Final answer:
The Black–Scholes–Merton model assumes stock prices follow a log-normal distribution, reflecting a random walk with a trend due to normally distributed percentage changes in stock prices. It allows for risk-free hedging and results in a partial differential equation to price the options.
Step-by-step explanation:
The Black–Scholes–Merton model is a cornerstone in modern financial theory and is used for the pricing of European stock options. It assumes that stock prices follow a log-normal distribution because the model applies the random walk hypothesis combined with the assumption of normally distributed percentage changes in stock price.
Specifically, in the context of option pricing, the model posits that the underlying stock price follows a continuous geometric Brownian motion.
The distribution of the stock price in one year under the Black–Scholes–Merton frame is log-normal, which implies that while stock prices can only be positive, there's a skew because stock prices can theoretically rise to any level, but can only fall as low as zero.
Under this model, the motion of the stock price is characterized by two parameters: the expected return (drift) and the volatility (standard deviation of the returns). These parameters contribute to the 'trend' and 'random walk' nature of stock price movements respectively.
According to the model, in the absence of arbitrage opportunities, it’s possible to construct a risk-free portfolio by holding the stock and the option in certain proportions.
This concept allows the use of hedging to eliminate risk and leads to a partial differential equation that the price of the option must satisfy. The equation can then be solved, given the boundary conditions, to find the theoretical price of the option.