Integrating Local Variance Against a Risk-Neutral Density

Summary

The document presents an integral for expected local variance under a risk-neutral distribution. It defines log price relative to its initial value, identifies a probability density over that state and time, and weights local variance by the density before integrating. The author reports obtaining negative results when integrating from zero to each sampled state, especially for negative log-price values, despite both the density and local variance being positive.

The example points to a distinction between a cumulative integral with a signed endpoint and an expectation over the full state domain. Integrating from zero to a negative endpoint reverses orientation, so the result can be negative even when the integrand is positive. The document itself asks how to address this and does not provide a resolution. It also leaves details of the density’s arguments, integration limits, and numerical implementation unspecified, so the displayed expression and procedure should be checked against the intended risk-neutral expectation before drawing conclusions.

Key ideas

• An expected local variance is formed by weighting local variance with a risk-neutral state density.
• A positive integrand can yield a negative signed integral when the upper limit lies below the lower limit.
• An expectation generally requires integration across the relevant state domain rather than to each sampled state from zero.
• The document identifies a numerical or setup issue but leaves the intended integration limits and implementation unresolved.

Tags

# compute the risk-neutral expected variance


# compute the risk-neutral expected variance












The formula (3.8) on page 30 of the book THE VOLATILITY SURFACE by Gatheral(2006) introduces a method for computing the expected variance under the risk neutral measure. By denoting $x_t = log(S_t/S_0)$, and $q(x_t,t;x_T,T)$ and $\sigma^2_{loc}(x_t,t)$ represent the $pdf$ of $x_t$ and local variance, respectively: $$ E[\sigma^2_{K,T}(t)]= \int dx_t\cdot q(x_t,t;x_T,T) \cdot \sigma^2_{loc}(x_t,t) $$

I use this formula to compute the expected variance and it can return the negative value. My approach is:

- simulate a sequence of $x_t$;

- use my function to compute $q(x_t,t;x_T,T)$ for each point of $x_t$ and they are positive;

- generate a sequence of local variance which are positive;

- integrate from $0$ to each point of $x_t$.Then, I get a sequence of expected variance. When $x_t$ is negative, then the integrated result is negative. How can I overcome this problem?

Thank you!

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