## Risk-neutral vs. real distribution of asset returns

Before we continue down this path, I think we must remember that the stylized facts that Cont lists are for the *observed/”real” time series*, and need not necessarily hold for the time series generated by a stochastic volatility model – which is in the risk neutral world. In fact, there is no reason to expect it would.

I raised this on Wilmott’s [CQF] forum:

Q: In the Black-Scholes world the implied risk-neutral (RN) distribution of asset returns is normal. As we go from local volatility Dupire type models to Heston-type stochastic volatility model, the implied RN distribution exhibit fatter tails and in any reasonable stochastic volatility model (with negative asset-volatility correlation), a significant negative skew. Obviously, the implied RN distribution from more sophisticated models is closer to what we observe in reality across asset classes.

Assuming that there are no tradables available for volatility (i.e. no VIX, no variance swap etc.), so use of any stochastic volatility model brings with itself a market price of volatility risk, does it make

sense–finance wise, that is, to make comparisons between the risk neutral and the real distribution? Or, is it that if we assume a ‘linear’ market price of risk (as in, e.g. Hull-White/Heston), it is ok to do so in an approximate sense?

This is what Paul Wilmott had to say:

A: It’s quite hard to find the Market Price of Vol Risk even with traded (vol) instruments!

So, the answer is that it doesn’t make a lot of sense.

Suddenly I am not sure if it’s even worthwhile to go on this path. However, as another Wilmott member in a different forum [Technical] points out:

I’m not sure if this is exactly what you are looking for, but Ait-Sahalia has done some nice stuff on real vs risk neutral distributions.

Of course, he was one of the first guys to address this question. I had totally forgotten about it.

But, then, his work, I think, is doing the reverse. He starts out with a one factor model for asset returns with an *unknown* diffusion component. He argues that because Girsanov’s theorem ensures that when we change measures the diffusion term remains unchanged, it is ok to extract the diffusion component from the real world data. Then, what just remains is drift adjustment, which he does suitably. In his conclusion he says that they plan to do so in a stochastic volatility set up, but am not sure he has anything published on that yet.

I think it’s time to take a step back and think through before jumping into econometrics.

**Addendum**: Since the drift term is not important (compared to diffusion) at very short time scales, Paul Wilmott suggests that it may not be all that meaningless to do a comparison using high frequency data and very short maturity options (though this raises its own problems) – but even then one’ll have to be very careful (ala Ait-Sahalia) in drawing comparisons between the dynamics implied by short maturity options vs. high frequency data.

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