Asymptotic arbitrage and large deviations
type: Item
instance of: scholarly article
title: Asymptotic arbitrage and large deviations
zbMATH Open document ID: 1153.91015
published in: Mathematics and Financial Economics
publication date: 2008-09-04T00:00:00Z
review text: Typical models of mathematical finance admit equivalent martingale measures up to any finite time horizon but not globally which means that arbitrage opportunities arise in the long run. In the present paper explicit estimates for asymptotic arbitrage are derived, and it is shown how they are related to large deviation estimates for the market price of risk. More precisely, let \((S_t)\) be an \(\mathbb{R}^d\)-valued diffusion given by \[ dS_t= \sigma(S_t)(dw_t+ \varphi(S_t)\,dt). \] Here, \((W_t)\) is an \(\mathbb{R}^N\)-valued standard Brownian motion, \(\sigma:\mathbb{R}^d\to \mathbb{R}^{d\times N}\) is a (deterministic, time-independent) volatility function and \(\varphi: \mathbb{R}^d\to \mathbb{R}^N\) is a market price of risk function (taking values in \(\ker(\sigma(S_t))^\perp\)). For a fixed finite time horizon \(T\) let \(K_T= \{(H\cdot S)_T: H\in{\mathcal H}\}\) denote the set of attainable contingent claims (\({\mathcal H}\) denoting the class of predictable \(S\)-integrable admissible processes). \(S= (S_t)\) is said to allow for strong asymptotic arbitrage if, for \(\varepsilon> 0\), there are \(T<\infty\) and \(X_T\in K_t\) satisfying \[ (i)\qquad X_t\geq -\varepsilon\text{ a.s., and }\quad (ii)\qquad P(X_t\geq \varepsilon^{-1})\geq 1-\varepsilon. \] (This concept is essentially due to \textit{Yu. M. Kabanov} and \textit{D. O. Kramkov} [Finance Stoch. 2, No. 2, 143--172 (1998; Zbl 0894.90020)].) \(S\) is said to have an average squared market price of risk above the threshold \(c> 0\) if \((\|\varphi_t\|)\) satisfies \[ \lim_{T\to\infty}\, P\Biggl({1\over T} \int^T_0 \|\varphi_t\|^2\,dt< c\Biggr)= 0.\tag{\(*\)} \] It is shown that this implies that, for \(\varepsilon> 0\), \(\gamma_1+ \gamma_2< c/2\), and for \(T\) large enough, there exists \(X_T\in K_T\) such that \[ X_T\geq - e^{-\gamma_1 T}\text{ a.s.,}\tag{1} \] and \[ P(X_T\geq e^{\gamma_2 T})\geq 1-\varepsilon\tag{2} \] implying (i) and (ii) above. It is shown that if \((*)\) is replaced by a ce…
reviewed by: Klaus Schürger
Mathematics Subject Classification ID: 91B16
Mathematics Subject Classification ID: 91B24
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60G48
zbMATH DE Number: 5320836
zbMATH Keywords: asymptotic arbitrage
zbMATH Keywords: utility maximization
zbMATH Keywords: large deviations
zbMATH Keywords: cost averaging
author: Hans Föllmer
author: Walter Schachermayer
MaRDI profile type: Publication
full work available at URL: s11579-008-0009-3
OpenAlex ID: W2143620550
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DOI: 10.1007/S11579-008-0009-3
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description: scientific article; zbMATH DE number 5320836