PORTFOLIO OPTIMIZATION
By
WANRUDEE SKULPAKDEE
A dissertation submitted in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
WASHINGTON STATE UNIVERSITY Department of Mathematics
MAY 2016
© Copyright by WANRUDEE SKULPAKDEE, 2016 All Rights Reserved
© Copyright by WANRUDEE SKULPAKDEE, 2016 All Rights Reserved
To the Faculty of Washington State University:
The members of the Committee appointed to examine the dissertation of WANRUDEE SKULPAKDEE find it satisfactory and recommend that it be accepted.
______Hong Ming Yin, Ph.D., Chair
______Haijun Li Ph.D.
______Charles Moore, Ph.D.
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ACKNOWLEDGMENT
Though only my name appears on the cover of this dissertation, a great many
people have contributed to its production. I owe my gratitude to all those people
who have made this dissertation possible and because of whom my graduate
experience has been one that I will cherish forever.
My deepest gratitude is to my advisor, Dr. Hong Ming Yin. I have been amazingly
fortunate to have an advisor who gave me a lot of useful suggestion. His patience and support helped me overcome many crisis situations and finish this dissertation.
Most importantly, none of this would have been possible without the love and patience of my husband, Mongkol Hunkrajok, and my family. Finally, I appreciate the support from my colleagues and the Department of Mathematics and Statistics,
Washington State University.
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PORTFOLIO OPTIMIZATION
ON JUMP DIFFUSION
Abstract
by Wanrudee Skulpakdee, Ph.D. Washington State University May 2016
Chair: Hong Ming Yin
Portfolio optimization problem is an important research topic in financial
applications. In this research, we focus on a consumption process and a terminal
wealth problem. A portfolio including a riskless asset, a zero coupon bond and a
stock is presented. All assets are modeled by continuous time dynamic. Bond is
modeled by a classical Vasicek’s model. A stock is modeled using Merton Jump
diffusion (MJD) model. This work is new because an economic inflation rate and consumption price index (CPI) are taken into consideration to evaluate a real value
of assets. A Hamilton-Jacobi-Bellman (HJB) equation that satisfies an optimal
solution is derived. Then, the solution is proved to exist and to be unique under
certain conditions. Finally, the solution is found by using numerical method.
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TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS...... iii
ABSTRACT...... iv
LIST OF TABLES...... vii
LIST OF FIGURES...... viii
CHAPTER
1. INTRODUCTION...... 1
1.1 Overview...... 1
1.2 Portfolio Optimization on Discrete time Model……………………………… 3
1.3 The Efficient Frontier………………………………………………………… 6
2. STOCHASTIC MODELS FORMULATION...... 9
2.1 Stochastic Processes and Filtrations…………………….……………………. 9
2.2 Continuous Time Models………………………………...…………………..11
2.3 Utilities Functions……………………………………………………………14
2.4 Portfolio Optimization…………………………….………………………… 16
2.5 Bond Models…………………………………………………………...... …. 21
3. PORTFOLIO OPTIMIZATION AND HJB EQUATION...... 28
3.1 Stock Models……………………………………………..…………………. 28
3.2 Ito Calculus…………………………………………….……………………. 33
3.3 Real Value of Assets…………………………………………………………35
3.4 Consumption and Terminal Wealth Problem……………………………….. 38
3.5 Wealth Process……………………………………………………………….39
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3.6 Hamilton-Jacobi-Bellman Equation for Consumption and Terminal Wealth
Problem……………….……………………………………………………. 40
3.7 A Model Reduction…………………………………..………………………43
4. EXISTENCE AND UNIQUENESS OF THE OPTIMAL SOLUTION…………….. 47
4.1 Existence of the Optimal Solution for Cauchy Problem……………..………47
4.2 Existence and Uniqueness of the Portfolio………………………..…………51
5. A COMPUTATIONAL SOLUTION FOR THE OPTIMIZATION PROBLEM….…56
5.1 Parameters Calibration………………………………………………….……56
5.2 A Numerical Solution for a HJB equation………………………………...…66
BIBLIOGRAPHY...... 73
APPENDIX
A. DATA AND MATERIALS...... 76
B. MATLAB CODE FOR NUMERICAL SOLUTION………………………...... 85
vi
LIST OF TABLES
1. The Reduction Model Parameters...... 71
2. Correlation Parameters...... 71
vii
LIST OF FIGURES
1. Figure1; Minimal variance portfolios (Markowitz bullet) on mean-variance plane...... 6
2. Figure 2; The efficient frontier on Markowitz bullet………………………………...... 7
3. Figure 3; A linear mixture between a risk-free and a risky asset...... 8
4. Figure 4; A graph of the U.S. monthly interest rate…………………………………….……..58
5. Figure 5; The forecasted interest rate………………………………………………….………58
6. Figure 6; The predicted zero coupon bond yield…………………………………….……….. 60
7. Figure 7; A nominal price and a real value price for 7-year maturity bond………………….. 61
8. Figure 8; A histogram of ln S ……………………………………………………..……….. 64
9. Figure 9; Nominal and real value of SP500 stock index…………………………...………… 66
10. Figure 10; A value function for HJB equation……………………………………………….72
11. Figure 11; An optimal consumption for the portfolio problem………………….……….…. 72
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CHAPTER ONE
INTRODUCTION
1.1 Overview
Portfolio optimization is an important research topic in financial applications. One question always asked is how to manage financial assets so that the portfolio performs well in comparison to general markets. We will answer this question of an investment problem under uncertainty to optimize investors’ return with reasonable risk. Techniques used separate this field of study into two major categories, discrete-time models and continuous-time models. Methods for discrete-time models are previously well developed. The mean-variance method is the pioneer work introduced by Harry Markowitz in 1952. By optimizing a portfolio mean of return while considering a portfolio variance as a level of uncertainty. This makes a one-period decision of assets allocation. His work was awarded the Nobel Prize in economic sciences in 1990.
However, his model itself is overly simplified in practice since parameters in the model often depends on time. Merton’s continuous-time approach was considered the continuous time model in 1969. R.C. Merton adopted stochastic control techniques to find an optimal utility of the portfolio combined with risky and riskless assets for both finite and infinite lifetime consumption problems. This method is required to solve Hamilton-Jacobi-Bellman equations (HJB equation) which are typically highly nonlinear partial differential equations (PDE). They rarely obtain explicit form solutions unless models are simplified. Nevertheless, his work earned him the
Nobel prize in economic sciences in 1997. The stochastic differential equation (SDE) model for the risky assets in Merton’s approach has been studied and refining to be close to reality including transaction cost, dividend, or special events. Other preliminary works should be regarded to put here included a use of stochastic distribution by E.F. Fama,1960-1980, the use of
1 stochastic integral by Harrison and Pliska,1981, the martingale approach by Karatzas,1986, and
Cox and Huang, 1989.
In this dissertation, we consider the portfolio optimization problem with Merton’s continuous time model. Unlike Merton’s and other related research, our model includes the risk assets with jump diffusions which are more realistic in the financial markets due to special events and corporations’ announcements. Moreover, our model also incorporates the changes of interest rate and consumption rate.
This dissertation is organized as follows. In the first chapter, we will review Markowitz’s discrete-time model. In the second chapter, classical stochastic models for risky assets and portfolio optimization using stochastic control are presented. In the third chapter, Jump diffusion model including economic inflation in consideration is introduced. A portfolio optimization problem including a riskless asset, a bond and a risky stock is illustrated. HJB equation satisfies a solution of the optimal problem is derived. In Chapter Four, we prove that a solution of the HJB equation exists and unique. Finally, in Chapter Five, a reduced model is served as an example of a numerical solution.
We would like to point out that the model does not include option values of risk assets. It is our plan that future research will include this topic in the model. See recent research papers in [10] and [11].
In this chapter, we mainly introduce Markowitz’s discrete-time models portfolio optimization as a fundamental technique for the problem.
2
1.2 Portfolio Optimization on Discrete time Model
A cornerstone technique of portfolio optimization on discrete time model is known as
Markowitz’s efficient frontier. Mean and variance of a portfolio is adopted to refer to a return and a risk of the investment choices.
1.2.1 Discrete time Portfolio Model of Multiple Assets
Suppose there are n risky assets in an investment market. Let us consider choices of investment in one period of time.
Definition 1.2.1 (Return of Portfolio and Risk)
Define a return of asset i in time interval (t, t+1) by
Pii( t 1) P ( t ) Rti () , Pti () is a price of asset i at time t. Pti ()
Then, an expected return of asset i is defined as
i i(t ) E [ R i ( t )] . (1.2.1)
T An expected return of all n assets are denoted in vector form as 12 ..... n .
Next, let ij denote as a covariance between Ri and Ri , defined by
ijCovRR(,) ij ER [( iijj )( R )] ERR [ ijij ] (1.2.2)
22 and iiER[( i i ) ] i , a variance of .
11 1n Denote as a covariance matrix of n assets. n1 nn
T Now, consider an investment portfolio[ 12 ... n ] , where i ,in 1,..., is a proportion of an investment in asset i.
3
n T An expected return of Portfolio is ii . (1.2.3) i1
nn 2 T The portfolio variance is i j ij . (1.2.4) ij11
Now assume that all investors are risk averse. For the underline return, we want to minimize risk which is measured by using a variance of portfolio. We are seeking the portfolio π that minimizes the variance.
1.2.2 Discrete-Time Portfolio Optimization
n T Define an attainable set , A, to be a set of portfolios that satisfy i 1 1. i1
A portfolio optimization problem can then be defined as
2 min (1.2.5) A
T subject to c 0 , where c is the underlining return
T and 1 1, where 01i .
The alternative optimization problem of (1.2.5) is
max (1.2.6) A
T 2 2 subject to c , where c is the maximum acceptable risk, and T 1 1, where .
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Theorem 1.2.2
Assume (a) 1 cn where 12 ... n .
(b) 1 cn where 12 ... n .
(1) If is a solution of (1.2.5) with then is also a solution of (1.2.6) with c
2 T . .
(2) If ˆ is a solution of (1.2.6) with 22 then ˆ is also a solution of (1.2.5) with c
ˆT .
Theorem 1.2.3
A solution of problem (1.2.5) can be represented explicitly as
11TT111 1 1 1'1 TT11 1 cc 1TT11 1 1 1TT11 where the denominator is not zero.
Proof: By using a technique of Lagrange Multiplier.
is the portfolio whose expected return equals c and variance is minimal on the attainable set. This optimal portfolio of the problem (1.2.5) can be illustrated on the mean- variance plane as following.
5
0.3
0.25
0.2
0.15
0.1
0.05
0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Figure1: Minimal variance portfolios (Markowitz Bullet) on mean-variance plane.
The curve is called a minimum variance line which known as “The Markowitz Bullet.”
1.3 The Efficient Frontier
In general, investors are able to choose to invest in a riskless asset with a riskless return as choices of preference. It reduces a risk of a portfolio.
Definition 1.3.1 (Dominating) [17]
Suppose there are two securities whose expected returns are 1 and 2 . Their variance
are 1 and 2 respectively.
If 12 and 12 , we say that the first security “dominates” the second security because a majority of investors, risk-averse type, prefers to invest in a security which has a lower risk, but higher return.
6
Definition 1.3.2 (The Efficient Frontier) [17]
A portfolio is called “efficient” if there are no other portfolio dominates it. A set of all efficient portfolios in the attainable set is called “the efficient frontier.”
0.3
0.25
0.2
0.15
0.1
0.05
0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Figure2: The efficient frontier on Markowitz Bullet represented by the dark curve.
Furthermore, suppose there exists a riskless asset 0 whose return is 0 in the market and
2 efficient portfolios e whose return is e and variance is e can be considered as the investment products.. A portfolio B combining a fraction of investment in asset 0 and in an efficient
e 0 2 2 portfolio e has a return in the form BB 0 2 where B is a variance of Portfolio B. e
Theorem 1.3.3 (Efficient Portfolios of a Risk-free and Risky Assets) [17]
If there is a risk-free asset in the market, all efficient portfolios including the risk-free asset is a linear mixture between a risk-free asset and a risky portfolio lies on the straight line tangent to the curve of efficient frontier of risky assets in the market.
7
0.3
0.25
0.2
0.15
0.1
0.05
0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Figure 3: A linear mixture between a risk-free and a risky asset represented by the red line.
The major problem of mean-variance portfolio optimization technique is that it over simplifies the actual applications. Since the model is discrete on time, the suggested portfolio should be recalculated in every period of time.
In the next chapter, continuous time optimal investment techniques will be discussed since it provides a suggestion of an investment plan on time domain. The techniques have been developed involving a use of stochastic control introduced by R. C. Merton. There is gathering of several assets models to optimize the investment in the limited resources.
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CHAPTER TWO
STOCHASTIC MODELS FORMULATION
In this chapter, the classical Merton’s continuous time approach for portfolio optimization is illustrated, in theoretical and in practical. A refined model of two financial products, bond and stock used in this research are introduced. The classical Vasicek’s model of bond is adopted. Stock price is simulated by using Jump-diffusion. These models are new because they are taken into account with economic inflation as well as a deterministic model for a riskless asset. Then, they will be cooperated in a portfolio optimization problem in Chapter
Three.
2.1 Stochastic Processes and Filtrations
d A stochastic process is a family of random variables XX t for tT in the measurable sample space, . (,,) FP denoted a complete probability space where F is a collection of events occurred with the probability measure P or sometime called Borel -field.
The time tT may be discrete or continuous. T is either finite TT[0, ] or infinite T [0, ) .
For any event , the mapping X():() t T Xt is called the path of the process for event .
Definition 2.1.1 (Stationary Stochastic Process)
Two main pieces information of stochastic process are represented in term of mean and variance for time t.
()[]t E Xt
2 and V( t ) Var ( t ) E [( Xt ( t )) ]
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Time-between information is represented as covariance
Ctt( , ) CovXX ( , ) EX [( ( tX ))( ( t ))] for t, t T . 1 2t1 t 2 t 1 1 t 2 2 12
We call a stochastic process X “stationary” when X t has the same distribution for all tT .
Definition 2.1.2 (Filtration)
We denote Ft the set of information up to time t. It is obtained by observing values of the vector process X up to time t.
Ft X , [0, t ]
Therefore, the information is not lost and for 0 tt12 , FFFFF0 tt . 12 0
Ft is also called a filtration which is adapted by time.
The stochastic process XX{}t is called “Cadlag” if it admits a right continuous with the left limit. It is called “Caglad” if it has left continuous with the right limit.
Definition 2.1.3 (Wiener Process)
Consider stochastic process, X, with stationary independent increments. That is XXt h t are independent and stationary for all t. This process is called “Wiener Process” or known as
“Brownian motion.”
