The fundamental theorem of asset pricing for unbounded stochastic processes.

*(English)*Zbl 0917.60048The fundamental theorem of asset pricing is a basic result in mathematical finance. Using the concept of martingale measures it characterizes the class of semimartingale models for financial assets which exclude arbitrage, i.e., riskless profits from dynamic trading. Beginning with the seminal work by J. M. Harrison and D. M. Kreps [J. Econ. Theory 20, 381–408 (1979; Zbl 0431.90019)] and J. M. Harrison and S. R. Pliska [Stochastic Processes Appl. 11, 215–260 (1981; Zbl 0482.60097)] several formulations of this theorem have been established. The article reviewed here states and proves this theorem in the following very general form: Let \(S=(S_t)_{t \geq 0}\) be an \(R^d\)-valued semimartingale defined on the stochastic base \((\Omega, \mathcal{F},(\mathcal{F}_t)_{t \geq 0},P)\). Then \(S\) satisfies the condition of no free lunch with vanishing risk (NFLVR) if and only if there exists a probability measure \(Q \sim P\) such that \(S\) is a sigma-martingale with respect to \(Q\).

The concept of NFVLR is a mild strengthening of the no-arbitrage property (see e.g., [the authors, Math. Ann. 300, No. 3, 463–520 (1994; Zbl 0865.90014)]). A sigma-martingale is defined as a semimartingale \(S=(S_t)_{t \geq 0}\) which can be written as a stochastic integral \(S=\varphi \cdot M\) where \(M\) is a martingale and \(\varphi\) an \(M\)-integrable predictable process with nonnegative values. The concept of a sigma-martingale is new in the context of mathematical finance. It is introduced to deal with possibly unbounded jumps of the asset price process \(S\). Indeed, under the assumption that \(S\) is locally bounded, previous versions of the fundamental theorem, e.g., Delbaen-Schachermayer [loc. cit.], yield equivalence of NFLVR to the existence of a local martingale measure. The authors provide an example of a sigma-martingale which does not admit an equivalent local martingale measure, illustrating that sigma-martingales are in fact unavoidable in the presence of unbounded jumps. Besides the proof of the fundamental theorem the article presents a general version of the dual characterization of super-hedging prices for contingent claims, first established by N. El Karoui and M.-C. Quenez [SIAM J. Control Optimization 33, No. 1, 29–66 (1995; Zbl 0831.90010)]. In addition, it yields a very general characterization of attainable claims, extending the results of S. D. Jacka [Math. Finance 2, No. 4, 239–250 (1992)].

The concept of NFVLR is a mild strengthening of the no-arbitrage property (see e.g., [the authors, Math. Ann. 300, No. 3, 463–520 (1994; Zbl 0865.90014)]). A sigma-martingale is defined as a semimartingale \(S=(S_t)_{t \geq 0}\) which can be written as a stochastic integral \(S=\varphi \cdot M\) where \(M\) is a martingale and \(\varphi\) an \(M\)-integrable predictable process with nonnegative values. The concept of a sigma-martingale is new in the context of mathematical finance. It is introduced to deal with possibly unbounded jumps of the asset price process \(S\). Indeed, under the assumption that \(S\) is locally bounded, previous versions of the fundamental theorem, e.g., Delbaen-Schachermayer [loc. cit.], yield equivalence of NFLVR to the existence of a local martingale measure. The authors provide an example of a sigma-martingale which does not admit an equivalent local martingale measure, illustrating that sigma-martingales are in fact unavoidable in the presence of unbounded jumps. Besides the proof of the fundamental theorem the article presents a general version of the dual characterization of super-hedging prices for contingent claims, first established by N. El Karoui and M.-C. Quenez [SIAM J. Control Optimization 33, No. 1, 29–66 (1995; Zbl 0831.90010)]. In addition, it yields a very general characterization of attainable claims, extending the results of S. D. Jacka [Math. Finance 2, No. 4, 239–250 (1992)].

Reviewer: Peter Bank (Berlin)

##### MSC:

60G44 | Martingales with continuous parameter |

46N30 | Applications of functional analysis in probability theory and statistics |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

91B25 | Asset pricing models (MSC2010) |

60H05 | Stochastic integrals |