For all I know, rainbow options(above 3 equities) have no the closed form solution. I can use MC to simulate the price, but I want a benchmark to check my answer. Does any idea you can provided? Or does any analytic approximation to rainbows price you can share? (such as,paper, articles or your opinions...) I deeply appreciated your kindness!

An approximation widely used for basket options (any number of equities) is to use a moment matching method. If, say, the basket option is a call on the average performances of the underlyings then you need to calculate the basket spot (forward) and the basket volatility. ie treat the payoff function as your new underlying then use a Black Scholes method.The spot, in this case, would be the current average performances. To get the volatility you'll need a correlation matrix and the individual volatilities. If I remember correcly the basket vol is calculated usingw'Rw where w is the 1xN matrix of each underlyings (vol*sqrt(weight)) and R is your NxN correlation matrix. The basket vol is then just the square root of this matrix multiplication

There is the article f*$%ing scary stuff, though. Firstly you have to derive the formulae (he does quite a bit of handwaving, but it probably is correct, but the formulae don't really get written down explicitly, if you see what I mean) and then you need n-variate cumulative normal distributions for n assets. For this, I think Mathematica is good, I don't know about Maple or Matlab. If you do fortran, the website of Alan Genz at Washington State Univ provides all you need. Otherwise, you need to translate his fortran to another more familiar language. (Good luck, I did n=2 (easy) and n=3 (had a headache for a couple of weeks). Can provide these if you want them.)

Thanks for Graeme's response. If I used VBA, do you suggest me giving up this idea? rainbow options(above 3 equities) is really a difficult task, especially hedging! (have a headache, too)

Well, certainly vb classes for the n-variate normal stuff. I think it can be done. As I recall, the n+1 case is reduced by genz to the n case using some clever manipulations, as low as n=3. (There are reasons why you have to stop there, can't remember.) So, a recursive evaluation of the integral, getting slower and slower for higher dimensions of course. This I might return to one day soon, having done the hard work (n=3). [I did this when I was a quant at a bank where they were going to trade rainbows TOMORROW - never did of course.] The hard work for the problem, of course, is to write down the option formulae correctly. As I have indicated, Johnson's paper is not especially 'coder friendly'. I haven't done it for n >= 3, for n=2 it is just Margrabe and Stulz of course.

I have a similar question on rainbow options. Ideally I would like to price a basket option on three assets with the final payoff being x1*ret1+x2*ret2+x3*ret3, where x1 is the ex-post best performing asset and x3 the worst. To the best of my knowledge there isn't any closed formula to price these kind of options (or a reasonable approx to use as a benchmark to compare with Monte Carlo results). I tried reading the Stultz paper but there he only deals with best of/worst of kind of options. At the moment the basket contains 3 assets but at this stage finding something for only 2 assets would be a great start anyway.Thanks,Andrea