May 19, 2026
In modern corporate reporting, option valuation is not just an abstract exercise in financial engineering; it is a regulatory and practical necessity. Under current accounting standards, companies must record the fair value of stock options—such as employee stock options or corporate warrants—directly on their financial statements. Whether evaluating compensation expenses on the income statement or analyzing liabilities and equity components on the balance sheet, accountants and analysts rely on precise mathematical models to satisfy financial reporting requirements. Consequently, understanding how these options are valued is essential for anyone involved in financial accounting (FA) or financial statement analysis and valuation (FSAV).
Beyond its accounting applications, option pricing stands as one of the most fundamental topics in financial mathematics. An option is a financial derivative whose value depends on the price of an underlying asset, such as a stock. For example, a European call option gives its holder the right, but not the obligation, to buy the underlying stock at a fixed price—known as the strike price—at a specified future date. Because the future stock price is uncertain, determining the fair value of this contract is not trivial. Therefore, we require a mathematical model that can capture the random fluctuations of stock prices to determine a fair option price.
The Black–Scholes model provides a powerful framework for solving this problem. The core idea of the model is to describe the stock price as a stochastic process, meaning that its value evolves randomly over time. To build a clear understanding of this framework, we begin with the concept of a Markov process, where the future depends only on the current state and not on the past history. We then introduce the Wiener process, also known as Brownian motion, which serves as the foundation for modeling continuous random shocks in financial markets.
After establishing this probabilistic foundation, we model the stock price using Geometric Brownian Motion (GBM). This allows us to apply Itô’s Lemma and derive the famous Black–Scholes partial differential equation (PDE). By solving this PDE under the specific payoff conditions of a European call option—which can only be exercised at a fixed expiration date—we obtain the standard Black–Scholes formula.
While American options allow for exercise at any time before expiration, we focus on the European framework because, for a stock that pays no dividends, it can be mathematically proven that the early exercise of a call option is never optimal. Since a European call option insurance value (time value) is always positive prior to expiration for a non-dividend-paying stock, selling the option in the open market always yields a higher return than exercising it early. Therefore, the European model serves as an excellent proxy for an American call option, allowing us to analyze the contract without the added complexity of early exercise boundaries.
The goal of this essay is to explain the valuation of a stock option step by step, guiding the reader from the underlying stochastic behavior of asset prices to the formal derivation of the Black–Scholes model. As an educator teaching financial accounting (FA), management accounting (MA), and financial statement analysis and valuation (FSAV), my objective is to bridge the gap between complex quantitative finance and practical financial analysis. While the discussion relies on quantitative methods, each step is carefully explained to ensure that students can grasp both the financial intuition and the mathematical reasoning behind the model.
A Markov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting its future path. Under this framework, the past history of the variable and the specific sequence of events that led to its present state are entirely irrelevant. This characteristic is commonly referred to as the memoryless property.
In quantitative finance, stock prices are widely assumed to follow a Markov process. For instance, suppose that the price of a stock is $100 today. If the stock price behaves according to a Markov process, our predictions for its future performance are unaffected by whether the price was higher or lower one week ago, one month ago, or one year ago. The only relevant piece of information for forecasting tomorrow's distribution is that the price is exactly $100 right now.
This assumption aligns closely with the Weak-Form Efficient Market Hypothesis (EMH), which states that all past market prices and historical data are already fully reflected in the current asset price. Consequently, technical analysis or chart-reading cannot be used to consistently predict future returns, as the current price contains all available information required to establish expectations for the next step.
Consider a variable that follows a Markov stochastic process. Suppose that its current value is \$100 and that the change in its value during a year is $\phi(0, 400)$, where $\phi(m, v)$ denotes a probability distribution that is normally distributed with mean $m$ and variance $v$. This variance of 400 corresponds to a standard deviation of \$20 ($\sqrt{400} = 20$), representing a realistic annual price volatility of 20% for a \$100 stock. What is the probability distribution of the change in the value of the variable during 2 years?
Because the variable follows a Markov process, the two probability distributions for each individual year are independent. When we add two independent normal distributions, the result is a normal distribution where the mean is the sum of the means and the variance is the sum of the variances.
Therefore, when Markov processes are considered, the variances of the changes in successive time periods are additive:
This yields a total 2-year change distribution of $\phi(0, 800)$.
