A Beginner’s Guide to Option Pricing – Part 7 (Gamma)

A Beginner’s Guide to Option Pricing

Part 7 – Gamma: How Fast Delta Itself Changes

What Is Gamma?

Part 6 ended with a question: if Delta is constantly changing, what determines how quickly it changes? The answer is Gamma, the next of the Greeks, and it’s a natural next stop, because Gamma isn’t really a new, unrelated concept — it’s simply a way of measuring how fast Delta moves.

Here’s the plain-language version. Delta tells you how much an option’s price is expected to change for a one-dollar move in the stock. Gamma tells you how much Delta itself is expected to change for that same one-dollar move. If Delta measures the option’s current sensitivity to the stock, Gamma measures how quickly that sensitivity changes.

This matters because, as we saw in Part 6, Delta isn’t fixed. A $205 call’s Delta today might be 0.65, but if NVDA rallies, that Delta could rise to 0.80 by tomorrow. Gamma is the number that tells you, in advance, roughly how much that shift is likely to be for a given move in the stock. A high Gamma means Delta is poised to change quickly. A low Gamma means Delta will barely move even if the stock does.

A Simple Example

Suppose the July 17 $207.50 NVDA call has a Delta of 0.56 and a Gamma of 0.058 (both real figures). A Gamma of 0.058 means Delta is expected to change by about 0.058 for the next one-dollar move in NVDA, assuming other factors remain roughly unchanged.

So if NVDA rises a dollar, the $207.50 call’s Delta doesn’t just stay at 0.56 — it moves up to roughly 0.56 + 0.058 = 0.618. That new, higher Delta then applies to the next dollar the stock moves. If NVDA falls a dollar instead, Delta drops to roughly 0.56 − 0.058 = 0.502.

Delta isn’t a single, stable multiplier you can apply over and over. Every time the stock moves, Delta itself recalculates, and Gamma is what tells you the size of that recalculation.

Why Gamma Is Highest Near the Stock Price

Just like Delta, Gamma isn’t the same across every strike on the chain — and understanding why reveals something genuinely useful about how options behave.

Let’s look at the real NVDA chain from Part 6, expanded to include Gamma:

Notice the shape. Gamma is small at both ends of the chain and largest right in the middle, near $207.50, which sits closest to NVDA’s actual price of roughly $208.

This makes intuitive sense once you think back to what Delta represents: the market’s rough estimate of the odds an option finishes in the money. The $200 call is already deep in the money, with a Delta of 0.875 — the market is quite confident it’ll stay there, so a one-dollar move in NVDA doesn’t change that confidence very much. Its Delta has little room left to grow (it’s already close to its ceiling of 1.00), so Gamma is low. The same logic applies at the other extreme: the $217.50 call is so far out of the money that a one-dollar move barely changes the market’s assessment of its odds, so its Delta barely moves either, and Gamma stays low there too.

But right around $207.50, the option is genuinely on the fence — a one-dollar move in NVDA can meaningfully shift the odds of finishing in or out of the money, so Delta responds forcefully to every tick, and Gamma is at its peak. This is the same “zone of greatest uncertainty” we identified in Part 6 when discussing Delta — it turns out that same zone is also where Delta is most volatile, which is exactly what Gamma is measuring.

Gamma Risk Near Expiration

Here’s a detail that explains a pattern that surprises many newer traders: Gamma isn’t just highest near the current stock price — it also grows more extreme the closer an option gets to expiration.

Think back to the NVDA options we’ve been using throughout this series, expiring just two days after the chain was captured. With so little time left, there’s very little room left for anything to change the outcome — so when the stock does move, the market has to revise its odds sharply and immediately, rather than gradually over weeks. That’s what makes Gamma spike near expiration for at-the-money options specifically: Delta can swing dramatically within a single trading day, simply because a modest stock move is now a much bigger deal to an option with almost no time left.

This is sometimes called “gamma risk,” and it’s a real, practical concern for anyone holding short-dated, at-the-money options — buyers and sellers alike. It cuts both ways for everyone involved: a position can look calm and manageable in the morning and become considerably larger, in either direction, by the afternoon, purely because Delta itself is accelerating. What counts as the favorable direction, and what counts as the dangerous one, depends on whether you’re buying or selling, and whether you’re holding a call or a put — which is worth walking through explicitly, one position at a time.

Gamma Risk for a Call Buyer

Take our $207.50 call again: price $3.15, Delta 0.556, Gamma 0.058, two days from expiration.

Suppose NVDA rises a dollar, to $209. Using Delta alone, we’d expect the option to gain about 56 cents, moving to roughly $3.71. But Gamma tells us Delta doesn’t stay at 0.556 for that move — it climbs to roughly 0.614 by the time the stock gets there. So the next dollar NVDA rises, the option gains roughly 61 cents instead of 56. Each additional dollar produces a bigger gain than the one before it, because Delta keeps climbing right along with the stock. That’s the favorable side of gamma risk for a call buyer: the option’s price isn’t just rising, its rate of rise is accelerating too.

