Introduction

Understanding the lognormal property of stock prices is fundamental when delving into the Black-Scholes-Merton Model. This chapter explores the characteristics of stock price distributions, the lognormal nature of these distributions, and how it all ties into the calculation of expected returns.

Lognormal Property

Stock prices are inherently uncertain and subject to fluctuations. These price changes, often referred to as returns, can be modeled using a lognormal distribution. The lognormal distribution is characterized by two key parameters: the mean return $(\mu)$ and the standard deviation of returns $(\sigma)$.

Mathematical Representation

The lognormal distribution is mathematically represented as:

$$P(S=s)=\frac{1}{s \sigma \sqrt{2 \pi}} e^\left(\frac{-(\ln (s)-\mu)^2}{2 \sigma^2}\right)$$

Where:

• $P(S=s)$ is the probability of the stock price being $s$.
• $s$ is the stock price.
• $\sigma$ is the standard deviation of returns.
• $\mu$ is the mean return.

Distribution of Rates of Return

Rates of return, denoted as $r$, are calculated as the natural logarithm of the ratio of two stock prices at different times:

$$r=\ln \left(\frac{S_t}{S_0}\right)$$

Here, $S_0$ represents the initial stock price, and $S_t$ represents the stock price at time $t$. Rates of return are assumed to be normally distributed. This assumption simplifies the modeling of stock prices and forms the basis for the Black-Scholes-Merton Model.

Calculation of Expected Return

The expected return $(E(r))$ is a crucial metric in financial modeling. It represents the average rate of return an investor can anticipate from holding a particular stock. The calculation of the expected return is based on the lognormal distribution:

$$E(r)=e^{\mu+\frac{\sigma^2}{2}}$$

Example: Let’s say you have historical data on a stock’s returns. The mean return $(\mu)$ is 0.10, and the standard deviation $(\sigma)$ is 0.20. Using the formula:

$$\begin{aligned}& E(r)=e^{0.10+\frac{0.20^2}{2}} \\ & E(r)=e^{0.10+0.02} \\ & E(r)=e^{0.12}\end{aligned}$$ Using the exponential function, you can calculate the expected return:

$$E(r) \approx 1.127$$

So, the expected return for this stock is approximately 11.27%.