Hi,
I'm trying to understand the effect that vol-of-vol of an underlying with stochastic vol would have on a VIX computed on the vol smile of that underlying.
Assuming the SPX follows a Heston dynamic, would the VIX capture the additional risk of an increase in vol-of-vol or a change in correlation between the two Wieners?
Putting it differently, since the VIX is a zero delta portfolio of constant dollar gammas, how would I hedge the vol-of-vol or price-vol-correlation-sensitivity?
Thanks!
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Re: Vol-of-Vol and the VIX
If the SPX followed the Heston model, neither the vol-of-vol nor price-vol-correlation would change.
In principle, you probably want VIX options (or related vol product options) to hedge the SPX vol-of-vol. Bid-ask spreads are quite wide, however. Also, the Heston model is a very poor model for those options, as it predicts smiles decreasing at larger strikes rather than increasing, the latter being the typical market pattern for VIX options.
In principle, you probably want VIX options (or related vol product options) to hedge the SPX vol-of-vol. Bid-ask spreads are quite wide, however. Also, the Heston model is a very poor model for those options, as it predicts smiles decreasing at larger strikes rather than increasing, the latter being the typical market pattern for VIX options.
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- AwesomeBanana443
Re: Vol-of-Vol and the VIX
Thanks Alan, that makes sense. I'm still struggling to understand it fully, so let me try to put my question in different words.
First, assume an underlying that follows a stochastic process, such as Heston (+ a specific set of parameters). Nature reveals itself, and I will observe a single realization of the process in the form of a time series of returns. This path has a certain squared deviation from its mean (i.e. volatility as property of the series).
Second, my stochastic model has parameters that determine the properties of the return series realization directly or indirectly. In the BS case, the sigma parameter is equal to the vol of the time series realization. In the Heston case, I can compute the variance as E[X^2]-E[X]^2 from the characteristic function.
Third, the VIX. I can use the stochastic model + parameters to compute option prices, plug those into the VIX formula, and get another variance for the time series of the underlying.
It is my understanding that those three should be, in a perfect world, identical values with different interpretations. But apparently, I'm wrong, since the second and third differ significantly, and the magnitude of the difference is dependent on the parameter values itself. Could you enlighten me on the difference of two and three, please?
First, assume an underlying that follows a stochastic process, such as Heston (+ a specific set of parameters). Nature reveals itself, and I will observe a single realization of the process in the form of a time series of returns. This path has a certain squared deviation from its mean (i.e. volatility as property of the series).
Second, my stochastic model has parameters that determine the properties of the return series realization directly or indirectly. In the BS case, the sigma parameter is equal to the vol of the time series realization. In the Heston case, I can compute the variance as E[X^2]-E[X]^2 from the characteristic function.
Third, the VIX. I can use the stochastic model + parameters to compute option prices, plug those into the VIX formula, and get another variance for the time series of the underlying.
It is my understanding that those three should be, in a perfect world, identical values with different interpretations. But apparently, I'm wrong, since the second and third differ significantly, and the magnitude of the difference is dependent on the parameter values itself. Could you enlighten me on the difference of two and three, please?
Re: Vol-of-Vol and the VIX
Yes, the second and third are different. From the characteristic function for [$]X_T[$], if you compute [$]E_0[X_T^2] - (E_0[X_T])^2[$], you are getting the variance of the log-stock-price distribution at time [$]T[$]. It should depend upon all the parameters of the model -- in particular the correlation [$]\rho[$].
[$]VIX_T^2[$], on the other hand, equals (in this ideal world) [$]E_0[\int_0^T V_s ds]/T[$], the time-average of the expected realized variance rate of the trajectory of [$]{X_t}[$] from 0 to [$]T[$]. (This is the fair strike of the variance swap). As you can compute this easily from the stand-alone variance SDE, you can see it cannot depend on [$]\rho[$]. Indeed, it only depends upon the variance SDE drift parameters (and the initial variance rate [$]V_0[$]).
The fact that one formula depends upon [$]\rho[$] and one doesn't should convince you they cannot be the same. Similarly, the second one depends upon the vol-of-vol parameter and the third does not (as it only depends upon the [$]V[$]-drift parameters). So they wouldn't agree even if [$]\rho[$] were zero.
There are many different types of 'volatilities' in stochastic volatility models. These are two different ones. The formulas are different and so, as you have already discovered, the numerical results are different.
[$]VIX_T^2[$], on the other hand, equals (in this ideal world) [$]E_0[\int_0^T V_s ds]/T[$], the time-average of the expected realized variance rate of the trajectory of [$]{X_t}[$] from 0 to [$]T[$]. (This is the fair strike of the variance swap). As you can compute this easily from the stand-alone variance SDE, you can see it cannot depend on [$]\rho[$]. Indeed, it only depends upon the variance SDE drift parameters (and the initial variance rate [$]V_0[$]).
The fact that one formula depends upon [$]\rho[$] and one doesn't should convince you they cannot be the same. Similarly, the second one depends upon the vol-of-vol parameter and the third does not (as it only depends upon the [$]V[$]-drift parameters). So they wouldn't agree even if [$]\rho[$] were zero.
There are many different types of 'volatilities' in stochastic volatility models. These are two different ones. The formulas are different and so, as you have already discovered, the numerical results are different.
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- AwesomeBanana443
Re: Vol-of-Vol and the VIX
Eye-opening comment, thank you very much!
Do you know of any approach to compute the expected realized variance rate from the characteristic function?
Do you know of any approach to compute the expected realized variance rate from the characteristic function?
Re: Vol-of-Vol and the VIX
You're welcome.Eye-opening comment, thank you very much!
Do you know of any approach to compute the expected realized variance rate from the characteristic function?
Re question.
You'd use a *different* characteristic function (well, mgf): [$]\phi_T(c) = E_0[e^{ -c \int_0^T V_s \, ds}][$], easily found in Heston model.
Actually, not so different, as you can set [$]\rho = 0[$] in the c.f. you've got, replace occurrences of [$](i z - z^2)/2[$] by [$]c[$] (assuming the c.f. parameter is 'z') and drop the cost-of-carries. Result should be this new one. Or something like that; probably missed a sign. Best to start from the PDE for [$]\phi_T(c)[$], however, to do this cleanly and confirm the replacements I just mentioned.