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- Applications of optimal portfolio management by Dimitrios Bisias Submitted to the Sloan School of Management in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2015 c Massachusetts Institute of Technology 2015. All rights reserved. Author .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sloan School of Management June 22, 2015 Certified by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Andrew W. Lo Charles E. and Susan T. Harris Professor of Finance Thesis Supervisor Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Patrick Jaillet Dugald C. Jackson Professor, Department of Electrical Engineering and Computer Science Co-director, Operations Research Center
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- Applications of optimal portfolio management by Dimitrios Bisias Submitted to the Sloan School of Management on June 22, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research Abstract This thesis revolves around applications of optimal portfolio theory. In the first essay, we study the optimal portfolio allocation among convergence trades and mean reversion trading strategies for a risk averse investor who faces Value- at-Risk and collateral constraints with and without fear of model misspecification. We investigate the properties of the optimal trading strategy, when the investor fully trusts his model dynamics. Subsequently, we investigate how the optimal trading strategy of the investor changes when he mistrusts the model. In particular, we assume that the investor believes that the data will come from an unknown member of a set of unspecified alternative models near his approximating model. The investor believes that his model is a pretty good approximation in the sense that the relative entropy of the alternative models with respect to his nominal model is small. Concern about model misspecification leads the investor to choose a robust optimal portfolio allocation that works well over that set of alternative models. In the second essay, we study how portfolio theory can be used as a framework for making biomedical funding allocation decisions focusing on the National Institutes of Health (NIH). Prioritizing research efforts is analogous to managing an invest- ment portfolio. In both cases, there are competing opportunities to invest limited resources, and expected returns, risk, correlations, and the cost of lost opportunities are important factors in determining the return of those investments. Can we apply portfolio theory as a systematic framework of making biomedical funding allocation decisions? Does NIH manage its research risk in an efficient way? What are the challenges and limitations of portfolio theory as a way of making biomedical funding allocation decisions? Finally in the third essay, we investigate how risk constraints in portfolio opti- mization and fear of model misspecification affect the statistical properties of the market returns. Risk sensitive regulation has become the cornerstone of international financial regulations. How does this kind of regulation affect the statistical properties of the financial market? Does it affect the risk premium of the market? What about the volatility or the liquidity of the market? 3
- Thesis Supervisor: Andrew W. Lo Title: Charles E. and Susan T. Harris Professor of Finance 4
- Acknowledgments I would like to express my gratitude to my advisor and mentor, Professor Andrew W. Lo, for his continuing support and advice over all the years I spent at MIT. His immense knowledge in diverse research areas, enthusiasm, hard work, outstanding leadership and motivation have been a source of inspiration. Working with him has been an honor and privilege and I could not have imagined having a better advisor and mentor for my Ph.D study. I would also like to thank the rest of my thesis committee: Professor Dimitri P. Bertsekas for comments that greatly improved this thesis and for his great books that made me love the field of optimization in the first place and Professor Leonid Kogan who provided his insight and expertise that greaty assisted this research. In addition I would like to thank Dr. James F. Watkins, MD for his invaluable help, insights and contribution to the second part of this research. Moreover, I would like to thank Dr. Paul Mende, Dr. Saman Majd and Dr. Eric Rosenfeld whom I had the fortune of being their teaching assistant in finance classes. Paul’s experience in quantitative trading made me realize what career I would like to follow and I am grateful for this. Being part of MIT and in particular the ORC and LFE communities has been a blessing and I consider myself very fortunate to be among very interesting and smart people. I will always remember my years at MIT with nostalgia and joy and I hope that I ’ll be able to express my gratitude in the future several times. My life at MIT would not be so complete and joyful if I didn’t have good lifelong friends to spend time and have productive discussions with. In particular, I would like to thank Nick Trichakis and his wife Lena, Christos and Elli Nicolaides, Markos and Sophia Trichas, Thomas and Anastasia Trikalinos, the golden coach George Pa- pachristoudis and Gerry Tsoukalas. Last but not least I would like to thank my parents Giorgo and Roula and my sister Katerina for their unconditional love and support. I owe to them everything and this thesis is dedicated to them. 5
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- Contents 1 Optimal trading of arbitrage opportunities under constraints 29 1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.2.4 Connection with Ridge and Lasso regression . . . . . . . . . . 46 1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.3.1 Convergence trades . . . . . . . . . . . . . . . . . . . . . . . . 47 1.3.2 Mean reversion trading opportunities . . . . . . . . . . . . . . 56 1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2 Optimal trading of arbitrage opportunities under model misspecifi- cation 57 2.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.2.1 Alternative models representation . . . . . . . . . . . . . . . . 61 2.2.2 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3.1 No fear of model misspecification . . . . . . . . . . . . . . . . 65 2.3.2 Fear of model misspecification no constraints . . . . . . . . . . 67 2.3.3 Fear of model misspecification with VaR and margin constraints 70 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7
- 2.4.1 Convergence trades without constraints . . . . . . . . . . . . . 73 2.4.2 Mean reversion trading strategies without constraints . . . . . 78 2.4.3 Convergence trades with constraints . . . . . . . . . . . . . . . 92 2.4.4 Mean reversion trading strategies with constraints . . . . . . . 111 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3 Estimating the NIH Efficient Frontier 131 3.1 NIH Background and Literature Review . . . . . . . . . . . . . . . . 132 3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.2.1 Funding Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.2.2 Burden of Disease Data . . . . . . . . . . . . . . . . . . . . . 139 3.2.3 Applying Portfolio Theory . . . . . . . . . . . . . . . . . . . . 142 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.3.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . 147 3.3.2 Efficient Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4 Impact of model misspecification and risk constraints on market 157 4.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.2.1 Model setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.2.2 Varying constraints . . . . . . . . . . . . . . . . . . . . . . . . 161 4.2.3 Varying risk aversions . . . . . . . . . . . . . . . . . . . . . . 165 4.2.4 Varying constraints and risk aversions . . . . . . . . . . . . . . 168 4.2.5 Varying fear of model misspecification . . . . . . . . . . . . . 168 4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 A Technical Notes 173 8
- List of Figures 1-1 Ellipsoids. Ellipsoids of poor investment opportunities for N=2 con- vergence trades at times t = 0.3, 0.6, 0.9. . . . . . . . . . . . . . . . . 42 1-2 Weights for the case of uncorrelated spreads and collateral constraint. Weights for the case of uncorrelated spreads. . . . . . . 45 1-3 VaR constraints, positive correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing VaR constraints (K=1). Initial wealth is $100. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1-4 VaR constraints, negative correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively cor- related (ρ = −0.5) convergence trades, while facing VaR constraints (K=1). Initial wealth is $100. . . . . . . . . . . . . . . . . . . . . . . 48 1-5 VaR constraints, positive correlations, tight constraints. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100. . . . . . . . . . . . . . . 49 1-6 VaR constraints, negative correlations, tight constraints. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100. . . . . . . . . . . . . . . 49 9
- 1-7 Wealth evolution under VaR constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under the VaR constraint. Initial wealth is $100. . . . . . . . . . . . . . . . 50 1-8 Relation between final wealth and frequency the VaR con- straint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence trades are low. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1-9 Margin constraints, positive correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively cor- related (ρ = 0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100. . . . . . . . . . . . . . . . . . 52 1-10 Margin constraints, negative correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively cor- related (ρ = −0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100. . . . . . . . . . . . . . . . . . 52 1-11 Margin constraints, positive correlations, more collateral needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing margin constraints (Collateral = 2). Initial wealth is $100. . . . . . . 53 1-12 Margin constraints, negative correlations, more collateral needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while fac- ing margin constraints (Collateral = 2). Initial wealth is $100. . . . . 53 1-13 Wealth evolution under margin constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under the margin constraint. Initial wealth is $100. . . . . . . . . . . . . . . 54 10
