trading strategy and a collateral constraint with L = 0.7. VS as a function of time at S = 1 for different values of the robustness multiplier for L =0.7. Drift distortion and its two terms as a function of time for S = 1 0.4 0.3 h h1 h2 0.2 0.1 Distortion 0 -0.1 -0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-45: Distortion drift terms for a single mean reversion trading strat- egy and a collateral constraint with L =0.7. Distortion drift terms as a function of time at S = 1 for ν = 2 and for L =0.7. The first term corresponds to a positive distortion drift that reduces the wealth of the investor, since the investor is short- ing the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. The first term is bounded above due to the collateral constraint. 113 Optimal weights as a function of time for S = 1 -0.5 -0.55 nu is: 1 L is: 0.7 nu is: 2 L is: 0.7 nu is: 3 L is: 0.7 -0.6 Weights -0.65 -0.7 -0.75 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-46: Optimal weight of a single mean reversion trading strategy with a collateral constraint with L =0.7. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier and for L =0.7. Vs as a function of time for S = 1 0.4 0.35 0.3 0.25 0.2 Vs 0.15 0.1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-47: Partial derivative of the value function with respect to S for a single mean reversion trading strategy with different collateral constraints. VS as a function of time at S = 1 for different values of the robustness multiplier and different collateral constraints. The solid line is for L = 7 and the dotted line for L =0.7. 114 Optimal weights as a function of time for S = 1 -0.5 -0.55 -0.6 -0.65 Weights -0.7 -0.75 -0.8 -0.85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-48: Optimal weight of a single mean reversion trading strategy with different collateral constraints. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier and different collateral constraints. The solid line is for L = 7 and the dotted line for L =0.7. 115 Let us now consider the case where we have N = 2 mean reversion trading strate- gies and again S¯ = 0. We solve numerically the HJB equation using the method of finite differences. We have assumed that rf = 0, T = 1, 2 0 Φ= 0 1 and 1 ρ Σ= ρ 1 Additionally we assume that we have a VaR constraint F T ΣF ≤ L. In Figures 2-49, 2-51 and 2-53 we plot the weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of L and different values of the robustness multiplier. At these values of S1 and S2 the drift is the same for both the strategies. We have assumed that there is no correlation between the two trading strategies. Figures 2-50,2-52 and 2-54 show F T ΣF as a function of time for the different values of the robustness multiplier. We observe the following: • First of all when there is no fear of model misspecification the weights of the two strategies are the same, since there is no asymmetry between them, and they do not change over time independent on whether the VaR constraint binds or not. • When the VaR constraint binds, and there is a fear of model misspecification we invest more on the spread with the higher φ coefficient, due to higher VS. As we saw in the unconstrained case (Figure 2-10), the ratio of the weights is reduced over time, since their asymmetry due to the reduction of the VS1 and VS2 is reduced over time. Therefore, when the VaR constraint binds, the difference in the weights of the two strategies have to go down, as we see in Figures 2-49 and 2-51 where L = 3 and L = 2 respectively. Moreover, the investment in the spread with the higher φ coefficient is higher than the corresponding investment when there is no fear of model misspecification. This is due to the fact that 116 2 2 in both cases we have the same VaR constraint F1 + F2 = L, but in the first case there is an asymmetry causing F1 to be higher and F2 to be lower than the corresponding weights in the second case. • When the VaR constraint does not bind the weights of both of the strategies go down just like in the unconstrained case as we see in Figures 2-49 and 2-53. Optimal weights as a function of time for S1 = 1 and S2 = 2 -1 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 -1.05 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -1.1 -1.15 -1.2 Weights -1.25 -1.3 -1.35 -1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-49: Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 3. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 3. 117 VaR Constraint as a function of time for S1 = 1 and S2 = 2 3.2 nu is: 1 nu is: 10 nu is: 100 3 2.8 2.6 Weights 2.4 2.2 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-50: Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 =1 and S2 =2 when L =3. Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 3. Optimal weights as a function of time for S1 = 1 and S2 = 2 -0.85 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 -0.9 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.95 -1 Weights -1.05 -1.1 -1.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-51: Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. 118 VaR Constraint as a function of time for S1 = 1 and S2 = 2 2.1 nu is: 1 nu is: 10 2.08 nu is: 100 2.06 2.04 2.02 2 Weights 1.98 1.96 1.94 1.92 1.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-52: Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 =1 and S2 =2 when L =2. Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. Optimal weights as a function of time for S1 = 1 and S2 = 2 -1 -1.1 Spread 1 nu is: 1 -1.2 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 -1.3 Spread 1 nu is: 100 Spread 2 nu is: 100 -1.4 -1.5 Weights -1.6 -1.7 -1.8 -1.9 -2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-53: Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 7. 119 VaR Constraint as a function of time for S1 = 1 and S2 = 2 8 Spread 1 nu is: 1 Spread 1 nu is: 10 Spread 1 nu is: 100 7 6 5 Weights 4 3 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-54: Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 =1 and S2 =2 when L =7. Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 7. 120 Figures 2-55 and 2-56 show the weights and the constraint for the case of positive correlation, while Figures 2-57 and 2-58 cover the case for negative correlations. The properties are similar with the ones for the uncorrelated case. Optimal weights as a function of time for S1 = 1 and S2 = 2 -0.6 -0.7 -0.8 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 -0.9 Spread 1 nu is: 100 Spread 2 nu is: 100 -1 Weights -1.1 -1.2 -1.3 -1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-55: Optimal weights of two positively correlated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0.5 and the rhs of the VaR constraint is L = 7. 121 VaR Constraint as a function of time for S1 = 1 and S2 = 2 5.5 5 4.5 Spread 1 nu is: 1 4 Spread 1 nu is: 10 Spread 1 nu is: 100 3.5 3 Weights 2.5 2 1.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-56: Value of the normalized wealth variance for two positively correlated mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 7. Value of the normalized wealth variance for two positively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ =0.5 and the rhs of the VaR constraint is L = 7. Optimal weights as a function of time for S1 = 1 and S2 = 2 -2 Spread 1 nu is: 1 Spread 2 nu is: 1 -2.1 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -2.2 -2.3 -2.4 -2.5 Weights -2.6 -2.7 -2.8 -2.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-57: Optimal weights of two negatively correlated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 7. 122 VaR Constraint as a function of time for S1 = 1 and S2 = 2 7.5 Spread 1 nu is: 1 Spread 1 nu is: 10 Spread 1 nu is: 100 7 6.5 6 Weights 5.5 5 4.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-58: Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 7. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 7. 123 So far we have examined the two trading strategies at values where the drifts are the same. We now choose S1 = 1 and S2 = 1 just like in the unconstrained case, since for these values the drifts differ for the two strategies. Figure 2-59 shows the weights of the strategies over time for different values of the robustness multiplier when there is no correlation between the two trading strategies and when L = 2, while Figure 2-60 shows the VaR constraint over time. We now observe the following: • First of all when there is no fear of model misspecification the ratio of the weights of the two strategies is the same as the ratio of their drifts normalized by their variances and it does not change over time independent on whether the VaR constraint binds or not. • When the VaR constraint binds and there is a fear of model misspecification the investor invests more on the spread with the higher drift. Since the constraint is 2 2 F1 + F2 = L if F1 increases (decreases) over time, then F2 decreases (increases) over time. In Figure 2-59 F1 increases over time, because the asymmetry be- tween the two strategies grows larger as we can see in the unconstrained case in Figure 2-15. • When the VaR constraint does not bind then the investor becomes more and more conservative in both the strategies over time just like in the N = 1 case. 124 Optimal weights as a function of time for S1 = 1 and S2 = 1 -0.5 -0.6 Spread 1 nu is: 1 Spread 2 nu is: 1 -0.7 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.8 -0.9 Weights -1 -1.1 -1.2 -1.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-59: Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 1 when L = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. VaR Constraint as a function of time for S1 = 1 and S2 = 1 2.1 2 nu is: 1 nu is: 10 1.9 nu is: 100 1.8 1.7 1.6 Weights 1.5 1.4 1.3 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-60: Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 =1 and S2 =1 when L =2. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. 125 It is interesting to note the weights when there is a positive correlation high enough to be long the second spread and use it as a hedge to the first one. In Figure 2-61 we plot the weights of the two trading strategies and we observe that they have different signs. Figure 2-62 shows the VaR value as a function of time. When the VaR constraint does not bind both the weights reduce in magnitude like in the unconstrained case. When the VaR constraint binds, both of the weights become larger in magnitude. This is due to the fact that the asymmetry between the two strategies is reduced towards the asymmetry in the case without fear of model misspecification, as we see in the unconstrained case (Figure 2-19). Finally Figures 2-63 and 2-64 show what happens when there is a negative corre- lation. The results are similar with the case of no correlation. Optimal weights as a function of time for S1 = 1 and S2 = 1 1.5 1 0.5 Spread 1 nu is: 1 Spread 2 nu is: 1 Spread 1 nu is: 10 0 Spread 2 nu is: 10 Spread 1 nu is: 100 Spread 2 nu is: 100 -0.5 Weights -1 -1.5 -2 -2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-61: Optimal weights of two positively correlated mean reversion trading strategies for S1 = 1 and S2 = 1 when L = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9 and the rhs of the VaR constraint is L = 2. 126 VaR Constraint as a function of time for S1 = 1 and S2 = 1 2.1 2 nu is: 1 nu is: 10 1.9 nu is: 100 1.8 1.7 1.6 Weights 1.5 1.4 1.3 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-62: Value of the normalized wealth variance for two positively correlated mean reversion trading strategies at S1 = 1 and S2 = 1 when L = 2. Value of the normalized wealth variance for two positively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ =0.9 and the rhs of the VaR constraint is L = 2. Optimal weights as a function of time for S1 = 1 and S2 = 1 -1.2 -1.4 -1.6 Spread 1 nu is: 1 -1.8 Spread 2 nu is: 1 Spread 1 nu is: 10 Spread 2 nu is: 10 Spread 1 nu is: 100 -2 Spread 2 nu is: 100 -2.2 Weights -2.4 -2.6 -2.8 -3 -3.