The Greeks

Medium5–6 hrsDeltaGammaVegaThetaHedging

Overview

The Greeks describe how an option's value responds to changes in each variable. Delta is the most important — it tells you how much your position moves like stock. Gamma tells you how quickly Delta changes (non-linearity). The key insight: long options are long gamma (you BENEFIT from large moves) and short theta (you PAY time decay). Short options are the opposite. This Gamma-Theta tradeoff is at the heart of options trading.

How to Recognize

→'How does option price change if stock moves $1?'
→Delta-hedging questions — how many shares to buy/sell
→P&L attribution: why did my options book make/lose money?
→'Which option is most sensitive to volatility?'

Step-by-Step Approach

1.Delta (Δ): sensitivity to stock price. 0 to 1 for calls, −1 to 0 for puts
2.Gamma (Γ): rate of change of delta. Largest for ATM options near expiry
3.Vega (ν): sensitivity to volatility. Always positive for long options
4.Theta (Θ): time decay. Negative for long options (you lose value as time passes)
5.Delta-hedge: hold −Δ shares per option to neutralize price risk

Key Formulas

01Call:

\Delta_C = \Phi(d_1)

; Put:

\Delta_P = \Phi(d_1)-1 = \Delta_C - 1

\Gamma = \frac{\phi(d_1)}{S\sigma\sqrt{T}}

(same for put and call)

03ATM shortcuts:

\Delta \approx 0.5

\Gamma \approx \phi(0)/(S\sigma\sqrt{T})

04P\&L

\approx \Delta\cdot\Delta S + \tfrac{1}{2}\Gamma(\Delta S)^2 + \nu\cdot\Delta\sigma + \Theta\cdot\Delta t

05Gamma-Theta tradeoff:

\Theta \approx -\tfrac{1}{2}\Gamma(\sigma S)^2

Worked Examples

0Quick Example

ATM call, Δ≈0.5, Γ≈0.02. Stock moves

2. P&L \approx 0.5\cdot2 + ½\cdot0.02\cdot4 = 1 + 0.04 =

1.04.

Problem

You're short 100 ATM calls on a $50 stock (Δ≈0.5). How do you delta-hedge? What happens when the stock moves to$ 52?

Solution

Being short 100 calls with Δ=0.5 each means: option portfolio has Δ = −100·0.5 = −50 (negative because short).

To delta-hedge: buy 50 shares of stock (portfolio Δ = −50 + 50 = 0).

Stock moves to $52 (+$ 2). Call delta increases (say Δ = 0.55 now, because option is more in-the-money).

New option portfolio Δ = −100·0.55 = −55. Stock position Δ = +50. Net Δ = −5.

You are now short 5 delta — need to buy 5 more shares to re-hedge.

This constant re-hedging is called 'gamma scalping' — you pay transaction costs but are delta-neutral.

Answer

Buy 50 shares initially. After $2 move, delta increases and you need to buy 5 more shares to stay neutral. Delta hedging requires continuous adjustment — this is the 'rebalancing' that Black-Scholes assumes.

Common Mistakes

!
ATM delta ≈ 0.5 is an approximation (exact value uses Φ(d₁) which depends on rates and vol). For quick interviews, 0.5 is fine.
!
Vega is NOT a Greek letter — it's named 'vega' but uses ν in some texts. Don't be confused.
!
P&L uses Γ(ΔS)²/2, not just Γ·ΔS. The factor of 1/2 comes from Taylor expansion.
!
Gamma is the same for puts and calls with the same strike/expiry (follows from PCP differentiating twice).

Practice Problems

Click "Show Answer" to reveal

You hold a delta-neutral straddle (long call + long put, same strike). What happens to your P&L if the stock makes a big move? What about if vol rises?

Hint: Delta neutral, long gamma, long vega.

Which is more gamma-sensitive: a 1-month ATM option or a 6-month ATM option (same stock, same vol)?

Hint: Γ ∝ 1/(S·σ·√T). What happens as T decreases?

An option has delta = 0.6. The stock falls by $1. Approximately what is the new delta?

Hint: Delta changes at rate Gamma. If Γ = 0.04 and stock falls $1, delta decreases by Γ·1.

Back to Study Plan

Put-Call Parity Black-Scholes Intuition