Do I need stock and option prices and calculate a vol beta or can I multiply the stock beta by delta? Sorry if the question is too simple.

If you're prepared to assume deterministic vol (e.g. constant as in BS or a function of share/strike price and time to expiry as in local vol models) then the beta of an option is equal to the beta of the share multiplied by the delta of the option.

Thanks Johnny. For now, I will use deterministic vol.

Actually, I've re-thought my answer to this. The beta of an option is equal to the beta of the share times delta times share price divided by option price. I'll post the explanation in a minute. ============================================================================================EDIT:Initially my logic was: The beta is the number of index contracts to sell short versus one shareThe delta is the number of shares to sell short versus one callTherefore the beta of the call must be the number of index contracts to sell short vs one call, i.e. share beta * deltaBut now I've thought some more about it:1. Use BS constant vol as an example. Assume the share price follows:dS = mu.S dt + sigmaS.S dW2. Want to write down an expression for the volatility of a call, something like:dC = muC.C dt + sigmaC. C dWwhere dW represents the same source of uncertainty as for the share price returns.3. We have C = C(S,t) so by Ito:dC = [~] dt + sigmaS.S dC/dS dW4. Match terms in dW between 2. and 3.sigmaC. C = sigmaS.S dC/dStherefore sigmaC = sigmaS. (dC/dS). (S/C)5. Now calculate beta as the number of index contracts that minimises the variance of a portfolio of 1 call and Beta index contractsVarPortfolio = (sigmaS^2) + (Beta^2).(SigmaIndex^2) + 2.Beta.Rho.SigmaS.SigmaIndexdifferentiate wrt Beta to find minimum variance portfolio ...dVarPort/dBeta = 2.Beta.(SigmaIndex^2) + 2.Rho.SigmaC.SigmaIndex = 0 for a minimumtherefore Beta = - Rho.SigmaC/SigmaIndex6. Remember from 4. that ...sigmaC = sigmaS. (dC/dS). (S/C)therefore ...CallBeta = - Rho.(SigmaS/SigmaIndex).(dC/dS).(S/C)compare with share beta ...ShareBeta = - Rho. (SigmaS/SigmaIndex)therefore ...CallBeta = ShareBeta.(dC/dS).(S/C)This is much more sensible, otherwise an (approximately ATM) call with a delta of 50% would have only half the beta of the underlying shares. Suppose (just for the sake of it) that the ratio of share price to call price is 6, then now the call beta is 3 times (50% * 6) more than the share beta. This makes more sense, doesn't it?

assuming everyone is talking about beta wrt market prices...to be clear, the vol also has a beta wrt market vol as well...

QuoteOriginally posted by: warheroassuming everyone is talking about beta wrt market prices...to be clear, the vol also has a beta wrt market vol as well...Which is why I wrote in my first post: "If you're prepared to assume deterministic vol"What are you doing? Adding irrelevant comments to lots of old threads without reading them carefully. I can think of more constructive ways to pass the time.

I would assume a normal distribution for the option to avoid having to divide with a 0 or near 0 call price. Plus the results behave better for stable correlations between underlying,option and the index. There is a typo in 5 whereby we have sigmaC, not sigmaS in the formulas. Rho is the correlation between C and index therefore it's not cancelled out in 6 with the other Rho which is the correlation between the stock and the index.Taking all this into account the option beta becomes: CallBeta = ShareBeta.(dC/dS).S.RHOcall/RHOShareYour derivation is a good starting point but there are two issues that need to be dealt in order to make it usable as I'm sure you are aware of. First, the correlations and how these should be estimated. Even a 10% change in the correlation can make the hedge less effective than simply not hedging. It would be ideal to estimate beta/correlation based on today's and tomorrow's estimated price and not on any other points. Intuitively this makes sense as we are looking at the P&L of the hedge between two days. Practically it can easily be shown in a spreadsheet that in cases where the correlation changes dramatically between one day and the next it's the correlation of only the last day's values that should be taken into account.The second point of course is what distribution to chose for the option and how to adjust it with time. Ideally we could find a balanced approximation between those two points. Any other ideas ?

There is a mistake in Apollon's CallBeta formula. It should read CallBeta=ShareBeta.(dC/dS).(S/C).(RHOcall/RHOshare).

In Jeremy Evnine and Andrew Rudd's article, "Option portfolio risk analysis" in the Journal of Portfolio Management (Winter 1984), they introduce a multifactor model for options. The general form of the factor model is r_c-r_f=intercept+eta.(r_s-r_f)+psi.(r_s-r_f)^2+nu.DELVAR+zeta.DELR+CONS+chi.DVRG+d_1OTM+d_2SHRT+d_3.LONG+residual,where, r_f is the risk free rate, r_c is the return on a call option, and r_s is the return on the underlying security. They further expand (r_s-r_f)^2=B_s^2MKTSQR+RES2. I will not explain explicitly what each factor represents because I am only interested in the (r_s-r_f)^2 term. I will give a brief description of some of the coefficients in the model as this is more relevant to my question. The eta coefficient is the option elasticity, which is equal to (dC/dS).(S/C)=Delta.(S/C). The nu and zeta are also elasticities but with respect to volatility and interest rates, respectively. They say that "psi is related to option gamma, just as eta is related to option delta." Eta is interpreted as the percentage change in the option price that is caused by a percentage change in the underlying. This is elasticity in economics. This can be seen by Eta=(dC/C)/(dS/S)=dC/dS.(S/C)=dlog S/dlogC. Based on their comment, I think psi is the percentage change in the option that is caused by the percentage change in the underlying squared. Thus, psi=(dC/C)/(dS/S)^2=(d^2C/dS^2).(S^2/C)=Gamma.(S^2/C). Does this sound right to everyone? Or am I doing something wrong? Thank you in advance.