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-63: Optimal weights of two negatively correlated mean reversion trading strategies for S1 = 1 and S2 = 1 when L = 8. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 8. 127 VaR Constraint as a function of time for S1 = 1 and S2 = 1 9 8 nu is: 1 7 nu is: 10 nu is: 100 6 Weights 5 4 3 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 2-64: Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies at S1 = 1 and S2 = 1 when L = 8. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 8. 128 2.5 Conclusions We investigated how the optimal trading strategy of a risk averse investor, facing arbitrage opportunities and risk constraints, changes when he is not confident of his model dynamics. In particular, we assumed that the investor believes that the data come from an unknown member of a set of unspecified alternative models near his approximating model. The investor believes that his model is a pretty good approximation in the sense that the relative entropy of the alternative models with respect to his nominal model is small. Concern about model misspecification leads the investor to choose a robust trading strategy that works well over that set of alternative models. We found what the optimal trading strategy is for the case of the convergence trades and mean reversion trading strategies with and without constraints by solving the corresponding Hamilton Jacobi Bellman equation. In all of our cases we dealt with diffusion processes and the alternative models distorted the conditional mean of the Brownian motion. An interesting extension of our work would be to assume that we have jump processes, where the misspecification is now about the dynamics of the jumps. This will be the topic of future work. 129 130 Chapter 3 Estimating the NIH Efficient Frontier The National Institutes of Health (NIH) is among the world’s largest and most im- portant investors in biomedical research. Its stated mission is to “seek fundamental knowledge about the nature and behavior of living systems and the application of that knowledge to enhance health, lengthen life, and reduce the burdens of illness and disability” (http://www.nih.gov/about/mission.htm). Some have criticized the NIH funding process as not being sufficiently focused on disease burden[69, 45, 43]. We consider a framework in which biomedical research allocation decisions are more directly tied to the risk/reward trade-off of burden-of-disease outcomes. Pri- oritizing research efforts is analogous to managing an investment portfolio—in both cases, there are competing opportunities to invest limited resources, and expected returns, risk, correlations, and the cost of lost opportunities are important factors in determining the return of those investments. Financial decisions are commonly made according to portfolio theory[55], in which the optimal trade-off between risk and reward among a collection of competing investments—known as the “efficient frontier”—is constructed via quadratic opti- mization, and a point on this frontier is selected based on an investor’s risk/reward preferences. Given a measure of “return on investment” (ROI), an “efficient portfo- lio” is defined to be the investment allocation that yields the highest expected return 131 for a given and fixed level of risk (as measured by return volatility), and the locus of efficient portfolios across all levels of risk is the efficient frontier. We recast the NIH funding allocation decision as a portfolio-optimization problem in which the objective is to allocate a fixed amount of funds across a set of disease groups to maximize the expected “return on investment” (ROI) for a given level of volatility. We define ROI as the subsequent improvements in years of life lost (YLL). We use historical time series data provided by the NIH and the Centers for Disease Control for each of 7 disease groups and we estimate the means, variances, and co- variances among these time series. These serve as inputs to the portfolio-optimization problem. Such an approach provides objective, systematic, transparent, and repeat- able metrics that can incorporate “real-world” constraints, and yields well-defined optimal risk-sensitive biomedical research funding allocations expressly designed to reduce the burden of disease. In the rest of this chapter, we will discuss the relevant literature review. Then we will discuss about the data and solution methods used, present our results and conclude with a discussion of our findings. 3.1 NIH Background and Literature Review The National Institutes of Health (NIH) was established in 1938 and has a budget of over $31 billion, of which 80% is awarded in competitive research grants to more than 325,000 researchers through nearly 50,000 competitive grants at over 3,000 uni- versities, medical schools, and other research institutions (http://www.nih.gov). The NIH allocates funding among competing priorities by assessing such priorities with re- spect to five major criteria[65]: (a) public needs; (b) scientific quality of the research; (c) potential for scientific progress (the existence of promising pathways and qualified investigators); (d) portfolio diversification; and (e) adequate support of infrastruc- ture (human capital, equipment, instrumentation, and facilities). This framework was supported, with some additional recommendations, by an Institute of Medicine (IoM) blue-ribbon panel in 1998 (see Table 3.1)[1]. 132 Criteria Processes Public Congress Recommendation 1. The Recommendation 5. In exercising Recommendation 7. NIH should Recommendation 10. The U.S. committee generally supports the the overall authority to oversee and establish an Office of Public Liaison Congress should use its authority to criteria that NIH uses for priority coordinate the priority-setting in the Office of the Director, and, mandate specific research programs, setting and recommends that NIH process, the NIH director should where offices performing such a establish levels of funding for them, continue to use these criteria in a receive from the directors of all the function are not already in place, in and implement new organizational balanced way to cover the full institutes and centers multiyear each institute. These offices should entities only when other approaches spectrum of research related to strategic plans, including budget document, in a standard format, their have proven inadequate. NIH should human health. scenarios, in a standard format on an public outreach, input, and response provide Congress with analyses of annual basis. mechanisms. The director's Office of how NIH is responding to requests for Public Liaison should review and such major changes and whether these evaluate these mechanisms and requests can be addressed within identify best practices. existing mechanisms. Recommendation 2. NIH should Recommendation 6. The director Recommendation 8. The director Recommendation 11. The director make clear its mechanisms for of NIH should increase the of NIH should establish and of NIH should periodically review and implementing its criteria for setting involvement of the Advisory appropriately staff a Director's report on the organizational structure priorities and should evaluate their Committee to the Director in the Council of Public Representatives, of NIH, in light of changes in science use and effectiveness. priority-setting process. The chaired by the NIH director, to and the health needs of the public. diversity of the committee's facilitate interactions between NIH membership should be increased, and the general public. particularly with respect to its public members. Recommendation 3. In setting Recommendation 9. The public Recommendation 12. Congress priorities, NIH should strengthen its membership of NIH policy and should adjust the levels of funding for analysis and use of health data, such program advisory groups should be research management and support so as burdens and costs of diseases, and selected to represent a broad range that the NIH can implement of data on the impact of research on of public constituencies. improvements in the priority-setting the health of the public. process, including stronger analytical, planning, and public interface capabilities. Recommendation 4. NIH should improve the quality and analysis of its data on funding by disease and should include direct and related expenditures. Table 3.1: IoM recommendations. 12 major recommendations of the 1998 In- stitute of Medicine panel in four large areas for improving the process of allocating research funds. 133 Despite this framework and the IoM endorsement, NIH funding has been criticized as not being aligned to disease burden and insufficiently effective[69, 45, 43]. For example, the impact of cancer has been estimated as only 5% of total direct cost but 23% of all deaths[20], while extramural spending by the National Cancer Institute (NCI) is about 15% of the total (http://report.nih.gov/). Sandler et al.[70] suggested that digestive diseases were relatively underfunded based on comparisons of disease burden as measured by direct and indirect cost. Gross[38] noted that NIH funding is reasonably predicted by some burden-of-disease metrics (disability-adjusted life- years or DALY, which are unavailable in time-series form)[57]. Earmarks or target funding levels for specific diseases and programs have been suggested by a number of policymakers[2]. Funding allocation decisions are not unique to the NIH; in a study similar to Gross et al.Gross, Curry et al. [27] has questioned the allocations of the Centers for Disease Control. NIH leaders have noted that funding basic research is itself a risky endeavor, involving trade-offs among all five of their funding criteria, and may also include unstated secondary objectives, e.g., actively “balancing out” spending by other agencies, charities, and the private sector[76]. Collectively, these factors impose significant challenges to determining an ideal allocation of research funds. Although the economic impact of biomedical research has been considered[23], the main focus has been on measuring value-added rather than determining optimal funding allocations. Murphy and Topel [63] estimate U.S. economic surplus from improved health on the order of $2.6 trillion annually, with benefits distributed un- equally across age and gender, and suggest that in some cases, incremental benefits may not exceed the cost of achieving them. Johnston et al.