The Option Greeks: Five Numbers That Tell You Where Your Risk Is

Lina holds 1,000 call options on a stock trading at $100. Her head of desk stops by at 5 pm with three questions: "What if the stock drops $2 tomorrow? What if volatility jumps? And what does it cost us just to wait a day?"

Lina doesn't reprice anything. She looks at five numbers on her screen and answers in under a minute. Those five numbers are the Greeks. Here's how each one works.

The option behind the numbers

Every figure below comes from one call option, priced with the Black-Scholes model:

• Stock price $S=\$100$, strike $K=\$100$, three months to expiry.
• Volatility 20% a year, risk-free rate 5%, no dividends.
• Call price: about $4.61.

Each Greek answers one question: if one input moves and everything else stays still, how much does the option's price move?

Delta: the stock moves

Delta is how much the option's price changes for a $1 move in the stock:

Lina's call has a delta of 0.57. If the stock rises $1, the call gains about 57 cents. If it falls $2, the call loses about $1.14.

• A call's delta is between 0 and 1. Deep in the money it approaches 1: the option moves like the stock. Far out of the money it approaches 0.
• A put's delta is between −1 and 0. Lina's matching put has a delta of −0.43.
• An at-the-money call sits a little above 0.5 (here 0.57), because with no dividends the interest rate and the volatility push $d_1$ above zero, and a call's delta is $N(d_1)$.

Delta is also a hedge ratio. Lina's 1,000 calls behave like about 570 shares, so selling 570 shares makes her position delta-neutral: small stock moves no longer change its value.

Gamma: delta itself moves

Delta doesn't stay put. Gamma is how much delta changes for a $1 move in the stock:

Lina's call has a gamma of 0.039. If the stock rises from $100 to $105, delta grows from 0.57 to about 0.75. That is why her 570-share hedge stops being right as soon as the stock moves: she has to rebalance.

Gamma is highest for at-the-money options close to expiry (near expiry, the gamma of options far from the strike drops toward zero), and it is positive for anyone who owns options, calls and puts alike. Positive gamma works in your favour: when the stock rises your gains speed up, and when it falls your losses slow down.

With gamma, the price change for a stock move $\delta S$ becomes:

Here $\delta$ means "a small change in", so it doesn't get mixed up with delta, $\Delta$.

Vega: volatility moves

Vega is how much the option's price changes for a 1 percentage point change in volatility. Lina's call has a vega of about $0.20: if volatility goes from 20% to 21%, the call gains about 20 cents.

Owners of options, calls or puts, have positive vega: more uncertainty makes the right to buy or sell more valuable. Vega is highest for at-the-money options with a lot of time left.

(Vega isn't actually a Greek letter. It is usually written with the symbol ν, the Greek letter nu.)

Theta: time passes

Theta is how much the option's price changes as one day passes, with nothing else changing. Lina's call has a theta of about −$0.029 per day. On 1,000 options, waiting a day costs her about $29.

For most options theta is negative (exceptions include deep in-the-money European puts, and deep in-the-money calls on high-dividend assets): part of an option's price pays for the chance that something good happens before expiry, and every day there is less time for it. Theta and gamma are two sides of the same trade. If you own options you have positive gamma and pay for it with negative theta. If you sell options you collect theta and carry negative gamma.

Rho: interest rates move

Rho is the change in price for a 1 percentage point change in the interest rate. Lina's call has a rho of about $0.13. For calls it is positive, for puts negative. For short-dated options it is usually the least important Greek, which is why it rarely gets much attention.

Putting them together

Back to the head of desk's three questions, all at once: the stock falls $2, volatility rises from 20% to 22%, and one day passes. For one call:

Repricing the option with the Black-Scholes model gives a change of −$0.698. The Greeks got within a fraction of a cent. Delta alone would have said −$1.14 and missed the volatility jump completely.

In one line, the approximation Lina used is:

$\delta \sigma$ is in volatility points (2 for 20% to 22%) because $\nu$ is per point, and $\delta t$ is in days because $\Theta$ is per day.

For her 1,000 calls: a loss of about $700.

Try it yourself

This is Lina's call. Move the stock, the volatility and the time left, switch to a put, and compare delta alone with the real change when the stock moves.

Where the Greeks show up on the exams

• CFA Level I only covers what drives option value in words; the Greeks themselves start at Level II.
• CFA Level II (Derivatives, Valuation of Contingent Claims): interpreting all five Greeks, delta hedging and why it needs rebalancing, gamma risk, and implied volatility.
• CFA Level III: using delta, gamma, vega and theta to read the risk of option strategies such as spreads, straddles and collars.
• FRM Part I (Valuation and Risk Models, Option Sensitivity Measures: The Greeks): all five Greeks, dynamic delta hedging, delta- and gamma-neutral positions, and how each Greek behaves with moneyness and time.

The four mistakes that cost marks

1. Mixing up vega units. Vega is usually quoted per 1 percentage point of volatility. If a question gives it per 1.00 (100 points), divide by 100.

2. Using annual theta as daily. Theta from the formula is often per year. Divide by 365 (or by the number of trading days, if the question says so).

3. Getting the put delta sign wrong. Put deltas are negative. A delta-neutral hedge of a long put means buying shares, not selling them.

4. Thinking gamma is negative for puts. Long calls and long puts both have positive gamma. Short options have negative gamma.

Back to 5 pm

Lina answered the three questions in a minute: about $1,140 lost to the stock move, about $390 won back from the volatility jump, a little help from gamma, and $29 for waiting a day. She knew where her risk was before anything had happened. That's what the Greeks are for.

Related reading: options build on the simpler contracts in Forwards vs Futures. To turn the Greeks into one risk number for a whole book, see Value at Risk.

Want to test yourself? The derivatives questions in the JephAi Questions Bank include delta hedging, gamma and vega calculations, each with a worked explanation.