In this paper, we combine the theory of stochastic process and techniques of machine learning with the regression analysis, first proposed by [1] to solve for American option prices, and apply the new methodologies on financial derivatives pricing. Rigorous convergence proofs are provided for some of the methods we propose. Numerical examples show good applicability of the algorithms. More applications in finance are discussed in the Appendices.
Longstaff, F. and Schwartz, E. (2001) Valuing American Options by Simulation: A Simple Least—Square Approach. The Review of Financial Studies, 14, 113-147. https://doi.org/10.1093/rfs/14.1.113
El Karoui, N., Peng, S. and Quenez, M.C. (1997) Backward Stochastic Differential Equations in Finance. Mathematical Finance, 7, 1-71. https://doi.org/10.1111/1467-9965.00022
Adrian, T., Crump, R. and Vogt, E. (2018) Nonlinearity and Flight-to-Safety in the Risk-Return Trade-Off for Stocks and Bonds. Forthcoming in Journal of Finance, 74, 1931-1973.
Fama, E. and French, K. (1993) Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33, 3-56. https://doi.org/10.1016/0304-405X(93)90023-5
Fama, E. and French, K. (2015) A Five-Factor Asset Pricing Model. Journal of Financial Economics, 116, 1-22.
Zhu, S. and Pykhtin, M. (2008) A Guide to Modeling Counterparty Credit Risk. Working Paper. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1032522
Aydogdu, M. (2018) Predicting Stock Returns Using Neural Networks. Working Paper. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3141492 https://doi.org/10.2139/ssrn.3141492
Voshgha, H. (2008) Early Detection of Defaulting Firms: Artificial Neural Network Application; Australian Context. Working Paper. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2130505
Hutchinson, J., Lo, A. and Poggio, T. (1994) A Nonparametric Approach to Pricing and Hedging Derivative Securities via Learning Networks. Journal of Finance, 49, 851-889. https://doi.org/10.1111/j.1540-6261.1994.tb00081.x
Hahn, J.T. (2013) Option Pricing Using Artificial Neural Networks: The Australian Perspective. Ph.D. Thesis, Bond University, Queensland.
Kohler, M., Krzyzak, M. and Todorovic, N. (2010) Pricing of High-Dimensional American Options by Neural Networks. Mathematical Finance, 20, 383-410. https://doi.org/10.1111/j.1467-9965.2010.00404.x
Dugas, C., Bengio, Y., Bélisle, F., Nadeau, C. and Garcia, R. (2009) Incorporating Functional Knowledge in Neural Networks. Journal of Machine Learning Research, 10, 1239-1262.
Eckstein, S., Kupper, M. and Pohl, M. (2018) Robust Risk Aggregation with Neural Networks. Quantitative Finance, 1-40. https://arxiv.org/abs/1811.00304
Giovanis, E. (2010) Applications of Neural Network Radial Basis Function in Economics and Financial time Series. SSRN Electronic Journal. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1667442 https://doi.org/10.2139/ssrn.1667442
Kopitkov, D. and Indelman, V. (2018) Deep PDF: Probabilistic Surface Optimization and Density Estimation. Computer Science, 1-18. https://arxiv.org/abs/1807.10728
Luo, R., Zhang, W., Xu, X. and Wang, J. (2017) A Neural Stochastic Volatility Model. Computer Science, 1-11. https://arxiv.org/pdf/1712.00504.pdf
Sasaki, H. and Hyvarinen, A. (2018) Neural-Kernelized Conditional Density Estimation. Statistics, 1-12. https://arxiv.org/abs/1806.01754
Weissensteiner, A. (2009) AQ-Learning Approach to Derive Optimal Consumption and Investment Strategies. IEEE Transactions on Neural Networks, 20, 1234-1243. https://doi.org/10.1109/TNN.2009.2020850
Casgrain, P. and Jaimungal, S. (2016) Trading Algorithms with Learning in Latent Alpha Models. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.2871403 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2871403