[46] found a return to soci- ety in the form of averted treatment costs and public health benefits divided by cost of trial expenditures of 46% for clinical trials at the National Institute of Neurological Disorders and Stroke (NINDS), where the returns or net savings were generated by four of the 28 trials examined, and collectively exceeded the costs of not only the clinical, but the entire program of research at NINDS during the study period. Cut- ler and McClellan[28] computed returns of technological advances for five conditions 134 and found net benefit for four and costs equal to benefits in the fifth. Fleurence and Togerson[31] suggested that research should be allocated to provide the most health benefits to the population, subject to equity considerations, and observed that sub- jective, burden-of-disease, and payback methods all failed this test to some degree. Instead, they argue that a method of information valuation is superior. Modern financial portfolio theory—in which the expected return, risk (as measured by volatility), and correlations of a collection of investment opportunities are taken as inputs, and the set of all portfolio weights with the highest expected return for a given level of risk is the output—produces rational allocations of limited resources among competing priorities. For developing this method in 1952, Markowitz shared the Nobel Memorial Prize in Economic Sciences in 1990. The theory has had extensive applications among mutual funds, pension funds, endowments, and sovereign wealth funds[55, 56, 24]. More recently, portfolio theory has been proposed as a means for conducting risk- sensitive cost-benefit analysis for health-care budgeting decisions[11, 66, 19, 18, 71, 72]. The motivation for these studies is the observation that typical cost-effectiveness studies of healthcare programs ignore the uncertainty of realized costs, which can be addressed by applying portfolio theory to balance the risks against the rewards of spe- cific budget allocations. These studies present simplified frameworks for incorporating risk into the healthcare budgeting process, e.g., two-security examples (although[71] does contain 11 hypothetical cost/effect distributions) and do not contain full-scale empirical applications to realistic budgeting tasks. As the authors note, applying portfolio theory to large public healthcare reimbursement problems can be challeng- ing. Patients may have differing and non-constant utility functions, and some argue that the manager/administrator should only consider expected returns, allowing the patient and physician to consider risk trade-offs at individual treatment levels, in which case the aggregate utility function is implicit. Despite the growing interest in measuring the return on biomedical research[67, 51], and the fact that portfolio theory has already been applied to healthcare bud- geting decisions, some sceptics continue to argue against the use of any quantita- 135 tive metrics in this domain. For example, Black[14] states categorically that “[t]he biomedical ‘payback’ approach is certainly inappropriate and attempts to impose it should be strenuously resisted. Instead, a qualitative approach should be applied that takes into account the ‘slow-burning fuse’ and avoids simple attribution of cause and effect”. While such a response may be acceptable for certain types of funding, it is becoming increasingly untenable with respect to public funds and government support, which, by law, almost always require some form of cost/benefit analysis, performance attribution, and oversight. 3.2 Methods 3.2.1 Funding Data The NIH has 27 Institutes and Centers, of which we identified 10 with research missions clearly tied to specific disease states, and which account for $21 billion of funding in 2005 or 74% of the total (see Table 3.2 for the disease classification scheme used and Figure 3-1 for the procedure for constructing the appropriation time series). The National Institute of Allergies and Infectious Diseases (NIAID) spending has been split to account for HIV, which is presented separately (see HIV discussion below). These Institutes and the basic research they fund have inevitable overlap and effect beyond their charter; we treat all spending for any given Institute as being directed toward the corresponding disease states, and account for spillover effects by considering the correlations in the lessening of the burden of disease in other groups. For example, molecular biology funded by the NCI may be relevant to infectious diseases but, like the entire NCI budget, would be assumed for modeling purposes to be directed at cancer; the hypothetical infectious-disease improvement would appear in the correlation between the decrease in years of life lost for cancer and that of infectious diseases. 136 Analytic ICD 9 ICD 10 NIH Group Chapter(s) Codes Chapter(s) Blocks Institute(s) AID Infectious and Parasitic Diseases 001-139 Certain infectious and parasitic diseases A00-B99 NIAID NCI Neoplasms 140-239 Neoplasms C00-D48 NCI DDK Endocrine, nutritional and metabolic diseases, and 240-279; Endocrine, nutritional and metabolic diseases; E00-E88; NIDDK immunity disorders; Diseases of the digestive system; 520-579; Diseases of the digestive system; K00-K92; Diseases of the genitourinary system 580-629 Diseases of the genitourinary system N00-N98 HLB Diseases of the blood and blood-forming organs: 280-289; Diseases of the blood and blood-forming organs and D50-D89; NHLBI certain disorders involving the immune mechanism; Diseases of the circulatory system; 390-459; Diseases of the circulatory system; I00-I99; Diseases of the respiratory system 460-519 Diseases of the respiratory system J00-J98 NMH Mental disorders 290-319 Mental and behavioural disorders F01-F99 NIMH CNS Diseases of the nervous system and sense organs 320-389 Diseases of the nervous system; G00-G98; NINDS Diseases of the eye and adnexa; H00-H57; NEI Diseases of the ear and mastoid process H60-H93 NIDCD CHD Complications of pregnancy, childbirth, and the 630-676; Pregnancy, childbirth and the puerperium; O00-O99; NICHD puerperium; Congenital anomalies; 740-759; Certain conditions originating in the perinatal period; P00-P96; Certain conditions originating in the perinatal period 760-779 Congenital malformations, deformations and Q00-Q99 chromosomal abnormalities AMS Diseases of the skin and subcutaneous tissue; 680-709; Diseases of the skin and subcutaneous tissue; L00-L98; NIAMS Diseases of the musculoskeletal system and connective 710-739 Diseases of the musculoskeletal system and connective M00-M99 tissue tissue LAB Symptoms, signs, and ill-defined conditions 780-799 Symptoms, signs and abnormal clinical and laboratory R00-R99 findings, not elsewhere classified EXT External causes of injury and poisoning E800- Codes for special purposes; External causes of morbidity U00-U99; E999 and mortality V01-Y89 Table 3.2: ICD mapping. Classification of ICD-9 (1978–1998) and ICD-10 (1999– 2007) Chapters and NIH appropriations by Institute and Center to 7 disease groups: oncology (ONC); heart lung and blood (HLB); digestive, renal and endocrine (DDK); central nervous system and sensory (CNS) into which we placed dementia and un- specified psychoses to create comparable series as there was a clear, ongoing migration noted from NMH to CNS after the change to ICD-10 in 1999; psychiatric and sub- stance abuse (NMH); infectious disease, subdivided into estimated HIV (HIV) and other (AID); maternal, fetal, congenital and pediatric (CHD). The categories LAB and EXT are omitted from our analysis. 137 Figure 3-1: NIH time series flowchart. Flowchart for the construction of NIH ap- propriations time series. “NIH Approp.” denotes NIH appropriations; “PHS Gaps” denotes Institute funding by the U.S. Public Health Service; “Complete Approp.” denotes the union of these two series; “FY Change” allows for the change in govern- ment fiscal years; “4Q FY” time series refers to the resulting series in which all years are treated as having four quarters of three months each. 25 AID HIV AMS CHD 20 CNS DDK HLB ONC NMH 15 10 Funding (in billions of $) 5 0 1940 1950 1960 1970 1980 1990 2000 Year Figure 3-2: Appropriations data. NIH appropriations in real (2005) dollars, cate- gorized by disease group. 138 3.2.2 Burden of Disease Data Because of its simplicity, availability, breadth, and long history, years of life lost (YLL) was chosen as the measure of burden of disease to be used in constructing the esti- mated return on investment from NIH-funded research. The CDC Wide-ranging On- line Data for Epidemiologic Research (WONDER) database (http://wonder.cdc.gov/) was queried for the underlying cause of death at the Chapter level (except for mental disorders, where dementia and unspecified psychoses were all placed in CNS for consis- tency with CDC coding after 1998) for International Classification of Diseases (ICD) categories ICD-9 (for 1979–1998) and ICD-10 (for 1999–2007). The two datasets for pre- and post-1998 were joined into one continuous series, data were stratified into groups by age at death, and YLL were computed by comparing the midpoint of the age ranges with the World Health Organization’s (WHO) year-2000 U.S. life table (http://www.who.int/whosis/en/). Years of life lost were then tabulated by Chapter annually, and adjusted for population growth to remove what would otherwise be a systematic downward bias in realized health improvements. This process yielded YLL series for 9 distinct disease groups. Using 2005 as the base year, the raw YLL observations were adjusted in other years to be comparable to the 2005 population: raw POP2005 YLLt ≡ YLLt × , POPt ≡ U.S. population in year t . (3.1) POPt The procedure for assembling the YLL time series is summarized in Figure 3-3, and the resulting series, both raw and normalized for population growth, are shown in Figure 3-4. The change in burden of disease was measured by taking first differences. These first differences were used to compute the “return on investment” on which the mean- variance optimizations were based (see the “Methods” section below). Three disease areas required special consideration: HIV, AMS, and dementia. AMS and HIV have shorter histories, which is problematic for estimating parameters based on historical returns that are lagged by typical FDA approval times plus 4 years. 139 Figure 3-3: YLL time series flowchart. Flowchart for the construction of years of life lost (YLL) time series. “WONDER Chapter Age Group” refers to a query to the CDC WONDER database at the chapter level, stratified by age group at death; “US Pop.” is the United States population from census data as expressed in the WONDER dataset; and “US GDP” denotes U.S. gross domestic product. 140 (a) YLL Gross (b) YLL Normalized Figure 3-4: YLL data. Panel (a): Raw YLL categorized by disease group. Panel (b): Population-normalized YLL (with base year of 2005), categorized by disease group. Both panels are based on data from 1979 to 2007. 141 Dementia, including Alzheimer’s disease and unspecified psychoses, was reclassified with the change from ICD-9 to ICD-10 from mental and behavioral disorders to diseases of the nervous system; we placed all dementia YLL in the CNS group to avoid a transition-point artifact at the juncture between ICD-9 and ICD-10, and then performed a sensitivity analysis with and without the dementia YLL. HIV poses a special challenge given its extreme returns after the introduction of protease inhibitors, which are outliers that are likely to be non-stationary and would heavily bias the parameter estimates on which the portfolio optimization is based. To address this outlier, HIV spending and its corresponding YLL were omitted from those of other infectious diseases—the component of NIAID spending directed at HIV was estimated by straight-line interpolation from published figures, and this HIV spending was treated as a separate entity and subtracted from reported NIAID appropriations; a similar procedure was followed for the estimation of HIV-related YLL, and WONDER was queried at the subchapter level to implement this separa- tion. Because of their unique characteristics, these two groups are omitted from our main empirical results. 