A fast numerical method for the valuation of American lookback put options

Abstract A fast and efficient numerical method is proposed and analyzed for the valuation of American lookback options. American lookback option pricing problem is essentially a two-dimensional unbounded nonlinear parabolic problem. We reformulate it into a two-dimensional parabolic linear complementary problem (LCP) on an unbounded domain. The numeraire transformation and domain truncation technique are employed to convert the two-dimensional unbounded LCP into a one-dimensional bounded one. Furthermore, the variational inequality (VI) form corresponding to the one-dimensional bounded LCP is obtained skillfully by some discussions. The resulting bounded VI is discretized by a finite element method. Meanwhile, the stability of the semi-discrete solution and the symmetric positive definiteness of the full-discrete matrix are established for the bounded VI. The discretized VI related to options is solved by a projection and contraction method. Numerical experiments are conducted to test the performance of the proposed method.

[1]  Haiming Song,et al.  Front-fixing FEMs for the pricing of American options based on a PML technique , 2015 .

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

[3]  Cornelis W. Oosterlee,et al.  The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation , 2014 .

[4]  Ken Seng Tan,et al.  Pricing Options Using Lattice Rules , 2005 .

[5]  T Zhang.  THE NUMERICAL METHODS FOR AMERICAN OPTION PRICING , 2002 .

[6]  Pierre L'Ecuyer,et al.  Efficient Monte Carlo and Quasi - Monte Carlo Option Pricing Under the Variance Gamma Model , 2006, Manag. Sci..

[7]  Lishang Jiang.  Mathematical Modeling and Methods of Option Pricing , 2005 .

[8]  Haiming Song,et al.  Weak Galerkin finite element method for valuation of American options , 2014 .

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

[10]  G. Barone-Adesi,et al.  Efficient Analytic Approximation of American Option Values , 1987 .

[11]  Zhu,et al.  FINITE DIFFERENCE APPROXIMATION FOR PRICING THE AMERICAN LOOKBACK OPTION , 2009 .

[12]  Min Dai,et al.  American Options with Lookback Payoff , 2004 .

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

[14]  Rajae Aboulaich,et al.  Simulation of European Lookback Options , 2012 .

[15]  B. He.  A class of projection and contraction methods for monotone variational inequalities , 1997 .

[16]  Peter A. Forsyth,et al.  Convergence of numerical methods for valuing path-dependent options using interpolation , 2002 .

[17]  Song‐Ping Zhu.  An exact and explicit solution for the valuation of American put options , 2006 .

[18]  Min Dai.  A Modified Binomial Tree Method for Currency Lookback Options , 2000 .

[19]  T. Lai,et al.  Exercise Regions And Efficient Valuation Of American Lookback Options , 2004 .

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

[21]  S. H. Babbs.  Binomial valuation of lookback options , 2000 .

[22]  Xiaonan Wu,et al.  A Fast Numerical Method for the Black-Scholes Equation of American Options , 2003, SIAM J. Numer. Anal..

[23]  Georgios Foufas.  Valuing European, Barrier, and Lookback Options using the Finite Element Method and Duality Techniques , 2004 .

[24]  A. Conze,et al.  Path Dependent Options: The Case of Lookback Options , 1991 .

[25]  J. Barraquand,et al.  PRICING OF AMERICAN PATH‐DEPENDENT CONTINGENT CLAIMS , 1996 .

[26]  Yue Kuen Kwok,et al.  Early exercise policies of American floating strike and fixed strike lookback options , 2001 .

[27]  Fabien Heuwelyckx.  CONVERGENCE OF EUROPEAN LOOKBACK OPTIONS WITH FLOATING STRIKE IN THE BINOMIAL MODEL , 2014 .

[28]  M. Goldman,et al.  Path Dependent Options: "Buy at the Low, Sell at the High" , 1979 .

[29]  Hongtao Yang,et al.  A Front-Fixing Finite Element Method for the Valuation of American Options , 2008, SIAM J. Sci. Comput..

[30]  J. Hull.  Fundamentals of Futures and Options Markets , 2001 .

[31]  Toshikazu Kimura,et al.  American Fractional Lookback Options: Valuation and Premium Decomposition , 2011, SIAM J. Appl. Math..

[32]  Peng Liu,et al.  Numerical Methods For American Option Pricing , 2008 .

[33]  P. Carr,et al.  ALTERNATIVE CHARACTERIZATIONS OF AMERICAN PUT OPTIONS , 1992 .