## A limit theorem for financial markets with inert investors (2003)

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Venue: | Mathematics of Operations Research |

Citations: | 16 - 2 self |

### BibTeX

@ARTICLE{Bayraktar03alimit,

author = {Erhan Bayraktar and Ulrich Horst and Ronnie Sircar},

title = {A limit theorem for financial markets with inert investors},

journal = {Mathematics of Operations Research},

year = {2003},

pages = {33--54}

}

### Years of Citing Articles

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### Abstract

We study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time. It turns out that semi-Markov processes are tailor made for modeling inert investors. With a suitable scaling, we show that when the price is driven by the market imbalance, the log price process is approximated by a process with long range dependence and non-Gaussian returns distributions, driven by a fractional Brownian motion. Consequently, investor inertia may lead to arbitrage opportunities for sophisticated ‘third parties’. The mathematical contributions are a functional central limit theorem for stationary semi-Markov processes, and approximation results for stochastic integrals of continuous semimartingales with respect to fractional Brownian motion.

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Citation Context ...milar approaches. Applying an invariance principle to a sequence of suitably defined discrete time models, they derived a diffusion approximation for the logarithmic price process. Duffie and Protter =-=[22]-=- also provided a mathematical framework for approximating sequences of stock prices by diffusion processes. All the aforementioned models assume that the agents trade the asset in each period. At the ... |

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Citation Context ...tors by trading frequently. We discuss a simple combination of both inert and active traders in Section 2.3. Evidence of long-range dependence in financial data is discussed in [19]. Bayraktar et al. =-=[5]-=- studied an asymptotically efficient wavelet-based estimator for the Hurst parameter, and analyzed high frequency S&P 500 index data over the span of 11.5 years (1989-2000). It was observed that, alth... |

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Citation Context ...wal function with a slowly decaying function plus a term which has asymptotically, i.e., for t → ∞, a vanishing effect compared to the first term; see Lemma 3.2 below. We will then apply results from =-=[31]-=- and [34] to analyze the tail structure of the convolution term. Let 2 We thank Chris Rogers for Example 3.1. { ∞∑ } R(i, j, t) := E 1 ∣ {ξn=j,Tn≤t} x0 = ξ0 = i n=0Bayraktar, Horst, Sircar: Investor ... |

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Citation Context ...ls n n H Ψ dX to ΨdB . In addition, we obtain a stability result for the integral of a fractional Brownian motion with respect to itself. These results may be viewed as an extension of Theorem 2.2 in =-=[38]-=- beyond the semimartingale setting. The remainder of this paper is organized as follows. In Section 2, we describe the financial market model with inert investors and state the main result. Section 3 ... |

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Citation Context ...bserved ([50]). A number of microeconomic models study investor caution with regard to model risk, which is termed uncertainty aversion. Among others, Dow and Werlang ([21]) and Simonsen and Werlang (=-=[51]-=-) considered models of portfolio optimization where agents are uncertain about the true probability measure. Their investors maximize their utility with respect to nonadditive probability measures. It... |

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