An iterative splitting method for pricing European options under the Heston model

In this paper, we propose an iterative splitting method to solve the partial differential equations in option pricing problems. We focus on the Heston stochastic volatility model and the derived two-dimensional partial differential equation (PDE). We take the European option as an example and conduct numerical experiments using different boundary conditions. The iterative splitting method transforms the two-dimensional equation into two quasi one-dimensional equations with the variable on the other dimension fixed, which helps to lower the computational cost. Numerical results show that the iterative splitting method together with an artificial boundary condition (ABC) based on the method by Li and Huang (2019) gives the most accurate option price and Greeks compared to the classic finite difference method with the commonly-used boundary conditions in Heston (1993).

[1]  P. Glasserman,et al.  Monte Carlo methods for security pricing , 1997 .

[2]  Guido Germano,et al.  Fluctuation identities with continuous monitoring and their application to the pricing of barrier options , 2018, Eur. J. Oper. Res..

[3]  Jinghong Shu,et al.  PRICING S&P 500 INDEX OPTIONS UNDER STOCHASTIC VOLATILITY WITH THE INDIRECT INFERENCE METHOD , 2004 .

[4]  Jari Toivanen,et al.  Operator splitting methods for pricing American options under stochastic volatility , 2009, Numerische Mathematik.

[5]  Hongshan Li,et al.  Artificial boundary method for the solution of pricing European options under the Heston model , 2019, ArXiv.

[6]  Alan G. White,et al.  Valuing Derivative Securities Using the Explicit Finite Difference Method , 1990, Journal of Financial and Quantitative Analysis.

[7]  S. Heston A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[8]  E. Stein,et al.  Stock Price Distributions with Stochastic Volatility: An Analytic Approach , 1991 .

[9]  Marc Yor,et al.  Time Changes for Lévy Processes , 2001 .

[10]  P. Forsyth,et al.  PDE methods for pricing barrier options , 2000 .

[11]  G. Papanicolaou,et al.  MEAN-REVERTING STOCHASTIC VOLATILITY , 2000 .

[12]  E. Ghysels,et al.  A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation , 2000 .

[13]  S. Ross,et al.  Option pricing: A simplified approach☆ , 1979 .

[14]  D. Shanno,et al.  Option Pricing when the Variance Is Changing , 1987, Journal of Financial and Quantitative Analysis.

[15]  Carl Chiarella,et al.  The evaluation of barrier option prices under stochastic volatility , 2012, Comput. Math. Appl..

[16]  L. Bachelier,et al.  Théorie de la spéculation , 1900 .

[17]  T. Chan Pricing contingent claims on stocks driven by Lévy processes , 1999 .

[18]  P. Carr,et al.  Option valuation using the fast Fourier transform , 1999 .

[19]  Peter A. Forsyth,et al.  A finite element approach to the pricing of discrete lookbacks with stochastic volatility , 1999 .

[20]  M. Yor,et al.  Stochastic Volatility for Lévy Processes , 2003 .

[21]  K. I. '. Hout,et al.  ADI finite difference schemes for option pricing in the Heston model with correlation , 2008, 0811.3427.

[22]  P. Boyle A Lattice Framework for Option Pricing with Two State Variables , 1988, Journal of Financial and Quantitative Analysis.

[23]  James B. Wiggins Option values under stochastic volatility: Theory and empirical estimates , 1987 .

[24]  Bo Wang,et al.  Unified Approach for the Affine and Non-affine Models: An Empirical Analysis on the S&P 500 Volatility Dynamics , 2019 .

[25]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[26]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[27]  Bruno Dupire Pricing with a Smile , 1994 .

[28]  Karol Mikula,et al.  Diamond--cell finite volume scheme for the Heston model , 2015 .

[29]  Gunter Winkler,et al.  Valuation of Options in Heston's Stochastic Volatility Model Using Finite Element Methods 1 , 2001 .

[30]  Eduardo S. Schwartz,et al.  Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis , 1977 .

[31]  P. Hagan,et al.  MANAGING SMILE RISK , 2002 .

[32]  Robert C. Merton,et al.  Applications of Option-Pricing Theory: Twenty-Five Years Later , 1997 .

[33]  Cornelis W. Oosterlee,et al.  A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions , 2008, SIAM J. Sci. Comput..

[34]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[35]  A Gu,et al.  Breaking down barriers , 2018, Nature Astronomy.

[36]  P. Boyle Options: A Monte Carlo approach , 1977 .

[37]  Liming Feng,et al.  PRICING DISCRETELY MONITORED BARRIER OPTIONS AND DEFAULTABLE BONDS IN LÉVY PROCESS MODELS: A FAST HILBERT TRANSFORM APPROACH , 2008 .

[38]  Guido Germano,et al.  Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options , 2016, Eur. J. Oper. Res..

[39]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[40]  Jari Toivanen,et al.  Operator splitting methods for American option pricing , 2004, Appl. Math. Lett..