- 1-14 Relation between final wealth and frequency the margin con- straint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence trades are low. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1-15 Positions evolution under VaR constraints. Typical path of the positions in two convergence trading opportunities under VaR con- straints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1-16 Positions evolution under margin constraints. Typical path of the positions in two convergence trading opportunities under margin constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2-1 Partial derivative of the value function with respect to S for a single convergence trade. VS as a function of time at S = 1 for different values of the robustness multiplier for a single convergence trade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2-2 Distortion drift for a single convergence trade. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier for a single convergence trade. . . . . . . . . . . . . . . . . 76 2-3 Distortion drift terms for a single convergence trade. Distor- tion drift terms as a function of time at S = 1 for ν = 1 for a single convergence trade. The first term corresponds to a positive distor- tion drift that reduces the wealth of the investor since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. . . . . 77 2-4 Optimal weight of a single convergence trade. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. . . . . . . . . . . . . . . . . . . . 77 2-5 Partial derivative of the value function with respect to S for a single mean reversion trading strategy. VS as a function of time at S = 1 for different values of the robustness multiplier. . . . . 80 11
- 2-6 Distortion drift for a single mean reversion trading strategy. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. . . . . . . . . . . . . . . . . . . . . . . . . . 81 2-7 Distortion drift terms for a single mean reversion trading strategy. Distortion drift terms as a function of time at S = 1 for ν = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor, since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. . . . . . . . . . . 81 2-8 Optimal weight of a single mean reversion trading strategy. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier. . . . . . . . . . 82 2-9 Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2-10 Ratio of the optimal weights. Ratio of the optimal weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2-11 Partial derivative of the value function with respect to S1 and S2 at S1 = 1 and S2 = 2 when ρ = 0. Partial derivative of the value function with respect to S1 and S2 as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 12
- 2-12 Optimal weights of two positively correlated mean reversion trading strategies for S1 = 1 and S2 = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2-13 Optimal weights of two negatively correlated mean reversion trading strategies for S1 = 1 and S2 = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2-14 Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 1. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 2-15 Ratio of the optimal weights at S1 = 1 and S2 = 1 when ρ = 0. Ratio of the optimal weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. . . . . . . 88 2-16 Partial derivative of the value function with respect to S1 and S2 at S1 = 1 and S2 = 1 when ρ = 0. Partial derivative of the value function with respect to S1 and S2 as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2-17 Optimal weights of two positively correlated mean reversion trading strategies for S1 = 1 and S2 = 1. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 13
- 2-18 Partial derivative of the value function with respect to S1 and S2 at S1 = 1 and S2 = 1 when ρ = 0.9. Partial derivative of the value function with respect to S1 and S2 as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.9. . . . . . . . . . . . . . . . . . . 90 2-19 Ratio of the optimal weights at S1 = 1 and S2 = 1 when ρ = 0.9. Ratio of the magnitude of the optimal weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2-20 Optimal weights of two negatively correlated mean reversion trading strategies for S1 = 1 and S2 = 1. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2-21 Partial derivative of the value function with respect to S for a single convergence trade when L = 0.1 and L = 100. VS as a function of time at S = 1 for different values of the robustness multiplier. The solid line is when L = 100 and the dotted line is for L = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2-22 Partial derivative of the value function with respect to S for a single convergence trade when L = 0.1. VS as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F| ≤ 0.1. . . . . . . . . . . . . . . . . . . . . . 95 2-23 Optimal weight of a single convergence trade when L = 0.1. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F| ≤ 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 95 14
- 2-24 Optimal weight of a single convergence trade when L = 1. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F| ≤ 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2-25 Distortion drift for a single convergence trade when L = 0.1. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F| ≤ 0.1. . . . 96 2-26 Distortion drift for a single convergence trade when L = 1. Distortion drift as a function of time at S = 1 for different values of the robustness multiplier. The collateral constraint is |F| ≤ 1. . . . . 97 2-27 Distortion drift terms for a single convergence trade when L = 0.1. Distortion drift terms as a function of time at S = 1 for ν = 1 and L = 0.1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor and it is bounded above due to the collateral constraint, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. 97 2-28 Distortion drift terms for a single convergence trade when L = 1. Distortion drift terms as a function of time at S = 1 for ν = 1 and L = 1. The first term corresponds to a positive distortion drift that reduces the wealth of the investor and it is bounded above due to the collateral constraint, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. 98 2-29 Optimal weight of a single convergence trade when L = 0.1 and L = 100. Weight of the convergence trading strategy as a function of time at S = 1 for different values of the robustness multiplier. The solid line is when L = 100 and the dotted line is for L = 0.1. . . . . . 98 15
- 2-30 Optimal weights of two uncorrelated convergence trades for S1 = 1 and S2 = 2 when L = 0.5. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 0.5. . . . . . . . . . . . . . . . . . . . . 100 2-31 Value of the normalized wealth variance for two uncorrelated convergence trades at S1 = 1 and S2 = 2 when L = 0.5. Value of the normalized wealth variance for two uncorrelated convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 0.5. . . . . . . . . . . . . . . . . 101 2-32 Optimal weights of two uncorrelated convergence trades for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . . . . . . 101 2-33 Value of the normalized wealth variance for two uncorrelated convergence trades at S1 = 1 and S2 = 2 when L = 0.05. Value of the normalized wealth variance for two uncorrelated convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . 102 2-34 Optimal weights of two positively correlated convergence trades for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.5 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . 103 16
- 2-35 Value of the normalized wealth variance for two positively cor- related convergence trades at S1 = 1 and S2 = 2 when L = 0.05. Value of the normalized wealth variance for two positively correlated convergence trades as a function of time at S1 = 1 and S2 = 2 for dif- ferent values of the robustness multiplier. The correlation coefficient is ρ = 0.5 and the rhs of the VaR constraint is L = 0.05. . . . . . . . 104 2-36 Optimal weights of two negatively correlated convergence trades for S1 = 1 and S2 = 2 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . 104 2-37 Value of the normalized wealth variance for two negatively correlated convergence trades at S1 = 1 and S2 = 2 when L = 0.05. Value of the normalized wealth variance for two nega- tively correlated convergence trades as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The cor- relation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2-38 Optimal weights of two uncorrelated convergence trades for S1 = 1 and S2 = 1 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . . . . . . 107 2-39 Value of the normalized wealth variance for two uncorrelated convergence trades at S1 = 1 and S2 = 1 when L = 0.05. Value of the normalized wealth variance for two uncorrelated convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . 107 17
- 2-40 Optimal weights of two positively correlated convergence trades for S1 = 1 and S2 = 1 when L = 0.05. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.8 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . 108 2-41 Value of the normalized wealth variance for two positively cor- related convergence trades at S1 = 1 and S2 = 1 when L = 0.05. Value of the normalized wealth variance for two positively correlated convergence trades as a function of time at S1 = 1 and S2 = 1 for dif- ferent values of the robustness multiplier. The correlation coefficient is ρ = 0.8 and the rhs of the VaR constraint is L = 0.05. . . . . . . . 109 2-42 Optimal weights of two negatively correlated convergence trades for S1 = 1 and S2 = 1 when L = 8. Weights of the convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.8 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . 109 2-43 Value of the normalized wealth variance for two negatively correlated convergence trades at S1 = 1 and S2 = 1 when L = 0.05. Value of the normalized wealth variance for two nega- tively correlated convergence trades as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The cor- relation coefficient is ρ = −0.8 and the rhs of the VaR constraint is L = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2-44 Partial derivative of the value function with respect to S for a single mean reversion trading strategy and a collateral con- straint with L = 0.7. VS as a function of time at S = 1 for different values of the robustness multiplier for L = 0.7. . . . . . . . . . . . . . 113 18
