ON PORTFOLIO MANAGEMENT AND DERIVATIVES HEDGING

In this thesis, efforts are devoted to modeling and addressing practical problems arising in portfolio management, derivatives pricing, and quantitative trading. Optimal portfolio selection problems with multiple risk constraints, vulnerable European options pricing, and pairs trading strategies under a mean-variance framework are investigated.

Expected utility maximization problems with initial-time and intermediate-time Value-at-Risk (VaR) constraints on terminal wealth are studied. The closed-form solutions to the two-VaR problem are first derived, which are optimal among all feasible controls at the initial time, i.e., precommitted strategies. Moreover, we find that the precommitted strategies are optimal at the intermediate time for “bad” market states as well. A contingent claim on Merton’s portfolio is constructed to replicate the optimal portfolio. Risk management with intermediate-time risk constraints turns out to be prudent in hedging “bad” intermediate market states and outperforms the single terminal-wealth risk constrained solutions under the relative loss ratio measure. We then investigated an extended problem with joint initial-time and intermediate-time VaR regulations and a portfolio insurance constraint on terminal wealth. It is a real-world problem faced by defined-contribution pension fund managers, whose objective is to maximize the expected utility of terminal wealth over the minimum guaranteed annuity by allocating assets in a risk-free bond, an indexed bond and a stock. We apply the martingale method to transform the dynamic optimization problem into a static pointwise optimization problem and derive the closed-form precommitted strategies. The numerical results show that an intermediate-time VaR constraint can significantly change the tail distribution of optimal solutions and greatly reduce the risk of loss in “bad” market conditions.

A pricing model is proposed for vulnerable European options, where the price dynamics of the underlying and counterparty assets follow two jump-diffusion processes with fast mean-reverting stochastic volatility. The pricing problem is transformed into solving a partial differential equation by the Feynman-Kac formula. We then approximate the solution by pricing formulas with constant volatility obtained from a two-dimensional Laplace transform via a multiscale asymptotic analysis. Thus, an analytic approximation formula for vulnerable European options is derived in our setting.

This thesis also studied the static and dynamic optimal pairs trading strategies under the mean-variance criterion. The spread of the entity pairs is assumed to be mean-reverting and follows an Ornstein-Uhlenbeck process. Considering the constrained optimal control problem, we adopt the Lagrange multiplier technique to transform the primal problem into a family of linear-quadratic optimal control problems that can be solved by the classical dynamic programming principle. Solutions to static and dynamic optimal pairs trading problems are derived and discussed. We show that the “static and dynamic optimality” is a viable approach to the time-inconsistent control problem. Furthermore, numerical experiments of optimal pair trading strategies are presented.

[1] Basak, S. and Chabakauri, G. (2010). Dynamic mean-variance asset allocation. The Review of Financial Studies, 23(8):2970–3016.

[2] Basak, S. and Shapiro, A. (2001). Value-at-risk-based risk management: optimal policies and asset prices. The Review of Financial Studies, 14(2):371–405.

[3] Battocchio, P. and Menoncin, F. (2004). Optimal pension management in a stochastic framework. Insurance: Mathematics and Economics, 34(1):79–95.

[4] Black, F. and Scholes, M. (2019). The pricing of options and corporate liabilities. In World Scientific Reference on Contingent Claims Analysis in Corporate Finance: Volume 1: Foundations of CCA and Equity Valuation, pages 3–21. World Scientific.

[5] Blake, D., Wright, D., and Zhang, Y. (2013). Target-driven investing: Optimal investment strategies in defined contribution pension plans under loss aversion. Journal of Economic Dynamics and Control, 37(1):195–209.

[6] Blake, D., Wright, D., and Zhang, Y. (2014). Age-dependent investing: Optimal funding and investment strategies in defined contribution pension plans when members are rational life cycle financial planners. Journal of Economic Dynamics and Control, 38:105–124.

