Everything has been leading here. The
risk-neutral price
of a European call is the discounted expected payoff,
C = e^{-rT}\,\mathbb{E}_{\mathbb{Q}}\!\big[(S_T - K)^+\big],
and under \mathbb{Q} the terminal stock price is
lognormal,
S_T = S_0\,\exp\!\Big(\big(r - \tfrac12\sigma^2\big)T + \sigma\sqrt{T}\,Z\Big), \qquad Z \sim N(0, 1).
That is a single Gaussian integral. We will evaluate it and out will drop a closed form so
clean it won
Itô's successors a Nobel Prize — two cumulative
normals, \Phi(d_1) and \Phi(d_2), the
Bachelier dream made exact.
Evaluating the expectation, line by line
The call pays only when it finishes in the money, S_T > K. We split
the expectation into the two pieces hiding inside
(S_T - K)^+ and handle each.
Step 1 — split the payoff. On the event
\{S_T > K\} the payoff is S_T - K; elsewhere
it is zero. Writing \mathbf{1} for the indicator,
C = e^{-rT}\,\mathbb{E}_{\mathbb{Q}}\!\big[(S_T - K)\,\mathbf{1}_{\{S_T > K\}}\big] = e^{-rT}\Big(\underbrace{\mathbb{E}_{\mathbb{Q}}[S_T\,\mathbf{1}_{\{S_T > K\}}]}_{\text{term I}} - K\,\underbrace{\mathbb{Q}(S_T > K)}_{\text{term II}}\Big).
Step 2 — solve S_T > K for
Z. Substitute the lognormal form and take logs. The inequality
S_0 e^{(r - \frac12\sigma^2)T + \sigma\sqrt{T}Z} > K becomes, after
\ln and rearranging for Z:
Z > -\,\frac{\ln(S_0/K) + (r - \tfrac12\sigma^2)T}{\sigma\sqrt{T}} = -\,d_2, \qquad d_2 := \frac{\ln(S_0/K) + (r - \tfrac12\sigma^2)T}{\sigma\sqrt{T}}.
Step 3 — compute term II, \mathbb{Q}(S_T > K).
It is now \mathbb{Q}(Z > -d_2), and by the symmetry of the standard
normal, \mathbb{P}(Z > -d_2) = \mathbb{P}(Z < d_2) = \Phi(d_2):
\mathbb{Q}(S_T > K) = \Phi(d_2).
So \Phi(d_2) is the risk-neutral probability of exercise.
Step 4 — state term I,
\mathbb{E}_{\mathbb{Q}}[S_T\,\mathbf{1}_{\{S_T > K\}}].
Multiplying by S_T = S_0 e^{(r - \frac12\sigma^2)T + \sigma\sqrt{T}Z}
before integrating shifts the effective mean of the Gaussian by
\sigma\sqrt{T} (the computation is in the vignette), turning the
threshold -d_2 into -d_1 with
d_1 = d_2 + \sigma\sqrt{T}:
\mathbb{E}_{\mathbb{Q}}[S_T\,\mathbf{1}_{\{S_T > K\}}] = S_0\,e^{rT}\,\Phi(d_1), \qquad d_1 := d_2 + \sigma\sqrt{T} = \frac{\ln(S_0/K) + (r + \tfrac12\sigma^2)T}{\sigma\sqrt{T}}.
Step 5 — combine. Put terms I and II back into Step 1 and watch the
e^{rT} from term I cancel the discount
e^{-rT}:
C = e^{-rT}\Big(S_0\,e^{rT}\,\Phi(d_1) - K\,\Phi(d_2)\Big) = S_0\,\Phi(d_1) - K\,e^{-rT}\,\Phi(d_2).
That is the Black–Scholes formula. The price is the stock weighted by
\Phi(d_1), less the discounted strike weighted by the exercise
probability \Phi(d_2).
A European call and put on a non-dividend stock, with spot
S_0, strike K, expiry
T, rate r, volatility
\sigma, are worth:
-
Call:
C = S_0\,\Phi(d_1) - K\,e^{-rT}\,\Phi(d_2);
-
Put:
P = K\,e^{-rT}\,\Phi(-d_2) - S_0\,\Phi(-d_1);
-
d_1 = \dfrac{\ln(S_0/K) + (r + \tfrac12\sigma^2)T}{\sigma\sqrt{T}};
-
d_2 = d_1 - \sigma\sqrt{T} = \dfrac{\ln(S_0/K) + (r - \tfrac12\sigma^2)T}{\sigma\sqrt{T}}.
Here is the integral promised in Step 4. Write
S_T = S_0\,e^{(r - \frac12\sigma^2)T + \sigma\sqrt{T}z} and
integrate against the standard normal density
\varphi(z) = \tfrac{1}{\sqrt{2\pi}}e^{-z^2/2} over the exercise
region z > -d_2:
\mathbb{E}_{\mathbb{Q}}[S_T\,\mathbf{1}_{\{S_T > K\}}] = \int_{-d_2}^{\infty} S_0\,e^{(r - \frac12\sigma^2)T + \sigma\sqrt{T}z}\,\frac{1}{\sqrt{2\pi}}\,e^{-z^2/2}\,dz.
Pull S_0 e^{(r - \frac12\sigma^2)T} out and combine the two
exponentials, completing the square in the exponent
\sigma\sqrt{T}z - \tfrac12 z^2:
\sigma\sqrt{T}\,z - \tfrac12 z^2 = -\tfrac12\big(z - \sigma\sqrt{T}\big)^2 + \tfrac12\sigma^2 T.
The leftover constant \tfrac12\sigma^2 T combines with the
prefactor: e^{(r - \frac12\sigma^2)T}\,e^{\frac12\sigma^2 T} = e^{rT}.
So
\mathbb{E}_{\mathbb{Q}}[S_T\,\mathbf{1}_{\{S_T > K\}}] = S_0\,e^{rT}\int_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}}\,e^{-\frac12 (z - \sigma\sqrt{T})^2}\,dz.
Substitute w = z - \sigma\sqrt{T}. The lower limit moves from
-d_2 to -d_2 - \sigma\sqrt{T} = -d_1,
and the integrand is a clean standard normal:
= S_0\,e^{rT}\int_{-d_1}^{\infty}\varphi(w)\,dw = S_0\,e^{rT}\,\mathbb{P}(W > -d_1) = S_0\,e^{rT}\,\Phi(d_1).
The shift of the mean by \sigma\sqrt{T} — convexity again, the same
mechanism behind the lognormal mean — is exactly what turns d_2
into d_1.
The formula's two halves are not interchangeable lookalikes; each
\Phi means something concrete.
-
\Phi(d_2) = \mathbb{Q}(S_T > K) is the risk-neutral
probability the option is exercised — it fell straight out of Step 3.
-
\Phi(d_1) = \partial C / \partial S_0 is the
delta, the number of shares to hold to hedge one call — the
Greek
that links this formula back to replication. (It is also the exercise probability under a
different, stock-numéraire measure — hence the family resemblance.)
So C = S_0\Phi(d_1) - Ke^{-rT}\Phi(d_2) reads: "hold
\Phi(d_1) shares, borrow the present value of the strike times the
chance you will need it." The replication portfolio and the formula are the same object.