Calibration of local-stochastic and path-dependent volatility models to vanilla and no-touch options

In this paper, we consider a large class of continuous semi-martingale models and propose a generic framework for their simultaneous calibration to vanilla and no-touch options. The method builds on the forward partial integro-differential equation (PIDE) derived by B. Hambly, M. Mariapragassam and C. Reisinger in their 2016 paper, “A forward equation for barrier options under the Brunick & Shreve Markovian projection”; this allows fast computation of up-and-out call prices for the complete set of strikes, barriers and maturities. We also use a novel two-state particle method to estimate the Markovian projection of the variance onto the spot and the running maximum. We detail a step-by-step procedure for a Heston-type local-stochastic volatility model with local volatility-of-volatility, as well as two path-dependent volatility models where the local volatility component depends on the running maximum.