The Black–Scholes formula
gives a single number — the price. But a trader who has sold an option lives or dies by how that
price moves when the market moves. The Greeks are the price's
sensitivities — its
partial derivatives
with respect to spot, volatility, time, and rate — each named after a Greek letter. They are the
dials of risk management, and the most important of them, delta, is exactly the
hedge ratio V_S we met when deriving the
PDE.
Throughout, \varphi is the standard-normal density and
\Phi its CDF, with
d_1, d_2 as in the formula. We will derive delta in full and record
the rest.
Delta, derived line by line
Delta is \Delta = \partial C / \partial S, the
change in the option price per unit change in the stock — the shares to hold against one call.
Differentiating C = S\,\Phi(d_1) - K e^{-rT}\,\Phi(d_2) looks
fearsome because d_1 and d_2 both depend on
S — but a beautiful cancellation rescues us.
Step 1 — differentiate by the product and chain rules. The first term is a
product; the two \Phi's need the chain rule (with
\Phi' = \varphi):
\frac{\partial C}{\partial S} = \Phi(d_1) + S\,\varphi(d_1)\,\frac{\partial d_1}{\partial S} - K e^{-rT}\,\varphi(d_2)\,\frac{\partial d_2}{\partial S}.
Step 2 — note the derivatives of
d_1, d_2 are equal. Since
d_2 = d_1 - \sigma\sqrt{T} and
\sigma\sqrt{T} is constant in S,
\frac{\partial d_2}{\partial S} = \frac{\partial d_1}{\partial S} = \frac{1}{S\,\sigma\sqrt{T}}.
Step 3 — invoke the key identity
S\,\varphi(d_1) = K e^{-rT}\,\varphi(d_2). This is the
load-bearing lemma (proved in the vignette): the two density terms are equal. Hence
the last two terms of Step 1, which share the common factor
\partial d_1/\partial S, cancel:
S\,\varphi(d_1)\,\frac{\partial d_1}{\partial S} - K e^{-rT}\,\varphi(d_2)\,\frac{\partial d_2}{\partial S} = \big(S\,\varphi(d_1) - K e^{-rT}\,\varphi(d_2)\big)\frac{\partial d_1}{\partial S} = 0.
Step 4 — collect. Only the very first term of Step 1 survives:
\Delta = \frac{\partial C}{\partial S} = \Phi(d_1).
Clean as a whistle. Delta is just \Phi(d_1) — a number between
0 and 1, the fraction of a share to hold.
Deep in the money \Delta \to 1 (the call behaves like the stock);
deep out, \Delta \to 0 (it behaves like cash).
For a European call C = S\,\Phi(d_1) - K e^{-rT}\,\Phi(d_2), the
first-order sensitivities are:
-
Delta
\Delta = \dfrac{\partial C}{\partial S} = \Phi(d_1) — shares to
hold per call (the hedge ratio).
-
Gamma
\Gamma = \dfrac{\partial^2 C}{\partial S^2} = \dfrac{\varphi(d_1)}{S\,\sigma\sqrt{T}}
— how fast delta itself moves; always positive, peaked near the money.
-
Vega
\mathcal{V} = \dfrac{\partial C}{\partial \sigma} = S\,\varphi(d_1)\sqrt{T}
— sensitivity to volatility.
-
Theta
\Theta = \dfrac{\partial C}{\partial t} — time decay (usually
negative for a long option).
-
Rho
\rho = \dfrac{\partial C}{\partial r} = K T e^{-rT}\,\Phi(d_2) —
sensitivity to the interest rate.
Everything hinged on the claim that the two density terms are equal. Here is why. Work with
the ratio and use \varphi(x) = \tfrac{1}{\sqrt{2\pi}}e^{-x^2/2}:
\frac{S\,\varphi(d_1)}{K e^{-rT}\,\varphi(d_2)} = \frac{S}{K e^{-rT}}\,\exp\!\Big(-\tfrac12 d_1^2 + \tfrac12 d_2^2\Big).
Factor the exponent as a difference of squares, using
d_2 = d_1 - \sigma\sqrt{T} so that
d_1 - d_2 = \sigma\sqrt{T}:
\tfrac12(d_2^2 - d_1^2) = \tfrac12 (d_2 - d_1)(d_2 + d_1) = -\tfrac12\,\sigma\sqrt{T}\,(d_1 + d_2).
Now d_1 + d_2 = 2 d_2 + \sigma\sqrt{T}, and
\sigma\sqrt{T}\,d_2 = \ln(S/K) + (r - \tfrac12\sigma^2)T straight
from the definition of d_2. So
\tfrac12(d_2^2 - d_1^2) = -\Big(\ln\tfrac{S}{K} + (r - \tfrac12\sigma^2)T\Big) - \tfrac12\sigma^2 T = -\ln\tfrac{S}{K} - rT.
Exponentiating, \exp(\tfrac12(d_2^2 - d_1^2)) = \tfrac{K}{S}\,e^{-rT}.
Substitute back into the ratio:
\frac{S\,\varphi(d_1)}{K e^{-rT}\,\varphi(d_2)} = \frac{S}{K e^{-rT}}\cdot\frac{K}{S}\,e^{-rT} = 1.
The ratio is exactly 1, so
S\,\varphi(d_1) = K e^{-rT}\,\varphi(d_2). This same identity is
what makes vega and gamma share the single factor \varphi(d_1).
Delta tells you how many shares to hold now; gamma tells you how quickly that number
goes stale. A delta-hedged book holds
\Delta = \Phi(d_1) shares against each short call, so a small move
in S leaves the portfolio value unchanged to first order — exactly
the riskless portfolio of the PDE derivation, rebalanced continuously.
But delta drifts as S moves, at rate
\Gamma. High gamma means frequent, costly rebalancing; a trader who
wants to be insulated from that too will gamma-hedge with a second
option to flatten \Gamma, leaving the book stable over larger
moves. The Greeks are, quite literally, the control surface of an options desk.
Delta and gamma against spot
Below, \Delta = \Phi(d_1) climbs smoothly from
0 (deep out of the money) to 1 (deep in),
crossing near the strike; \Gamma = \varphi(d_1)/(S\sigma\sqrt{T}) is
the bell-shaped slope of delta, tallest right around the money where the option is most
sensitive. Cutting \sigma or shortening T
sharpens both — delta steepens into a step, gamma into a spike. Slide and watch.
Note the bell \Gamma is exactly the rate at which the
\Delta curve rises — that is why a near-the-money option needs the
most frequent rehedging.