Option Greeks Explained: Delta, Gamma, Theta, Vega and Rho

The option Greeks measure how an option's price responds to changes in its inputs. Delta is the response to the share price, gamma is how fast delta itself changes, theta is the response to time passing, vega to implied volatility, and rho to interest rates. Each holds the other inputs fixed.

Delta

Delta is the change in the option price for a $1 rise in the share price. Call deltas run from 0 to 1, put deltas from −1 to 0. At-the-money options sit near 0.5 or −0.5; deep in-the-money options approach 1 or −1, far out-of-the-money options 0. A call with a delta of 0.5 behaves, for small moves, like half a share.

Gamma

Gamma is the change in delta for a $1 rise in the share price. It is positive for calls and puts alike, largest at the money, and grows as expiration approaches: a short-dated at-the-money option's delta can swing from near 0 to near 1 over a small move. See gamma exposure.

Theta

Theta is the change in the option price as one day passes, everything else unchanged. It is usually negative, because extrinsic (time) value shrinks to zero at expiration, and largest in size for at-the-money options near expiration. Some tools quote it per calendar day, others per trading day.

Vega

Vega is the change in the option price for a one-point rise in implied volatility, for example from 20% to 21%. It is positive for calls and puts, largest at the money, and larger for longer-dated options.

Rho

Rho is the change in the option price for a one-percentage-point rise in the interest rate: positive for calls, negative for puts, and most noticeable for long-dated options.

Black–Scholes–Merton Greeks Formulas

With S the share price, K the strike, t the time to expiry in years, r the interest rate, q the carry (dividend) yield and σ the implied volatility:

d1 = [ln(S/K) + (r − q + σ²/2) t] ÷ (σ√t), d2 = d1 − σ√t

Call delta = e^(−qt) N(d1); put delta = e^(−qt) [N(d1) − 1]
Gamma = e^(−qt) φ(d1) ÷ (S σ √t)
Vega = S e^(−qt) φ(d1) √t, per 1.00 of volatility (divide by 100 for one point)

N is the standard normal cumulative distribution and φ its density.

Illustrative example: a 30-day at-the-money call on a $100 stock with 20% implied volatility, and r and q set to zero, is worth about $2.29. Its delta is about 0.51, gamma about 0.07 per $1, theta about −$0.038 per calendar day, vega about $0.11 per volatility point and rho about $0.04 per percentage point.

How Vyreon Computes Greeks

Vyreon computes its own Greeks rather than taking them from a data vendor, so each is consistent with the implied volatility it solved:

Model: Black–Scholes–Merton.
Implied volatility: solved by Vyreon for each contract from its closing quote.
Share price: the session's unadjusted closing price.
Time: calendar days to expiry ÷ 365.
Rate: the federal funds rate.
Carry: inferred for each expiry from put–call parity in the same option chain, so expected dividends are reflected without a separate dividend forecast; where parity cannot be read, a carry of zero is used (the number of expiries with a parity carry is recorded with the 30-day implied volatility).

Vyreon uses delta and gamma, plus vanna (how delta changes with implied volatility) and charm (how delta changes as time passes), in its exposure measures. Greeks from different providers differ with the rate, dividend treatment, time convention and quote used.

What The Greeks Do Not Tell You

They describe small changes, one input at a time. Large moves change the Greeks themselves.
They are not probabilities or forecasts. Delta is sometimes read as a rough probability of finishing in the money, but it is not the same quantity.
They do not show who holds an option or how it is hedged.

Sources

Fischer Black and Myron Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy (1973).
Robert C. Merton, "Theory of Rational Option Pricing", Bell Journal of Economics and Management Science (1973): the model with a continuous dividend (carry) yield.
John C. Hull, Options, Futures, and Other Derivatives (Pearson, many editions): definitions and formulas for delta, gamma, theta, vega and rho.
Espen Gaarder Haug, The Complete Guide to Option Pricing Formulas (2nd ed., 2007): formulas for second-order Greeks, including vanna and charm.