## 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 ... that γ(t) ∼ c 2 H(2H − 1)t 2H−2 L(t) as t → ∞. (24)10 Bayraktar, Horst, Sircar: Investor Inertia Mathematics of Operations Research xx(x), pp. xxx–xxx, c○200x INFORMS Proof. By Proposition 1.5.8 in =-=[8]-=-, every slowly varying function L which is locally bounded on R+ satisfies ∫ t τ 0 β L(τ)dτ ∼ tβ+1L(t) β + 1 if β > −1. Applying this proposition to the slowly varying function we conclude and so our ... |

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Citation Context ...ion 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 Inertia M... |

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Citation Context ... dynamical systems. From numerical simulations, they showed that financial price fluctuations can exhibit chaotic behavior if the effects of technical trading become too strong. Föllmer and Schweizer =-=[27]-=- took the probabilistic point of view, with asset prices arising from a sequence of temporary price equilibria in an exogenous random environment of investor sentiment; see [25], [32] or [26] for simi... |

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Citation Context ...g interest in agent-based models of financial markets. These models are capable of explaining, often through simulations, many facts like the emergence of herding behavior [41], volatility clustering =-=[42]-=- or fat-tailed distributions of stock returns [17] that are observed in financial data. Brock and Hommes [10, 11] proposed models with many traders where the asset price process is described by determ... |

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Citation Context ...tors, the dynamics of the asset price process can be approximated in law by a stochastic integral with respect to a superposition, BH + δW, of a fractional and a regular Brownian motion. It is known (=-=[12]-=-) that BH + δW is a semimartingale for any δ ̸= 0, if H > 3 4 , that is, if α < 3 2 , but not if H ∈ (1 2 , 3 4 ]. Thus, no arbitrage opportunities ariseBayraktar, Horst, Sircar: Investor Inertia Mat... |

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Citation Context ...informed small investors. We assume that x is temporally homogeneous under the measure P, that is, P{ξn+1 = j, Tn+1 − Tn ≤ t ∣ ∣ξn = i} = Q(i, j, t) (4) is independent of n ∈ N. By Proposition 1.6 in =-=[15]-=-, this implies that {ξn}n∈N is a homogeneous Markov chain on E whose transition probability matrix P = (pij) is given by pij = lim t→∞ Q(i, j, t). Clearly, x is an ordinary temporally homogeneous Mark... |

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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 ... Our approach may thus be viewed as a microeconomic foundation for these models. A recent paper that proposes entirely different economic foundations for models based on fractional Brownian motion is =-=[36]-=-. An approximation result for 1 A 401k retirement plan is a special type of account funded through pre-tax payroll deductions. The funds in the account can be invested in a number of different stocks,... |

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Citation Context ...ing online, investors trade more actively, more speculatively and less profitably than before”. Similar empirical findings were recently reached, using a completely different statistical technique in =-=[6]-=-. Thus, the dramatic fall in the estimated Hurst parameter in the late 1990s can be thought of as a posteriori validation of the link our model provides between investor inertia and long-range depende... |

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Citation Context ...ds does not depend on the current state; for an extension to the case of heavy-tailed rewards, see [40]. A recent paper [45] studies the binary case under a different limit taking mechanism; see also =-=[28]-=-. Binary state spaces are natural for modelling internet traffic, but for many applications in Economics or Queueing Theory, it is clearly desirable to have more flexible results that apply to general... |