Fundamental theorem of asset pricing
The fundamental theorems of asset pricing (also: of arbitrage, of finance), in both financial economics and mathematical 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.[1] Though arbitrage opportunities do exist briefly in real life, it has been said that any sensible market model must avoid this type of profit.[2]: 5 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 contingent claim can be replicated. Though this property is common in models, it is not always considered desirable or realistic.[2]: 30
Discrete markets
[edit]In a discrete (i.e. finite state) market, the following hold:[2]
- The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space is arbitrage-free if, and only if, there exists at least one risk neutral probability measure that is equivalent 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 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
[edit]When stock price returns follow a single Brownian motion, 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 (NFLVR) must be used to describe these opportunities in an infinite dimensional setting.[3]
In continuous time, a version of the fundamental theorems of asset pricing reads:[4]
Let be a d-dimensional semimartingale market (a collection of stocks), the risk-free bond and the underlying probability space. Furthermore, we call a measure an equivalent local martingale measure if and if the processes are local martingales under the measure .
- The First Fundamental Theorem of Asset Pricing: Assume is locally bounded. Then the market satisfies NFLVR if and only if there exists an equivalent local martingale measure.
- The Second Fundamental Theorem of Asset Pricing: Assume that there exists an equivalent local martingale measure . Then is a complete market if and only if is the unique local martingale measure.
See also
[edit]- Arbitrage pricing theory
- Asset pricing
- Financial economics § Arbitrage-free pricing and equilibrium
- Rational pricing
References
[edit]Sources
- ^ Varian, Hal R. (1987). "The Arbitrage Principle in Financial Economics". Economic Perspectives. 1 (2): 55–72. doi:10.1257/jep.1.2.55. JSTOR 1942981.
- ^ a b c Pascucci, Andrea (2011) PDE and Martingale Methods in Option Pricing. Berlin: Springer-Verlag
- ^ Delbaen, Freddy; Schachermayer, Walter. "What is... a Free Lunch?" (PDF). Notices of the AMS. 51 (5): 526–528. Retrieved October 14, 2011.
- ^ Björk, Tomas (2004). Arbitrage Theory in Continuous Time. New York: Oxford University Press. pp. 144ff. ISBN 978-0-19-927126-9.
Further reading
- Harrison, J. Michael; Pliska, Stanley R. (1981). "Martingales and Stochastic integrals in the theory of continuous trading". Stochastic Processes and Their Applications. 11 (3): 215–260. doi:10.1016/0304-4149(81)90026-0.
- Delbaen, Freddy; Schachermayer, Walter (1994). "A General Version of the Fundamental Theorem of Asset Pricing". Mathematische Annalen. 300 (1): 463–520. doi:10.1007/BF01450498.
