Introduction

In this chapter, we will explore how to compute the value of a European option using the Black-Scholes-Merton model for various types of underlying assets. While the Black-Scholes-Merton model is widely used for pricing European options on stocks without dividends, we will extend our understanding to include dividend-paying stocks, futures contracts, and foreign exchange rates. Each of these assets comes with its own unique characteristics that affect option pricing.

European Options on Dividend-Paying Stocks

European options on dividend-paying stocks require a modification of the Black-ScholesMerton model to account for the impact of dividends. The modified formula for calculating the value of a European call option on a dividend-paying stock is as follows:

$$C=S_0 e^{-q t} N\left(d_1\right)-K e^{-r t} N\left(d_2\right)$$

Where:

• $C$ is the call option price
• $S_0$ is the current stock price
• $K$ is the strike price
• $r$ is the risk-free interest rate
• $t$ is the time to expiration
• $q$ is the continuous dividend yield
• $N\left(d_1\right)$ and $N\left(d_2\right)$ are cumulative probability functions

Example: Suppose a stock is trading at USD 100 with a strike price of USD 95, an annual risk-free rate of 5%, and an annual dividend yield of 2%. The option expires in 6 months. Using the Black-Scholes-Merton model, we can calculate the call option price.

$$C=100 e^{-0.02 \times 0.5} N\left(d_1\right)-95 e^{-0.05 \times 0.5} N\left(d_2\right)$$

European Options on Futures Contracts

European options on futures contracts involve pricing options on the future price of an underlying asset, such as commodities or financial instruments. The formula for calculating the value of a European call option on a futures contract is similar to the Black-ScholesMerton model, with some adjustments:

$$C=e^{-r T}\left(F_0 N\left(d_1\right)-K N\left(d_2\right)\right)$$

Where:

• $C$ is the call option price
• $F_0$ is the current futures price
• $K$ is the strike price
• $r$ is the risk-free interest rate
• $T$ is the time to expiration
• $N\left(d_1\right)$ and $N\left(d_2\right)$ are cumulative probability functions

Example: Consider a futures contract with a current price of USD 105, a strike price of USD 110, a risk-free rate of 4%, and an expiration time of 3 months. Using the Black-Scholes-Merton model, we can calculate the call option price.

$$C=e^{-0.04 \times 0.25}\left(105 N\left(d_1\right)-110 N\left(d_2\right)\right)$$

European Options on Foreign Exchange Rates

European options on foreign exchange rates are used to hedge against currency fluctuations. The formula for calculating the value of a European call option on a foreign exchange rate is as follows:

$$C=S e^{-q t} N\left(d_1\right)-K e^{-r t} N\left(d_2\right)$$

Where:

• $C$ is the call option price
• $S$ is the current exchange rate (spot rate)
• $K$ is the strike price
• $r$ is the domestic risk-free interest rate
• $t$ is the time to expiration
• $q$ is the foreign risk-free interest rate
• $N\left(d_1\right)$ and $N\left(d_2\right)$ are cumulative probability functions

Example: If the current exchange rate is 1.20 (1 USD = 1.20 EUR), the strike price is 1.25, the domestic risk-free rate is 3%, the foreign risk-free rate is 2%, and the option expires in 9 months, we can calculate the call option price.

$$C=1.20 e^{-0.02 \times 0.75} N\left(d_1\right)-1.25 e^{-0.03 \times 0.75} N\left(d_2\right)$$