Numeric example to understand the effect of option gamma

Question

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 1

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 2

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.