[7] Bodie, Z., Detemple, J. B., Otruba, S., and Walter, S. (2004). Optimal consumption-portfolio choices and retirement planning. Journal of Economic Dynamics and Control, 28(6):1115–1148.

[8] Bodie, Z., Merton, R. C., and Samuelson, W. F. (1992). Labor supply flexibility and portfolio choice in a life cycle model. Journal of Economic Dynamics and Control, 16(3-4):427–449.

[9] Bollinger, J. (1992). Using bollinger bands. Stocks and Commodities, 10(2):47–51.

[10] Boulier, J.-F., Huang, S., and Taillard, G. (2001). Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund. Insurance: Mathematics and Economics, 28(2):173–189.

[11] Boyle, P. and Tian, W. (2007). Portfolio management with constraints. Mathematical Finance, 17(3):319–343.

[12] Cairns, A. J., Blake, D., and Dowd, K. (2006). Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans. Journal of Economic Dynamics and Control, 30(5):843–877.

[13] Chen, A., Nguyen, T., and Stadje, M. (2018a). Optimal investment under var-regulation and minimum insurance. Insurance: Mathematics and Economics, 79:194–209.

[14] Chen, A., Nguyen, T., and Stadje, M. (2018b). Risk management with multiple VaR constraints. Mathematical Methods of Operations Research, 88(2):297–337.

[15] Chen, Z., Li, Z., Zeng, Y., and Sun, J. (2017). Asset allocation under loss aversion and minimum performance constraint in a dc pension plan with inflation risk. Insurance: Mathematics and Economics, 75:137–150.

[16] Cochrane, J. H. (2009). Asset pricing: Revised edition. Princeton universitypress.

[17] Cox, J. C. and Huang, C. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. Journal of Economic Theory, 49(1):33–83.

[18] Cuoco, D., He, H., and Isaenko, S. (2008). Optimal dynamic trading strategies with risk limits. Operations Research, 56(2):358–368.

[19] Cuoco, D. and Liu, H. (2006). An analysis of var-based capital requirements. Journal of Financial Intermediation, 15(3):362–394.

[20] Cvitanic, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. The Annals of Applied Probability, 767-818.

[21] Deelstra, G., Grasselli, M., and Koehl, P.-F. (2003). Optimal investment strategies in the presence of a minimum guarantee. Insurance: Mathematics and Economics, 33(1):189–207.

[22] Deelstra, G., Grasselli, M., and Koehl, P.-F. (2004). Optimal design of the guarantee for defined contribution funds. Journal of Economic Dynamics and Control, 28(11):2239–2260.

[23] Dong, Y. and Zheng, H. (2020). Optimal investment with s-shaped utility and trading and value at risk constraints: An application to defined contribution pension plan. European Journal of Operational Research, 281(2):341–356.

[24] Dumas, B. and Luciano, E. (1991). An exact solution to a dynamic portfolio choice problem under transactions costs. The Journal of Finance, 46(2):577–595.

[25] Elliott, R. J., Aggoun, L., and Moore, J. B. (2008). Hidden Markov models: Estimation and control, volume 29. Springer Science and Business Media.

[26] Engle, R. F. and Granger, C. W. (1987). Co-integration and error correction: representation, estimation, and testing. Econometrica: Journal of the Econometric Society, pages 251–276.

[27] Feynman, R. P. (2005). Space-time approach to non-relativistic quantum mechanics. In Feynman’s Thesis: A New Approach To Quantum Theory, volume 71–109. World Scientific.

[28] Fouque, J.-P., Papanicolaou, G., and Sircar, K. R. (1998). Asymptotics of a two-scale stochastic volatility model. Equations aux derivees partielles et applications, in honour of Jacques-Louis Lions, pages 517–525.

[29] Fouque, J.-P., Papanicolaou, G., Sircar, R., and Sølna, K. (2011). Multiscale stochastic volatility for equity, interest rate, and credit derivatives. Cambridge University Press.

[30] Gatev, E., Goetzmann, W. N., and Rouwenhorst, K. G. (2006). Pairs trading: Performance of a relative-value arbitrage rule. The Review of Financial Studies, 19(3):797–827.

