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Just discovered this website while trying to find some info on forward volatility.Not wishing to sound glib but there are certain basic problems here.On the fundamental level.I thought some bright Austrian bloke called Kurt proved that all this theory unification thing was a bunch of bunk.Is it not the completeness/consistency problem?Complete formal systems are theoretically of infinite scope and of no real use .Consistent formal systems are of limited application but lead to paradoxes, inconsistencies real quickly.A formal system cannot be both complete and consistent no matter how hard we try to make them.We just have to get on with solving problems as we see them, tidying up every so often.On the practical level.Is this not what markets are for?We can always price something - say a contingent claim - relative to something but this requires dependent factors and an assumption of behaviour or a whole set of usually quite questionable axioms.I am a firm believer that this is all back to front.Hence my statement that "prices determine processes and not vice versa."Until someone tells me how to value something as 'simple' as a Microsoft share, why should I be worried about some rinky-dink contingent claim on it.There is a fundamental problem with a formal systems approach to finance, as there is to physics and mathematics, which I would guess are more tractable to this approach.--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Anyone know where I can find out about approaches in pricing forward equity options, and strongly path-dep payouts.Assuming that there is significant skew and term structure effects in the pricing volatility of vanilla european options.I read something about a model-free approach (hence market-specified process) and was wondering about looking at extending this to forward-starting things.thx--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Martingales unify finance in the sense that we say "Look, all financial data can be modelled in terms of stochastic processes, and assuming market efficiency, these processes are martingales or semi-martingales (I'm trying to allow for jumps), and on sufficiently large scales all we need to consider are Gaussian/Poisson processes and their children.". Risk premia are there (I think, I think) because certain instruments are not replicable, so they are not fully hedgeable, and "investor preferences" need to be parametrized. It's just as in physics: One can speak about a unified theory of stong and electroweak interactions, but there are about 23 parameters that cannot be computed and need to be measured.If you say "No, I want to be able to compute everything", then that's a different story.I'm playing the Devil's advocate here a bit, but I'm also half-convinced that there is nothing really deep in finance that can be found using mathematics. What remains, most probably, requires a better understanding of the behavioural sciences.

LongThetaOn the other hand (playing devil's advocate to the devil's advocate) prior to 1973 you might have had the view that pricing options was not possible (so I haven't given up hope!) without the insight about volatility.What I see as the objective of a unified theory of finance is not getting true prices (whatever that means). Perhaps instead if we are able to bring the disparate models for pricing the various assets and derivatives into the one framework, we may be able to reduce some the of the inefficiencies that exist in how assets are currently priced. That is, by understanding how different assets classes relate to each other, we may be able to incorporate more information into pricing models. This might move us towards the next generation of pricing models.

Quote What I see as the objective of a unified theory of finance is not getting true prices (whatever that means). Perhaps instead if we are able to bring the disparate models for pricing the various assets and derivatives into the one framework, we may be able to reduce some the of the inefficiencies that exist in how assets are currently priced.But there is only one model: Asset prices are (semi-)martingales. Discrepancies in pricing, e.g. caps and swaptions, are really practical issues: If you try to price both correctly, you end up with PDE's that you cannot solve exactly. It's not like people don't understand what's going on.

A short review of "Asset Pricing" is here:http://papers.ssrn.com/sol3/papers.cfm? ... =447620The book certainly tries to address unified pricing (through a single pricing equation using a stochastic discount factor). Seems very relevant to the topic at hand. Pozabee??

