Candidate no. 900228 Trading Bond Convexity - A Model Agnostic Approach A thesis submitted in partial fulfillment of the MSc in Mathematical Finance April 7, 2015 Candidate no. 900228 This thesis is dedicated to those multitude of Indian children who do not have the means to afford the luxury of education. 2 Candidate no. 900228 Acknowledgements I would like to express my sincere gratitude to my advisor Dr. Riccardo Rebonato for introducing me to this fascinating topic and for coaching me throughout. This thesis would not have been possible without his relentless patience in answering my many trivial questions and in explaining to me the key concepts. I have learned a lot, both from his classes at the Mathematical Institute and from our discussions over the phone and email. I have been extremely lucky to have interacted with Dr.Vladimir Putyatin throughout the course of this thesis. His insights, advice and review of my calculations have gone a long way in helping me to complete the analysis and write this thesis. The interactions with the faculty members at the Mathematical Institute have been extremely beneficial to my overall understanding of the nuances of Mathematical Finance and I would like to thank them for their quality teaching and feedback. My experience on each of the 7 visits to the Mathematical Institute has been very satisfying and it would not have been possible without the hard work that the staff of the Mathematical Institute put in prior to every visit. A big thank you to them. As we are all aware, life is much more easier to live when one has a supportive and caring family. This is where I have been blessed with wonderful parents who have been encouraging throughout and my wife Padma who has been a phenomenal source of positive energy in my life. My deepest gratitude to all of them. 3 Candidate no. 900228 In this essay, we study bond portfolio Convexity and we do so from three different perspectives. First, we introduce a model based representation of what the portfolio convexity should be using a simple Vasicek setting followed by a general multi-factor Affine set up. Second, we derive a novel model agnostic approach to extract the value of portfolio convexity in terms of portfolio \Carry" and \Roll-Down". Finally, we develop a trading strategy which employs the model agnostic representation of portfolio convexity to exploit discrepancies in implied and realized convexity using the Treasury data provided by the US Federal Reserve[10] for the period 1987-2014. Our intention to focus on portfolio convexity is ultimately linked to the belief that mis-pricing in the fair value of convexity exist in today's markets. These mis-pricings provide us with a trading opportunity and motivates us to develop a model agnostic approach to monetize convexity. The trading strategy is relatively easy to implement and is overall profitable conditional on the quality of the estimates of future yield volatility. Furthermore, we show that the profitability of trading strategy is not due to uncontrolled residual exposure to level, slope or curvature of the yields but is purely due to the ability of the strategy to tap into the mis-pricings in convexity. 4 Candidate no. 900228 Contents 1 Introduction 1 2 Convexity 3 2.1 Monetizing Convexity ........................... 5 2.2 The Vasicek Setting ............................ 6 2.3 A General Affine Model ......................... 8 3 A Model Agnostic Approach 13 3.1 The Weights ................................ 16 4 The Trading Strategy 18 5 Strategy Implementation 21 5.1 Estimation of Weights .......................... 22 5.2 Estimation of Volatilities ......................... 24 5.3 Implementation Steps ........................... 30 6 Results 32 6.1 Estimated Weights ............................ 32 6.2 Estimated Volatilities ........................... 34 6.3 Portfolio Profit & Loss .......................... 39 7 Conclusion 41 8 Appendix 1 - Estimation of Portfolio Weights: MATLAB Code 46 9 Appendix 2 - Estimation of Volatilities: R Code 48 References 51 i Candidate no. 900228 List of Figures 2.1 Bond Price vs. Yield ........................... 4 5.1 ACF plot for 10 year bond at t = 9th Feb 1987 ............. 26 5.2 PACF plot for 10 year bond at t = 9th Feb 1987 ............ 26 5.3 ACF plot for 20 year bond at t = 2nd Oct 2014 ............. 27 5.4 PACF plot for 20 year bond at t = 2nd Oct 2014 ............ 28 5.5 ACF plot for 30 year bond at t = 12th May 2014 ............ 28 5.6 PACF plot for 30 year bond at t = 12th May 2014 ........... 29 6.1 Estimated weights for the 10 (Blue line) and 20 (Orange line) year bonds for the portfolio 10/20/30 ..................... 33 6.2 Estimated weights for the 5 (Blue line) and 10 (Orange line) year bonds for the portfolio 5/10/30 ......................... 33 6.3 Estimated weights for the 5 (Blue line) and 15 (Orange line) year bonds for the portfolio 5/15/30 ......................... 