Take the tour ×  Stock price has a classic model based on GBM:
$$dS = \mu S dt + \sigma S dW$$  based on this call options
values could be solve -- Black-Scholes formula.  But, what is
the solution for the Stock price itself? is it  $$S(t) = S(0)
e^{\mu t + \sigma W}$$ ?  share|improve this question add
comment  2 Answers  up vote 3 down vote accepted  Your guess
would have been true if $dW$ were just an ordinary differential.
But this is not the case. $dW$ behaves somewhat different from
the ordinary differential, in that intuitively we have $dW
\approx \sqrt{dt}$ unlike to ordinary case. That's why the Ito,s
formula is so important.  Now let us make a heuristic
calculation. Let $Z_t = f(S_t) = \log S_t$, where $f(x) = \log
x$. Then by the Ito's formula, we have  $$ Z_t - Z_0 = f(S_t) -
f(S_0) = \int_{0}^{t} f'(S_s) \; dS_s + \int_{0}^{t} \frac{1}{2}
f''(S_s) \; dS_s^2,$$  where $dS^2$ is a formal symbol given by
the product rule  $$dW_s^2 = ds \quad \text{and} \quad dW_s ds =
ds dW_s = 0 = ds^2.$$  Thus $d