- 2-45 Distortion drift terms for a single mean reversion trading strategy and a collateral constraint with L = 0.7. Distortion drift terms as a function of time at S = 1 for ν = 2 and for L = 0.7. The first term corresponds to a positive distortion drift that reduces the wealth of the investor, since the investor is shorting the spread, while the second term corresponds to a negative distortion drift that points to worse investment opportunities. The first term is bounded above due to the collateral constraint. . . . . . . . . . . . . . . . . . . 113 2-46 Optimal weight of a single mean reversion trading strategy with a collateral constraint with L = 0.7. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier and for L = 0.7. . . . . . . . . . . 114 2-47 Partial derivative of the value function with respect to S for a single mean reversion trading strategy with different collat- eral constraints. VS as a function of time at S = 1 for different values of the robustness multiplier and different collateral constraints. The solid line is for L = 70 and the dotted line for L = 0.7. . . . . . . 114 2-48 Optimal weight of a single mean reversion trading strategy with different collateral constraints. Weight of the mean reversion trading strategy as a function of time at S = 1 for different values of the robustness multiplier and different collateral constraints. The solid line is for L = 70 and the dotted line for L = 0.7. . . . . . . . . . . . 115 2-49 Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 3. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 3. . . . . 117 19
- 2-50 Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 3. Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 3. . . . . 118 2-51 Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. . . . . 118 2-52 Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 2. Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. . . . . 119 2-53 Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 7. . . . . 119 2-54 Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 7. Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 7. . . . . 120 20
- 2-55 Optimal weights of two positively correlated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.5 and the rhs of the VaR constraint is L = 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 2-56 Value of the normalized wealth variance for two positively correlated mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 7. Value of the normalized wealth variance for two positively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.5 and the rhs of the VaR constraint is L = 7. . . . . . . . . . . . . . . . . . . . . . . . . . 122 2-57 Optimal weights of two negatively correlated mean reversion trading strategies for S1 = 1 and S2 = 2 when L = 7. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 2-58 Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies at S1 = 1 and S2 = 2 when L = 7. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 2 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 7. . . . . . . . . . . . . . . . . . . . . . . . . . 123 21
- 2-59 Optimal weights of two uncorrelated mean reversion trading strategies for S1 = 1 and S2 = 1 when L = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. . . . . 125 2-60 Value of the normalized wealth variance for two uncorrelated mean reversion trading strategies at S1 = 1 and S2 = 1 when L = 2. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0 and the rhs of the VaR constraint is L = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2-61 Optimal weights of two positively correlated mean reversion trading strategies for S1 = 1 and S2 = 1 when L = 2. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.9 and the rhs of the VaR constraint is L = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2-62 Value of the normalized wealth variance for two positively correlated mean reversion trading strategies at S1 = 1 and S2 = 1 when L = 2. Value of the normalized wealth variance for two positively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = 0.9 and the rhs of the VaR constraint is L = 2. . . . . . . . . . . . . . . . . . . . . . . . . . 127 22
- 2-63 Optimal weights of two negatively correlated mean reversion trading strategies for S1 = 1 and S2 = 1 when L = 8. Weights of the mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2-64 Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies at S1 = 1 and S2 = 1 when L = 8. Value of the normalized wealth variance for two negatively correlated mean reversion trading strategies as a function of time at S1 = 1 and S2 = 1 for different values of the robustness multiplier. The correlation coefficient is ρ = −0.5 and the rhs of the VaR constraint is L = 8. . . . . . . . . . . . . . . . . . . . . . . . . . 128 3-1 NIH time series flowchart. Flowchart for the construction of NIH appropriations time series. “NIH Approp.” denotes NIH appropria- tions; “PHS Gaps” denotes Institute funding by the U.S. Public Health Service; “Complete Approp.” denotes the union of these two series; “FY Change” allows for the change in government fiscal years; “4Q FY” time series refers to the resulting series in which all years are treated as having four quarters of three months each. . . . . . . . . . 138 3-2 Appropriations data. NIH appropriations in real (2005) dollars, categorized by disease group. . . . . . . . . . . . . . . . . . . . . . . 138 3-3 YLL time series flowchart. Flowchart for the construction of years of life lost (YLL) time series. “WONDER Chapter Age Group” refers to a query to the CDC WONDER database at the chapter level, strati- fied by age group at death; “US Pop.” is the United States population from census data as expressed in the WONDER dataset; and “US GDP” denotes U.S. gross domestic product. . . . . . . . . . . . . . . 140 23
- 3-4 YLL data. Panel (a): Raw YLL categorized by disease group. Panel (b): Population-normalized YLL (with base year of 2005), categorized by disease group. Both panels are based on data from 1979 to 2007. 141 3-5 Efficient frontiers. Efficient frontiers for (a) all groups except HIV and AMS, γ = 0; (b) all groups except HIV and AMS, γ = 5; (c) all groups except HIV and AMS without the dementia effect, γ = 0; and (d) all groups except HIV and AMS without the dementia effect, γ =5; based on historical ROI from 1980 to 2003. . . . . . . . . . . . . . . . 148 4-1 Price of the risky asset as a function of the aggregate market supply under varying constraints. We assume that we have 5 agents with the same risk aversion coefficients. The red plot assumes the same L = 30 for all the agents, while the blue assumes L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 = 50. 163 4-2 Price of the risky asset as a function of the aggregate market supply under tightening constraints. We assume that we have 5 agents with the same risk aversion coefficients. The blue plot assumes L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 = 50 and the red assumes that each Li is reduced by 20%. . . . 164 4-3 Price of the risky asset as a function of the aggregate market supply with less variable constraints. We assume that we have 5 agents with the same risk aversion coefficients. The blue plot assumes L to be different across the agents L1 = 10, L2 = 20, L3 = 30, L4 = 40, L5 = 50 and the red assumes that L1 = 20, L2 = 25, L3 = 30, L4 = 35, L5 = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4-4 Price of the risky asset as a function of the aggregate market supply with constraints and varying risk aversions. We assume that we have 5 agents with same constraints but different risk aversion coefficients. The blue plot assumes L = 30 for each agent, while the red line assumes that the agents are unconstrained. . . . . . . . . . . 166 24
- 4-5 Price of the risky asset as a function of the aggregate market supply with tightening constraints and varying risk aversions. We assume that we have 5 agents with same constraints but different risk aversion coefficients. The blue plot assumes L = 30 for each agent, while the red line assumes that L = 20 for each agent. . . . . . . . . . 167 25
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- List of Tables 3.1 IoM recommendations. 12 major recommendations of the 1998 Institute of Medicine panel in four large areas for improving the process of allocating research funds. . . . . . . . . . . . . . . . . . . . . . . . 133 3.2 ICD mapping. Classification of ICD-9 (1978–1998) and ICD-10 (1999– 2007) Chapters and NIH appropriations by Institute and Center to 7 disease groups: oncology (ONC); heart lung and blood (HLB); diges- tive, renal and endocrine (DDK); central nervous system and sensory (CNS) into which we placed dementia and unspecified psychoses to create comparable series as there was a clear, ongoing migration noted from NMH to CNS after the change to ICD-10 in 1999; psychiatric and substance abuse (NMH); infectious disease, subdivided into estimated HIV (HIV) and other (AID); maternal, fetal, congenital and pediatric (CHD). The categories LAB and EXT are omitted from our analysis. 137 3.3 Return summary statistics. Summary statistics for the ROI of disease groups, in units of years (for the lag length) and per-capita- GDP-denominated reductions in YLL between years t and t+4 per dollar of research funding in year t−q, based on historical ROI from 1980 to 2003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.4 ROI example. An example of the ROI calculation for HLB from 1986. 147 3.5 Portfolio weights. Benchmark, single- and dual-objective optimal portfolio weights (in percent), based on historical ROI from 1980 to 2003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 27
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- Chapter 1 Optimal trading of arbitrage opportunities under constraints In financial economics, an arbitrage is an investment opportunity that is too good to be true when there are no market frictions. In actual financial markets however, there are frictions and even if there are arbitrage opportunities the investors may not be able to fully exploit them due to the constraints they face. We will explore two kinds of risky arbitrage opportunities when there are market frictions. The first one is a case of a textbook arbitrage, a convergence trade strategy. The second is a case of a statistical arbitrage, a mean reversion trading strategy. These two strategies are two of the most popular trading strategies that hedge funds follow, so studying them in detail when there are market frictions is a valuable exercise. A convergence trade is a trading strategy consisting of long/short positions in two similar assets, where we buy the cheap asset, we short the expensive asset and we wait until the prices of the two assets to converge which we know it will happen for sure some particular time in the future. An example of this trade involves the difference in price between the on-the-run and the most recent off-the-run security. An on- the-run security is the most recently issued, and hence most liquid, of a periodically issued security. Since an on-the-run security is more liquid it trades at a premium to off-the-run securities [29]. A convergence trade involves taking a long position in the most recent off-the-run security and shorting the on-the-run security. The on- 29
