Learn Gamma and delta sensitivity

Overview

Gamma measures how fast delta changes when the underlying stock price moves. Delta tells you your directional exposure right now, but gamma tells you how unstable that exposure is. High gamma means your delta swings wildly—you can go from hedged to massively exposed in a small price move.

WHY this definition? Delta quantifies directional risk. If you own a call with $\Delta = 0.6$, you gain $0.60 for every$1 the stock rises per share. Since one standard option contract controls 100 shares, that call gains $60 per$1 move at the contract level. It's your equivalent stock position: owning that call is like owning 0.6 shares (or 60 shares per contract).

WHY the second derivative? Because risk isn't linear. Your hedge ratio (delta) shifts as the market moves. Gamma captures that curvature. High gamma = your hedge breaks down quickly.

Derivation from First Principles

Start with the Black-Scholes call value: $C = S N(d_1) - K e^{-rT} N(d_2)$

where $d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ and $d_2 = d_1 - \sigma\sqrt{T}$.

Step 1: Find delta by differentiating with respect to $S$.

$\Delta_{\text{call}} = \frac{\partial C}{\partial S} = N(d_1) + S N'(d_1) \frac{\partial d_1}{\partial S} - K e^{-rT} N'(d_2) \frac{\partial d_2}{\partial S}$

WHY? Product rule on $S N(d_1)$ and chain rule on the cumulative normals.

Compute the partials: $\frac{\partial d_1}{\partial S} = \frac{1}{S \sigma \sqrt{T}}, \quad \frac{\partial d_2}{\partial S} = \frac{1}{S \sigma \sqrt{T}}$

The standard normal PDF is $N'(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$. Using the Black-Scholes putcall relationship, the terms involving $N'(d_1)$ and $N'(d_2)$ cancel:

$S N'(d_1) = K e^{-rT} N'(d_2)$

WHY do they cancel? Because $d_2 = d_1 - \sigma\sqrt{T}$, and substituting into the normal PDF with the discount factor gives this identity. This is Black-Scholes symmetry.

Thus: $\Delta_{\text{call}} = N(d_1)$

For a put (by put-call parity or direct differentiation): $\Delta_{\text{put}} = N(d_1) - 1 = -N(-d_1)$

Step 2: Find gamma by differentiating delta.

$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial N(d_1)}{\partial S} = N'(d_1) \frac{\partial d_1}{\partial S}$

Substitute: $\Gamma = \frac{N'(d_1)}{S \sigma \sqrt{T}} = \frac{1}{S \sigma \sqrt{T}} \cdot \frac{1}{\sqrt{2\pi}} e^{-d_1^2/2}$

WHY is gamma positive? $N'(d_1) > 0$ always. Whether you own a call or put, gamma is the same: long options gain convexity, short options lose it.

Gamma (same for calls and puts): $\Gamma = \frac{N'(d_1)}{S \sigma \sqrt{T}}$

Gamma peaks at-the-money because $d_1 \approx 0$ when $S \approx K$, maximizing $N'(d_1)$.

Delta Sensitivity: How Position Delta Changes

Consider a portfolio with $n$ options, each with delta $\Delta_i$ and gamma $\Gamma_i$. If the stock moves by $\Delta S$:

$\Delta_{\text{new}} \approx \Delta_{\text{old}} + \Gamma \cdot \Delta S$

WHY? Taylor expansion: $\Delta(S + \Delta S) \approx \Delta(S) + \Gamma(S) \cdot \Delta S$. This is a linear approximation; higher-order greks (speed, etc.) refine it.

Scenario: Stock moves to $101.

$\Delta_{\text{new}} \approx 0.50 + 0.08 \cdot 1 = 0.58$

WHY this matters: Per contract (100 shares), you were hedged with 50 shares short. Now you need 58 shares short. In a volatile market, rebalancing every $1 move is expensive (bid-ask spread, commissions). High gamma = high transaction costs for delta hedgers.

Scenario: Stock moves to $125.

$\Delta_{\text{new}} \approx 0.92 + 0.01 \cdot 5 = 0.97$

WHY? Delta is already near 1. Gamma is small because the option is far from the strike—it's almost certain to be exercised, so it moves nearly 1:1 with the stock.

Stock rises $2. New delta per share:$0.50 + 0.05 \cdot 2 = 0.60$. Portfolio delta =$100 \cdot 0.60 = 60$.

WHY did the hedge break? You're now short 60 deltas but only own 50 shares. You're short 10 deltas—exposed to further upside. You must buy 10 more shares.

**If the stock falls $2:** Delta per share =$0.50 - 0.05 \cdot 2 = 0.40$. Portfolio delta = 40. You're now long 10 deltas (you own 50 shares but are only short 40 deltas). You must sell 10 shares.

Gamma forces you to buy high, sell low—the enemy of profitability.

Why Gamma Matters: The Greeks in Action

1. Market Makers are short gamma (they sell options to provide liquidity). In volatile markets, they must constantly rebalance, creating feedback loops: buying as the market rises, selling as it falls. This amplifies moves—"gamma squeeze."

2. Volatility Traders want long gamma. If you buy a straddle (ATM call + ATM put), you have positive gamma. Large moves in either direction increase your delta, letting you profit if rebalance.

3. Expiration Risk: Gamma explodes as expiry approaches, especially ATM. A small move can flip an option from worthless to ITM. This is why "gamma risk" dominates in the last days before expiration.

Why it's wrong: Delta changes with every price move (that's gamma). A static hedge only works for tiny moves. The larger the gamma, the faster your hedge deteriorates.

The fix: Dynamic hedging. Rebalance periodically or when delta shifts exceed your risk tolerance. Accept that high-gamma positions = high hedging costs.

Why it's wrong: Any portfolio with options (including convertibles, warants, structured products) has gamma. Even if you're a "stock investor" with a covered call, you have negative gamma—your hedge (the short call) works against you in large moves.

The fix: Understand your portfolio's greks. If you sold calls for income, you're short gamma. Know the risk.

Connections

• Delta hedging strategies — How to rebalance dynamically
• Vega and gamma relationship — High vega often accompanies high gamma ATM
• Gamma scalping — Profiting from rebalancing a long-gamma position
• Pin risk at expiration — Extreme gamma risk when options expire exactly ATM
• Implied volatility smile — ATM options have highest gamma, affects IV structure

#flashcards/stock-market

What does delta measure? :: The rate of change of option value with respect to stock price; your directional exposure (equivalent stock position). A delta of 0.6 = $0.60 gain per$1 move per share (= $60 per standard 100-share contract).

Concept Map

Hinglish (regional understanding)