  Pricing options is an important problem in financial engineering. In many
scenarios of practical interest, financial option prices associated to an
underlying asset reduces to computing an expectation w.r.t.~a diffusion
process. In general, these expectations cannot be calculated analytically, and
one way to approximate these quantities is via the Monte Carlo method; Monte
Carlo methods have been used to price options since at least the 1970's. It has
been seen in Del Moral, P. \& Shevchenko, P.V. (2014) `Valuation of barrier
options using Sequential Monte Carlo' and Jasra, A. \& Del Moral, P. (2011)
`Sequential Monte Carlo for option pricing' that Sequential Monte Carlo (SMC)
methods are a natural tool to apply in this context and can vastly improve over
standard Monte Carlo. In this article, in a similar spirit to Del Moral, P. \&
Shevchenko, P.V. (2014) `Valuation of barrier options using sequential Monte
Carlo' and Jasra, A. \& Del Moral, P. (2011) `Sequential Monte Carlo for option
pricing' we show that one can achieve significant gains by using SMC methods by
constructing a sequence of artificial target densities over time. In
particular, we approximate the optimal importance sampling distribution in the
SMC algorithm by using a sequence of weighting functions. This is demonstrated
on two examples, barrier options and target accrual redemption notes (TARN's).
We also provide a proof of unbiasedness of our SMC estimate.
