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## Gosper's Algorithm Meets Wall Street Formulas124

Posted by Unknown Lamer
from the hypergeometric-overlords dept.
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."
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## Gosper's Algorithm Meets Wall Street Formulas

• #### Hurh? (Score:2)

Iz deoznt understandz dis

• #### Re: (Score:2)

Indeed, I realize that this is /., and that /. doesn't have any editors, but this is pretty ridiculous. At least link to something that has some information if you can't be arsed to create an informative summary.

• #### Re:Hurh? (Score:5, Funny)

on Wednesday March 02, 2011 @03:16PM (#35360972)

It's very simple. What part of "We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach" you didn't understand?

• #### Re: (Score:2)

It's very simple. What part of "We set a lower bound on the complexity of options pricing formulae in the lattice metric by proving that no general explicit or closed form (hypergeometric) expression for pricing vanilla European call and put options exists when employing the binomial lattice approach" you didn't understand?

The vanilla part, of course. After all, why should it matter if the options come in vanilla or chocolate flavour? :-)

• #### Re: (Score:2, Funny)

Why is this modded "informative"? It should have been modded funny.
• #### Re: (Score:2)

Why is this modded "informative"? It should have been modded funny.

For the fun of it.

• #### Re: (Score:2)

The part where a lost all my money.

Aaaand it's gone.
• #### Re:Hurh? (Score:4, Informative)

by Anonymous Coward on Wednesday March 02, 2011 @03:30PM (#35361184)

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? (Score:4, Insightful)

on Wednesday March 02, 2011 @03:47PM (#35361426) Journal
Isn't it just a bet on the variance of the underlying stock? Skimming over those models, they look like they're really all just different ways of mutating your choice of initial assumption about the distribution of possible futures.
• #### how dare you (Score:3, Interesting)

as "Devil takes the hindmost" (by Edward Chancellor) points out, many traders will be offended by your vulgar terminolgoy.

they are 'hedging', they are 'creating efficiencies', they are 'earning', they are absolutey not, in no way, gambling.

• #### Probability Theory began with games of chance. (Score:1)

It absolutely is gambling and any investor who denies this is either dishonest or not very well informed. I infer from your sarcastic quote marks that I'll enjoy Edward Chancellor's book -- sometime after I enjoy Taleb's, and Reich's, and ...
• #### Re: (Score:1)

The refined financial term is "speculation". It only becomes "gambling" in the business sense when you crater something.

While I sense a great deal of sarcasm in this thread, it is worth noting that speculation actually can result in positive return over long periods of time. Some people are very good trader/gamblers and their talents shouldn't be denied. But the problem comes when you have high leverage amplifying all the risks of the market and the business.

High leverage where one borrows staggering
• #### Re: (Score:1)

While I sense a great deal of sarcasm in this thread, it is worth noting that speculation actually can result in positive return over long periods of time. Some people are very good trader/gamblers and their talents shouldn't be denied. But the problem comes when you have high leverage amplifying all the risks of the market and the business.

Totally agree. I almost said the same thing, but just wanted to be sure not to say too much. I tend to go on a bit, you see. :-)

But since somebody is interested, I'll say that a very important distinction is knowing the odds and playing them smartly, versus guessing the odds skillfully/luckily. Over time, luck always runs out eventually. So says the Strong Law of Large Numbers.

High leverage where one borrows staggering amounts of money (something like 50 borrowed dollars to 1 dollar of assets was common in the recent real estate crash) will guarantee a crash even for very stable investments.

In other words, a "black swan" is sure to come by, eventually. So only fools construct systems in which one "black swan" will

• #### Re: (Score:2)

I'll put it this way... Wall Street does more gambling than Las Vegas Blvd: at least the casinos know the odds, and don't bet unless they are in their favor.
• #### Re: (Score:2)

There's a small but significant difference.

Gambling= when stuff goes really bad you lose all your money and then some.

"Investment banking" = when stuff goes really bad you lose everyone else's money but keep your bonuses, fees and salaries. Gamble with other people's money, get paid really big bucks when it goes well, get paid big bucks when it goes poof. Privatise the profits, socialize the losses.

So clearly investment banking is a smarter choice for intelligent sociopaths. And they can even use lots of fa
• #### Re: (Score:1)

"they are 'hedging', they are 'creating efficiencies', they are 'earning', they are absolute not, in no way, gambling."

Ok what are they producing of value then?

• #### Re: (Score:2)

An USA landowner grows a particular type of wheat. He sells that wheat in USD to pay his workers in USD and make a profit.

A German biscuit maker makes a biscuit that he sells in EUR to pay his workers in EUR. So far everything is simple

The German biscuit maker needs to buy the particular wheat he uses from the USA landowner. But his incoming cash is in EUR while (some of) his expenses are now in USD.

The biscuit maker is now exposed to the USD/EUR exchange rate and the price of wheat. He wants to expand - if

• #### Re: (Score:2)

Hedging - and, in a sense, a lot of the issues around option pricing - are the opposite of gambling. They're more like running a bookmaker's. The idea is to limit your risk exposure whilst making a nice profit from the actual gamblers.

