Derivation of the Black-Scholes-Merton PDE › The Model Assumptions

Before deriving the PDE, we must clearly state the assumptions underlying the Black-Scholes-Merton framework. These assumptions define an idealized market that, while not perfectly realistic, captures enough structure to produce useful results. Each assumption plays a specific role in the derivation, and understanding why each is needed helps you appreciate both the power and the limitations of the resulting framework.

The most critical assumption is that the stock price $S_t$ follows geometric Brownian motion. This mathematical model captures the essential features we observe in real stock prices: they fluctuate randomly, they exhibit larger absolute movements when prices are higher, and they cannot become negative. The geometric Brownian motion specification takes the form:

where:

• $S_t$: stock price at time $t$
• $\mu$: expected return (drift) of the stock
• $\sigma$: volatility of stock returns
• $dW_t$: increment of a standard Wiener process (random shock)
• $dt$: small time increment

To understand this equation intuitively, think of the stock price as experiencing two simultaneous influences over each infinitesimally small time interval. The first influence, captured by $\mu S_t \, dt$, represents the deterministic drift: on average, the stock price tends to grow at rate $\mu$. This drift is proportional to the current price, which reflects the multiplicative nature of returns. The second influence, captured by $\sigma S_t \, dW_t$, represents the random shocks that buffet the price. The Wiener increment $dW_t$ is the mathematical formalization of "white noise," introducing unpredictable fluctuations. The volatility parameter $\sigma$ scales these fluctuations, and multiplying by $S_t$ ensures that absolute price movements scale with the price level.

As we discussed in our chapter on Brownian Motion and Random Walk Models, this model ensures stock prices remain positive and that returns are normally distributed. The positivity follows from the multiplicative structure: since we're multiplying by $S_t$, the process never crosses zero.

Let $V(S, t)$ denote the price of a derivative security that depends on the stock price $S$ and time $t$. Our goal is to determine what equation $V$ must satisfy. We make no assumptions about the specific form of $V$: it could represent a call option, a put option, or any other derivative whose payoff depends on the stock price at maturity. This generality is one of the beautiful features of the BSM framework: rather than deriving separate formulas for each type of derivative, we derive a single equation that all derivatives must satisfy.