Hi, For options on more than 2 assets, close-form solutions are usually given in the form of multi-dimensional cummulative normal distribution function. Most softwares like Excel use approximations (typically a polynomial combined with exponential functions) for the 1-D cummulative normal distribution and approximations for the usual 1-D normal distribution are very accurate (compared w.r.t. MC). Some old books do give some approximations of cummulative bi-normal, but they don't seem to be accurate particularly at stress cases. I once remember a university prof mentioned in class about a paper on function approximation of a 5-dim cummulative normal distribution and he had tested it to find it bad. I want to know about the latest (state of the art) on approximation functions for cummulative bi-normal (i.e. 2 dim), cummulative tri-normal (i.e. 3 dim) and/or higher orders - ofcourse, i.e. apart from integration of the taylor expansion to a fixed number of terms. Can anyone direct me to some relevant papers. Thanks.

no one? is it that my question is not clear ... or is there genuinely no one ...

http://www.isye.gatech.edu/apps/events/ ... ?id=295The speaker in this talk may have some papers. Missed the talk, though.

Hull gives the Drezner approximation for bivariate cumulative normal distribution, which is okay to a point.Alan Genz has done some better stuff (in Fortran but it's translatable to C with effort). The URL for his homepage is below. Take a look at the function BVND in his TVPACK software.http://www.sci.wsu.edu/math/faculty/genz/homepage

If you don't mind translating FORTRAN into another language that you might actually use, you can definitely find papers including code in the ACM Transactions on Mathematical Software (TOMS). Start from http://www.acm.org/.Good luck.

Alan Genz's routine workes fine. However, if you want more than 100 dimensions you need to do a part (or even all) of the creative programming yourself.How large is your problem?Regards,R.

Hi rector, DavidJN, tonged, KOthanks a lot. my problem is small (2D and 3D), but needed very good accuracy. 100 D is really large, are there any places where 100 D derivatives being offered or used ?

Last edited by pb273 on December 3rd, 2002, 11:00 pm, edited 1 time in total.

You can try:R. Divgi, Annals of Statistics 1979 Vol 7 No 4 903-910for bivariate case

If you are dealing with a low dimension such as the trivariate normal distribution, the task is simple. A good scientific software must have a reliable built-in three-dimensional quadrature routine. For example, Maple 8 has a quick and accurate routine called Tripleint (I am NOT affiliated with Maple in any way !). All you have to do is enter the trivariate normal density function and use it as an input to the Tripleint routine to get the cumulative function.As far as paper references are concerned, you could check out the classical paper by R.Geske and H.E.Johnson ("The American Put Option Valued Analytically", The Journal of Finance, Vol. XXXIX, n°5, December 1984) : they find clever tricks to reduce the dimensionality of the problem (p.1520). From an applied mathematician standpoint, the best overview is probably the one done by A.Genz, already mentioned in this thread.In an option pricing context, the problem is also tackled by T.Guillaume in "Analytical Valuation of Options on Joint Minima and Maxima" (2001), Applied Mathematical Finance, 8, 209-233, as well as in a forthcoming paper in Review of Derivatives Research.

is there a chance that calculations(of CDF) by genz's routine might give different outputs for sameinputs(same cutoff and correlations), As there is a random number array involved in calculations.

- Martinghoul
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I have implemented Drezner's popular method for the bi-variate bugger in Excel... Works fine for me, i.e. it's accurate enough. I think I might have a paper knocking around somewhere, if you want it...