# 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?

• 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. Commented May 1, 2023 at 15:18
• @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" Commented May 1, 2023 at 17:42
• Hmmm... emanuelderman.com/wp-content/uploads/1999/02/… What is the page and equation number? Commented May 1, 2023 at 18:22
• The expecation of the log can never be positive (in theory) as $$\log S_T/S_t = -\frac12 \int_t^T \sigma_u^2 du + \int_t^T \sigma_u dW_u$$ Commented May 2, 2023 at 7:39

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.