Fundamental Theorem of Asset Pricing: Tautological Meeting of Mathematical Martingale and Financial Arbitrage by the Measure of Probability.

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The Fundamental Theorem of Asset Pricing (FTAP hereafter) has two broad tenets, viz.

1. A market admits no arbitrage, if and only if, the market has a martingale measure.

2. Every contingent claim can be hedged, if and only if, the martingale measure is unique.

The FTAP is a theorem of mathematics, and the use of the term ‘measure’ in its statement places the FTAP within the theory of probability formulated by Andrei Kolmogorov (Foundations of the Theory of Probability) in 1933. Kolmogorov’s work took place in a context captured by Bertrand Russell, who observed that

It is important to realise the fundamental position of probability in science. . . . As to what is meant by probability, opinions differ.

In the 1920s the idea of randomness, as distinct from a lack of information, was becoming substantive in the physical sciences because of the emergence of the Copenhagen Interpretation of quantum mechanics. In the social sciences, Frank Knight argued that uncertainty was the only source of profit and the concept was pervading John Maynard Keynes’ economics (Robert Skidelsky Keynes the return of the master).

Two mathematical theories of probability had become ascendant by the late 1920s. Richard von Mises (brother of the Austrian economist Ludwig) attempted to lay down the axioms of classical probability within a framework of Empiricism, the ‘frequentist’ or ‘objective’ approach. To counter–balance von Mises, the Italian actuary Bruno de Finetti presented a more Pragmatic approach, characterised by his claim that “Probability does not exist” because it was only an expression of the observer’s view of the world. This ‘subjectivist’ approach was closely related to the less well-known position taken by the Pragmatist Frank Ramsey who developed an argument against Keynes’ Realist interpretation of probability presented in the Treatise on Probability.

Kolmogorov addressed the trichotomy of mathematical probability by generalising so that Realist, Empiricist and Pragmatist probabilities were all examples of ‘measures’ satisfying certain axioms. In doing this, a random variable became a function while an expectation was an integral: probability became a branch of Analysis, not Statistics. Von Mises criticised Kolmogorov’s generalised framework as un-necessarily complex. About a decade and a half back, the physicist Edwin Jaynes (Probability Theory The Logic Of Science) champions Leonard Savage’s subjectivist Bayesianism as having a “deeper conceptual foundation which allows it to be extended to a wider class of applications, required by current problems of science”.

The objections to measure theoretic probability for empirical scientists can be accounted for as a lack of physicality. Frequentist probability is based on the act of counting; subjectivist probability is based on a flow of information, which, following Claude Shannon, is now an observable entity in Empirical science. Measure theoretic probability is based on abstract mathematical objects unrelated to sensible phenomena. However, the generality of Kolmogorov’s approach made it flexible enough to handle problems that emerged in physics and engineering during the Second World War and his approach became widely accepted after 1950 because it was practically more useful.

In the context of the first statement of the FTAP, a ‘martingale measure’ is a probability measure, usually labelled Q, such that the (real, rather than nominal) price of an asset today, X0, is the expectation, using the martingale measure, of its (real) price in the future, XT. Formally,

X0 = EQ XT

The abstract probability distribution Q is defined so that this equality exists, not on any empirical information of historical prices or subjective judgement of future prices. The only condition placed on the relationship that the martingale measure has with the ‘natural’, or ‘physical’, probability measures usually assigned the label P, is that they agree on what is possible.

The term ‘martingale’ in this context derives from doubling strategies in gambling and it was introduced into mathematics by Jean Ville in a development of von Mises’ work. The idea that asset prices have the martingale property was first proposed by Benoit Mandelbrot in response to an early formulation of Eugene Fama’s Efficient Market Hypothesis (EMH), the two concepts being combined by Fama. For Mandelbrot and Fama the key consequence of prices being martingales was that the current price was independent of the future price and technical analysis would not prove profitable in the long run. In developing the EMH there was no discussion on the nature of the probability under which assets are martingales, and it is often assumed that the expectation is calculated under the natural measure. While the FTAP employs modern terminology in the context of value-neutrality, the idea of equating a current price with a future, uncertain, has ethical ramifications.

The other technical term in the first statement of the FTAP, arbitrage, has long been used in financial mathematics. Liber Abaci Fibonacci (Laurence Sigler Fibonaccis Liber Abaci) discusses ‘Barter of Merchandise and Similar Things’, 20 arms of cloth are worth 3 Pisan pounds and 42 rolls of cotton are similarly worth 5 Pisan pounds; it is sought how many rolls of cotton will be had for 50 arms of cloth. In this case there are three commodities, arms of cloth, rolls of cotton and Pisan pounds, and Fibonacci solves the problem by having Pisan pounds ‘arbitrate’, or ‘mediate’ as Aristotle might say, between the other two commodities.

Within neo-classical economics, the Law of One Price was developed in a series of papers between 1954 and 1964 by Kenneth Arrow, Gérard Debreu and Lionel MacKenzie in the context of general equilibrium, in particular the introduction of the Arrow Security, which, employing the Law of One Price, could be used to price any asset. It was on this principle that Black and Scholes believed the value of the warrants could be deduced by employing a hedging portfolio, in introducing their work with the statement that “it should not be possible to make sure profits” they were invoking the arbitrage argument, which had an eight hundred year history. In the context of the FTAP, ‘an arbitrage’ has developed into the ability to formulate a trading strategy such that the probability, under a natural or martingale measure, of a loss is zero, but the probability of a positive profit is not.

