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James
Topic Author
Posts: 1557
Joined: January 23rd, 2002, 2:24 pm

### Alternating-sign matrix conjecture

Can nyone explain this physics/math area and its applications to finance? I am looking for omething a bit more rigoros than just Chlesky fatorization, etc. Thanks. - James

Posts: 477
Joined: September 22nd, 2002, 9:08 pm

### Alternating-sign matrix conjecture

James, given that it’s quite difficult implementing applications of Asm conjecture to financial markets, an alternating sign matrix is a matrix of 0’s, 1’s and –1’s with the non zero elements in each row and column alternating between 1 and –1 and beginning with 1. The conjecture, proven by Zeilberger and Bressoud, is that you could calculate the number of alternative sign matrices given by: An= prod [j=0, n-1] (3j+1)!/(n+j)!One possible connection with finance could be lattices. Usually CIR, Black Derman and Toy and Fabozzi research refer to classic binomial lattices. There’s a particular kind of lattices, the osculating lattices, where being {y alfa} with alfa=1…n we find y1<y2….<yn such that no three paths in sequence are equal if y (alfa)= y (alfa+1) then y (alfa- 1)< y (alfa) and y (alfa+1)<y (alfa+2), a couple-pair of discrete points y (alfa)=y(alfa+1) is called osculation. Osculating lattice paths, are substantially paths where there is an inter path interaction Some physicians demonstrated that the number of total osculating lattice paths could be explained through the alternating sing matrices conjecture (see Melbourne Math Dept. by R. Brak 2001).The basic hypothesis for application derivatives may be supposing an underlying asset price following an osculating lattice set of discrete points instead of a continuous process. Actually I don’t know if someone has studied this application. With reference to Cholesky factorization I do not really think it’s not rigorous or too little rigorous; on the other side, I do not find a useful way to apply Cholesky factorization to A.s.matrices in finance, and if you have one please suggest it to me. You use Cholesky to compute square root of the matrix. Rgds,

Omar
Posts: 1552
Joined: August 27th, 2001, 12:17 pm

### Alternating-sign matrix conjecture

QuoteCan anyone explain this physics/math area and its applications to finance? James,What makes you think ASM's have anything to do with finance? PS The ASM conjecture was proven in 1992 by Zeliberger, then there came a 2nd, simpler proof by Kuperberg a few years later. The relationship to paths is explained in Bressoud's book (Proofs and Confirmations, Cambridge, 1999). But these are very special paths, and all ASM's have the same weight. If you want something less trivial, you have to look into the the stat mech model (used by Kuperberg) to prove the conjecture. I still don't see any relevance to finance. One can think of each ASM as a generalised Brownian bridge: N paths start from the left boundary, each moves only to the right or up, and all end up at the upper boundary. Each ASM corresponds to one way that the paths make it from start to finish. Each ASM can be assigned a weight. But so what?It would be great if this sort of thing has a concrete application in finance, but I don't see it.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### Alternating-sign matrix conjecture

ASMs are (trival or end point) solutions to Nonius' 'odd linear algebra problem' when you have 1 to n companies that can default. You can see that thecomplexity gets high when there are more than a few companies that are correlated.James, into CDOs???

Johnny
Posts: 2431
Joined: October 18th, 2001, 3:26 pm

### Alternating-sign matrix conjecture

This looks like it would be useful in any problem involving path-dependency. Which means almost all problems in finance.
Last edited by Johnny on August 1st, 2003, 10:00 pm, edited 1 time in total.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### Alternating-sign matrix conjecture

This looks like it would be useful in any problem involving path-dependency. Which means almost all problems in finance. Johnny,Well not exactly. Problems involving path-dependency have nothing to do with ASM. In fact, problems with path-dependency only arise when one doesn't know any better math. However, you are correct when you say ASM is relevant to all problems in finance.N

LongTheta
Posts: 1504
Joined: August 3rd, 2003, 6:06 am

### Alternating-sign matrix conjecture

N, In what way are alternating sign matrices related to all problems in finance? Thank you,LT

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### Alternating-sign matrix conjecture

LongTheta,As I mentioned below, ASM are solutions to Nonius' odd linear algebra problem. That is, if you have the combined volatility forfailures for a set of correlated businesses, how are the individual business failure volatilities related as the correlationvaries.End point solutions for correlation matrix.one business [1]two businesses100101100-1-1 0three, there are 21 combinations of 0, 1 and -1.In what way are alternating sign matrices related to all problems in finance? All problems in finance involve constraints, correlation and volatility.N

LongTheta
Posts: 1504
Joined: August 3rd, 2003, 6:06 am

### Alternating-sign matrix conjecture

0 -1-1 0is not an alternating sign matrix. Each row and each column must start and end with +1. Read the conditions, as stated correctly by MrBadGuy earlier in the thread.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### Alternating-sign matrix conjecture

is not an alternating sign matrix. Each row and each column must start and end with +1. Read the conditions, as stated correctly by MrBadGuy earlier in the thread. Incorrect. Look at n=3, where -1 is at the start of columns. In fact, MrBadGuy is defining a subset of ASM.
Last edited by N on August 2nd, 2003, 10:00 pm, edited 1 time in total.

LongTheta
Posts: 1504
Joined: August 3rd, 2003, 6:06 am

### Alternating-sign matrix conjecture

N,You may go ahead and define anything any odd way you wish, but when people talk about these objects, they mean exactly the objects defined correctly earlier in the thread.LT

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### Alternating-sign matrix conjecture

Hey pal, There are ASMs and "Robins and Rumsey" alternating sign matrices. Your def is Robins and Rumsey.ASMs are the result of the Yang-Baxter equation and resulting braid group relations. Your definition throws out fermionic statistics.

LongTheta
Posts: 1504
Joined: August 3rd, 2003, 6:06 am

### Alternating-sign matrix conjecture

N, I'm afraid you're wasting my time. LT.

N
Posts: 2808
Joined: May 9th, 2003, 8:26 pm

### Alternating-sign matrix conjecture

LT,You wouldn't happen to be one of those 'mentally challenged' dudes from ETH? The school that ensures ignorance of diff geometry.Tell me why Gaussian curvature in ASM must always be 1 (Hilbert space) for certain covariances? Any elements in ASM could have a cov surface curvature of -1, thatway, you're not forcing weid constraints on problems. Actually, the ASM just contains endpoints of the relevant exp maps. Exp(iTheta) extends from +1 to -1 whichcharacterize the 'fractional' statistics for bosons and fermions, respectively. N
Last edited by N on August 2nd, 2003, 10:00 pm, edited 1 time in total.

LongTheta
Posts: 1504
Joined: August 3rd, 2003, 6:06 am

### Alternating-sign matrix conjecture

N,You're an idiot. Others have probably told you the same already.LT.

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