The same mechanic works against the buyer in reverse. If NVDA falls instead, Delta drops toward zero as the option moves out of the money, and the option can shed most of its remaining value surprisingly quickly in the final day or two, since there’s so little time left for the stock to recover. A call buyer’s worst-case outcome — the stock reversing hard in the last two days — is exactly when Gamma is at its most extreme.

Gamma Risk for a Put Buyer

The logic is a mirror image, but worth walking through separately since puts gain value when the stock falls, not rises.

Take the $207.50 put: price $2.29, Delta -0.448, Gamma 0.058. Suppose NVDA falls a dollar, to $207. Delta alone suggests the put gains about 45 cents, moving to roughly $2.74. But Gamma tells us Delta becomes more negative as the put moves further into the money — it moves to roughly -0.506. So the next dollar NVDA falls, the put gains about 51 cents instead of 45. Just like the call buyer benefiting from a rally, the put buyer benefits from a decline that accelerates in their favor: bigger gains on each successive dollar the stock keeps falling.

And just like the call buyer, the put buyer faces the unfavorable version too. If NVDA rises instead of falls, the put’s Delta shrinks toward zero as it moves out of the money, and the put can lose most of its remaining value quickly in these final days, since there’s so little time left for the stock to reverse back in the buyer’s favor.

Gamma Risk for a Call Seller

Selling that same $207.50 call flips the position — and when the stock is rising, Gamma now works against the seller rather than for them.

If NVDA rises, the short call’s Delta grows just as it did for the buyer — but since the seller is on the other side, a growing Delta means growing losses, and those losses grow faster with each additional dollar the stock climbs, for exactly the same reason the buyer’s gains accelerated. A position that looked like a modest, well-defined risk that morning can turn into a considerably larger one by the afternoon if NVDA keeps rallying, purely because Delta is increasing against the seller.

If NVDA falls instead, the news is good for the call seller: Delta shrinks toward zero, the option loses value (which is exactly what a seller wants), and that erosion can happen quickly in the final days before expiration.

Gamma Risk for a Put Seller

Selling the $207.50 put creates the mirror-image risk to selling a call.

If NVDA falls, the short put’s Delta grows more negative, and — since the seller is on the losing side of that move — losses accelerate the same way they would for a call seller facing a rally. A modest, manageable short put position can turn into something considerably more painful within a single session if NVDA drops sharply, especially this close to expiration.

If NVDA rises instead, that’s the favorable outcome for the put seller: Delta shrinks toward zero, the put loses value, and the seller benefits from that decay.

The Common Thread

In all four cases, Gamma is doing the same underlying thing: making Delta move faster than a simple straight-line estimate would suggest, in whichever direction the stock happens to move. For buyers, that acceleration is a source of leveraged upside when the stock cooperates, and a source of fast losses when it doesn’t. For sellers, it’s the reverse — a smaller, more manageable risk that can escalate quickly if the stock moves against the position, and a source of accelerating profit (through faster time and value decay) if it doesn’t. This asymmetry, a help or a hindrance depending on which side of the trade you’re on, is worth keeping in mind as we look at option strategies with more than one leg later in this series, since combining a long and short position is one of the main tools traders use to manage exactly this kind of exposure.

What Gamma Doesn’t Tell You

Like Delta, Gamma is a snapshot, not a permanent property of the contract. It changes constantly as the stock price moves, as time passes, and as implied volatility shifts. A Gamma of 0.058 today doesn’t mean that number stays fixed tomorrow. In fact, as we just discussed, Gamma itself tends to grow as expiration approaches for at-the-money options, making Delta increasingly sensitive to movements in the underlying stock during the final days of a contract’s life.

Gamma measures how Delta changes as the stock price moves. It does not directly measure the effect of time passing with the stock unchanged, nor does it directly measure how changes in implied volatility affect an option’s price. Those are the jobs of two other Greeks entirely (Theta and Vega).

Looking Ahead

We’ve now looked at how an option’s price responds to a move in the stock (Delta) and how quickly that response itself changes (Gamma). But there’s a third force acting on every option’s price, even when the stock doesn’t move at all: the simple passage of time.

We touched on this idea earlier in the series when discussing time value and time decay, but we haven’t yet given it a formal name or explored it in detail. In Part 8, we’ll examine Theta, the Greek that measures how much an option’s value changes simply because another day has passed. For investors who spend more time selling options and collecting time premium than buying options and racing against the clock, Theta is arguably one of the most important numbers in the entire option chain.