See A bit Stochastic Calculus .
For,
$$ dS_t = \mu S_t \ dt +\sigma S_t \ dW_t $$
- In calculating d f(S_t), we would get, (by Taylor Expansion)
$$ df(S_t) = \bigg( \frac{\partial f}{\partial t} + \frac{\partial f}{\partial S_t}\mu S_t +\frac{1}{2}\frac{\partial^2 f}{\partial S_t^2}\sigma^2 S_t^2 \bigg)dt + \frac{\partial f}{\partial S_t}\sigma S_t \ dW_t $$
- A Special Form of f(\cdot) is f(S) = log(S),
$$ d\ log(S_t) = \bigg( \mu – \frac{1}{2}\sigma^2 \bigg)dt + \sigma \ dW_t $$
We get Y_t = log S_t is the price of a derivative security with respect to S_t and t and then,
$$ dY_t= \bigg( \frac{\partial Y_t}{\partial t} + \frac{\partial Y_t}{\partial S_t}\mu S_t +\frac{1}{2}\frac{\partial^2 Y_t}{\partial S_t^2}\sigma^2 S_t^2 \bigg)dt + \frac{\partial Y_t}{\partial S_t}\sigma S_t \ dW_t $$
Consider a portfolio \Pi is constructed with (1) short one derivative, and (2) long some fraction of stocks, \Delta, such that the portfolio is risk natural. (\Delta = \frac{\partial Y}{\partial S})
$$ \Pi_t = -Y +\Delta \ S_t $$
Differentiate it,
$$ d\Pi_t = -dY_t +\frac{dY}{dS}dS_t $$
Subtitute dY_t and dS_t into the above equation, we would then get the stochastic process of portfolio, by Ito’s Lemma.
$$ d\Pi_t =-\bigg( \frac{\partial Y}{\partial t} + \frac{1}{2}\frac{\partial^2 Y}{\partial S^2} \sigma^2 S_t^2 \bigg) dt $$
The diffusion term dW_t disappears, and that means the portfolio is riskless during the interval dt. Under a no arbitrage assumption, this portfolio can only earn the riskless return, r.
$$ d\Pi_t =r\Pi_t \ dt $$
- Subtitute d\Pi_t and \Pi_t into, we would get the Partial Differential Equation (PDE) / Black-Scholes equation,
$$ – \bigg( \frac{\partial Y}{\partial t} + \frac{1}{2}\frac{\partial^2 Y}{\partial S^2} \sigma^2 S_t^2 \bigg) dt = r\bigg(- Y_t + \frac{\partial Y}{\partial S}S_t \bigg)dt $$
$$ rY_t = \frac{\partial Y}{\partial t} + \frac{1}{2}\frac{\partial^2 Y}{\partial S^2} \sigma^2 S_t^2 + \frac{\partial Y}{\partial S}S_t $$
Then, we guess (where U(.) is a function of S_t at time t=T),
$$ Y_t = e^{-r(T-t)} U(S_T) $$
For a European call with strike price, K, U(S_T) would be the payoff at maturity,
$$ U(S_T) = max( S – K , 0 ) $$
Finally, through a series of process to find a specific solution of the PDE, we can solve the value of call, (\Phi(\cdot) is the cumulative standard normal distribtion)
$$ c = S_t\Phi(d_1) – K e^{-r (T-t)}\Phi(d_2) $$
with,
$$ d_1 =\frac{ log(S_t/K) + (r-\frac{1}{2}\sigma^2)(T-t)}{\sigma \sqrt{T-t}} $$
$$ d_1 = \sigma \sqrt{T-t} $$