Conditional Expected Variance from a Joint Risk-Neutral Density
Summary
This exchange explains how to calculate the conditional expectation of a variance variable given a particular terminal asset price, using a joint probability density under a chosen measure. The derivation applies the conditional density identity: divide the integral of variance times the joint density by the marginal density of the conditioning price. The marginal is obtained by integrating the joint density over all admissible variance values.
This produces the ratio of two integrals, rather than simply taking the reciprocal of the weighted integral. A discrete approximation follows the same normalization principle, so common cell-width factors cancel when the joint-density values represent probability mass consistently. The result is an expression for the conditional expected variance; taking its inverse is a separate operation after computing that expectation. The discussion assumes continuous variables, a valid joint density, and a nonzero marginal density at the specified price. It provides a probability calculation, not an options-pricing application or empirical validation.
Key ideas
- A conditional expectation is calculated using the conditional density of variance given the terminal price.
- The conditional density equals the joint density divided by the marginal density of the conditioning variable.
- The marginal price density is found by integrating the joint density over variance.
- A discrete approximation must normalize the variance-weighted sum by the corresponding probability sum.
- The specified price must have a nonzero marginal density for the ratio to be defined.
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Full text
# How to express the conditional expected variance under the risk neutral measure?
# How to express the conditional expected variance under the risk neutral measure?
The conditional expection of variance under risk neutral measure is
$$\mathbb{E}^Q[V_T |S_T=K]$$
where $S_T$ and $K$ represent the spot price at maturity and strike price, respectively.
Assume I know the risk neutral density $q(V,S,t)$, I want to calculate the inverse of conditional expectation of $V_T$. Should I use this formula $$\frac{1}{\mathbb{E}^Q[V_T|S_T=K]} = \frac{1}{\int V_t \cdot q(v,s,t)\cdot dV}$$ or another formula
$$\frac{1}{\mathbb{E}^Q[V_T|S_T=K]} = \frac{\int q(v,s,t)\cdot dV}{\int V_t \cdot q(v,s,t)\cdot dV}$$
Further, if $$g(v,s,t)\approx q(v,s,t)\cdot dv \cdot ds$$ I wonder if we can get $$\Bbb{E}^\Bbb{Q}\left[ V_T \vert S_T = s \right] = \frac{\int_\Omega v\,q_{V_T,S_T}(v, s,T) dv}{\int_\Omega q_{V_T,S_T}(v, s,T) dv} \approx \frac{\sum v\cdot q_{V_T,S_T}(v,s,t) \cdot dv }{\sum q_{V_T,S_T}(v,s,t) \cdot dv}$$
$$= \frac{\sum v \cdot \frac{g(v,s,t)}{dv \cdot ds} \cdot dv }{\sum \frac{g(v,s,t)}{dv \cdot ds} \cdot dv} = \frac{\sum v \cdot g(v,s,t)}{\sum g(v,s,t)}$$
Thanks!
## Answer by Quantuple (score 3, accepted)
https://quant.stackexchange.com/a/31170
Assume that $q_{V_T,S_T}(v,s,T)$ represents the known joint pdf of 2 continuous random variables $V_T$ and $S_T$ under some probability measure $\Bbb{Q}$. Further assume that the definition domain of $V_T$ is $\Omega$.
By definition of the expectation operator + conditional probability density function $$ \Bbb{E}^\Bbb{Q}\left[ V_T \vert S_T = s \right] = \int_\Omega \, q_{V_T \vert S_T}(v, s, T) dv $$
From the product rule of probability $$ q_{V_T \vert S_T}(v, s, T) q_{S_T}(s,T) = q_{V_T,S_T}(v, s,T) $$ hence, going back to the expectation calculation $$ \Bbb{E}^\Bbb{Q}\left[ V_T \vert S_T = s \right] = \frac{\int_\Omega v\, q_{V_T,S_T}(v, s,T) dv}{q_{S_T}(s,T)} $$ Finally using the sum rule to express the marginal $q_{S_T}$ from the joint pdf $q_{V_T,S_T}$ one gets: $$ \Bbb{E}^\Bbb{Q}\left[ V_T \vert S_T = s \right] = \frac{\int_\Omega v\, q_{V_T,S_T}(v, s,T) dv}{\int_\Omega q_{V_T,S_T}(v, s,T) dv} $$ which is equivalent to the second formula you mention.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.