[31] Geske, R. (1977). The valuation of corporate liabilities as compound options. Journal of Financial and Quantitative Analysis, 541–552.

[32] Granger, C. W. (1981). Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 16(1):121–130.

[33] Grauer, R. R. and Hakansson, N. H. (1993). On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: A comparison of returns and investment policies. Management Science, 39(7):856–871.

[34] Gundel, A. and Weber, S. (2007). Robust utility maximization with limited downside risk in incomplete markets. Stochastic Processes and Their Applications, 117(11):1663–1688.

[35] Gundel, A. and Weber, S. (2008). Utility maximization under a shortfall risk constraint. Journal of Mathematical Economics, 44(11):1126–1151.

[36] Hakansson, N. H. (1971). Multi-period mean-variance analysis: Toward a general theory of portfolio choice. The Journal of Finance, 26(4):857–884.

[37] Han, N.-w. and Hung, M.-w. (2012). Optimal asset allocation for dc pension plans under inflation. Insurance: Mathematics and Economics, 51(1):172–181.

[38] He, W.-H., Wu, C., Gu, J.-W., Ching, W.-K., and Wong, C.-W. (2022). Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial and Management Optimization, 18(3):2077.

[39] Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2):327–343.

[40] Hull, J. and White, A. (1995). The impact of default risk on the prices of options and other derivative securities. Journal of Banking and Finance, 19(2):299–322.

[41] Jang, B. G. and Park, S. (2016). Ambiguity and optimal portfolio choice with value-at-risk constraint. Finance Research Letters, 18:158–176.

[42] Jarrow, R. A. and Turnbull, S. M. (1995). Pricing derivatives on financial securities subject to credit risk. The Journal of Finance, 50(1):53–85.

[43] Johnson, H. and Stulz, R. (1987). The pricing of options with default risk. The Journal of Finance, 42(2):267–280.

[44] Kac, M. (1949). On distributions of certain wiener functionals. Transactions of the American Mathematical Society, 65(1):1–13.

[45] Karatzas, I. (1989). Optimization problems in the theory of continuous trading. SIAM Journal on Control and Optimization, 27(6):1221–1259.

[46] Karatzas, I., Lehoczky, J. P., and Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM Journal on Control and Optimization, 25(6):1557–1586.

[47] Karatzas, I., Lehoczky, J. P., Shreve, S. E., and Xu, G.-L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal on Control and Optimization, 29(3):702–730.

[48] Karatzas, I. and Shreve, S. (2012). Brownian motion and stochastic calculus, volume 113. Springer Science & Business Media.

[49] Klein, P. (1996). Pricing black-scholes options with correlated credit risk. Journal of Banking and Finance, 20(7):1211–1229.

[50] Klein, P. and Inglis, M. (2001). Pricing vulnerable european options when the option’s payoff can increase the risk of financial distress. Journal of Banking and Finance, 25(5):993–1012.

[51] Korn, R. (1997). Optimal portfolios: stochastic models for optimal investment and risk management in continuous time. World scientific.

[52] Kou, S., Peng, X., and Heyde, C. C. (2013). External risk measures and basel accords. Mathematics of Operations Research, 38(3):393–417.

[53] Kraft, H. and Steffensen, M. (2013). A dynamic programming approach to constrained portfolios. European Journal of Operational Research, 229(2):453–461.

[54] Kwok, Y. K. (2003). Compound options. Technical report.

[55] Lakner, P. and Ma Nygren, L. (2006). Portfolio optimization with downside constraints. Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics, 16(2):283–299.

[56] Ma, C., Yue, S., and Ren, Y. (2018). Pricing vulnerable european options under levy process with stochastic volatility. Discrete Dynamics in Nature and Society, 2018.

[57] Ma, Q.-P. (2011). On “optimal pension management in a stochastic framework” with exponential utility. Insurance: Mathematics and Economics, 49(1):61–69.

