Lesson 5.3: Gamma (Γ): The "Acceleration" of an Option

Welcome to Lesson 5.3. In our last lesson, we mastered Delta (Δ), the 'speed' of our option. We also learned how to 'delta-hedge' to create a risk-free portfolio. But we were left with a huge, unanswered problem: our 'perfect' hedge is only perfect for a single instant. The moment the stock price St moves, our option's Delta also moves, and our hedge is broken. We have a 'risk of our risk.' We need a new tool to measure the instability of our Delta. That tool is Gamma.

Part 1: What is Gamma? (The "Acceleration" Analogy)

If Delta is the "speed" of your option, Gamma is its "acceleration."

Definition 1 (The Calculus): Gamma is the Second Derivative

Gamma ( $\Gamma$ ) is the second partial derivative of the option's price ( $V$ ) with respect to the stock's price ( $S$ ). It is the "derivative of the Delta."

\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}

The Intuitive Meaning

"If the stock price moves up by $1, how much will my Delta change?"

Low Gamma (like a cruise ship): A $1 change in $S$ barely changes your $\Delta$ . Your "speed" is very stable and your hedge is easy to manage.
High Gamma (like a race car): A $1 change in $S$ causes a huge change in your $\Delta$ . Your "speed" is "twitchy" and unstable, and your hedge breaks constantly.

Visually, Gamma is the "curvature" or "convexity" of the option's price curve.

A "flatter" curve (low $\Gamma$ ) means a stable Delta.
A "sharply bent" curve (high $\Gamma$ ) means an unstable Delta.

Part 2: The First "Aha!" Moment (The P&L of Curvature)

This should sound very familiar. Where did we see this "curvature" term $\frac{\partial^2 V}{\partial S^2}$ before?

It was the "Itô Correction Term" from Lesson 3.2!

When we derived the full Itô's Lemma, we found the option's drift had a "magic" extra term:

\text{Itô Correction Term} = \frac{1}{2}\sigma^2 S_t^2 \left( \frac{\partial^2 V}{\partial S^2} \right)

Now we can replace the math with its new financial name, $\Gamma = \frac{\partial^2 V}{\partial S^2}$ :

\text{Itô Correction Term} = \frac{1}{2}\sigma^2 S_t^2 \Gamma

This is the "Gamma P&L (Profit & Loss)". This is not just theory; this is a real dollar amount that a trader's portfolio earns or loses every second.

Aha! Moment #1: The P&L of Curvature

This formula, $\frac{1}{2}\sigma^2 S_t^2 \Gamma$ , tells you the predictable profit/loss you get just for "being curved."

$b^2$ (or $\sigma^2 S_t^2$ ) is the "jiggliness" of the stock.
$\Gamma$ is the "curvature" of your option.

This term is the physical meaning of the "skateboard in a half-pipe" analogy from Lesson 3.1. It is the "profit" you get from "jiggling" ( $\sigma^2$ ) on a "curved" path ( $\Gamma$ ).

If you buy a call option, you are "long Gamma" ( $\Gamma > 0$ ). You are "long" the smile. This term is positive, and you make money from volatility (jiggling).
If you sell a call option, you are "short Gamma" ( $\Gamma < 0$ ). You are "short" the smile. This term is negative, and you lose money from volatility, every single second.

Part 3: The Second "Aha!" Moment (The Formula for Gamma)

So, how do we calculate Gamma? We would have to take the second derivative of the Black-Scholes formula.

This would be a nightmare. But, just like with Delta, the math simplifies beautifully.

Aha! Moment #2: The Formula for Gamma

The formula for Gamma (for a call or put) is:

\Gamma = \frac{N'(d_1)}{S_t \sigma \sqrt{T-t}}

Let's deconstruct this. It's not as scary as it looks.

$N'(d_1)$ : This is the "probability *density* function." It's not the *area* under the bell curve ( $N(d_1)$ ), it's the *height* of the bell curve at point $d_1$ .
$S_t \sigma \sqrt{T-t}$ : This is the stock price, volatility, and the square root of time.

Intuitive Meaning: This formula tells us *when* Gamma is high or low.

When is Gamma HIGH?
- Gamma is high when $N'(d_1)$ (the numerator) is high. The bell curve is highest at its center, which is when the option is "at-the-money" ( $S_t \approx K$ ).
- Gamma is high when $\sqrt{T-t}$ (in the denominator) is small. This means time to expiration is very short.
When is Gamma LOW?
- Gamma is low when $N'(d_1)$ is low (i.e., the option is far "in-the-money" or "out-of-the-money").
- Gamma is low when $\sqrt{T-t}$ is large (i.e., there is a long time to expiration).

Part 4: Two Concrete Examples (The Trader's Risk)

This brings us to the most important practical lesson about Gamma.

Example 1: The "Cruise Ship" vs. The "Race Car"

Situation: You buy two "at-the-money" call options ( $\Delta=0.5$ ).
Option A (Cruise Ship): Expires in 1 Year. From our formula, $T-t$ is large, so Gamma is very low.
Option B (Race Car): Expires in 1 Hour. From our formula, $T-t$ is near zero, so Gamma is infinitely high.

The Risk: You "delta-hedge" both by selling 0.5 shares of stock. Both portfolios start at 0 Delta. Now, the stock price moves up by $1.

Portfolio A (Low $\Gamma$ ): The stock moves $1. The option's Delta *barely changes*. Maybe it moves from 0.50 to 0.51. Your hedge is still almost perfect. Your portfolio is stable and safe.
Portfolio B (High $\Gamma$ ): The stock moves $1. The option's Delta *explodes*. It might jump from 0.50 to 0.90 *instantly*. Your "hedge" of 0.5 shares is now catastrophically wrong. Your portfolio is no longer hedged and is massively exposed to risk. This is called "Pin Risk."

Example 2: The "Gamma Scalper" (How to Profit)

The Situation: You are a trader who wants to *profit* from Gamma, not just fear it. You buy the high-Gamma "Race Car" option (Option B) and delta-hedge it (sell 0.5 shares).
Your Position: You are "Long Gamma" (you own the "smile") and "Delta-Neutral" (you have no "speed").
Your P&L Formula: Your profit/loss is dominated by two terms from the Black-Scholes PDE:

\text{Your P\&L} \approx \underbrace{\left( \frac{1}{2}\sigma^2 S_t^2 \Gamma \right)dt}_{\text{Gamma 'Profit'}} + \underbrace{\left( \frac{\partial V}{\partial t} \right)dt}_{\text{Theta 'Decay'}}

The Trade: You are *making* money every second from the Gamma term (your "volatility profit") but *losing* money every second from the Theta term (your "time decay").

How you win: You win if the stock *actually jiggles around* (realized volatility) *more* than the "time decay" you are paying. This is called "Gamma Scalping." You are "long convexity" and you profit as long as the market moves, in *either* direction.

What's Next? (The 'Hook')

We have now mastered the risks of the stock's price $S_t$ (Delta) and the *instability* of that risk, $\Gamma$ .

But our entire derivation, and the formulas for $C$ , $\Delta$ , and $\Gamma$ , all depend on one "magic" number that we just assumed we knew: $\sigma$ (volatility).

What is $\sigma$ ? It's not written on a screen. It's a *guess* about the future.

What if *our guess* is wrong?
What happens to our option's price if the *market's opinion* of $\sigma$ changes (like in a panic)?

This leads us to our next lesson on the "jiggle risk" itself: Lesson 5.4: Vega ( $\mathcal{V}$ ).

Up Next: Lesson 5.4: Vega (ν)