Our basic logic is,
$$ \text{Option Value} =\text{Discounted Expectation of the Payoff} $$
However, there is not a perfect correlation between the Option and the Underlying Asset. That fact encourage us to do the risk-netural method for valuing an option.
Geometric Brownian Motion
- Non-dividend paying stock, (S_0>0)
$$ dS_t = \mu S_t\ dt+\sigma S_t \ dW_t $$
- Money market, or bank account, (B_0 = 1)
$$ dB_t = r B_t\ d_t $$
and,
$$ \{ W_t \}_{t\in [0,T]} : \mathbb{P} \sim Brownian \ Motion $$
$ \{S_t\} $ is not a martingals under \mathbb{P} because \mu S_t is not zero.
So, we apply a transformation to S_t and to make it be a martingal (eliminate the drift term).
We here look at \{ \frac{S_t}{B_t} \},
$$ d\big( \frac{S_t}{B_t} \big)=-r\frac{S_t}{B_t}dt +\frac{1}{B_t} dS_t +\frac{1}{2}dS_t (-r\frac{1}{B_t}dt) $$
$$ d\big( \frac{S_t}{B_t} \big)=-r\frac{S_t}{B_t}dt +\bigg( \frac{1}{B_t}-\frac{1}{2}r\frac{1}{B_t}dt \bigg)\bigg( \mu S_t\ dt +\sigma S_t \ dW_t \bigg) $$
The cross terms dt & dW would decay to zero quickly in the stochastic integration. We therefore would get,
$$ d\big( \frac{S_t}{B_t} \big) = \sigma ( \frac{S_t}{B_t} )\underbrace{\bigg( \frac{\mu – r}{\sigma} dt + dW_t \bigg)}_{d\tilde{W_t}} $$
The drift disappears, instead we get d\big( \frac{S_t}{B_t} \big)d\tilde{W_t}
$$ d\big( \frac{S_t}{B_t} \big)d\tilde{W_t} $$
$$ d\tilde{W_t} = \frac{\mu -r}{\sigma} dt +dW_t $$
by Girsanov’s Theorem that \{a_t \}_{t\in [0,T]} be and adapted \{ \mathcal{F} \}_{t\in [0,T]} Ito Process, so that \mathbb{Q} is an equivalent measure on \mathbb{P} such that \tilde{W_t} is a Brownian Motion, \tilde{W_t} = W_t +\int_0^t a_u du.
$$ dS_t = rS_t \ d_t +\sigma S_t \ d\tilde{W_t} $$
Implication
The implication is that,
We can construct a relicating portfolio (S_t, B_t) with value process \{ V_t \}_{t\in[0,T]}. Under \mathbb{Q} the discounted value process exists that is a martingale \{ e^{-rt}V_t \}.
$$ e^{-rt} C_t =\mathbb{E}_{\mathbb{Q}}[ e^{-rT} C_T | \mathcal{F}_t ] $$
$$ =\mathbb{E}_{\mathbb{Q}}[ e^{-rT} V_T | \mathcal{F}_t ] = e^{-rt} V_t $$
Risk Neutral Expectation Pricing Formula
We therefore get,
$$ \frac{C_t}{B_t} = \mathbb{E}_{\mathbb{Q}}\bigg[ \frac{C_T}{B_t} \bigg| \mathcal{F}_t \bigg] $$
$$ B_T=1,\quad and \quad B_t=exp\{ \int_0^t r_s ds \} $$