3.2.3 Applying Portfolio Theory To apply portfolio theory, the concept of a “return on investment” (ROI) must first be defined. Although YLL has already been chosen as the metric by which the impact of research funding is to be gauged, there are at least two issues in determining the relation between research expenditures and YLL that must be considered. The first is whether or not any relation exists between the two quantities. While the objectives of pure science do not always include practical applications that impact YLL, the fact that part of the NIH mission is to “enhance health, lengthen life, and reduce the burdens of illness and disability” suggests the presumption—at least by the NIH—that there is indeed a non-trivial relation between NIH-funded research and burden of disease. For the purposes of this study, and as a first approximation, we assume that YLL improvements are proportional to research expenditures. Of course, factors other than NIH research expenditures also affect YLL, including research 142 from other domestic and international medical centers and institutes, spending in the pharmaceutical and biotechnology industries, public health policy, behavioral patterns, prosperity level and environmental conditions. Therefore, the YLL/NIH- funding relation is likely to be noisy, with confounding effects that may not be easily disentangled. The Discussion section contains a more detailed discussion of this assumption and some possible alternatives. The second issue is the significant time lag between research expenditures and observable impact on YLL. For example, Mosteller[62] cites a lag of 264 years, starting in 1601, for the adoption of citrus to prevent scurvy by the British merchant marine. More contemporary examples[26, 35, 39] cite lags of 17 to 20 years. We use shorter lags in this study both because of data limitations (our entire dataset spans only 29 years), and also to reduce the impact of factors other than research expenditures on our measure of burden of disease (YLL). Any attempt to optimize appropriations to achieve YLL-related objectives must take this lag into account, otherwise the resulting optimized appropriations may not have the intended effects on subsequent YLL outcomes. The impact of NIH-funded research on disease burden is likely to be spread out over several years after this intervening lag, given the diffusion-like process in which research results are shared in the scientific community. For simplicity, the same duration (p = 5 years) of the diffusion-like impact for all the disease groups was hypothesized. The lag q for each disease group was estimated by running linear regressions associating improvements in YLL over p = 5 years with NIH funding q years earlier and real income and choosing the lag between 9 and 16 years ( beyond which data limitations and other factors make it impossible to distinguish the impact of research funding from other confounding factors affecting YLL) that maximizes the R2 and the corresponding lags are shown in Table 3.3. This procedure is, of course, a crude but systematic heuristic for relating research funding to YLL outcomes. Alternatives include using a single fixed lag across all groups, simply assuming particular values for group-specific lags based on NIH man- dates and experience, computing a time-weighted average YLL for each group with 143 a weighting scheme corresponding to an assumed or estimated knowledge-diffusion rate for that group, or constructing a more accurate YLL return series by tracking individual NIH grants within each group to determine the specific impact on YLL (through new drugs, protocols, and other improvements in morbidity and mortality) from the award dates to the present. While the choice of lag is critical in determining the characteristics of the YLL return series and deserves further research, it does not effect the applicability of the overall analytical framework. While our procedure is surely imperfect, it is a plausible starting point from which improvements can be made. Assuming constant impact of research funding on YLL over the duration of p years, the measure of the ROI that accrues to funds allocated in year t is then given by: 1 p−1 p i=0 (YLLt+q+i − YLLt+q+i−1) × GDPt+q+i Rt+q ≡ − (3.2) P Appropriationt where the minus sign reflects the focus on decreases in YLL, and the multiplier GDPt+q is per capita real gross domestic product (GDP) in year t + q , which is included to convert the numerator to a dollar-denominated quantity to match the denominator. This ratio’s units are then comparable to those of typical investment returns: date-(t+q) dollars of return per date-t dollars of investment. Given the definition in equation (3.2) for the ROI of each of the disease groups, the “optimal” appropriation of funds among those groups must be determined, i.e., the appropriation that produces the best possible aggregate expected return on total ′ research funding per unit risk. Denote by R ≡ [ R1 R2 ··· Rn ] the vector of returns of all n groups for a given appropriation date t (where time subscripts have been suppressed for notational simplicity), and denote by µ and Σ the vector of expected returns and the covariance matrix, respectively. If the weights of the budget allocation among the groups are ω, the ROI for the entire portfolio of grants, ′ ′ denoted by Rp, is given by Rp = ω R, and its expected value and variance are ω µ and ω′Σω, respectively. The objective function to be optimized is then given by 144 the expected value minus some multiple of the variance which reflects risk tolerance, and this quadratic function of ω is maximized using standard quadratic optimization techniques, subject to the constraint that the weights sum to 1. In the mean-variance framework, we seek to find the best trade-offs between risk and expected return by varying the portfolio weights ω to trace out the locus of mean-variance combinations that cannot be improved upon, i.e., that are “efficient”. This set of efficient portfolios, also known as the “efficient frontier” is formally defined as the curve in mean-variance (or mean-standard deviation) space corresponding to all portfolios with the highest level of expected return for a given level of variance. This efficient frontier defines the set of allocations that cannot be improved upon from a mean-variance perspective, and the optimal allocation is a single point on this frontier that is determined by the investor’s desired volatility level or risk tolerance. More formally, an investor with standard mean-variance preferences is assumed to prefer portfolios with greater expected return and lower variance, with diminishing returns in each (so that progressively greater increments of expected return must be offered to the investor to induce him to accept increases in the same increment of risk as the level of risk rises). This type of preferences generates so-called “indifference curves” (non-intersecting curves in mean-variance space that trace out combinations of mean and variance for which an individual is indifferent) that are upward sloping and convex. The optimal portfolio for a given set of indifference curves is the tangency point T of the efficient frontier with the most upper-left indifference curve. To compute the efficient frontier, the following optimization problem must be solved (we maintain the following notational conventions: (1) all vectors are column vectors unless otherwise indicated; (2) matrix transposes are indicated by a prime superscript, hence ω′ is the transpose of ω; and (3) vectors and matrices are always typeset in boldface, i.e., X and µ are scalars and X and µ are vectors or matrices): Minimize ω′Σω ω (3.3) ′ ′ subject to ω µ ≥ µo , ω ι =1 , ω ≥ 0 145 where ι is an (n × 1)-vector of 1’s and µo is an arbitrary fixed level of expected return. By varying µo between a range of values and solving the optimization problem for each value, all the efficient allocations ω∗ may be tabulated, and the locus of points in mean-standard-deviation space corresponding to these efficient allocations is the efficient frontier. This so-called Markowitz portfolio optimization problem involves minimizing a quadratic objective function with linear constraints, which is a standard quadratic programming (QP) problem that can easily be solved analytically in some cases[58], and numerically in all other cases by a variety of efficient and stable solvers[37, 36]. One additional refinement to address the well-known issue of “corner solutions” (in which several components of ω∗ are 0) that often arise in the standard portfolio- optimization framework is proposed. While such extreme allocations may, indeed, be optimal with respect to the mean-variance criterion, they are more often the result of estimation error and outliers in the data[8]. Moreover, even in the absence of es- timation error, mean-variance optimality may not adequately reflect other objectives such as social equity across disease groups or distance from current status quo in al- location. To incorporate such considerations, a “regularization” technique is applied in which the objective function is penalized for allocations that are far away from the average allocation policy. Specifically, we consider the following regularized version of the standard portfolio-optimization problem: ′ 2 Minimize ω Σω + γ kω − ωNIH k ω (3.4) ′ ′ subject to ω µ ≥ µo , ω ι =1 , ω ≥ 0 This formulation is essentially a dual-objective optimization problem in which the first objective is to minimize the portfolio’s variance (ω′Σω), and the second objective 2 is to minimize the difference from the average NIH allocation policy (kω − ωNIH k ) and the non-negative parameter γ determines the relative importance of these two objectives. Larger values of γ yield optimal weights that are closer to average NIH al- 146 Group Lag Mean SD Min Med Max Skewness Kurtosis AID 10 -0.9 1.3 -4.3 -0.8 2.4 -0.3 5.2 CHD 12 5.1 3.8 0.2 4.4 11.1 0.1 1.4 CNS 10 -1.7 0.8 -3.1 -1.7 -0.4 0.1 1.9 DDK 11 -0.8 1.5 -3.8 -0.8 2.9 0.2 3.6 HLB 16 9.8 3.4 4.3 9.6 18.6 0.7 3.6 ONC 16 0.5 1.3 -2.3 1.2 2.1 -0.7 2.2 NMH 9 0.0 0.3 -0.8 0.0 0.6 -0.1 3.2 Table 3.3: Return summary statistics. Summary statistics for the ROI of disease groups, in units of years (for the lag length) and per-capita-GDP-denominated re- ductions in YLL between years t and t+4 per dollar of research funding in year t−q, based on historical ROI from 1980 to 2003. Year 1985 1986 1987 1988 1989 1990 YLL 19,741,993 19,380,387 19,015,838 18,951,220 18,086,872 17,670,665 ∆ 361,605 364,549 64,617 864,348 416,207 GDP/Capita ($) 29,443 30,115 31,069 31,877 32,112 GDP-Weighted ∆ ($) 10,646,748,753 10,978,408,090 2,007,589,676 27,552,827,900 13,365,233,032 Mean GDP-Weighted YLL ∆ ($) 12,910,161,490 Lag (years) 16 Funding year 1970 Appropriation $ 695,809,705 ROI 18.6 Table 3.4: ROI example. An example of the ROI calculation for HLB from 1986. location but which correspond to portfolios with greater volatility, and smaller values of γ yield optimal weights that may be more concentrated among a smaller subset of groups, but which imply lower portfolio volatility. 3.3 Results 3.3.1 Summary Statistics Summary statistics of the ROI for the period 1980–2003 are presented in Table 3.3. In Table 3.4 we provide an example of the ROI calculation for HLB for 1986, when the return was 18.6. Large differences in mean ROI for different Institutes are evident in Table 3.3, 147 10 10 Efficient 8 8 AID CHD 6 6 CNS DDK 4 4 HLB ONC 2 2 NMH 0 0 NIH Avg 1/n −2 −2 NIH−Var 0 1 2 3 4 0 1 2 3 4 NIH−Mean (a) With Alzheimer effect, gamma = 0 (b) With Alzheimer effect, gamma = 5 Min−Var Eff−25% 10 10 Eff−50% Eff−75% 8 8 6 6 4 4 2 2 0 0 −2 −2 0 1 2 3 4 0 1 2 3 4 (c) Without Alzheimer effect, gamma = 0 (d) Without Alzheimer effect, gamma = 5 Figure 3-5: Efficient frontiers. Efficient frontiers for (a) all groups except HIV and AMS, γ = 0; (b) all groups except HIV and AMS, γ = 5; (c) all groups except HIV and AMS without the dementia effect, γ =0; and (d) all groups except HIV and AMS without the dementia effect, γ =5; based on historical ROI from 1980 to 2003. ranging from small negative values (e.g., −1.7 for CNS) to large positive values (e.g., 9.8 for HLB). Large differences in standard deviation also exist, ranging from 0.3 for NMH to 3.8 for CHD. 3.3.2 Efficient Frontiers In Figure 3-5 , efficient frontiers for the single- and dual-objective optimization prob- lems are plotted in mean-standard deviation space for the 7-group cases with and without taking into account the dementia effect. 