American Options Under Stochastic Volatility

The problem of pricing an American option written on an underlying asset with constant price volatility has been studied extensively in literature. Real-world data, however, demonstrate that volatility is not constant, and stochastic volatility models are used to account for dynamic volatility changes. Option pricing methods that have been developed in literature for pricing under stochastic volatility focus mostly on European options. We consider the problem of pricing American options under stochastic volatility, which has had relatively much less attention from literature. First, we develop a transformation procedure to compute the optimal-exercise policy and option price and provide theoretical guarantees for convergence. Second, using this computational tool, we explore a variety of questions that seek insights into the dependence of option prices, exercise policies, and implied volatilities on the market price of volatility risk and correlation between the asset and stochastic volatility. The speed and accuracy of the procedure are compared against existing methods as well.

[1]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

[2]  Mark Broadie,et al.  A Primal-Dual Simulation Algorithm for Pricing Multi-Dimensional American Options , 2001 .

[3]  Sunil Kumar,et al.  MULTIDIMENSIONAL PORTFOLIO OPTIMIZATION WITH PROPORTIONAL TRANSACTION COSTS , 2006 .

[4]  Gurdip Bakshi,et al.  Delta-Hedged Gains and the Negative Market Volatility Risk Premium , 2001 .

[5]  I. Kim.  The Analytic Valuation of American Options , 1990 .

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

[7]  Elias N. Houstis,et al.  Front-Tracking Finite Difference Methods for the Valuation of American Options , 1998 .

[8]  Jari Toivanen,et al.  COMPONENTWISE SPLITTING METHODS FOR PRICING AMERICAN OPTIONS UNDER STOCHASTIC VOLATILITY , 2007 .

[9]  P. Glasserman,et al.  A Sotchastic Mesh Method for Pricing High-Dimensional American Options , 2004 .

[10]  P. Glasserman,et al.  Pricing American-style securities using simulation , 1997 .

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

[12]  S. Ikonen,et al.  Efficient numerical methods for pricing American options under stochastic volatility , 2008 .

[13]  G. Papanicolaou,et al.  Derivatives in Financial Markets with Stochastic Volatility , 2000 .

[14]  T. Björk.  Arbitrage Theory in Continuous Time , 2019 .

[15]  C. Cryer.  The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[16]  Antonio Roma,et al.  Stochastic Volatility Option Pricing , 1994, Journal of Financial and Quantitative Analysis.

[17]  M. Broadie,et al.  American options with stochastic dividends and volatility: A nonparametric investigation , 2000 .

[18]  R. Chou,et al.  ARCH modeling in finance: A review of the theory and empirical evidence , 1992 .

[19]  Eduardo S. Schwartz,et al.  The Valuation of American Put Options , 1977 .

[20]  Peter A. Forsyth,et al.  Penalty methods for American options with stochastic volatility , 1998 .

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

[22]  Cornelis W. Oosterlee,et al.  On multigrid for linear complementarity problems with application to American-style options. , 2003 .

[23]  Francis A. Longstaff,et al.  Valuing American Options by Simulation: A Simple Least-Squares Approach , 2001 .

[24]  Kevin Parrott,et al.  Multigrid for American option pricing with stochastic volatility , 1999 .

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

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

[27]  Louis O. Scott.  Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application , 1987, Journal of Financial and Quantitative Analysis.

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

[29]  S. Jacka.  Optimal Stopping and the American Put , 1991 .

[30]  K. Muthuraman.  A Moving Boundary Approach to American Option Pricing , 2007 .

[31]  M. Rubinstein..  Implied Binomial Trees , 1994 .

[32]  Mark Broadie,et al.  ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications , 2004, Manag. Sci..

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

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

[35]  S. Shreve,et al.  Methods of Mathematical Finance , 2010 .

[36]  V. Henderson,et al.  ANALYTICAL COMPARISONS OF OPTION PRICES IN STOCHASTIC VOLATILITY MODELS , 2005 .

[37]  V. Yakovenko,et al.  Probability distribution of returns in the Heston model with stochastic volatility , 2002, cond-mat/0203046.

[38]  R. Myneni.  The Pricing of the American Option , 1992 .

[39]  Sunil Kumar,et al.  A Numerical Method for Solving Singular Stochastic Control Problems , 2004, Oper. Res..

[40]  M. Broadie,et al.  Option Pricing: Valuation Models and Applications , 2004 .

[41]  Jari Toivanen,et al.  Operator Splitting Methods for Pricing American Options with Stochastic Volatility , 2004 .