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Article Excerpt
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 modelling 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 market participants. 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.
Key words: semi-Markov processes; fractional Brownian motion; functional central limit theorem; market microstructure; investor inertia
MSC2000 subject classification: Primary: 60F13, 60G15; secondary: 91B28 OR/MS subject classification: Primary: probability; secondary: stochastic model applications History: Received December 10, 2004; revised December 23, 2005.
1. Introduction and motivation. We prove a functional central limit theorem for stationary semi-Markov processes in which the limit process is a stochastic integral with respect to fractional Brownian motion. Our motivation is to develop a probabilistic framework within which to analyze the aggregate effect of investor inertia on asset price dynamics. We show that, in isolation, such infrequent trading patterns can lead to long-range dependence in stock prices and arbitrage opportunities for other more sophisticated traders.
1.1. Market microstructure models for financial markets. In mathematical finance, the dynamics of asset prices are usually modelled by trajectories of some exogenously specified stochastic process defined on some underlying probability space ([OMEGA], F, P). Geometric Brownian motion has long become the canonical reference model of financial price fluctuations. Since prices are generated by the demand of market participants, it is of interest to support such an approach by a microeconomic model of interacting agents.
In recent years, there has been increasing 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 (Lux [40]), volatility clustering (Lux and Marchesi [41]), or fat-tailed distributions of stock returns (Cont and Bouchaud [18]) that are observed in financial data. Brock and Hommes [10, 11] proposed models with many traders where the asset price process is described by deterministic dynamical systems. From numerical simulations, they showed that financial price fluctuations can exhibit chaotic behavior if the effects of technical trading become too strong.
Follmer and Schweizer [26] 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 Follmer [25], Horst [31], or Follmer et al. [27] for similar approaches). Applying an invariance principle to a sequence of suitably defined discrete time models, they derived a diffusion approximation for the logarithmic price process. Duffle 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 end of each trading interval, the agents update their expectations for the future evolution of the stock price and formulate their excess demand for the following period. However, small investors are not so efficient in their investment decisions: they are typically inactive and actually trade only occasionally. This may be because they are waiting to accumulate sufficient capital to make further stock purchases; or they tend to monitor their portfolios infrequently; or they are simply scared of choosing the wrong investments; or they feel that as long-term investors, they can defer action; or they put off the time-consuming research necessary to make informed portfolio choices. Long uninterrupted periods of inactivity may be viewed as a form of investor inertia. The focus of this paper is the effect of such investor inertia on asset prices in a model with asynchronous order arrivals. See Kruk [36] for an alternative microstructure model with asynchronous trading.
1.2. Inertia in financial markets. Investor inertia is a common experience and is well documented. The New York Stock Exchange (NYSE)'s survey of individual shareownership in the United States, "Shareownership2000" [45], demonstrates that many investors have very low levels of trading activity. For example, they find that "23 percent of stockholders with brokerage accounts report no trading at all, while 35 percent report trading only once or twice in the last year" (see New York Stock Exchange [45, pp. 58-59]). The NYSE survey (e.g., New York Stock Exchange [45, Table 28]) also reports that the average holding period for stocks is long, for example, 2.9 years in the early 1990s.
Empirical evidence of inertia also appears in the economic literature. For example, Madrian and Shea [42] looked at the reallocation of assets in employees' individual 401(k) (retirement) plans (1) and found "a status quo bias resulting from employee procrastination in making or implementing an optimal savings decision" (p. 1177). A related study by Hewitt Associates (a management consulting firm) found that in 2001, four out of five plan participants did not do any trading in their 401(k)s. Madrian and Shea explain that "if the cost of gathering and evaluating the information needed to make a 401k savings decision exceeds the short-run benefit from doing so, individuals will procrastinate" [42, p. 1177]). The prediction of Prospect Theory (Kahneman and Tversky [34]) that investors tend to hold onto losing stocks too long has also been observed (Shefrin and Statman [49]).
A number of microeconomic models study investor caution regarding model risk, which is termed uncertainty aversion. Among others, Dow and Werlang [21] and Simonsen and Werlang [50] 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 turns out that uncertainty aversion leads to inertia: the agents do not trade the asset unless the price exceeds or falls below a certain threshold.
