Introduction

In this chapter, we will delve into the heart of option pricing with the Black-Scholes-Merton Model. Specifically, we will focus on computing the value of a European option on a non-dividend-paying stock using this widely recognized model. The Black-Scholes-Merton Model is a mathematical framework that aids in the valuation of options. It takes into account various factors such as the current stock price, the option’s strike price, the time to expiration, and market volatility. By understanding and applying this model, we can determine the fair market value of European options with precision.

Option Pricing with Black-Scholes-Merton Model

The Black-Scholes-Merton Model provides a formula for calculating the theoretical price of European call and put options. For a European call option, the formula is as follows:

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

Where:

• $C$ is the call option price.
• $S_0$ is the current stock price.
• $N\left(d_1\right)$ is the cumulative standard normal distribution function of $d_1$.
• $K$ is the option’s strike price.
• $r$ is the risk-free interest rate.
• $t$ is the time to expiration (in years).
• $e$ is the base of the natural logarithm.
• $N\left(d_2\right)$ is the cumulative standard normal distribution function of $d_2$. For a European put option, the formula is as follows:

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

Where:

• $P$ is the put option price
• $N\left(-d_1\right)$ and $N\left(-d_2\right)$ are the cumulative standard normal distribution functions of $-d_1$ and $-d_2$ respectively

Key Variables

$d_1$ Calculation:

$$d_1=\frac{\ln \left(\frac{S_0}{K}\right)+\left(r+\frac{\sigma^2}{2}\right) t}{\sigma \sqrt{t}}$$

• $S_0$ is the current stock price.
• $K$ is the option’s strike price.
• $r$ is the risk-free interest rate.
• $\sigma$ is the implied volatility of the stock price.
• $t$ is the time to expiration (in years).

$d_2$ Calculation:

$$d_2=(d_1-\sigma \sqrt{t})$$

Example: Suppose you want to calculate the value of a European call option on a non-dividend-paying stock. The current stock price $(S_0)$ is USD 100, the option’s strike price $(K)$ is USD 105, the risk-free interest rate $(r)$ is 5%, the implied volatility $(\sigma)$ is USD 20%, and the time to expiration $(t)$ is 0.5 years.

Using the Black-Scholes-Merton Model, you can calculate $d_1$, $d_2$, and the call option price (C).

• $d_1$ Calculation $$d_1=\frac{\ln \left(\frac{100}{105}\right)+\left(0.05+\frac{0.2^2}{2}\right) \cdot 0.5}{0.2 \sqrt{0.5}} \approx 0.2$$

• $d_2$ Calculation $$d_2=(0.2-0.2 \sqrt{0.5}) \approx-0.09$$

• Calculating $N(d_1)$ and $N(d_2)$ using standard normal distribution tables:

  • $N(d_1) \approx 0.5793$
  • $N(d_2) \approx 0.4641$

Now, apply the call option pricing formula:

$$C = 100 \cdot 0.5793 - 105 \cdot e^{-0.05 \cdot 0.5} \cdot 0.4641 \approx \text{USD 5.97}$$

So, the calculated value of the European call option is approximately USD 5.97.