# 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:

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

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

• 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. – AlRacoon Mar 3 '19 at 17:25
• @AlRacoon yes it has methodology as I attached to this post. But it doesn’t involve derivation. – TryingtobeQuant Mar 3 '19 at 17:45
• It is clear that the $\Delta K_i$ come discretizing the integral - is it? – Ric Mar 3 '19 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 ? – TryingtobeQuant Mar 3 '19 at 19:32
• @TryingtobeQuant yes, it is the second order approximation of the logarithm of 1+x as discussed below in the answer – Ric Mar 4 '19 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.