Models with time-dependent parameters using transform methods: application to Heston's model

This paper presents a methodology to introduce time-dependent parameters for a wide family of models preserving their analytic tractability. This family includes hybrid models with stochastic volatility, stochastic interest-rates, jumps and their non-hybrid counterparts. The methodology is applied to Heston's model. A bootstrapping algorithm is presented for calibration. A case study works out the calibration of the time-dependent parameters to the volatility surface of the Eurostoxx 50 index. The methodology is also applied to the analytic valuation of forward start vanilla options driven by Heston's model. This result is used to explore the forward skew of the case study.

[1]  A. W. Kemp,et al.  Kendall's Advanced Theory of Statistics. , 1994 .

[2]  S. Mikhailov,et al.  Heston ’ s Stochastic Volatility Model Implementation , Calibration and Some , 2003 .

[3]  Gurdip Bakshi,et al.  Empirical Performance of Alternative Option Pricing Models , 1997 .

[4]  N. Shephard.  Numerical integration rules for multivariate inversions , 1991 .

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

[6]  R. Lord,et al.  Why the Rotation Count Algorithm Works , 2006 .

[7]  Mark Broadie,et al.  Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes , 2006, Oper. Res..

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

[9]  T. Alderweireld,et al.  A Theory for the Term Structure of Interest Rates , 2004, cond-mat/0405293.

[10]  Neil Shephard,et al.  From Characteristic Function to Distribution Function: A Simple Framework for the Theory , 1991, Econometric Theory.

[11]  Vladimir V. Piterbarg.  Stochastic Volatility Model with Time‐dependent Skew , 2005 .

[12]  Mark Britten-Jones,et al.  Option Prices, Implied Price Processes, and Stochastic Volatility , 2000 .

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

[14]  Louis O. Scott.  Pricing Stock Options in a Jump‐Diffusion Model with Stochastic Volatility and Interest Rates: Applications of Fourier Inversion Methods , 1997 .

[15]  Roger Lee.  Option Pricing by Transform Methods: Extensions, Unification, and Error Control , 2004 .

[16]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[17]  R. Davies.  Numerical inversion of a characteristic function , 1973 .