No abstract

We study the Heston-Cox-Ingersoll-Ross++ stochastic-local volatility model in the context of foreign exchange markets and propose a Monte Carlo simulation scheme which combines the full truncation Euler scheme for the stochastic volatility component and the stochastic domestic and foreign short interest rates with the log-Euler scheme for the exchange rate. We establish the exponential integrability of full truncation Euler approximations for the Cox-Ingersoll-Ross process and find a lower bound on the explosion time of these exponential moments. Under a full correlation structure and a realistic set of assumptions on the so-called leverage function, we prove the strong convergence of the exchange rate approximations and deduce the convergence of Monte Carlo estimators for a number of vanilla and path-dependent options. Then, we perform a series of numerical experiments for an autocallable barrier dual currency note.

We propose a novel and generic calibration technique for four-factor foreign-exchange hybrid local-stochastic volatility models (LSV) with stochastic short rates. We build upon the particle method introduced by Guyon and Henry-Labordère [Nonlinear Option Pricing, Chapter 11, Chapman and Hall, 2013] and combine it with new variance reduction techniques in order to accelerate convergence. We use control variates derived from: a calibrated pure local volatility model; a two-factor Heston-type LSV model (both with deterministic rates); the stochastic (CIR) short rates. The method can be applied to a large class of hybrid LSV models and is not restricted to our particular choice of the diffusion. However, we address in the paper some specific difficulties arising from the Heston model, notably by a new PDE formulation and finite element solution to bypass the singularities of the density when zero is attainable by the variance. The calibration procedure is performed on market data for the EUR-USD currency pair and has a comparable run-time to the PDE calibration of a two-factor LSV model alone.

Abstract. We derive a forward equation for arbitrage-free barrier option prices in continuous semimartingale 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. Alternatively, a forward partial-integro differential equation (PIDE) allows computation of up-and-out call prices under such a model, for the complete set of strikes, barriers and maturities from a single equation. In the same way as the vanilla forward Dupire PDE, the above-named forward PIDE can serve as a building block for an efficient calibration routine including barrier option quotes. We propose a discretisation scheme for the PIDE as well as a numerical validation.