Scott:Nice to have you back, and thanks for your kind words. I know you went fishing... did you catch anything big in the process?What I have in mind in this thread is nothing like cementing QFT and GR or some kind of "theory of everything" in finance rather the premises of this thread is:(1) A theory that links the valuation of basic securities (e.g. bond, stock) to the valuation of the underlying on them. My sense is that type of problem requires one to use infinite dimensional ideas to provide a rich and consistent content, which would lead us back, hopefully, to simpler and “implementable” results in finite dimension – a promising miracle of modern finance. (2) The basis for such a theory has to proceed from a generalized asset valuation equation (GAVE), and such a fundamental equation has to be supported by duality argument (I believe), which basically allows one to draw a concrete connection between the primal and dual space. The trick to the problem, it seems to me, is how to preclude arbitrage from the primal space that could be supported in equilibrium. How does one preclude arbitrage in the primal space, and still appeal to say, stochastic volatility models? How does the role of information ties into this? Is the space of “information continuum” possible in arbitrage equilibrium? Indeed, what we have here is Alice Restaurant problem: “One fundamental question and too many answers.” My thought was to recollect some known and “not too known” results about the notion of equilibrium. It is instructive to not that theories in finance are generally supported by partial or general equilibrium arguments, and for example, the “no arbitrage condition” prevalent in financial models has a rich content anchored by the claim that in finite dimensional space, the monotonicity of preferences implies uniform properness which taken together with the joint continuity of the evaluation map leads naturally to valuation operator with nice viability properties. This is a natural embedding problem into the partial equilibrium framework.By the way, I look through one of your working papers – “TRACKING BLACK HOLES IN NUMERICAL RELATIVITY” that is some interesting stuff!Cheers

Our foray into to the land that many have considered impossible or a “type of mission to Karla” has taken us afar into two foundational topics, namely, compensated competitive equilibrium, and valuation equilibrium. The nice things from this detour are:(1) We now know that robust result in economic science is always cleverly anchored on the notion of economic equilibrium. And the reason for that is the fundamental insights provided by this theory -- that use prices in defining equilibria and in resolving the social conflict of interest -- are elegantly encapsulated in the first and second theorems of welfare economics. The first theorem states that, any competitive economic equilibrium is Pareto efficient, and the second theorem say, in a finite-dimensional convex economy, any Pareto efficient allocation can be supported as compensated competitive equilibrium.(2) Valuation equilibrium allows one to directly characterize price as a continuous linear functional in a norm topology on L (L is a linear space whose elements are real numbers). The implication of this is that we can extend this result further to account for the interplay of time and uncertainty in a way that lend credence to the notion of price per unit of probability for k commodity. Applying coherence principle in subjective probability, we can interpret 1_A as the price of a contingent claim that pays a unit of account (say a dollar) if event A occurs and zero unit of account otherwise… which ties in nicely with Arrow-Debreu security. The confluence of naysayers notwithstanding, our next stop is a précis of arbitrage equilibrium.

I'm back...We turn to the notion of arbitrage equilibrium, and as before we take the axiomatic route. And the focus of the argument we will be following here is that arbitrage opportunities cannot be consistent with economic equilibrium where arbitrage opportunities are situations where economic agents can earn "simple free lunches" (that is opportunity to get something for nothing). A market valuation system admits no arbitrage opportunity if no agent has an arbitrage opportunity. In this view, arbitrage equilibrium is an economic equilibrium supported by the absence of arbitrage prices. The thrust of arbitrage argument in financial modeling is the nontriviality of pricing operator that assigns a unique additive measure to each contingent claims in a complete market.The key to the development of arbitrage equilibrium is to endow the contingent claims space with a vector lattice operation, which allows us to formulate the notion of monotone preferences and positive prices. In finite dimensional space, the monotonicity of preferences implies uniform properness which taken together with the joint continuity of the evaluation map leads naturally to valuation operator with nice viability properties. For a quick digression on the notion of viability, we recall from Harrison and Kreps (1979) that a price system (M* n) is said to be viable if there exist a continuous, strictly monotone, convex preference relation ~= on IR x L^2(Omega, F_T, P) such that for (alpha, g) in IR x M*, alpha + n(g) < = 0 implies (alpha, g) ~= (0, 0).We note that uniform proper preference expresses the economic intuition that any loss along direction determined by a vector of uniform properness cannot be recovered by a small bundle. See Aliprantis, Brown and Burkinshaw (1989) on the idea of uniform properness.To be continued...

Kreps (1981) working with infinite dimensional commodity space has shown that "arbitrage depends on the degree of the continuity of agents' preferences," and that arbitrage restriction on security market model is necessary condition for economic equilibrium. Clark (1992) following axiomatic approach, and modeling the contingent claims space as a Hausdorff, locally convex topological vector lattice, provided necessary and sufficient conditions for continuous, strictly positive linear functional to characterize arbitrage equilibrium.We now present some basic conditions, which describe arbitrage opportunities. Consider a linear security market, denoted by X* (a real linear space, locally convex, and endow with Hausdroff topology T*). Furthermore, we endow X* with topological vector lattice operation. Simply, a vector lattice operation enables us to formulate the notion of monotonic preference and positive prices. A topological vector lattice is a topological vector space equipped with a (uniformly) continuous lattice operation v = X x X ---> X. The lattice operation v creates a natural order >= on X defined by the condition x_1 >= x_2 if and only if (x_1 - x_2) v 0 = (x_1 - x_2). The positive cone is given by X+ = {x in X : x >= 0}. A strictly postive cone is given by X++ = {x in X : x > 0}. The vector order >= is relexive, transitive, and antisymetric.