The martingale approach is more than a mathematical apparatus. It starts with a very well-defined, and refutable, hypothesis: Asset prices are (semi-)martingales. Where does that come from? Well, it's the CLT all over again. If you accept that, then the mathematical apparatus follows. Which is wonderful. Imagine we had a hypothesis, but didn't have the mathematics to compute using it. There are examples of that in physics: String theory is a wonderful hypothesis/framework, but we have no way, so far, to compute anything using it. Now martingales may not be the most realistic working hypothesis, and there may be deeper, finer ones, but so far, it's the best "working, unifying" hypothesis/theory that we've got. Once you say "Martingales", all good things follow, people can actually compute numbers. It's like in physics, when you say "affine Lie algebras". Wow -- good things flow.PS I must admit I'm thinking about this as I write about it. I don't hav any firm ideas on the matter. I'm not one of those

What I don't like about your notions about martingales, LT, is that with it I don't think that you can say that any single price is wrong. At most, you can say that one or more prices are inconsistent, but that's it. And I mean this absolutely -- I can tell you that something is worth a penny or that it is worth a million dollars, and (other than for strictly limited cash flow processes -- if there is even the remotest possibility that it will be worth more than a million dollars in the future, you cannot rule out the possibility that its proper price today is a million dollars) all you can do is ask to see some other prices. And I don't think it's generally asking very much of a pricing model to insist that it must be able to say that either a penny or a million dollars is not a reasonable price for something.

Marsden,Okay, I see what you are saying. Let me think. We could be talking about two different things.

I just got back from traveling...and I would respond to everyone by Wednesday.