Let Z Zt ,0 t be an F-adapted stochastic process with continuous sample path for
which Z0 0 and Zt h Z t ~ (0, h ) , a Gaussian distribution with zero mean and variance h.
Z is defined as a “Standard Wiener Process.”
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Wiener process itself is not a stationary process since it depends on the known initial value.
However, according to its initial set up we get
EZEZZZEZEZZ[][t0 ( t 0 )][][ 0 t 0 ]0
2 2 2 and VarZ()[][][]t EZ t EZ t EZ t t .
2.2 Continuous Time Models
R.C. Merton brought a new technique for portfolio optimization by applying stochastic control to optimize the utility function of the life-time consumption and the consumption and terminal wealth problem on continuous time models [23]. Since then, his idea has been adapted and extended to many refined models because the result can suggest investment choices on time domain and under a variety of conditions.
2.2.1 Risk-Free Asset Model
Consider a risk-free asset i.e. money in the bank account with the deterministic rate of
return rt(). Suppose a principle for the investment at time 0 equals P0 . A total amount at time t can be written in dynamic form as
dP()()() t P t r t dt . (2.2.1)
t The explicit form solution can be solved as P( t ) P exp r ( ) d 0 0
2.2.2 Risky Asset Model
In this subsection, a risky asset model introduced in Merton’s work is presented. Let
Sti () be a price of risky asset i i.e. stock at time t obey a stochastic differential equation (SDE).
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m dSti()()()()() St i i tdt ij tdZt j (2.2.2) j1
where Sti ( ) 0 is a price at time t of stock i.
T ()t 12 () t ().... t n () t is a vector of functions of time representing drift of the stocks.
()(),t t m n is the matrix represents diffusive volatility of n stocks with m ij nm
factors. ij ()t is a volatility of stock i effected by factor j.
T Z( t ) Z12 ( t ) Z ( t ) ..... Zm ( t ) is a vector of m independent Wiener process defined
on a complete probability space (,,) FP. Ft is a market information at time t. rt(), ()t and ()t are F-adapted and uniformly bounded on (t , S ) [0, T ]
()()ttT are required to be uniformly positive definite. nn
The explicit solution for (2.2.2) is
tt1 mm S()(0) t S () 2 () d ()() dZ (2.2.3) i i i ij ij j 002 jj11
2.2.3 Wealth Process
Let T be a fixed time horizon. Suppose Asset 0 is a riskless investment and Asset 1 to n
are risky stocks where 12 ... n .
Definition 2.2.1 (Trading Strategies)
T Define ()t 01 () t ()... t n () t to be a number of share invested on each asset at time t with F-adapted such that
12
T ()() Pd , 00 0
nmT 2 and ()()() Sd . i i ij ij110
()t is called “the trading strategies.”
A total wealth at time t equals
n W()()()()() t00 t P t ii t S t , (2.2.4) i1
where Ww(0) 0 is an initial wealth and Pt0 ( ) 1.
Definition 2.2.2 (Consumption Process, Self Financing and Portfolio Process)
Define Ct() a non-negative, adapted process called “consumption rate process.”
t It satisfies Cd(). 0
A trading strategy and a consumption rate process (,) C is called “self-financing” if the wealth process W satisfies
n dWt()()()()()()00 tdPt ii tdSt Ctdt i1
nm 0 ()()()()()()()trtdt i t i tdt ij tdZt j Ctdt (2.2.5) ij11
The explicit form for wealth process can rewrite in the form
tn t t W( t ) W (0) ( ) P ( ) d ( ) S ( ) d C ( ) d (2.2.6) 00 ii 0i1 0 0
13
T Let ()t 12 () t ()... t n () t be a “portfolio process” where
()()t S t ()t ii for in1,..., , (2.2.7) i Wt()
n n where 0i (t ) 1 and i (t ) 1 , thus 0 (tt ) 1 i ( ) . i0 i1
Then the wealth process can be derived in the form
nm dWt()()()()()()()()() Wt0 trtdt i t i tdt ij tdZt j Ctdt ij11
n n m Wt()1 i ()() trtdt i ()() t i tdt ij ()() tdZt j Ctdt () i1 i 1 j 1
TTT Wt()1 () t1 rtdt () ()() t tdt ()()() t tdZt Ctdt () (2.2.8)
2.3 Utilities Functions
Utility function was first introduced in the 18th century by Daniel Bernoulli. The function is designed to fit the investor style of investment and to optimize an investor preference. It is used to define a level of investors’ preference in obtaining a total wealth.
Let W be a total wealth. The utility function is defined by UW(): .
Investors can be categorized by their preference of investment into 3 cases.
(a) Risk aversion: prefer safe to risk
(b) Risk neutral: no preference on different level of risk but the same level of return
(c) Risk seeking: prefer risk to safe
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Assumption 2.3.1
(1) U is an increasing function, UW( ) 0.
(2) For a risk aversion, U is concave, UW( ) 0 .
For a risk neutral, U is linear,UW( ) 0 .
For a risk seeking, U is convex,UW( ) 0 .
The function U( t , W ), t [0, T ] is also called a utility function if U(,) t W is a utility function on
W for every fixed tT[0, ]. is continuous on t for every fixed W .
An example of a utility function on time space is a discount utility function
U(,)() t W et U W , where et is a discount factor with the discount rate 0 .
An important class of utility functions in finance is Hyperbolic Absolute Risk Aversion
(HARA function). This is defined by
1 UWW() 1
W on the domain 0 where 0, 1 and 0. 1
The class is named by the model of its risk averse measure known as Arrow-Pratt measure.
1 UWW() It is expressed as ARA() W UW( ) 1
The model is hyperbolic on W.
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Some special cases of HARA functions:
Quadratic utility
2 When 2, UWW() . 2
The function is increasing up to the bliss pointUW( ) 0.
Exponential utility (Constant Absolute Risk Averse, CARA)
Let 1 as ,
1 then U() W eW , where ARA() W .
The function is lognormal distributed and it provides normal distributed risk.
Power utility (Constant Relative Risk Averse, CRRA)
1 Let 0, 1 ,
W when 1 and 0 , UW() .
Logarithm utility
Let 0 as 0 UWW( ) ln( )
2.4 Portfolio Optimization
2.4.1 Consumption and Terminal Wealth Problem
Consider a terminal wealth and cumulative consumption at the fixed time T obtained by
the initial wealth w0 with the portfolio π and a consumption process Ct() for tT[0, ].
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T W(;,,)() T w C C d 0 0
A function represents an expected utility of the total wealth associated to the strategy (π,C) as a present value at time 0 is
T Jw(0, , , C ) EUTWT [ ( , ( )) UC ( , ( )) d ] 0 0
T E[ eT U ( W ( T )) e U ( C ( )) d ] (2.4.1) 0 where 0 and U is a utility function.
Then, the portfolio optimization problem is addressed as
maxJ (0, w0 , , C ) (2.4.2) ( ,C ) A (0, w0 ) where A(t,w)={(π,C) such that the wealth process is self-financing, W(t)=w andC( ) 0 for all
[,]tT and 01i } is called a set of admissible strategies.
The constraint of the problem is the wealth process
dWt() Wt ()[((1 () tTTT1 )() rt () t ()) tdt () t () tdZt ()] Ctdt ()
[((1 ()tTTT1 )() rt () t ()) tWt () CtdtWt ()] ()() t () tdZt () (2.4.3)
Problem (2.4.2) can be solved by the stochastic control method sometimes known as
“Merton’s approach”. It considers the problem in terms of the Hamilton-Jacobi-Bellman (HJB) equation. An existance of the verification theorem concludes that a solution of HJB equation is a solution of the optimization problem.
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2.4.2 Stochastic Control
Consider the optimization problem with a control u in general form as
T maxJwu (0,0 , ) max EfWudgWT ( , ( ), ( )) ( ( )) (2.4.4) u A(0, w ) u A (0, w ) 000 subject to a dynamic of a stochastic variable Wt(),
dWt() (, tWtutdt (),()) (, tWtutdZt (),()) () . (2.4.5)
Introduce a value function
T Vtw(,) sup Jtwu (,,) sup EfW (, (),()) udgWT (()) . (2.4.6) u A(,)(,) t w u A t w t
By the Bellman principle, a discrete version of the value function can be presented in the form
Vtw(,) max ftwuhEVthWth (,,)tt [( ,( ))|() Wt wuu , ] ut A(,) t w
** ftwuh(,,)tt EVt [( hWt ,( h ))|() Wt wu , u ],
** so ftwuh(,,)tt EVt [( hWt ,( h ))|() Wt wu , u ] Vtw (,)0 .
Thus, EVthWth[( ,( ))] EVthWth [( ,( ))|() Wt wu , ut ]
EVthWth[( ,( ))] Vtw (,) EVthWth [( ,( ))|() Wt wu , ut ] Vtw (,)
EVt[( hWt ,( h ))] Vtw (,) ftwuh (,,)t
EVt[( hWt ,( h ))|() Wt wu , utt ] Vtw (,) ftwuh (,,) ,
or EVt[( hWt ,( h )) Vtw (,)] ftwuh (,,)t 0 .
Divide both sides by h and take the limit as h 0 , to get
DV( t , w ) f ( t , w , ut ) 0
The inequality becomes equality when the control obtains the supremum.
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Definition 2.4.1 (Hamilton-Jacobi-Bellman (HJB) Equation) [3]
HJB equation for problem (2.4.4) that satisfies (2.4.5) is defined to be
supLVtw (,) sup[ DVtw (,) ftwu (,,)] 0 (2.4.7) u A(,)(,) t w u A t w
1 2 where DVtw(,)(,)(,,)(,)(,,)(,) Vtw twuVtw twu V tw t w2 ww
A solution of the HJB equation that satisfies a boundary and terminal condition
T B.C. V( t ,0) f ( ,0) d g (0) (2.4.8) t
T.C. V( T , W ( T )) g ( W ( T )) (2.4.9) is the optimal solution for (2.4.4).
Theorem 2.4.2 (The Verification Theorem) [22]
Let V*(,) t w be a polynomially-bounded solution to the HJB equation satisfying
V*( t , w ) C 1,2 ([0, T ], ) and continuous on[0,T ] ; then we have
(a) V*(,)(,) t w V t w for all (t , w ) [0, T ] .
(b) If there exists an admissible control u* satisfying
u* arg max LV ( t , w ) u A(,) t w where w satisfies the wealth process.
Then V**(,)(,,) t w J t w u for all (t , w ) [0, T ] .
Thus, is the optimal control of the problem (2.4.4).
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2.4.3 Solving HJB equation
Returning back to the optimal consumption and terminal wealth problem (2.4.2) with the wealth process (2.4.3), the problem is generally solved by two steps according to [22].
Step1 Solving an unconstrained optimization problem.
Define a value function as V(,) t w sup J (,,,) t w C (,)(,) C A t w
T supE [ e()() t U ( C ( )) d e T t U ( W ( T ))] (,)(,) C A t w t
V(,) t w is a maximum possible expected utility for the problem which starts at time t where
W() t w and terminates at time T.
We want to find a general form solution for the optimal problem
supLVtwC (,,,) sup[ DVtwCUCt (,,,) (()) V ] (2.4.10) (,)(,)(,)(,)C A t w C A t w
1 where DVtwCV(,,,) [((1 ())() trt1 () tTTT ()) twCtV ()] () t ()() tt () twV2 . t w2 ww
(2.4.10) is maximized when
LV 0 (2.4.11)
and LVC 0 (2.4.12)
T 2 From (2.4.11), (()())t r t1 wVw ()()() t t t w V ww 0
V or *1()(()())(()())t t tT t r t 1 x . (2.4.13) wVww
From (2.4.12), UCV( )w 0
*1 C()() t U Vw . (2.4.14)
Then, (,) **C is the regular control for the general solutionV * .
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Step2 Solving HJB equation
The nonlinear HJB equation for (2.4.2) is defined to be
supLV (,) t w LV*** (,;,)0 t w C . (,)(,) C A t w
Substitute (,) **C to (2.4.10). The equation becomes
2 1 11 TT 1 Vw 0UUV ( ())w V t VUvvrtwV () x x () w (()())(()())(()()) trt 11 tt trt 2 Vww (2.4.15)
T ()()() t T t 11 T t The boundary condition is V( t ,0) e U (0) d e U (0) e U (0) . t
The terminal condition is VTWTUWT( , ( )) ( ( )) .
Then, solve the PDE with B.C. and T.C. for V .
2.5 Bond Models
Bond is debt financial product in which an investor loans money to an entity which borrows the funds for a certain period of time at a fixed interest rate. A loan fund must be returned at maturity date with a promising value which is called face value. The market prices of bond depend on the market speculating. It normally opposes risk-free interest rate.
2.5.1 Term Structure of Bond
Consider a term structure of an instantaneous interest rate (spot rate) as
drt()r (,()) trtdt r (,()) trtdZ r (2.5.1)
where r (t , r ( t )) an expected value of the interest rate at time t,
2 and r (t , r ( t )) variance of the interest rate at time t.
21
Theorem 2.5.1 (Term Structure of Bond) [27]
Let B B(,) t r be a zero coupon bond price at time t with maturity date T.
Bond can be modeled in dynamic form as
dB(,) t r (,)(,)()(,)(,)trdt trdZ rt trdt trdZ (2.5.2) B(,) t r B B r B B r
where B (,)tr is an expected return of bond
B (,)tr r is a volatility of returns of bond B Br
B r B(,)tr r B is a risk premium of bond and r (,)tr is a market price of risk. B
Then, a term structure of bond is illustrated as
BBB 2 2 r rB 0 (2.5.3) tr r r r2 r 2
Proof: Assume that an investment market has no arbitrage.
Suppose the model for bond is illustrated in the form
dB utTdt(,)(,) vtTdZ (2.5.4)
To accomplish a term structure for bond, the derivation includes three major steps.
Step1 Expand Taylor’s series for bond differential in term of T and r.
BBBBB 11 2 2 2 dB dt dr dt2 dr 2 dtdr O() dt 3 t r22 t22 r t r
BBB 1 2 dt dt dZ dt 2 t rr r r 2 t 2
22 1 BB2 dt dZ dt dZ dt 2 r2 r r r t r r r r
22
2 2 BBBB r r 2 dt r dZ r t r2 r r
BBB 2 2 B Therefore, u(,) t T r and v(,) t T . tr r2 r 2 r r
Step2 Use -Hedging to eliminate volatility.
Define a portfolio that includes selling an amount 1 of bond with maturity time T1 , and
buying an amount 2 of bond with maturity time T2 ,
The portfolio total value equals WBB2 2 1 1 .
