Local Variance as Conditional Expected Instantaneous Variance
Summary
The document discusses a question about Gatheral’s derivation connecting local variance to the conditional expectation of instantaneous variance. The central issue is why a stochastic differential term involving the underlying price and a strike disappears, and which filtration the conditional expectation uses.
The responses appeal to martingale properties and the zero conditional mean of future Brownian increments. They outline conditioning on information available at a pricing time and argue that the increment contributes no expected drift under the stated assumptions. However, the explanations are informal and contain potentially questionable conditioning steps: in particular, multiplying conditional expectations is not generally valid without suitable independence or measurability conditions. The exchange gives intuition rather than a complete derivation, so readers should verify the precise filtration and time indexing in the original argument.
Key ideas
- A martingale has zero expected future increments conditional on its current information.
- The derivation concerns the relationship between local variance and conditional expected instantaneous variance.
- Conditional expectations must be taken with respect to a clearly specified filtration.
- The responses offer intuition, but their factorization steps require assumptions that are not fully established.
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# Local variance derivation by Gatheral
# Local variance derivation by Gatheral
I've bought Gatheral's book on Local Volatility and I have troubles with understanding a part where he shows that local variance is a conditional expectation of instantaneous variance.
Why in the second equation from the bottom he just skips the term $\theta (S_T-K)dS_T$? He says that it's because $F_{t,T}$ is a martingale. I see that $F_{t,T}$ is a martingale, but don't know how this helps. Also , what "condiational expectations" is he talking about? The notation looks a bit sloppy. Thanks for help.
http://www.math.ku.dk/~rolf/teaching/ctff03/Gatheral.1.pdf
## Answer by KT8 (score 0)
https://quant.stackexchange.com/a/78146
I think the reason for that term to vanish is that you can write the expectation of the $dS_T = dF_{T,T}$ in term of conditional expectations. First, recall the tower property of conditional expectations
$$\mathbb{E}[X]=\mathbb{E}[\mathbb{E}[X|Y]].$$
Then $$ \begin{align} \mathbb{E}\left[ \theta (S_T - K) dS_T \right] & = \mathbb{E}\left[ \theta (F_{T,T} - K) dF_{T,T} \right] = \mathbb{E}\left[ \mathbb{E}\left[ \theta (F_{T,T} - K) dF_{T,T} \Big \vert \mathcal{F}_T\right] \right] \\ & = \mathbb{E}\left[ \theta (S_{T} - K) \mathbb{E}\left[ dF_{T,T} \Big \vert \mathcal{F}_T\right] \right] \end{align} $$ and since $F_{t,T}$ is a martingale, $\mathbb{E}\left[ dF_{T,T} \Big \vert \mathcal{F}_T\right] = 0$.
If someone finds that my reasoning has some "muddy" steps, I'd appreciate the correction, so I also get this right.
## Answer by siou0107 (score 0)
https://quant.stackexchange.com/a/78152
Conditional expectations is with respect to the filtration of your pricing date $t \leq T$. Because $S$ is a martingale, you have $$ \mathbb{E} \left[\theta \left(S_T - K\right) dS_T \middle\vert \mathcal{F}_t\right] = \mathbb{E} \left[\theta \left(S_T - K\right) \middle\vert \mathcal{F}_t\right] \mathbb{E} \left[d \, S_T\middle\vert \mathcal{F}_t\right] $$ Remember that $S_T$ is $\mathcal{F}_T$-measurable, and increment $d\,S_T$ has to be dependent on $\mathcal{F}_T$ only through current value of the process $S_T$ ; to convince yourself, just rewrite $$\theta \left(S_T - K\right) dS_T = \theta \left(S_T - K\right) \sigma_T S_T dW_T$$ and remember that Brownian motion increments are independent, hence $\mathbb{E} \left(dW_T \middle \vert \mathcal{F}_T\right) = \mathbb{E} \left(dW_T \middle \vert \mathcal{F}_t\right) = 0$.
That is why that term vanishes. The rest is straightforward.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.