# How can one compute the Greeks on VIX Futures

I am guessing the short answer to this question is "use the chain rule and linearity of the derivative," but I am looking for more specific advice on how to compute the derivatives of a VIX futures contract. Roughly speaking, the VIX index represents a rolling weighted position in a bunch of S&P call & put options at different strikes. Generally speaking, one cannot buy and sell spot VIX, but there are VIX futures contracts. I would like to be able to compute the sensitivity of the price of the futures contract to: the price of the underlying (in this case the S&P index), volatility of the underlying, change in volatility of the underlying, the risk-free rate, and time.

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## 3 Answers

Short Answer : Futures don't have Greeks

Long Answer : Assuming a non strictly mathematical (i.e. false) point of view.

Well, having Greeks on VIX Futures is not relevant, VIX value is itself a "Greek" (and Futures don't have Greeks).

Sensitivity to

• Price of the Underlying : Insensitive (ν = 0)
• Volatility of the Underlying : Delta Δ = 1 (to Volatility of S&P Option Combination used to compute VIX)
• Change in Volatility of the Underlying : Gamma Γ = 1
• Risk Free Rate : Same than any other kind of future
• Time : 0 (Θ = 0)

Volatility of VIX should be Vega of S&P Option Combination used to compute VIX Vega of VIX should be Vomma of S&P Option Combination used to compute VIX

You can compute the Greeks on the VIX Options, it would be more relevant, but don't expect relationships with the S&P Volatility/Prices, VIX is much more complex than simple plain Volatility of S&P.

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The "underlying" in @shabbychef's question is the S&P 500, not spot VIX. Perhaps, then, it would have been more appropriate to call them "Greek-like" exposures, but I believe you did not answer the question. – Tal Fishman Sep 7 '11 at 15:20
Well there is no direct mathematical relationship between VIX and plain S&P500 (the index). There is only relationship between VIX and S&P500 options. – Lliane Sep 7 '11 at 15:57

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"VIX" is just shorthand for a portfolio of S&P 500 options, so any relationship between S&P 500 and S&P 500 options carries over to VIX. The question is how does this carry over to futures on VIX. I actually think the answer to @shabbychef's question is just to take the weighted average of the Greeks on each option that forms the VIX. I'm not 100% sure this is right, so not posting as an answer. – Tal Fishman Sep 7 '11 at 16:11

VIX is calculated from a basket of SPX options, and VIX futures expire into following expiration, e.g. September VIX futures that will expire next Wednesday will use SPX October options chain to calculate settlement value. If $B$ is the value of the basket then VIX value at expiration is $\sqrt{ B }$. Then VIX futures price is the expectation of the basket $VIX _{F} = E[\sqrt{ B }]$. Delta of the VIX futures price with respect to the basket would be $$\ \frac{\partial VIX _{F}}{\partial B} = \frac{\partial E[\sqrt{ B }]}{\partial B}$$

As you can see that taking that expectation is not simple, since there is no simple connection between VIX futures greeks and SPX options greeks because of the expectation and square root. So "use the chain rule and linearity of the derivative" approach would not get you anywhere. But that does not mean that such derivative is 0. Such derivative can be calculated in Malliavin sense, but that is probably not what you're looking for.

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hmm. I think I get it. Although the expectation is of the future value of the basket, and the derivative is with respect to spot basket value, right? In that case, I think I can use the 'Delta method' (after genuflecting before some regularity conditions) and get a good approximation. (Well, I view Delta method as Taylor's theorem plus expectation magic.) – shabbychef Sep 20 '11 at 22:55

I would calculate implied greeks based on the expectations for the underlying S&P option basket. Unfortunately, these are just expectations as I have a database of 17 years of S&P options pricing that will tell you your expectations are likely wrong more often than not. If you had a pricing model that had these expectations built in, I think the resulting greeks would be as correct as your pricing model.

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