Joint Characteristic Function of Stock Log-Price and Squared Volatility in the Bates Model and Its Asset Pricing Applications

Oleksandr Zhylyevskyy

doi:10.4236/tel.2012.24074

Abstract

The model of Bates specifies a rich, flexible structure of stock dynamics suitable for applications in finance and economics, including valuation of derivative securities. This paper analytically derives a closed-form expression for the joint conditional characteristic function of a stock’s log-price and squared volatility under the model dynamics. The use of the function, based on inverting it, is illustrated on examples of pricing European-, Bermudan-, and American-style options. The discussed approach for European-style derivatives improves on the option formula of Bates. The suggested approach for American-style derivatives, based on a compound-option technique, offers an alternative solution to existing finite-difference methods.

Highlights

Stochastic volatility and jump-diffusion are standard tools of modeling asset price dynamics in finance research
Popularity of stochastic volatility models, such as the continuous-time model of Heston (1993) [2], is partly due to their ability to account for several aspects of stock price data that are not captured by analytically simpler geometric Brownian motion dynamics
This paper aims to fill in the gap by deriving a closed-form expression for the function, which is an analytically challenging task

Summary

Introduction

Stochastic volatility and jump-diffusion are standard tools of modeling asset price dynamics in finance research (see Aït-Sahalia and Jacod, 2011 [1]). Popularity of stochastic volatility models, such as the continuous-time model of Heston (1993) [2], is partly due to their ability to account for several aspects of stock price data that are not captured by analytically simpler geometric Brownian motion dynamics These models can help to account for an empirically relevant “leverage effect,” which refers to an increase in the volatility of a stock when its price declines, and a decrease in the volatility when the price rises. I analytically derive and provide examples for the use of a closed-form expression for the joint conditional characteristic function of a stock’s log-price and squared volatility under the dynamics of the Bates model.

The Bates Model

Dynamics of Log-Price

Martingale Property and Dynamics of Joint Characteristic Function

Closed-Form Solution for Joint Characteristic Function

Applications of Joint Characteristic Function

Conclusion