Calculating Option Beta from Underlying Beta and Delta

Summary

The document asks how to calculate beta-weighted delta and gamma for a portfolio of options on different stocks. Its answer gives a relationship for an individual option’s beta: multiply the underlying price by the option’s delta and the underlying asset’s beta, then divide by the option price. The variables are identified as the underlying price, option value, the usual first derivative of option value with respect to the underlying, and the underlying beta.

The answer suggests rearranging this relationship if an option beta has been measured empirically, and differentiating with respect to the underlying to obtain a gamma-related expression. It also cautions that this extension may not answer the portfolio question as posed. No worked portfolio aggregation, treatment of differing betas, or empirical validation is supplied, so the formula is a starting point rather than a complete method for portfolio beta-weighted Greeks.

Key ideas

• The option beta relationship scales the underlying beta by underlying price times delta divided by option value.
• Delta is the sensitivity of option value to the underlying price.
• The answer proposes differentiating a rearranged relationship to explore gamma, but does not work through the result.
• The document does not provide a complete portfolio aggregation formula or empirical validation.

Tags

# What is the formula for beta weighted delta and gamma?


# What is the formula for beta weighted delta and gamma?












I am trying to calculate the beta weighted delta and gamma for a portfolio of options of different underlying stocks, but I can't seem to find the correct formula.

Can someone point me to it or a book that contains it?

## Answer by GoneAsync (score 4)

https://quant.stackexchange.com/a/15141

I've started thinking about this, too. My gedanken conclusion turned out to be too simple once I found what I was after: http://www.investment-and-finance.net/derivatives/o/option-beta.html, which I've confirmed in Black & Scholes (1973) p10 (eq 15).

In short: $$ \beta_{\text{option}} = \frac{S\cdot\Delta}{O}{\beta_S} $$ where $S$ is the underlying price ($x$ in the B&S paper), $O$ is the option price ($w_1$ in B&S), $\Delta$ is the usual $\partial{O}/\partial{S}$, and $\beta_S$ is $\beta$ for the underlying.

Regarding your question, you'd just have to re-arrange this to use an empirically measured option $\beta$, and differentiate for Gamma. I'm not sure that gets you where you want to go, though.

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