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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?

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    $\begingroup$ Can you check this formula for typos? it seems to be missing a $(\cdot)^2$ which is required to make the log positive by squaring it. $\endgroup$
    – nbbo2
    Commented May 1, 2023 at 15:18
  • $\begingroup$ @nbbo2 pretty sure this is it. This is a result you get from deriving the fair value of a variance swap, which defines the VIX. See Derman's paper on "More than you ever wanted to know about variance swaps" $\endgroup$ Commented May 1, 2023 at 17:42
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    $\begingroup$ Hmmm... emanuelderman.com/wp-content/uploads/1999/02/… What is the page and equation number? $\endgroup$
    – nbbo2
    Commented May 1, 2023 at 18:22
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    $\begingroup$ The expecation of the log can never be positive (in theory) as $$ \log S_T/S_t = -\frac12 \int_t^T \sigma_u^2 du + \int_t^T \sigma_u dW_u$$ $\endgroup$
    – Frido
    Commented May 2, 2023 at 7:39

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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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