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"Financi ...
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, 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.
[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 Black� ...
). 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 R ...
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 ...
is
arbitrage-free 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
*'' Equival ...
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 ti ...
''B'' is
complete 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 ins ...
, 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-martingal ...
or
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.
See also
*
Arbitrage pricing theory
In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976, it is widely bel ...
*
Asset pricing
In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models.
There have been many models developed for different situations, but correspon ...
*
*
Rational pricing
Rational pricing is the assumption in financial economics that asset prices - and hence asset pricing models - will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is us ...
References
Sources
Further reading
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*
External links
* http://www.fam.tuwien.ac.at/~wschach/pubs/preprnts/prpr0118a.pdf
{{DEFAULTSORT:Fundamental Theorem Of Arbitrage-Free Pricing
Financial economics
Mathematical finance
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