Show that the formula in equation (17.12) for a put option...

Show that the formula in equation (17.12) for a put option to sell one unit of currency A for currency B at strike price $K$ gives the same value as equation (17.11) for a call option to buy $K$ units of currency B for currency A at strike price $1 / K$.

Key Concepts

Duality in Option Pricing

Duality in option pricing refers to the symmetric relationship between certain option contracts where flipping the roles of the currencies (or underlying assets) leads to equivalent values. In this particular case, the put option to sell currency A for B at strike rate K is shown to have the same value as the call option to buy currency B using currency A with a reciprocal strike of 1/K. This duality is a powerful tool in understanding and deriving relationships between seemingly disparate option contracts.

Reciprocal Strike Price Concept

The reciprocal strike price concept arises in the domain of currency options because the exchange rate inherently involves two directions. When viewing the option from the perspective of the other currency, the strike price must be inverted (i.e., 1/K) to maintain the equivalence of contracts. This inversion is crucial in establishing the equivalence between the put and call options as it correctly reflects the exchange relationship between the two currencies involved.

Currency Options

Currency options are derivative instruments that grant the holder the right, but not the obligation, to exchange one currency for another at a predetermined exchange rate on or before a specified date. They are essential in managing foreign exchange risk and are priced by taking into account factors such as the underlying exchange rate, interest rate differentials between the two currencies, volatility, and time to expiration.

Put-Call Parity in Currency Options

Put-call parity is a fundamental relationship in options pricing that shows how the prices of puts and calls are connected. In the context of currency options, this parity must account for the fact that the underlying asset is a currency, meaning that interest rates in both the domestic and foreign economies influence option prices. This relationship ensures that two options with appropriately related strike prices and positions can be used to replicate identical payoffs, thus preventing arbitrage opportunities.

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