[58] Markowitz, H. (1952). The utility of wealth. Journal of Political Economy, 60(2):151–158.

[59] Melino, A. and Turnbull, S. M. (1990). Pricing foreign currency options with stochastic volatility. Journal of Econometrics, 45(1-2):239–265.

[60] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time case. The Review of Economics and Statistics, 247–257.

[61] Merton, R. C. (1973). Theory of rational option pricing. The Bell Journal of Economics and Management Science, 141–183.

[62] Merton, R. C. (1975). Optimum consumption and portfolio rules in a continuous-time model. Stochastic Optimization Models in Finance, 621–661.

[63] Morgenstern, O. and Von Neumann, J. (1953). Theory of games and economic behavior. Princeton university press.

[64] Mudchanatongsuk, S., Primbs, J. A., and Wong, W. (2008). Optimal pairs trading: A stochastic control approach. In 2008 American control conference, pages 1035–1039. IEEE.

[65] Pedersen, J. L. and Peskir, G. (2017). Optimal mean-variance portfolio selection. Mathematics and Financial Economics, 11(2):137–160.

[66] Pennacchi, G. G. (2008). Theory of asset pricing. Pearson/Addison-Wesley Boston.

[67] Ramm, A. G. (2001). A simple proof of the fredholm alternative and a characterization of the fredholm operators. The American Mathematical Monthly, 108(9):855–860.

[68] Samuelson, P. A. (1975). Lifetime portfolio selection by dynamic stochastic programming. Stochastic Optimization Models in Finance, 517–524.

[69] Shi, Z. and Werker, B. J. (2012). Short-horizon regulation for long-term investors. Journal of Banking and Finance, 36(12):3227–3238.

[70] Shreve, S. E. et al. (2004). Stochastic calculus for finance II: Continuous-time models, volume 11. Springer.

[71] Wu, C., Gu, J., and Ching, W.-K. (2021). Precommitted strategies with initial-time and intermediate-time var constraints. Available at SSRN 3943822.

[72] Wu, C., Gu, J., Ching, W.-K., and So, C. C. (2022). Optimal investment for a dc pension plan under var regulations and minimum insurance with inflation risk. Available at SSRN 4005410.

[73] Xu, G. L. and Shreve, S. E. (1992). A duality method for optimal consumption and investment under short-selling prohibition. II. Constant market coefficients. . The Annals of Applied Probability, 314-328.

[74] Yang, S.-J., Lee, M.-K., and Kim, J.-H. (2014). Pricing vulnerable options under a stochastic volatility model. Applied Mathematics Letters, 34:7–12.

[75] Yao, H., Yang, Z., and Chen, P. (2013). Markowitz’s mean-variance defined contribution pension fund management under inflation: A continuous-time model. Insurance: Mathematics and Economics, 53(3):851–863.

[76] Yiu, K.-F. C. (2004). Optimal portfolios under a value-at-risk constraint. Journal of Economic Dynamics and Control, 28(7):1317–1334.

[77] Yu, F., Ching, W.-K., Wu, C., and Gu, J. (2021). Optimal pairs trading strategies: a stochastic mean-variance approach. Available at SSRN 3977911.

[78] Zeng, Y., Li, D., Chen, Z., and Yang, Z. (2018). Ambiguity aversion and optimal derivative-based pension investment with stochastic income and volatility. Journal of Economic Dynamics and Control, 88:70–103.

[79] Zhang, A. and Ewald, C.-O. (2010). Optimal investment for a pension fund under inflation risk. Mathematical Methods of Operations Research, 71(2):353–369.

[80] Zhou, X. Y. and Li, D. (2000). Continuous-time mean-variance portfolio selection: A stochastic lq framework. Applied Mathematics and Optimization, 42(1):19–33.

[81] Zhu, D.-M., Gu, J.-W., Yu, F.-H., Siu, T.-K., and Ching, W.-K. (2021). Optimal pairs trading with dynamic mean-variance objective. Mathematical Methods of Operations Research, 94(1):145–168.