Wednesday, November 14, 2018

volatility - 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)=2TlnF0S2T[F0S1]+2T[SK=01K2eRTp(K)dK+K=S1K2eRTc(K)dK]


to VIX Formula σ2=2TiKiK2ieRTQ(Ki)1T[FK01]2



Answer



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


ln(1+x)xx22.


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


2TlnF0S=2Tln(1+(F0S1))2T((F0S1)12(F0S1)2).


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


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