Using Gamma for a Second-Order Option Value Estimate
Summary
The document clarifies how option gamma improves a delta-only estimate of an option’s value after a small underlying price move. Delta gives the first-order change in option value, while gamma measures how delta itself changes with the underlying. The standard second-order Taylor approximation adds half of gamma times the squared price change to the initial value plus the delta contribution. In the example, this produces an estimate close to the Black–Scholes value for a one-unit move in the underlying.
An alternative explanation treats gamma as a slope adjustment and uses a midpoint delta, yielding the same second-order estimate in this example. The response notes that higher derivatives can refine the approximation further. These are local estimates: accuracy generally declines for larger underlying moves, and the calculation holds other model inputs fixed. The numerical example is illustrative rather than evidence of performance across options, market conditions, or changes in volatility and time to expiry.
Key ideas
- Delta estimates the first-order change in option value as the underlying moves.
- Gamma measures the change in delta per unit move in the underlying.
- A second-order Taylor estimate adds half the gamma times the squared underlying price change.
- The approximation is local and can become less accurate for larger moves or changing model inputs.
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# Numeric example to understand the effect of option gamma
# Numeric example to understand the effect of option gamma
Gamma of an option is the second partial derivative of the theoretical value of an option wrt the underlying. It should be the rate of change of Delta wrt to a small change on the underlying. However many textbook (e.g. Trading option greeks, Passarelli) says that gamma is conventionally Stated in terms of Delta per dollar move.
Let’s suppose we want to use the B&S Model for a call option on non divident paying stock. Let’s also suppose that:
- S = 52 (the underlying)
- K = 50 (the strike)
- tau = 0.25 (the time to maturity)
- r = 0.12 (the risk free rate)
- sigma = 0.3 ( the volatility of the underlying)
Then we have: Call = 5.057387 Delta = 0.7041836 Gamma = 0.04429147
I want to estimate How the option value will be if the stock change from 52 to 53 ( in this situation the B&S model would give as exact answer Call = 5.783055).
As the first approximation (Delta) i would do: Call = 5.057387 + (53 -52)0.7041836 = 5.761571 (which is not equal to 5.783055) Then of i want to be more precise, i could use gamma as well: The new Delta should be 0.7041836 + 0.04429147 (gamma stated ad Delta per dollar move) or 0.7041836(1+0.04429147), i.e. Rate of change of Delta. Why?
## Answer by Kevin (score 9, accepted)
https://quant.stackexchange.com/a/49648
Using our good friend Taylor, we know that \begin{align*} C(S+\Delta_S)\approx C(S)+\Delta_C\Delta_S+\frac{1}{2}\Gamma_C(\Delta_S)^2, \end{align*} where $\Delta_C$ and $\Gamma_C$ are the call's sensitivities and $\Delta_S$ a small change in the price of the underlying asset. In your example, $\Delta_S=1$ and thus, \begin{align*} C(52+1) &\approx 5.057387 + 0.7041836 + \frac{1}{2}0.04429147 \\ &=5.783716335. \end{align*}
Of course, the smaller the change in the price of the underlying asset ($\Delta_S\to0$), the lower the influence of gamma (and delta). You could even improve the above approximative polynomial and include higher derivatives (the third derivative is sometimes called ``Speed'').
## Answer by Alex C (score 3)
https://quant.stackexchange.com/a/49649
I am not sure what you are trying to do, but I think you are trying to use the Modified Euler Method to find the option value.
If the Delta at $S=52$ is $0.7041836$
the Delta at $S=53$ can be approximated as $0.7041836+(53-52)0.04429147=0.74847507$
The Delta to be used in the modified Euler method (or Heun Method) is half-way between these i.e. $(0.7041836+0.7484751)/2=0.72632935$ (sometimes called the mid-interval estimate of slope)
The estimate of option value at $S=53$ according to Modified Euler is then $5.057387+(53-52)0.72632935 = 5.78371635$ which is quite close to the correct value.
However this is not the way Gamma is usually used in option calculations, rather the Taylor Series method described by KeSchn is usually used.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.