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.
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