Pricing a Six-Month American Put Option on the Swiss Franc

Introduction

In the world of finance, options play a crucial role in managing risk and speculation. Among the various types of options, American put options give the holder the right to sell a specific currency or asset at a predetermined price (strike price) before or on a specified expiration date. In this scenario, we will explore how to price a six-month American put option on the Swiss Franc, assuming a rational market. The spot exchange rate is $0.7008, the six-month forward rate is $0.6954, and the annualized six-month Eurodollar rate is 3.5 percent.

Understanding the Basics

Before delving into the pricing of the American put option, let’s clarify some essential concepts:

Spot Exchange Rate ($0.7008): This is the current exchange rate at which one currency can be exchanged for another.

Six-Month Forward Rate ($0.6954): The forward rate represents the future exchange rate agreed upon today for a transaction to occur in the future.

American Put Option: An American put option gives the holder the right (but not the obligation) to sell a currency at a specified strike price at any time before or on the option’s expiration date.

Strike Price ($0.6804): This is the price at which the holder of the put option can sell the Swiss Franc.

Annualized Six-Month Eurodollar Rate (3.5 percent): This interest rate is used to calculate the present value of future cash flows.

Pricing the American Put Option

To determine the minimum price that a six-month American put option with a striking price of $0.6804 should sell for in a rational market, we can use the concept of arbitrage-free pricing. The key idea is that the option’s price should eliminate the possibility of risk-free profits, ensuring a fair market value.

Here are the steps to calculate the option’s minimum price:

Calculate the Intrinsic Value: The intrinsic value of an American put option is the greater of zero or the difference between the strike price and the current spot rate. In this case, the intrinsic value is max($0.6804 – $0.7008, 0) = $0.0204.

Calculate the Present Value of Intrinsic Value: Since the option expires in six months, we need to calculate the present value of the intrinsic value using the annualized six-month Eurodollar rate. The formula for present value (PV) is: PV = Intrinsic Value / (1 + Eurodollar Rate)^t

PV = $0.0204 / (1 + 0.035)^0.5 ≈ $0.01963

Determine the Minimum Option Price: The minimum price for the American put option should be at least equal to the present value of its intrinsic value to prevent arbitrage opportunities. Therefore, the option should sell for approximately $0.01963.

Conclusion

In a rational market, a six-month American put option with a striking price of $0.6804 should sell for a minimum price of approximately $0.01963. This price ensures that no risk-free profit opportunities exist, aligning with the principles of arbitrage-free pricing. It’s essential to note that market conditions, volatility, and other factors can influence option prices, but this calculation serves as a fundamental starting point for pricing such options in the foreign exchange market.