Gosper's Algorithm Meets Wall Street Formulas

peter.hill.1980 writes "Wall Street's money making formulas need to be as explicit as possible for efficiency purposes. An old, existing and famous formula — binomial options pricing formula — has now been scrutinized for theoretical optimality in a forthcoming paper by Evangelos Georgiadis of MIT using Gosper's Algorithm, proving that no general explicit or closed form expression exists for pricing."

European not American option pricing

[milgram]

It seems, after reading through the paper (to the extent my non-MIT mind understood things) that this is based upon a pricing model of European options [slashdot.org]. European options can only be exercise on the expiry date, American options can be exercised any time before that date.

Re:Hurh?

[Anonymous Coward]

I'm an analyst for a large financial firm, and this is actually old news for us soul-sellers. There is no good options pricing model; they all have problems.

These articles should help clear things up:

http://en.wikipedia.org/wiki/Binomial_options_pricing_model [wikipedia.org]
http://en.wikipedia.org/wiki/Black-Scholes [wikipedia.org]
http://en.wikipedia.org/wiki/Monte_Carlo_option_model [wikipedia.org]

Re:Hurh?

[emurphy42]

To the Wikipedia-mobile, Geek Wonder!

• Option [wikipedia.org], e.g. "I pay you $100 and you agree to (sell to me / buy from me) 1,000 shares of XYZ at a locked-in price of $50 apiece whenever I decide to exercise my option" (I may decide not to exercise it at all, and I may have a time limit)
• Binomial options pricing model [wikipedia.org], a formula for how valuable an option is in practice
• Closed-form expression [wikipedia.org], pertaining to a method that gets values out of a formula without resorting to brute-force approximation or other such PITA methods
• Gosper's algorithm [wikipedia.org], pertaining to proving that there ain't no such method for this model

The actual Deal, If anyone cares

[cb123]

The naive CRR (Cox, Ross, Rubinstein) method for pricing options is O(n^2) where n is the number of levels in a recombinant binomial pricing lattice. That is, a lattice like a binary tree, but where you have cross links connecting nodes. The naive approach requires visiting each one of these nodes and hence O(n^2) and the error of the produced option goes down only proportional to the node spacing. For at least 15 years this problem has been converted to "linear time" (really the important relation is between the price error and the CPU time) by means of a variety of extrapolation methods (this began with Richardson extrapolation) using evaluation with two trees to get a much smaller error. There are in fact numerical methods that for special options can do slightly better than this. Broadie 1996 is one reference. While pretty fast and very easy to understand, there are yet faster methods using adaptive mesh crank-nicolson PDE solvers that do a bit better. Just a couple of years ago, Dai, et al. published a paper showing how to get linear time an entirely different approach involving combinatorial sums. This may have improved performance bounds for some exotic options, but did NOT do much for improving real-world implemented algorithmic performance of pricing the European and American options that are so commonly traded on exchanges, in the US and worldwide. So, at least for the most important class of options Dai et al was kind of a snoozer. The paper referenced in the summary above is entirely a follow-up paper to Dai, et al 2008. This new paper merely shows that there is no "short cut" in evaluating the relevant sums with hypergeometric functions, a kind of special function common in mathematical physics. So, in short, all this says is that the already "non fastest method" cannot be made faster by one numerical methods approach. It is certainly deserving of publication and dissemination, but changes the world not at all.