Deriving a Minimum-Variance Hedge from Itô’s Lemma

Summary

The document considers hedging a spot asset with a forward whose value depends on the spot price and time. Applying Itô’s lemma expresses the forward’s change as a time-dependent drift term plus its sensitivity to the spot change. Writing that sensitivity as the derivative of forward value with respect to spot makes the instantaneous portfolio variance depend on the hedge position and this sensitivity.

Using the diffusion rules for the spot process, the answer derives the variance rate of the hedged portfolio as proportional to the square of one plus the hedge position times the forward’s spot sensitivity. It also explains why the squared expected change contributes no term at this instantaneous order: the expectation is of order dt, so its square vanishes in the differential variance calculation. The response does not explicitly solve for the minimizing hedge position, and its result concerns local variance under the stated diffusion assumptions rather than finite-horizon hedge risk.

Key ideas

• Itô’s lemma writes the forward’s change as a drift component and a spot exposure component.
• The forward’s local spot exposure is represented by its derivative with respect to the underlying price.
• The portfolio’s instantaneous variance is proportional to the square of its combined spot exposure.
• The derivation is local and does not itself state the hedge position that minimizes variance.

Tags

# minimum variance hedge with stochastic processes


# minimum variance hedge with stochastic processes












Problem set up:

asset S: $$\frac{dS}{S} = \mu dt+\sigma dz$$ Hedged using a forward contract: $F = F(S,t).$ Hedge portfolio: $$P = S+nF$$ I want to find the variance of $dP$, and then minimize that with respect to $n$, to calculate the optimal number of forward contracts.

$$dP = dS + ndF;$$ $dF$ uses Ito's Lemma The variance of the change in the portfolio is defined as follows: then for $$V(dP) = EdP^2 - (EdP)^2$$ where V stands for Variance and E stands for Expectation, of the Portfolio P My goal is to find the Variance and then minimize it with respect to n. Does anyone have experience using the concept of minimum variance hedge ratio in a set up like this? Any guidance would be appreciated. Thanks

## Answer by spaceisdarkgreen (score 1, accepted)

https://quant.stackexchange.com/a/35890

Ito's lemma gives $$dF = \left(\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2F}{\partial S^2}\sigma^2 S^2\right)dt + \frac{\partial F}{\partial S}dS = adt + bdS $$

Using the usual rules, e.g. $dz^2 = dt$, we get $$ dS^2 = \sigma^2S^2dt,$$ $$dF^2 = b^2dS^2 = b^2\sigma^2S^2dt,$$ and $$dSdF = bdS^2 = b \sigma^2S^2dt,$$ so this gives $$dP^2 = dS^2 + n^2dF^2 + 2ndFdS = \sigma^2S^2 (1+n^2b^2+2nb)dt = \sigma^2S^2(nb+1)^2dt.$$

This is the same as the expected value $E(dP^2)$.

Then for the other term, you know that $E (dP)$ is going to be of the form $c dt$ so that $(E(dP))^2 = 0.$

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