Why does VIX need to calculate the Forward term?

Question

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

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.

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

Answer 1

The answer lies in the derivation of the VIX, as implemented by the CBOE.

The basic derivation of the VIX was done by Demeterfi et al. (1999), where they used a "basket" of options to replicate expected future variance. This yields the formula: $$\begin{aligned} \mathbb{E}[\mathbb{V}] =& \frac{2}{T} \left[ rT - \left( \frac{S_0 e^{rt}}{S_\star} - 1 \right) - \ln\left(\frac{S_\star}{S_0} \right) \right] \\ &+ e^{rT} \int_0^{S_\star} \frac{1}{K^2} P_0(K) dK\\ &+ e^{rT} \int_{S_\star}^\infty \frac{1}{K^2} C_0(K) dK\\ \end{aligned}$$

With risk-free-rate $r$, time to expiration $T$, $S_0$ the initial stock price, $S_\star$ a boundary price and $P$ and $C$ representing put and call options with strike price $K$ respectively.

The CBOE then approximates the first line by $$ -\frac{1}{T} \left( \frac{F_0}{K_0} - 1 \right)^2 $$ as shown by Jiang and Tian (2007). Using numerical integration, the integrals turn into the sums seen in the CBOE formula.

So in a purely technical way the answer is that you use the forward price in order to get rid of the first term, making the calculation feasible.

For a full derivation of the CBOE VIX with steps you can look up Appendix A in my paper No Model No Cry? on SSRN.

Answer 2

forward index level 是用

call - put = forward算出来的，

call payoff - put payoff = forward payoff,

所以两边价格应该相等,

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),

等式右边是forward的定价公式，

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.