Heaton, J., Polson, N. and Witte, J. (2016) Deep Learning for Finance: Deep Portfolios. Applied Stochastic Models in Business and Industry, 33, 3-12. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2838013 https://doi.org/10.2139/ssrn.2838013
Samo, Y. and Vernuurt, A. (2016) Stochastic Portfolio Theory: A Machine Learning Perspective. Quantitative Finance, 1-9. https://arxiv.org/pdf/1605.02654.pdf
Jiang, Z., Xu, D. and Liang, J. (2017) A Deep Reinforcement Learning Framework for the Financial Portfolio Management Problem. Computational Finance, 1-31. https://arxiv.org/pdf/1706.10059.pdf
Deng, Y., Bao, F., Kong, Y., Ren, Z. and Dai, Q. (2017) Deep Direct Reinforcement Learning for Financial Signal Representation and Trading. IEEE Transactions on Neural Networks and Learning Systems, 28, 653-664. https://doi.org/10.1109/TNNLS.2016.2522401
Halperin, I. (2017) QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds. Quantitative Finance, 1-34. https://arxiv.org/abs/1712.04609v2 https://doi.org/10.2139/ssrn.3087076
Ritter, G. (2017) Machine Learning for Trading. Working Paper. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3015609 https://doi.org/10.2139/ssrn.3015609
Xing, F., Cambrida, E., Malandri, L. and Vercellis, C. (2018) Discovering Bayesian Market Views for Intelligent Asset Allocatio. https://arxiv.org/pdf/1802.09911.pdf
Becker, S., Cheridito, P. and Jentzen, A. (2018) Deep Optimal Stopping. Mathematics, arXiv: 1804. 05394. https://arxiv.org/abs/1804.05394
Gu, S., Kelly, B. and Xiu, D. (2018) Empirical Asset Pricing via Machine Learning. 31st Australasian Finance and Banking Conference 2018, Sydney, 13-15 December 2018. https://doi.org/10.3386/w25398 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3159577
Weinan, E., Han, J. and Jentzen, A. (2017) Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations. Mathematics, 1-39. https://arxiv.org/pdf/1706.04702.pdf
Weinan, E., Hutzenthaler, M., Jentzen, A. and Kruse, T. (2017) On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations. Mathematics, 1-25. https://arxiv.org/pdf/1708.03223.pdf
Han, J., Jentzen, A. and Weinan, E. (2017) Overcoming the Curse of Dimensionality: Solving High-Dimensional Partial Differential Equations Using Deep Learning. Mathematics, 1-14. https://arxiv.org/pdf/1707.02568.pdf
Khoo, Y., Lu, J. and Ying, L. (2017) Solving Parametric PDE Problems with Artificial Neural Networks. Mathematics, 1-17. https://arxiv.org/pdf/1707.03351.pdf
Beck, C., Weinan, E. and Jentzen, A. (2017) Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-Order Backward Stochastic Differential Equations. Mathematics, 1-56. https://arxiv.org/pdf/1709.05963.pdf
Sirignano, J. and Spiliopoulos, K. (2017) DGM: A Deep Learning Algorithm for Solving Partial Differential Equations. Mathematics, 1-31. https://arxiv.org/pdf/1708.07469.pdf
Long, Z., Lu, Y. and Ma, X. (2018) PDE-Net: Learning PDEs from Data. Mathematics, 1-17. https://arxiv.org/pdf/1710.09668.pdf
Long, Z. and Lu, Y. (2018) PDE-Net 2.0: Learning PDEs from Data with a Numeric Symbolic Hybrid Deep Network. Computer Science, 1-16. https://arxiv.org/pdf/1812.04426.pdf
Haehnel, P., Marecek, J. and Monteil, J. (2018) Scaling up Deep Learning for PDE-Based Models. Computer Science, 1-39. https://arxiv.org/pdf/1810.09425.pdf
Berg, J. and Nystrom, K. (2018) Data-Driven Discovery of PDEs in Complex Datasets. Statistics, 1-22. https://arxiv.org/pdf/1808.10788.pdf
Rudy, S., Alla, A., Brunton, S. and Nathan Kutz, J. (2018) Data-Driven Identification of Parametric Partial Differential Equations. Mathematics, 1-17. https://arxiv.org/pdf/1806.00732.pdf
Detemple, J., Lorig, M., Rindisbacher, M. and Zhang, L. (2018) An Analytical Expansion Method for Forward Backwards to Chastic Differential Equations with Jumps.
