Deep Learning of Transition Probability Densities for Stochastic Asset Models with Applications in Option Pricing

References

• Aït-Sahalia Y (2002) Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70:223–262.Crossref, Google Scholar
• Aït-Sahalia Y (2008) Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36:906–937.Crossref, Google Scholar
• Aït-Sahalia Y, Kimmel R (2007) Maximum likelihood estimation of stochastic volatility models. J. Financial Econom. 83(2):413–452.Crossref, Google Scholar
• Aït-Sahalia Y, Yu J (2006) Saddlepoint approximations for continuous-time Markov processes. J. Econometrics 134(2):507–551.Crossref, Google Scholar
• Al-Aradi A, Correia A, Naiff D, Jardim G, Saporito Y (2018) Solving nonlinear and high-dimensional partial differential equations via deep learning. Preprint, submitted November 21, https://arxiv.org/abs/1811.08782.Google Scholar
• Andricopoulos AD, Widdicks M, Duck PW, Newton DP (2003) Universal option valuation using quadrature methods. J. Financial Econom. 67:447–471.Crossref, Google Scholar
• Andricopoulos AD, Widdicks M, Newton DP, Duck PW (2007) Extending quadrature methods to value multi-asset and complex path dependent options. J. Financial Econom. 83:471–499.Crossref, Google Scholar
• Antoulas AC, Ionutiu R, Martins N, ter Maten EJW, Mohaghegh K, Pulch R, Rommes J, et al. (2015) Model order reduction: Methods, concepts and properties. Günther M, ed. Coupled Multiscale Simulation and Optimization in Nanoelectronics (Springer, Berlin), 159–265.Crossref, Google Scholar
• Applebaum D (2009) Lévy Processes and Stochastic Calculus (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• Baron AR (1994) Approximation and estimation bounds for artificial neural networks. Machine Learn. 14(1):115–133.Crossref, Google Scholar
• Bates DS (1996) Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche Mark options. Rev. Financial Stud. 9(1):69–107.Crossref, Google Scholar
• Beskos A, Roberts GO (2005) Exact simulation of diffusions. Ann. Appl. Probability 15(4):2422–2444.Crossref, Google Scholar
• Black F, Scholes M (1973) The pricing of options and corporate liabilities. J. Political Econom. 81:637–654.Crossref, Google Scholar
• Chen N, Huang Z (2013) Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Oper. Res. 38(3):591–616.Link, Google Scholar
• Chen D, Härkönen HJ, Newton DP (2014) Advancing the universality of quadrature methods to any underlying process for option pricing. J. Financial Econom. 114:600–612.Crossref, Google Scholar
• Cox JC (1996) The constant elasticity of variance option pricing model. J. Portfolio Management 23:15–17.Crossref, Google Scholar
• Dissanayake M, Phan-Thien N (1994) Neural-network-based approximations for solving partial differential equations. Comm. Numerical Methods Engrg. 10:195–201.Crossref, Google Scholar
• Figlewski S (2009) Estimating the implied risk-neutral density for the US market portfolio. Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle (Oxford University Press, Oxford, UK).Google Scholar
• Figlewski S (2018) Risk-neutral densities: A review. Annu. Rev. Financial Econom. 10:329–359.Crossref, Google Scholar
• Filipović D, Mayerhofer E, Schneider P (2013) Density approximations for multivariate affine jump-diffusion processes. J. Econometrics 176:93–111.Crossref, Google Scholar
• Floc’h L, Kennedy GJ (2014) Finite difference techniques for arbitrage free SABR. Preprint, submitted May 8, https://dx.doi.org/10.2139/ssrn.2402001.Google Scholar
• Freidlin MI (1985) Functional Integration and Partial Differential Equations (Princeton University Press, Princeton, NJ).Google Scholar
• Garcke J, Griebel M (2013) Sparse Grids and Applications (Springer, Berlin).Crossref, Google Scholar
• Gardiner CW (2004) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin).Crossref, Google Scholar
• Gatheral J (2011) The Volatility Surface: A Practitioner’s Guide, vol. 357 (John Wiley & Sons, New York).Google Scholar
• Geist M, Petersen P, Raslan M, Schneider R, Kutyniok G (2020) Numerical solution of the parametric diffusion equation by deep neural networks. Preprint, submitted April 25, https://arxiv.org/abs/2004.12131.Google Scholar
• Gichman II, Skorochod AV (1972) Stochastic Differential Equations (Springer, Berlin).Crossref, Google Scholar
• Giesecke K, Schwenkler G (2019) Simulated likelihood estimators for discretely observed jump–diffusions. J. Econometrics 213(2):297–320.Crossref, Google Scholar
• Giesecke K, Smelov D (2013) Exact sampling of jump diffusions. Oper. Res. 61(4):894–907.Link, Google Scholar
• Glasserman P (2003) Monte Carlo Methods in Financial Engineering (Springer, Berlin).Crossref, Google Scholar
• Goodfellow I, Bengio Y, Courville A (2016) Deep Learning (MIT Press, Cambridge, MA).Google Scholar
• Guay F, Schwenkler G (2021) Efficient estimation and filtering for multivariate jump–diffusions. J. Econometrics 223(1):251–275.Crossref, Google Scholar
• Hagan PS, Kumar D, Lesniewski AS, Woodward DE (2002) Managing smile risk. Best Wilmott 1:249–296.Google Scholar
• Hagan PS, Kumar D, Lesniewski A, Woodward D (2014) Arbitrage-free SABR. Wilmott Magazine 2014:60–75.Crossref, Google Scholar