Then dW2 dB 2 1 dB 1
2utTdt(,)(,)(,)(,) 2 vtTdZ 2 1 utTdt 1 vtTdZ 1
2utT(,)(,)(,)(,) 2 1 utT 1 dt 2 vtT 2 1 vtT 1 dZ .
By using -Hedging, choose 1 and 2 to be
v(,) t T2 v(,) t T1 1 and 2 , v(,)(,) t T12 v t T v(,)(,) t T12 v t T
Thus, the volatility term is 2v( t , T 2 ) 1 v ( t , T 1 ) 0 ,
vtTutT(,)(,)(,)(,) vtTutT and dW 1 2 2 1 dt . (2.5.5) v(,)(,) t T12 v t T
Step3 Eliminate arbitrage.
Following the arbitrage-free assumption,
dW r( t ) Wdt r ( t )(2 B 2 1 B 1 ) dt
v(,)(,) t T B v t T B r() t1 2 2 1 dt (2.5.6) v(,)(,) t T12 v t T
23
Set (2.5.5) = (2.5.6)
vtTutT(,)(,)(,)(,)(,)()(,)()1 2 vtTutT 2 1 vtTBrt 1 2 vtTBrt 2 1
utT(,)()(,)(,)()(,)2 rtBvtT 2 1 utT 1 rtBvtT 1 2
utT(,)()(,)()2 rtB 2 utT 1 rtB 1 r v(,)(,) t T21 v t T
u(,)() t T r t B and since T and T are arbitrary, (,)rt , the market price of risk. 1 2 r v(,) t T
Therefore utT(,)()(,)(,) rtBr rtvtT
2 2 BBBB r r 2 rB r r , t r2 r r
BBB 2 2 or r rB 0 . tr r r r2 r 2
This is a term structure of bond.
The bond model can now be represented in the form
dB 11 rB vtT(,)(,) dt vtTdZ BBBrr
dB or r dt dZ B B B r
B where r and a risk premium. B Br B r B
It satisfies a terminal condition BTT( , ) 1.
24
2.5.2 Mean Reverting Vasicek’s Model
In this section, we consider a term structure of an interest rate which satisfies the mean reverting return introduced by O. Vasicek [19] in 1971. Let an instantaneous interest rate be modeled as
dr() r dt rr dZ (2.5.7)
and are interest rate long-term mean and the degree of mean reverting respectively.
From (2.5.3), the term structure of bond is rewritten into
BBB 2 2 ( r ) r rB 0 trr r2 r 2
2 2 BBBr r r ( ) r 2 rB 0 t r2 r
BBB 2 2 M r r rB 0 , (2.5.8) t r2 r 2
where M rr.
Theorem 2.5.2 (A Solution of Vasicek’s Bond Model) [19]
An exact solution for (2.5.8) is B( t , r ) exp a ( t ) b ( t ) r (2.5.9)
1 where b( t ) 1 eTt , (2.5.10)
2 and a()()()()() t R T t b t r b t 2 . (2.5.11) 4
22 RM() r r r r . 2222
25
Proof: By using a conjecture, let the explicit form solution for the PDE be in the form
B( t , r ) exp a ( t ) b ( t ) r
The condition B( T , r ) 1 implies aT( ) 0 and bT( ) 0.
B Then a b r ea br a b r B t
B bea br bB r
2B and b22 ea br b B . r 2
Substitute the derivatives into (2.5.8),
2 a brB MrbB r bBrB2 0 2
2 a b r M r b r b2 r 0 2
2 2 b b 10 r a Mb r b , 2 so bb 10 , (2.5.12)
2 and a Mb r b2 0 . (2.5.13) 2
From (2.5.12), by using the integrating factor technique,
b et be t e t
bett e
26
T 1 bTTe(,)(,)T btTe t ed e T e t t
1 b( t , T ) 1 eTt .
2 From (2.5.13), a Mb r b2 2
TT 2 a()() T a t M bd r b2 d tt2
TT 2 a( t ) M 1 eT t d r 1 2 e T e 2 T d 2 tt2
2 1T t 2 T t 1 2 T t M( T t ) 1 e r ( T t ) 1 e 1 e 2 22 2 11T t T t M( T t ) 1 e r ( T t ) 1 e 2 2
2 11T t 2 T t r 11 ee 2 22
22 1 T t T t 2 T t M rr( T t ) 1 e 1 2 e e 23 24
22 rr2 M 2 ()()(,) T t b t b t T 24
2 R()()()() T t b t r b t 2 . 4
27
CHAPTER THREE
PORTFOLIO OPTIMIZATION AND HJB EQUATION
We consider that a portfolio is comprised of investing in a riskless investment, bond and a stock. Bond is modeled by using Vasicek’s model. Stock is modeled by Jump-diffusion.
Consumer price index(CPI) and economics inflation are taken into account to evaluate real values of assets. An optimal portfolio optimization problem is to optimize a portfolio of real value. Thus, the problem is novel. Then stochastic control or Merton’s approach is applied to derive the corresponding HJB equation.
3.1 Stock Models
In an actual stock market, sometimes there are special which lead to a jump of stock price. These events depend on a number of factors. From the classical diffusion stock model in subsection 2.2.2, now consider a model including a jump of a stock return. It is added to the model so that the model features are more close related to the actual stock price.
3.1.1 Poisson Process
In theory, Jump situations have Poisson distribution. The process that runs by Poisson random variable is called the Poisson process.
Definition 3.1.1 (Poisson Process)
Let N Nt ,0 t be a Poisson process with expected intensity 0 .
It is a piecewise constant process with a stationary independent increment where N0 0 and
28
ehhk() P(N N k ) t h t k! for k = 0,1,2,… t 0 and h 0 a time increment.
Mean and variance of the process equal
()[]t E Nt t ,
and V()() t Var Nt t .
When an intensity ()t depends on time t in more general cases, the Poisson process can be
k t h t h exp ( )dd ( ) defined by P(N N k ) tt t h t k!
Next, suppose that the mark Ji is defined for any JNit0,1,... .
Nt Then the process YJti is defined as a compound Poisson process. i1
3.1.2 Jump Diffusion Stock Model
Let S(t) be a price of stock at time t that obey a jump-diffusion stochastic differential equation(SDE).
dS() t (,())tStdt (,()) tStdZt () Jt (,()) St dNt () (3.1.1) St() S S s
where St( ) 0 and SS(0) 0
S (,)tS is a function of time representing drift of the stock
S (,)tS is the diffusive volatility
ZtS () is a continuous one dimensional Wiener process
29
J(,) t S is the relative jump amplitude of stock prices
Nt()is a discontinuous one dimensional Poisson process with the rate N 0 .
Poisson process is the right-continuous counting process with the increment density
() dt k P(()) dN t kN eN dt . k!
The jump term JdN can be represented in more appropriate form as the Poisson measure
dN() t JtStdNt(,()) () JtStqNdtdq (,(),)(, ) JtStQ (,(), ) , i i1
0 in which J( t , S ( t ), Qi ) 0 . i1
q is defined on , the mask space for the jump amplitude associated with the random variable Q. 1 Jq ( ) is included to avoid a negative stock price.
An expected value for the Poisson measure density is E[ N ( dt , dq )] NQ dtf ( q ) dq where
fqQ ()is the mask density distribution.
The first and the second moment for the jump processes are
EJtStQdNt[(,(),) ()] dtJtStqf (,(),) () qdq (,()) tSt dt , NQN and VarJtStQdNt[(,(),) ()] dtJ22 (,(),) tStqf () qdq EJ [ (,(),)] tStQ dt , NQN where (t , S ( t )) is an expected jump mark at time t.
Thus (3.1.1) should be more properly modeled in the compensated form as
dS() t ( (,())tSt (,())) tSt dt (,()) tStdZt () JtSt (,()) dNt () (3.1.2) St() S N S s
30
Equation (3.1.2) is transformed into log-diffusion form as
11 dStln( ( )) dSt ( )c 2 dt [ln St ( ) ln St ( )] (3.1.3) St( ) 2 S
1 [ (,)tS (,) tS 2 (,)] tSdt (,) tSdZt ()ln(1 JtSQdNt (,,)) () SNSS2
(3.1.4) dS() t c is the continuous part of dS() t
An explicit form solution for (3.1.2) and (3.1.3) is
t 1 (,()) S (,) t S 2 (,()) S d Nt() SNS2 0 S() t S0 e (1(,(),)) J t S t Qi , (3.1.5) i1
tt Nt() where it is required (,)(,)(,,) S d S2 dZ J t S Q . S S i 00 i1
3.1.3 Existence and Uniqueness of SDE with Jumps.
To guarantee that the SDE solution exists, the integral term in (3.1.2) should be well defined.
Theorem 3.1.2 (A Unique Strong Solution for SDE with Jump) [18]
Suppose that a(.), b(.) and c(.) coefficients for SDE
dXt() atXt (,()) dt btXt (,()) dZ ctXt (,(),)(, qNdtdq )
Satisfy Lipchit condition, Linear growth and the initial condition as follow,
Lipschitz Conditions a(,)(,) t X a t Y C1 X Y
b(,)(,) t X b t Y C2 X Y
ctXq(,,)(,,)() ctYq22 f qdq CX Y . Q 3
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Linear Growth Conditions a( t , X ) K1 1 X
b( t , X ) K2 1 X
c( t , X , q )22 f ( q ) dq K 1 X , Q 3 for all tT[0, ] and XY, .
2 Initial Condition EX[]0 .
Then, the SDE admits a unique strong solution. Moreover, the solution satisfies an estimation
22 E[sup X ( t ) ] C (1 E [ X 0 ]) 0tT with T and C both finite positive constants.
Assume that the coefficient functions for the SDE with jumps (3.1.2) satisfy Lipschitz conditions that are
SS(,)(,)()()t S1 t S 2 C 1 S 1 t S 2 t ,
SS(,)(,)()()t S1 t S 2 C 2 S 1 t S 2 t , and JtSq(,,)(,,)()()() JtSq22 fqdq CSt St for all tT[0, ]. 1 2Q 3 1 2
They also satisfy the linear growth conditions as well.
S (t , S ) K1 1 S ( t )
S (t , S ) K2 1 S ( t )
JtSq( , , )22 fqdq ( ) K 1 St ( ) . Q 3
2 Assume an initial value S0 satisfies ES[]0
32
Then (2.6.2) has a unique strong solution. The solution satisfies estimation
22 E[sup S ( t ) ] C (1 E [ S0 ]) . 0tT
3.2 Ito Calculus
Ito Calculus is the extended techniques in Calculus to stochastic processes. It is named by the founder, Kiyoshi Ito in 1944. Ito’s lemma, or stochastic version of the chain rule is widely used in financial applications. The formula are derived by expanding Taylor’s series up to second order derivatives, then identifying the square of an increment in the Wiener process with an increment in time.
3.2.1 One Dimensional Continuous Ito Formula [5]
Let Xt() be a continuous stochastic process which obeys Ito differential
dXt()()()() atdt btdZt (3.2.1)
For t 0 and the initial value Xx(0) 0 .
Consider f :[0, ) which is differentiable with respect to time t and twice continuously differentiable on X. The change in f( t , X ( t )) is defined by Ito formula as
f f1 2 f dftXt(,()) dtat () dX dXt [()] t X2 X 2
2 f f1 2 f f at()()()() bt2 dtbt dZt (3.2.2) t X2 X X where d[ X ( t )] 22 ( t ) dt b ( t ) dt is the quadratic variation process.
33
3.2.2 Multi-dimensional Continuous Ito Formula [5]
1 d T Let X( t ) x ( t ),..., x ( t ) be a d-dimensional stochastic vector process where the k-th component xtk () is defined by the Ito differential
m dxtk()()()() atdt k b k, j tdZt j , (3.2.3) j1
T 1 d for t 0and given an initial value X( t ) x00 ( t ),..., x ( t ) .
ff Consider a function f :[0, ) d that has continuous partial derivative , and txk
2 f for all k, l 1,2,..., d . xxkl
Then the stochastic f satisfies Ito formula
fdd f1 2 f dftxt( ,1 ( ),..., xtd ( )) dt dx k dxtxt [ k ( ), l ( )] k k l tk1 x2 k , l 1 x x where d[ xkl ( t ), x ( t )] ( t ) a covariance between xk and x j at time t. xxkl,
It can be rewritten in the form
fd f1 m d 2 f m d f dftxx( ,1 ,...,d ) a k bb k , j l , j dt b k , j dZ j t t tk t t k l t k t tk1 x2 j 1 k , l 1 x x j 1 k 1 x
(3.2.4)
3.2.3 Ito Formula for Jump Process [5]
Let be a d-dimensional stochastic process including jump process which obeys the model
mn dxtk()()()()(,)(,) atdt k b k,, j tdZt j c k r tqNdtdq r . (3.2.5) jr11
34
Then the Ito formula for f has the form
fd f1 m d 2 f m d f dftxtxt(,(),...,())1d at k () btbt k , j ()() l , j dt bt k , j () dZt j () k k l k tk1 x2 j 1 k , l 1 x x j 1 k 1 x n 11d d r ftxt( , ( ),..., xt ( )) ftxt ( , ( ),..., xt ( )) Ndtdq ( , ) r1
(3.2.6)
3.3 Real Value of Assets
Both stock and bond actual price represent their nominal price. Their value included an economic inflation. It is crucial to consider the inflation rate to calculate the real value of assets.
[2]
3.3.1 Inflation Rate Model
Suppose the nominal price of consumption goods at time t, denoted as ()t , follows the SDE
dt () ()()()t dt t dZ t (3.3.1) ()t where ()t refers to an economic inflation rate which satisfies the dynamic SDE
d()(,)(,)() t t dt t dZ t . (3.3.2)
(,)t and (,)t are drift and diffusion of an economic inflation rate.
35
3.3.2 Real Values of Assets
A real value of assets can be found by degrading their nominal value using the price of consumption goods.
Anominalprice Arealprice (3.3.3) Anominal price of consumption good
Pt() Bt() Let a real value of bank deposit, bond and stock in (2.2.1), (2.5.2) and (3.1.2) equal , ()t ()t
St() and respectively. ()t
A stochastic model for a real value of risk-free asset, bond and stock that follow SDE (2.2.1),
(2.5.2) and (3.1.2) respectively is represented as follows.
A dynamic of the real value of a risk-free asset
p 2 d( r ( t ) ( t ) ( t )) dt ( t ) dZ (3.3.4) P
A dynamic of the real value of bond
B 2 d(() rt B (,) tr () t () t B ()) tdt B (,) trdZ r () tdZ (3.3.5) B
BBB(,)(,)()t r t r t is a covariance between bond return rate and nominal price of
consumption goods. Br corr(,) Z Z .
