Why Return-Based Hedge Ratios May Not Minimize Price Variance

Summary

The document asks whether the minimum-variance hedge ratio derived from returns also minimizes variance at the price level. It highlights that a hedge ratio chosen to reduce variation in changes or returns may not optimize variation in price levels, especially when modeling lognormally distributed prices from normally distributed returns.

The question proposes comparing hedge portfolios built from simulated spot and futures paths using the return-based ratio. It gives no derivation, simulation results, or answer establishing whether the two objectives coincide. The practical lesson is to distinguish the variable being hedged and the variance being minimized: return variance and price-level variance are separate objectives, so a ratio optimized for one should not be assumed optimal for the other. The issue remains open in this document, and its setup does not specify a model, horizon, or precise price-level objective.

Key ideas

• A minimum-variance hedge ratio is defined relative to the variable whose variance is being minimized.
• A ratio optimized for returns does not automatically minimize the variance of price levels.
• Transforming normally distributed returns into lognormal prices can change distributional properties.
• Monte Carlo paths could be used to compare hedge ratios under clearly specified return and price-level objectives.

Tags

# Minimum variance hedge ratio price difference vs. log-returns


# Minimum variance hedge ratio price difference vs. log-returns












So from my understanding Hull (2012) f.e. shows that the optimal hedge ratio minimizes the variance of the returns. But what happens to the variance of the prices? Is the Minimum variance hedge portfolio also the one, which has the lowest variance at the price level? For example, if I compute the optimal hedge ratio for the returns and then I simulate the spot and futures path with Monte-Carlo and use the optimal hedge ratio to construct the hedge portfolio, will it in the end also be the one with the lowest variance? I was wondering because when going from the returns (which are by assumption normally distributed) to the price level (log-normal) the distribution changes so this might not hold anymore.

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