Measuring instantaneous variance of SPX with VIX and the Heston stochastic volatility model

[stochastix]

This paper attached provides a relatively simple and accurate formula for the VIX futures based on the CIR model (square root diffusion)

https://en.wikipedia.org/wiki/Cox–Ingersoll–Ross_model

so here,
S[t] is the value of the SPX index,
μ is the mean daily return
θ is the long-run mean value of the instantaneous variance V[t] of S[t]
σ is the variance of the variance of S[t]

then they make a change-of-measure and introduce the risk premium λ . This parameter has large estimation errors and I've seen some literature where they just set this value by hand to λ=-0.87.

Now we see that the value of the VIX index right now is the expected value, under the risk-neutral measure, of the integral of the instantaneous variance V of SPY over s=now to s=one month from now

so here we see that we can solve for the instantaneous variance V[t]=(VIX^2-A)/B.

in the attached paper the authors posit that the transition density of V[t] follows a Cox-Ingersoll-Ross model.

Now to estimate the parameters.

I don't understand Equation 12, where did A go? fP is a probability distrubution on V[t] , i dont see how dividing a probability measure by B will create another probability measure for VIX^2

 

[Dazz]

the underlying problem with variance measurement is the correction needed due to lack of normal distribution. Variance determined by a non-parametric method is superior to parametric statistics. And instantaneous is no more predictive of future movements than say 20 sec delayed.

 

[stochastix]

the underlying problem with variance measurement is the correction needed due to lack of normal distribution. Variance determined by a non-parametric method is superior to parametric statistics. And instantaneous is no more predictive of future movements than say 20 sec delayed.

Huh? Read the paper dude ,it has nothing to do with normal distribution. Whats this trend of all these knee jerk reactions of people yelling about normal dist without reading .. too reactive I guess. It's not about predicting anything its about modeling and logic . I get it. You fear that you are slow. And you are trying to bolster your idea of surviving on lagged data . variance is a concept that applies to all random variables, not just normally distributed random variables

 

[ffs1001]

@stochastix , nice.

Honest question - does this formula help you make money?

If 'No', then let it lie dormant in the bowels of academia, where it may provide useful (if non-practical) intellectual stimulus to those inclined to such things.

If 'Yes', then I'd love to know how it can be used profitably.

Happy trading.

 

[stochastix]

@stochastix , nice.

Honest question - does this formula help you make money?

If 'No', then let it lie dormant in the bowels of academia, where it may provide useful (if non-practical) intellectual stimulus to those inclined to such things.

If 'Yes', then I'd love to know how it can be used profitably.

Happy trading.

@ffs1001 Well, it allows for instance to compute the probability of a VIX option culminating in the money or not. My first attempt I got 88% which was the same odds IB's software was showing me on this short strangle, computed from implied vols. I think I calibrated it incorrectly but since I got the same percentage I thought perhaps it wasnt an error.

I tried fitting a CIR process to VIX with https://www.mathworks.com/matlabcen...ersoll-ross-process-the-matlab-implementation and then I checked the parameters to make sure they made sense.. they did.

After reading the paper again, I'm wondering, is the calibration supposed to be made on VIX, VIX^2, or V ? the best Log-lik scores are gotten from fitting to VIX.. and one needs to know the parameters to transform VIX to V so that wouldnt make sense..

 

[stochastix]

the underlying problem with variance measurement is the correction needed due to lack of normal distribution. Variance determined by a non-parametric method is superior to parametric statistics. And instantaneous is no more predictive of future movements than say 20 sec delayed.

It depends what you mean by superior. Yeah the joint SPX/VIX calibration problem was solved in discrete-time nonparametrically but the computational cost is quite large. There is a continuous time model that came afterwards, that fits like 99.9% as well, but is extremely fast and parsimonious and allows to reason about economic interpretation of parameters.

 

[traderking007]

seems like no one can disagree with you..ever. lol

 

[Scataphagos]

seems like no one can disagree with you..ever. lol

I disagreed with him once... he didn't like it.

(Please don't bother to disagree with me, either. I also don't like it. Thankfully, I'm righteous about my indignation.... I actually know my shit!)

KISS, baby. As always.

 

[stochastix]

seems like no one can disagree with you..ever. lol

I'm a genius and I'm always right, and people like to spout off without thinking or analyzing around here. Sometimes i do get clumsy and hit the wrong button tho or put something in the wrong field

 

[tommcginnis]

It depends what you mean by superior. Yeah the joint SPX/VIX calibration problem was solved in discrete-time nonparametrically but the computational cost is quite large. There is a continuous time model that came afterwards, that fits like 99.9% as well, but is extremely fast and parsimonious and allows to reason about economic interpretation of parameters.

Hence, EOD data.

I appreciate the elegant nuance of the calculus, but remembering that the primary root of differential equations is difference -- that really knocks things down to a tidy bunch of spreadsheets. I hate to make this analogy, but it's as if the pdes were the scenic route, while some difference matrices were the interstate. I'd *rather* take the scenic route. But spotty (market) data, and doubt in variables' underlying distributions + sampling, ...: "Meh!"

I'll take the interstate and get there safely.

(Yes, this does mean that my ability to translate/map to intraday data/markets gets less helpful as the anticipated trade length shortens. But if what most of what I'm trading is a day or better? (and, even a week or three?). Then I'm good with it.

Nice paper. I *think* I have it. But I'm going to copy the url/pdf just in case.