- the-run will become off-the-run upon the issue of a newer security and then there will be almost no difference between the two securities in our trade so their prices will converge. Another example involves investing in Treasury STRIPS with identical maturity dates but different prices. A mean reversion trading strategy involves investing in an asset or a portfolio of assets whose value is a mean reverting process. Since most price series in the equity space follow random walk, this strategy most commonly involves investing in a port- folio of non-mean-reverting assets whose value is a stationary mean reverting series. These price series that can be combined in such a way are called cointegrating. A classic statistical arbitrage example is the pairs trading, which is the first type of algorithmic mean reversion trading strategy invented by institutional traders, report- edly by the trading desk of Nunzio Tartaglia at Morgan Stanley [64]. The statistical arbitrage pairs trading strategy bets on the convergence of the prices of two similar assets whose prices have diverged without a fundamental reason for this. These arbitrage opportunities are risky under market frictions. In particular the first case is exposed to the “divergence risk”, i.e. the fact that the pricing differential between the two similar assets can diverge arbitrarily far from 0 prior to its conver- gence at some particular time in the future. The second case is exposed both to the “divergence risk” and to the “horizon risk”, in other words the fact that the times at which the spread will converge to its long run mean are uncertain. We will explore the optimal portfolio allocation of a risk averse investor who invests in N convergence trades or mean reverting trading strategies, while facing constraints. In particular, we will study the optimal trading strategy when he faces VaR constraints or collateral constraints. Risk sensitive regulation, such as the VaR constraint, has lately become a central component of international financial regu- lations. Collateral or margin constraints, where the investor has to have sufficient wealth to secure the liabilities taken by short positions, have been ubiquitous in the financial transactions for centuries and margin calls have been behind several crises including the LTCM debacle[54]. In the rest of this chapter, we will discuss the relevant literature review. Then we 30
- will discuss about the setup of the model and the constraints, we will find the optimal trading strategy of the investor and finally we will explore the characteristics of this optimal strategy. 1.1 Literature review Merton studied the problem of optimal portfolio allocation in a continuous time set- ting without any market frictions [59]. The optimal portfolio involves two terms: a market timing term and a hedging demand term. The first term is a myopic term that represents the optimal allocation if you were interested at each time instant t only for an horizon dt ahead. The second term represents the investor’s additional demand due to the covariance of the wealth process with the attractiveness of the available investment opportunities. Although Merton gives an analytical general solution this is expressed in terms of the partial derivatives of the value function and additional work is needed to derive the solution in terms of the model parameters. Additionally it assumes that there are no market frictions. Optimal trading of mean reversion trading strategies have been studied by both Boguslavsky and Boguslavskaya [15] and Jurek and Yang [48]. They have found analytical solutions for the optimal weight of a single mean reverting trading strategy for risk averse CRRA investors. Their analysis is similar with the one in Kim and Omberg [49], where they assume that there is a risk free asset with a constant risk- free rate and a single risky asset with a mean reverting risk premium, which implies a mean reverting instantaneous Sharpe ratio. In all the cases they have assumed that there are no market frictions whatsoever. Longstaff and Liu [53] have studied the problem of optimal trading of a single convergence trade under a margin constraint. For the single convergence trade case both VaR constraints and margin constraints collapse in the same constraint and the problem is significantly easier. In addition by studying only convergence trades they have taken out one important dimension of risk, the horizon risk, keeping only the divergence risk. Brennan and Schwarz [17] have also studied the problem of optimal 31
- trading of a single convergence trade including transaction costs when the arbitrage potential is restricted by position limits. The literature is rich with papers that study the existence of an equilibrium where there exists mispricings. This persistence of mispricings is typically attributed to agency problems, frictions or some kind of risk. Unlike textbook arbitrages, which generate riskless profits and require no capital commitments, exploiting real-world mispricings requires the assumption of some kind of risk. Shleifer and Vishny [74] emphasized that risks such as the uncertainty about when the pricing differential will converge to 0 and the possibility of a divergence of the mispricing prior to its elimination may play a role in limiting the size of positions that arbitrageurs are willing to take, contributing to the persistence of the arbitrage in equilibrium. Basak and Choitoru [6] also showed that arbitrage can persist in equilibrium when there are frictions. They study dynamic models with log utility and heterogeneous beliefs in the presence of margin requirements and other portfolio constraints. With respect to the constraints, Basak and Shapiro [7] study the problem of opti- mal trading strategy of a risk averse investor who faces finite horizon VaR constraints in a complete markets setting using the martingale representation approach [4]. Here again there are no constraints in the optimal portfolio allocation at each time t but there is only one constraint in the wealth at some finite horizon. Finally, Geanakoplos [33] studies the collateral constraints, how these determine an equilibrium leverage and how this leverage changes over time, the so-called leverage cycles. Let us now discuss about the setup of the model and the constraints and find the optimal trading strategy of the investor. 1.2 Analysis We assume we have a risk averse investor maximizing the expected continuously compounded rate of return or equivalently the expected logarithm of his final wealth E(lnWT ). There are two cases to consider. In the first case, the investor can invest in a risk-free asset and N non-redundant convergence trades, modeled as correlated 32
- Brownian bridges. In the second case, the investor can invest in a risk-free asset and N non-redundant mean reversion trading strategies, modeled as a multivariate Ornstein-Uhlenbeck (OU) process. The investor faces two kinds of constraints: VaR constraints or collateral constraints. We determine the optimal trading strategy and its characteristics in both cases. 1.2.1 Models As we mentioned already, a convergence trade is a trading strategy consisting of long/short positions in two similar assets, where we buy the cheap asset, we short the expensive asset and we wait until the prices of the two assets to converge which we know it will happen for sure some time in the future. The spread of the convergence trade can be modeled as a Brownian bridge driven by K Brownian motions, which has the property that the spread will converge to 0 almost surely at some determined time in the future. The stochastic differential equation governing the spread of the trade is given by: dSt = − aSt T − t dt + K X k=1 σkdZkt (1.1) where St is the spread of the trade, a is a parameter controlling the rate of the mean reversion to 0, T is the horizon of the investor which is also the time at which the spread goes to 0 with probability 1 and Zt is a Brownian motion in RK . We can see that the reversion to 0 grows stronger as t → T. Therefore, the investment opportunities get better as the spread gets larger and t → T, since then the drift term pushing the spread towards 0 gets larger. A mean reversion trading strategy involves investing in a stationary portfolio of non-mean reverting assets, whose value is a mean reverting process. The value of the portfolio can be modeled as an Ornstein-Uhlenbeck (OU) process. The stochastic differential equation governing it is given by: dSt = −φ(St − S̄)dt + K X k=1 σkdZkt 33
- In our case we have N of these mean reverting processes and we assume that they are modeled as a multivariate Ornstein-Uhlenbeck process, which is defined by the following stochastic differential equation: dSt = −Φ(St − S̄)dt + σdZt (1.2) Above Φ is a N-by-N square transition matrix that characterizes the deterministic portion of the evolution of the process, S̄ is the vector representing the unconditional mean of the process, σ is a N-by-K matrix that drives the dispersion of the process and Zt is a Brownian motion in RK . The Ornstein-Uhlenbeck process has the nice property that its conditional distri- bution is normal at all times, with mean equal to Et[St+τ ] = S̄ + e−Φτ (St − S̄) and covariance matrix independent of St [60]. We assume that Φ has eigenvalues with positive real part, so that the conditional expectation approaches to S̄ as t → ∞. The Ornstein-Uhlenbeck process captures the two important dimensions of risk in all relative value trades: the “horizon risk”, in other words the fact that the times at which the spread will converge to its long run mean are uncertain and the “divergence risk”, i.e. the fact that the pricing differential can diverge arbitrarily far from its long run mean prior to its convergence. The Brownian bridge captures only the “divergence” risk, since by its definition we assume that the investor has perfect information about the magnitude of the mispricing at some future date T, i.e. we assume that the date T on which the mispricing will be eliminated is known ahead with certainty. 1.2.2 Constraints We consider two kinds of constraints: VaR and collateral constraints. The VaR constraint is a widely used statistical risk measure, adopted both by the regulators 34
