A foreign currency is currently worth 1.50. The domestic...

A foreign currency is currently worth $$\$ 1.50$$. The domestic and foreign risk-free interest rates are $5 \%$ and $9 \%$, respectively. Calculate a lower bound for the value of a six-month call option on the currency with a strike price of $\$ 1.40$ if it is (a) European and (b) American.

Key Concepts

Option Lower Bound Pricing

This concept refers to the minimum value an option should have to avoid arbitrage. For a call option, it is often derived based on the intrinsic value, which is the underlying asset’s current price minus the present value of the strike price. The lower bound ensures that the option’s price reflects the time value of money and the possibility of a favorable payoff at expiration.

Risk-Free Interest Rates in Option Pricing

Risk-free rates are crucial in discounting future cash flows in option pricing. In currency options, two different risk-free rates—the domestic and the foreign—are used to adjust for the cost of carry and to determine the present value of cash flows, such as the strike price. This concept is central to arbitrage pricing models and ensures consistency in valuation when different currencies and interest environments are involved.

Foreign Currency Option Pricing

Pricing options on foreign currencies entails specific considerations compared to standard asset options because the underlying asset is itself a currency. This situation requires incorporating both the domestic and the foreign risk-free interest rates. The interplay of these rates, together with the current exchange rate, determines the lower bound and overall option valuation under a no-arbitrage framework.

American vs European Options

The key difference between American and European options is the timing of exercise. European options can only be exercised at expiration, whereas American options allow for early exercise. This flexibility can lead to a higher premium on American options if early exercise is advantageous, although for options on non-dividend paying assets (or currencies with certain interest dynamics), the difference in premium may be minimal.

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