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Expected Market Returns, Risk-Neutral Variance, and VIX Squared

Article Quant Q&A · Author: Mats Lind
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Summary

The document examines the relationship between required returns, variance, stock-index levels, and expected volatility measures. The response points to a decomposition of the market’s expected excess return that includes risk-neutral variance and additional terms. It argues that VIX squared is not generally the same as risk-neutral variance: the equivalence depends on restrictive distributional assumptions, such as the standard lognormal case.

It contrasts the weighting used to construct a variance swap with the strike weighting embedded in VIX squared. The response describes a swap-style measure using equal weights across puts and calls, while VIX squared weights option prices by the inverse square of strike; it gives formulas for both measures. This offers a reason that a chart of index levels against VIX may not match a simple variance-based relationship. The exchange does not provide empirical tests of linear or quadratic relationships, and the result depends on the pricing assumptions and option data used.

Key ideas

  • Expected market excess returns can be related to risk-neutral variance, alongside other components.
  • VIX squared is not generally identical to risk-neutral variance outside particular distributional assumptions.
  • Variance-swap and VIX calculations weight option prices differently across strikes.
  • The exchange gives a theoretical explanation but no empirical test of index-level relationships.

Tags

OptionsVolatilityStatistics
Full text
# Risk, required return and expected volatility - what is the relationship?


# Risk, required return and expected volatility - what is the relationship?












Return required from risk averse agents from risky investments are proportional to expected return variance. That is from the textbook, you take the portfolio with the highest return to standard deviation, and then you lever or dilute it to fit the return to variance requirement of your investor.

Now, shouldn't you expect future volatility indicators like the VIX to have a quadratic relationship to stock index levels? A quick look at a chart suggests more of a linear relation. Or is it so that the stock-index has a linear relation to expected volatility, but that the VIX have a quadratic relation to longer term expected volatility?

Any references to literature or explanations on this would be more than helpful!

## Answer by fni (score 7, accepted)

https://quant.stackexchange.com/a/30913

I think you may be interested in this QJE forthcoming article by Ian Martin. The key idea of the article (page 5) is that the expected return on the market can be decomposed as $E_t[R_{t+1}]-R_f = \frac{1}{R_f}Var^Q(R_{t+1}) + \text{extra terms}$ As you correctly pointed out the expected return should be related with the risk neutral variance. The issue with VIX$^2$ is that it doesn’t measure risk neutral variance, unless we are in the standard lognormal case. As you can see from page 15 the correct way to construct a variance swap is by using a portfolio with the same weights on all puts and calls while the VIX$^2$ has weights proportional to the inverse of the squared strike, i.e. $SVIX^2=\frac{1}{(T-t)R_f^2}Var^Q(R_{t+1})=\frac{2}{S_t^2}\left[\int_0^F put(K)dK+\int_F^\infty call(K)dK\right]$ $VIX^2=\frac{2R_f}{T-t}\left[\int_0^F \frac{1}{K^2}put(K)dK+\int_F^\infty \frac{1}{K^2}call(K)dK\right]$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.