Calibration to vanilla and barrier options with the Gyöngy and Brunick--Shreve Markovian projections

In this thesis, we present a novel approach to the calibration of diffusion models to vanilla and barrier options with the Gyöngy and BrunickâShreve Markovian projection results. Firstly, we derive a forward equation for arbitrage-free barrier option prices in continuous semi-martingale models, in terms of Markovian projections of the stochastic volatility process. This leads to 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. Secondly, we treat the problem of long-dated foreign-exchange option pricing and propose a novel and generic calibration technique to vanilla options for four-factor foreign-exchange hybrid local-stochastic volatility models with stochastic short rates. We build upon the particle method introduced by Guyon and Labordère and combine it with new variance reduction techniques in order to accelerate convergence. Finally, we derive the necessary and sufficient condition for the exact calibration to up-and-out call options and provide a step-by-step procedure to calibrate a BrunickâShreve volatility model, a Heston-type local-stochastic volatility model with local vol-of-vol and a path-dependent local-maximum-stochastic volatility model. While the BrunickâShreve model is calibrated with our forward PIDE for barriers, both stochastic volatility models require an interesting two-dimensional extension of the particle method. We then derive and prove the self-consistency condition for perfect calibration to barrier options for path-dependent models with stochastic domestic and foreign short rates, where techniques from our previous work can be combined, in order to price accurately long-dated derivatives with barrier feature.