Marsden:My intention is not to obfuscate the subject matter by introducing complex or foreign notions. The notion you highlighted in your “post;” Infinite dimensional,State price density,Linear pricing rule,Duality argument – primal and dual space,Spanning,Radnerian equilibrium,Arrow’s Impossibility Theorem,are well known and used in the first principle of financial economics. The context they were used seems to have given sufficient clues as to their meanings. And I identify below those contexts, with Radernian equilibrium and Arrow’s Impossibility theorem used in response to Kr, and Johnny queries respectively.Here is additional note to help clarify these terms: (1) Infinite dimensional – what I meant is that a robust unified theory of finance would need to be treated in the context of infinite dimensional space. The result achieved in such analysis would hopefully be simple enough and “implementable” in finite dimension space. The role of linear analysis is security market problems is fundamental in the sense that arbitrage pricing theory is based on the notion that marketed claims and prices can be combined in a linear fashion to produce a portfolio and their prices. These prices (by arbitrage argument) define a linear functional on the spaces of marketed portfolios. I would recall that the fundamental theorem of asset pricing, namely, absence of arbitrage, the existence of linear pricing rule, the existence of state price density, the existence of a risk neutral probability measure, and existence of an optimal demand for some rational economic agent, has been shown to obtain in finite-dimensional space. By finite dimensional space, I mean a vector space that has finite Hamel basis. The number of elements in a Hamel basis of finite-dimensional vector space is unique, and it defines the dimension of the vector space.(2) Linear Pricing Rule – one of the fundamental theorems of asset pricing (FTAP) says that there is no arbitrage if only and only if there exists a consistent linear pricing rule (think of it as linear pricing functional). Elementary proof of this result requires Farkas’ Lemma of Alternatives from duality problem in linear programming (or Farkas-Minkowski Lemma). To read about Farkas-Minkowski Lemma, I would suggest you read David Gale “The Theory of Linear Economic Model” The University of Chicago Press, 1960; and Richard B. Holmes “Geometric Functional Analysis and its Applications” Springer Verlag, 1975.(3) State price density - one of the fundamental theorems of asset pricing (FTAP), which says that price is the expectation of the quantity times the state price density, that is, the state price per unit of probability. I recall this from one of my post in this thread:++++++We quickly extend the notion of valuation equilibrium to economy where there is interplay of time and uncertainty. To do so, we introduce a normed linear space L* define on probability space (Omega, F, P), t (t in [0, 1]), z_t ^k(w) in L*, for every t, w in Omega and k = 0, 1 … In particular, we writeL* = {[ z_t ^1(w),…, z_t ^k(w)]\ z_t ^k(w) is real valued, F_t measurable for k, and the norm of z_t is less than infinity},where, z_t ^k(w) may be thought of as the quantity of a commodity (or a payoff of contingent claim) k in time t if the state w occurs. The nature of the problem here is to find some F_t measurable function q* for every k and for every t such that p(z_t) = Sum_k =1 ^K Integral_Omega q*_t ^k(w) z_t ^k(w)Pd(w) = Sum_k =1 ^K E[q*_t ^k(w) z_t ^k].For elementary event say A generated by partition h_t of Omega, we haveIntegral_A q*_t ^k(w) Pd(w) = E[q*_t ^k 1_A],and q*_t ^k is called the price per unit of probability for k commodity. Applying coherence principle (see for example De Finetti (1970, or 1974 in English) 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. Interesting to note that the development here foreshadows the integral representation of prices, which is the heart of arbitrage theory.++++++(4) Duality argument (primal and dual space) - What I have in mind here is that one can consider a partial equilibrium economy as the primal space, and if one prohibit any linear combination of assts from producing arbitrage profit, Farkas-Minkowski Lemma asserts that there must exist a function f* representing a solution to a dual problem which turn out to be formulated in the valuation space. Consult the following books by David G. Luenberger: (1) “Optimization by Vector Space Methods,” Wiley, 1969, and (2) “Linear and Nonlinear Programming,” Addison Wesley, 1984 for more on duality. I highly recommend #1.(5) Spanning – Let S be the possible states of the economy, one which will be revealed as true. Before the true state is revealed, n securities are traded. Security number j is a vector d_j in R^S, which represent a claim to d_(sj) unit of account in state s in S.Spanning is defined as span({d_j : 1 < = j< = n}) = R^S.Arrow-Debreu security payoff a unit of account (say $1) if a particular sate of nature (the economy) occurs and otherwise zero. The payoff structure of Arrow-Debreu security has the following representation(0,…, 1,…,0),where 1 occurs in state s. Arrow-Debreu security is sometimes called primitive securities in the sense that it forms the building blocks in the construction of other securities.Recall that a set of vectors {x_1,…, x_n} is said to “spans” the vector space X if each vector in X is a linear combination of the vectors {x_1,…, x_n}. A finite-dimensional space is one “spanned” by a finite set of vectors {x_1,…, x_n}. The set of vectors {x_1,…, x_n} is called “basis” if the vector can be uniquely be expressed as a linear combination of x_1,…, x_n. Alternatively we say that x_1,…, x_n “spans” the space and x_1,…, x_n are linear independent.Consequently, the link between Arrow-Debreu securities and basis set is obvious one. Just as basis is the foundation for linear space, Arrow-Debreu securities form the most basic securities for which different pattern of payoff structures can be constructed. I would recommend D. Duffie’s “ Dynamic Asset Pricing Theory” as place to learn the role of spanning in the theory of financial markets. Also, if you can get your hand on Jonathan E. Ingersoll, Jr. working paper titled “Spanning in Financial Markets” that would be helpful.