Robust calibration of financial models using Bayesian estimators

We consider a general calibration problem for derivative pricing models, which we reformulate into a Bayesian framework to attain posterior distributions for model parameters. It is then shown how the posterior distribution can be used to estimate prices for exotic options. We apply the procedure to a discrete local volatility model and work in great detail through numerical examples to clarify the construction of Bayesian estimators and their robustness to the model specification, number of calibration products, noisy data and misspecification of the prior.

[1]  Marco Avellaneda.  Quantitative Analysis in Financial Markets: Collected Papers of the New York University Mathematical Finance Seminar , 2002 .

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

[3]  A. Stuart,et al.  MCMC methods for sampling function space , 2009 .

[4]  Wolfgang J. Runggaldier,et al.  The volatility of the instantaneous spot interest rate implied by arbitrage pricing - A dynamic Bayesian approach , 2006, Autom..

[5]  A. V. D. Vaart,et al.  Convergence rates of posterior distributions for non-i.i.d. observations , 2007, 0708.0491.

[6]  Bruno Dupire.  Pricing with a Smile , 1994 .

[7]  Stéphane Crépey,et al.  Calibration of the Local Volatility in a Generalized Black-Scholes Model Using Tikhonov Regularization , 2003, SIAM J. Math. Anal..

[8]  A. Stuart,et al.  Sampling the posterior: An approach to non-Gaussian data assimilation , 2007 .

[9]  C. Chiarella,et al.  The Calibration of Stock Option Pricing Models Using Inverse Problem Methodology , 2000 .

[10]  Endre Süli,et al.  Computation of Deterministic Volatility Surfaces , 1998 .

[11]  Rama Cont.  Model Uncertainty and its Impact on the Pricing of Derivative Instruments , 2004 .

[12]  Robert A. Jarrow,et al.  Bayesian analysis of contingent claim model error , 2000 .

[13]  Henri Berestycki,et al.  Asymptotics and calibration of local volatility models , 2002 .

[14]  P. Mykland.  Financial options and statistical prediction intervals , 2003 .

[15]  Terry Lyons,et al.  Uncertain volatility and the risk-free synthesis of derivatives , 1995 .

[16]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[17]  Ben G. Fitzpatrick,et al.  Bayesian analysis in inverse problems , 1991 .

[18]  Yves Achdou,et al.  Numerical Procedure for Calibration of Volatility with American Options , 2005 .

[19]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[20]  Lea Fleischer,et al.  Regularization of Inverse Problems , 1996 .

[21]  Eric R. Ziegel,et al.  Practical Nonparametric and Semiparametric Bayesian Statistics , 1998, Technometrics.

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

[23]  Marco Avellaneda,et al.  Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model , 1996 .

[24]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[25]  S. Ben Hamida,et al.  Recovering Volatility from Option Prices by Evolutionary Optimization , 2004 .

[26]  Stanley Osher,et al.  A technique for calibrating derivative security pricing models: numerical solution of an inverse problem , 1997 .

[27]  T. Coleman,et al.  Reconstructing the Unknown Local Volatility Function , 1999 .

[28]  A. Gupta,et al.  Model Uncertainty and its Impact on Derivative Pricing ∗ , 2010 .

[29]  M. Monoyios.  Optimal hedging and parameter uncertainty , 2007 .

[30]  Marco Avellaneda,et al.  Calibrating Volatility Surfaces Via Relative-Entropy Minimization , 1996 .

[31]  H. Engl,et al.  Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates , 2005 .