The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both

financial economics Financial economics, also known as finance, is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade".William F. Sharpe"Financial ...

and

mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...

, provide necessary and sufficient conditions for a market to be

arbitrage-free In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between the ...

, and for a market to be

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

. 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.Pascucci, Andrea (2011) ''PDE and Martingale Methods in Option Pricing''. Berlin:

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

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 The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Blac ...

). A complete market is one in which every

contingent claim In finance, a contingent claim is a derivative whose future payoff depends on the value of another “underlying” asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit Ri ...

can be replicated. 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 In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...

(\Omega, \mathcal, P)

if, and only if, there exists at least one risk neutral probability measure that is

equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...

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 bond A risk-free bond is a theoretical bond that repays interest and principal with absolute certainty. The rate of return would be the risk-free interest rate. It is primary security, which pays off 1 unit no matter state of economy is realized at tim ...

''B'' is

if and only if there exists a unique risk-neutral measure that is equivalent to ''P'' and has numeraire ''B''.

In more general markets

When stock price returns follow a single

Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...

, there is a unique risk neutral measure. When the stock price process is assumed to follow a more general

sigma-martingale In mathematics and information theory of probability, a sigma-martingale is a semimartingale with an integral representation. Sigma-martingales were introduced by C.S. Chou and M. Emery in 1977 and 1978. In financial mathematics, sigma-martingale ...

semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the l ...

, then the concept of arbitrage is too narrow, and a stronger concept such as

no free lunch with vanishing risk No free lunch with vanishing risk (NFLVR) is a no- arbitrage argument. We have ''free lunch with vanishing risk'' if by utilizing a sequence of time self-financing portfolios, which converge to an arbitrage strategy, we can approximate a self-fin ...

must be used to describe these opportunities in an infinite dimensional setting.

References

Sources Further reading * *

External links

* http://www.fam.tuwien.ac.at/~wschach/pubs/preprnts/prpr0118a.pdf {{DEFAULTSORT:Fundamental Theorem Of Arbitrage-Free Pricing Financial economics Mathematical finance Corporate development

Discrete markets

In more general markets

See also

References

External links