The Jacobi Stochastic Volatility Model

TL;DR

This paper introduces a new stochastic volatility model based on the Jacobi process, providing closed-form series for option pricing and demonstrating efficient, accurate approximations for various financial derivatives.

Contribution

It presents the Jacobi stochastic volatility model, extending the Heston model, with explicit series representations for option prices and practical numerical methods.

Findings

Series representations enable accurate option pricing.

Model generalizes the Heston model as a limit case.

Efficient truncation yields precise numerical approximations.

Abstract

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the joint density of any finite sequence of log returns admits a Gram-Charlier A expansion with closed-form coefficients. We derive closed-form series representations for option prices whose discounted payoffs are functions of the asset price trajectory at finitely many time points. This includes European call, put, and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical analysis we show that option prices can be accurately and efficiently approximated by truncating their series representations.