Valuation of Barrier Options using Sequential Monte Carlo

TL;DR

This paper introduces a Sequential Monte Carlo (SMC) method for pricing barrier options, demonstrating improved efficiency and accuracy over standard Monte Carlo techniques, especially in complex or multi-asset scenarios.

Contribution

The paper presents the first application of SMC methods to barrier option pricing, showing significant variance reduction and efficiency gains over traditional Monte Carlo approaches.

Findings

SMC provides unbiased, convergent estimates for barrier options.

SMC reduces variance and maintains efficiency with increasing time steps.

Implementation effort for SMC is minimal compared to standard Monte Carlo.

Abstract

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