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1.
We give an example of a subspace K of such that , where denotes the closure with respect to convergence in probablity. On the other hand, the cone C ≔ K − L ∞ + is dense in L ∞ with respect to the weak-star topology σ( L ∞ , L 1 ) . This example answers a question raised by I. Evstigneev. The topic is motivated by the relation of the notion of no arbitrage and the existence of martingale measures in Mathematical Finance. 相似文献
2.
The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time 总被引:5,自引:0,他引:5
We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov's modeling of foreign exchange markets under transaction costs. The financial market is described by a d × d matrix-valued stochastic process (Π t ) T t =0 specifying the mutual bid and ask prices between d assets. We introduce the notion of "robust no arbitrage," which is a version of the no-arbitrage concept, robust with respect to small changes of the bid-ask spreads of (Π t ) T t =0 . The main theorem states that the bid-ask process (Π t ) T t =0 satisfies the robust no-arbitrage condition iff it admits a strictly consistent pricing system. This result extends the theorems of Harrison-Pliska and Kabanov-Stricker pertaining to the case of finite Ω, as well as the theorem of Dalang, Morton, and Willinger and Kabanov, Rásonyi, and Stricker, pertaining to the case of general Ω. An example of a 5 × 5 -dimensional process (Π t )2 t =0 shows that, in this theorem, the robust no-arbitrage condition cannot be replaced by the so-called strict no-arbitrage condition, thus answering negatively a question raised by Kabanov, Rásonyi, and Stricker. 相似文献
3.
4.
Christa Cuchiero Walter Schachermayer Ting‐Kam Leonard Wong 《Mathematical Finance》2019,29(3):773-803
Cover's celebrated theorem states that the long‐run yield of a properly chosen “universal” portfolio is almost as good as that of the best retrospectively chosen constant rebalanced portfolio. The “universality” refers to the fact that this result is model‐free, that is, not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory: the market portfolio is taken as the numéraire, and the rebalancing rule need not be constant anymore but may depend on the current state of the stock market. By fixing a stochastic model of the stock market this model‐free result is complemented by a comparison with the numéraire portfolio. Roughly speaking, under appropriate assumptions the asymptotic growth rate coincides for the three approaches mentioned in the title of this paper. We present results in both discrete and continuous time. 相似文献
5.
We examine Kreps' conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete‐time economies that “approach” the BSM economy in a natural sense: The nth discrete‐time economy is generated by a scaled n‐step random walk, based on an unscaled random variable ζ with mean 0, variance 1, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that . 相似文献
6.
The mutual fund theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. The main results are:
Financial support from the Austrian Science Fund (FWF) under the grant P19456, from Vienna Science and Technology Fund (WWTF)
under Grant MA13 and by the Christian Doppler Research Association (CDG) is gratefully acknowledged by the first author. The
research of the second author was partially supported by the National Science Foundation under Grant DMS-0604643. 相似文献
(i) | Let N be the wealth process of the numéraire portfolio (i.e., the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading only in N and the risk-free asset, then the MFT holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton’s classical result on Black–Merton–Scholes markets as well as the work of Chamberlain in the framework of Brownian filtrations (Chamberlain in Econometrica 56:1283–1300, 1988). Conversely, under a supplementary weak completeness assumption, we show that the validity of the MFT for general utility functions implies the replicability property for options on the numéraire portfolio described above. |
(ii) | If for a given class of utility functions (i.e., investors) the MFT holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz. |
7.
A version of the fundamental theorem of asset pricing is proved for continuous asset prices with small proportional transaction
costs. Equivalence is established between: (a) the absence of arbitrage with general strategies for arbitrarily small transaction
costs ${\varepsilon > 0}${\varepsilon > 0}, (b) the absence of free lunches with bounded risk for arbitrarily small transaction costs ${\varepsilon > 0}${\varepsilon > 0}, and (c) the existence of e{\varepsilon}-consistent price systems—the analogue of martingale measures under transaction costs—for arbitrarily small ${\varepsilon > 0}${\varepsilon > 0}. The proof proceeds through an explicit construction, as opposed to the usual separation arguments. The paper concludes comparing
numéraire-free and numéraire-based notions of admissibility, and the corresponding martingale and local martingale properties
for consistent price systems. 相似文献
8.
W. Schachermayer 《Mathematical Finance》1994,4(1):25-55
Let ( St ) tεI be an Rd -valued adapted stochastic process on (Ω, , ( t ) tεI , P ). A basic problem occurring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on equivalent to P such that ( St ) tεI is a martingale with respect to Q. It is known (see the fundamental papers of Harrison and Kreps 1979; Harrison and Pliska 1981; and Kreps 1981) that there is an intimate relation of this problem with the notions of "no arbitrage" and "no free lunch" in financial economics. We introduce the intermediate concept of "no free lunch with bounded risk." This is a somewhat more precise version of the notion of "no free lunch." It requires an absolute bound of the maximal loss occurring in the trading strategies considered in the definition of "no free lunch." We give an argument as to why the condition of "no free lunch with bounded risk" should be satisfied by a reasonable model of the price process ( St ) tεI of a securities market. We can establish the equivalence of the condition of "no free lunch with bounded risk" with the existence of an equivalent martingale measure in the case when the index set I is discrete but (possibly) infinite. A similar theorem was recently obtained by Delbaen (1992) for continuous-time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous-time case when the process ( St ) t εR+ is bounded and, roughly speaking, the jumps occur at predictable times. In the infinite horizon setting, the price process has to be "almost a martingale" in order to allow an equivalent martingale measure. 相似文献
9.
Walter Schachermayer 《Mathematical Finance》1993,3(2):217-229
We construct a continuous bounded stochastic process ( S t ,) 1E[0,1] which admits an equivalent martingale measure but such that the minimal martingale measure in the sense of Föllmer and Schweizer does not exist. This example also answers (negatively) a problem posed by Karatzas, Lehozcky, and Shreve as well as a problem posed by Strieker. 相似文献
10.
Weighted norm inequalities and hedging in incomplete markets 总被引:1,自引:0,他引:1
Freddy Delbaen Pascale Monat Walter Schachermayer Martin Schweizer Christophe Stricker 《Finance and Stochastics》1997,1(3):181-227
Let be an -valued special semimartingale on a probability space with canonical decomposition . Denote by the space of all random variables , where is a predictable -integrable process such that the stochastic integral is in the space of semimartingales. We investigate under which conditions on the semimartingale the space is closed in , a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient
conditions for the closedness of in . Most of these conditions deal with BMO-martingales and reverse H?lder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities,
we also extend previous results on the F?llmer-Schweizer decomposition. 相似文献