- and the private sector. It is the cornerstone of the capital regulations adopted by Basel regulations. Both the 1996 market risk amendment of the original 1988 Basel accord and the Basel II regulations have been built on the notion of Value-at-Risk [47]. The Value at risk (VaR) at α-level is defined as the threshold value such that the probability of losses greater than the threshold is less than α. In our case we consider instantaneous VaR constraints which amount for determining an upper bound in the wealth volatility, since locally the diffusion processes have normal distributions. Therefore, the instantaneous VaR constraints are given by: θT Σθ ≤ LW2 where θ is a N by 1 vector of positions, Σ is the instantaneous covariance matrix of the spreads, L is some proportionality constant that determines the tightness of the constraint and W is the investor’s wealth. Collateral or margin constraints have been ubiquitous in the financial transactions for centuries. Even Shakespeare in the “Merchant of Venice” points out the importance of the collateral, as Shylock charged Antonio no interest rate but he asked for a pound of flesh as a collateral. The collateral constraints provide protection against mark-to-market losses whenever an investor generates a liability by shorting an asset. Therefore, they require that the investor’s wealth is bounded below by the collateral necessary to secure the liabilities. They are given by: N X i=1 λi|θi| ≤ W where λi is the collateral necessary to secure the liability in spread i. In our work, each unit of arbitrage should be understood as being relative to a fixed face or notional amount and therefore each λi is a percentage of this fixed face value or notional amount. 35
- 1.2.3 Solution Let us now find the optimal trading strategy of a risk averse investor who maximizes the expected logarithm of his final wealth E(lnWT ). We consider two cases: • The investor invests in the risk free asset and in N correlated convergence trades. • The investor invests in the risk free asset and in N correlated mean reversion trading strategies. For both cases our analysis is similar. For both cases we have: Wt = N X i=1 θitSit + θ0tB0t ∀t ∈ [0, T] (1.3) where θit is the investor’s position in opportunity i for i = 1, · · · , N, θ0t is the in- vestor’s position in the risk free asset, Sit is the spread of the convergence trade or the value of the mean reverting portfolio and B0t is the price of the risk free asset. The process θt is adapted to the filtration generated by the Brownian motion Zt. The investor solves the following problem: maximizeθ∈Θ E(lnWT ) subject to dWt = PN i=1 θitdSit + θ0tdB0t dSt = µ(S, t)dt + σ(S, t)dZt (1.4) where Θ is the set of admissible trading strategies. Let us first define ∀t ∈ [0, T] Ft = θt/Wt ∈ RN . 36
- For the convergence trades case, investor’s wealth satisfies the following stochastic differential equation: dWt = Wtrdt + N X i=1 θitSit(− ai T − t − r)dt + θT σdZt dWt Wt = rdt + N X i=1 FitSit(− ai T − t − r)dt + FT σdZt By applying Ito’s Lemma we have that: d(ln(Wt)) = rdt + N X i=1 FitSit(− ai T − t − r)dt − 1/2FT t ΣFtdt + FT t σdZt Therefore it is: ln(WT ) = ln(Wt) + Z T t rs ds + Z T t N X i=1 FisSis(− ai T − t − rs) − 1 2 FT s ΣFs ! ds + Z T t FT s σdZs (1.5) Assuming constant interest rate, we have: Et(ln(WT )) = ln(Wt) + r(T − t) + Et Z T t N X i=1 FisSis(− ai T − t − rs) − 1 2 FT s ΣFs ! ds ! + Et( Z T t FT s σdZs) (1.6) 37
- For the mean reversion trading strategies case, investor’s wealth satisfies the following stochastic differential equation: dWt = Wtrdt + N X i=1 θit(−ΦT i (St − S̄) − rSit)dt + θT σdZt dWt Wt = rdt + N X i=1 Fit(−ΦT i (St − S̄) − rSit)dt + FT t σdZt where Φi is the i’th row of the transition matrix Φ. By applying Ito’s Lemma we have that: d(ln(Wt)) = rdt + N X i=1 Fit(−ΦT i (St − S̄) − rSit)dt − 1/2FT t ΣFtdt + FT t σdZt Therefore it is: ln(WT ) = ln(Wt) + Z T t rs ds + Z T t N X i=1 Fis(−ΦT i (Ss − S̄) − rSis) − 1 2 FT s ΣFs ! ds + Z T t FT s σdZs (1.7) Assuming constant interest rate we have: Et(ln(WT )) = ln(Wt) + r(T − t) + Et Z T t N X i=1 Fis(−ΦT i (Ss − S̄) − rSis) − 1 2 FT s ΣFs ! ds ! + Et( Z T t FT s σdZs) (1.8) Under VaR constaints it is: FT t ΣFt ≤ L < ∞ ∀t 38
- Under the margin constraints it is: N X i=1 λi|Fit| ≤ 1 ∀t FT t ΣFt = N X i=1 N X j=1 FitFjtσij ≤ N X i=1 N X j=1 λiλj|Fit||Fjt| σij λiλj < C < ∞ ∀t Therefore, for both the cases and both the constraints the integrand of the stochastic integral belongs in H2 , which is a sufficient condition for the stochastic integral to be a martingale. Consequently, Et( R T t FT s σdZs) is equal to 0. Maximizing Et(ln(WT )) is equivalent to maximizing the third term is equations (1.6), (1.8) for both the cases respectively. Let’s now stydy in detail the solution for both cases for both the constraints. VaR constraint Maximizing Et(ln(WT )) under the VaR constraint is equivalent to solving ∀t the following QCQP: minimize FT t µt + 1 2 FT t ΣFt subject to FT t ΣFt ≤ L (1.9) where µt = S1t( a1 T−t + r) . . . SNt( aN T−t + r) (1.10) for the convergence trades case and µt = ΦT 1 (St − S̄) + rS1t . . . ΦT N (St − S̄) + rSNt (1.11) for the mean reversion trading strategies case. 39
- We can easily solve the problem 1.9 by applying the KKT conditions or by ge- ometry (see Appendix). Fopt t , λopt t are optimal iff they satisfy the following KKT conditions ([10]): • Primal feasibility: FT opt t ΣFopt t ≤ L • Dual feasibility: λopt t ≥ 0 • Complementary slackness: λopt t (FT opt t ΣFopt t − L) = 0 • Minimization of the Lagrangean: Fopt t = argmin L(Ft, λopt t ) By solving the KKT conditions (see Appendix for details) we find that: θopt t = −Σ−1 µtWt if µT t Σ−1 µt ≤ L − Σ−1µtWt r µT t Σ−1µt L if µT t Σ−1 µt ≥ L This is equivalently written as: θopt t = − Σ−1 µtWt 1 + λopt t where 1 + λopt t = max 1, r µT t Σ−1µt L ! Let’s now discuss more the properties of the solution. The investor has logarithmic preferences. Therefore, he is a myopic optimizer - there is no hedging demand [59]. At each time t he looks dt ahead and decides how to trade in an optimal way. There are two cases to consider: • Case 1: At time t: µT t Σ−1 µt ≤ L In this case, the optimal solution is the unconstrained myopic optimal solution, since it satisfies the VaR constraint. For the convergence trades case, this is equivalent to the spread St being in the ellipsoid Et = {S | ST (AtΣ−1 At)S ≤ L} where At = diag( a1 T−t + r, · · · , aN T−t + r). 40
- The volume of the ellipsoid Et is shrinking as t → T, since vol(E) = QN i=1( a1 T−t + r)−1 p det(Σ)vol(B(0, 1)) where B(0, 1) is the unit sphere. Figure 1-1 shows this shrinking ellipsoid at three time instants. For the mean reversion trading strategies case, this is equivalent to the spread or value of the trade being inside the convex set C = {S | (S−S̄)T ((Φ+rI)T Σ−1 (Φ+ rI))(S − S̄) + 2rS̄T Σ−1 (Φ + rI))(S − S̄) ≤ L − r2 S̄T Σ−1 S̄}, which in the case of r = 0 is the ellipsoid C = {S | (S − S̄)T (ΦT Σ−1 Φ)(S − S̄) ≤ L. If S̄ = 0 this convex set is also an ellipsoid. These ellipsoids characterize poor opportunities where the constraints are not active. What constitutes poor investment opportunities changes over time for the case of convergence trades, while it remains invariant for the mean reversion trades case. For the case of convergence trades, the same spreads initially can be considered poor investment opportunities, where the investor does not bind the constraint, he is more conservative, but after some time they can be considered good opportunities and the investor becomes more aggressive and binds the constraint. Informally, when the investment opportunities are poor, the spreads are more likely to widen which then would lead to mark-to-market losses and the investor would not have sufficient wealth to take advantage the better investment oppor- tunities and simultaneously satisfy the VaR constraints. Therefore, the investor is more conservative. • Case 2: At time t: µT t Σ−1 µt > L Now the unconstrained myopic optimal solu- tion does not satisfy the VaR constraint. This case is equivalent to the spread St being outside the shrinking ellipsoid Et for the convergence trades case or the set C for the mean reversion trades case. Now the investment opportunities are good. The investor wants to invest the unconstrained optimal trading strategy, but due to the VaR constraint invests in the proportion of this optimal trading strategy necessary to satisfy the VaR constraint. 41
- −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 Ellipsoids of poor investment opportunities for t=0.3, 0.6, 0.9. Spread 1 Spread 2 Figure 1-1: Ellipsoids. Ellipsoids of poor investment opportunities for N=2 conver- gence trades at times t = 0.3, 0.6, 0.9. Margin constraint Maximizing Et(ln(WT )) under the margin constraint is equivalent to solving ∀t the following convex program: minimize FT t µt + 1 2 FT t ΣFt subject to PN i=1 λi|Fit| ≤ 1 (1.12) µt = S1t( a1 T−t + r) . . . SNt( aN T−t + r) for the convergence trades case and µt = ΦT 1 (St − S̄) + rS1t . . . ΦT N (St − S̄) + rSNt 42
- for the mean reversion trading strategies case. Let’s apply the KKT conditions. Fopt t , νopt t are optimal iff they satisfy the KKT conditions: • Primal feasibility: PN i=1 λi|Fit| ≤ 1 • Dual feasibility: νopt t ≥ 0 • Complementary slackness: νopt t ( PN i=1 λi|Fit| − 1) = 0 • Minimization of the Lagrangean Fopt t = argmin L(Ft, λopt t ) This program cannot be solved analytically in general. Again there are two cases to consider. • Case 1: At time t: kΛΣ−1 µtk1 ≤ 1 where Λ = diag(λ1, · · · , λN ) In this case, the optimal solution is the unconstrained myopic optimal solution, since it satisfies the margin constraint. For the convergence trades, this is equivalent to having at time t: kΛΣ−1 AtSk1 ≤ 1 where Λ = diag(λ1, · · · , λN ) and At = diag( a1 T−t + r, · · · , aN T−t + r). In this case we have that St is inside a “diamond” in N dimensional space, which shrinks as t → T. For the mean reversion trades, this is equivalent to having at time t: kΛΣ−1 (Φ(St− S̄) + rSt)k1 ≤ 1 where Λ = diag(λ1, · · · , λN ). Informally again, when the investment opportunities are poor, the spreads are more likely to widen which then would lead to mark-to-market losses and the in- vestor would not have sufficient wealth to take advantage the better investment opportunities and have enough wealth for the collateral necessary to secure the liabilities. 43
- • Case 2: At time t: kΛΣ−1 µtk1 ≥ 1 where Λ = diag(λ1, · · · , λN ). Now the investment opportunities are good, the unconstrained myopic optimal solution does not satisfy the collateral constraint and the constraint binds at the optimal solution. Uncorrelated opportunities There is a special case when the trading opportunities are uncorrelated, where we can solve analytically the KKT conditions (see Appendix for details). In that case the optimal positions are given by: θopt it = sign(−µit)(|µit λi | − νopt t )+ σ2 i λi Wt (1.13) We observe the following: • First of all for the convergence trades, in case the spread is positive we short the spread as we would expect and in case it is negative we are long the spread. For the mean reversion trades, the sign is the opposite of the sign of ΦT i (St−S̄)+rSit. • Second, if µt is high relative to the collateral then the magnitude of the position is higher. • Third, if the variability of the opportunity is high the magnitude of the corre- sponding position is low. • Finally the more interesting property of the solution is that it has a cutoff value, the dual variable, and if the absolute value of µt over the collateral is greater than the dual variable the position is different from zero otherwise the position is 0. It is: νopt t = 0 if N X i=1 |λiµit| σ2 i ≤ 1 44
