Understand why directional exposure changes as the underlying moves.
Learn Options and GreeksAdvanced4 min
Illustrative; other inputs held constant
Plain English
The idea
Gamma measures how delta changes when the underlying price changes. If delta is speed, gamma is the change in speed.
High gamma means the option exposure can change quickly. This often matters near the strike and near expiry, where a small underlying move can shift the option from unlikely to likely to finish in the money.
Gamma is not good or bad by itself. It describes how unstable the first-order exposure can be.
Delta is the slope of option value against underlying price. Gamma measures how that slope changes when the underlying moves: it is the curvature of the value relationship. Its units are change in delta per currency unit of the underlying, so it should not be read as another direct price sensitivity.
For identical inputs, long European calls and puts have the same positive gamma in the non-dividend Black–Scholes model. A short position reverses the position's gamma sign.
Worked Example
Why stable delta is not guaranteed
Suppose an option has a delta of 0.50 and high gamma. A small move in the underlying may push delta towards 0.60 or 0.40 quickly.
A reader who only looks at the starting delta may underestimate how quickly the position can become more directional.
Gamma is the warning that the risk reading itself can move.
Add an assumed gamma of 0.04 per pound to the hypothetical starting delta of 0.50. For a £0.50 underlying rise, the local change in delta is approximately 0.04 × 0.50 = 0.02. The updated delta estimate is therefore 0.52, holding the other inputs fixed.
The corresponding second-order option-value estimate is different: 0.50 × £0.50 plus one half of 0.04 times £0.50 squared, giving £0.255 per option unit. The extra £0.005 is the curvature contribution in this illustrative local expansion.
Starting delta
0.50
Gamma per pound
0.04
Underlying move
+0.50 pounds
Approximate new delta
0.52
Second-order value change
+0.255 pounds per unit
Reading the result
Curvature matters in both directions
The squared price move in the second-order value term is positive whether the underlying rises or falls. For the long option with positive gamma, that curvature term sits above the initial straight-line tangent. This describes the shape of the model value curve; it does not imply a guaranteed trading profit.
Gamma can make a previously small directional exposure change quickly. It is therefore useful when explaining why a position that looked locally balanced earlier can have a different sensitivity after a price move. Re-estimating the sensitivities changes the measurement, but any actual adjustment also faces costs and market constraints.
Limits and assumptions
A local expansion is not a full revaluation
The example uses starting delta and gamma and a deliberately small move. Both sensitivities can change during the move, and simultaneous volatility or time changes introduce effects the calculation leaves out. A full model revaluation and a tradable market price are also different things.
The optional formula requires positive price, strike, annualised decimal volatility and time in years. It assumes European exercise, no dividends and constant model rates and volatility. Near expiry, sensitivity around the strike can become sharp; broad statements about gamma increasing everywhere as expiry approaches would be misleading. The formula is not evaluated directly at expiry.
Common Mistake
Thinking gamma is only for professionals
Gamma can look technical, but the plain idea is accessible. It tells the reader whether the delta shown now may remain stable or move sharply.
That matters for anyone trying to understand why an option price can start moving faster as the underlying approaches the strike.
Multiplying gamma by the price move estimates a delta change. Treating that quantity as a price change loses a unit and omits the one-half and squared-move terms needed in the second-order value approximation.
Formula Summary
Gamma from Black-Scholes delta
European exercise, no dividends, positive S, K, sigma and T; time T is in years, sigma is an annualised decimal and r is a continuously compounded annual rate. Rates and volatility are constant model inputs; jumps and trading frictions are excluded. Do not apply these expressions directly at expiry or zero volatility.
N is the cumulative standard normal distribution; n is its density. The formulas describe long-option model values; signed positions change the exposure.
Gamma
Gamma equals normal density d one divided by S times sigma times square root of T.
Second sensitivity of option value to the underlying price.
Standard normal density at d1.
Current underlying price.
Annualised volatility assumption.
Time to expiry in years.
The expression is largest when n(d1) is high and the denominator is small.
Optional derivationExplore the derivation
The shortest clean derivation starts from delta, then applies the chain rule.
01
Start from delta
For the non-dividend call, delta is N(d1). The put delta is N(d1) minus 1.
Starting Point
Call delta equals cumulative normal d one.
02
Differentiate delta with respect to S
The derivative of N(d1) is n(d1) times the derivative of d1.
Chain Rule
Gamma equals derivative of delta with respect to S, which equals normal density d one times derivative of d one with respect to S.
03
Substitute the d1 derivative
Only ln(S/K) inside d1 changes with S, giving the compact denominator.
Input Derivative
Derivative of d one with respect to S equals one divided by S times sigma times square root of T.
04
Read the result
Call and put gamma match because put delta differs from call delta by the constant 1.
Gamma Result
Call gamma equals put gamma equals normal density d one divided by S times sigma times square root of T.
Self-check
Check your understanding
What is the updated delta estimate in the worked example?
0.52: starting delta 0.50 plus gamma 0.04 per pound times the £0.50 move. It is a local estimate with other inputs fixed.
Why is the curvature price contribution £0.005 rather than £0.02?
A value expansion uses one half × gamma × the squared underlying move. Gamma × the move instead estimates a change in delta, which has different units.
Does positive gamma guarantee a positive investment return?
No. The option has a price, other inputs change, and costs and time matter. Gamma describes curvature, not the profitability of a strategy.
General learning context
Where this helps a public investor
Gamma helps readers see why an option risk reading can become more directional as the underlying price approaches important strike levels.
Disclaimer
Educational Use Only
This article is for informational and educational purposes only. Options involve risk and are not suitable for every investor. Nothing here is a recommendation to buy, sell, write, or trade an option.
