Deriving a Minimum-Variance Hedge for Two Correlated Assets

Summary

The document concerns the derivation of a minimum-variance hedge for a portfolio involving two underlying assets, in the setting of a load-serving deal. The author is trying to reproduce an equation from a cited energy risk paper and asks whether taking partial derivatives of portfolio value with respect to the two asset prices outside an expectation is valid. They also note that they did not start with Itô’s lemma and are unsure whether the shortcut is justified.

The underlying topic is stochastic hedging: portfolio sensitivities to the asset prices and their covariance structure determine hedge positions that minimize variance. A sound derivation typically models changes in portfolio value, uses stochastic calculus where prices follow diffusion processes, and optimizes the variance of the hedged change. Moving derivatives through an expectation requires suitable regularity and integrability conditions; random sensitivities generally cannot be treated as fixed constants without justification. The document supplies no attached equations or data and does not resolve the derivation, so its assumptions and the exact hedge formula cannot be assessed from the text alone.

Key ideas

• A minimum-variance hedge chooses positions to reduce the variance of portfolio changes.
• The hedge depends on asset sensitivities and the covariance between underlying price changes.
• Itô’s lemma can express the portfolio’s stochastic change when its value depends on asset prices.
• Taking random partial derivatives outside an expectation requires conditions that justify the operation.
• The document does not include the referenced equations or a completed derivation.

Tags

# Min variance Hedge II


# 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

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