- and νopt t > 0 if N X i=1 |λiµit| σ2 i > 1 The dual variable is 0 when the investment opportunities are poor. It is easy to see that when the margin constraint binds we have: F̃it opt = sign(−µit)(|µit λi | − νopt t )+ σ2 i λ2 i andkF̃k1 = 1 (1.14) 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Weights in different arbitrage opportunities. Weights Figure 1-2: Weights for the case of uncorrelated spreads and collateral con- straint. Weights for the case of uncorrelated spreads. In Figure 1-2 we can see an example of how we invest in different convergence trades when there is no correlation among them, with λ = 1 and volatilities equal to 1 for all the opportunities. The height of each bar is the absolute value of µit and we invest only in those spreads where the µit is larger than the dual variable. If PN i=1 |µit| < 1, then the dual variable is 0, we invest in all the opportunities and the collateral constraint does not bind. If PN i=1 |µit| > 1 as is in the figure then the margin constraint binds, the dual variable is positive and we can find it as follows. We start from the maximum µit and then reduce it until the sum of the weights is equal to 1 where each weight is the distance between the absolute value of µit and ν. 45
- 1.2.4 Connection with Ridge and Lasso regression Before we explore further the properties and the results of the optimal trading strate- gies, it would be interesting to digress for a while and see what connection there is between our problems and the regularized regressions. In the basic form of regularized regression, the goal is not only to have a good fit, but also regression coefficients that are “small”. Two of the most common forms of regularized regressions are the Ridge and Lasso regression. Ridge regression shrinks the regression coefficients by imposing a penalty on their size [42]. Equation 1.15 is one of the ways to write the Ridge problem. minimize PN i=1(yi − β0 − Pp j=1 xijβj)2 subject to Pp j=1 β2 j ≤ t (1.15) The Ridge regression coefficients solution is similar to the optimal trading strategy followed by a risk averse investor with logarithmic preferences, who can choose among N diffusion processes and faces VaR constraints. In both cases we have this propor- tional shrinkage where we reduce all the weights by a constant. Lasso regression is another common form of a regularized regression. It can be used as a heuristic for finding a sparse solution. It does a kind of continuous subset selection [16]. Equation 1.16 is one of the ways to write the Lasso problem. minimize PN i=1(yi − β0 − Pp j=1 xijβj)2 subject to Pp j=1 kβjk ≤ t (1.16) The Lasso regression coefficients solution is similar to the optimal trading strategy followed by a risk averse investor with logarithmic preferences, who can choose among N diffusion processes and faces margin constraints. Therefore, we can expect that in this case we will have a sparse solution where the weights of several of the opportu- nities will be 0. 46
- 1.3 Results Let us move on now to the results first for the convergence trades and then for the mean reversion trading strategies. 1.3.1 Convergence trades VaR constraints. We have simulated the optimal trading strategy for N = 2 correlated convergence trading opportunities under VaR constraints. We find the following: • It is often optimal for an investor to underinvest i.e. not to bind the constraint. • The investor typically experiences losses early before locking at a profit as we can see in Figures 1-3, 1-4, 1-5, 1-6. • Tighter constraints lead to less variability and less skewness in the distribution of wealth. They also lead to less final wealth as we can see in Figures 1-5, 1-6. • The wealth is higher when the opportunities hedge each other, as we can see in Figures 1-4, 1-6. This makes sense because when the constraints are binding we care more about losing money which would then lead surely to liquidation when the investment opportunities are better and therefore we prefer the op- portunities to hedge each other. Figure 1-7 shows a typical path for the wealth evolution using the same noise process for positive and negative correlation under the VaR constraint. We see clearly this hedging effect where negative correlation leads to more wealth. • When the initial values of the convergence trades are low, the constraints bind for a small percentage of time and final wealth is negatively correlated to the percentage of time the constraints bind. Figure 1-8 shows this effect. • The final portfolio wealth is highly positively skewed as it is obvious in Figures 1-3, 1-4, 1-5, 1-6 47
- For all the simulations we used: σ1 = σ2 = 1, a1 = a2 = 1, S[0] = [1; 1], rf = 0.06, number of steps = 1000. 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Distribution of wealth Time 0.25 rho 0.5 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Distribution of wealth Time 0.5 rho 0.5 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Distribution of wealth Time 0.75 rho 0.5 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Distribution of wealth Time 1 rho 0.5 Figure 1-3: VaR constraints, positive correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing VaR constraints (K=1). Initial wealth is $100. 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.25 rho −0.5 K 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.5 rho −0.5 K 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.75 rho −0.5 K 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 1 rho −0.5 K 1 Figure 1-4: VaR constraints, negative correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing VaR constraints (K=1). Initial wealth is $100. 48
- 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.25 rho 0.5 K 0.25 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.5 rho 0.5 K 0.25 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.75 rho 0.5 K 0.25 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 1 rho 0.5 K 0.25 Figure 1-5: VaR constraints, positive correlations, tight constraints. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively cor- related (ρ = 0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100. 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.25 rho −0.5 K 0.25 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.5 rho −0.5 K 0.25 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.75 rho −0.5 K 0.25 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 1 rho −0.5 K 0.25 Figure 1-6: VaR constraints, negative correlations, tight constraints. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing VaR constraints (K=0.25). Initial wealth is $100. 49
- 0 200 400 600 800 1000 1200 0 200 400 600 800 1000 1200 1400 Simulation step Final wealth rho 0.5 rho −0.5 Figure 1-7: Wealth evolution under VaR constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under the VaR constraint. Initial wealth is $100. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 1200 1400 1600 Frequency the constraint binds Final wealth Figure 1-8: Relation between final wealth and frequency the VaR constraint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence trades are low. 50
- Margin constraints. We have also simulated the optimal trading strategy for N = 2 correlated convergence trading opportunities under margin constraints using the same noise process as with the VaR constraints. We have similar results with the case of VaR constraints as we see in Figures 1-9, 1-10, 1-11, 1-12, 1-13 with the following important differences: • When the constraints bind, it is often the case that the position in one of the convergence trades is 0, i.e. we have less diversification, sparse solution. Figure 1-15 shows a typical path of the positions in two convergence trading opportu- nities under VaR constraints, where we see that they tend to be different than 0. Figure 1-16 shows the evolutions of the positions in two convergence trading opportunities under margin constraints for the same exactly asset processes as before. We clearly see that often we invest only in one position, as we expected due to the similarity of the positions with the Lasso regression coefficients. • The final wealth is less skewed and smaller with respect to the case of VaR constraints. 51
- 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.25 rho 0.5 Collateral 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.5 rho 0.5 Collateral 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.75 rho 0.5 Collateral 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 1 rho 0.5 Collateral 1 Figure 1-9: Margin constraints, positive correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100. 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.25 rho −0.5 Collateral 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.5 rho −0.5 Collateral 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.75 rho −0.5 Collateral 1 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 1 rho −0.5 Collateral 1 Figure 1-10: Margin constraints, negative correlations. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing margin constraints (Collateral = 1). Initial wealth is $100. 52
- 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.25 rho 0.5 Collateral 2 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.5 rho 0.5 Collateral 2 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.75 rho 0.5 Collateral 2 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 1 rho 0.5 Collateral 2 Figure 1-11: Margin constraints, positive correlations, more collateral needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two positively correlated (ρ = 0.5) convergence trades, while facing margin con- straints (Collateral = 2). Initial wealth is $100. 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.25 rho −0.5 Collateral 2 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.5 rho −0.5 Collateral 2 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 0.75 rho −0.5 Collateral 2 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 Time 1 rho −0.5 Collateral 2 Figure 1-12: Margin constraints, negative correlations, more collateral needed. Wealth distribution at t = 0.25, 0.5, 0.75, 1 for an investor who invests in two negatively correlated (ρ = −0.5) convergence trades, while facing margin con- straints (Collateral = 2). Initial wealth is $100. 53
