Admissible Strategies in Semimartingale Portfolio Selection

📝 Abstract

The choice of admissible trading strategies in mathematical modelling of financial markets is a delicate issue, going back to Harrison and Kreps (1979). In the context of optimal portfolio selection with expected utility preferences this question has been a focus of considerable attention over the last twenty years. We propose a novel notion of admissibility that has many pleasant features - admissibility is characterized purely under the objective measure; each admissible strategy can be approximated by simple strategies using finite number of trading dates; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity nor differentiability of the utility function are necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility. For utility functions finite on the whole real line, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function.

💡 Analysis

The choice of admissible trading strategies in mathematical modelling of financial markets is a delicate issue, going back to Harrison and Kreps (1979). In the context of optimal portfolio selection with expected utility preferences this question has been a focus of considerable attention over the last twenty years. We propose a novel notion of admissibility that has many pleasant features - admissibility is characterized purely under the objective measure; each admissible strategy can be approximated by simple strategies using finite number of trading dates; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity nor differentiability of the utility function are necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility. For utility functions finite on the whole real line, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function.

📄 Content

arXiv:0910.3936v5 [q-fin.CP] 11 Dec 2010 ADMISSIBLE STRATEGIES IN SEMIMARTINGALE PORTFOLIO SELECTION∗ SARA BIAGINI† AND ALEˇS ˇCERN´Y‡ Dedicated to Walter Schachermayer on the occasion of his 60th birthday. Abstract. The choice of admissible trading strategies in mathematical modelling of financial markets is a delicate issue, going back to Harrison and Kreps [HK79]. In the context of optimal portfolio selection with expected utility preferences this question has been the focus of considerable attention over the last twenty years. We propose a novel notion of admissibility that has many pleasant features – admissibility is characterized purely under the objective measure P ; each admissible strategy can be approximated by simple strategies using finite number of trading dates; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity nor differentiability of the utility function are necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility. For utility functions finite on R, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function. Key words. utility maximization, non locally bounded semimartingale, incomplete market, σ-localization and I-localization, σ-martingale measure, Orlicz space, convex duality AMS subject classifications. primary 60G48, 60G44, 49N15, 91B16; secondary 46E30, 46N30 JEL subject classifications. G11, G12, G13

1. Introduction. A central concept of financial theory is the notion of a self- financing investment strategy H, whose discounted wealth is expressed mathemati- cally by the stochastic integral x + H · St := x + Z (0,t] HsdSs, where S is a semimartingale process on a stochastic basis (Ω, (Ft)0≤t≤T , P), repre- senting discounted prices of d traded assets, and x is the initial wealth. Stochastic integration theory formulates minimal requirements for the integral above to exist, see Protter [Pr05]. The class of predictable processes H for which the integral exists is denoted by L(S; P) or simply L(S). However, the whole of L(S) is not appropriate for financial applications. Specifically, Harrison and Kreps [HK79] noted that when all processes in L(S) are allowed as trading strategies, arbitrage opportunities arise even in the standard Black-Scholes model. This is not a problem of the model S – the reason is that the theory of stochastic integration operates with a set of integrands far too rich for such applications. The solution proposed by the subsequent no-arbitrage literature, see [Sch94, DS98], is to restrict attention to a subset Hb ⊆L(S) of strategies whose wealth is bounded uniformly from below by a constant. ∗To appear in SIAM J. Control Optim. †University of Pisa (sara.biagini@ec.unipi.it). Part of this research was conducted while visiting Collegio Carlo Alberto in Moncalieri, Turin, Italy in Spring 2009. Warm hospitality and financial support of the Collegio are gratefully acknowledged. ‡Cass Business School, City University London (Ales.Cerny.1@city.ac.uk). 1 2 S. BIAGINI AND A. ˇCERN´Y Now consider a concave non decreasing utility function U and an agent who wishes to maximize the expected utility of her terminal wealth, E[U(x + H · ST )]. In this context, A ⊆L(S) will be a good set of trading strategies if the utility maximization over H ∈A is well posed and if A contains the optimizer, U(+∞) > sup H∈A E[U(x + H · ST )] = max H∈A E[U(x + H · ST )]. Historically, the search for a good definition of admissibility has proved to be a difficult task and it has evolved in two streams. For utility functions finite on a half-line, for example a logarithmic utility, there is a natural definition: admissible strategies are again those in Hb, see [KS99, CSW01, KS03]. Remarkably, this theo- retical framework is valid for any arbitrage-free S. For utility functions finite on the whole R, the situation is more complicated. The definition of admissibility via Hb works only to a certain extent. Here S has to be locally bounded (or σ-bounded) to ensure that Hb is sufficiently rich for a duality framework to work, cf. [Sch01]. Moreover, the class Hb will typically fail to contain the optimizer – this happens, for example, in the classical Black-Scholes model under exponential utility. A possible choice in this situation is to consider all strategies whose wealth is a martingale under all suitably defined pricing measures (see §3.1). This approach works well for exponential utility, cf. [Dal02, KSt02]. However, the seminal work of Schachermayer [Sch03] shows that, for general utilities, the martingale class is too narrow to catch the optimizer. The optimal strategy only exists among strategies whose wealth is a supermartingale under all pricing measures. For th

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