Quant Finance Foundations, Part 9. Previously: Black-Scholes Line by Line.
The option Greeks are what a trader actually watches, because prices alone tell you what things are worth rather than what happens next. Each Greek answers one question: if this input moves, how much does my position move?
In short, they are the bridge between the Black-Scholes formula and the trading floor. Moreover, they are where option theory stops being theory and becomes a job.
The option Greeks at a glance
| Greek | Measures sensitivity to | Sign for a long option |
|---|---|---|
| Delta | Underlying price | + for calls, − for puts |
| Gamma | Change in delta | Positive |
| Vega | Implied volatility | Positive |
| Theta | Passage of time | Negative |
| Rho | Interest rates | + for calls, − for puts |
Delta: how much you move with the underlying
Delta is the change in the option’s value per one-unit change in the underlying. For a call it runs from 0 to 1; for a put, −1 to 0.
Concretely: a call with delta 0.6, on a stock that rises by 1, gains about 0.6. Holding 100 such contracts on 100 shares each means the position behaves like 6,000 shares.
This is not new — it is Δ from the binomial tree in Part 6, and N(d₁) in Black-Scholes. The same object under three names.
Delta is what a hedger neutralises. Sell 100 calls with delta 0.6, buy 6,000 shares, and small moves in the stock leave the combined position roughly unchanged. That is delta hedging, and it is the mechanical enactment of replication.
Traders also read delta as a rough probability of finishing in the money. A 0.25-delta option is loosely a one-in-four shot. Not exactly true, but close enough for desk conversation.
Gamma: how fast delta changes
The trouble with delta hedging is that delta does not stay put. Gamma measures the rate at which it moves as the underlying moves.
Return to the 6,000-share hedge. The stock rallies, delta rises to 0.75, and the position now needs 7,500 shares. You are under-hedged and must buy 1,500 more. The stock falls back, delta drops, and you sell. Gamma is what forces this continuous rebalancing.
The economics of that rebalancing determine which side of the trade you want to be on.
Long gamma — you own options. Rebalancing means buying low and selling high automatically: the hedge requires you to sell shares as the stock rises and buy as it falls. Every round trip banks a small profit. You want movement.
Short gamma — you have sold options. Rebalancing forces you to buy high and sell low. Every move costs money, and a large sudden move costs a great deal. You want stillness.
This is the trade-off that defines option market making. The seller collects premium up front and pays it back through hedging losses in a volatile market. The whole business is the bet that the premium collected exceeds the hedging cost incurred — which is to say, that implied volatility exceeds realised volatility.
Gamma is largest for at-the-money options near expiry. That combination is where hedging is hardest and where losses concentrate.
Vega: sensitivity to volatility
Vega is the change in value per one percentage point change in implied volatility. It is always positive for a long option, calls and puts alike.
A position with vega 500 gains 500 if implied volatility rises from 20% to 21%, everything else held constant. That “everything else held constant” is doing real work — in practice volatility usually spikes precisely when the underlying moves sharply.
Vega is largest for at-the-money, long-dated options. A one-month option has little vega; the uncertainty simply does not have time to matter. A two-year option has a great deal.
The name is a small joke in the industry: vega is not a Greek letter.
Theta: the cost of waiting
Theta is the change in value per day of elapsed time, and for a long option it is negative. Options decay.
Theta and gamma are two views of one thing. Long gamma means you profit from movement; theta is the rent you pay for that privilege. Short gamma means you collect that rent while carrying the risk of a large move. On a well-hedged book, the daily question is simply whether realised movement earned more than theta cost.
Theta accelerates as expiry approaches, which is why a short-dated at-the-money option can lose a meaningful fraction of its value in a single quiet day.
Rho: usually the last one anybody checks
Rho is sensitivity to interest rates. Positive for calls, negative for puts, and small for short-dated equity options.
It stops being an afterthought for long-dated products and rate-sensitive instruments. On a book of multi-year structured notes or a swaption portfolio, rho is a first-order concern.
What a desk actually watches
A trader does not manage individual option Greeks. They aggregate across the entire book, because the Greeks are additive — a portfolio’s delta is the sum of its positions’ deltas.
The typical morning view is a grid: net delta, gamma, vega, theta per underlying, per maturity bucket, per strike bucket. What matters is the aggregate exposure, not any single trade.
A representative situation: the book is delta-neutral, short 2,000 gamma, long 8,000 theta. Translation — flat on direction, collecting 8,000 a day in decay, and losing money if the market moves sharply in either direction. The trader’s job is to judge whether the coming session will be quiet enough to keep that 8,000, and to buy back gamma if it will not.
Where the Greeks mislead
Every Greek is a local derivative. It describes behaviour for a small move, and its accuracy degrades as the move gets larger.
Delta plus gamma gives a second-order approximation that holds well for a few percent. For a 20% gap it does not. This is why every serious risk system also runs full revaluation across a grid of scenarios rather than relying on sensitivities alone — the Greeks tell you about the neighbourhood, and a crisis is not in the neighbourhood.
The second trap is cross-effects. The Greeks are computed holding other inputs fixed, and markets do not oblige. In a sell-off, spot falls and volatility rises together, so the delta loss and the vega gain arrive simultaneously. Second-order Greeks such as vanna and volga exist to capture exactly these interactions.
Try it yourself
Compute the Greeks numerically, in this order:
- Extend the Black-Scholes spreadsheet from the previous post.
- Reprice the option with spot at S+0.01.
- Subtract the price at S and divide by 0.01 — that is delta.
- Apply the same difference to delta itself to get gamma.
- Repeat for volatility and time to obtain vega and theta.
This finite-difference approach is how many production systems compute Greeks for products that have no closed form.
Then simulate a delta hedge. Generate 60 days of random stock prices, rebalance daily to stay delta-neutral against a short call, and track cumulative profit. Run it with realised volatility below the implied volatility you sold at, and then above it. The first case profits, the second loses — and you will have demonstrated the entire economics of option market making in one spreadsheet.
Frequently asked questions
What is delta hedging?
It is holding enough of the underlying to offset the option’s delta, so small moves leave the combined position unchanged. Essentially, it is replication carried out in practice.
Why does gamma force continuous rebalancing?
Because delta itself changes as the underlying moves. Consequently the hedge that was correct this morning is wrong by the afternoon.
Is being long or short gamma better?
Neither, inherently. Long gamma profits from movement but pays theta; short gamma collects theta but loses when markets move sharply.
When do the Greeks stop being reliable?
They are local derivatives, so they describe small moves well and large ones badly. For that reason risk systems also run full revaluation across scenario grids.
Next in this series: historical versus implied volatility, and why the smile exists. The binomial tree post explains where delta originally comes from.