• Henry-Labordère P (2008) Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (CRC Press, Boca Raton, FL).Crossref, Google Scholar
• Henry-Labordere P, Tan X, Touzi N (2017) Unbiased simulation of stochastic differential equations. Ann. Appl. Probability 27(6):3305–3341.Crossref, Google Scholar
• Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Stud. 6:327–343.Crossref, Google Scholar
• Heston SL (1997) A simple new formula for options with stochastic volatility. Preprint, submitted September 19, https://dx.doi.org/10.2139/ssrn.86074.Google Scholar
• Hinds PD, Tretyakov MV (2023) Neural variance reduction for stochastic differential equations. J. Comput. Finance 27(3):1–41.Crossref, Google Scholar
• Hochreiter S, Schmidhuber J (1997) Long short-term memory. Neural Comput. 9:1735–1780.Crossref, Google Scholar
• Hutchinson JM, Lo AW, Poggio T (1994) A nonparametric approach to pricing and hedging derivative securities via learning networks. J. Finance 49:851–889.Crossref, Google Scholar
• Khoo Y, Lu J, Ying L (2017) Solving parametric PDE problems with artificial neural networks. Preprint, submitted July 11, https://arxiv.org/abs/1707.03351.Google Scholar
• Kingma DP, Ba J (2014) Adam: A method for stochastic optimization. Preprint, submitted December 22, https://arxiv.org/abs/1412.6980.Google Scholar
• Kou SG (2002) A jump-diffusion model for option pricing. Management Sci. 48(8):1086–1101.Link, Google Scholar
• Kutyniok G, Petersen P, Raslan M, Schneider R (2019) A theoretical analysis of deep neural networks and parametric PDEs. Preprint, submitted March 31, https://arxiv.org/abs/1904.00377.Google Scholar
• Ladyženskaja OA, Solonnikov VA, Ural’ceva NN (1968) Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23 (AMS, Providence, RI).Google Scholar
• Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neural Networks 9:987–1000.Crossref, Google Scholar
• Lagaris IE, Likas AC, Papageorgiou DG (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans. Neural Networks 11:1041–1049.Crossref, Google Scholar
• Lee H, Kang IS (1990) Neural algorithm for solving differential equations. J. Comput. Phys. 91:110–131.Crossref, Google Scholar
• Lewis AL (2016) Option Valuation Under Stochastic Volatility II (Finance Press, Newport Beach, CA).Google Scholar
• Lord R, Fang F, Bervoets F, Oosterlee CW (2008) A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes. SIAM J. Sci. Comput. 30:1678–1705.Crossref, Google Scholar
• Malliaris M, Salchenberger L (1993) Beating the best: A neural network challenges the Black-Scholes formula. Proc. 9th IEEE Conf. Artificial Intelligence Appl. (IEEE, New York), 445–449.Google Scholar
• Merton RC (1973) Theory of rational option pricing. Bell J. Econom. Management Sci. 4:141–183.Crossref, Google Scholar
• Milstein GN, Tretyakov MV (2003) The simplest random walks for the Dirichlet problem. Theory Probability Appl. 47:53–68.Crossref, Google Scholar
• Milstein GN, Tretyakov MV (2021) Stochastic Numerics for Mathematical Physics, 2nd ed. (Springer, Cham, Switzerland).Crossref, Google Scholar
• O’Sullivan C (2005) Path dependant option pricing under Lévy processes. Preprint, submitted February 25, https://dx.doi.org/10.2139/ssrn.673424.Google Scholar
• Rackauckas C, Ma Y, Martensen J, Warner C, Zubov K, Supekar R, Skinner D, et al. (2020) Universal differential equations for scientific machine learning. Preprint, submitted January 13, https://arxiv.org/abs/2001.04385.Google Scholar
• Rebonato R, McKay K, White R (2009) The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives (John Wiley & Sons, Hoboken, NJ).Google Scholar
• Schmidhuber J (2015) Deep learning in neural networks: An overview. Neural Networks 61:85–117.Crossref, Google Scholar
• Sirignano J, Spiliopoulos K (2018) DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375:1339–1364.Crossref, Google Scholar
• Smolyak S (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Doklady 4:240–243.Google Scholar
• Srivastava RK, Greff K, Schmidhuber J (2015) Training very deep networks. Preprint, submitted July 22, https://arxiv.org/abs/1507.06228.Google Scholar
• Steen N, Byrne G, Gelbard E (1969) Gaussian quadratures for the integrals ∫0∞exp(−x2)f(x)dx and ∫0bexp(−x2)f(x)dx. Math. Comput. 23:661–671.Google Scholar
• Su H, Newton DP (2020) Widening the range of underlyings for derivatives pricing with QUAD by using finite difference to calculate transition densities: Demonstrated for the no-arbitrage SABR model. J. Derivatives 28(2):22–46.Crossref, Google Scholar
• Su H, Chen D, Newton DP (2017) Option pricing via QUAD: From Black-Scholes-Merton to Heston with jumps. J. Derivatives 24:9–27.Crossref, Google Scholar
• van Milligen BP, Tribaldos V, Jiménez J (1995) Neural network differential equation and plasma equilibrium solver. Phys. Rev. Lett. 75:3594.Crossref, Google Scholar
• Yadav N, Yadav A, Kumar M (2015) An Introduction to Neural Network Methods for Differential Equations (Springer, Berlin).Crossref, Google Scholar
• Yu J (2007) Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese yuan. J. Econometrics 141(2):1245–1280.Crossref, Google Scholar
• Zhou Y (2022) Option trading volume by moneyness, firm fundamentals, and expected stock returns. J. Financial Marketing 58:100648.Crossref, Google Scholar