Numerical Analysis of Novel Finite Difference Methods

The core target of this chapter is numerical analysis and computing of novel finite difference methods related to several different option pricing models, including jump-diffusion, regime switching and multi-asset options. A special attention is paid to positivity, consistency and stability of the proposed methods. The consideration of jump processes leads to partial integro-differential equation (PIDE) for the European option pricing problem. The problem is solved by using quadrature formulas for the approximation of the integrals and matching the discretization of the integral and differential part of the PIDE problem. More complicated model under assumption that the volatility is a stochastic process derives to a PIDE problem where the volatility is also an independent variable. Such a problem is solved by introducing appropriate change of variables. Moreover, American options are considered proposing various front-fixing transformations to treat a free boundary. This free boundary challenge can be treated also by a recent rationality parameter approach that takes into account the irrational behavior of the market. Dealing with multidimensional problems the core difficulty is the appearance of the cross derivative terms. Appropriate transformations allow eliminating the cross derivative terms and reduce of the computational cost and the numerical instabilities. After using a semidiscretization approach, the time exponential integration method and appropriate quadrature integration formulas, the stability of the proposed method is studied independent to the problem dimension.

[1]  Y. Kwok.  Mathematical models of financial derivatives , 2008 .

[2]  Rama Cont,et al.  A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models , 2005, SIAM J. Numer. Anal..

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

[4]  Jari Toivanen,et al.  Numerical Valuation of European and American Options under Kou's Jump-Diffusion Model , 2008, SIAM J. Sci. Comput..

[5]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[6]  Peter A. Forsyth,et al.  Numerical convergence properties of option pricing PDEs with uncertain volatility , 2003 .

[7]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Thephlx Deutschemark Options , 1993 .

[8]  Curt Randall,et al.  Pricing Financial Instruments: The Finite Difference Method , 2000 .

[9]  Lucas Jódar,et al.  A new efficient numerical method for solving American option under regime switching model , 2016, Comput. Math. Appl..

[10]  M.-C. Casabán,et al.  Removing the Correlation Term in Option Pricing Heston Model: Numerical Analysis and Computing , 2013 .

[11]  G. Smith,et al.  Numerical Solution of Partial Differential Equations: Finite Difference Methods , 1978 .

[12]  Paul Garabedian,et al.  Partial Differential Equations , 1964 .

[13]  José Vicente Romero,et al.  Double Discretization Difference Schemes for Partial Integrodifferential Option Pricing Jump Diffusion Models , 2012 .

[14]  Lina von Sydow,et al.  Iterative Methods for Pricing American Options under the Bates Model , 2013, ICCS.

[15]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[16]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[17]  K. J. in 't Hout,et al.  Stability of ADI schemes for multidimensional diffusion equations with mixed derivative terms , 2012 .

[18]  J. Bunch,et al.  Some stable methods for calculating inertia and solving symmetric linear systems , 1977 .

[19]  J. Crank.  Free and moving boundary problems , 1984 .

[20]  Daniel J. Duffy,et al.  The Alternating Direction Explicit (ADE) Method for One‐Factor Problems , 2011 .

[21]  C.C.W. Leentvaar.  Pricing multi-asset options with sparse grids , 2008 .

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

[23]  Daniel Sevcovic,et al.  Analysis of the free boundary for the pricing of an American call option , 2001, European Journal of Applied Mathematics.

[24]  Iris R. Wang,et al.  Robust numerical valuation of European and American options under the CGMY process , 2007 .

[25]  J. Helsing.  Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial , 2012, 1207.6737.

[26]  R. H. Liu,et al.  Pricing American options under multi-state regime switching with an efficient L- stable method , 2015, Int. J. Comput. Math..

[27]  N. Higham.  Stability of the Diagonal Pivoting Method with Partial Pivoting , 1997, SIAM J. Matrix Anal. Appl..

[28]  D. Duffy,et al.  Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems , 2009 .

[29]  R. Company,et al.  Positive finite difference schemes for a partial integro-differential option pricing model , 2014, Appl. Math. Comput..

