For the Black-Scholes market model where the price of the riskless asset (bond) satisfies $$dB_t=rB_tdt, B_0 = 1$$ for some $r>0$ and the stock price evolves according to $$dS_t = µS_tdt + σS_tdW_t, S_0 = 1,$$ where $µ, σ > 0$ constants and $W_t$ is a (standard) Brownian motion. With fixed time horizon $T > 0$, and fixed a constant $K>0$. How can we find the price of the European call option $$G=f(S_T ) $$ where $f(x) = (x − K)_+$.
I was wondering if anyone could help me? Thanks.
First of all you confuse the payoff of the option with its price. The payoff of this option is $C_T = \max\{S_T-K, 0\}$, you wish to find its price, that is, $C_0$. So the price of the stock at maturity (expiry) $S_T$ is given.
In order to find $C_0$ you have to use Black-Scholes formula \begin{align} C_t &= N(d_1)S_t - N(d_2) Ke^{-r(T - t)}, \\ \end{align} where, \begin{align} d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S_t}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\ d_2 &= d_1 - \sigma\sqrt{T - t}. \\ \end{align} and $x \mapsto N(x)$ is the cumulative distribution function of the standard normal distribution $\mathcal{N}(0,1)$. For the current price of the option set $t=0$.
If you wish to prove this formula (in particular you need the one when $t=0$), a nice approach is the martingale approach, that is, we have to find a unique measure $Q$ (so called risk neutral measure), under the assumption that our market is complete, equivalent to $P$ (measure from the definition of $(W_t)$) such that the discounted stock market is a martingale with respect to this measure and the Brownian filtration $(\mathcal{F}_t)$. Then $$ C_0 = e^{-rT}\mathbb{E}_Q[C_T].$$ To find this measure we have to apply Girsanov theorem to construct a Brownian motion $(W_t^Q)$ such that
$$ dS_t = rS_t dt + \sigma S_t dW^Q_t.$$
An alternative approach (the original Black-Scholes one) of finding $C_t$ is to assume that $C_t = g(t, S_t)$ for some "nice" function $g$ and use self-financing strategies and Ito's lemma. This will lead to the famous Black-Scholes PDE $$ C_t + \frac{1}{2}\sigma s^2C_{ss} + r(sC_s-C)=0.$$ Solving this PDE will give you the Black-Scholes formula.
Very good lecture notes for this subject can be found here http://www.ntu.edu.sg/home/nprivault/indext.html