Exponential L\'evy-type models with stochastic volatility and stochastic jump-intensity

TL;DR

This paper develops an advanced exponential Lévy-type model with stochastic volatility and jump-intensity, providing explicit pricing formulas and demonstrating improved calibration to market implied volatility surfaces.

Contribution

It extends multiscale stochastic volatility models to include Lévy jumps and stochastic jump-intensity, with explicit pricing formulas and empirical calibration results.

Findings

Extended models better fit implied volatility surfaces.

Explicit Fourier-based pricing formulas derived.

Calibration to S&P 500 data shows significant improvement.

Abstract

We consider the problem of valuing a European option written on an asset whose dynamics are described by an exponential L\'evy-type model. In our framework, both the volatility and jump-intensity are allowed to vary stochastically in time through common driving factors -- one fast-varying and one slow-varying. Using Fourier analysis we derive an explicit formula for the approximate price of any European-style derivative whose payoff has a generalized Fourier transform; in particular, this includes European calls and puts. From a theoretical perspective, our results extend the class of multiscale stochastic volatility models of \citet*{fpss} to models of the exponential L\'evy type. From a financial perspective, the inclusion of jumps and stochastic volatility allow us to capture the term-structure of implied volatility. To illustrate the flexibility of our modeling framework we extend five exponential L\'evy processes to include stochastic volatility and jump-intensity. For each of the extended models, using a single fast-varying factor of volatility and jump-intensity, we perform a calibration to the S&P500 implied volatility surface. Our results show decisively that the extended framework provides a significantly better fit to implied volatility than both the traditional exponential L\'evy models and the fast mean-reverting stochastic volatility models of \citet{fpss}.