A dynamic of the real value of stock
S 2 dSNSSS(,)(,)()()()(,)()(,()) tS tS t t tdt tSdZ tdZJtStdN S (3.3.6)
where SSS(,)(,)()t S t S t refers to the covariance between the stock return rate and the
nominal price of consumption goods, where SS corr(,) Z Z .
36
Proof: By using Ito formula, the dynamics for the assets’ real value are the following:
From the model of P and in (2.2.1) and (3.3.1), we derive
P 11 d Pd dP PP
d 2 P dt 1 dP 2 P
d dP 2 dt P
2 dt dZ dt rdt
2 ()r dt dZ .
From the model of B and in (2.5.2) and (3.3.1), we get
B 1 1 1 d Bd dB dBd BB
dd 22 B dt 1 dB dB dt 22 B
d dB dB d 2 dt BB
2 ( dt dZ ) dt (( r B ) dt B dZ r ) B dt
2 ()r B B dt B dZ r dZ .
Finally, from the model of S and in (3.1.2) and (3.3.1) ,we get
37
S 1 1cc 1 1 d Sd dSdSd StSt()() SS
dd 22 S dt 11 dSdScc dt JtStStdN ,()() 22 S cc d2 dS dS d dt J t,() S t dN SS
2 dt dZ dt
((,)SNSSStS (,)) tSdt (,) tSdZ dtJtSt ,() dN
2 SNSSS(,)(,)(,)tS tS dt tSdZ dZ
J( t , S ( t )) dN .
3.4 Consumption and Terminal Wealth Problem
Consider a consumption and terminal wealth problem. Define a non-negative Ct(), a
T consumption rate process and C() t dt . Let [,,] be a portfolio strategy at time t, s B P 0
where 01i ,and s B P 1.
T Let Jw(0, , , C ) EeUC [ ( ( )) d eUWT T ( ( ))], (3.4.1) 0 0 where 0 is a discount factor and Ux()is a utility function.
An objective function for portfolio optimization of consumption and terminal wealth problem is
defined as maxJ (0, w0 , , C ) . (3.4.2) ( ,C ) A (0, w0 )
A is called an attainable set. A(,) t w { (,) C such that it is self-financing, W() t w and
W( ) 0 for all [,]tT }.
38
3.5 Wealth Process
Let W(t) be a self-financing wealth process of the portfolio including a bank deposit, a
bond, and a stock, with the weight PB, and S respectively.
Let WPPBB( t ) W ( t ), W ( t ) W ( t ) and WSS()() t W t be an amount of wealth invested in bank deposit, bond and stock at time t. Then the real values are
WWWPPP2 d r dt dZ ,
WWWBBB2 d r B B dt B dZ r dZ ,
WWWWSSSS2 dSNSSS dt dZ dZ JdN .
Hence, a dynamic for the real value of wealth is illustrated as following.
WWC2 d()()() r SSNSBBB r dt dt
W SW SBW 1 dZ JdN , (3.5.1)
2 S SB S dZS where 22 and . BS B B dZWr dZ 2 SB dZ
B r is a covariance between a stock return and a bond return SB BS Sr S B Sr S Br
rate and Sr is a correlation between ZS and Zr .
W C and are real values of the total wealth and consumption rate at time t.
39
W C Let W and C then (3.5.1) is rewritten into
dWt()(WWWW Wt ()()) Ctdt WtdZ () JWtdN () , (3.5.2)
2 where W (,,,)()()()(,)(,)()(,)tSr rt t t SSNS tS tSrt tS
BBB (,)(,)t r t r ,
2 S SB S S 22(t , S , r ) [ 1] W S B BS B B B 2 SB 1
2 2 2 2 2 SS(,)tS BB (,) tr ()2 t SBSB (,,)2 tSr SS (,)2 tS BB (,) tr ,
and JW ( t , S ( t )) S J ( t , S ( t )) .
3.6 Hamilton-Jacobi-Bellman Equation for Consumption and Terminal Wealth Problem
Define the value function for the above problem as
T VtWSr(,,,,) max JtWC (,,,) sup Ee [()() t UCde (()) T t UWT (())] (,)(,) C A t W (,)(,) C A t W t
(3.6.1)
W , S, r and satisfies the following SDEs respectively.
dWt()(WWW Wt ()()) Ctdt WtdZ () JWtdN ()
dSt()((,)SNSS tS (,))() tSStdt (,)() tSStdZ JtSt (,())() StdN
drt()(,)(,)r trdt r trdZ r
d()(,)(,) t t dt t dZ
The Hamilton-Jacobi-Bellman (HJB) equation that satisfies the optimal portfolio is presented as
sup[UCt ( ( )) VDV ] UCt (*** ( )) V DV 0 . (3.6.2) (,) C
40
A term DV * is a differential of the function value V * and is written as
****** DVtWSr(,,,,)()() Vt WW WCV S N SV S r V r V
1 1 1 1 2WVSVVV 2 * 2 2 * 2 * 2 * 2W WW 2S SS 2 r rr 2
****** WSWSV WS Wr WV Wr W WV W Sr SV Sr S SV S r V r
EV(* 1 JtSt (,()) Wt (),1 JtSt (,())() St S
* V( W ( t ), S ( t ))) dN .
The expected value for the change of value function occurred by jumps can be written in a term of integration on Poisson measure as
EV**1 JtSt (,()) Wt (),1 JtSt (,()() St VWtSt (),() dN S
V* 1 JtStq (,(),) Wt (),1 JtStqSt ,(), () NS
V* W( t ), S ( t ) f ( q ) dq . Q
Thus, an integro-differential equation represents HJB equation in the form
******** 0UCt ( ( )) VV t ( WW WCtV ( )) S N SV S r V r V
1 1 1 1 2WVSVVV 2 * 2 2 * 2 * 2 * 2W WW 2S SS 2 r rr 2
****** WSWSV WS Wr WV Wr W WV W Sr SV Sr S SV S r V r
V**1 JtStq (,(),) Wt (),1 JtStqSt ,(), () NS
V* W( t ), S ( t ) f ( q ) dq . (3.6.3) Q
41
1 or 0UCt (******** ( )) VV ( WCtV ( )) SV V V aV tWW S N S r r ij xij x 2 ij,
V**1 JtStq (,(),) Wt (),1 JtStqSt ,(), () NS
V* W( t ), S ( t ) f ( q ) dq , (3.6.4) Q
22W WS W W W WS Wr W SW 22 S S S where x,{,,,} x W S r , and Aa[] SW S Sr S . ij ij WS 2 rW rS r r WS 2 W Sr
Elements of A are as follows.
2 WS S S B SB S , W SSBB
2 B where r Wr S Sr B Br r Br Br B r Br
Sr Sr S r , SSS and r r r
Note that an optimal solution for (3.6.4) requires A to be positive definite.
2 WW0 0 0 W WS Wr W 0 0 0 2 0SS 0 0 0 0 0 T Consider AB SW S Sr S ΣB 0 0 1 0 2 0 0 1 0 rW rS r r 0 0 0 1 2 0 0 0 1 W Sr
dW dS where is a covariance matrix between , , dr and d so it can be written in the form W S
T E (X m )( X m ) .
T T dW dS dW dS where X dr d and m E E E dr E d . WS WS
42
TTTT 2 Thus, for any vector v, vAv vBEE( XmXm )( ) Bv vBXm ( ) 0 .
So A is semi definite, It can be concluded that A is positive definite in the sense that all elements in X are random variables whose variance is not zero. Therefore (3.6.4) has an optimal solution.
It is also required to satisfy a boundary and terminal condition,
T ()()()() t T tU(0) T t T t B.C. VtSr( ,0, , , ) eUdeU (0) (0) 1 e Ue (0) t
(3.6.5)
T.C. V( T , W , S , r , ) U ( W ( T )) . (3.6.6)
Then the regular controls * and C* are given implicitly in (3.6.7)-(3.6.9)
** U( C ( t )) VW (3.6.7)
* * 2 * 2 * 0S (,)tS N (,)() tSrt S (,) tSWV W S S (,) tS B SB S WV WW
Wt() V** 1 JtSt (,(),) q Wt (),1 JtSt ,(), qSt ()(,(),)()] JtSt qfqdq NSQ W
(3.6.8)
* * 2 * 2 * 0 (B B )WVW ( B B S SB B ) W V WW (3.6.9)
3.7 A Model Reduction
In this section, we consider a reduced-form model as a simple example to solve for a solution.
3.7.1 Models Formulation
x Consider a power utility function Ux() where x 0 and 01 .
Let a stock price, an interest rate, and an economic inflation satisfy the following SDEs.
43
dS() t ()() dt dZ J q dN (3.7.1) St() SNSS
S , S , N and are positive constants. dZS is a standard Wiener process.
q 2 Define J( q ) e 1 ,a relative jump amplitude, in which q (,) qq is independent and identically normally distribution (iid) so the mark space .
Next, we adopt Versicek’s mean reverting model for interest rate and bond.
dr( t ) ( r ( t )) dt rr dZ (3.7.2)
, and r are positive constants.
Then, a dynamic of bond is
dB(,) t r r() t dt dZ (3.7.3) B(,) t r B B r
B r B and B are positive constants.
Define an economic inflation model by using mean reversion as well.
d( t ) ( ( t )) dt dZ (3.7.4)
, and are positive constants.
Next, from the above assumption, we can see the dynamic of real value of wealth turn into the form
dW(WWW W C ( t )) dt WdZ JSS ( q ) WdN ( q ) (3.7.5)
2 where W (,,)(()())(())()t r r t t SSSBBB r t
2 2 2 2 2 2 W SS BB 2 SBSB 2 SS 2 BB
S,,,,, B SB S B and B are constant,
where SB SBSBS,, SS B BB .
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3.7.2 HJB Equation for Consumption and Terminal Wealth Problem
The consumption and terminal wealth optimal problem is defined as
T CWT()() T maxJ (0, w0 , , C ) sup E e d e (3.7.6) ( ,C ) A (0, w ) 0 ( ,C ) A (0, w0 ) 0
T CWT()() Let VtWSr(,,,,) max(,,,) JtWC sup[ Ee()() t de T t ]. (,)(,) C A t W (,)(,) C A t W t
From (3.6.3), a HJB equation for (3.7.6) is represented by
Ct* () 0 VV******* ( WCtV ())( )()() SV rV V tWW S N S r
1 1 1 1 2WVSVVV 2 * 2 2 * 2 * 2 * 2W WW 2S SS 2 r rr 2
****** WSWSV WS Wr WV Wr W WV W Sr SV Sr S SV S r V r
V***1 Jq () Wt (),(1 JqSt ())() VWtSt (),() () qdq (3.7.7) NS with B.C. V*( t ,0, S , r , ) 0 (3.7.8)
WT() T.C. V* (,,,,) T W S r (3.7.9)
2 WS S S B SB S , Wr S Sr B Br r , W SSBB
Sr Sr S r , SSS and r r r .
3.7.3 A Solution of Reduction Form
From (3.7.7), * and Ct* () are regular control which can present in the form
45
* 1 * From (3.6.6), C() t VW
1 ** 1 C() t VW . (3.7.10)
* * 2 * 2 * From (3.6.7), 0 S N r S WVW S S B SB S W V WW
Wt() V** 1 Jq () Wt (),1 JqSJqqdq () ()()] . NS W
(3.7.11)
* * 2 * 2 * From (3.6.8), 0 (B B )WVW ( B B S SB B ) W V WW . (3.7.12)
Then (3.7.11) and (3.7.12) can rewritten as
0 r 2 WV * * 2 2 2 2 W 2 V * SN SB BBSB W SBSSB SBBSB WW
2Wt() V * 1 Jq () * Wt (),1 JqSJqqdq () ()()] (3.7.13) NBS W
VW B S SB BBB 2 2 2 (3.7.14) BBBWVWW
Since the optimal controls (3.7.10-(3.7.14) represented in the implicit form, and the HJB equation (3.7.7) is non linear, we then find the optimal solution by using techniques of numerical solving. Next, an existence and uniqueness of the solution for (3.6.3) is proved in Chapter Four.
The numerical solution for the reduced form problem (3.7.6) is presented in Chapter Five.
46
CHAPTER FOUR
EXISTENCE AND UNIQUENESS OF THE OPTIMAL SOLUTION
In this chapter we shall prove that the HJB equation (3.3.3) derived from the necessary condition for the dynamic optimal portfolio problem has a unique solution under certain mild conditions. We will indicate that these conditions are financially reasonable. The existence of the problem relies on a priori estimates for the partial differential equation and Schauder’s fixed- point theorem. The uniqueness is proved from the solution representation.
4.1 Existence of the Optimal Solution for Cauchy Problem
Consider the following Cauchy problem
n ut Lu f(,) x t x,0 t T (4.1.1)
u( x ,0) g ( x ) x n (4.1.2)
2uu where Luaxtij(,)(,)(,) bxt i cxtu . xi x j x i
Assumption 4.1.1 Suppose the coefficients of operator L satisfy the following conditions:
22n a01 aij i j a ij,
n biT(,)() x t L Q QTT (0, ]
c(,)() x t L QT
Assumption 4.1.2 suppose f(,) x t and gx() satsfies
, 2 nn f(,)()() x t L QTT C Q and g()()() x L C
47
Lemma4.1.3
Under the assumption 4.1.1-4.1.2, the Cauchy problem (4.1.1)-(4.1.2) has a unique solution.
2 ,1 , 22 u(,)()() x t C QTT C Q
Moreover, the solution can be expressed by the following representation
t uxt(,) (,,,0)() xyt gydy (,,,)(,) xyt fy dyd . nn0 where (,,,)x y t is the fundamental solution of the parabolic equation
ut Lu (,) x y t
See the monograph [20] for the proof.
Lemma 4.1.4
The fundamental solution (,,,)x y t of (4.1.1) satisfies the following estimate for any nonnegative integers r and s
2rs 2 xy Drs D(,,,) x y t C( t )2 exp C , tx 12 t
where C1 depends only on known data and is the Holder exponent of the coefficients of operator L. See reference [20].
Lemma 4.1.5
There exists constants C3 and C4 such that
t uCC f d , LL()()nn34 0 where and depend only on known data.
