A primal-dual algorithm for BSDEs. (English) Zbl 1423.91008
Summary: We generalize the primal-dual methodology, which is popular in the pricing of early-exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was precomputed, e.g., by least-squares Monte Carlo, this methodology enables us to construct a confidence interval for the unknown true solution of the time-discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two 5-dimensional nonlinear pricing problems where tight price bounds were previously unavailable.
MSC:
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 65C05 | Monte Carlo methods |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
References:
| [1] | Alanko, S., and M.Avellaneda (2013): Reducing Variance in the Numerical Solution of BSDEs, C. R. Math. Acad. Sci. Paris351, 135-138. · Zbl 1269.65004 |
| [2] | Andersen, L., and M.Broadie (2004): A Primal‐Dual Simulation Algorithm for Pricing Multidimensional American Options, Manage. Sci. 50, 1222-1234. |
| [3] | Bally, V., and G.Pagès (2003): A Quantization Algorithm for Solving Multidimensional Discrete‐Time Optimal Stopping Problems, Bernoulli9, 1003-1049. · Zbl 1042.60021 |
| [4] | Belomestny, D. (2011): Pricing Bermudan Options by Nonparametric Regression: Optimal Rates of Convergence for Lower Estimates, Finance Stoch. 15, 655-683. · Zbl 1303.91166 |
| [5] | Belomestny, D. (2013): Solving Optimal Stopping Problems via Empirical Dual Optimization, Ann. Appl. Probab. 23, 1988-2019. · Zbl 1298.60049 |
| [6] | Belomestny, D., C.Bender, and J.Schoenmakers (2009): True Upper Bounds for Bermudan Products via Non‐Nested Monte Carlo, Math. Finance19, 53-71. · Zbl 1155.91376 |
| [7] | Belomestny, D., F.Dickmann, and J.Schoenmakers (2013): Multilevel Dual Approach for Pricing American Style Derivatives, Finance Stoch. 17, 717-742. · Zbl 1310.91142 |
| [8] | Bender, C., and R.Denk (2007): A Forward Scheme for Backward SDEs, Stoch. Process. Appl. 117, 1793-1812. · Zbl 1131.60054 |
| [9] | Bender, C., and J.Steiner (2012): Least‐Squares Monte Carlo for BSDEs, in Numerical Methods in Finance, R. A. Carmona, P. Del Moral, P. Hu, and N. Oudjane, eds. Berlin: Springer, 257-289. · Zbl 1269.91098 |
| [10] | Bender, C., and J.Steiner (2013): A‐Posteriori Estimates for Backward SDEs, SIAM/ASA J. Uncertainty Quantification1, 139-163. · Zbl 1282.65019 |
| [11] | Bergman, Y. Z. (1995): Option Pricing with Differential Interest Rates, Rev. Financ. Stud. 8, 475-500. |
| [12] | Bouchard, B., and J.‐F.Chassagneux (2008): Discrete‐Time Approximation for Continuously and Discretely Reflected BSDEs, Stoch. Process. Appl. 118, 2269-2293. · Zbl 1158.60030 |
| [13] | Bouchard, B., and R.Elie (2008): Discrete‐Time Approximation of Decoupled Forward‐Backward SDE with Jumps, Stoch. Process. Appl. 118, 53-75. · Zbl 1136.60048 |
| [14] | Bouchard, B., and N.Touzi (2004): Discrete‐Time Approximation and Monte Carlo Simulation of Backward Stochastic Differential Equations, Stoch. Process. Appl. 111, 175-206. · Zbl 1071.60059 |
| [15] | Brown, D. B., J. E.Smith, and P.Sun (2010): Information Relaxations and Duality in Stochastic Dynamic Programs, Oper. Res. 58, 785-801. · Zbl 1228.90062 |
| [16] | Chassagneux, J.‐F., and A.Richou (2016): Numerical Simulation of Quadratic BSDEs, Ann. Appl. Probab. 26(1), 262-304. · Zbl 1334.60129 |
| [17] | Cheridito, P., M.Kupper, and N.Vogelpoth (2015): Conditional Analysis on \(\mathbb{R}^d\), in Set Optimization and Applications in Finance—The State of the Art, A. Hamel, F. Heyde, A. Löhne, B. Rudloff, and C. Schrage, eds. Berlin: Springer. · Zbl 1338.49028 |
| [18] | Cheridito, P., and M.Stadje (2013): BSΔEs and BSDEs with Non‐Lipschitz Drivers: Comparison, Convergence and Robustness, Bernoulli19, 1047-1085. · Zbl 1306.60069 |
| [19] | Cohen, S. N., and R. J.Elliott (2010): A General Theory of Finite State Backward Stochastic Difference Equations, Stoch. Process. Appl. 120, 442-466. · Zbl 1205.60111 |
| [20] | Crépey, S., R.Gerboud, Z.Grbac, and N.Ngor (2013): Counterparty Risk and Funding: The Four Wings of the TVA, Int. J. Theor. Appl. Finance16, 1350006. · Zbl 1266.91115 |
