Hull and White and Al\`os type formulas for barrier options in stochastic volatility models with nonzero correlation

TL;DR

This paper introduces two new closed-form formulas for barrier option pricing in stochastic volatility models with nonzero correlation, extending existing methods to more realistic market conditions.

Contribution

It presents a Hull and White type formula and an Alòs-like decomposition formula specifically for barrier options under correlated stochastic volatility models.

Findings

Derivation of a Hull and White type formula for barrier options.

Development of an Alòs-like decomposition formula.

Provision of a model-free approximation method.

Abstract

Two novel closed-form formulas for the price of barrier options in stochastic volatility models with zero interest rate and dividend yield but nonzero correlation between the asset and its instantaneous volatility are derived. The first is a Hull and White type formula, and the second is a decomposition formula similar in form to the Al\`os decomposition for vanilla options. A model-free approximation is also given.