The concept of absence of opportunities for free lunches is one of the pillars in the economic theory of financial markets. This natural assumption has proved very fruitful and has lead to great mathematical, as well as economical, insights in Quantitative Finance. Formulating rigorously the exact definition of absence of opportunities for riskless profit turned out to be a highly non-trivial fact that troubled mathematicians and economists for at least two decades. The purpose of this note is to give a quick (and, necessarily, incomplete) account of the recent work aimed at providing a simple and intuitive no-free-lunch assumption that would suffice in formulating a version of the celebrated Fundamental Theorem of Asset Pricing.
In the process of building realistic mathematical models of financial markets, absence of opportunities for riskless profit is considered to be a minimal normative assumption in order for the market to be in equilibrium state. The reason is quite obvious. If opportunities for riskless profit were present in the market, every economic agent would try to reap them. Prices would then instantaneously move in response to an imbalance between supply and demand. This sudden pricemovement would continue as long as opportunities for riskless profit are still present in the market.
Therefore, in market equilibrium, no such opportunities should be possible.
The aforementioned simple and very natural idea has proved very fruitful and has lead to great mathematical, as well as economical, insight in the theory of Quantitative Finance. Formulating rigorously the exact definition of “absence of opportunities for riskless profit” turned out to be a highly non-trivial fact that troubled mathematicians and economists for at least two decades 1 . As the road unfolded, the valuable input of the theory of stochastic analysis in financial theory was obvious; in the other direction, the development of the theory of stochastic processes benefited immensely from problems that emerged purely from these financial considerations.
Since the late seventies, it has been a folklore fact that there is a deep connection between absence of opportunities for riskless profit and the existence of a risk-neutral measure 2 , that is, a probability that is equivalent to the original one under which the discounted asset price processes has some kind of martingale property. Existence of such measures are of major practical importance, since they open the road to pricing illiquid assets or contingent claims in the market. The above folklore result has been called the Fundamental Theorem of Asset Pricing.
The easiest and most classical way to formulate the notion of riskless profit is via the so-called arbitrage strategy An arbitrage is a combination of positions in the traded assets that requires Key words and phrases. Free lunch, arbitrage, Fundamental Theorem of Asset Pricing, separating hyperplane theorem, Equivalent Martingale Measures.
The exact market viability definition is still sometimes the source of debate. zero initial capital and results in nonnegative outcome with a strictly positive probability of the wealth being strictly positive at a fixed time-point in the future (after liquidation has taken place, of course). Naturally, the previous formulation of an arbitrage presupposes that a probabilistic model for the random movement of liquid asset prices has been set up. In [5], a discrete-state-space, multi-period discrete time financial market was considered. For this model, the authors showed the equivalence between the economical “No Arbitrage” (NA) condition and the mathematical stipulation of existence of an equivalent probability that makes the discounted asset-price processes martingales.
Crucial in the proof of the result in [5] was the separating hyperplane theorem in finite-dimensional Euclidean spaces. One of the convex sets to be separated is the class of all terminal outcomes resulting from trading and possible consumption starting from zero capital; the other is the positive orthant. The NA condition is basically the statement that the intersection of these two convex sets consists of only the zero vector.
After the publication of [5], a saga of papers followed that were aimed, one way or another, at strengthening the conclusion by considering more complicated market models. It quickly became obvious that the previous NA condition is no longer sufficient to imply the existence of a risk-neutral measure; it is too weak. In infinite-dimensional spaces, separation of hyperplanes, made possible by means of the geometric version of the Hahn-Banach theorem, requires the closedness of the set C of all terminal outcomes resulting from trading and possible consumption starting from zero capital.
The simple NA condition does not imply this in general. This has lead Kreps in [7] to define a free lunch as a generalized, asymptotic form of an arbitrage.
Essentially, a free lunch is a possibly infinite-valued random variable f with P[f ≥ 0] = 1 and P[f > 0] > 0 that belongs to the closure of C. Once an appropriate topology is defined on L 0 , the space of all random variables, in order for the last closure (call it C) to make sense, the “No Free
Lunch” (NFL) condition states that 3 C ∩ L 0 + = {0}. Kreps, in [7], used this idea with a very weak topology on locally convex spaces and showed the existence of a separating measure 4 . However, apart from trivial cases, this topology does not stem from a metric, which means that closedness cannot be described in terms of convergence of sequences. This makes the definition of a free lunch quite nonintuitive.
After [7], there were lots of attempts to introduce a condition closely relate
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