Asymptotic arbitrage and large deviations
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 certain stronger large deviation estimate, then the probability of falling short of the exponential lower bound in (2) above decays exponentially in time. Furthermore it is shown how strong asymptotic arbitrage relates to dynamic portfolio optimization for a certain class of utility functions. Finally, the authors obtain for a geometric Ornstein-Uhlenbeck a process the optimal trading strategies and the resulting optimal growth rates of expected utility for all HARA utilities.
scientific article; zbMATH DE number 5320836
scientific article