Valuation of barrier options using sequential Monte Carlo

Pavel V Shevchenko and Pierre Del Moral

Abstract

ABSTRACT

Sequential Monte Carlo (SMC) methods have been used successfully in many applications in engineering, statistics and physics. However, they are seldom used in financial option pricing literature and its practice. We present an SMC method for pricing barrier options with continuous and discrete monitoring of the barrier condition. With our method, simulated asset values rejected due to the barrier condition are resampled from asset samples that do not breach the barrier condition, improving the efficiency of the option price estimator, while with the standard Monte Carlo method many simulated asset paths can be rejected by the barrier condition, making it harder to estimate the option price accurately. We compare the SMC with the standard Monte Carlo method and demonstrate that there is little extra effort required to implement the former compared with the latter, while the improvement in price estimation can be significant. Both methods result in unbiased estimators for the price converging to the true value as 1 / √M , where M is the number of simulations (asset paths). However, the variance of the SMC estimator is smaller and does not grow with the number of time steps, unlike standard Monte Carlo. In this paper, we demonstrate that the SMC can successfully be used for pricing barrier options. SMC methods can also be used for pricing other exotic options and for cases with many underlying assets and additional stochastic factors, such as stochastic volatility; we provide general formulas and references.