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Fundamental theorem of asset pricing

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In a general sense, the fundamental theorem of arbitrage/finance is a way to relate arbitrage opportunities with risk neutral measures that are equivalent to the original probability measure.

Discrete markets

In a discrete (i.e. finite state) market, the following hold[1]:

  1. The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space (Ω, , P) 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.
  2. 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

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 semi-martingale (see Delbaen and Schachermayer), 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.

See also

We believe

References

  1. ^ Pascucci, Andrea. Pde and Martingale Methods in Option Pricing. Berlin: Springer, 2011.
  • Harrison, J. Michael (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. {{cite journal}}: Cite has empty unknown parameter: |month= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
  • Delbaen, Freddy (1994). "A General Version of the Fundamental Theorem of Asset Pricing". Mathematische Annalen. 300 (1): 463–520. doi:10.1007/BF01450498. {{cite journal}}: Cite has empty unknown parameter: |month= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
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