From the reference, the Vix Whitepaper of CBOE, I found the formula of VIX.

VIX's generalized formula

There are two terms. The first one is focusing on the info from Option contracts. And the second one is focusing on the relationship between the forward index and strike price.

The Forward index formula

In addition, there is a part to illustrate the forward index.

I am confused with the purpose of the second term and the forward index.

I appreciate any help to explain that!

**Reference : https://www.cboe.com/micro/vix/vixwhite.pdf

  • $\begingroup$ Possibly this question coud be helpful quant.stackexchange.com/questions/44388/… $\endgroup$
    – Alex C
    Aug 14 '19 at 12:48
  • 2
    $\begingroup$ The ideal replication is based on two integrals over out of the money options based on the Carr-Madan decomposition. The second term corrects for there being no market strike equal to the forward. Instead the closest strike below, $K_0$ in your first equation, is used. $\endgroup$ Nov 6 '20 at 7:11

forward index level 是用

call - put = forward算出来的,

call payoff - put payoff = forward payoff,


用c=call price,

p=put price,

F 假设是T的 index level,

K是strike price,

那么应该有 c - p = e^{-RT} (F-K),


两边乘一下e^{RT}就得到那个forward term了

Google Translation:

Forward index level is used

Call - put = forward calculated,

Call payoff - put payoff = forward payoff,

So the prices on both sides should be equal,

Use c=call price,

p=put price,

F is assumed to be the index level of T,

K is the strike price,

Then there should be c - p = e^{-RT} (F-K),

The right side of the equation is the pricing formula for forward.

Multiply e^{RT} on both sides to get the forward term.
  • 1
    $\begingroup$ This answers (in Chinese) HOW to calculate F, it does not answer WHY we use F in the formula for VIX. $\endgroup$
    – noob2
    Nov 7 '20 at 11:24

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