• #### Re: (Score:2)

While this is not my area of expertise I actually know of an example. Several years ago when jet fuel prices skyrocketed and all the airlines started charging fees on everything to stay afloat Southwest Airlines didn't even raise their fares because they had hedged [nytimes.com] long term fuel oil at ~\$50 barrel. At the time they set up the hedge the median price was much lower so it meant they were paying more than any of the other airlines for their fuel but it also meant that when the price shot up to >\$90 barrel t

• #### Re: (Score:1)

But was there a mathematical proof? Or just 'Street common wisdom & anecdote, that no sure-fire formula exists?

The summary: "... proving that no general explicit or closed form expression exists for pricing."

There is something new here. I get your point that Wall Street will not change its ways because you and every other trader have already been assuming what this paper proves, and I don't doubt what you said one bit, but for the rest of us this is significant because formal, incontrovertible mat
• #### Re: (Score:2)

inherently irrational

I hope that's a really clever pun. Says Wolfram Mathworld on Gosper's algorithm

The algorithm treats sums whose successive terms have ratios which are rational functions.

• #### Re: (Score:1)

No pun intended. Any "funny" mods for that pun are rightfully yours.
• #### Re:Hurh? (Score:5, Informative)

on Wednesday March 02, 2011 @03:33PM (#35361228) Homepage

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
• #### Re: (Score:2)

Btw, if you are really interested in the algorithms for solving hypergeometric identities check out the ebook by Marko Petkovsek, Herbert Wilf and Doron Zeilberger:

http://www.math.upenn.edu/~wilf/AeqB.html [upenn.edu]

It is one of the best free mathematics ebooks out there...

• #### What? (Score:1)

I don't really understand this... does this mean that it is not determinable whether an option pricing formula for European puts/calls is optimal or what??
• #### Georgiadis (NOT Georgiaids) (Score:1)

by Anonymous Coward

• #### Basic Form: (Score:1)

The basic form of the algorithm (according to *AA groups) is as such: \$Max_Payable_Price times (\$Total_World_Population - \$Steenking_Pirates). *AA's obviously want to minimize the \$Steenking_Pirates, especially the ones who simply don't listen to their music in the first place.

Many lawmakers agree with this, with the agreement being proportional to the money they receive from the *AA's.

And yes, I know that people who don't listen to music shouldn't need to pay, but I dare you to tell the RIAA that. It's eve

• #### And this means? (Score:1)

And what does this have to do with option pricing? It just proves that there is no closed form. From the quick little research I did on closed forms, all this means is that you can't use limits or integrals, which are used as solutions for a slew of real world problems.
• #### MIT^3 (Score:1)

by Anonymous Coward

ahh see there Gosper was MIT so is John Cox (creator of binomial options pricing model). interesting how the flow of ideas concentrates and propagates ... MIT^cubed

• #### European not American option pricing (Score:3, Informative)

on Wednesday March 02, 2011 @03:19PM (#35361024)

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: (Score:3)

More specifically, the paper proves that there is no closed form expression for the *binomial* options pricing model on a European put or call.

There's a closed form for European options pricing, under certain assumptions, which is of course the Black Scholes formula. The paper notes this obvious fact in footnote 7.

The binomial model is generally more flexible, and allows the tweaking of assumptions (dividend payments, etc.). As a result, it's used in practice to value certain types of options (exotics, st

• #### Re: (Score:3)

"And the existence or lack thereof of a closed form expression"

I am a modelier (a fancy pants way of "I configure and run models") though not in Economics. This has *huge* implications for the practical application of models. Now, what no closed form solution means is that there may be a number of different paths that a solution can be achieved. You can converge from a number of different directions, and be "right for the wrong reasons". "Great!", you say, "if I am wrong on one of my parameters, this means

• #### Re: (Score:1)

Although I've only looked at the paper briefly I think that what it proves is that there is no closed-form expression for the partial sums in the Cox-Ross-Rubenstein model. Such a closed form expression would be neat, but ultimately one only cares about the limit, where one has, as you write, the Black-Scholes formula.

I don't work with this, so this might be entirely academic- but isn't convergence in the binomial model extremely slow for barrier options?

Also, there is a nice explicit formula for all
• #### Re: (Score:3)

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. European options can only be exercise on the expiry date, American options can be exercised any time before that date.

I'm not sure I follow. An "American option" as you call it has two dates - one is the vesting date (the first day the option may be exercised) and an expiry date (the date the option will no longer be valid). Sometimes the vesting date can b

• #### Re: (Score:2)

I'm not sure I follow. An "American option" as you call it has two dates - one is the vesting date (the first day the option may be exercised) and an expiry date (the date the option will no longer be valid).

It sounds like you are describing the options that are given out as an employment benefit. There is a different kind, traded on an exchange just like stocks are.

