Chapter 19: The Greek Letters

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Overview

The Greeks measure the sensitivity of an option's price to changes in underlying parameters. They are essential tools for hedging and risk management.

Delta ( $Δ\Delta$ )

Delta measures the rate of change of the option price with respect to the underlying asset price:

$Δ=∂f∂S\Delta = \frac{\partial f}{\partial S}$

Option Type	Delta Range	Delta Formula (BSM)
Long call	0 to 1	$N(d_1)$
Short call	0 to −1	$N(d_1)$
Long put	−1 to 0	$N(d_1) - 1$
Short put	0 to 1	$1 - N(d_1)$

At-the-money call delta ≈ 0.5
Deep in-the-money call delta → 1 (behaves like the stock)
Deep out-of-the-money call delta → 0

Delta Hedging

A position is delta-neutral when its overall delta is zero. For a portfolio:

$Δportfolio=∑wiΔi\Delta_{\text{portfolio}} = \sum w_i \Delta_i$

To make a position delta-neutral with $N$ shares held and $N_O$ options:

$N=−NO⋅ΔO\Delta_{\text{portfolio}} = N + N_O \cdot \Delta_O = 0 \implies N = -N_O \cdot \Delta_O$

Example: A bank sells 100,000 call options ( $Δ=0.6\Delta = 0.6$ ) for $300, 000. D e l t a - n e u t r a l h e d g e : b u y$ 100,000 \times 0.6 = 60,000 $s ha r es . I f t h es t oc k p r i cer i ses b y$ 1, the options lose $60, 000 b u tt h es ha r es g ain$ 60,000.

Dynamic Hedging

Delta changes as the stock price changes, so the hedge must be rebalanced periodically. The cost of delta hedging an option position converges to the Black–Scholes–Merton price as rebalancing frequency increases.

Theta ( $Θ\Theta$ )

Theta measures the rate of change of the option price with respect to time:

$Θ=∂f∂t\Theta = \frac{\partial f}{\partial t}$

Option	Theta Sign	Meaning
Long call (no div)	Negative	Loses value as time passes
Long put (no div)	Usually negative	Loses value as time passes

For a European call:

$Θ=−S0N′(d1)σ2T−rKe−rTN(d2)\Theta = -\frac{S_0 N'(d_1) \sigma}{2\sqrt{T}} - rK e^{-rT} N(d_2)$

Theta is usually expressed per calendar day (divide yearly theta by 365) or per trading day (divide by 252). Time decay accelerates as expiration approaches.

Gamma ( $Γ\Gamma$ )

Gamma measures the rate of change of delta with respect to the stock price:

$Γ=∂2f∂S2=N′(d1)S0σT\Gamma = \frac{\partial^2 f}{\partial S^2} = \frac{N'(d_1)}{S_0 \sigma \sqrt{T}}$

Gamma is highest for at-the-money options
Gamma is highest for short-dated options
Gamma is always positive for long positions (good for holder)
Gamma is always negative for short positions (creates hedging difficulties)

High positive gamma: Delta increases as the stock rises, decreases as it falls → favorable for the holder ("buying low, selling high" when rebalancing).

High negative gamma: Opposite effect → unfavorable, requires buying high and selling low.

Relationship Between Delta, Theta, and Gamma

For a portfolio $Π\Pi$ of derivatives on a non-dividend-paying stock:

$Θ+rSΔ+12σ2S2Γ=rΠ\Theta + rS\Delta + \frac{1}{2}\sigma^2 S^2 \Gamma = r\Pi$

For a delta-neutral portfolio:

$Θ+12σ2S2Γ=rΠ\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma = r\Pi$

This shows that theta and gamma have opposite signs: a large positive gamma implies a large negative theta (and vice versa). If gamma is large and positive, time decay is large, offsetting the gamma benefit in expected terms.

Vega ( $ν\nu$ )

Vega measures sensitivity to volatility:

$ν=∂f∂σ=S0TN′(d1)\nu = \frac{\partial f}{\partial \sigma} = S_0 \sqrt{T} N'(d_1)$

Vega is positive for long call and long put positions
Vega is highest for at-the-money options
Vega increases with time to maturity (longer options are more sensitive to volatility)
Vega is highest when the option is near the money (approximately at-the-money)

A position with high positive vega benefits from volatility increases. Traders holding volatility-sensitive positions are said to be long volatility or short volatility.

Rho ( $ρ\rho$ )

Rho measures sensitivity to the risk-free rate:

$ρ=∂f∂r\rho = \frac{\partial f}{\partial r}$

For a European call: $ρ=KTe−rTN(d2)\rho = K T e^{-rT} N(d_2)$ For a European put: $ρ=−KTe−rTN(−d2)\rho = -K T e^{-rT} N(-d_2)$

Rho is generally the least important Greek for short-dated options (except for very large rate changes or very long-dated options like equity warrants).

Summary of Greek Signs

Position	Delta	Gamma	Theta	Vega	Rho
Long call	+	+	−	+	+
Short call	−	−	+	−	−
Long put	−	+	+/−	+	−
Short put	+	−	+/−	−	+

Scenario Analysis

Beyond Greeks, risk managers use scenario analysis: compute the portfolio value under specific market moves (e.g., "equity down 20%, vol up 10%, rates down 50bp"). This captures non-linear effects that Greeks miss for large moves.

Portfolio Insurance

Portfolio insurance creates a synthetic put on a stock portfolio by dynamically trading between stocks and cash (or futures). When markets fall, the strategy sells stocks → accelerates market declines. Portfolio insurance was widely blamed for amplifying the 1987 stock market crash.