Pricing TARNs Using a Finite Difference Method

Target accumulation redemption notes (TARNs) have become very popular products among Asian foreign exchange investors due to their flexibility to be structured to suit any foreign exchange outlook. TARN payoff is path dependent, and typically practitioners use the Monte Carlo method to evaluate TARN prices. This article describes a finite difference scheme for pricing a TARN option. Key steps in the proposed scheme involve tracking multiple one-dimensional finite difference solutions, applying jump conditions at each cash flow exchange date, and a cubic spline interpolation of results after each jump. Since a finite difference scheme for TARNs 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 using 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 than the Monte Carlo method by an order of magnitude while achieving the same accuracy in price. The finite difference method can be even more efficient in the case of a local volatility model because the Monte Carlo method requires a significantly larger number of time steps. In terms of robust and accurate estimation of Greeks, the advantage of the finite difference method is even more pronounced.