Greeks in the Black–Scholes Model¶

Recall (see § Delta, Gamma, Vega, Theta, Rho): Greeks are partial derivatives of the pricing map \(V(t,S;\sigma,r,\dots)\). We now specialize these definitions to the Black–Scholes model and compute the Greeks explicitly from the closed-form European call/put formulas.

Recall (see § Black–Scholes formula): \(C = SN(d_1) - Ke^{-r\tau}N(d_2)\) — this section differentiates that formula.

Notation and Black–Scholes parameters¶

Let

• \(S\): current underlying price (at time \(t\))
• \(K\): strike
• \(r\): constant risk-free rate
• \(\sigma\): volatility
• \(T\): maturity
• \(t\): current time
• \(\tau := T-t\): time to maturity
• \(N(\cdot)\): standard normal CDF
• \(N'(x) = \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}\): standard normal PDF

Define

The Black–Scholes prices are

Preliminary: Derivatives of d₁ and d₂¶

Before computing Greeks, we establish the key derivatives:

A crucial identity linking \(N'(d_1)\) and \(N'(d_2)\):

Proof. Since \(d_1 - d_2 = \sigma\sqrt{\tau}\), we have

Now \(d_1 + d_2 = 2d_1 - \sigma\sqrt{\tau}\), and using \(d_1 = \frac{\ln(S/K) + (r + \frac12\sigma^2)\tau}{\sigma\sqrt{\tau}}\):

Thus \(\frac{N'(d_1)}{N'(d_2)} = \frac{K e^{-r\tau}}{S}\), giving \(S N'(d_1) = K e^{-r\tau} N'(d_2)\). \(\square\)

Delta¶

Derivation for a call. Differentiate \(C = SN(d_1) - Ke^{-r\tau}N(d_2)\) with respect to \(S\):

Since \(\frac{\partial d_1}{\partial S} = \frac{\partial d_2}{\partial S} = \frac{1}{S\sigma\sqrt{\tau}}\):

Using the identity \(SN'(d_1) = Ke^{-r\tau}N'(d_2)\), the last two terms cancel:

• Call:

• Put (by put-call parity \(P = C - S + Ke^{-r\tau}\)):

Delta lies in \((0,1)\) for calls and \((-1,0)\) for puts.

Gamma¶

Derivation. Differentiate \(\Delta_{\text{call}} = N(d_1)\) with respect to \(S\):

For both calls and puts,

Gamma is always positive (convexity in \(S\)) and becomes large near maturity (small \(\tau\)). Note that \(\Gamma_{\text{call}} = \Gamma_{\text{put}}\) since they differ by a linear function of \(S\).

Vega¶

Derivation for a call. Differentiate \(C = SN(d_1) - Ke^{-r\tau}N(d_2)\) with respect to \(\sigma\):

Using \(\frac{\partial d_1}{\partial \sigma} = -\frac{d_2}{\sigma}\) and \(\frac{\partial d_2}{\partial \sigma} = -\frac{d_1}{\sigma}\):

Using \(SN'(d_1) = Ke^{-r\tau}N'(d_2)\) and \(d_1 - d_2 = \sigma\sqrt{\tau}\):

For both calls and puts,

Vega is typically largest near-the-money and for moderate maturities.

Theta¶

Convention. Some texts define theta as \(-\partial_t V\) (time-to-maturity derivative).
Here we use calendar-time theta \(\Theta=\partial_t V\). Since \(\tau = T - t\), we have \(\frac{\partial}{\partial t} = -\frac{\partial}{\partial \tau}\).

Derivation for a call. Recall (see § BS PDE structure): from the Black–Scholes PDE,

Substituting the explicit forms:

• Call:

• Put:

Theta is typically negative (time decay), with exceptions in certain regimes (e.g., deep ITM puts where the interest earned on the strike exceeds time decay).

Rho¶

Derivation for a call. Differentiate \(C = SN(d_1) - Ke^{-r\tau}N(d_2)\) with respect to \(r\):

Since \(\frac{\partial d_1}{\partial r} = \frac{\partial d_2}{\partial r} = \frac{\sqrt{\tau}}{\sigma}\), the terms involving \(N'\) cancel:

• Call:

• Put:

Rho reflects the present-value effect of the strike and is usually small for short maturities.

Symmetries and notes¶

• Gamma and Vega are identical for calls and puts: they depend on the diffusion of \(S\), not the payoff direction.
• Delta, Theta, and Rho differ between calls and puts due to payoff asymmetry and the discounting term.
• All expressions above follow by differentiating the Black–Scholes closed forms for \(C\) and \(P\).

Conceptual interpretation (typical signs)¶

Summary¶

This section shows how the abstract Greeks reduce, in the Black–Scholes model, to explicit closed-form expressions. These formulas are the baseline for practical risk management, hedging, and calibration, and they also highlight where Black–Scholes is too restrictive (e.g., constant \(\sigma\) across strikes/maturities).

Higher-Order Greeks¶

For completeness, we present three commonly used second-order cross-Greeks:

Charm (delta decay):

For a call:

where \(q\) is the dividend yield (set \(q=0\) for non-dividend-paying stocks). Charm measures how delta changes as time passes.

Vanna (delta-vol sensitivity):

Vanna measures how delta changes with volatility, or equivalently how vega changes with spot.

Volga (vega convexity):

Volga is important for volatility surface dynamics and smile risk management. Long volga positions benefit from volatility-of-volatility.

Forward reference. Deferred — see § Vega and the smile for the role of vanna/volga in smile dynamics, and § Greek asymptotics for limiting behavior.

Exercises¶

Exercise 1. Using the identity \(SN'(d_1) = Ke^{-r\tau}N'(d_2)\), verify that vega is the same for a European call and a European put by directly computing \(\partial P / \partial \sigma\) from the put formula \(P = Ke^{-r\tau}N(-d_2) - SN(-d_1)\).

Exercise 2. For a European call with \(S = 100\), \(K = 105\), \(\tau = 0.5\), \(r = 0.03\), \(\sigma = 0.25\), compute \(d_1\), \(d_2\), and all five Greeks: \(\Delta\), \(\Gamma\), \(\nu\), \(\Theta\), \(\rho\). Verify the theta-gamma identity \(\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + rS\Delta - rC = 0\).

Exercise 3. Prove that the peak of gamma occurs near \(S = K\) for short maturities by setting \(d_1 = 0\) and solving for \(S\) in terms of \(K\), \(r\), \(\sigma\), and \(\tau\). Show that as \(\tau \to 0\), the peak location converges to \(S = K\).

Exercise 4. Using the closed-form expression for charm,

with \(q = 0\), compute the rate at which delta changes over one day for an ATM call with \(S = K = 100\), \(\sigma = 0.20\), \(r = 0.05\), \(\tau = 30/365\). Interpret the sign of charm in terms of the hedger's rebalancing needs.

Exercise 5. Show that volga \(= \nu \cdot d_1 d_2 / \sigma\) is negative for options that are near-the-money (where \(d_1 > 0\) and \(d_2 < 0\)) and positive for options that are sufficiently far from the money. At what moneyness does volga change sign?

Exercise 6. Using put-call parity \(P = C - S + Ke^{-r\tau}\), derive the put formulas for \(\Delta_{\text{put}}\), \(\Theta_{\text{put}}\), and \(\rho_{\text{put}}\) from the corresponding call formulas without differentiating the put pricing formula directly.