 Solution

A standard Brownian motion $W_t$ starts at zero, has continuous sample paths, and has independent increments $W_t-W_s\sim N(0,t-s)$ for $0\le s<t$. The Black-Scholes model assumes a frictionless market permitting continuous, self-financing trading and short selling, a bank account $B_t=e^{rt}$ at constant interest rate, and a non-dividend-paying assetwith constant coefficients and no arbitrage.

For an option price $C(S,t)$, Itô's formula gives diffusion coefficient $\sigma S{C}_S$. Holding $C_S$ units of the asset hedges this diffusion. The hedged portfolio has value $C-S{C}_S$ and instantaneous gain $(C_t+\frac{1}{2}{\sigma }^2{S}^2{C}_{SS})dt$, so no arbitrage requiresEquivalently, the risk-neutral measure makes $e^{-rt}{S}_t$ a martingale and gives the solution $C(S_0,0)=e^{-rT}{\mathbb{E}}_Q(S_T-K{)}^{+}$. Under that measure,Put $d_2=[\log (S_0/K)+(r-{\sigma }^2/2)T]/(\sigma \sqrt{T})$ and $d_1=d_2+\sigma \sqrt{T}$. Then ${\mathbb{P}}_Q(S_T>K)=\Phi (d_2)$, and completing the square in the normal integral gives ${\mathbb{E}}_Q[S_T{1}_{S_T>K}]=S_0{e}^{rT}\Phi (d_1)$. Therefore the Black-Scholes formula isHere $K>0$; a zero strike has value $S_0$ directly.

For the forward-start call option, condition at the strike-fixing time $t$. Given ${\mathcal{F}}_t$, the remaining asset-price ratio has the same lognormal law over $\tau =T-t$, independently of past increments. Thus its time-$t$ value isSince ${\mathbb{E}}_Q(e^{-rt}{S}_t)=S_0$, conditioning once more yieldsThe strike is random before $t$; inserting an unknown $S_t$ as a fixed strike at time zero would miss the conditioning step.