Monte Carlo Calibration Method of Stochastic Volatility Model with Stochastic Interest Rate

Implied volatility skew and smile are ubiquitous phenomena in the financial derivative market especially after the Black Monday 1987 crash. Various stochastic volatility models have been proposed to capture volatility skew and smile in derivative pricing and hedging. Almost 30 years after the advent of the first type of stochastic volatility model calibrating them to the market volatility surface still remains challenging, especially when stochastic interest rate has to be also taken into account for long-dated options. Many techniques have been applied to tackle this problem, including Fast Fourier Transform, singular perturbation expansion, heat kernel expansion, Markovian projection, to name a few. Although they have achieved some success in deriving either a close-form solution for a specific type of model or asymptotic solution in more general, none of them can really solve the calibration problem satisfying our need in term of both efficiency and accuracy. Monte Carlo method is a flexible numerical pricing method but has not been considering for calibration because of its slow convergence. However, with the great advance in computational power, in particular, parallel computation and the invention of other variance reduction techniques fast and accurate calibration using Monte Carlo becomes possible. This paper presents a Monte Carlo calibration method for stochastic volatility models with stochastic interest rate, which reduces simulation dimension by conditional expectation and further improves speed by vectorization. Numerical experiments show that both the calibration speed and accuracy of this generic method are satisfactory for almost all applications.