  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.
