# 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

• I edited to typeset with mathjax for readability. Click on the edit to see how to do it. It's unclear to me what $V,$ $E,$ and lower-case $p$ are (although perhaps lowercase $p$ is just a typo for uppercase). And also it's unclear as a result what you're trying to do in the last line. – spaceisdarkgreen Sep 2 '17 at 23:38
• Thanks! I'll check ti out. I'll also make some edits along the lines you suggested to clarify the question better. – Chet Sep 2 '17 at 23:41

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 $(E(dP))^2 = 0.$

• Man thanks a ton! I was applying Ito's incorrectly, somewhere. I followed your steps to attempt again and got to the right point. My final solution for the minimum variance hedge is then n = -1/b. Are you able to confirm that you get the same? – Chet Sep 3 '17 at 0:22
• @user24392 Yeah that would just be hedging on whatever the delta is between the forward and the underlying is, so it seems like that has to minimize the one-period variance. – spaceisdarkgreen Sep 3 '17 at 0:37
• thanks. I need to rethink my problem set up though...I should be getting n = -b, not 1/b. I'll go back to the drawing board for now. thanks again! – Chet Sep 3 '17 at 0:41
• @Chet Yes, it is because $dP^2$ is not random. – spaceisdarkgreen Jan 16 '18 at 20:49
• @Chet yes, that is right – spaceisdarkgreen Jan 16 '18 at 22:22