Derive the Black– Scholes formula for the European call option.

Question

Consider the standard Black–Scholes model. Derive the Black– Scholes formula for the European call option.

thanks for help.

Answer 1

The equation $dS(t)=rS(t)dt+\sigma S(t)dW(t)$ is not the Black-Scholes formula. It is a stochastic differential equation for geometric Brownian motion, which is one of the assumptions made in the derivation of the Black-Scholes-Merton pricing formula for an option.

The stock price, $S(t)$, at any future time, is a lognormal random variable -- under the assumption of geometric Brownian motion.

For a call option with strike $K$ that expires at time $T$, the option price at some earlier time $t$ is the expected payoff under the risk-neutral measure (where a more general drift $\mu$ can be replaced by a risk-free rate $r$ as you have already shown.) This takes the form

$$C[S(t),t] = E_t \{\max[S(T)-K,0]\},$$

where the expectation is conditioned on the information known at time $t$.

Using a dynamic hedging argument, it can be shown that the option price satisfies the Black_Scholes partial differential equation

$$\frac{\partial C}{\partial t} +rS\frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}-rC=0,$$

which can be solved in closed form (under suitable assumpions such as constant volatility) for the Black-Scholes option pricing formula.