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)=2TlnF0S∗−2T[F0S∗−1]+2T[∫S∗K=01K2eRTp(K)dK+∫∞K=S∗1K2eRTc(K)dK]
to VIX Formula σ2=2T∑i△KiK2ieRTQ(Ki)−1T[FK0−1]2
Answer
The piece you are missing is an approximation via the Taylor formula of the logarithm:
ln(1+x)≈x−x22.
Apply this to the first term in the final formula of the technical paper:
2TlnF0S∗=2Tln(1+(F0S∗−1))≈2T((F0S∗−1)−12(F0S∗−1)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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