Stochastic Volatility Jump Diffusion Calibration, Dynamics & Implementation
Document ID: 2008-1260
Published on: 30th December 2008
Author: Graeme Lawson
Stochastic Volatility Jump Diffusion (SVJD) (also known as the Bates Model), is a mixture of two well known models of quantitative finance – Heston’s Stochastic Volatility Model and Merton’s Jump Diffusion Model.
The Heston model, which has been described in detail in Technical Note 2008/08, consists of two Stochastic Differential Equations (SDE’s). The first describes the evolution of the asset price process, and the second describes the evolution of the stochastic variance process.
The Merton Jump Diffusion Model consists of one SDE describing the evolution of the asset price process, which can be broken into two independent parts – a continuous part and a discontinuous part.
The continuous part is modelled using a Brownian motion scaled by a constant ‘volatility’ parameter, as in the seminal Black-Scholes-Merton model. The discontinuous part (jumps) is modelled using a compound Poisson process with independent random variables (jumps) drawn from the Log-Normal distribution.
