Log-Return Moments in a Local Volatility Model

Summary

The document derives an expression for the expected log return over a time interval under a local volatility stochastic differential equation. Applying Itô’s lemma gives a drift contribution and a negative half-variance contribution; the expected stochastic integral vanishes under the usual integrability conditions. The remaining expectation of squared local volatility depends on the distribution of the stock process, so it generally cannot be reduced using only the deterministic drift and the local volatility function as written.

It also expands the log-return variance into the variance of the stochastic integral, a covariance term with integrated squared volatility, and the variance of that integrated quantity. It identifies the first term through Itô’s isometry and asks how to handle the others. Constant volatility recovers the familiar variance proportional to elapsed time. The text is an open derivation rather than a completed general formula, and it offers no numerical or empirical validation.

Key ideas

• Itô’s lemma expresses log returns as an integrated drift adjustment plus a stochastic integral.
• The expected log return depends on the expected squared local volatility along the stock process.
• Itô’s isometry evaluates the stochastic integral’s second moment as integrated expected squared volatility.
• The remaining variance terms involve dependence between the stochastic integral and integrated squared volatility.
• With constant volatility, log-return variance reduces to volatility squared times the interval length.

Tags

# Formula for variance of European call/put in Black Scholes


# Formula for variance of European call/put in Black Scholes












I have a quite basic question, but I can't find a reference with it.

Recall that we can use the Black-Scholes formula to price a European call or put for a market consisting when:

- the underlying asset following geometric Brownian motion;

- the risk free interest rate is considered constant;

- the volatility of the underlying asset returns is constant.

In deriving this, one writes the call/put as expectations of discounted payoffs, e.g. $C=E^Q[\exp^{-rT}(S_T-K)_+]$ for call ($Q=$risk neutral prob.), where $(S_t)$ is follows geometric Brownian motion.

My question is : what is the variance of what lies in the bracket ? I ask this for calls and puts.

## Answer by math (score 2, accepted)

https://quant.stackexchange.com/a/8351

I just sketch how you can derive the formula. In your setting we have $S_T=s_0e^{\sigma W_T-\frac{1}{2}\sigma^2T}$. W.l.o.g we put $s_0=1$. Hence $S_T$ is lognormal distributed and $S_T^2=e^{2\sigma W_T-\sigma^2T}$, which is still lognormal. $Var(X)=E[X^2]-E[X]^2$. We just have to compute $E[X^2]$. Let $A:=\{S_T>K\}$

$$E[X^2]=E[(S_T-K)_+^2]=E[S_T^2\mathbf1_A]-2KE[S_T\mathbf1_A]+K^2Q[A]$$

Now the two last terms you have already calculated in the usual derivation of Black Scholes. For the first one you use the same techniques as for $E[S_T\mathbf1_A]$ with a different lognormal distribution.

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