| [21] | Crisan, D., and K.Manolarakis (2012): Solving Backward Stochastic Differential Equations Using the Cubature Method: Application to Nonlinear Pricing, SIAM J. Financ. Math. 3, 534-571. · Zbl 1259.65005 |
| [22] | Desai, V. V., V. F.Farias, and C. C.Moallemi (2012): Pathwise Optimization for Optimal Stopping Problems, Manage. Sci. 58, 2292-2308. |
| [23] | Di Nunno, G., B.Øksendal, and F.Proske (2009): Malliavin Calculus for Lévy Processes with Applications in Finance, Berlin: Springer. · Zbl 1528.60001 |
| [24] | Duffie, D., M.Schroder, and C.Skiadas (1996): Recursive Valuation of Defaultable Securities and the Timing of Resolution of Uncertainty, Ann. Appl. Probab. 6, 1075-1090. · Zbl 0868.90008 |
| [25] | El Karoui, N., S.Peng, and M.‐C.Quenez (1997): Backward Stochastic Differential Equations in Finance, Math. Finance7, 1-71. · Zbl 0884.90035 |
| [26] | Fahim, A., N.Touzi, and X.Warin (2011): A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs, Ann. Appl. Probab. 21, 1322-1364. · Zbl 1230.65009 |
| [27] | Glasserman, P., and B.Yu (2004): Simulation for American Options: Regression Now or Regression Later?, in Monte Carlo and Quasi‐Monte Carlo Methods 2002, H. Niederreiter, ed. Berlin: Springer, pp. 213-226. · Zbl 1062.91033 |
| [28] | Gobet, E., and C.Labart (2007): Error Expansion for the Discretization of Backward Stochastic Differential Equations, Stoch. Process. Appl. 117, 803-829. · Zbl 1117.60058 |
| [29] | Gobet, E., and A.Makhlouf (2010): L^2‐Time Regularity of BSDEs with Irregular Terminal Functions, Stoch. Process. Appl. 120, 1105-1132. · Zbl 1195.60079 |
| [30] | Guyon, J., and P.Henry‐Labordère (2011): Uncertain Volatility Model: A Monte Carlo Approach, J. Comput. Finance14, 37-71. |
| [31] | Haugh, M., and L.Kogan (2004): Pricing American Options: A Duality Approach, Oper. Res. 52, 258-270. · Zbl 1165.91401 |
| [32] | Henry‐Labordère, P. (2012): Cutting CVA’s Complexity, Risk Mag. 25(7), 67-73. |
| [33] | Johnson, H. (1987): Options on the Maximum or the Minimum of Several Assets, J. Financ. Quant. Anal. 22, 277-283. |
| [34] | Joshi, M. S. (2014): Analyzing the Bias in the Primal‐Dual Upper Bound Method for Early Exercisable Derivatives: Bounds, Estimation and Removal, SSRN eLibrary. |
| [35] | Joshi, M. S., and R.Tang (2014): Effective Sub‐Simulation‐Free Upper Bounds for the Monte Carlo Pricing of Callable Derivatives and Various Improvements to Existing Methodologies, J. Econ. Dyn. Control40, 25-45. · Zbl 1402.91893 |
| [36] | Laurent, J.‐P., P.Amzelek, and J.Bonnaud (2014): An Overview of the Valuation of Collateralized Derivative Contracts, Rev. Deriv. Res. 17, 261-286. · Zbl 1300.91050 |
| [37] | Lemor, J.‐P., E.Gobet, and X.Warin (2006): Rate of Convergence of an Empirical Regression Method for Solving Generalized Backward Stochastic Differential Equations, Bernoulli12, 889-916. · Zbl 1136.60351 |
| [38] | Ma, J., and J.Zhang (2005): Representations and Regularities for Solutions to BSDEs with Reflections, Stoch. Process. Appl. 115, 539-569. · Zbl 1076.60049 |
| [39] | Nualart, D. (2006): The Malliavin Calculus and Related Topics, 2nd ed., Berlin: Springer. · Zbl 1099.60003 |
| [40] | Pallavicini, A., D.Perini, and D.Brigo (2012): Funding, Collateral and Hedging: Uncovering the Mechanics and the Subtleties of Funding Valuation Adjustments, SSRN eLibrary. |
| [41] | Rockafellar, R. T. (1970): Convex Analysis, Princeton, NJ: Princeton University Press. · Zbl 0202.14303 |
| [42] | Rogers, L. C. G. (2002): Monte Carlo Valuation of American Options, Math. Finance12, 271-286. · Zbl 1029.91036 |
| [43] | Schoenmakers, J., J.Zhang, and J.Huang (2013): Optimal Dual Martingales, Their Analysis, and Application to New Algorithms for Bermudan Products, SIAM J. Financ. Math. 4, 86-116. · Zbl 1282.91346 |
| [44] | Zhang, J. (2004): A Numerical Scheme for BSDEs, Ann. Appl. Probab. 14, 459-488. · Zbl 1056.60067 |
| [45] | Zhang, G., M.Gunzburger, and W.Zhao (2013): A Sparse‐Grid Method for Multi‐Dimensional Backward Stochastic Differential Equations, J. Comput. Math. 31, 221-248. · Zbl 1289.65011 |
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