To understand the connection between the financial concept of arbitrage and the mathematical idea of a martingale measure, consider the most basic case of a single asset whose current price, X0, can take on one of two (present) values, XTD < XTU, at time T > 0, in the future. In this case an arbitrage would exist if X0 ≤ XTD < XTU: buying the asset now, at a price that is less than or equal to the future pay-offs, would lead to a possible profit at the end of the period, with the guarantee of no loss. Similarly, if XTD < XTU ≤ X0, short selling the asset now, and buying it back would also lead to an arbitrage. So, for there to be no arbitrage opportunities we require that

XTD < X0 < XTU

This implies that there is a number, 0 < q < 1, such that

X0 = XTD + q(XTU − XTD)

= qXTU + (1−q)XTD

The price now, X0, lies between the future prices, XTU and XTD, in the ratio q : (1 − q) and represents some sort of ‘average’. The first statement of the FTAP can be interpreted simply as “the price of an asset must lie between its maximum and minimum possible (real) future price”.

If X0 < XTD ≤ XTU we have that q < 0 whereas if XTD ≤ XTU < X0 then q > 1, and in both cases q does not represent a probability measure which by Kolmogorov’s axioms, must lie between 0 and 1. In either of these cases an arbitrage exists and a trader can make a riskless profit, the market involves ‘turpe lucrum’. This account gives an insight as to why James Bernoulli, in his moral approach to probability, considered situations where probabilities did not sum to 1, he was considering problems that were pathological not because they failed the rules of arithmetic but because they were unfair. It follows that if there are no arbitrage opportunities then quantity q can be seen as representing the ‘probability’ that the XTU price will materialise in the future. Formally

X0 = qXTU + (1−q) XTD ≡ EQ XT

The connection between the financial concept of arbitrage and the mathematical object of a martingale is essentially a tautology: both statements mean that the price today of an asset must lie between its future minimum and maximum possible value. This first statement of the FTAP was anticipated by Frank Ramsey when he defined ‘probability’ in the Pragmatic sense of ‘a degree of belief’ and argues that measuring ‘degrees of belief’ is through betting odds. On this basis he formulates some axioms of probability, including that a probability must lie between 0 and 1. He then goes on to say that

These are the laws of probability, …If anyone’s mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning better and would then stand to lose in any event.

This is a Pragmatic argument that identifies the absence of the martingale measure with the existence of arbitrage and today this forms the basis of the standard argument as to why arbitrages do not exist: if they did the, other market participants would bankrupt the agent who was mis-pricing the asset. This has become known in philosophy as the ‘Dutch Book’ argument and as a consequence of the fact/value dichotomy this is often presented as a ‘matter of fact’. However, ignoring the fact/value dichotomy, the Dutch book argument is an alternative of the ‘Golden Rule’– “Do to others as you would have them do to you.”– it is infused with the moral concepts of fairness and reciprocity (Jeffrey Wattles The Golden Rule).

FTAP is the ethical concept of Justice, capturing the social norms of reciprocity and fairness. This is significant in the context of Granovetter’s discussion of embeddedness in economics. It is conventional to assume that mainstream economic theory is ‘undersocialised’: agents are rational calculators seeking to maximise an objective function. The argument presented here is that a central theorem in contemporary economics, the FTAP, is deeply embedded in social norms, despite being presented as an undersocialised mathematical object. This embeddedness is a consequence of the origins of mathematical probability being in the ethical analysis of commercial contracts: the feudal shackles are still binding this most modern of economic theories.

Ramsey goes on to make an important point

Having any definite degree of belief implies a certain measure of consistency, namely willingness to bet on a given proposition at the same odds for any stake, the stakes being measured in terms of ultimate values. Having degrees of belief obeying the laws of probability implies a further measure of consistency, namely such a consistency between the odds acceptable on different propositions as shall prevent a book being made against you.

Ramsey is arguing that an agent needs to employ the same measure in pricing all assets in a market, and this is the key result in contemporary derivative pricing. Having identified the martingale measure on the basis of a ‘primal’ asset, it is then applied across the market, in particular to derivatives on the primal asset but the well-known result that if two assets offer different ‘market prices of risk’, an arbitrage exists. This explains why the market-price of risk appears in the Radon-Nikodym derivative and the Capital Market Line, it enforces Ramsey’s consistency in pricing. The second statement of the FTAP is concerned with incomplete markets, which appear in relation to Arrow-Debreu prices. In mathematics, in the special case that there are as many, or more, assets in a market as there are possible future, uncertain, states, a unique pricing vector can be deduced for the market because of Cramer’s Rule. If the elements of the pricing vector satisfy the axioms of probability, specifically each element is positive and they all sum to one, then the market precludes arbitrage opportunities. This is the case covered by the first statement of the FTAP. In the more realistic situation that there are more possible future states than assets, the market can still be arbitrage free but the pricing vector, the martingale measure, might not be unique. The agent can still be consistent in selecting which particular martingale measure they choose to use, but another agent might choose a different measure, such that the two do not agree on a price. In the context of the Law of One Price, this means that we cannot hedge, replicate or cover, a position in the market, such that the portfolio is riskless. The significance of the second statement of the FTAP is that it tells us that in the sensible world of imperfect knowledge and transaction costs, a model within the framework of the FTAP cannot give a precise price. When faced with incompleteness in markets, agents need alternative ways to price assets and behavioural techniques have come to dominate financial theory. This feature was realised in The Port Royal Logic when it recognised the role of transaction costs in lotteries.

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