48
2 xy Proof : By using Lemma 4.4 for rs0 and the fact that t 2 et dy C , n thus (,,,)Cx y t .
tt Therefore, u (,,,0)() xytgydy (,,,)(,) xytfydyd C C f d . LL()()nn 34 nn00
Lemma 4.1.6
There exists constants C5 and C6 such that
t f n (a) udCC L () x L ()n 56 0 t
(b) uf2,1 C6 p WQLQp()() T T where W2,1()()() Q uu LQandupp LQ . p T xij x T t T
Proof : From the solution representation, we see
t uxt(,) (,,,0)() xyt gydy (,,,)(,) xyt fydyd , in1,..., . xi x i x i nn0
Let I( x , y , t ,0) g ( y ) dy xi n
(x , y , t ,0) g ( y ) dy since (,,,)x y t is symmetric on x and y variables yi n
(x , y , t ,0) g ( y ) dy yi n
So Ig C 1 x L ()n
t f L ()n Then, we use Lemma 4.3 for rs0, 1 to obtain unnCC g d . xxLL()()12 0 t
49
Lemma 4.1.7
There exists a constant C78 ,C such that
t f n L () , u2 CC7 8 d C 9 f n t CQL()()t 0 t where depends only on known data
, 0, Proof: Consider u2 u u 2 CQCQ()()t CQ()t
u()() x12 u x supu sup x Q x12, x Q xx 2 xx12 12
t From lemma 4.1.5, uCC f d LL()()nnab 0
t f n For t 1, udCC L () CQ() ab 0 t
, 2,1 2 Since WQpT()↪CQ()T for any (0,1) provided p is large.
By Sobolev embedding for large p,
, u2 C f CQLQ()()TT
Cmn C f p . LQ()T
Therefore,
, , 2 2 because CQ()T ↪CQ()T since .
50
4.2 Existence and Uniqueness of the Portfolio
Now, consider the HJB equation (3.6.3)
******** 0UCt ( ( )) VV t ( WW WCtV ( )) S N SV S r V r V
1 1 1 1 2WVSVVV 2 * 2 2 * 2 * 2 * 2W WW 2S SS 2 r rr 2
****** WSWSV WS Wr WV Wr W WV W Sr SV Sr S SV S r V r
V***1 JtStq (,(),) Wt (),1 JtStqSt ,(), () VWtSt (),() fqdq () NSQ
(4.2.1)
We want to write (4.2.1) in form of Cauchy problem.
Introduce the new variables, V**(,,,,)(,,,,) t W S r u y x r . where xS ln( ) , yW ln( ) , Tt, and qJln(1 ).
11 So dy d(ln W ) dWc 2 dt ln(1 J ) dN W 2 WW
121 WW CW WWSS dt dZ ln(1 J ) dN , 2
11 and dx d(ln S ) dSc 2 dt ln(1 J ) dN S 2 SS
1 2 SNSSSS dt dZ ln(1 J ) dN . 2
Then, (4.2.1) becomes
51
2 2 0UCT (******** ( )) uu CTe ( ) y WS u uuu W y S N x r r 22
1 1 1 1 2*u 2* u 2*2* u u u * u * u * u * u * u * 2Wyy 2 S xx 2 r rr 2 WS yx Wr yr W y Sr xr S x r r
uyq**** ln( (1 ) exquyxfqdqq ), , ( ) (4.2.2) SSQ or in the Cauchy form as
*1 2* 1 2*2*2* 1 1 * * * * * * uuuuuuuuuuuyy S xx r rr yx yr y Sr xr S x r r 2W 2 2 2 WS Wr W
2 2 C******() T e y WS u u u u u W y S N x r r 22
uyq***** ln( (1 ) eq ), xquyxfqdqUCT , ( ) ( ( )) (4.2.3) NSSQ with I.C. u*(0, y , x , r , ) U ( ey ) . (4.2.4)
Assumptions 4.2.1 Basic Assumptions for (4.2.3) and (4.2.4)
2 2 2 2 1) Consider WW(,,,),tSr S (,), tS r (,), tr (,), t (,,), tSr S (,), tS r (,), tr (,), t
WS(,,),tSr Wr (,,), tSr W (,,,), tSr Sr (,,), tSr S (,,), tS r (,,) tr LQ ( T )
They satisfy the Lipchitz condition, and the linear growth condition.
2) Consider
2 W WS Wr W 2 A SW S Sr S 2 rW rS r r 2 W Sr
A is positive definite so that the optimal solution exists.
52
These assumptions make sense in the actual situation because there is not any asset in the market that provide infinite return. For example, a return of stock will never be infinite which
2 implies that an expected return S , the stock return diffusion S , and the stock return jump J must be bounded.
dW dS The matrix A is semi definite because it is covariance matrix between , , dr and d . W S
They are stochastic random variables whose variance is not zero, so vAvT 0 .
Theorem 4.2.2 (Schauder’s Fixed-Point Theorem) [4]
Let K be a compact convex set in a Banach space V and let M be a continuous mapping of
K into itself. Then, M has fixed point, that is, Mx = x for some xK .
Theorem 4.2.3
Under the assumption(4.1.1)-(4.1.2), the problem (4.2.3)-(4.2.4) has a unique solution
2 ,1 , 22 u C()() QTT C Q
, 2 Proof: Let K v C( Q ), v , K where K is a constant to be chosen. T CQ2 () 0 0 T
For each vK , consider the problem (4.2.3)-(4.2.4) as a Cauchy problem
** u Lu f(,) v
u*(0, y , x , r , ) g ( y ) where fv(,) is defined as the right hand side of (4.2.3) and g()() y U ey , yW ln( ) .
53
By solution representation,
t u* (,,,,)()()(,,,,)(,)(,,,,) yxr Mv gy yxr dy fv yxrdyd . nn0
t , fv() 2 CQ() By Lemma 4.1.7, we have u , CC d C f t CQL2 ()()1 2 3 n 0 t
t v , CC CQ2 ()d C f t 1 4 3 L ()n 0 t
CC1 4Kt 0 , min , 2
1 1 Choose t to be small. Let t and K C1. Thus, uK , 01 CQ2 () 0 C40K
Next, consider [TT00 ,2 ], by the same fashion, we obtain
t , fv() 2 CQ() , u2 CCCC d K t T and . CQ() 1 2 t 1 4 0 0 T0
For any n 0,1,2,...such that [nT00 ,( n 1) T ], .
Therefore for all [0,T ] .
So by Theorem 4.2.2, the mapping M has a fixed point.
t u***(,,,,)()()(,,,,)(,)(,,,,) yxr Mu gy yxr dy fu yxrdyd nn0
There exists a solution of the problem (4.2.3)-(4.2.4).
54
Next, we have remaining to show that the solution is unique.
Let u(,)(,)(,) x t u12 x t u x t where u1 and u2 are both the solution of a problem (4.2.3)-(4.2.4)
So u Lu f(,)(,) u12 f u with I.C. ux( ,0) 0
By the solution representation, we get
t uxt(,)(,)(,)(,,,,) fu fu yxrdyd 12 0 n
t
u fn u(,,,,) y x r dyd LL u 0 n
t C u d L n 0
By Gronwall’s Inequality, u( x , t ) 0 so u12(,)(,) x t u x t . The solution is unique.
55
CHAPTER FIVE
A COMPUTATIONAL SOLUTION FOR THE OPTIMIZATION PROBLEM
5.1 Parameters Calibration
5.1.1 Interest Rate Model Calibration
Let consider a mean reverting interest rate model of the form
dr( r ( t )) dt r dZ
A solution of the dynamic interest is
t r( t ) r e()()t t00 (1 e t t ) e()t dZ ( ) 0 t0
Given an initial condition r() t00 r
For rt() whose distribution is Gaussian , it appears that
E r( t ) F r e()()t t00 (1 e t t ) t0 0
2 V r( t ) Fr (1 e2 (tt0 ) ) t0 2
Now consider dr()()() t dt r t dt r dZ t
as y()()() t C12 C r t t , (5.1.1)
where C1 dt , C2 dt and ()()t rr dZ t Z dt . (5.1.2)
We wish to determine the model parameter , and r .
First of all, consider
V() t V Z dt 22 dtV Z dt r r r
56
Vt () Thus (5.1.3) r dt
Next, we fit the model to the existing data by using a linear least square method.
So (5.1.1) is considered as Y RC E (5.1.4)
T where Y dr( t12 ) dr ( t ) .... dr ( tn )
T 1 1 ... 1 R r( t12 ) r ( t ) ... r ( tn )
T C CC12
T E (t12 ) ( t ) .... ( tn )
n 2 By the least square method, the error, Et ()i is minimized. i1
So we estimate Y Yˆ RC. (5.1.5)
Then CRRRY ()()TT1 , (5.1.6) and E Y Yˆ Y RC . (5.1.7)
Therefore, the parameter estimations are done by
C2 CC11 V E , and r (5.1.8) dt dt C2 dt
By using The U.S. interest rate during January, 2009-May, 2015, (see Appendix A),
we got 1.0075 , 0.0011 and r 0.00077 .
Figure 4 show a graph of the U.S. interest rate during January, 2009-May, 2015 and the modeled value by Vasicek’s interest rate model.
57
Spot Interest Rate 0.35 real value modeled value 0.3
0.25
0.2
Rate (%) Rate 0.15
0.1
0.05
0 0 10 20 30 40 50 60 70 80 month
Figure 4: A graph of the U.S. monthly interest rate during January, 2009-May, 2015 and the
modeled value by Vasicek’s interest rate model.
Forcasted Interested Rate r(t) 0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003 Forcasted Interested Rate(%) 0.002
0.001
0 0 5 10 15 20 25 30 35 40 months in the future
Figure 5: The forecasted interest rate for January, 2016-December, 2018.
58
5.1.2 A Calibration of the Market Price of Risk
Consider an explicit form solution for bond of the form
B(,) t T ebtTrt(,)() atT (,) e YttTt (,)( ) . (5.1.9)
So YtTT(,)( t ) btTrt (,)() atT (,)
YtTT(,)( t ) btTrt (,)() atT (,) , where T is a maturity time.
b(0, T ) r (0) a (0, T ) At t 0, YT(0, ) . T
1 From (2.5.10) and (2.5.11), b(0, T ) 1 eT ,
22 rr2 a(0,) T M 2 T b (0,) T b (0,) T , 24
and M rr.
We approximate the market price of risk, r , by minimizing the difference between
b(0, T )(0) r a (0, T ) YT(0, ) and . T
22 rr2 or YTTbTr0()()()()() 0 0 2 TbT 0 bT 0 TbTM 0 24 a or YX M (5.1.10)
22 where Y YTTbTr()()()() rr TbT bT 2 , 0 0 0242 0 0
and X T b0 () T .
Thus M ()()XXXYTT1 . (5.1.11)
59
and r M (5.1.12) r
We used U.S. Treasury bonds yield during January, 2009-December, 2015, including one-month, three-month, six-month, one-year, two-year, three-year, five-year, seven-year, ten-year, twenty-
year and thirty-year maturity. We obtained r 5.5506 . Thus the predicted bond yield for the next three years can be seen in Figure 6.
Forcasted Bond Yields
0.03
0.025
0.02
0.015 Yields 0.01
0.005
0 30 40 20 30 10 20 10 0 0 Time to Matuarity months in the future
Figure 6 : The predicted zero-coupon bond yield during January, 2016-December, 2018.
60
5.1.3 A Bond Model with Economic Inflation
Consider a SDE model of a nominal price of consumption goods as the form (3.3.1)
dt () ()()()t dt t dZ t ()t where an economic inflation is formed by a mean reverting model,.
d()() t dt dZ .
By applying the similar calibration technique as an interest rate model to an economic inflation
during January, 2009 to December, 2015, we got 1.5588 , 0.0142 and 0.0217 .
Then by using a similar time interval for consumer price index data, 0.0088.
Figure 7 demonstrates a seven-year maturity bond beginning at January 2009 and ending at
December 2015 in nominal value and real value.
Seven year maturity bond 1
0.95
0.9
Bond Bond price 0.85
0.8
nominal value real value 0.75 0 10 20 30 40 50 60 70 80 90 month
Figure 7: A nominal price and a real value price for seven-year maturity bond during January
2009 - December 2015.
61
5.1.4 A Calibration of the Stock Model with Jump
Consider a stock model by SDE with jump.
dS S()()SNN dt S t dZ SJdN ,
2 or dln( S ) ( ) dt ( t ) dZ ln(1 J ) dN , SNN2
2 where J~ iid Lognormal or ln(1 J ) ~ N(,)qq.
2 q So E J e q 2 1.
Suppose x dln S has Lognormal-diffusion and Lognormal-jump distribution.
2 S 2 Then x~dln S ( xPdtNx ) 0 ( N ) ; S dtdt , S 2
2 S 22k Pk();,;, N dtNx S dt S dtNx q q k1 2 as dt 0 , this can be approximated as
22 SS2 2 2 , dln S()(1);x N dt N x S dt , S dt (); N dt N x S dt , S dt N x ;, q q 22
(5.1.13)
()x 2 e 2 2 where Nx(;,)2 is a normal distribution, 2
e x and Px(;) is a Poisson distribution. x!
62
Characteristics of x generate cumulants as follow [10]
2 S cumulant1S N ( N q ) (5.1.14) 2
2 2 2 cumulant2S N ( q q ) (5.1.15)
23 cumulant33N q q q (5.1.16)
4 2 2 4 cumulant4N 3 q q 6 q q q . (5.1.17)
Then E dln( S ) cumulant 1
V dln( S ) cumulant 2
cumulant3 Skewness dln( S ) 3 cumulant2 2
cumulant4 Kurosis dln( S ) 4 . cumulant2 2
22 We wish to find a set of parameters S,,,, S q q N in order to distinguish a feature of jumps to normal diffusion. The data including jump tend to have a longer tail on the negative side compared to normal distribution. Statistics of the realistic SP500 data during the same period
ac ac ac obtained Ex 0.0104 , Var x 0.0011, Skew x 0.6557 and
ac Kurtosis x 5.3240 .
63
Histogram of dlnS 14
12
10
8
frequency 6
4
2
0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 dlnS
Figure 8: A histogram of an actual data of ln S .
We fit the distribution of dSln to a proper weight so they satisfy a least-square fit.
N bin 22th ac wi() f i f i , (5.1.18) i1
where wi is a weight of the i-th bin of the histogram
ac fi is the i-th empirical bin frequency data
th fi is the i-th theoretical Jump-diffusion frequency
N bin is a number of bin in a histogram
N bin ac ac and Nf i is a number of samples. i1
2 th We use a searching technique to find five parameters that minimize . wi and fi in
(5.1.18) can be found by the following method.
64
We can approximate an empirical frequency of the histogram by simulating x. [9]
N sim times of a N sp -samples size data are simulated where the five parameters,
22 S,,,, S q q N are given.
thsim Let fi be the i-th bin frequency of a histogram of x
Then th thsim , (5.1.19) fii E f
th 2 2 thsim simfi th and th Var f N1 f , (5.1.20) f iisim i N where the i-th expected bin frequency after N sim times simulation is
xi1 fth N sim th () x dx . ii xi
The bin weight are chosen as
1 2 th fi wi (5.1.21) N bin 1 2 j1 th f j for iN1,..., bin .
