A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection
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.
Keywords
Cite
@article{arxiv.1411.3618,
title = {A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection},
author = {Ben Hambly and Matthieu Mariapragassam and Christoph Reisinger},
journal= {arXiv preprint arXiv:1411.3618},
year = {2016}
}
Comments
20 pages, Quantitative Finance Volume 16, 2016 - Issue 6