Valuation of options for hedging against exchange rate exposure

Abstract

The risk associated with currency exposure is one of the main sources of risk in terms of internationally diversi ed portfolios. Controlling the risk is important for improving the performance of international investments. One approach to hedging against exchange rate exposure is by employing financial derivatives, particularly, foreign currency options. Currency options provide insurance against unfavorable exchange rate fluctuation, but also make provision to lock in a pro t when the exchange rate fluctuation are favorable. However, these instruments cannot be traded or managed without the relevant valuation techniques. In this dissertation we discuss one of the approaches to cover the risk associated with currency exposure. In particular, we focus on the partial differential equation (PDE) valuation of currency options by employing various finite difference schemes. We commence by introducing the mathematical tools required for the valuation of financial derivatives. Thereafter we study the valuation of European options. This involves deriving the famous Black-Scholes PDE for pricing options on stocks that do not yield dividends. Using the Black-Scholes PDE we derive the Black-Scholes formula for pricing European options. This derivation involves transforming the Black-Scholes PDE into the heat equation and by solving the heat equation we obtain the Black-Scholes formula. After completing the pricing of European options we now move to the pricing of American options. The early exercise facility associated with American options, leads to a free boundary problem which makes the pricing process of American options a challenging task. As in the case of the European options, we first derive the Black-Scholes inequality for American options and then transform this inequality for application to the heat equation to value American options. In the absence of an explicit formula for pricing American options we use numerical methods. Thus, we discuss the finite difference methods quite extensively with a focus on the implicit and Crank-Nicholson finite difference methods.