148 For each of these frontiers, the mean-standard deviation points for the following funding allocations are also plotted: (i) historical average NIH allocation for years 1996–2005; (ii) equal-weighted (1/n) allocation; (iii) minimum-variance allocation; (iv) the allocation on the efficient frontier that has the same mean as the average NIH allocation (the “NIH-mean” allocation); (v) the allocation on the efficient frontier which has the same variance as the average NIH allocation (the “NIH-var” allocation); (vi) the allocation on the efficient frontier that is 25% of the distance from the minimum variance allocation to the maximum expected-return allocation; (vii) the allocation on the efficient frontier that is 50% of the distance from the minimum variance allocation to the maximum expected-return allocation; (viii) the allocation on the efficient frontier that is 75% of the distance from the minimum variance allocation to the maximum expected-return allocation. The region bounded by (i), (iv), (v) and the efficient frontier is of special interest because all portfolios in this region offer lower variance, higher expected return, or both when compared to the average NIH allocation, hence from a mean-variance perspective such allocations are unambiguously preferable. These allocations are called “dominating” portfolios relative to the average NIH allocation (i). Figure 3-5a shows that a number of the disease groups appear to be concentrated in a relatively low-risk sector of the risk/reward universe, which may be evidence of active variance-minimization strategies by various stakeholders. A sensitivity analysis is conducted by estimating the efficient frontier with (Fig- ure 3-5a) and without the dementia effect (Figure 3-5c). Table 3.5 contains the portfolio weights corresponding to Figures 3-5a and 3-5c respectively. 149 Benchmarks Single-Objective Portfolios (in %) Dual-Objective Portfolios (γ = 5) (in %) Group NIH NIH- NIH- Min- Eff- Eff- Eff- NIH- NIH- Min- Eff- Eff- Eff- 1/n Avg Var Mean Var 25% 50% 75% Var Mean Var 25% 50% 75% All Groups: AID 8 14 0 0 0 0 0 0 0 3 5 0 0 0 CHD 7 14 24 11 0 13 27 36 18 11 7 18 28 34 CNS 14 14 0 0 25 0 0 0 7 15 19 8 0 0 DDK 10 14 0 0 0 0 0 0 0 5 8 0 0 0 HLB 17 14 23 11 0 13 32 55 24 14 9 23 39 59 ONC 27 14 53 28 16 33 42 9 34 33 32 34 31 7 NMH 16 14 0 50 58 41 0 0 17 20 21 17 2 0 Without Dementia: AID 8 14 0 0 0 0 0 0 0 3 5 0 0 0 CHD 7 14 23 11 0 14 28 36 17 11 8 18 28 34 CNS 14 14 2 32 41 27 0 0 12 18 19 11 0 0 DDK 10 14 0 0 0 0 0 0 0 5 8 0 0 0 HLB 17 14 23 12 0 16 34 55 23 13 9 24 40 60 ONC 27 14 52 37 17 43 39 9 33 32 31 33 31 6 NMH 16 14 0 8 41 0 0 0 15 19 20 14 1 0 Table 3.5: Portfolio weights. Benchmark, single- and dual-objective optimal port- folio weights (in percent), based on historical ROI from 1980 to 2003. The top left sub-panel of Table 3.5 shows that the single-objective optimization does yield sparse weights as expected. For example, the minimum-variance portfolio allocates to only three groups: 58% to NMH, 25% to CNS, and 16% to ONC. By min- imizing variance, irrespective of the mean, this portfolio allocates funding to groups with least variability in YLL improvements. The efficient-25% portfolio allocates non-zero weights in four groups (41% to NMH, 33% to ONC, 13% to HLB, and 13% to CHD), and yields 26% better expected return with 28% less risk. With still more emphasis on expected return, the efficient-50% portfolio gives non-zero weights only to three successful groups: 42% to ONC, 32% to the higher risk, higher expected- return HLB, and 27% to CHD. This portfolio has 172% higher expected return but only 27% more risk than the NIH portfolio. The efficient-75% portfolio gives an even higher weight of 55% to HLB, 36% to CHD, and 9% to ONC, yielding 318% higher expected return and 148% more risk, a diminishing risk-adjusted expected return as compared to portfolios with lower volatility. Given the greater emphasis on expected return for this portfolio, it is not surprising to see HLB getting a bigger role due to its apparent historical success in reducing YLL. Of course, whether or not past suc- 150 cess is indicative of comparable future success hinges on the science and associated translational efforts underlying the diseases covered by HLB. This underscores the importance of incorporating research and clinical insights into the funding allocation process, especially within a systematic framework such as portfolio theory. However, the dementia effect may underestimate the performance of the CNS disease group, hence the lower panel of Table 3.5 reports corresponding optimal- portfolio results without the dementia effect. In the single-objective case, the efficient- 50% and 75% portfolios are still sparse, with non-zero weights in 3 groups, while the lower risk efficient-25% portfolio is less concentrated with non-zero weights to 4 groups and significant weight (27%) to the CNS group. Table 3.5 also contains the optimal portfolios for the dual-objective case (with γ = 5) in the right sub-panels (see Figures 3-5b and 3-5d). These cases correspond to portfolios that trade off closeness to the average NIH allocation policy with better risk-adjusted expected returns. Now we observe that for both upper and lower sub- panels corresponding to the 7-group with/without the dementia effect optimization, respectively, the weights are less concentrated than in the single-objective case. For example, the minimum-variance portfolio without the dementia effect now allocates funding to all the groups, with weights ranging from 5% to 31%. However, even in this case, the efficient-75% portfolio is still extreme, allocating weights only to HLB, CHD and ONC. Therefore, special care must be exercised in selecting the appropriate point on the efficient frontier. We also observe from the NIH-var or NIH-mean portfolios that slight changes to the average NIH policy apparently yield superior performance in mean-standard deviation space (28% to 89% relative improvement, depending on the assumptions). 3.4 Discussion Portfolio theory provides a systematic framework for determining optimal research funding allocations based on historical return on investment, variance, and correlation between appropriations and reductions in disease burden. The optimization results 151 suggest that significant YLL improvements with respect to a mean-variance criterion may be possible through funding re-allocation. To our knowledge, this is the first time such an approach has been empirically implemented in this domain. However, our findings must be qualified in at least three respects: (1) YLL as a measure of burden of disease, which is clearly incomplete and less than ideal; (2) the definition of ROI and the challenges of relating research expenditures to subsequent outcomes such as burden of disease; and (3) the known limitations of portfolio the- ory. While each of these qualifications can be addressed to varying degrees through additional data and analysis, the empirical conclusions are likely to depend critically on the nature of their resolution. In this section, we provide a short synopsis of these qualifications, and also consider other objections to this framework and directions for future research. YLL captures only the most extreme form of disease burden, and other measures such as disability-adjusted or quality-adjusted life years are clearly preferable. How- ever, time series histories for such measures are currently unavailable; hence YLL is the most natural starting point for gauging the impact of biomedical research funding, and is directly aligned with the NIH mission to “lengthen life”. As a measure of disease burden, YLL captures only lethal illness by definition; chronic illness enters the optimization process only indirectly, mortality in the young is more heavily weighted than that of the elderly, and quality-of-life is not captured at all. The choice of YLL is motivated by several factors: long time-series observations of YLL are readily available, they cover a large population, and they address the entire spectrum of diagnoses categorized under the ICD. Broader measures of burden of disease such as disability adjusted life years (DALY)[38] and quality-adjusted life years (QALY)[68, 22] have been proposed, but historical time series for such measures are not yet available. As better measures are developed (e.g., incidence, prevalence, physician visits, hospitalization, DALY, QALY), portfolio-optimization methods may be applied to them as well through appropriately defined “returns”. Should datasets covering not only age and cause of death but also ante-mortem symptoms become available, mean-variance-efficient allocations would likely place significant weight on 152 improvements in the care of less-lethal chronic diseases. Even if YLL is an appropriate measure of disease burden, our definition of ROI can also be challenged as being imprecise and ad hoc in several respects. NIH funding is typically focused on basic research rather than translational efforts, therefore, NIH spending may not be as directly related to subsequent YLL improvements. We have not accounted for other expenditures that may also affect YLL, and to the extent that NIH appropriations are systematically used to complement private spending to allocate total funding across diseases more fairly[76], the relation between NIH funding and subsequent YLL improvements may be even noisier, and may require modelling private-sector expenditures as a separate but complementary portfolio- optimization problem with an objective function and constraints that are linked to those of the NIH. Also, the standard portfolio-optimization framework implicitly as- sumes a constant multiplicative relation between dollars invested today and dollars returned tomorrow (so that doubling the investment will typically double the ROI of that investment), whereas the return to biomedical investments may be non-linear. In addition, translational research takes time and significant non-NIH resources, fur- ther blurring the relation between NIH allocations and subsequent changes in YLL. Finally, other factors may contribute to YLL improvements, including changes in cul- tural norms (including consumption of alcohol and cigarettes), economic conditions (such as recessions vs. expansions), and public policy (such as vaccine programs and mandates for automobile, home, and workplace safety). While all of these qualifica- tions have merit, they are not insurmountable obstacles and can likely be addressed through additional data collection and more sophisticated metrics, perhaps along the lines of Porter[67] or Lane and Bertuzzi[51]. Moreover, the portfolio-optimization approach provides a useful conceptual framework for formulating funding allocation decisions systematically, even if its empirical implications are imprecise. The estimates of q were an initial attempt to link appropriation with outcome in a systemic and non-discretionary