American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics

American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate

Multi-scale control variate methods for uncertainty quantification in kinetic equations

Kinetic equations play a major rule in modeling large systems of interacting particles. Uncertainties may be due to various reasons, like lack of knowledge on the microscopic interaction details or incomplete informations at the boundaries. These uncertainties, however, contribute to the curse of dimensionality and the development of efficient numerical methods is a challenge. In this paper we consider the construction of novel multi-scale methods for such problems which, thanks to a control variate approach, are capable to reduce the variance of standard Monte Carlo techniques

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationComment: 25 page

Parallel random number generators in Monte Carlo derivative pricing: An application-based test

Parallel pseudorandom number generators (PPRNG) that satisfy classical statistical tests may still demonstrate intra-stream and inter-stream correlations in real life applications. In order to investigate the suitability of a PPRNG for use in Monte Carlo pricing of financial derivatives, an application-based test is proposed to evaluate the bias and the standard error of the mean (SE) associated with the PPRNG as a gauge of intrastream and inter-stream correlations respectively. This test involves estimating the price of a vanilla European call option via Monte Carlo simulation, where the asset price at maturity is estimated by propagating the Black-Scholes stochastic differential equation via the Euler-Maruyama discretization scheme. The mean and SE profiles of the numerical results based on three PPRNG libraries (RNGSTREAM, TRNG and SPRNG) that implement parallel random numbers via sequence splitting strategies (RNGSTREAM and TRNG) and parameterization strategy (SPRNG) are compared. In terms of the bias and SE profiles, the best performing PPRNG constructed using the sequence splitting strategy is comparable to that constructed using parameterization, both use multiple recursive generators in their kernel. © de Gruyter 2012