# Min variance Hedge II

In a paper from Energy Risk - "Delta hedging the load serving deal", the author shows how to calculate the min variance hedge for a portfolio of two underlying assets. I've added a picture of the relevant section (I don't think I can upload the paper here):

My attempted solution is also attached:

I "sort of" get to the solution...but not quite. I did not apply Ito's Lemma to begin with. The main shortcut I took is - I took the partial derivatives of V with respect to P and L (the two assets) out of the expectations operator (see note 1 at the bottom of the second image). I'm not sure why this operation is allowed (or is it allowed at all)? So while I match the equation 6 from the paper, I'm not convinced that I've done it correctly.Has anyone seen a result like this before? Any guidance is appreciated. Thanks

• Hint: at time $t$ the rebalancing is performed and the hedge ratios are left untouched until the next rebalancing at $t+dt$. – Quantuple Feb 2 '18 at 8:25
• @Quantuple - so basically for that instant dt, the partials can be treated as constants? Very intuitive..thanks sir! But to dig a little deepr (just curious, I'm no quant, just trying to learn), is this anything to do with Ito Isosymmetry? Could you point me to a good reference that discusses the principle in more detail? Right now, I'm going from one text to another and running around in circles..... – Chet Feb 2 '18 at 14:37
• That's right! This is related to the concept of self-financing portfolio. This is the same reason why one can consider $\Delta = \Delta(t,S_t)$ constant in the BS derivation, see here quant.stackexchange.com/questions/34574/… and references therein. You could write the self-financing condition in terms of quadratic variations as well hence linking to "Itô isometry" of sorts see math.stackexchange.com/questions/1828876/…, but the key concept is the self-financing ppty. – Quantuple Feb 2 '18 at 14:56
• @Quantuple - Thanks again....got a LOT of reading to do! – Chet Feb 2 '18 at 16:44