[30]  Peter A. Forsyth,et al.  Negative coefficients in two-factor option pricing models , 2003 .

[31]  Christof Heuer,et al.  High-Order Compact Schemes for Parabolic Problems with Mixed Derivatives in Multiple Space Dimensions , 2014, SIAM J. Numer. Anal..

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

[33]  R. Cont,et al.  Financial Modelling with Jump Processes , 2003 .

[34]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[35]  Vera N. Egorova,et al.  Solving American Option Pricing Models by the Front Fixing Method: Numerical Analysis and Computing , 2014 .

[36]  Lucas Jódar,et al.  Positive Solutions of European Option Pricing with CGMY Process Models Using Double Discretization Difference Schemes , 2013 .

[37]  J. L. Pedersen,et al.  Rationality Parameter for Exercising American Put , 2014 .

[38]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations, Second Edition , 2004 .

[39]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[40]  Lucas Jódar,et al.  Constructing positive reliable numerical solution for American call options: A new front-fixing approach , 2016, J. Comput. Appl. Math..

[41]  J. Teugels,et al.  Lévy processes, polynomials and martingales , 1998 .

[42]  Aslak Tveito,et al.  Penalty methods for the numerical solution of American multi-asset option problems , 2008 .

[43]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[44]  M. Yor,et al.  Representing the CGMY and Meixner Lévy processes as time changed Brownian motions , 2008 .

[45]  Muddun Bhuruth,et al.  High-order computational methods for option valuation under multifactor models , 2013, Eur. J. Oper. Res..

[46]  Domingo Tavella.  Quantitative Methods in Derivatives Pricing , 2002 .

[47]  S. Cox,et al.  Exponential Time Differencing for Stiff Systems , 2002 .

[48]  Lucas Jódar,et al.  Solving partial integro-differential option pricing problems for a wide class of infinite activity Lévy processes , 2016, J. Comput. Appl. Math..

[49]  Matthias Ehrhardt,et al.  A FAST, STABLE AND ACCURATE NUMERICAL METHOD FOR THE BLACK–SCHOLES EQUATION OF AMERICAN OPTIONS , 2008 .

[50]  Gabriel Wittum,et al.  Efficient Hierarchical Approximation of High-Dimensional Option Pricing Problems , 2007, SIAM J. Sci. Comput..

[51]  A. Tveito,et al.  Penalty and front-fixing methods for the numerical solution of American option problems , 2002 .

[52]  C.-H. Lai,et al.  Moving boundary transformation for American call options with transaction cost: finite difference methods and computing , 2017, Int. J. Comput. Math..

[53]  S. Ross,et al.  The valuation of options for alternative stochastic processes , 1976 .

[54]  Svetlana Boyarchenko,et al.  OPTION PRICING FOR TRUNCATED LÉVY PROCESSES , 2000 .

[55]  H. G. Landau,et al.  Heat conduction in a melting solid , 1950 .

[56]  Fazlollah Soleymani,et al.  A mixed derivative terms removing method in multi-asset option pricing problems , 2016, Appl. Math. Lett..

[57]  Bertram Düring,et al.  High-Order Compact Finite Difference Scheme for Option Pricing in Stochastic Volatility Models , 2010, J. Comput. Appl. Math..

[58]  Ansgar Jüngel,et al.  Convergence of a High-Order Compact Finite Difference Scheme for a Nonlinear Black-Scholes Equation , 2004 .

[59]  Daniel Sevcovic,et al.  Transformation Methods for Evaluating Approximations to the Optimal Exercise Boundary for Linear and Nonlinear Black-Scholes Equations , 2008, 0805.0611.

[60]  Tim Sauer,et al.  Computational solution of stochastic differential equations , 2013 .

[61]  Peter A. Forsyth,et al.  Numerical solution of two asset jump diffusion models for option valuation , 2008 .

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

[63]  Mark Bowen,et al.  ADI schemes for higher-order nonlinear diffusion equations , 2003 .