# Derivation of the Gamma approximation formula from the Delta approximation formula

The formula of $$Gamma = (Vplus + Vminus - 2V0)/(V0 * dS^2)$$, where

• $$V$$ is the contract value,
• $$S$$ is the stock price.

We also know that $$Gamma = (Dplus-Dminus)/(Splus-Sminus)$$, where

• $$D$$ is the contract Delta,
• $$Splus = S0 + dS$$,
• $$Sminus = S0 - dS$$,
• $$Dplus = (Vplus - V0)/(V0 * dS)$$,
• $$Dminus = (V0 - Vminus)/(V0 * dS)$$.

Substituting $$Dplus$$, $$Dminus$$, and replacing $$Splus-Sminus$$ with $$2dS$$ we get:
$$(Vplus + Vminus - 2V0)/(V0 * dS)/2dS = (Vplus + Vminus - 2V0)/(V0 * 2 * dS^2)$$

I must be doing something wrong, because 1/2 is not present in the book formula.
Could you, please, help me with the derivation?

(Welcome to Quant SE. It looks like you haven't made up your mind on whether your shock is $$dS$$ or $$S_0dS$$. Also, $$V_0$$ doesn't belong in the denominator. You probably mean $$S_0$$. Please try to use Latex on this site next time you visit.)

It's better to start with Taylor's theorem with remainder to convince yourself of the validity of these finite difference schemes:

$$V(S+dS)= V(S)+V'(S)dS+\boxed{\frac{1}{2}V''(S)(dS)^2}+ \frac{1}{6}V'''(S_1)(dS)^3$$

for some $$S_1\in (S, S+dS)$$ $$V(S-dS)= V(S)-V'(S)dS+\boxed{\frac{1}{2}V''(S)(dS)^2}-\frac{1}{6}V'''(S_2)(dS)^3$$ for some $$S_2\in (S-dS, S)$$

We then get (note that the two halves in the boxes will make sure there is no $$2$$ in the final denominator):

$$\frac{V(S+dS)+V(S-dS)-2V(S)}{(dS)^2} = V''(S)+ \frac{1}{6}(V'''(S_1) + V'''(S_2))dS$$

and finally we let $$dS \rightarrow 0$$.

Replacing $$dS$$ by $$SdS$$ everywhere, we also get:

$$\frac{V(S+SdS)+V(S-SdS)-2V(S)}{S^2(dS)^2} \approx V''(S),$$

for small $$dS$$.

• Many thanks for the derivation! The original equations come from FRM Part I 2012 preparation by Kaplan and reference GARP Reading - Tuckman, Chapter 5. Commented Jun 15, 2021 at 22:11
• @Ragewave No problem. Consider accepting one of the answers.
– ir7
Commented Jun 15, 2021 at 22:15
• Ok, done! Thanks for the tip! Commented Jun 15, 2021 at 22:19

Also, I do not think Gamma = (Dplus-Dminus)/(Splus-Sminus) is correct, rather it is Gamma = (Dplus-Dminus)/(dS). You must use the increment dS throughout, for consistency. By making the denominator be a 2*dS step you are not correctly differentiating with respect to the variable "S".