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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?

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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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