An optimization for (5.1.18) is subject to constraints
2 thS sp cumulant1S N ( N q ) E x , 2
th2 2 2 sp and cumulant2S N ( q q ) Var ( x ).
We set N bin 100 , N sim 100 and N sp 83 from the monthly SP500 during January 2009-
December 2015. By using “fminsearch”-an optimization toolbox in Matlab, we got a set of
22 parameters as follows. S0.0104, S 0.0011, q 0.0104, q 0.0331,and N 0.2003 .
65
The set of parameters obtained the minimal least square 2 0.0895. By this set of parameters,
th th th we get its statistics as Ex 0.0104, Var x 0.0011, Skew x 0.3005, and
th Kurtosis x 10.8787 .
SP500 2200
2000
1800
1600
1400 Stock price 1200
1000
800 nominal value real value 600 0 10 20 30 40 50 60 70 80 90 month
Figure 9: Nominal and real value of SP500 stock index during January 2009-December 2015.
5.2 A Numerical Solution for a HJB Equation
Consider the reduction form in section 3.7.2. Regular control Ct(), S and B for the
HJB equation (3.7.7) satisfy (3.7.10) , (3.7.13) and (3.7.14) respectively. Since the HJB itself and the control are nonlinear, we have to find a solution by a numerical method. Here, we choose a
Crank-Nicolson Finite Difference to discretize the equations.
66
5.2.1 Finite Difference for Differential Discretization
The HJB equation is discretized by backward in time since the problem has a terminal condition. It is discretized on space domain by regular grid. Suppose we look for a solution in
time interval t[,] t0 T and a bounded space domain W[0, Wmax ], S [0, S max ], r [0, r max ] and
[,] min max .
tk T ( k 1) t for kN1,..., t where t ( T t0 ) /( Nt 1)
Wj ( j 1) W for jN1,..., W where WWN (max 0)/(W 1)
Sl ( l 1) S for lN1,..., S where SSN (max 0)/(S 1)
rm ( m 1) r for mN1,..., r where r ( rmax 0) /( Nr 1)
n (n 1) for nN1,..., where ( max min )/(N 1)
Then we discretize differential terms in (3.7.7) as follows.
V(,,,,) tk W j S l r m n V k.... j l m n
VV V(,,,,) t W S r k1. j . l . m . n k . j . l . m . n t k0.5 j l m n t
VV V(,,,,) t W S r k. j 1. l . m . n k . j 1. l . m . n W k j l m n 2W
VVV2 V(,,,,) t W S r kj.1... lmn kjlmn .... kj .1... lmn WW k j l m n ()W 2
VVVV V(,,,,) t W S r kjlmn. 1. 1. . kjlmn . 1. 1. . kjlmn . 1. 1. . kjlmn . 1. 1. . WS k j l m n 4WS
67
VS , Vr and V are discretied in the same way as VW .
VSS , Vrr and V are discretied in the same way as VWW .
VWr , VW , VS and Vr are discretied in the same way as VWS .
5.2.2 Discretization for Jump Integration
One method to approximate jump integration is by using Gauss-Statistical Rule.[7]. We
generate Gaussian weight, wi to imitate jump mark probability density function for NQ points of
jump amplitude Qi , iN1,..., Q . These weights satisfy a precision of polynomial degree NQ 1.
NQ Then wQd E Q d q d () q dq for dN0,...,2 1. i i i Q Q i1 Q
Let VJtWSr(,,,,)kjlmn VJ kjlmn.... Vt k ,(1 JtQ (,)),(1 kiSj W JtQSr (,)),, kilmn .
VJ is a value function including jump.
NQ So VJtWSr(,,,,)(,,,,)() VtWSr qdq wVJ V . k jlmn k jlmn Q i kjlmn...... kjlmn Q i1
Since VJ is implicit on W and S, we have to approximate VJ by using interpolation from the neighbor state node.
Let (1J ( t , Q ) ) W W W W k i S j j pi i
W where J(,)() tk Q i S W j p i i W
J(,) tk Q i S W j pi , a quotient of the division W
J(,) t Q W and W k i S j p , a remainder of the division. iiW
68
Let (1J ( t , Q )) S S S S k i l l ppi i
S where J(,)() tk Q i S l pp i i S .
J(,) tk Q i S l ppi , a quotient of the division S
J(,) t Q S and S k i l pp , a remainder of the division. iiS
Thus, VJ( Q ) (1 SWW ) (1 ) V V kjlmn.... i i i kjplppmn ....i i i kjp .1... i lppmn i
SWW(1 )VV . i i kjplpp..i i 1.. mn i kjp . i 1. lpp i 1.. mn
5.2.3 Numerical Solving for Optimal Solution
* Regular controls, Ct(), S and B , for HJB equation (3.7.7) are represented in implicit form in (3.7.10), (3.7.13) and (3.7.14) as follow.
1 ** 1 C() t VW (3.7.10)
0 r 2 WV * * 2 2 2 W 2 V * SN SBBBSB W SSSB SBBSB WW
2Wt() V * 1 Jq () * Wt (),1 JqSJqqdq () ()()] (3.7.13) NBS W
VW B S SB BBB 2 2 2 (3.7.14) BBBWVWW
The integral term in (3.7.13) can be approximated by using Gaussian weights as well.
NQ VJ()() Q V Q Vt(,(1 JtQ (, ) ) W ,(1 JtQSr (, )),, )(, JtQ ) () qdq w kjlmn. 1. . . i kjlmn . 1. . . i Wk kiSj kilmnkiQ i Q l1 2W
69
* By a terminal condition, we start from k 1. We search for Sk()t that satisfy (3.7.13). Then
* * substitute it to (3.7.14) for Bk()t . We also get Ct()k from (3.7.10). Next, plug the regular controls into the HJB equation (3.7.7).
Suppose (3.7.7) is represented in the form
Vt LV 0 so Vt LV ,
VV or kk1 LV . t
Then Vkk1 V LV t .
For a given initial wealth, Ww(0) 0 , we need to start an iteration by setting W() T w0 .
Parameters for a stock model (3.7.1), an interest rate model(3.7.2), a bond model (3.7.3), an economic inflation (3.7.4) and consumer price index model (3.3.1) obtained by calibrating them using real data during January 2009 to December 2015 downloaded from [27] are shown in
Table 1 and Table 2.
70
Table1: Parameters calibration result for stock, interest rate, bond, economic inflation and
consumer price index.
0.0104 Economic inflation variance 4 Stock return mean ()S 4.708 10
2 ()
2 0.0011 Interest rate drift( ) 1.0075 Stock return variance () S
Log normal jump mean( -0.0104 Interest rate longterm mean( ) 0.0011
()q )
Log normal jump variance 0.0331 2 7 Interest rate variance () r 5.929 10
2 () q
0.2003 5.5506 Jump mean ( N ) Market price of risk( r )
0.0062 2 7 Stock return jump mean(N ) Bond return variance () B 2.708 10
Economic inflation drift ( ) 1.5588 Consumer price index variance 7.744 105
2 ()
Economic inflation long 0.0142 term mean ( )
Table 2: Correlation between stock, bond, interest rate, consumer price index, and economics
inflation series.
SB 0.0279 Sr 0.2136 S 0.1641
S 0.2209 Br 0.0431 B 0.1839
B 0.0767 r 0.1451 0.5712
71
We set a discount rate as 0.005. A power utility function where 0.5 is adopted.
Then the value function for the last ten months of 2015 is illustrated in Figure10.
Value Function
1000
800
600 V 400
200 10 0 0 2 4 5 6 8 10 12 0 W months
Figure10: A value function for HJB equation (3.7.11)
Consumption Function
10
8
6 C 4
2 10
0 8 0 6 2 4 4 6 8 2 10 12 0 W months
Figure11: An optimal consumption for the portfolio problem
72
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75
APPENDIX A
FINANCIAL DATA AND MATERIALS
1. U.S. Interest Rate data During January 2009 to December 2015 yy-mm Int Rate yy-mm Int Rate yy-mm Int Rate yy-mm Int Rate
2009-01 0.15 2010-10 0.21 2012-07 0.14 2014-04 0.05
2009-02 0.28 2010-11 0.2 2012-08 0.14 2014-05 0.05
2009-03 0.22 2010-12 0.19 2012-09 0.12 2014-06 0.06
2009-04 0.22 2011-01 0.19 2012-10 0.14 2014-07 0.07
2009-05 0.22 2011-02 0.19 2012-11 0.14 2014-08 0.08
2009-06 0.18 2011-03 0.17 2012-12 0.13 2014-09 0.06
2009-07 0.18 2011-04 0.14 2013-01 0.12 2014-10 0.06
2009-08 0.17 2011-05 0.11 2013-02 0.12 2014-11 0.07
2009-09 0.14 2011-06 0.11 2013-03 0.1 2014-12 0.11
2009-10 0.12 2011-07 0.09 2013-04 0.08 2015-01 0.09
2009-11 0.13 2011-08 0.11 2013-05 0.07 2015-02 0.08
2009-12 0.14 2011-09 0.08 2013-06 0.07 2015-03 0.08
2010-01 0.13 2011-10 0.09 2013-07 0.06 2015-04 0.07
2010-02 0.13 2011-11 0.1 2013-08 0.05 2015-05 0.08
2010-03 0.15 2011-12 0.1 2013-09 0.05 2015-06 0.09
2010-04 0.19 2012-01 0.09 2013-10 0.07 2015-07 0.08
2010-05 0.21 2012-02 0.12 2013-11 0.05 2015-08 0.1
2010-06 0.19 2012-03 0.13 2013-12 0.06 2015-09 0.13
2010-07 0.2 2012-04 0.13 2014-01 0.05 2015-10 0.11
2010-08 0.2 2012-05 0.13 2014-02 0.06 2015-11 0.11
2010-09 0.21 2012-06 0.14 2014-03 0.06 2015-12 0.28
77
2. U.S. Treasury Bond Data During January 2009 to December 2015
time 1month 3month 6month 1year 2year 3year 5year 7year 10year 20year 30year 2009- 01 0.05 0.13 0.3 0.44 0.81 1.13 1.6 1.98 2.52 3.46 3.13 2009- 02 0.22 0.3 0.46 0.62 0.98 1.37 1.87 2.3 2.87 3.83 3.59 2009- 03 0.1 0.22 0.43 0.64 0.93 1.31 1.82 2.42 2.82 3.78 3.64 2009- 04 0.1 0.16 0.35 0.55 0.93 1.32 1.86 2.47 2.93 3.84 3.76 2009- 05 0.14 0.18 0.3 0.5 0.93 1.39 2.13 2.81 3.29 4.22 4.23 2009- 06 0.1 0.18 0.31 0.51 1.18 1.76 2.71 3.37 3.72 4.51 4.52 2009- 07 0.15 0.18 0.28 0.48 1.02 1.55 2.46 3.14 3.56 4.38 4.41 2009- 08 0.12 0.17 0.27 0.46 1.12 1.65 2.57 3.21 3.59 4.33 4.37 2009- 09 0.06 0.12 0.21 0.4 0.96 1.48 2.37 3.02 3.4 4.14 4.19 2009- 10 0.04 0.07 0.16 0.37 0.95 1.46 2.33 2.96 3.39 4.16 4.19 2009- 11 0.05 0.05 0.15 0.31 0.8 1.32 2.23 2.92 3.4 4.24 4.31 2009- 12 0.03 0.05 0.17 0.37 0.87 1.38 2.34 3.07 3.59 4.4 4.49 2010- 01 0.02 0.06 0.15 0.35 0.93 1.49 2.48 3.21 3.73 4.5 4.6 2010- 02 0.06 0.11 0.18 0.35 0.86 1.4 2.36 3.12 3.69 4.48 4.62 2010- 03 0.12 0.15 0.23 0.4 0.96 1.51 2.43 3.16 3.73 4.49 4.64 2010- 04 0.15 0.16 0.24 0.45 1.06 1.64 2.58 3.28 3.85 4.53 4.69 2010- 05 0.15 0.16 0.22 0.37 0.83 1.32 2.18 2.86 3.42 4.11 4.29 2010- 06 0.08 0.12 0.19 0.32 0.72 1.17 2 2.66 3.2 3.95 4.13 2010- 07 0.16 0.16 0.2 0.29 0.62 0.98 1.76 2.43 3.01 3.8 3.99 2010- 08 0.15 0.16 0.19 0.26 0.52 0.78 1.47 2.1 2.7 3.52 3.8 2010- 09 0.12 0.15 0.19 0.26 0.48 0.74 1.41 2.05 2.65 3.47 3.77
78
2. U.S. Treasury Bond Data During January 2009 to December 2015 (continued)
