Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets
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Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets Martin Le Doux Mbele Bidima · Miklós Rásonyi
Received: 16 October 2012 / Accepted: 11 June 2014 © Springer Science+Business Media Dordrecht 2014
Abstract Consider a discrete-time infinite horizon financial market model in which the logarithm of the stock price is a time discretization of a stochastic differential equation. Under conditions different from those given in (Mbele Bidima and Rásonyi in Ann. Oper. Res. 200:131–146, 2012), we prove the existence of investment opportunities producing an exponentially growing profit with probability tending to 1 geometrically fast. This is achieved using ergodic results on Markov chains and tools of large deviations theory. Furthermore, we discuss asymptotic arbitrage in the expected utility sense and its relationship to the first part of the paper. Keywords Asymptotic exponential arbitrage · Markov chains · Large deviations · Expected utility
1 Introduction In the classical theory of financial markets, absence of arbitrage (riskless profit) is characterized by the existence of suitable “pricing rules”: risk-neutral (i.e. equivalent martingale) measures for the discounted price process of the risky asset. This result is often referred to as “the fundamental theorem of asset pricing”. Further developments of arbitrage theory encompass the so-called “large financial markets” (see [5, 7] and the references therein). In these papers the following common feature
The authors thank the referee and the associate editor for extremely constructive and helpful reports. M.L.D. Mbele Bidima University of Yaoundé I, Yaoundé, Cameroon e-mail: [email protected]
B
M. Rásonyi ( ) MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary e-mail: [email protected] M. Rásonyi University of Edinburgh, Edinburgh, UK
M.L.D. Mbele Bidima, M. Rásonyi
of numerous models is highlighted: on each finite time horizon T > 0, there is no arbitrage opportunity but when T tends to infinity, one may realize riskless profit in the long run. Such trading opportunities are referred to as “asymptotic arbitrage”. An important tool that can be used for the study of asymptotic arbitrage is the theory of large deviations (see [2]), as proposed in [5]. More recently, in [10] we presented the discrete-time versions of some results in [5] about asymptotic arbitrage and, in this framework, we extended them by studying “asymptotic exponential arbitrage with geometrically decaying probability of failure”, i.e. we discussed the possibility for investors to realize an exponentially growing profit on their long-term investments while controlling (at a geometrically decaying rate) the probability of failing to achieve such a profit. Some of these results were subsequently proved for continuous-time models in [4]. In the present paper we prove results similar to Theorem 5 of [10] using different arguments and technical tools (the large deviation results of [9] instead of those in [8]). In this way
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