# Analytic solutions of variance swaps for Heston models with stochastic long-run mean of variance and jumps

## TL;DR

This paper provides new formulas for pricing variance swaps using an advanced Heston model that includes jumps and a changing long-term variance mean.

## Contribution

The paper introduces a novel pricing formula for variance swaps with a Heston model that includes stochastic long-run mean and jumps.

## Key findings

- A partial integro-differential equation is derived for the joint moment-generating function.
- A series pricing formula is developed for discretely sampled variance swaps.
- Numerical simulations confirm the formula's accuracy and analyze parameter impacts.

## Abstract

This paper presents the pricing formulas for variance swaps within the Heston model that incorporates jumps and a stochastic long-term mean for the underlying asset. By leveraging the Feynman-Kac theorem, we derive a partial integro-differential equation (PIDE) to obtain the joint moment-generating function for the aforementioned model. Furthermore, we provide a series pricing formula for discretely sampled variance swap, derived through the use of this joint moment-generating function. Additionally, we discuss the limiting properties of the pricing formula for discretely sampled variance swap, namely, the pricing formula for continuously sampled variance swap. Finally, to demonstrate the efficacy of the pricing formula, we conduct several numerical simulation experiments, including comparisons with Monte Carlo (MC) simulation results and an analysis of the impact of parameter variations on the strike price of variance swaps.