how would you go about pricing a European vanilla call with strike K where the underlying is an arithmetic average of two stocks 1/2*(S1+S2) given that the sum of two lognormal r.v. is not lognormal? is it a safe assumption that it is approximately lognormal (if one is trying to come up with a closed-form solution rather than use MC)?

I am not sure about the closed form solution ... but in my humble opinion ... the monte carlo ( witht the cholesky decomposition )approach is better - by simulating the two stocks separately , you can find out the value of the basket at maturity and hence the option value .For a closed form solution, even if we assume that the average of two stocks follows a lognormal distribution, I am not sure what values are you going to take for the dividend yields and the variance - my best guess would be and average of the two yields and combined variance = sqrt(var1sqr + 1var2sqr + 2*sqrt(var1*var2) *correlation ) ... What are your views ?

I think you can additionally do a variance reduction thing on your monte carlo simulations. For example I think you can compute an appropriate weighted sum of two call options, each on just one of the stocks, which hopefully can be done analytically. Then when you do simulations, you tweak your realized input for the basket call. You can also create a fictitious asset created from the other two and price a call on that one analytically to get more variance reduction. This is probably done in a lot of textbooks, but I've seen it in Cont and Tankov.

Sorry ignore wat I wrote here. My brain was frazzled this morinng from the heat and I realised later it was utter tosh.

Last edited by ACD on June 30th, 2009, 10:00 pm, edited 1 time in total.

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