# Derivation of VIX Formula

I've read a lot of derivations about VIX formula. I can say it is -adjusted- fair strike of variance swap. But I can't see how it goes from variance swap rate to VIX formula. In particular I can't see the last part of VIX formula hosted here on page 4.

Could you please lead me from Hull Technical Note 22:

$$$$\ E(V)= \frac{2}{T}ln\frac{F_{0}}{S^{*}} - \frac{2}{T}\left[ \frac{F_{0}}{S^{*}}-1\right] +\frac{2}{T}\left[\int_{K=0}^{S^{*}} \frac{1}{K^{2}}e^{RT}p(K)dK + \int_{K=S^{*}}^{\infty} \frac{1}{K^{2}}e^{RT}c(K)dK\right]$$$$

to VIX Formula $$$$\sigma^{2}= \frac{2}{T}\sum_i^{}\frac{\triangle K_{i}}{K_{i}^{2}}e^{RT}Q(K_{i}) - \frac{1}{T}\left[ \frac{F}{K_{0}}-1\right]^{2}$$$$

• The determination of Vix is quite involved, using the implied volatility of listed options and weighs them by how far they are OTM. The CBOE website has the calculation methodology. Mar 3, 2019 at 17:25
• @AlRacoon yes it has methodology as I attached to this post. But it doesn’t involve derivation. Mar 3, 2019 at 17:45
• It is clear that the $\Delta K_i$ come discretizing the integral - is it? Mar 3, 2019 at 19:29
• @Richard Yes it is. What about first part of Hull's equation and last term in VIX white? Can we say it is just an approximation ? Mar 3, 2019 at 19:32
• @TryingtobeQuant yes, it is the second order approximation of the logarithm of 1+x as discussed below in the answer Mar 4, 2019 at 10:43

## 1 Answer

The piece you are missing is an approximation via the Taylor formula of the logarithm:

$$\ln(1+x) \approx x-\frac{x^2}{2} \; .$$

Apply this to the first term in the final formula of the technical paper:

$$\frac{2}{T}\ln\frac{F_{0}}{S^{*}} = \frac{2}{T}\ln\left(1+\left(\frac{F_{0}}{S^{*}}-1\right)\right) \approx \frac{2}{T}\left(\left(\frac{F_{0}}{S^{*}}-1\right) - \frac{1}{2}\left(\frac{F_{0}}{S^{*}}-1\right)^2\right) \;.$$

Now, the first term of this approximation cancels with the second term of the technical paper formula. You're left with the quadratic term.