An improved least squares Monte Carlo valuation method based on heteroscedasticity

Longstaff–Schwartz’s least squares Monte Carlo method is one of the most applied numerical methods for pricing American-style derivatives. We examine the algorithms regression step, demonstrating that the OLS regression is not the best linear unbiased estimator because of heteroscedasticity. We prove the existence of heteroscedasticity for single-asset and multi-asset payoffs numerically and theoretically, and propose weighted-least squares MC valuation method to correct for it. An extensive numerical study shows that the proposed method produces significantly smaller pricing bias than the Longstaff–Schwartz method under several well-known price dynamics. An empirical pricing exercise using market data confirms the advantages of the improved method.

[1]  John N. Tsitsiklis,et al.  Regression methods for pricing complex American-style options , 2001, IEEE Trans. Neural Networks.

[2]  Lars Stentoft.  Assessing the Least Squares Monte-Carlo Approach to American Option Valuation , 2004 .

[3]  Arunachalam Chockalingam,et al.  The implication of missing the optimal-exercise time of an American option , 2015, Eur. J. Oper. Res..

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

[5]  Alfredo Ibáñez,et al.  The Sensitivity of American Options to Suboptimal Exercise Strategies , 2009 .

[6]  Frank J. Fabozzi,et al.  An improved method for pricing and hedging long dated American options , 2016, Eur. J. Oper. Res..

[7]  Denis Belomestny,et al.  Pricing Bermudan Options via Multilevel Approximation Methods , 2015, SIAM J. Financial Math..

[8]  Kumar Muthuraman,et al.  An approximate moving boundary method for American option pricing , 2015, Eur. J. Oper. Res..

[9]  R. Carmona,et al.  Valuation of energy storage: an optimal switching approach , 2010 .

[10]  Manuel Moreno,et al.  On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives , 2007 .

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

[12]  Philip Protter,et al.  An analysis of a least squares regression method for American option pricing , 2002, Finance Stochastics.

[13]  Nelson Areal,et al.  On improving the least squares Monte Carlo option valuation method , 2008 .

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

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

[16]  Lars Stentoft,et al.  Refining the Least Squares Monte Carlo Method by Imposing Structure , 2013 .

[17]  Jérôme Detemple.  American-Style Derivatives : Valuation and Computation , 2005 .

[18]  James Jackson,et al.  American-Style Derivatives: Valuation and Computation , 2009 .

[19]  David S. Bates.  Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options , 1998 .

[20]  Daniel Z. Zanger.  CONVERGENCE OF A LEAST‐SQUARES MONTE CARLO ALGORITHM FOR AMERICAN OPTION PRICING WITH DEPENDENT SAMPLE DATA , 2018 .

[21]  M. Kohler.  A Review on Regression-based Monte Carlo Methods for Pricing American Options , 2010 .

[22]  Lars Stentoft,et al.  Value Function Approximation or Stopping Time Approximation: A Comparison of Two Recent Numerical Methods for American Option Pricing using Simulation and Regression , 2008 .

[23]  Daniel Z. Zanger.  Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing , 2013, Finance Stochastics.

[24]  Haitao Li,et al.  Reduced-Form Valuation of Callable Corporate Bonds: Theory and Evidence , 2006 .

[25]  M. Broadie,et al.  American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods , 1996 .

[26]  J. Carriére.  Valuation of the early-exercise price for options using simulations and nonparametric regression , 1996 .

[27]  Lars Stentoft,et al.  Convergence of the Least Squares Monte Carlo Approach to American Option Valuation , 2004, Manag. Sci..

[28]  M. Schweizer,et al.  Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes , 2005, math/0505208.

[29]  M. Kohler,et al.  A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options , 2007, 0710.3640.

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

[31]  Denis Belomestny,et al.  Pricing Bermudan options using nonparametric regression: optimal rates of convergence for lower estimates , 2009, 0907.5599.

[32]  Anthony Brockwell,et al.  Sequential Monte Carlo Pricing of American-Style Options under Stochastic Volatility Models , 2009, 1010.1372.

[33]  Malay Ghosh,et al.  On the robustness of least squares procedures in regression models , 1980 .

[34]  W. Greene,et al.  计量经济分析 = Econometric analysis , 2009 .

[35]  Vadim Linetsky,et al.  Pricing Multi-Asset American Options: A Finite Element Method-of-Lines with Smooth Penalty , 2007, J. Sci. Comput..

[36]  R. Caflisch,et al.  Pricing and Hedging American-Style Options: A Simple Simulation-Based Approach , 2009 .

[37]  Robert A. Jarrow,et al.  Specification Tests of Calibrated Option Pricing Models , 2013 .

[38]  G. Meyer,et al.  The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines , 2008 .

[39]  P. Glasserman,et al.  Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes , 2004, math/0505208.

[40]  Oleksii Mostovyi.  On the stability the least squares Monte Carlo , 2013, Optim. Lett..