Briand, P. and Labart, C. (2012) Simulation of BSDEs by Wiener Chaos Expansion. The Annals of Applied Probability, 24, 1129-1171. https://doi.org/10.1214/13-AAP943
Geiss, C. and Labart, C. (2015) Simulation of BSDEs with Jumps by Wiener Chaos Expansion. Mathematics, arXiv: 1502.05649. http://arxiv.org/abs/1502.05649
Gnameho, K., Stadje, M. and Pelsser, A. (2017) A Regression-Later Algorithm for Backward Stochastic Differential Equations. Mathematics, 1-33. https://arxiv.org/pdf/1706.07986
Gobet, E. and Labart, C. (2007) Error Expansion for the Discretization of Backward Stochastic Differential Equations. Stochastic Processes and Their Applications, 117, 803-829. https://doi.org/10.1016/j.spa.2006.10.007
Takahashi, A. and Yamada, T. (2016) An Asymptotic Expansion for Forward-Backward SDEs: A Malliavin Calculus Approach. Asia-Pacific Financial Markets, 23, 337-373.
Takahashi, A. and Yamada, T. (2015) On the Expansion to Quadratic FBSDEs.
Gobet, E. and Pagliarani, S. (2014) Analytical Approximations of BSDEs with Non-Smooth Driver. SIAM Journal on Financial Mathematics, 6, 919-958. https://doi.org/10.2139/ssrn.2448691
Fujii, M. and Takahashi, A. (2012) Analytical Approximation for Non-Linear FBSDEs with Perturbation Scheme. International Journal of Theoretical and Applied Finance, 15, Article ID: 1250034. https://doi.org/10.1142/S0219024912500343
Fujii, M. and Takahashi, A. (2012) Perturbative Expansion of FBSDE in an Incomplete Market with Stochastic Volatility. The Quarterly Journal of Finance, 2, 1-22. https://doi.org/10.2139/ssrn.1999137
Fujii, M. and Takahashi, A. (2015) Asymptotic Expansion for Forward-Backward SDEs with Jumps. Quantitative Finance, 1-39. https://doi.org/10.2139/ssrn.2672890 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2672890
Fujii, M. and Takahashi, A. (2016) Quadratic-Exponential Growth BSDEs with Jumps and Their Malliavin’s Differentiability. Working Paper. https://doi.org/10.2139/ssrn.2705670 http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2705670
Fujii, M. and Takahashi, A. (2016) Solving Backward Stochastic Differential Equations by Connecting the Short-Term Expansions. Quantitative Finance, 1-41. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2795490
Detemple, J. and Rindisbacher, M. (2005) Closed-Form Solutions for Optimal Portfolio Selection with Stochastic Interest Rate and Investment Constraints. Mathematical Finance, 15, 539-568. https://doi.org/10.1111/j.1467-9965.2005.00250.x
Hansen, L. and Richard, S. (1987) The Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models. Econometrica, 55, 587-613. https://doi.org/10.2307/1913601
Jiang, J. and Tian, W. (2018) Semi-Nonparametric Approximation and Index Options. Annals of Finance, 1-38. https://doi.org/10.1007/s10436-018-0341-4
Tian, W. (2014) Spanning with Indexes. Journal of Mathematical Economics, 53, 111-118. https://doi.org/10.1016/j.jmateco.2014.06.007
Tian, W. (2018) The Financial Market: Not as Big as You Think. Mathematics and Financial Economics, 51, 1-19.
Bolcskei, H., Grohs, P., Kutyniok, G. and Petersen, P. (2018) Optimal Approximation with Sparsely Connected Deep Neural Networks. Computer Science, 1-36. https://arxiv.org/abs/1705.01714
Henry-Labordere, P. (2015) Exact Simulation of Multi-Dimensional Stochastic Differential Equations. Working Paper, 1-28. https://doi.org/10.2139/ssrn.2598505 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2598505
Prater, A. (2012) Discrete Sparse Fourier Hermite Approximations in High Dimensions. Doctoral Thesis, Syracuse University, New York.
