Conditional Expectations as Dirac-Weighted Ratios

Summary

The note explains how a conditional expectation at a specific asset level can be expressed as a ratio of expectations weighted by the Dirac delta function. For a terminal asset value and a random variance, the numerator weights variance by the event density at the strike, while the denominator supplies the corresponding marginal density. Their ratio gives the variance conditional on the asset ending at that level.

The derivation uses joint and marginal probability densities, then represents conditioning at the point with a delta function inside integrals. This identity is relevant to derivations such as the Dupire local volatility equation. It assumes suitable densities and treats conditioning on an exact value through density-based notation; for a continuous random variable, the event of equaling one exact value has probability zero, so the formula is understood in this density or generalized-function sense. The note provides a derivation, not a discussion of regularity conditions or cases where densities may fail to exist.

Key ideas

• A conditional expectation at a fixed asset level can be written as a ratio of delta-weighted expectations.
• The numerator combines the quantity of interest with the density concentrated at the conditioning level.
• The denominator normalizes by the marginal density at that level.
• The derivation uses joint and marginal probability densities.
• Point conditioning for a continuous variable is interpreted through densities rather than an ordinary positive-probability event.

Tags

# Conditional expectation and Dirac delta function


# Conditional expectation and Dirac delta function












In the proof of `Dupire equation` we end up with an identity involving the `Dirac delta function`.

How to prove that $$\dfrac{E[\sigma_T^2\delta(S_T-K)]}{E[\delta(S_T-K)]}=E[\sigma_T^2|S_T = K].$$

where $\delta(x)$ is the `Dirac delta function.` $S_T$ is a random variable, and $\sigma_T$ also.

## Answer by Quantuple (score 3, accepted)

https://quant.stackexchange.com/a/36613

Slightly abusing notations \begin{align} \Bbb{E}\left[ \sigma^2_T \vert S_T = K \right] &= \int_{0}^{+\infty} \sigma^2_T \, p( \sigma^2_T \vert S_T = K) d\sigma^2_T \\ &= \int_{0}^{+\infty} \sigma^2_T \frac{p(\sigma^2_T, S_T=K)}{p(S_T=K)} d\sigma^2_T \\ &= \frac{\int_{0}^{+\infty} \sigma^2_T p(\sigma^2_T, S_T=K) d\sigma^2_T }{p(S_T=K)} \\ &= \frac{\int_{0}^{\infty} \int_{0}^{+\infty} \sigma^2_T \delta(S_T-K) p(\sigma^2_T, S_T) d\sigma^2_T dS_T }{\int_{0}^{+\infty} \delta(S_T-K) p(S_T) dS_T} \\ &= \frac{\Bbb{E}\left[ \sigma^2 \delta(S_T-K) \right]}{\Bbb{E}\left[ \delta(S_T-K) \right]} \end{align}

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