Calibration of Jump-Diffusion Option Pricing Models: A Robust Non-Parametric Approach

We present a non-parametric method for calibrating jump-diffusion models to a finite set of observed option prices. We show that the usual formulations of the inverse problem via nonlinear least squares are ill-posed and propose a regularization method based on relative entropy. We reformulate our calibration problem into a problem of finding a risk neutral jump-diffusion model that reproduces the observed option prices and has the smallest possible relative entropy with respect to a chosen prior model. Our approach allows to conciliate the idea of calibration by relative entropy minimization with the notion of risk neutral valuation in a continuous time model. We discuss the numerical implementation of our method using a gradient based optimization algorithm and show via simulation tests on various examples that the entropy penalty resolves the numerical instability of the calibration problem. Finally, we apply our method to datasets of index options and discuss the empirical results obtained.

[1]  Nicole El Karoui,et al.  Pricing Via Utility Maximization and Entropy , 2000 .

[2]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[3]  Dominick Samperi,et al.  Calibrating a Diffusion Pricing Model with Uncertain Volatility: Regularization and Stability , 2002 .

[4]  Andrew W. Lo,et al.  Nonparametric estimation of state-price densities implicit in financial asset prices , 1995, Proceedings of 1995 Conference on Computational Intelligence for Financial Engineering (CIFEr).

[5]  Cecilia Mancini,et al.  Disentangling the jumps of the diffusion in a geometric jumping Brownian motion , 2001 .

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

[7]  Jesper Andreasen,et al.  Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Pricing , 1999 .

[8]  R. Wolpert.  Lévy Processes , 2000 .

[9]  Koponen,et al.  Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

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

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

[13]  M. Frittelli.  The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets , 2000 .

[14]  E. Eberlein,et al.  New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model , 1998 .

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

[16]  Rama Cont,et al.  Dynamics of implied volatility surfaces , 2002 .

[17]  Robert Buff,et al.  WEIGHTED MONTE CARLO: A NEW TECHNIQUE FOR CALIBRATING ASSET-PRICING MODELS , 2001 .

[18]  Y. Miyahara.  Minimal Entropy Martingale Measures of Jump Type Price Processes in Incomplete Assets Markets , 1999 .

[19]  Ludger Rüschendorf,et al.  Minimax and minimal distance martingale measures and their relationship to portfolio optimization , 2001, Finance Stochastics.

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

[21]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.