Pricing joint claims on an asset and its realized variance under stochastic volatility models

TL;DR

This paper develops a general pricing equation for European claims depending on an asset's terminal value and realized variance within stochastic volatility models, providing new formulas and numerical insights.

Contribution

It introduces a unified pricing framework for joint asset and variance payoffs in stochastic volatility models, including formulas for Greeks and new payoff types.

Findings

Derived a general pricing equation for joint claims

Provided formulas for Delta and Gamma of these claims

Numerical comparisons with vanilla derivatives

Abstract

In a stochastic volatility framework, we find a general pricing equation for the class of payoffs depending on the terminal value of a market asset and its final quadratic variation. This allows a pricing tool for European-style claims paying off at maturity a joint function of the underlying and its realised volatility/variance. We study the solution under different stochastic volatility models, give a formula for the computation of the Delta and Gamma of these claims, and introduce some new interesting payoffs that can be priced through this equation. Numerical results are given and compared to those from plain vanilla derivatives.