time 1month 3month 6month 1year 2year 3year 5year 7year 10year 20year 30year 2010- 10 0.14 0.13 0.18 0.23 0.38 0.57 1.18 1.85 2.54 3.52 3.87 2010- 11 0.13 0.14 0.18 0.25 0.45 0.67 1.35 2.02 2.76 3.82 4.19 2010- 12 0.09 0.14 0.19 0.29 0.62 0.99 1.93 2.66 3.29 4.17 4.42 2011- 01 0.14 0.15 0.18 0.27 0.61 1.03 1.99 2.72 3.39 4.28 4.52 2011- 02 0.11 0.13 0.17 0.29 0.77 1.28 2.26 2.96 3.58 4.42 4.65 2011- 03 0.06 0.1 0.16 0.26 0.7 1.17 2.11 2.8 3.41 4.27 4.51 2011- 04 0.03 0.06 0.12 0.25 0.73 1.21 2.17 2.84 3.46 4.28 4.5 2011- 05 0.02 0.04 0.09 0.19 0.56 0.94 1.84 2.51 3.17 4.01 4.29 2011- 06 0.02 0.04 0.1 0.18 0.41 0.71 1.58 2.29 3 3.91 4.23 2011- 07 0.04 0.04 0.08 0.19 0.41 0.68 1.54 2.28 3 3.95 4.27 2011- 08 0.02 0.02 0.06 0.11 0.23 0.38 1.02 1.63 2.3 3.24 3.65 2011- 09 0.01 0.01 0.04 0.1 0.21 0.35 0.9 1.42 1.98 2.83 3.18 2011- 10 0.01 0.02 0.05 0.11 0.28 0.47 1.06 1.62 2.15 2.87 3.13 2011- 11 0.01 0.01 0.05 0.11 0.25 0.39 0.91 1.45 2.01 2.72 3.02 2011- 12 0 0.01 0.05 0.12 0.26 0.39 0.89 1.43 1.98 2.67 2.98 2012- 01 0.02 0.03 0.07 0.12 0.24 0.36 0.84 1.38 1.97 2.7 3.03 2012- 02 0.06 0.09 0.12 0.16 0.28 0.38 0.83 1.37 1.97 2.75 3.11 2012- 03 0.06 0.08 0.14 0.19 0.34 0.51 1.02 1.56 2.17 2.94 3.28 2012- 04 0.07 0.08 0.14 0.18 0.29 0.43 0.89 1.43 2.05 2.82 3.18 2012- 05 0.07 0.09 0.15 0.19 0.29 0.39 0.76 1.21 1.8 2.53 2.93 2012- 06 0.05 0.09 0.15 0.19 0.29 0.39 0.71 1.08 1.62 2.31 2.7
79
2. U.S. Treasury Bond Data During January 2009 to December 2015 (continued)
time 1month 3month 6month 1year 2year 3year 5year 7year 10year 20year 30year 2012- 07 0.07 0.1 0.15 0.19 0.25 0.33 0.62 0.98 1.53 2.22 2.59 2012- 08 0.09 0.1 0.14 0.18 0.27 0.37 0.71 1.14 1.68 2.4 2.77 2012- 09 0.08 0.11 0.14 0.18 0.26 0.34 0.67 1.12 1.72 2.49 2.88 2012- 10 0.11 0.1 0.15 0.18 0.28 0.37 0.71 1.15 1.75 2.51 2.9 2012- 11 0.12 0.09 0.14 0.18 0.27 0.36 0.67 1.08 1.65 2.39 2.8 2012- 12 0.04 0.07 0.12 0.16 0.26 0.35 0.7 1.13 1.72 2.47 2.88 2013- 01 0.05 0.07 0.11 0.15 0.27 0.39 0.81 1.3 1.91 2.68 3.08 2013- 02 0.08 0.1 0.12 0.16 0.27 0.4 0.85 1.35 1.98 2.78 3.17 2013- 03 0.08 0.09 0.11 0.15 0.26 0.39 0.82 1.32 1.96 2.78 3.16 2013- 04 0.05 0.06 0.09 0.12 0.23 0.34 0.71 1.15 1.76 2.55 2.93 2013- 05 0.02 0.04 0.08 0.12 0.25 0.4 0.84 1.31 1.93 2.73 3.11 2013- 06 0.03 0.05 0.09 0.14 0.33 0.58 1.2 1.71 2.3 3.07 3.4 2013- 07 0.02 0.04 0.07 0.12 0.34 0.64 1.4 1.99 2.58 3.31 3.61 2013- 08 0.04 0.04 0.07 0.13 0.36 0.7 1.52 2.15 2.74 3.49 3.76 2013- 09 0.02 0.02 0.04 0.12 0.4 0.78 1.6 2.22 2.81 3.53 3.79 2013- 10 0.11 0.05 0.08 0.12 0.34 0.63 1.37 1.99 2.62 3.38 3.68 2013- 11 0.05 0.07 0.1 0.12 0.3 0.58 1.37 2.07 2.72 3.5 3.8 2013- 12 0.02 0.07 0.1 0.13 0.34 0.69 1.58 2.29 2.9 3.63 3.89 2014- 01 0.02 0.04 0.07 0.12 0.39 0.78 1.65 2.29 2.86 3.52 3.77 2014- 02 0.05 0.05 0.08 0.12 0.33 0.69 1.52 2.15 2.71 3.38 3.66 2014- 03 0.05 0.05 0.08 0.13 0.4 0.82 1.64 2.23 2.72 3.35 3.62
80
2. U.S. Treasury Bond Data During January 2009 to December 2015 (continued)
time 1month 3month 6month 1year 2year 3year 5year 7year 10year 20year 30year 2014- 04 0.02 0.03 0.05 0.11 0.42 0.88 1.7 2.27 2.71 3.27 3.52 2014- 05 0.03 0.03 0.05 0.1 0.39 0.83 1.59 2.12 2.56 3.12 3.39 2014- 06 0.02 0.04 0.06 0.1 0.45 0.9 1.68 2.19 2.6 3.15 3.42 2014- 07 0.02 0.03 0.06 0.11 0.51 0.97 1.70 2.17 2.54 3.07 3.33 2014- 08 0.03 0.03 0.05 0.11 0.47 0.93 1.63 2.08 2.42 2.94 3.20 2014- 09 0.01 0.02 0.04 0.11 0.57 1.05 1.77 2.22 2.53 3.01 3.26 2014- 10 0.02 0.02 0.05 0.10 0.45 0.88 1.55 1.98 2.30 2.77 3.04 2014- 11 0.04 0.02 0.07 0.13 0.53 0.96 1.62 2.03 2.33 2.76 3.04 2014- 12 0.03 0.03 0.11 0.21 0.64 1.06 1.64 1.98 2.21 2.55 2.83 2015- 01 0.02 0.03 0.08 0.20 0.55 0.90 1.37 1.67 1.88 2.20 2.46 2015- 02 0.02 0.02 0.07 0.22 0.62 0.99 1.47 1.79 1.98 2.34 2.57 2015- 03 0.02 0.03 0.11 0.25 0.64 1.02 1.52 1.84 2.04 2.41 2.63 2015- 04 0.02 0.02 0.09 0.23 0.54 0.87 1.35 1.69 1.94 2.33 2.59 2015- 05 0.01 0.02 0.08 0.24 0.61 0.98 1.54 1.93 2.20 2.69 2.96 2015- 06 0.01 0.02 0.09 0.28 0.69 1.07 1.68 2.10 2.36 2.85 3.11 2015- 07 0.03 0.03 0.12 0.30 0.67 1.03 1.63 2.04 2.32 2.77 3.07 2015- 08 0.04 0.07 0.22 0.38 0.70 1.03 1.54 1.91 2.17 2.55 2.86 2015- 09 0.01 0.02 0.18 0.37 0.71 1.01 1.49 1.88 2.17 2.62 2.95 2015- 10 0.01 0.02 0.11 0.26 0.64 0.93 1.39 1.76 2.07 2.50 2.89 2015- 11 0.07 0.13 0.33 0.48 0.88 1.20 1.67 2.02 2.26 2.69 3.03 2015- 12 0.17 0.23 0.50 0.65 0.98 1.28 1.70 2.04 2.24 2.61 2.97
81
3. SP500 Stock Data During January 2009 to December 2015 yy-mm SP500 yy-mm SP500 yy-mm SP500 yy-mm SP500
2009-01 865.58 2010-10 1171.58 2012-07 1359.78 2014-04 1864.26
2009-02 805.23 2010-11 1198.89 2012-08 1403.44 2014-05 1889.77
2009-03 757.13 2010-12 1241.53 2012-09 1443.42 2014-06 1947.09
2009-04 848.15 2011-01 1282.62 2012-10 1437.82 2014-07 1973.10
2009-05 902.41 2011-02 1321.12 2012-11 1394.51 2014-08 1961.53
2009-06 926.12 2011-03 1304.49 2012-12 1422.29 2014-09 1993.23
2009-07 935.82 2011-04 1331.51 2013-01 1480.40 2014-10 1937.27
2009-08 1009.72 2011-05 1338.31 2013-02 1512.31 2014-11 2044.57
2009-09 1044.55 2011-06 1287.29 2013-03 1550.83 2014-12 2054.27
2009-10 1067.66 2011-07 1325.18 2013-04 1570.70 2015-01 2028.18
2009-11 1088.07 2011-08 1185.31 2013-05 1639.84 2015-02 2082.20
2009-12 1110.38 2011-09 1173.88 2013-06 1618.77 2015-03 2079.99
2010-01 1123.58 2011-10 1207.22 2013-07 1668.68 2015-04 2094.86
2010-02 1089.16 2011-11 1226.41 2013-08 1670.09 2015-05 2111.94
2010-03 1152.05 2011-12 1243.32 2013-09 1687.17 2015-06 2099.28
2010-04 1197.32 2012-01 1300.58 2013-10 1720.03 2015-07 2094.14
2010-05 1125.06 2012-02 1352.49 2013-11 1783.54 2015-08 2039.87
2010-06 1083.36 2012-03 1389.24 2013-12 1807.78 2015-09 1944.40
2010-07 1079.80 2012-04 1386.43 2014-01 1822.36 2015-10 2024.81
2010-08 1087.28 2012-05 1341.27 2014-02 1817.03 2015-11 2080.62
2010-09 1122.08 2012-06 1323.48 2014-03 1863.52 2015-12 2054.08
82
4. US Consumer Price Index (CPI) Data. The based year: 1982-1984(100%) yy-mm CPI yy-mm CPI yy-mm CPI yy-mm CPI
2009-01 211.933 2010-10 219.035 2012-07 228.475 2014-04 236.495
2009-02 212.705 2010-11 219.590 2012-08 229.844 2014-05 236.803
2009-03 212.495 2010-12 220.472 2012-09 230.987 2014-06 237.016
2009-04 212.709 2011-01 221.187 2012-10 231.655 2014-07 237.259
2009-05 213.022 2011-02 221.898 2012-11 231.278 2014-08 237.163
2009-06 214.790 2011-03 223.046 2012-12 231.272 2014-09 237.510
2009-07 214.726 2011-04 224.093 2013-01 231.641 2014-10 237.651
2009-08 215.445 2011-05 224.806 2013-02 233.005 2014-11 237.261
2009-09 215.861 2011-06 224.806 2013-03 232.313 2014-12 236.464
2009-10 216.509 2011-07 225.395 2013-04 231.856 2015-01 234.954
2009-11 217.234 2011-08 226.106 2013-05 231.895 2015-02 235.415
2009-12 217.347 2011-09 226.597 2013-06 232.357 2015-03 235.859
2010-01 217.488 2011-10 226.750 2013-07 232.749 2015-04 236.197
2010-02 217.281 2011-11 227.169 2013-08 233.249 2015-05 236.876
2010-03 217.353 2011-12 227.223 2013-09 233.642 2015-06 237.423
2010-04 217.403 2012-01 227.860 2013-10 233.799 2015-07 237.734
2010-05 217.290 2012-02 228.377 2013-11 234.210 2015-08 237.703
2010-06 217.199 2012-03 228.894 2013-12 234.847 2015-09 237.489
2010-07 217.605 2012-04 229.286 2014-01 235.436 2015-10 237.949
2010-08 217.923 2012-05 228.722 2014-02 235.621 2015-11 238.302
2010-09 218.275 2012-06 228.506 2014-03 235.897 2015-12 238.041
83
5. US Economic inflation Rate Data During January 2009 to December 2015 yy-mm Inf Rate yy-mm Inf Rate yy-mm Inf Rate yy-mm Inf Rate
2009-01 0 2010-10 1.2 2012-07 1.4 2014-04 2
2009-02 0.2 2010-11 1.1 2012-08 1.7 2014-05 2.1
2009-03 -0.4 2010-12 1.5 2012-09 2 2014-06 2.1
2009-04 -0.7 2011-01 1.6 2012-10 2.2 2014-07 2
2009-05 -1.3 2011-02 2.1 2012-11 1.8 2014-08 1.7
2009-06 -1.4 2011-03 2.7 2012-12 1.7 2014-09 1.7
2009-07 -2.1 2011-04 3.2 2013-01 1.6 2014-10 1.7
2009-08 -1.5 2011-05 3.6 2013-02 2 2014-11 1.3
2009-09 -1.3 2011-06 3.6 2013-03 1.5 2014-12 0.8
2009-10 -0.2 2011-07 3.6 2013-04 1.1 2015-01 -0.1
2009-11 1.8 2011-08 3.8 2013-05 1.4 2015-02 0
2009-12 2.7 2011-09 3.9 2013-06 1.8 2015-03 -0.1
2010-01 2.6 2011-10 3.5 2013-07 2 2015-04 -0.2
2010-02 2.1 2011-11 3.4 2013-08 1.5 2015-05 0
2010-03 2.3 2011-12 3 2013-09 1.2 2015-06 0.1
2010-04 2.2 2012-01 2.9 2013-10 1 2015-07 0.2
2010-05 2 2012-02 2.9 2013-11 1.2 2015-08 0.2
2010-06 1.1 2012-03 2.7 2013-12 1.5 2015-09 0
2010-07 1.2 2012-04 2.3 2014-01 1.6 2015-10 0.2
2010-08 1.1 2012-05 1.7 2014-02 1.1 2015-11 0.5
2010-09 1.1 2012-06 1.7 2014-03 1.5 2015-12 0.7
84
APPENDIX B
MATLAB IMPLEMENTATION CODING
1. Interest Rate and Bond Model Calibration
% Vasicek Bond price modeling
%plot historical interest rate data old_r = xlsread('US Treasury Yield.xlsx', 1, 'F86:F169'); %historical yield old_r = old_r*0.01; figure(1) plot(1:length(old_r), old_r) xlabel('month'), ylabel('Yield (%)') title('2-Year Matuarity Yield')
%Calibrate parameters S, L, sigma %Interest rate model dr=S(L-r)dt + sigma dW %Linear least square fitting dr=c1+c2*r+error X = [ones(length(old_r) - 1, 1) old_r(1:end-1)]; Y=diff(old_r); coefficients = inv(X'*X)*(X'*Y); Y_hat = X*coefficients; residuals = Y-Y_hat; % or [coefficients, intervals, residuals] = regress(Y, X); dt = 1/12; % time increment = 1 month S = -coefficients(2)/dt; %c1=SLdt L = -coefficients(1)/coefficients(2); %c2=-Sdt sigma = std(residuals)/sqrt(dt); %var(residual)=sigma^2*dt*var(Z)
%Caribrate the market price of risk, lamda (or b) %For bond price model dP/dt+a(b-r)dP/dr+sigma^2/2(d^2P/dr^2)-rP=0 %and a=S, b=L-lamda*sigma/S , the solution is %P(t,T)=exp(A(t,T)r(t)+B(t,T)) %where yield(t=0)~= -(A(0)r(0)+B(0))/T
T=[1/12,3/12,6/12,1,2,3,5,7,10,20,30]; yield_0 = xlsread('US Treasury Yield.xlsx', 1, 'B169:L169')'*0.01; r_0 = yield_0(1); a=S; A_0=-(1-exp(-a*T'))./a; Y=yield_0.*T'+A_0*r_0+(sigma^2/(2*a^2))*(A_0+T')-(sigma^2/(4*a))*(A_0.^2); X=A_0+T'; b=inv(X'*X)*(X'*Y); lamda=S/sigma*(L-b);
%Generate forcasted interested rate r(t) maxt=36; %months to forcast r_forcast= [r_0; NaN(maxt,1)]; for k=2:maxt+1 dr=S*(L-r_forcast(k-1))*dt + sigma*randn(1)*sqrt(dt); r_forcast(k)=r_forcast(k-1)+dr; end figure(2) plot(1:length(r_forcast), r_forcast) xlabel('months in the future'), ylabel('Forcasted Interested Rate(%)') title('Forcasted Interested Rate r(t)')
86
% Predict bond prices and calculate yields