(6) Radnerian (or Radner) equilibrium - is an equilibrium of plans, prices and prices expectation, that is, “ each agent has model of what prices will be contingent on date-event pairs, and has undertaken a plan that maximizes his preferences given that model, the models all agree, and plans are consistent with each other at every date and every state. Papers to read on this subject are as follows: (1) Roy Radner (1972) “Existence of Equilibrium of Plans, Prices, and Price Expectations in Sequence of Markets,” Econometrica, Vol. 40, 289 – 303; (2) Oliver Hart (1975) “On the Optimality of Equilibrium When the Market Structure is Incomplete,” Journal of Economic Theory, Vol. 11, 418 – 443; (3) David Kreps (1987) “Three Essay on Capital Markets” GSB, Stanford University (and I believe the paper later appeared in El Revista); and (4) Darrell Duffie and Chi-Fu Huang (1985) “Implementing Arrow-Debreu Equilibria By Continuous Trading of Few Long-Lived Securities, Econometrica, Vol. 53, 1337 – 1356. (7) Arrow’s Impossibility Theorem (AIT) - In the context I used AIT; I was only highlighting the problem one accounts in aggregation of preference functions. In general, one encounters AIT in the theory of social choice or welfare economics. The general impossibility theorem has the following interpretation: “Suppose there is a social choice procedure, capable of making choices from any finite number of alternatives, which uses only ordinal information on individual preferences and satisfies the conditions of independence of irrelevant alternatives, positive response, non-imposition, and non-dictatorship. Then there will be some set of individual preferences such that the resulting social preference relation is not an ordering.” For more on the topics see Kenneth J. Arrow (1983) “ Social Choice and Justice – Collected Papers of Kenneth J. Arrow, vol. 1, The Belknap Press of Harvard University Press, and Amartya K. Sen (1970) “Collective Choice and Social Welfare” Holden-Day, Inc.***Note: Context in the thread where these terms where used is reproduced below:_________________________It appears to me that there is an overarching need for unified theory that links the valuation of primitive 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. The fundamental theorem of asset pricing, namely, absence of arbitrage, the existence of linear pricing rule, the existence of state price density, the existence of a risk neutral probability measure, and existence of an optimal demand for some rational economic agent, would have to be reworked or dropped for a unified theory to be had. Why this fundamental theorem provides the basis for valuation of contingent claims, however, it appears to be stifling progress in the development of unified theory.The basis for unified 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 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.” Perhaps, that is why people make a “ton of dough” in this business. Question: Is unified theory of finance possible?_________________________________________It is instructive to note that financial engineering (if you prefer quantitative finance or mathematical finance), in theory and practice captures the spanning role of security market as first convinced by Arrow (1953). In a precursor work to this paper Arrow had incorporated uncertainty into the general equilibrium framework, and to do that Arrow admitted that he was led “by the Wald-Savage viewpoint to consider an elementary decision as one that took value for one state of nature and zero elsewhere, thus, all general decisions could be regard as a bundle of elementary decisions.” As a side note to Arrow’s line of reasoning, I would ask that you look at contingent claims as proceeding hierarchically from a binary or digital option – more on this later. Arrow’s argument, I would call “reduction principle.”Another principle that gave birth to financial engineering is the invariance principle of Modigliani and Miller (1958) (hereafter M&M), which in essence presage the arbitrage argument that is fundamental to the pricing of contingent claim.So, with Arrow’s reduction principle, and M&M invariance principle we have the foundation to B-S-M, which provided strong incentive for academics and market professionals to embrace spanning, and to apply it in pricing of securities. Indeed, large part of quantitative finance is devoted to valuing derivative securities in a mechanical way. Which leads me to remark that, essentially, quantitative finance, financial engineering or mathematical finance is concerned with (1) the creation of larger sets of derivative payoffs by combining a given set of primitive securities, and (2) financial innovation, which involves the process of introducing new set of securities in the market.______________________________________Kr:Excellent points that institutional arrangement in the market need to be factored, somehow, in any form of unified theory. I would suggest that you look into the field of finance called Market Microstructure, and instructively, I would note that much of the works in that area derive their foundation from the Robert Lucas or Roy Radner school of rational expectation. I promised to connect the Radnerian equilibrium to Arrow-Debreu world later in this debate. I would ask that you keep in mind my remark about binary or digital option as we move on in this thread.The fundamental theorem of asset pricing, namely, absence of arbitrage, the existence of linear pricing rule, the existence of state price density, the existence of a risk neutral probability measure, and existence of an optimal demand for some rational economic agent, would have to be reworked or dropped for a unified theory to be had. Why this fundamental theorem provides the basis for valuation of contingent claims, however, it appears to be stifling progress in the development of unified theory.___________________________________Johnny :Unfortunately, you run into aggregation problem w.r.t "unified theory of how people react to risk." We probably are back to some form of Arrow’s Impossibility theorem!______________________________________

ScottCaveney:John Cochrane's Asset Pricing is an excellent book, and I would recommend it. Cochrane work always attempt to blend theory with empirical findings. When time permits, I would discuss some of the issues raised in the book.