Hybrid Equity, FX, and Interest Rate Models with Stochastic Volatility and Jump Diffusion

Abstract

We construct Hybrid Equity and multi-currency models with stochastic volatility and jump diffusion and correlated stochastic interest rates with a full matrix of correlations. We present a jump diffusion extension of Hybrid Models developed by Grzelak and Oosterlee [GO10], [GO10b] and use modeling techniques mentioned there to convert non-affine PIDE to solvable form. We model the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. The spot Equity or FX rate is governed by stochastic volatility and jump diffusion with stochastic intensity poisson process with any general jump size distribution. We provide semi-closed form approximations which lead to efficient calibration of the Equity and cross-currency models.

Keywords: Foreign-Exchange (FX);Equities, Stochastic Volatility, Heston Model, Jump Diffusion, Stochastic Interest Rates, Interest Rate Smile, Forward Characteristic Function, Hybrids, Affine Diffusion, Efficient Calibration, BGM, LIBOR Market Model