34 6.4 Estimated volatilitiesσ ^5 (Yellow line),σ ^10 (Orange line) andσ ^15 (Green line). .................................... 35 6.5 Estimated volatilitiesσ ^20 (Blue line) andσ ^30 (Orange line). ...... 36 6.6 P&L for the portfolios .......................... 39 7.1 Signal Strength vs. Average P&L for 10/20/30 ............ 42 7.2 Signal Strength vs. Average P&L for 5/15/30 ............. 42 7.3 Signal Strength vs. Average P&L for 5/10/30 ............. 43 ii Candidate no. 900228 Chapter 1 Introduction It is well known that the forces that shape the yield curve1 manifest themselves through the (i) Expectations around future one period interest rates, (ii) Term pre- mia and (iii) Convexity[7]. The expectations hypothesis asserts that yields on long term bonds must be equal to the expected future one period interest rates. Term pre- mia is the compensation that an investor hopes to receive for bearing duration risk and finally, Convexity arises from the nonlinear relationship between yields and bond prices. The overall shape of the yield curve is, in reality, a trade off between these three competing effects. The three factors have specific maturity ranges in which they are most active - for example the Expectations component is material at the short end of the yield curve, Term premia is material at the medium term maturities and Convexity, ultimately, dominates at the long end of the yield curve. In this essay, we study bond portfolio Convexity and we do so from three different perspectives. First, we introduce a model based representation of what the portfolio convexity should be using a simple Vasicek setting followed by a general multi-factor Affine set up. Second, we derive a novel model agnostic approach to extract the value of portfolio convexity in terms of portfolio \Carry" and \Roll-Down". Finally, we develop a trading strategy which employs the model agnostic representation of portfolio convexity to exploit discrepancies in implied and realized convexity using the Treasury data provided by the US Federal Reserve[10] for the period 1987-2014. Our intention to focus on portfolio convexity is ultimately linked to the belief that mis-pricing in the fair value of convexity exist in today's markets. This mis-pricing can be a result of the sudden changes in volatilities after the 2008 financial crisis or due to a net demand for long term bonds by institutions like the pension funds2. 1There are a number of other factors that can also affect a bond's yield. For example credit risk could affect yields of defaultable bonds. Illiquidity is another factor affecting yields[7]. 2Pension funds or Insurance companies may have negatively convex long-dated liabilities and may want to match their long-dated liabilities with long-dated assets. Their desire to match the maturity profile of their liabilities, and to reduce the associated negative convexity, creates a net institutional demand for these long-dated assets[21]. 1 Candidate no. 900228 Whatever be the reason, these mis-pricings provide us with a trading opportunity and motivate us to develop a model agnostic approach to monetize convexity. However, harnessing this value of convexity requires an active dynamic strategy, similar to that of `gamma' trading that an option trader engages in. Whether we make a profit will depend on the interplay between the market's perception of today's convexity and the future realized convexity or in other words, on the interplay between realized and implied yield volatilities, much like `gamma' trading for options. Academic literature that discuss the possibility of monetizing convexity is prac- tically non-existent. It is, beyond doubt, true that many financial institutions, like the hedge funds for example, employ proprietary trading strategies that attempt to exploit the mis-pricings in convexity however information about those strategies are largely unavailable publicly. This study is unique in that respect and attempts to introduce a trading strategy to monetize convexity using a novel model agnostic representation of portfolio convexity. The trading strategy is relatively easy to im- plement and is overall profitable conditional on the quality of the estimates of future yield volatility. Furthermore, we show that the profitability of trading strategy is not due to uncontrolled residual exposure to level, slope or curvature of the yields but is purely due to the ability of the strategy to tap into the mis-pricings in convexity. The rest of this essay is organized as follows: Chapter 2 introduces model de- pendent expressions of portfolio convexity using a Vasicek model and a multi-factor Affine model. The main contributions of this essay are in Chapter 3 and Chapter 4 where we introduce the model agnostic representation of portfolio convexity and the corresponding trading strategy. Chapter 5 and Chapter 6 discuss the implementation steps and the results of running the strategy on US Treasury data for the period 1987-2014.