Fonseca, Y., Medeiros, M., Vasconcelos, G. and Veiga, A. (2018) Boost: Boosting Smooth Trees for Partial Effect Estimation in Nonlinear Regressions. Statistics, 1-30. https://arxiv.org/pdf/1808.03698.pdf
Detemple, J. (2006) American-Style Derivatives: Valuation and Computation. Chapman and Hall/CRC, New York. https://doi.org/10.1201/9781420034868
Guyon, J. and Henry-Labordere, P. (2014) Nonliner Option Pricing. Chapman and Hall, New York. https://doi.org/10.1201/b16332
Detemple, J., Garcia, R. and Rindisbacher, M. (2005) Representation Formulas for Malliavin Derivatives of Diffusion Processes. Finance and Stochastics, 9, 349-367. https://doi.org/10.1007/s00780-004-0151-6
Detemple, J. and Rindisbacher, M. (2005) Asymptotic Properties of Monte Carlo Estimators of Derivatives. Management Science, 51, 1657-1675. https://doi.org/10.1287/mnsc.1050.0398
Detemple, J. (2014) Optimal Exercise for Derivative Securities. Annual Review of Financial Economics, 6, 459-487. https://doi.org/10.1146/annurev-financial-110613-034241
Fujii, M., Sato, S. and Takahashi, A. (2012) An FBSDE Approach to American Option Pricing with an Interacting Particle Method. Quantitative Finance, 1-18. https://arxiv.org/abs/1211.5867 https://doi.org/10.2139/ssrn.2180696
Chassagneux, J., Elie, R. and Kharroubi, I. (2010) A Note on Existence and Uniqueness for Solutions of Multidimensional Reflected BSDEs. Electronic Communications in Probability, 16, 120-128. https://doi.org/10.1214/ECP.v16-1614
Collin-Dufresne, P. and Goldstein, R. (2003) Generalizing the Affine Framework to HJM and Random Field Models. SSRN Electronic Journal. https://doi.org/10.2139/ssrn.410421 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=410421
Carmona, R. and Delarue, F. (2015) Forward-Backward Stochastic Differential Equations and Controlled McKean-Vlasov Dynamics. Annals of Probability, 43, 2647-2700. https://doi.org/10.1214/14-AOP946
Bianchi, D., Büchner, M. and Tamoni, A. (2019) Bond Risk Premia with Machine Learning. USC-INET Research Paper No. 19-11. https://doi.org/10.2139/ssrn.3400941 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3232721
Chen, L., Pelger, M. and Zhu, J. (2019) Deep Learning in Asset Pricing. Quantitative Finance, 1-89. https://arxiv.org/abs/1904.00745 https://doi.org/10.2139/ssrn.3350138
Feng, G., Polson, N. and Xu, J. (2019) Deep Learning in Asset Pricing. Statistics, 1-33. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3350138
Yang, Q., Ye, T. and Zhang, L. (2018) A General Framework of Optimal Investment. Working Paper. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3136708
Yu, P., Lee, J., Kulyatin, I., Shi, Z. and Dasgupta, S. (2019) Model-Based Deep Reinforcement Learning for Dynamic Portfolio Optimization. Computer Science, 1-21. https://arxiv.org/abs/1901.08740
Kingma, D. and Ba, J.L. (2014) Adam: A Method for Stochastic Optimization. Computer Science, 1-15. https://arxiv.org/abs/1412.6980
Heston, S. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6, 327-343. https://doi.org/10.1093/rfs/6.2.327
Dupire, B. (1994) Pricing with a Smile. Risk. http://www.risk.net/data/risk/pdf/technical/2007/risk20_0707_technical_volatility.pdf
Homescu, C. (2014) Local Stochastic Volatility Models: Calibration and Pricing. Working Paper. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2448098 https://doi.org/10.2139/ssrn.2448098
Broadie, M., Chernov, M. and Johannes, M. (2007) Model Specification and Risk Premia: Evidence from futures Options. Journal of Finance, 62, 1453-1490. https://doi.org/10.1111/j.1540-6261.2007.01241.x
Guennon, H. (2016) Local Volatility Models Enhanced with Jumps. Working Paper, 1-11. https://papers.ssrn.com/abstract=2781102 https://doi.org/10.2139/ssrn.2781102
Buehler, H., Gonon, L., Teichmann, J. and Wood, B. (2018) Deep Hedging. Working Paper. https://doi.org/10.2139/ssrn.3120710 https://arxiv.org/abs/1802.03042
Halperin, I. (2018) The QLBS Q-Learner Goes NuQLear: Fitted Q Iteration, Inverse RL, and Option Portfolios. Quantitative Finance, 1-18. https://arxiv.org/abs/1801.06077 https://doi.org/10.2139/ssrn.3102707
