Hedging Options with Stock Delta and Index Beta

Summary

This note addresses minimum-variance hedging for vanilla options in a binomial-model setting. When the hedging stock is the option's own underlying, the response recommends using the option position's net delta to determine the stock hedge. The binomial parameters in the question are not used to derive a separate regression estimate; the answer instead points to the standard delta hedge.

For an option portfolio on individual stocks hedged with an index, the response suggests scaling each stock's delta by that stock's beta to the index. It frames beta as analogous to delta: both describe a hedge relationship, with beta measuring a stock's relationship to the index. The answer cautions that beta is less precise than delta. It provides no calculation, data, or conditions under which this approximation performs well, so the method is a concise conceptual guide rather than a full minimum-variance derivation.

Key ideas

• For options hedged with their own underlying stock, use the portfolio's net delta to size the stock hedge.
• When hedging individual-stock exposure with an index, the response proposes scaling delta by the stock's index beta.
• Beta plays a role analogous to delta by describing a hedge relationship between a stock and an index.
• The beta-based index hedge is less precise than a direct delta hedge.

Tags

# Minimum Variance Hedge Ratio in Binomial Framework


# Minimum Variance Hedge Ratio in Binomial Framework












In order to find the minimum variance hedge ratio when holding a portfolio of vanilla call options and hedging with stock, you can do an OLS regression.

In a binomial model framework, given parameters So, K, sigma, r, T, and the number of periods in the tree, how can you calculate the minimum variance hedge ratio?

## Answer by user3264325 (score 1)

https://quant.stackexchange.com/a/12599

If the stock you'd like to hedge with is the same as the option's underlying obviously just find the net delta and hedge with that amount of stock.

If you have different types of stocks and would like to hedge with an index you can multiply the delta with the beta of each stock versus the index.

Beta is analogous to delta in a way. With delta we describe how to hedge an option with its underlying stock. Beta similarily is the minimum variance hedge between a stock on its index.

Obviously beta is a less accurate measure than delta however.

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