If a particular stock 'foo' is trading at \$60 today but you think it is going to go up in the future, you can buy a "June \$65 call" for some small amount of money. That option then allows you to buy a share of foo for \$65 any time between the purchase date and the June expiration date (American style), regardless of what the stock price happens to b

• #### Re: (Score:2)

Ah, shoot. I forgot they trade options on the opposite side of the street than we do. Drats.

• #### !algorithm (Score:1)

by Anonymous Coward

no algorithm exists for stupidity either

• #### abstract text (Score:5, Funny)

on Wednesday March 02, 2011 @03:29PM (#35361172) Homepage
This is the abstract used ( not really) to get teh funding grant for this research.

Two fundamentally different but complementary transition metal catalyzed chemo-, regio-,diastereo-, enantio-, and grantproposalo-selective approaches to the synthesis of a library of biologically significant nano- and pico-molecules will be presented with the focus on reaction mechanism and egocentric effects. The role of the nature of the metal, ligand, solvent, temperature, time, microwave, nanowave, picowave, ultrasound, hypersound, moon phase, and weather in this catalytic, sustainable, cost-effective, and eco-friendly technology will be discussed in detail.
• #### proof (Score:2)

As if we needed fancy mathematics to tell us that the formulas used by Wall Street traders don't work. Let me offer you Exhibit A [rollingstone.com].
• #### Re: (Score:1)

Judging by the number of millionaires who work on Wall Street I'd say their formulas work pretty damn well
• #### Re: (Score:2)

You make a good point -- perhaps they are Working As Intended, but not economically optimally.
• #### Re: (Score:2)

I would say that the Quant methods worked very damn well. (The book "Yhe Quants" is fascinating BTW - also "Too Big to Fail" - books on CD = excellent driving amusement). In fact their effect can be seen very well as a classic 'technology bubble'. So also the 'Mortgage Bubble' where a combination of new technology (capitalization of home loans), combined with some regulatory mistakes and a large dose of people-taking-advantage on all sides.

• #### "Bribe Congress for bailouts" is not a "formula." (Score:1)

It's criminal conspiracy.
• #### Re: (Score:1)

It is when you pull it off as masterfully as Goldman Sachs did
• #### Re: (Score:2)

Let me offer you Exhibit A.

Yes, because Rolling Stone is the perfect place to look to understand the goings on of the world economy. Matt Taibbi is an idiot as is anyone who reads what he says and doesn't laugh.

• #### An attempt at an explanation (Score:2)

When pricing options the bionomial way, one creates a sort of decision tree for movements the underlying value makes. (scroll down on http://software.intel.com/en-us/articles/high-performance-computing-with-binomial-option-pricing-part-1/ [intel.com] to see such a tree).

This paper seems to prove that there is no easy formula short cut for the tree: if one wants to know the answer, one really needs to build the entire tree.

• #### Re: (Score:1)

To be more exact, there is no easy formula of the hypergeometric kind, which is a formula "involving binomial coefficients, factorials, rational functions, and power functions" according to http://mathworld.wolfram.com/HypergeometricIdentity.html [wolfram.com]. It would thus theoretically still be possible that an easy formula exists, but it must involve constructions more exotic than that.

• #### Neither does most of physics (Score:2)

OK, so there is no exact solution to the formula. Do you need one? Or will a Monte Carlo simulation be good enough, the way it is for (say) the physicists building nuclear bombs or the engineers designing airplanes?

Closed-form solutions are nice for proving things with arbitrary precision, but they're often not necessary in the real world, where a few decimal places often suffice.

• #### Re: (Score:2)

Not to imply that the work isn't interesting. I'm sure it's got all sorts of implications with respect to the way economists analyze the algorithms. But commenters so far seem to want to jump from "no closed form exists" to "Wall Street is fundamentally unsound", which seems, uh, unsound.

• #### EConned by Yves Smith. (Score:1)

there is plenty of evidence that a large amount of the theory behind modern academic economics is based on payola and a religious belief, not on empiricism or on any sort of scientific rigor

• #### Re: (Score:2)

Sure. But the lack of a closed form for this formula doesn't go to demonstrate that.

• #### Re: (Score:2)

More like, it's a complex subject distorted by the fact that many of the participants can make large amounts of money. And Yves is something of a rabble-rouser - her analyses are just as bad as the ones she goes to town on.
• #### Re: (Score:2)

A good model can take into account corruption.

• #### Re: (Score:2)

Yes. A better answer would be, "economies are complex adaptive systems" (CAS is a term that essentially means 'living systems') - like neural networks, ecosystems, any biological system, etc.

• #### An Old and Famous (and not very good method) (Score:1)

The binomial model is common in textbooks because it's intuitively appealing, but if you only apply it to basic European (exercisable at expiry) options then there really are better ways of getting a closed form solution i.e. the Black-Scholes (or Bachelier-Thorp ....) formula. If you want half decent pricing methods for more general cases then you'll end up with Finite difference or Monte-Carlo methods depending on dimensionality, at which point you've already given up on a closed form solution. One of t
• #### The actual Deal, If anyone cares (Score:4, Informative)

on Wednesday March 02, 2011 @05:25PM (#35362636)
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.

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