A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection

TL;DR

This paper develops a forward equation for barrier option prices using Markovian projections, extending local volatility concepts and enabling efficient calibration and pricing of barrier options in continuous semi-martingale models.

Contribution

It introduces a Dupire-type formula and a forward PIDE for barrier options, providing new tools for pricing and calibration in stochastic volatility models.

Findings

Derived a forward equation for barrier options in Markovian projection models.

Introduced a forward PIDE for barrier options with numerical validation.

Provided a discretisation scheme for the PIDE.

Abstract

We derive a forward equation for arbitrage-free barrier option prices, in terms of Markovian projections of the stochastic volatility process, in continuous semi-martingale models. This provides a Dupire-type formula for the coefficient derived by Brunick and Shreve for their mimicking diffusion and can be interpreted as the canonical extension of local volatility for barrier options. Alternatively, a forward partial-integro differential equation (PIDE) is introduced which provides up-and-out call prices, under a Brunick-Shreve model, for the complete set of strikes, barriers and maturities in one solution step. Similar to the vanilla forward PDE, the above-named forward PIDE can serve as a building block for an efficient calibration routine including barrier option quotes. We provide a discretisation scheme for the PIDE as well as a numerical validation.