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?
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1$\begingroup$ What is beta weighted? Is it under Black Scholes model? $\endgroup$– emcorCommented Oct 22, 2014 at 20:21
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$\begingroup$ By beta weighting I mean, using the beta of the underlying stock to use with the individual options that make up a portoflio to come up with a delta of entire portfolio. $\endgroup$– CptanPanicCommented Oct 23, 2014 at 11:35
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1 Answer
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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.
