# 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.

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

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. Mar 3 '19 at 17:25
• @AlRacoon yes it has methodology as I attached to this post. But it doesn’t involve derivation. 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 ? 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

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