  Typically options with a path dependent payoff, such as Target Accumulation
Redemption Note (TARN), are evaluated by a Monte Carlo method. This paper
describes a finite difference scheme for pricing a TARN option. Key steps in
the proposed scheme involve tracking of multiple one-dimensional finite
difference solutions, application of jump conditions at each cash flow exchange
date, and a cubic spline interpolation of results after each jump. Since a
finite difference scheme for TARN has significantly different features from a
typical finite difference scheme for options with a path independent payoff, we
give a step by step description on the implementation of the scheme, which is
not available in the literature. The advantages of the proposed finite
difference scheme over the Monte Carlo method are illustrated by examples with
three different knockout types. In the case of constant or time dependent
volatility models (where Monte Carlo requires simulation at cash flow dates
only), the finite difference method can be faster by an order of magnitude than
the Monte Carlo method to achieve the same accuracy in price. Finite difference
method can be even more efficient in comparison with Monte Carlo in the case of
local volatility model where Monte Carlo requires significantly larger number
of time steps. In terms of robust and accurate estimation of Greeks, the
advantage of the finite difference method will be even more pronounced.