- 0 200 400 600 800 1000 1200 0 100 200 300 400 500 600 700 Simulation step Final wealth rho 0.5 rho −0.5 Figure 1-13: Wealth evolution under margin constraint. Typical path of the wealth evolution for an investor investing in two convergence trades using the same noise process for positive and negative correlation under the margin constraint. Initial wealth is $100. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 200 400 600 800 1000 1200 1400 1600 Frequency the constraint binds Final wealth Figure 1-14: Relation between final wealth and frequency the margin con- straint binds. Final wealth is negatively correlated to the percentage of time the constraints bind when the initial values of the convergence trades are low. 54
- 0 100 200 300 400 500 600 700 800 900 1000 −800 −600 −400 −200 0 200 400 600 800 Simulation step Position Convergence trade 1 Convergence trade 2 Figure 1-15: Positions evolution under VaR constraints. Typical path of the positions in two convergence trading opportunities under VaR constraints. 0 100 200 300 400 500 600 700 800 900 1000 −400 −300 −200 −100 0 100 200 300 400 Simulation step Position Convergence trade 1 Convergence trade 2 Figure 1-16: Positions evolution under margin constraints. Typical path of the positions in two convergence trading opportunities under margin constraints. 55
- 1.3.2 Mean reversion trading opportunities Similar results we get also by simulating the optimal trading strategy for N = 2 correlated mean reversion trading opportunities under VaR and margin constraints. This makes sense since without loss of generality, we have assumed that S̄ = 0 and that Φ is a diagonal matrix, which makes the mean reversion trades case similar to the convergence trades case. They are different in that the drift term of the convergence trades for the same spreads S gets better and better as t → T, while for the mean reversion trades it remains constant. 1.4 Conclusions We explored the optimal portfolio allocation of a risk averse investor who invests in N convergence trades or mean reverting trading strategies, while facing constraints. In particular, we studied the optimal trading strategy when he faces VaR constraints or collateral constraints. The optimal trading strategy is found by solving at each time t a convex program for both cases, we characterized the solution of the convex program and we found the properties of the optimal trading strategy. In all the chapter, we have assumed that the investor completely trusts his model and he is certain about the dynamics of the opportunities he faces. What happens if the model is just an approximation? What happens if the investor believes that opportunities’ dynamics come from an unknown member of a set of unspecified models near an approximating model? Concern about model misspecification will change the optimal trading strategy of the investor and this is the topic of the next chapter. 56
- Chapter 2 Optimal trading of arbitrage opportunities under model misspecification A decision maker maximizes a utility function subject to a model. Standard control theory helps a decision maker to make optimal decisions when his model is correct. Robust control theory helps him to make good decisions when his model is approxi- mately correct. In this chapter we will use methods of robust control theory to find the optimal portfolio allocation of a risk averse investor, who invests in convergence trades or mean reverting trading strategies, and is not completely confident about the dynamics of his models. In particular, we assume that the investor believes that the data comes from an unknown member of a set of alternative models near his nominal model. These alternative models are statistically difficult to distinguish from the nominal model. The investor believes that his model is a pretty good approximation in the sense that the discrepancies between the alternative models and his nominal model are small. We will use the relative entropy to characterize the discrepancies between different models. Concern about model misspecification leads the investor to choose a trading strategy that is robust over the alternative models. Three questions come naturally at this point: 57
- • What does it mean to have a robust trading strategy? A robust trading strategy is a strategy that works well over the set of alternative probability models. We evaluate the worst performance of a given strategy over that set of alternative probability models and we pick the one that maximizes this worst case performance. It is essentially a “max-min” problem, a two-player game in which a maximizing player chooses the best response to a malevolent player who can disturb the stochastic model within limits. • Why would we be interested in a robust decision rule over alternative models? Why don’t we take a Bayesian approach, where we put a prior distribution over the set of alternative probability models? This could be another approach, but this set of alternative models may be too large or too difficult for the investor to come up with a well behaved, plausible prior distribution. In addition, we might want our solution to work well over any kind of prior distribution [41]. • Why do we use the relative entropy to measure the discrepancy between an alternative and the nominal model? There are other ways to measure discrepancies between alternative probability models, like Prokhorov distance [9] but the relative entropy with respect to a measure P has nice properties and it is more tractable. It is given by: D(Q) = Z log( dQ dP )dQ and it is a convex function of the measure Q. In the rest of this chapter, we will review the relevant literature. Then we will discuss about the set of the alternative models, the relative entropy and equivalent ways to formulate our problem of the optimal robust portfolio allocation of the in- vestor. Subsequently, we will find the optimal robust trading strategy of the investor and finally we will explore the characteristics of this robust strategy. 58
- 2.1 Literature review Whittle [79], [80] has studied mathematical methods for answering the question of how to make decisions when you don’t fully trust your model. Lars Hansen and Thomas Sargent in [41] have studied how to make economic decisions in the face of model misspecification by modifying and extending aspects of robust control theory. Their work revolves mostly around the linear-quadratic regulator framework, where there is a certainty equivalence principle that allows a deterministic presentation of the control theory. Gilboa and Schmeidler in [34] have studied the max-min expected utility problem where the decision maker has multiple priors and maximizes his expected utility as- suming that nature chooses a probability measure to minimize his expected utility. The minimization is over a closed and convex set of finitely additive probability mea- sures. Their axiomatic treatment views this set of non-unique priors as an expression of the agent’s preferences and the priors are not cast as distortions to a nominal model. Lars Hansen et al. [40] have studied robust decision rules when the agent fears that the data are generated by a statistical perturbation of an approximating model that is either a controlled diffusion process or a control measure over continuous functions of time. They describe how stochastic formulations of robust control “constraint problems” can be viewed in terms of Gilboa and Schmeidler’s max-min expected utility model. They show the connection between the penalty robust control problem and the constraint robust control problem, two closely related problems and formulate the Hamilton Jacobi Bellman equations for various two-player zero sum continuous time games that are defined in terms of a Markov diffusion process. We extend their framework to the problem of optimal robust trading rules for a risk averse investor who does not trust his model dynamics, believes that his nominal model is a good approximation to the real model and invests in arbitrage opportunities. Fleming and Souganidis [77] present how the Bellman-Isaacs condition defines a Bellman equation for a two-player zero-sum game in which both players decide at time 59
- 0 or recursively. In other words, they show that the freedom to exchange orders of maximization and minimization guarantees that equilibria of games where the choices are done under mutual commitment at time 0 and of games where the choices are done sequentially by both agents coincide. Anderson et al. [3] show how the set of perturbed models in our formulations is difficult to distinguish statistically from the approximating model given a finite sample of timeseries observations. Jacobson [44] and Whittle [78] studied risk sensitive optimal control in the context of discrete-time linear quadratic regulator decision problems. They showed how the risk-sensitive control law can be computed by equivalently solving a robust penalty problem. We will now discuss first how to represent the alternative probability models over which we want our decision rules to be robust and how relative entropy can be used to describe their discrepancies from the nominal model. We will formulate two closely related nonsequential problems and the corresponding recursive HJB equations and finally we will find the optimal robust portfolio allocation for a risk averse investor who is not confident about the dynamics of his models and wants to invest in convergence trades or mean reversion trading strategies. 2.2 Analysis In Chapter 1 we saw that in the case when there is no model misspecification, the investor wants to find the optimal portfolio allocation that solves the following prob- lem: maximizeθ∈Θ E(lnWT ) subject to dWt = θtdSt + θ0tdBt dSt = µ(S, t)dt + σ(S, t)dZt dBt = rBdt (2.1) 60
- Here we have St ∈ RN and we have studied the following special cases: dSit = − aiSit T − t dt + K X k=1 σikdZkt dSt = −Φ(St − S̄)dt + σdZt and Θ is the set of admissible trading strategies: Θ = θ |θT Σθ ≤ LW2 for the case of VaR constraints and Θ = ( θ | N X i=1 λi|θi| ≤ W ) for the case of the margin constraints. In this Chapter, the investor doubts his model dSt = µ(S, t)dt + σ(S, t)dZt. To capture this doubt of the investor, we surround the approximating model with a cloud of models that are statistically difficult to distinguish and we add a malevolent agent who picks the worst possible model. The investor wants to find the optimal trading strategy that solves the following problem: maxθ∈Θ minQ∈Q EQ(lnWT ) (2.2) where Θ is the set of admissible trading strategies and Q is the set of alternative probability models. Problem 2.2 fits the max-min expected utility model of Gilboa and Schmeidler [34], where Q is a set of multiple different priors. Let’s now discuss how we represent the set of alternative probability models. 2.2.1 Alternative models representation We use martingales to represent perturbations to the probability models and relative entropy to measure the discrepancy between our nominal model and the alternative 61
