The current value of the British pound is 1.60 and the...

The current value of the British pound is $$\$ 1.60$$ and the volatility of the pound/dollar exchange rate is $15 \%$ per annum. An American call option has an exercise price of $$\$ 1.62$$ and a time to maturity of 1 year. The risk-free rates of interest in the United States and the United Kingdom are $6 \%$ per annum and $9 \%$ per annum, respectively. Use the explicit finite difference method to value the option. Consider exchange rates at intervals of 0.20 between 0.80 and 2.40 and time intervals of 3 months.

Key Concepts

Explicit Finite Difference Method

This is a numerical technique used to solve partial differential equations, particularly those arising in option pricing. The method discretizes both time and space into a grid and then approximates the derivative terms using finite differences, allowing for the evolution of the option's value step-by-step across the grid.

American Options

American options differ from European options in that they can be exercised at any time before expiration. This feature requires careful handling of the early exercise premium in the pricing model, typically by checking at each time step whether exercising the option is more beneficial than holding it.

Option Pricing Partial Differential Equation (PDE)

The option pricing problem can be expressed as a PDE under the risk-neutral valuation framework. This PDE incorporates factors such as the underlying asset’s volatility, risk-free interest rates, and dividend yields or foreign interest rates, providing a continuous description of the option’s price dynamics.

Grid Discretization

The process of dividing the continuous ranges of time and the underlying asset's value into discrete intervals, or grid points, is known as grid discretization. In the explicit finite difference method, choosing appropriate step sizes for both time and asset price is crucial to maintain stability and accuracy of the numerical solution.

Foreign Exchange Options and Interest Rate Differentials

When pricing options on currency pairs, it is important to account for the fact that each currency has its own risk-free rate. The difference between these domestic and foreign interest rates affects the cost-of-carry and, consequently, the option's value. This aspect is essential when applying risk-neutral valuation to currencies.

Volatility

Volatility measures the degree of variation in the price of the underlying asset – in this case, the exchange rate – and is a critical input in option pricing models. Higher volatility generally increases the option's value because it raises the probability of favorable movements in the underlying asset’s price.

Transcript

Please give Ace some feedback