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Why the Risk-Neutral Expected Log Return Keeps VIX Real

Article Quant Q&A · Author: Sinbad The Sailor
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Summary

The discussion examines a Monte Carlo pricing expression that represents VIX through the risk-neutral expectation of the log of the S&P 500’s future level relative to its current level. The question is why simulated values might make the expression under the square root negative, given that the index can rise over the horizon.

The response invokes the risk-neutral martingale property, assuming interest rates are ignored. Since the expected future index level then equals its current level, Jensen’s inequality bounds the expected log return above by zero. This explains why the population expectation in the formula should not be positive under those assumptions. A finite simulation estimate can still be positive through sampling error or implementation issues; the brief answer does not discuss those possibilities or model details.

Key ideas

  • With interest rates ignored, the S&P 500 is treated as a martingale under the risk-neutral measure.
  • Jensen’s inequality implies that the expected log of the future-to-current index ratio is no greater than zero.
  • A positive estimate from finite Monte Carlo draws may reflect sampling variability or a modeling or implementation issue.
  • The explanation relies on the stated zero-interest-rate assumption.

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OptionsVolatilityDerivatives pricingStatistics
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# Pricing VIX derivatives using Monte Carlo


# Pricing VIX derivatives using Monte Carlo












I am looking at pricing VIX options and futures using Monte Carlo simulation. Most papers recognise that VIX can be written in terms of the S&P500 index itself, namely as:

$$ VIX_t = \sqrt{-\frac{2}{\Delta}\mathbb{E}^{\mathbb{Q}}\Big[\ln \frac{S_{t+\Delta}}{S_t}\Big | \mathcal{F}_t \Big]}, $$

if we disregard interest rates and in which $\Delta$ is 30 days.

My question is, how to avoid negative values in the square-root? When running simulations I often get the expectation of the log to be positive, making the VIX go imaginary. Is it not reasonable to imagine $S_{t+\Delta}$ to be larger than $S_t$ and thus the mean of the log always become positive?

## Answer by Achrbot (score 1, accepted)

https://quant.stackexchange.com/a/75412

If we disregard interest rates, $S_t$ is a Martingale under $\mathbb{Q}$. So by Jensen's inequality, the expectation has an upper bound of 0.

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