Let M be a subspace of X*, think of x in X* as financial claims, and take {m_i} and i in I as the set of financial claims that can be brought and sold. The total marketed claims available for trading are denoted by M; which is a linear span of {m_i} and i in I. Notice that m in M can be thought of as a portfolio of long and short available for trade in the security market. In this connection, we write m = Sum_ i in I lambda_i m_i, where lambda_i is the shares of each financial claim held. We constraint lambda_i of m_i to be non-zero in some finite security markets. By definition of linear span, it easily seen thatM ={ x in X* : x = Sum_ i in I lambda_i m_i, for { lambda_i}_i in I}.

Let p_i be the market value assign to m_i for every i in I. A price correspondence p : M ---> IR (Real) is defined asp(m) = { Sum_ i in I lambda_i p_i : m = Sum_ i in I lambda_i m_i, for { lambda_i}_i in I },for every m in M. The idea here is that p(m) represents all the possible prices for which m in M can be brought by using a portfolio strategy lambda_i. The feasible set of trades is a collection G* = {There is some q in p(m), such that q < = 0}.

It has been shown that no arbitrage condition implies that there does not exist a feasible claim in the strictly positive cone of X* (i.e., X_++ = {x in X : x > 0}. This condition can be formulated as follows; for any x in M, let p(x) > 0 denote the condition that q > 0 for every q in p(x). Then, the condition of "no simple free lunches" can be stated axiomatically as follows:{For every x in M, x ~ 0 implies that p(x) > 0},where ~ is strict a preference ordering. To derive the no arbitrage condition from the axiomatic postulate, it suffices to show that we can find a financial claim m_0 such that m_0 ~ 0, and p is a linear functional. The claim m_0 is called the riskless security, and prototypical proxy for such a security is a default-tree zero-coupon bond. Applying this setup arbitrage equilibrium is a strictly positive linear functional that maps the total available claims to a real line; that is, p : M ---> IR. Under some mild technical hypothesis, the continuity of a price system for a choice of relative topology can be shown such that approximate arbitrage opportunities are eliminated. This argument is beyond the scope of this thread, we only record the result as follows: p* : M* ---> IR is called continuous arbitrage equilibrium, whenever p* is continuous, strictly positive linear functional.

HiTo NI´m sure you are very smart and know a lot of things about differential geometry.I´m sure too that you can use theories with 11 dimensions to understand the big bang and particles dinamycs.But the financial prices... I don´t think so.A men sell stocks because an earthquake destroy all he have.. or because he likes a new ferrari..... (so he need cash) or because George Sorus sell that stocks.... So if you really don´t think that the social behavior is the most importan fact.... you have problems.I really believe in mathematics, but it´s the way to reduce uncertainty because is an aproximation. But if you think that you can find how to describe with manifolds why the people doesn´t believe in Internet stocks like they do many years ago....... do not price my options.

Hi ScottLike in the case of insurance company, the math models reduce uncertanty through the use of probabilities and expectation.But remember.... If you create today an life insurance company, there is a probability that all your clients die tomorrow so your company will be broke.So... the models reduce uncertanty, but you can´t believe that the math models gave you an exact behavior model, like M believe with his manifolds.

So... the models reduce uncertanty, but you can´t believe that the math models gave you an exact behavior model, like M believe with his manifolds.javgome--I agree, M is off the wall with his manifold sheah. How could there possibly be a Deterministic Walk Down Wall Street?

Quotewhy don't insurance companies blow up?don't they? how 'bout all those loudly announced shorts of MBI?who can name the notorious UK insurance operation that was trying to renegotiate their gteed contracts... can't remember it off the top of my head, but surely someone on this board can...underfunded pension trusts... isn't this similar? if the company goes bankrupt with substantial pension liabs, and their pension setup gets put to PBGC, isn't that really a blow-up in disguise?finally we're getting all this press about SEC taking on this and that, Spitzer, etc. ... did you ever see anybody who was part of the insurance regulatory, complaining about how things were run?um, the US Social Security system... isn't that a little like the FNM/Freddie story, just a little anyway?Royal & Sun mergerfest... don't weak banks do this kind of thing? Lloyd's?did you follow the story about GE Employer's Ins? My sources tell me that their HY portfolio got so moldy that they had to take some drastic management actions...basically I think that insurance accounting is probably the worst of the bunch, worse than Enron in terms of allowing the numbers to behave in a non-martingale-type way... and for better or worse, they don't generally attract the aggressive wall st. types who know how to "find" money. Given the demographic drift, one can guess why things haven't done so badly up to now, but in the future... I wouldn't touch the stuff - at least not the bigger players.

QuoteOriginally posted by: krQuotewhy don't insurance companies blow up?who can name the notorious UK insurance operation that was trying to renegotiate their gteed contracts... can't remember it off the top of my head, but surely someone on this board can...Prudential.

no, not that one... I had a look at the bonds back in late '00, '01it's a name that's not really known in the US

Equitable Life?