# Determine the error term of SKEW-calculation

I am trying to recreate the CBOE's SKEW Index in Python. I need to calculate the errors terms that are adjustment terms for the differences between the atm strike and the forward.

My formula's

error_term_1 = -1 * (1 + np.log(forward / atm_strike) - (forward / atm_strike))
error_term_2 = 2 * np.log(atm_strike / forward) * ((forward / atm_strike) - 1) + ((np.log(atm_strike / forward) ** 2) / 2)
error_term_3 = 3 * (np.log(atm_strike / forward) ** 2) * ((np.log(atm_strike / forward) / 3) - 1 + (forward / atm_strike))


According to the example when the forward is equal to 1106.85 and the atm_strike is equal to 1105, the correct values are:

epilson_1 = 1.40E-06
epilson_2 = -4.2E-0.6
epilson_3 = 1.176E-11


My first and second formula gives the correct answer. My third formula doesn't however. What do I need to change?

I see in different source that the definition is as follows:

Still I am getting a different result when I remove 3.. Please help!

## 1 Answer

In the third equation they somehow did not put the 1/3 in front of the second ln-term.

Without the 1/3, the equation looks at follows:

error_term_3 = 3 * (np.log(atm_strike / forward) ** 2) * (np.log(atm_strike / forward) - 1 + (forward / atm_strike))


Filling in F0 = 1106.85 and K0 = 1105 gives Epsilon3 = 1.1752170209137409e-11, which is the answer you are looking for.