S(t)-S_{0}=\mu\int_{0}^{t}S(t)\;dt+\sigma\int_{0}^{t}S(t)\;dW(t).
In "differential" form,
dS_{t}=\mu S_{t}dt+\sigma S_{t}dW_{t}.
This contract is known as a binary call option. We seek to determine its price V at the initial time t=0.
The Black-Scholes theory implies (through its connection to parabolic partial differential equations and the Feyman-Kac formula) that this price is equal to the risk-neutral expected payoff \mathbb{E}_{\mathbb{Q}} discounted at the risk-free rate r. That is,
V=e^{-rT}\mathbb{E}_{\mathbb{Q}}[B].
A simple computation reveals that this is equal to e^{-rT}\mathbb{Q}(S_{T}>K). In detail,
e^{rT}V=\mathbb{E}_{\mathbb{Q}}[B]=1\cdot\mathbb{Q}(B=1)+0\cdot\mathbb{Q}(B=0)=\mathbb{Q}(B=1)=\mathbb{Q}(S_{T}>K).
Now, as covered in the Black-Scholes theory, an application of Girsonov's theorem leads us to the equivalent process for S
dS=rSdt+\sigma Sd\tilde{W_{t}}
where \tilde{W} is a Brownian motion under the equivalent martingale measure (risk-neutral measure) \mathbb{Q}. We now proceed as usual taking the natural logarithm to get
d\log S=(r-\frac{1}{2}\sigma^{2})dt+\sigma d\tilde{W_{t}},
which, in light of the risk-neutral probability computation above, immediately leads us to
\mathbb{E}_{\mathbb{Q}}[B]=N(d_{2}),
where N=\Phi(0,1), the standard normal density, and
d_{2}=\frac{\ln(S_{0}/K)+(r-\frac{1}{2}\sigma^{2})T}{\sigma\sqrt{T}}.
Thus,
V=e^{-rT}N(d_{2}).
The price of the put can be determined by following the steps above, but where the contract has a payoff of 1 when S(T)\leq K. The result is
V=e^{-rT}N(d_{1}),
or equivalently
V=e^{-rt}N(-d_{2}).
As an aside from this, we can compute the price of this option using only the known prices of European call options. Observe that the payoff function P of this option is given by
P=\frac{\partial C_{K}}{\partial S}
where C is the payoff function for European call option with strike price K.
This observation leads us to the conclusion
V=\lim_{\delta\to0}\frac{{U(K-\delta})-U(K)}{\delta}=-\frac{\partial U}{\partial S},
where U(K) is the value of the call option with strike K.