Halperin, I. (2018) QLBS: Q-Learner in the Black-Scholes(-Merton) Worlds. Quantitative Finance, 1-34. https://arxiv.org/abs/1712.04609 https://doi.org/10.2139/ssrn.3087076
Schroder, M. and Skiadas, C. (2008) Optimality and State Pricing in Constrained Financial Markets with Recursive Utility under Continuous and Discontinuous Information. Mathematical Finance, 18, 199-238. https://doi.org/10.1111/j.1467-9965.2007.00330.x
Detemple, J. and Zapatero, F. (1991) Asset Prices in an Exchange Economy with Habit Formation. Econometrica, 59, 1633-1657. https://doi.org/10.2307/2938283
Karatzas, I., Lehoczky, J., Shreve, S. and Xu, G. (1991) Martingale and Duality Methods for Utility Maximization in a Incomplete Market. SIAM Journal on Control and Optimization, 29, 702-730. https://doi.org/10.1137/0329039
He, H. and Pearson, N. (1991) Consumption and Portfolio Policies with Incomplete Markets and Short-Sale Constraints: The Infinite Dimensional Case. Journal of Economic Theory, 54, 259-304. https://doi.org/10.1016/0022-0531(91)90123-L
Karatzas, I. and Cvitanic, J. (1992) Convex Duality in Constrained Portfolio Optimization. Annals of Applied Probability, 2, 767-818. https://doi.org/10.1214/aoap/1177005576
Detemple, J., Garcia, R. and Rindisbacher, M. (2003) A Monte Carlo Method for Optimal Portfolios. Journal of Finance, 58, 401-446. https://doi.org/10.1111/1540-6261.00529
Detemple, J., Garcia, R. and Rindisbacher, M. (2005) Intertemporal Asset Allocation: A Comparison of Methods. Journal of Banking and Finance, 29, 2821-2848. https://doi.org/10.1016/j.jbankfin.2005.02.004
Detemple, J. and Rindisbacher, M. (2010) Dynamic Asset Allocation: Portfolio Decomposition Formula and Applications. The Review of Financial Studies, 23, 25-100. https://doi.org/10.1093/rfs/hhp040
Detemple, J. (2012) Portfolio Selection: A Review. Journal of Optimization Theory and Applications, 161, 1-21. https://doi.org/10.1007/s10957-012-0208-1
Matoussi, A. and Xing, H. (2016) Convex Duality for Stochastic Differential Utility. Quantitative Finance, 1-22. http://arxiv.org/pdf/1601.03562.pdf https://doi.org/10.2139/ssrn.2715425
Kraft, H., Seiferling, T. and Seifried, F. (2015) Optimal Consumption and Investment with Epstein-Z in Recursive Utility. Working Paper. https://doi.org/10.2139/ssrn.2444747 http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2424706
Ait-Sahalia, Y. (2008) Closed-Form Likelihood Expansions for Multivariate Diffusions. Annals of Statistics, 36, 906-937. https://doi.org/10.1214/009053607000000622
Filipovic, D., Mayerhofer, E. and Schneider, P. (2013) Density Approximations for Multivariate Affine Jump Diffusion Processes. Journal of Econometrics, 176, 93-111. https://doi.org/10.1016/j.jeconom.2012.12.003
Van Handel, R. (2008) Hidden Markov Models. Princeton Lecture Notes.
Markowitz, H. (1952) Portfolio Selection. Journal of Finance, 7, 77-91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
Schneider, P. and Trojani, F. (2018) (Almost) Model Free Recovery. Forthcoming in Journal of Finance, 74, 323-370. https://doi.org/10.1111/jofi.12737
Chabakauri, G. (2013) Dynamic Equilibrium with Two Stocks, Heterogeneous Investors, and Portfolio Constraints. The Review of Financial Studies, 26, 3104-3141. https://doi.org/10.2139/ssrn.2221073 http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2221073
Chabakauri, G. (2015) Asset Pricing with Heterogeneous Preferences, Beliefs, and Portfolio Constraints. Journal of Monetary Economics, 75, 21-34.
Kardaras, C., Xing, H. and Zitkovic, G. (2015) Incomplete Stochastic Equilibria for Dynamic Monetary Utility. Mathematics, 1-33. https://arxiv.org/abs/1505.07224
Halle, J.O. (2010) Backward Stochastic Differential Equations with Jumps. Master Thesis, University of Oslo, Oslo, Norway.
Dalderop, J. (2016) Nonparametric State-Price Density Estimation Using High Frequency Data. Working Paper. https://doi.org/10.2139/ssrn.2718938