Fundamental theorem of arbitrage-free pricing
Encyclopedia
The fundamental theorems of arbitrage/finance provide necessary and sufficient conditions for a market to be arbitrage free
and for a market to be complete
. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss. Though arbitrage opportunities do exist briefly in real life, it has been said that any sensible market model must avoid this type of profit. The first theorem is important in that it ensures a fundamental property of market models. Completeness is a common property of market models (for instance the Black–Scholes model). A complete market is one in which every European option can be replicated
. Though this property is common in models, it is not always considered desirable or realistic.
, there is a unique risk neutral measure. When the stock price process is assumed to follow a more general sigma-martingale
or semimartingale
, then the concept of arbitrage is too narrow, and a stronger concept such as no free lunch with vanishing risk must be used to describe these opportunities in an infinite dimensional setting.
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
and for a market to be complete
Complete market
In economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets without friction. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs...
. An arbitrage opportunity is a way of making money with no initial investment without any possibility of loss. Though arbitrage opportunities do exist briefly in real life, it has been said that any sensible market model must avoid this type of profit. The first theorem is important in that it ensures a fundamental property of market models. Completeness is a common property of market models (for instance the Black–Scholes model). A complete market is one in which every European option can be replicated
Replicating strategy
In Finance a Replicating Strategy of a particular financial instrument is a set of liquid, usually exchange-traded assets with the same net payoff....
. Though this property is common in models, it is not always considered desirable or realistic.
Discrete markets
In a discrete (i.e. finite state) market, the following hold :- The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space (Ω,
, P) is arbitrage-freeArbitrageIn economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
if and only if there exists at least one risk neutral probability measureRisk-neutral measureIn mathematical finance, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted...
that is equivalentEquivalence (measure theory)In mathematics, and specifically in measure theory, equivalence is a notion of two measures being "the same".-Definition:Let be a measurable space, and let μ, ν : Σ → R be two signed measures. Then μ is said to be equivalent to ν if and only if each is absolutely continuous with respect to the other...
to the original probability measure, P. - The Second Fundamental Theorem of Asset Pricing: An arbitrage-free market (S,B) consisting of a collection of stocks S and a risk-free bondBond (finance)In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
B is completeComplete marketIn economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets without friction. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs...
if and only if there exists a unique risk-neutral measure that is equivalent to P and has numeraireNuméraireNuméraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
B.
In more general markets
When stock price returns follow a single Brownian motionBrownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
, there is a unique risk neutral measure. When the stock price process is assumed to follow a more general sigma-martingale
Sigma-martingale
In the mathematical theory of probability, sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. It should be noted that not every local martingale is a sigma-martingale...
or semimartingale
Semimartingale
In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....
, then the concept of arbitrage is too narrow, and a stronger concept such as no free lunch with vanishing risk must be used to describe these opportunities in an infinite dimensional setting.
External links
- http://www.fam.tuwien.ac.at/~wschach/pubs/preprnts/prpr0118a.pdf