manner, but they were derived heuristically from regulatory, appropriation, and epidemiological data which may not be stationary or predictive. For example, if the Food and Drug Administration’s capacity for reviewing 153 new-drug applications is held constant and applications double, substantial increases in regulatory queuing would be expected, even with the added resources generated by the Prescription Drug User Fee Act. Finally, in converting changes in YLL to dollar amounts, per-capita real GDP was used as the “conversion factor” irrespective of age, despite the fact that children and retired individuals are economically less active. While these caveats highlight the imprecision with which the impact of research spending is measured, they also provide direction for developing better metrics. In particular, the underlying science of each grant implies a particular set of dynamics for translation and YLL impact, and with more sophisticated models of such dynamics, the returns to fundamental research should be measurable with greater accuracy. Even within the exact domain for which it was developed, portfolio theory has several well-known limitations, of which the most obvious is the possibility that the mean-variance criterion may not, in fact, be the appropriate objective function to be optimized. While there is little disagreement that higher expected ROI is prefer- able to its alternative, the trade-off between expected ROI and risk is fraught with subtleties involving specific psychological, perceptual, and behavioural mechanisms of individuals and groups. Because of these considerations, mean-variance analysis is often considered an approximation to a much more complicated reality—a starting point for investment allocation decisions, not the final answer. Another known limitation of portfolio theory is the fact that the input parameters (µ, Σ) must be estimated from historical data, and estimation error in these param- eter estimates can lead to portfolios that are unstable and sub-optimal[55]. One common approach to addressing this problem in the financial context is to employ prior information regarding the input parameters, thereby reducing the dependence on historical data. Using Bayesian methods, expert opinions regarding the statisti- cal properties of the individual asset returns can be incorporated into the portfolio optimization process[13, 5] [12]. One limitation that is unique to the current application is the fact that portfolio theory is silent on which mean-variance-optimal portfolio to select. In the financial context, the existence of a riskless investment (e.g., U.S. Treasury bills) implies that 154 one unique portfolio on the efficient frontier will be desired by all investors—the so- called “tangency” portfolio[73]. Because there is no analog to a riskless investment in biomedical research, the notion of a tangency portfolio does not exist in this context. Therefore, decision makers must first determine society’s collective preferences for risk and return with respect to changes in YLL before a unique solution to the portfolio- optimization problem can be obtained, i.e., they must agree on a societal “utility function” for trading off the risks and rewards of biomedical research. This critical step is a pre-requisite to any formal analysis of funding allocation decisions, and underscores the need for integration of basic science with biomedical investment performance analysis and science policy. Such integration will require close and ongoing collaboration between scientists and policymakers to determine the appropriate parameters for the funding allocation process, and to incorporate prior information and qualitative judgments[14] regarding likely research successes, social priorities, policy objectives and constraints, and hidden correlations due to non-linear dependencies not captured by the data. In particular, it is easy to imagine contexts in which funding objectives can and should change quickly in response to new envi- ronmental threats or public-policy concerns. However, such pressing needs must be balanced against the disruptions—which can be severe due to the significant adjust- ment costs implicit in biomedical research[32]—caused by large unanticipated positive or negative shifts in research funding. Although the end result of collaborative discus- sion may fall short of a well-defined objective function that yields a clear-cut optimal portfolio allocation, the portfolio-optimization process provides a transparent and ra- tional starting point for such discussions, from which several insights regarding the complex relation between research funding and social outcomes are likely to emerge. Any repeatable and transparent process for making funding allocation decisions— especially one that involves criteria other than peer-review-based academic excellence— will, understandably, be viewed with some degree of suspicion and contempt by the scientific community. However, if one of the goals of biomedical research is to reduce the burden of disease, some tension between academics and public policy may be unavoidable. Moreover, in the absence of a common framework for evaluating the 155 trade-offs between academic excellence and therapeutic potential, other approaches such as political earmarking[2] are being proposed, which may be even less palatable from the scientific perspective. In an environment of tightening budgets and increasing oversight of appropria- tions, portfolio theory offers scientists, policymakers, and regulators—all of whom are, in effect, research portfolio managers—a rational, systematic, transparent, and reproducible framework in which to explicitly balance and trade off expected benefits with potential risks while accounting for correlation among multiple research agen- das and real-world constraints in allocating scarce resources. Most funding agencies and scientists have already been making such trade-offs informally and heuristically; there may be additional benefits to making such decisions within an explicit frame- work based on standardized and objective metrics. One of the most significant benefits from adopting such a framework may be the reduction of uncertainty surrounding future funding-allocation decisions, which would greatly enhance the ability of funding agencies and scientists to plan for the future and better manage their respective budgets, research agendas, and careers. By approach- ing funding decisions in a more analytical fashion, it may be possible to improve their ultimate outcomes while reducing the chances of unintended consequences. 156 Chapter 4 Impact of model misspecification and risk constraints on market In Chapters 1, 2 we studied the optimal trading strategy of a risk averse investor who faces risk constraints and model misspecification. In this Chapter we will study how risk constraints and fear of model misspecification affect the statistical properties of the market returns. In particular, we will study their effect on the risk premium, the volatility and liquidity of the market. We find that the statistical properties of the market change. In particular, vari- ability of the risk constraints leads to increasing risk premium, increasing volatility and increasing illiquidity of the market. In addition, tightening of these constraints leads also to increasing risk premium, increasing volatility and increasing illiquidity. Moreover, we find that variability in risk aversions along with risk constraints also lead to a more concave pricing function of the aggregate supply for the market, imply- ing increasing risk premium, increasing volatility and increasing illiquidity. Finally, we explore how the properties of the asset returns change when the investors do not completely trust their models. We find that model misspecification is another source of increasing risk premium, endogenous volatility and increasing illiquidity. In the rest of this chapter, we will discuss the relevant literature review. Then we will discuss about the setup of the model, and we will analyze the impact on the market returns of varying risk constraints across agents, varying risk aversions across 157 agents and varying degrees of fear of model misspecification across agents. Finally, we will conclude with all of our results. 4.1 Literature review In the literature there are papers that assume heterogeneity along three dimensions: risk aversion coefficients of the agents, constraints the agents face and beliefs of the agents. Danielsson and Zigrand [81] assume that the agents have the same beliefs and face the same constraints but they differ in their risk aversion coefficients. They study the economic implications of a Value-at-risk based regulatory system by analyzing a two period multi-asset general equilibrium model with agents heterogeneous in risk preferences and wealth. They assume that the agents have CARA utilities and they argue that there will be endogenous volatility and increasing risk premium due to the fact that “... risk will have to be transferred from the more risk-tolerant to the more risk-averse”. As we will prove this is not true and not necessary for having increasing risk premium, endogenous volatility and increasing illiquidity. Kogan and Uppal [50] show how to analyze the equilibrium prices and policies in an economy with incomplete financial markets and stochastic investment opportunity set, where the agents face portfolio constraints. They study a general equilibrium exchange economy with multiple agents, who differ in the risk aversion coefficients and face borrowing constraints, while having the same beliefs. Brumm et al. [21] consider a general equilibrium infinite-horizon economy with the agents having heterogeneous risk preferences and facing the same constraints, while having the same beliefs. They find that the presence of collateral constraints leads to strong excess volatility and a regulation of margin requirements potentially has stabilizing effects. Then, there are papers with different prior beliefs not due to asymmetric infor- mation among the agents. Geanakoplos [33] assumes that the agents have different priors (optimists, pessimists) but same risk aversion coefficients and wealth and they 158 face identical collateral constraints. He studies how these constraints determine an equilibrium leverage and how this leverage changes over time leading to crashes and boom periods, the so-called leverage cycles. Chen, Hong and Stein [25] study what happens to the price of a risky asset, when there are investors with heterogeneous priors who face short sales constraints. The idea that short sales constraints increase the prices of risky assets when the investors have heterogeneous beliefs is due to Lintner [52] and Miller [61]. Chen, Hong and Stein show that greater dispersion of beliefs leads to even higher prices. Finally, Hansen and Sargent [41] study a framework where the agents have a common approximating model, but they differ in the degree of mistrust of the model. They find that agent’s caution in responding to concerns about model misspecification can raise prices assigned to macroeconomic risks. We will see how risk constraints affect the statistical properties of the market, in particular the risk premium, volatility and liquidity of the market. We will first study the case where the investors differ in the constraints they face and/or their risk aversion coefficients and then the case where they mistrust the model of asset payoffs and the mistrust varies among the investors. 