% T_t=ones(maxt+1,1)*T-(1:maxt+1)'*dt*ones(1,length(T)); T_t=ones(maxt+1,1)*T; % T-t is time to matuarity, row is future time(t) , column is matuarity time T T_t(T_t<0)=0; A=-(1-exp(-a*T_t))./a; B=(-A-T_t)*(b-sigma^2/(2*a^2))-(sigma^2/(4*a))*(A.^2); P=exp(A.*(r_forcast*ones(1,length(T)))+B); yield=-log(P)./T_t;
% [TT,tt] = meshgrid(T,1:maxt+1); [tt,TT] = ndgrid(1:maxt+1,T); figure(3) mesh(tt,TT,yield) xlabel('months in the future'), ylabel('Time to Matuarity'),zlabel('Yields') title('Forcasted Bond Yields')
%7 year maturity bond with real value yield7 = xlsread('US Treasury Yield.xlsx', 1, 'I86:I169'); yield7 = yield7*0.01; T=7*ones(size(yield7)); t=1/12:1/12:7; BB=exp(-yield7.*(T-t')); yield7_dt=yield7*(1/12); sigmaB=std(yield7_dt);
%real value bond old_CPI = xlsread('CPIAUCSL.xls', 1, 'B799:B882'); old_CPI = old_CPI./old_CPI(1); BB_r=BB./old_CPI; figure(4) plot(1:length(BB), BB,'b',1:length(BB_r), BB_r,'r') xlabel('month'), ylabel('Bond price') title('Seven year maturity bond') legend('nominal value','real value',4)
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2. Inflation Rate Model Calibration
%inflation rate old_infla = xlsread('Eco_Inflation.xlsx', 1, 'B97:M103'); old_infla = old_infla'; old_infla = old_infla (:); old_infla = old_infla*0.01;
%Calibrate parameters S, L, sigma %Interest rate model df=S(L-f)dt + sigma dW %Linear least square fitting df=c1+c2*r+error X = [ones(length(old_infla) - 1, 1) old_infla(1:end-1)]; Y=diff(old_infla); coefficients = inv(X'*X)*(X'*Y); Y_hat = X*coefficients; residuals = Y-Y_hat; % or [coefficients, intervals, residuals] = regress(Y, X); dt = 1/12; % time increment = 1 month S = -coefficients(2)/dt; %c1=SLdt L = -coefficients(1)/coefficients(2); %c2=-Sdt sigma = std(residuals)/sqrt(dt); %var(residual)=sigma^2*dt*var(Z)
%CPI based year=2009 old_CPI = xlsread('CPIAUCSL.xls', 1, 'B799:B882'); old_CPI = old_CPI./old_CPI(1); figure(1) plot(1:length(old_CPI), old_CPI) Y=diff(old_CPI)./old_CPI(1:end-1); residuals = Y - old_infla(1:end-1)*dt; sigma = std(residuals)/sqrt(dt);
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3. Correlation Calibration
%interest rate dr r = xlsread('US Int rate monthly.csv', 1, 'B151:B234'); r = r*0.01; dr=diff(r);
%7year bond dB/B yield7 = xlsread('US Treasury Yield.xlsx', 1, 'I86:I168'); yield7 = yield7*0.01; yield7 =yield7*(1/12);
%stock DS/S SP500 = xlsread('SP500.xls', 1, 'B72:B155'); S_rate = diff(SP500)./SP500(1:end-1);
%inflation rate D(infla) infla = xlsread('Eco_Inflation.xlsx', 1, 'B97:M103'); infla = infla'; infla = infla (:); infla = infla*0.01; dinfla = diff(infla);
%CPI DCPI/CPI CPI = xlsread('CPIAUCSL.xls', 1, 'B799:B882'); CPI = CPI./CPI(1); CPI_rate = diff(CPI)./CPI(1:end-1); corr_SB = corr(S_rate,yield7); corr_SCPI = corr(S_rate,CPI_rate); corr_BCPI = corr(yield7,CPI_rate); corr_Sr = corr(S_rate,dr); corr_Br = corr(yield7,dr); corr_CPIr = corr(CPI_rate,dr); corr_Sinf = corr(S_rate,dinfla); corr_Binf = corr(yield7,dinfla); corr_CPIinf = corr(CPI_rate,dinfla);
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4. Stock Model Calibration
%real value stock old_SP500 = xlsread('SP500.xls', 1, 'B72:B155'); old_CPI = xlsread('CPIAUCSL.xls', 1, 'B799:B882'); old_CPI = old_CPI./old_CPI(1); old_SP500_r=old_SP500./old_CPI; figure(1) plot(1:length(old_SP500), old_SP500,'b',1:length(old_SP500_r), old_SP500_r,'r') xlabel('month'), ylabel('Stock price') title('SP500') legend('nominal value','real value',4)
%Calculate Cumulant of dlnS N_bin=100; lnS=log(old_SP500); dlnS=lnS(2:end)-lnS(1:end-1); N_sp = length(dlnS); x_min=-0.2; x_max=0.2; dx= (x_max - x_min)/(N_bin-1); x=x_min:dx:x_max; f_ac=hist(dlnS,x); figure(2) hist(dlnS,N_bin) xlabel('dlnS'), ylabel('frequency') title('Histogram of dlnS') mean_dlnS = mean(dlnS); var_dlnS = var(dlnS); skew_dlnS = skewness(dlnS); kur_dlnS = kurtosis(dlnS);
% find parameters muS, sigmaS, muQ, sigmaQ, lamda options = optimset('Display','iter','TolFun',1e-8); para = fminsearch(@(para) chisquare_fit(para,f_ac,N_sp,x),[mean(dlnS),std(dlnS),mean(dlnS),std(dlnS),0. 2],options); cumu1=para(1)-0.5*para(2)^2-para(5)*(exp(para(3)+0.5*para(4)^2)- 1)+para(5)*para(3); cumu2=para(2)^2+(para(4)^2)*para(5)+para(5)*(para(3)^2); cumu3=para(5)*(3*para(3)*(para(4)^2)+para(3)^3); cumu4=para(5)*(3*para(4)^4+6*(para(3)^2)*(para(4)^2)+para(3)^4); mean_th=cumu1; var_th=cumu2; skew_th=cumu3/(cumu2^(3/2)); kur_th=cumu4/(cumu2^2);
90 function chi=chisquare_fit(para,f_ac,N_sp,x) N_sim = 100; N_bin = length(f_ac); ff_th = NaN(N_sim,N_bin); for k=1:N_sim y = (para(1)-0.5*para(2)^2-para(5)*(exp(para(3)+0.5*para(4)^2)- 1))*ones(N_sp,1) + randn(N_sp,1)* para(2)+... poissrnd(para(5),N_sp,1).*(randn(N_sp,1)* para(4)+ones(N_sp,1)* para(3)); ff_th(k,:) = hist(y,x); end f_th = mean(ff_th); var_fth = var(ff_th); w =(1./var_fth)/sum(1./var_fth); chi=sum(w.*(f_th-f_ac).^2);
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5. Solving a HJB Equation
%load financial data------%all financial data are during Jan2009-Dec2015 %interest rate r = xlsread('US Int rate monthly.csv', 1, 'B151:B234'); r = r*0.01;
%stock DS/S SP500 = xlsread('SP500.xls', 1, 'B72:B155');
%inflation rate D(infla) infla = xlsread('Eco_Inflation.xlsx', 1, 'B97:M103'); infla = infla'; infla = infla (:); infla = infla*0.01;
% set up parameters------mu_S = 0.01; sigma_S = 0.1836; mu_q = 0.0102; sigma_q = 0.1797; lamda_N = 0.1990; eta = exp(mu_q+(sigma_q^2)/2)-1;
%interest rate model dr=K(theta-r)dt+r-sigma_r*dZ K=1.0075; theta = 0.0011; sigma_r = 0.00077;
%bond model dB/B = (r+lamda_B)dt + sigma_B*dZ lamda_r = 5.5506; sigma_B = 5.204e-4; lamda_B = lamda_r*sigma_B;
%inflation rate Dphi = beta(mu_phi-phi)dt + sigma_phi*dz beta = 1.5588; mu_phi = 0.0142; sigma_phi = 0.0217;
%CPI d(CPI)=infla*dt + sigma_CPI*dZ sigma_CPI = 0.0088;
%correlation corr_SB = -0.0279; corr_SCPI = 0.2209; corr_BCPI = 0.0767; corr_Sr = -0.2136; corr_Br = -0.0431; corr_CPIr = -0.1451; corr_Sinf = 0.1641; corr_Binf = 0.1839; corr_CPIinf = 0.5712;
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%------
% Discretize grid Tmax=24; dt=1/12; Wmax = 100; dW=1; gamma = 0.5; rho=0.005; piS=zeros(Tmax,Wmax); % weight for stock piB=zeros(Tmax,Wmax); CC= zeros(Tmax,Wmax); % consumption rate W = (1:Wmax); % wealth process V = zeros(Tmax,Wmax); % value function V(end,:)=((1:Wmax).^gamma)./gamma;
%Generate Jump & jump weight nQ = 50; Q = normrnd(mu_q,sigma_q,[1 nQ]); J = exp(Q)-ones(1,nQ); for k=1:Tmax-1 v = V(end-(k-1),:); Vw=([v(2:end) v(end)+(v(end)-v(end-1))*dW]-[ v(1)-(v(2)-v(1))*dW v(1:end- 1)])/(2*dW); Vww=([v(2:end) v(end)+(v(end)-v(end-1))*dW]-2*v-[v(1)-(v(2)-v(1))*dW v(1:end-1)])/(dW^2);
C=Vw.^(1/(gamma-1)); C(C<0)=0; CC(end-(k-1),:)=C; %consumption at time k
for j = 1:Wmax para = [j r(end-(k-1)) W(j) v J]; x = fminbnd(@(x) Sweight(x,para),0,1); % find sup(piS) that optimize (3.7.13) piS(end-(k-1),j) = x; end
piB(end-(k-1),:) = (corr_BCPI- lamda_B)*Vw./((sigma_B^2)*W.*Vww)+corr_BCPI/(sigma_B^2)*ones(size(W))- corr_SB*piS(end-(k-1),:)/(sigma_B^2); piB(piB<0)=0; piB(piB>1)=1;
%------mu_W = (r(end-(k-1))-infla(end-(k-1))+sigma_CPI^2)*ones(1,Wmax) + ... piS(end-(k-1),:)*(mu_S-lamda_N*eta-corr_SCPI-r(end-(k-1)))+ piB(end-(k-1),:)*(lamda_B-corr_BCPI);
var_W =(piS(end-(k-1),:).^2)*(sigma_S^2)+(piB(end-(k- 1),:).^2)*(sigma_B^2)+(sigma_CPI^2)*ones(1,Wmax)+... 2*corr_SB*piS(end-(k-1),:).*piB(end-(k-1),:)-2*corr_SCPI*piS(end- (k-1),:)-2*corr_BCPI*piB(end-(k-1),:);
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% Define integral term in HJB------WJ = (J'*piS(end-(k-1),:)+ ones(length(J),length(W))).*(ones(length(J),1)*W); %wealh including jump for all generate jump WJ(WJ>Wmax)=Wmax; WJp = floor(WJ); WJr = WJ - WJp;
VJ = zeros(length(J),length(W)); %valude function at wealth+jump
VJ(WJp
%------%HJB Equation(3.7.7) Vt + LV = 0 --> V(t-1)=V(t)+ LV LV=C.^gamma/gamma -rho*V(end-(k-1),:)+(mu_W.*W-C).*Vw + 0.5*var_W.*(W.^2).*Vww + ... lamda_N*mean(VJ - ones(length(J),1)*v); V(end-k,:)=V(end-(k-1),:)+LV*dt; end
[t,w] = ndgrid(1:12,1:10); figure(1) surf(t,w,V(end-11:end,1:10)) xlabel('months'), ylabel('W'),zlabel('V') title('Value Function') figure(2) surf(t,w,CC(end-11:end,1:10)) xlabel('months'), ylabel('W'),zlabel('C') title('Consumption Function')
94 function diffV_piS = Sweight(x, para)
%------mu_S = 0.01; sigma_S = 0.1836; mu_q = 0.0102; sigma_q = 0.1797; lamda_N = 0.1990; eta = exp(mu_q+(sigma_q^2)/2)-1;
%interest rate model dr=K(theta-r)dt+r-sigma_r*dZ K=1.0075; theta = 0.0011; sigma_r = 0.00077;
%bond model dB/B = (r+lamda_B)dt + sigma_B*dZ lamda_r = 5.5506; sigma_B = 5.204e-4; lamda_B = lamda_r*sigma_B;
%inflation rate Dphi = beta(mu_phi-phi)dt + sigma_phi*dz beta = 1.5588; mu_phi = 0.0142; sigma_phi = 0.0217;
%CPI d(CPI)=infla*dt + sigma_CPI*dZ sigma_CPI = 0.0088;
%correlation corr_SB = -0.0279; corr_SCPI = 0.2209; corr_BCPI = 0.0767; corr_Sr = -0.2136; corr_Br = -0.0431; corr_CPIr = -0.1451; corr_Sinf = 0.1641; corr_Binf = 0.1839; corr_CPIinf = 0.5712; %------%calculate value function for all jump J ------Wmax=100; dW=1; j = para(1); r= para(2); W= para(3); v = para(4:103); J = para(104:end);
Vw=([v(2:end) v(end)+(v(end)-v(end-1))*dW]-[ v(1)-(v(2)-v(1))*dW v(1:end- 1)])/(2*dW); Vww=([v(2:end) v(end)+(v(end)-v(end-1))*dW]-2*v-[v(1)-(v(2)-v(1))*dW v(1:end- 1)])/(dW^2);
WJ = (x*J + ones(size(J)) )*W; %wealh including jump WJ(WJ>Wmax)=Wmax; WJp = floor(WJ);
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WJr = WJ - WJp;
VJw = zeros(1,length(J)); %valude function at wealth+jump
VJw(WJp
%------diffV_piS = abs(((mu_S-lamda_N*eta-r-corr_SCPI)*sigma_B^2-(lamda_B- corr_BCPI)*corr_SB)*W*Vw(j) +... (x*((sigma_B^2)*(sigma_S^2)-(corr_SB^2))- corr_SCPI*(sigma_B^2) + corr_BCPI*corr_SB)*(W^2)*Vww(j) +... lamda_N*(sigma_B^2)*W*sum(VJw.*J)/length(J));
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