Crucially, the standard deviations of the changes in successive time periods are not additive. The new standard deviation is $\sqrt{800} \approx \$28.28$.
$$\sqrt{400 + 400} < \sqrt{400} + \sqrt{400}$$
$$\$28.28 < \$20 + \$20$$
The process followed by the variable we have been considering is known as a Wiener process. It is a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0 per year. It has been used in physics to describe the motion of a particle that is subject to a large number of small molecular shocks and is sometimes referred to as Brownian motion.
To simplify our framework and maintain consistency, we will use the stock price variable $S$ directly instead of a general variable $z$. A stock price starting from our example value of $S_0 = 100$ follows a Wiener process if it has the following two properties:
$$\Delta S = \epsilon \sqrt{\Delta t}$$
where $\epsilon$ has a standard normal distribution $\phi(0, 1)$.
Note: See the Appendix for the mathematical derivation of Property 1.
The second property implies that $S$ follows a Markov process. This means that the change in the stock price in any future time increment is completely independent of its past path, reinforcing the memoryless nature of the process as it evolves.
Now let us examine the total change in the stock price over a longer time horizon, $T$. We can consider this long interval as being composed of $N$ small, consecutive intervals of length $\Delta t$, such that:
$$T = N \Delta t$$
The total change in the stock price between time $0$ and time $T$ is the sum of the changes in each of the $N$ small intervals:
$$S(T) - S(0) = \sum_{i=1}^{N} \epsilon_i \sqrt{\Delta t}$$
where each $\epsilon_i$ ($i = 1, 2, \dots, N$) represents an independent standard normal random variable. Because the stock price follows a Markov process, these intervals are strictly independent, allowing us to find the parameters of the long-term total change distribution:
So far, our basic framework assumed that the expected value of a future stock price is exactly equal to its current price. While this works mathematically for a basic Wiener process, it is not a realistic assumption for financial markets. Investors expect a positive rate of return on their capital, meaning stock prices are generally expected to drift upward over time.
To create a more reasonable framework, we need to account for this expected trend by adding a drift component. The mean change per unit time for a stochastic process is known as the drift rate, and the variance per unit time is known as the variance rate.
The basic Wiener process, $dz$, has a drift rate of zero and a variance rate of 1.0. A drift rate of zero implies no expected growth. To model a realistic stock price $S$ that grows over time, we define a generalized Wiener process by introducing a continuous drift component alongside the market noise:
$$dS = a \, dt + b \, dz\tag{1.1}$$
where $a$ and $b$ are constants.
$$S = S_0 + aT $$
To conceptualize or simulate this continuous process across a small discrete time interval $\Delta t$, we can express the absolute change in our variable as $\Delta S$:
$$\Delta S = a \, \Delta t + b \epsilon \sqrt{\Delta t} $$
where $\epsilon$ is a random sample drawn from a standard normal distribution, $\epsilon \sim N(0, 1)$. Because $\Delta S$ is a linear transformation of this standard normal variable, the price change itself over a short window is normally distributed. Its parameters are derived directly from the components:
$$\text{mean of } \Delta S = a \, \Delta t$$ $$\text{standard deviation of } \Delta S = b \sqrt{\Delta t}$$ $$\text{variance of } \Delta S = b^2 \Delta t$$
The mathematical expression for the change in the variable $S$ over a small time interval $\Delta t$ is defined as:
$$\Delta S = \epsilon \sqrt{\Delta t}$$
where:
$$\epsilon \sim N(0, 1)$$
Because $\Delta S$ is a linear transformation of the standard normal random variable $\epsilon$, it is also normally distributed. Its parameters are derived as follows:
Mean
The expected value of the increment is zero:
$$E[\Delta S] = E[\epsilon \sqrt{\Delta t}] = \sqrt{\Delta t} \, E[\epsilon]$$
Since $E[\epsilon] = 0$:
$$E[\Delta S] = 0$$
Variance
The variance of the increment scales linearly with the time increment:
$$\text{Var}(\Delta S) = \text{Var}(\epsilon \sqrt{\Delta t}) = (\sqrt{\Delta t})^2 \, \text{Var}(\epsilon)$$
Since $\text{Var}(\epsilon) = 1$:
$$\text{Var}(\Delta S) = \Delta t$$
Standard Deviation
The standard deviation is the square root of the variance:
$$\sigma_{\Delta S} = \sqrt{\text{Var}(\Delta S)} = \sqrt{\Delta t}$$