4.2 Analysis 4.2.1 Model setup We assume we have H mean-variance single-period optimizers with heterogeneous risk aversions, risk constraints and wealth. Each agent can invest in the market and the risk free rate at t = 0. We assume that the risk-free rate is exogenously given and the market is modeled as a risky asset with stochastic payoff at t = 1. There are also noise traders. We do not model their utility explicitly, we only assume that they are hit by random liquidity shocks and they submit random market orders at time t = 0. Equivalently, the supply of the risky asset is stochastic. Each agent faces a risk constraint, a constraint in his wealth volatility of the form |θh| ≤ LhWh0, where 159 θh is the position of the h agent in the market, Wh0 is the initial wealth of agent h and Lh determines the tightness of the risk constraint that the h agent faces. Each agent’s wealth at time t=1 is given by: Wh1 = dθh +(Wh0 − qθh)Rf where d is the payoff of the risky asset (market), θh is the number of shares, q is the price of the risky asset and Rf is the risk-free rate. It is: µθˆ +(W − qθ )R E(R)= h h0 h f Wh0 whereµ ˆ is the expected payoff of the risky asset. We also have: 2 2 σˆ θh var(R)= 2 Wh0 whereσ ˆ is the volatility of the payoff of the risky asset. Each agent solves the following optimization problem: θh 1 2 θh 2 maximize (ˆµ − qRf ) − γhσˆ ( ) θh Wh0 2 Wh0 subject to |θh| ≤ LhWh0 where γh is the agent’s risk aversion coefficient. By solving the KKT conditions we find: µˆ − qR θopt = f (4.1) h γh 2 σˆ (1 + λh) Wh0 |µˆ−qRf | where 1 + λh = max(1, 2 ) Lhγhσˆ 160 The solution can also be written as: 2 2 µˆ−qRf µˆ−Lhγhσˆ µˆ+Lhγhσˆ γh 2 if ≤ q ≤ σˆ Rf Rf Wh0 opt 2 θ = µˆ−Lhγhσˆ (4.2) h LhWh0 if q ≤ Rf 2 µˆ+Lhγhσˆ −LhWh0 if q ≥ Rf A competitive equilibrium is a set of portfolios (θ1, ··· , θH ) and a price q such that: • Markets clear • Each agent’s portfolio is optimal The market clearing condition implies that: opt θh = θα ⇒ hX∈H µˆ − Ψˆσ2θ q = α Rf 1 1 where θα is the aggregate supply of the risky asset and ≡ γh Ψ h∈H (1+λh) Wh0 We will consider two special cases: P • Constraints vary across the agents • Risk aversion relative to wealth varies across the agents 4.2.2 Varying constraints In the first case, we assume that the agents face heterogeneous constraints. In partic- γh ular we assume that the parameter Lˆh ≡ LhWh0 varies across the agents, while Wh0 is constant equal to γ. Without loss of generality, we assume that Lˆ1 ≤ Lˆ2 ···≤ LˆH . µˆ−qRf From the market clearing condition, we have that: Ψθα = σˆ2 . In addition from 161 equation 4.1, we have that: µˆ − qRf opt γh 2 = θh (1 + λh) σˆ Wh0 opt = θh γ(1 + λh) opt Therefore, it is: Ψθα = θh γ (1 + λh) ∀h ∈ H. We perform a sensitivity analysis for two cases: • Keep Lˆh constant and change θα. By changing the aggregate supply, we find that: – Ψθα is a piecewise linear convex increasing function of the aggregate supply. γ – Its slope is given by H−i+1 till the constraint binds for agent i. In other words, at each point it is equal to γ over the number of agents for whom the constraint is not binding yet. – The constraint binds for agent 1 when θα,1 = HLˆ1 and for agent i when θα,i = θα,i−1 +(H − i +1)(Lˆi − Lˆi−1). These are the kink points in Figures 4-1, 4-2, 4-3. ˆ – When the aggregate supply is greater than θα = h∈H Lh there is no equi- librium, since in that case all the agents are constrainedP in their positions and they cannot buy any more shares and therefore the market cannot clear. 2 – Since q = µˆ−Ψˆσ θα , we see that the pricing function is a piecewise linear Rf concave decreasing function of the aggregate supply, as we see in Figures 4-1, 4-2, 4-3. – Therefore, variability of the constraints leads to increasing risk premium, increasing volatility and increasing illiquidity, since a small change in the aggregate supply, a small liquidity shock by the noise traders leads to a larger change in the price of the risky asset comparing to the case, where there is no variability in the constraints the different agents face. Figure 162 Price of the risky asset 1.6 Variability in constraints Same constraints 1.4 1.2 1 Price 0.8 0.6 0.4 0.2 0 50 100 150 Aggregate supply Figure 4-1: Price of the risky asset as a function of the aggregate market supply under varying constraints. We assume that we have 5 agents with the same risk aversion coefficients. The red plot assumes the same L = 30 for all the agents, while the blue assumes L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 = 50. 4-1 shows the pricing function when the agents face the same constraints and when they face variable constraints. Figure 4-3 also shows the pricing function of the risky asset when the agents face two sets of constraints with the same mean but with different variability. • Keep θα constant and change Lˆh. As we see in Figure 4-2 as we tighten the constraints, the pricing function becomes more concave. Therefore, tightening of the constraints leads to increasing risk premium, increasing volatility and increasing illiquidity. 163 Price of the risky asset 1.6 Different constraints Lh Constraints reduced by a factor 1/5 1.4 1.2 1 Price 0.8 0.6 0.4 0.2 0 50 100 150 Aggregate supply Figure 4-2: Price of the risky asset as a function of the aggregate market supply under tightening constraints. We assume that we have 5 agents with the same risk aversion coefficients. The blue plot assumes L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 = 50 and the red assumes that each Li is reduced by 20%. Price of the risky asset 1.6 Different constraints Lh Less variable constraints by a factor of 2 1.4 1.2 1 Price 0.8 0.6 0.4 0.2 0 50 100 150 Aggregate supply Figure 4-3: Price of the risky asset as a function of the aggregate market supply with less variable constraints. We assume that we have 5 agents with the same risk aversion coefficients. The blue plot assumes L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 = 50 and the red assumes that L1 = 20, L2 = 25, L3 = 30, L4 = 35, L5 = 40. 164 4.2.3 Varying risk aversions So far we have assumed that the agents have heterogeneous constraints but same normalized risk aversions. Now, we assume that the risk aversion relative to wealth γh γˆh ≡ varies across the agents, while the parameters Lˆh are constant equal to L. Wh0 Without loss of generality, we assume thatγ ˆ1 ≤ γˆ2 ··· ≤ γˆH . As we proved in the opt previous section, we have: Ψθα = θh γˆh (1 + λh) ∀h ∈ H. We perform a sensitivity analysis for two cases: • Keepγ ˆh constant and change θα. By changing the aggregate supply, we find that: – Ψθα is a piecewise linear convex increasing function of θα. – Slope is ( H 1 )−1 till the constraint binds for agent i. In other words, h=i γˆh at each pointP it is equal to the aggregate risk aversion of the agents whose constraint is not binding yet. H γˆi – The constraint binds for agent i when θα,i = iL + L . These are k=i+1 γˆk the kink points in Figures 4-4, 4-5. P – When the aggregate supply is greater than θα = HL there is no equilib- rium, since in that case all the agents are constrained in their positions, they cannot buy any more shares and therefore the market cannot clear. 2 – Since q = µˆ−Ψˆσ θα , we see that the pricing function is a piecewise linear Rf concave decreasing function of the aggregate supply, as we see in Figures 4-4, 4-5. – Therefore, variability of the risk aversion coefficients and constraints leads to increasing risk premium, increasing volatility and increasing illiquidity, since a small change in the aggregate supply, a small liquidity shock by the noise traders leads to a larger change in the price of the risky asset comparing to the case, where there is variability in the risk aversion co- efficients but no constraints. Figure 4-4 shows the pricing function of the 165 Price of the risky asset 1.6 Variability in risk aversions No constraints 1.4 1.2 1 Price 0.8 0.6 0.4 0.2 0 50 100 150 Aggregate supply Figure 4-4: Price of the risky asset as a function of the aggregate mar- ket supply with constraints and varying risk aversions. We assume that we have 5 agents with same constraints but different risk aversion coefficients. The blue plot assumes L = 30 for each agent, while the red line assumes that the agents are unconstrained. risky asset when the agents with different risk aversion coefficients face constraints and when they do not face any constraints. • Keep θα constant and change L. As we see in Figure 4-5 as we tighten the constraints, the pricing function becomes more concave. Therefore, tightening of the constraints leads to increasing risk premium, increasing volatility and increasing illiquidity. 166 Price of the risky asset 1.6 Variability in risk aversions Tighter constraints 1.4 1.2 1 Price 0.8 0.6 0.4 0.2 0 50 100 150 Aggregate supply Figure 4-5: Price of the risky asset as a function of the aggregate market supply with tightening constraints and varying risk aversions. We assume that we have 5 agents with same constraints but different risk aversion coefficients. The blue plot assumes L = 30 for each agent, while the red line assumes that L = 20 for each agent. 167 4.2.4 Varying constraints and risk aversions γh Now we assume that both the risk aversion relative to wealthγ ˆh ≡ and the Wh0 parameters Lˆh vary across the agents. Without loss of generality we assume that γˆ1Lˆ1 ≤ γˆ2Lˆ2 ···≤ γˆH LˆH . By changing the aggregate supply, we find that: • Ψθα is a piecewise linear convex increasing function of θα. • Slope is ( H 1 )−1 till the constraint binds for agent i. In other words, at each h=i γˆh point it isP equal to the aggregate risk aversion of the agents whose constraint is not binding yet. • The constraint binds increasing in the order ofγ ˆiLˆi = γiLi. It binds for agent i H γˆi i when θα,i = Lˆk + Lˆi . k=1 k=i+1 γˆk P P ˆ • When the aggregate supply is greater than θα = h∈H Lh there is no equilib- rium, since in that case all the agents are constrainedP in their positions, they cannot buy any more shares and therefore the market cannot clear. • The pricing function of the risky asset is a piecewise linear concave decreas- ing function of the aggregate supply. That means increasing risk premium, increasing volatility and increasing illiquidity. 4.2.5 Varying fear of model misspecification We now study how fear of model misspecification across the agents affects the sta- tistical properties of risky assets. We change a little our initial model setup. In particular, we assume we have H investors with CARA utilities, heterogeneous risk aversions and wealth. Each investor can invest in N risky assets and the risk free rate at t = 0. Each agent mistrusts the model that describes the payoff distribution of the risky assets. All the agents have a common nominal model but they have heterogeneous fears of model misspecification. We assume that the risk-free rate is exogenously given and the common approximating model describing the risky assets’ payoff distribution at t = 1 is N(ˆµ, σˆ2). There are also noise traders. We do not 168 model their utility explicitly, we only assume that they are hit by random liquidity shocks and they submit random market orders at time t = 0. Equivalently, the supply of the risky assets is stochastic. Each agent’s wealth at time t=1 is given by: T T Wh1 = d θh +(Wh0 − q θh)Rf N where d ∈ R describes the payoff of the N risky assets, θh describe the number of N shares of the assets, q ∈ R is the vector of prices of the risky assets and Rf is the risk-free rate. The common approximating model of the payoff distribution is d =µ ˆ+Σˆ 1/2ǫ where ǫ follows the standard multivariate Gaussian distribution. The alternative models alter the distribution of ǫ. In particular, similarly with the framework described in Chapter 2, they change the mean of the shocks and they assume that the distribution of the shock is N(h, I). Therefore, under the alternative models the payoff is d = µˆ + Σˆ 1/2(ǫ + h) i.e. d follows N(ˆµ + Σˆ 1/2h, Σ).