- models. To understand better our continuous time formulations, we digress for a while by borrowing an example from [41]. Let’s consider a discrete time approximating model and its innovations ǫt which are i.i.d Gaussian shocks. An alternative model alters the distribution of these shocks. We use martingales to represent distortions to the probabilities. Let π̂t(ǫ) be the alternative density of the shock ǫt+1 based on date t information. Then the random variable Mt = Qt j=1 mj, where mj = π̂j−1(ǫ) π(ǫ) and M0 = 1, is a martingale and is a ratio of the joint alternative density over the joint nominal density. We define the entropy of the alternative distribution associated with Mt as the expected likelihood ratio with respect to the distorted distribution E(Mtlog(Mt)). It has the property that it is always non-negative and it is equal to 0, only when there is no distortion to the nominal distribution. Similarly, in our continuous time formulations we will use martingales to represent distortions to the nominal probability model. We will construct an alternative model by replacing Zt in our model by Ẑt + R t 0 hsds, where Ẑt is a Brownian motion under the alternative measure Q and ht is an adapted process that models the distortion, such that the process ξt = e R t 0 hsdZs− 1 2 R t 0 hT s hsds is a martingale. Therefore, the nominal model is misspecified by allowing the conditional mean of the shock vector in the alternative models to feed back arbitrarily on the history up to date t. Since ξ0 = 1 we have that E(ξt) = 1. Since in addition, ξt 0, we can define a probability measure Q such that Q(A) = E[1AξT ], in other words ξT = dQ dP is the Radon-Nikodym derivative of Q with respect to P, where the measures Q and P are equivalent. In fact one can always define a process ht so that for any measure Q the Radon-Nikodym derivative of Q with respect to P, dQ dP , is given by the exponential martingale ξT . In this way, our distorted models are: • For the convergence trades, dSt = − aiSit T − t dt + K X k=1 σik(dẐkt + hktdt) (2.3) 62
- • For the mean reversion trades, dSt = −Φ(St − S̄)dt + σ(dẐt + htdt) (2.4) where Ẑt is a Brownian motion under Q. Why is it a Brownian motion under Q? The answer lies with the Girsanov theorem [30] that states that if a process ht is such that ξt is a martingale and ξT = dQ dP is the Radon-Nikodym derivative of Q with respect to P, then the process ˆ Z(t) = Zt − R t 0 hsds is a Brownian motion under measure Q. Therefore, we parameterize Q by the choice of the drift distortion adapted process ht. Similarly with the discrete time case, we measure the discrepancy between mea- sures Q and P as the relative entropy D(Q) (see Appendix for derivation), D(Q) = Z T 0 1 2 EQ[hT t ht]dt (2.5) . This is to be expected, since the relative entropy between a multivariate Gaussian distribution N(µ, I) and the multivariate standard normal distribution is D(Q) = 1 2 µT µ (See Appendix for derivation) and htdt is the conditional mean of the process dZt under the alternative probability measure Q. To express the notion that the nominal model is a good approximation to the real model that generate the spread dynamics, we either restrain the alternative models by D(Q) ≤ η or we penalize them with the magnitude of the entropy. 2.2.2 Model setup Having described the set of alternative distributions, we are ready to formulate the problem a risk averse investor faces who distrusts his model dynamics. As in [40] we define two closely related problems: 63
- • A multiplier robust control problem. maxθ∈Θ minQ EQ(lnWT ) + νD(Q) subject to dWt = θtdSt + θ0tdBt dSt = µ(S, t)dt + σ(S, t)(dẐt + htdt) dBt = rBdt dξt = ξthtdZt (2.6) where ξT = dQ dP and D(Q) is given by equation 2.5. Here in essence there is an implicit restriction manifested by the nonnegative penalty parameter ν. • A constrained robust control problem. maxθ∈Θ minQ EQ(lnWT ) subject to dWt = θtdSt + θ0tdBt dSt = µ(S, t)dt + σ(S, t)(dẐt + htdt) dBt = rBdt dξt = ξthtdZt D(Q) ≤ η (2.7) where ξT = dQ dP and D(Q) is given by equation 2.5. In both cases the minimizing malevolent agent chooses the distortion process ht taken θ as given and the maximizing investor chooses the optimal strategy taken ht as given. We index the family of multiplier robust control problems by ν and the family of constrained robust control problems by η. Obviously the two problems are related, since the robustness parameter ν can be interpreted as the Lagrange multiplier on the constraint D(Q) ≤ η. Actually we can show that if V (ν) is the optimal value of the multiplier robust problem and K(η) is the optimal value of the constrained robust problem then we have: K(η) = maxν≥0V (ν) − νη [40]. Therefore we will be only interested in finding V (ν). 64
- 2.3 Solution We will solve 2.6 by solving the corresponding Hamilton Jacobi Bellman (HJB) equa- tion. We will solve the HJB equation for the case that there are no constraints in the admissible trading strategies and when the trading strategies are constrained by VaR or collateral considerations. But first let’s digress for a while and solve the HJB equation for the case when there is no fear of model misspecification. 2.3.1 No fear of model misspecification For now we assume that the investor completely trusts the dynamics of his models dSt = µ(S, t)dt + σ(S, t)dZt. He chooses the trading strategy θ ∈ Θ that solves the problem 2.1 where Θ is the set of admissible trading strategies: Θ = θ|θT Σθ ≤ LW2 for the case of VaR constraints and Θ = ( θ| N X i=1 λi|θi| ≤ W ) for the case of the margin constraints. In this case the HJB equation is: max θ∈Θ Vt + VW (Wr + θT (µ(S, t) − rSt)) + V T S µ(S, t) + 1/2VW W θT Σθ + VW SΣθ + 1/2trace(ΣVSS) = 0 where Σ = σσT and V (W, S, t) is the value function of the investor subject to the terminal condition V (W, S, T) = ln(W). Due to the logarithmic preferences of the investor it is: V (W, S, t) = ln(W)+H(S, t), therefore VW S = 0, VW = 1 W and VW W = − 1 W 2 . We also define ∀t ∈ [0, T] Ft = θt/Wt ∈ RN . 65
- The HJB equation becomes: max F ∈F Vt + (r + FT (µ(S, t) − rSt)) + V T S µ(S, t) − 1/2FT ΣF + 1/2trace(ΣVSS) = 0 where F is the set of admissible trading strategies: F = F|FT ΣF ≤ L for the case of VaR constraints and F = ( F| N X i=1 λi|Fi| ≤ 1 ) for the case of the margin constraints. The optimal trading strategy is the solution to the following convex problem min F ∈F FT (−µ(S, t) + rSt) + 1/2FT ΣF (2.8) as we also proved with a different method in Chapter 1, where µt = −µ(S, t) + rSt and in particular it is: µt = S1t( a1 T−t + r) . . . SNt( aN T−t + r) for the convergence trades case and µt = ΦT 1 (St − S̄) + rS1t . . . ΦT N (St − S̄) + rSNt for the mean reversion trading strategies case. 66
- 2.3.2 Fear of model misspecification no constraints In this section we assume that there are no constraints in the trading strategies fol- lowed by the risk averse investor and the investor is not confident about the dynamics of his models. The Hamilton Jacobi Bellman equation for the problem 2.6 is given by: max θ min h Vt + VW (Wr + θT (µ(S, t) − rSt)) + V T S µ(S, t) + 1/2VW W θT Σθ + VW SΣθ + 1/2trace(ΣVSS) + VW θT σh + V T S σh + ν 2 hT h = 0 where Σ = σσT and V (W, S, t) is the value function of the investor subject to the terminal condition V (W, S, T) = ln(W). The malevolent agent picks the worst case distortion drift process ht and the investor maximizes against the worst case scenario. After defining ∀t ∈ [0, T] Ft = θt/Wt ∈ RN the HJB equation becomes: max F min h Vt + WVW (r + FT (µ(S, t) − rSt)) + V T S µ(S, t) +1/2W2 VW W FT ΣF+WVW SΣF+1/2trace(ΣVSS)+WVW FT σh+V T S σh+ ν 2 hT h = 0 The inner minimization problem is a convex quadratic problem. The first order conditions are: WVW σT F + σT VS + νh = 0 h = − σT (WVW F + VS) ν The optimal value of the inner minimization problem is: g(F) = − W2 V 2 W Ft ΣF + V T S ΣVS + 2WVW FT ΣVS 2ν 67
- Plugging this back into the HJB equation we have: max F Vt + WVW (r + FT (µ(S, t) − rSt)) + V T S µ(S, t) + 1/2W2 VW W FT ΣF + WVW SΣF + 1/2trace(ΣVSS) − W2 V 2 W Ft ΣF + V T S ΣVS + 2WVW FT ΣVS 2ν = 0 Due to the logarithmic preferences of the investor it is: V (W, S, t) = lnW + H(S, t) and in that case VW = 1 W VW W = − 1 W 2 VW S = 0, VS(W, S, t) = HS(S, t) and the minimizing drift distortion h = −σT (F +HS) ν independent of the wealth. The HJB equation now becomes: max F Vt + r + FT (µ(S, t) − rSt) + V T S µ(S, t) − 1/2FT ΣF + 1/2trace(ΣVSS) − FT ΣF + V T S ΣVS + 2FT ΣVS 2ν = 0 The optimal trading strategy is the solution to the following convex quadratic prob- lem: maximize FT (µ(S, t) − rSt − ΣVS ν ) − 1 2 (1 + 1 ν )FT t ΣFt (2.9) The first order conditions are: µ(S, t) − rSt − ΣVS(St, t) ν = (1 + 1 ν )ΣFopt t Fopt t = 1 1 + 1 ν Σ−1 (µ(S, t) − rSt − ΣVS(St, t) ν ) Fopt t = ν ν + 1 Σ−1 (µ(S, t) − rSt) − VS(St, t) ν + 1 We clearly see that as ν → ∞ the optimal trading strategy converges to the one where we have no fear of model misspecification. This is to be expected since at this case the problems 2.1 and 2.6 are equivalent. It is interesting to find the conditions under which these weights are equal to the weights when there is no fear of model misspecification. When there is a fear of model misspecification, the optimal weights are a convex 68
- combination of Σ−1 (µ(S, t)−rSt), i.e.the weights without model misspecification and −VS. Therefore these weights are equal to the weights when there is no fear of model misspecification, when Vs + Fopt = 0, which is equivalent to hmin = 0. Of course this is expected since in that case there would be no distortion drift and the HJB equation would be the same as the benchmark case of no model misspecification. After plugging in the optimal trading strategy to the HJB equation, it becomes: Vt + r + 1/2 trace(ΣVSS) + 1/2 1 1 + 1 ν (µ(S, t) − rSt)T Σ−1 (µ(S, t) − rSt) + V T S µ(S, t) − µ(S, t) − rS ν + 1 − 1/2 1 ν + 1 V T S ΣVS = 0 We can plug in the optimal trading strategy to h = −σT (W VW F +VS) ν to find that: hmin = − (σT Fopt + σT VS) ν hmin = − (σT 1 1+ 1 ν Σ−1 (µ(S, t) − rSt − ΣVS ν ) + σT VS) ν hmin = − σT (Σ−1 (µ(S, t) − rSt) + VS) ν + 1 We consider two cases: • Convergence trades. The optimal trading strategy is given by: θopt t = − ν ν + 1 Σ−1 AtSt + VS ν + 1 Wt (2.10) where At = diag( a1 T−t + r, · · · , aN T−t + r). The optimal trading strategy is a convex combination of the strategy without fear of model misspecification and −VS with weights ν ν+1 and 1 ν+1 . From the symmetry of the problem we have: H(S, t) = H(−S, t) from which we get VS(St, t) = −VS(−St, t). • Mean reversion trades. The optimal trading strategy is given by: θopt t = − ν ν + 1 Σ−1 (Φ(St − S̄) + rSt) + VS ν + 1 Wt (2.11) 69

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