ˆ The relative entropy of the alternative distributions with respect to the nominal distribution is given by:D(Q)=1/2hT h as we have shown in Chapter 2 and proved in the Appendix. Under the alternative distributions the certainty equivalence of the investors with CARA utilities is γ(W0 − T T 2 T T 1/2 q θ)Rf + γµˆ θ − 1/2γ θ Σˆθ + γθ Σˆ h. We assume each investor solves the following optimization problem: T T 2 T T 1/2 T max min γ(W0 − q θ)Rf + γµˆ θ − 1/2γ θ Σˆθ + γθ Σˆ h + λ1/2h h (4.3) θ h where γ is the risk aversion coefficient of the investor and λ is the multiplier that penalizes the relative entropy of the alternative distribution. By solving this optimization problem (see Appendix) we find that: ˆ −1 opt Σ (ˆµ − Rf q) θh = 1 (4.4) γh(1 + ) λh 169 The market clearing condition for the risky assets implies that (see Appendix): opt θh = θα hX∈H µˆ − ΨΣˆθ q = α Rf N 1 1 where θα ∈ R is the aggregate supply of the risky assets and ≡ 1 . Ψ h∈H γh(1+ ) λh In case the investors fully trusted their model dynamics we wouldP have µˆ − Ψ Σˆθ q = nr α (4.5) Rf but now 1 ≡ 1 . Ψnr h∈H γh So we see theP case when the agents mistrust their models is equivalent to the case where they fully trust their models but with increasing effective risk aversion 1 γeff = γ(1 + λ ). By changing the aggregate supply, we find that when the investors mistrust their models and make a robust decision rule then the slope of the pricing function of the assets is larger compared to the case when the investors fully trust their models’ dynamics, since Ψ > Ψnr. If we consider the case where N = 1 and the risky asset is the market, then we see that the risk premium, the volatility and the illiquidity go up, since a small change in the aggregate supply, a small liquidity shock by the noise traders leads to a larger change in the price of the risky asset comparing to the case where the investors fully trust their models. 4.3 Conclusions Risk sensitive regulations have become the cornerstone of international financial reg- ulations. They imply an upper bound on the wealth volatility for each investor. In this chapter we studied how risk constraints and model misspecification affect the statistical properties of the market returns. In particular, we studied their effect on the risk premium, the volatility and liquidity of the market. 170 We studied the following cases: • Variability of the risk constraints: This is the case where the agents face differ- ent risk constraints. The more variability there is in the risk constraints, the larger the risk premium, volatility and illiquidity of the market is. In addition tightening of the constraints leads also to more risk premium, volatility and illiquidity of the market. • Variability in risk aversions: This is the case where the agents face the same risk constraint but have different aversions to risk. Variability in risk aversions along with the risk constraints also lead to a more concave pricing function of the aggregate supply for the market, implying increasing risk premium, increasing volatility and increasing illiquidity. An interesting question here is the following. Do we have a concave pricing function due to the fact that “...risk will have to be transferred from the more risk-tolerant to the more risk-averse”? Well as we saw this is not the case. Any constraint that binds for an agent forces the discount and the slope of the pricing function to be larger so that the other agents are induced to absorb the excess supply and this is the mechanism that leads to a more concave decreasing pricing function with respect to the aggregate supply. • Model misspecification: This is the case where the agents do not fully trust their model dynamics, they believe that the real model is an unknown member of a set of alternative models near their nominal model and they make robust decision rules. We find that model misspecification is another source of increasing risk premium, endogenous volatility and increasing illiquidity. 171 172 Appendix A Technical Notes Proposition 1. The following QCQP: 1 minimize F T µ + F T ΣF t t 2 t t T subject to Ft ΣFt ≤ L has a solution given by: −1 T −1 −Σ µt if µt Σ µt ≤ L opt Ft = −1 Σ µt T −1 − if µ Σ µt ≥ L T Σ−1 t µt µt r L opt opt Proof. Let us apply the Karush Kuhn Tucker (KKT) conditions. Ft ,λt are optimal iff they satisfy the following KKT conditions: T opt opt • Primal feasibility: Ft ΣFt ≤ L opt • Dual feasibility: λt ≥ 0 opt T opt opt • Complementary slackness: λt (Ft ΣFt − L)=0 opt opt • Minimization of the Lagrangean: Ft = argmin L(Ft,λt ) 173 The Lagrangean is given by: 1 L(F,λ)= F T µ + F T ΣF + λ(1/2F T ΣF − 1/2L) t t 2 t t t t The first order conditions are: opt opt opt µt + ΣFt + λt ΣFt =0 ⇒ −1 opt Σ µt Ft = − opt 1+ λt T −1 opt T opt opt If µt Σ µt > L, then we cannot have λt = 0, since in that case Ft ΣFt > L. opt Therefore λt > 0 and from the CS condition T opt opt Ft ΣFt = L ⇒ µT Σ−1µ t t = L ⇒ opt 2 (1 + λt ) T −1 opt µt Σ µt (1 + λt )= > 1 r L −1 T −1 opt Σ µt Therefore, if µ Σ µt > L, then F = − . t t T Σ−1 µt µt r L T −1 T opt opt opt opt −1 If µt Σ µt < L, then Ft ΣFt < L, therefore λt = 0 and Ft = −Σ µt. T −1 opt Finally if µt Σ µt = L, then we cannot have λt > 0, since in that case we would T opt opt opt have Ft ΣFt < L and λt > 0 and the CS condition would be violated. Therefore, opt opt −1 λt = 0 and Ft = −Σ µt. Therefore we proved that: −1 T −1 −Σ µt if µt Σ µt ≤ L opt Ft = −1 Σ µt T −1 − if µ Σ µt ≥ L T Σ−1 t µt µt r L 174 We could also prove this result in another way. Let us make a change of variables where: 1/2 y = Σ Ft −1/2 Ft = Σ y Then our problem becomes: 1 minimize yT Σ−1/2µ + yT y t 2 subject to yT y ≤ L −1/2 The optimal solution is given by finding the projection ofµ ˜ = −Σ µt on the Euclidean ball yT y ≤ L. This projection is given by: µ˜ yopt = µ˜T µ˜ max(1, L ) q Therefore the optimal solution for the original problem is given by: opt −1/2 opt Ft = Σ y −1 opt Σ µt Ft = − T −1 µt Σ µt max(1, L ) q 175 Proposition 2. When Σ is a diagonal matrix, the following convex program: 1 minimize F T µ + F T ΣF t t 2 t t N subject to i=1 λi|Fit| ≤ 1 P has a solution given by: µit opt + sign(−µit)(| | − ν ) opt λi t Fit = 2 σi λi opt opt Proof. Let us apply the Karush Kuhn Tucker (KKT) conditions. Ft , νt are optimal iff they satisfy the KKT conditions: N • Primal feasibility: i=1 λi|Fit| ≤ 1 P opt • Dual feasibility: νt ≥ 0 opt N • Complementary slackness: νt ( i=1 λi|Fit| − 1)=0 P opt opt • Minimization of the Lagrangean Ft = argmin L(Ft, νt ) The Lagrangean is given by: 1 N L(F, ν)= F T µ + F T ΣF + ν( λ |F | − 1) t t 2 t t i it i=1 X 176 opt opt Ft = argmin L(Ft, νt ) N N N 2 2 opt = argmin 1/2 σi Fit + Fitµit + νt ( λi|Fit| − 1) i=1 i=1 i=1 XN XN X N 2 2 opt = argmin 1/2 σi |Fit| + |Fit|sign(−µit)µit + νt ( λi|Fit| − 1) i=1 i=1 i=1 XN XN X 2 2 opt = argmin 1/2 σi |Fit| + |Fit|(sign(−µit)µit + λiνt ) i=1 i=1 X X opt opt where Fit = |Fit |sign(−µit). We have: opt 0 if sign(−µit)µit + λiνt ≥ 0 |F opt| = it opt sign(−µit)µit+λiνt opt − 2 if sign(−µit)µit + λiνt ≤ 0 σi Therefore: 0 if sign(−µ )µ + λ νopt ≥ 0 opt it it i t Fit = µit opt − −sign(−µit)ν λi t opt 2 if sign(−µit)µit + λiνt ≤ 0 σi /λi which is equivalent to: opt 0 if sign(−µit)µit + λiνt ≥ 0 opt Fit = |µit| −νopt λi t opt 2 sign(−µit) σ if sign(−µit)µit + λiνt ≤ 0 i λi Since it is: opt sign(−µit)µit + λiνt ≤ 0 ⇒ opt −|µit| + λiνt ≤ 0 ⇒ µit opt | | − νt ≥ 0 λi 177 we have proved that: µit opt + sign(−µit)(| | − ν ) opt λi t Fit = 2 σi λi Proposition 3. The relative entropy of a multivariate Gaussian distribution N(µ,I) 1 T with respect to the multivariate standard Gaussian distribution is D(Q)= 2 µ µ Proof. It is: +∞ f(x) D(Q)= log( )f(x)dx g(x) Z−∞ 1 −1/2(x−µ)T (x−µ) 1 −1/2xT x where f(x)= 2 e and g(x)= 2 e (2π)N/ (2π)N/ +∞ +∞ D(Q)= − 1/2(x − µ)T (x − µ)f(x)dx + 1/2xT xf(x)dx Z−∞ Z−∞ The first term is: 1 1 − E[(X − µ)T (X − µ)] = − E[trace((X − µ)T (X − µ))] 2 2 1 = − E[trace((X − µ)(X − µ)T )] 2 1 = − trace(I) 2 = −N/2 The second term is: 1 1 1 E[XT X]= E[trace((X − µ)T (X − µ))] + E[X]T E[X] 2 2 2 1 = N/2+ µT µ 2 1 T Therefore, we have D(Q)= 2 µ µ. 178 Proposition 4. The relative entropy of probability measure Q with respect to P , t T 1 t T dQ 0 h dZs− 2 0 h hsds where = ξT and ξt = e s s is given by: dP R R T 1 D(Q)= E [hT h ]dt 2 Q t t Z0 Proof. The relative entropy of Q with respect to P is: dQ D(Q)= E [log( )] Q dP = EQ[log(ξT )] T 1 T = E [ hT dZ − hT h ds] Q s s 2 s s Z0 Z0 T 1 T = E [hT dZ ] − E [hT h ]ds Q s s 2 Q s s Z0 Z0 T 1 t = E [E [hT dZ |F ]] − E [hT h ]ds Q Q s s s 2 Q s s Z0 Z0 T 1 = E [hT h ]dt 2 Q t t Z0 since EQ[dZt|Ft]= htdt from Girsanov’s theorem. Proposition 5. The following QCQP: ΣH 1 1 minimize −F T (µ(S, t) − rS − S )+ (1 + )F T ΣF t ν 2 ν t t subject to F T ΣF ≤ L has a solution given by: 1 −1 T −1 1 2 1 1+ Σ µt if µt Σ µt ≤ L(1 + ν ) opt ν Ft = −1 Σ µt T −1 1 2 if µ Σ µt ≥ L(1 + ) T Σ−1 t ν µt µt r L ΣHS (St,t) where µt = µ(S, t) − rSt − ν . opt opt Proof. Let us apply the Karush Kuhn Tucker (KKT) conditions. Ft ,λt are optimal iff they satisfy the following KKT conditions: 179 T opt opt • Primal feasibility: Ft ΣFt ≤ L opt • Dual feasibility: λt ≥ 0 opt T opt opt • Complementary slackness: λt (Ft ΣFt − L)=0 opt opt • Minimization of the Lagrangean: Ft = argmin L(Ft,λt ) The Lagrangean is given by: 1 1 L(F,λ)= −F T µ + (1 + )F T ΣF + λ(1/2F T ΣF − 1/2L) t t 2 ν t t t t The first order conditions are: 1 −µ +(1+ )ΣF opt + λoptΣF opt =0 ⇒ t ν t t t −1 opt Σ µt Ft = 1 opt 1+ ν + λt T −1 1 2 opt T opt opt If µt Σ µt > L(1+ ν ) , then we cannot have λt = 0, since in that case Ft ΣFt > opt L. Therefore λt > 0 and from the CS condition T opt opt Ft ΣFt = L ⇒ µT Σ−1µ t t = L ⇒ 1 opt 2 (1 + ν + λt ) T −1 1 opt µt Σ µt 1 (1 + + λt )= > 1+ ν r L ν −1 T −1 1 2 opt Σ µt Therefore, if µ Σ µt > L(1 + ) , then F = . t ν t T Σ−1 µt µt r L −1 T −1 1 2 T opt opt opt opt Σ µt If µt Σ µt < L(1 + ν ) , then Ft ΣFt < L, therefore λt = 0 and Ft = 1 . 1+ ν T −1 1 2 opt Finally if µt Σ µt = L(1 + ν ) , then we cannot have λt > 0, since in that case we T opt opt opt would have Ft ΣFt < L and λt > 0 and the CS condition would be violated. −1 opt opt Σ µt Therefore, λt = 0 and Ft = 1 . 1+ ν 180 Proposition 6. The following convex program: T T 2 T T 1/2 T max min γ(W0 − q θ)Rf + γµˆ θ − 1/2γ θ Σˆθ + γθ Σˆ h + λ1/2h h (A.1) θ h has a solution given by: ˆ −1 opt Σ (ˆµ − Rf q) θ = 1 γ(1 + λ ) Proof. For each h the objective function is concave in θ. Therefore, when we take the minimum over h, we take the minimum of concave functions, which leads to a concave function. Therefore this problem maximizes a concave function of θ. The inner minimization problem is: min γθT Σˆ 1/2h + λ1/2hT h h This is a convex problem where we minimize a quadratic function of h. The first order conditions are: γΣˆ 1/2θ + λh =0 ⇒ γΣˆ 1/2θ h = − λ The optimal value of the inner optimization problem is: γΣˆ 1/2θ γ2θT Σˆθ V (θ)= −γθT Σˆ 1/2 + λ1/2 λ λ2 γ2θT Σˆθ = − 2λ The problem A.1 then becomes: 2 T ˆ T T 2 T γ θ Σθ max γ(W0 − q θ)Rf + γµˆ θ − 1/2γ θ Σˆθ − θ 2λ 181 The first order conditions of this concave problem are: 1 γ(ˆµ − R q) − γ2Σˆθopt(1 + )=0 f λ ˆ −1 opt Σ (ˆµ − Rf q) θ = 1 γ(1 + λ ) Therefore we proved our proposition. Proposition 7. The market clearing condition for the risky assets implies that opt θh = θα ⇒ hX∈H µˆ − ΨΣˆθ q = α Rf N 1 1 where θα ∈ R is the aggregate supply of the risky assets and ≡ 1 Ψ h∈H γh(1+ ) λh P Proof. 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