We provide a mathematical framework for modelling investor inertia in a simple microstructure model where asset prices result from the demand of a large number of small investors whose trading behavior exhibits inertia. To each agent a, we associate a stationary semi-Markov process [x.sup.a] = [([x.sup.a.sub.t]).sub.t[greater than or equal to]0] on a finite state space, which represents the agent's propensity for trading. The processes [x.sup.a] have heavy-tailed-sojourn times in some designated "inert" state, and relatively thin-tailed sojourn times in various other states. Semi-Markov processes are tailor made to model individual traders' inertia as they generalize Markov processes by removing the requirement of exponentially distributed, and therefore thin-tailed, holding times. In addition, we allow for a markewide amplitude process [PSI] that describes the evolution of typical trading size in the market. It is large on heavy trading days and small on light trading days. We adopt a non-Walrasian approach to asset pricing and assume that prices move in the direction of market imbalance. We show that in a model with many inert investors, long-range dependence in the price process emerges.
1.3. Long-range dependence in financial time series. The observation of long-range dependence (sometimes called the Joseph effect) in financial time series motivated the use of fractional Brownian motion as a basis for asset pricing models (see, for instance, Mandelbrot [43] or Cutland et al. [19]). By our invariance principle, the drift-adjusted logarithmic price process converges weakly to a stochastic integral with respect to a fractional Brownian motion with Hurst coefficient H > 1/2. 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 Kluppelberg and Kuhn [35]. An approximation result for fractional Brownian motion in the context of a binary market model is given in Sottinen [51].
As is well known, fractional Brownian motion processes are not semimartingales, and so these models may theoretically allow arbitrage opportunities. Explicit arbitrage strategies for various models were constructed in Rogers [48], Cheridito [13], and Bayraktar and Poor [3]. These strategies capitalize on the smoothness of fractional Brownian motion (relative to standard Brownian motion) and involve rapid trading to exploit the finescale properties of the process' trajectories. As a result, in our microstrncture model, arbitrage opportunities may arise for other sufficiently sophisticated, market participants who are able to take advantage of inert investors 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 Cutland et al. [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, although the Hurst parameter was significantly above the efficient markets value of H = 1/2 up through the mid-1990s, it started to fall to that level over the period 1997-2000 (see Figure 1). Bayraktar et al. [5] suggested that this behavior of the market might be related to the increase in Internet trading, which is documented, for example, in NYSE's Stockownership2000 [45], Barber and Odean [1], and Choi [14], who find that after 18 months of access, the Web effect is very large: trading frequency doubles. Barber and Odean [2] find that "after going online, investors trade more actively, more speculatively and less profitably than before" (p. 455). Similar empirical findings were recently reached, using a completely different statistical technique in Bianchi [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 dependence in stock prices.
[FIGURE 1 OMITTED]
We note the evidence of long memory in stock price returns is mixed. There are several papers in the empirical finance literature providing evidence for the existence of long memory, yet there are several other papers that contradict these empirical findings (see, e.g., Bayraktar et al. [5] for an exposition of this debate and references). However, long memory is a well-accepted feature in volatility (squared and absolute returns) and trading volume (see, e.g., Cont [17] and Ding et al. [20]). The mathematical results of this paper might also be seen as an intermediate step toward a microstructural foundation for this phenomenon.
1.4. Mathematical contributions. We establish a functional central limit theorem for semi-Markov processes (Theorem 2.1 below), which extends the results of Taqqu et al. [53], who proved a result similar to ours for on and off processes; that is, semi-Markov processes taking values in the binary state space {0, 1}. Their arguments do not carry over to models with more general state spaces. Our approach builds on Markov renewal theory. We also demonstrate (see Example 3.1) that there may be a different limit behavior when the semi-Markov processes are centered, a situation that cannot arise in the binary case. Taqqu and Levy [52] considered renewal reward processes with heavy-tailed renewal periods and independent and identically distributed rewards. They assume a general state space, but the distributions of the length of renewal periods does not depend on the current state; for an extension to the case of heavy-tailed rewards, see Levy and Taqqu [39]. A recent paper (Mikosch et al. [44]) studies the binary case under a different limit-taking mechanism (see also Galgalas and Kaj [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 semi-Markov processes on finite state spaces. In the context of a financial market model, it is natural to allow for both positive (buying), negative (selling), and a zero (inactive) state. Our results also have applications to complex multilevel queueing networks where the level-dependent holding time distributions are allowed to have slowly decaying tails. They may serve as a mathematical basis for proving heavy traffic limits in the network models studied in, e.g., Duffield and Whitt [23, 24] and Whitt [54].
We allow for limits, which are integrals with respect to fractional Brownian motion proving an approximation result for stochastic integrals of continuous semimartingales with respect to fractional Brownian motion. Specifically, we consider a sequence of good semimartingales {[[PSI].sup.n]} and a sequence of stochastic processes {[X.sup.n]} having zero quadratic variation and give sufficient conditions, which guarantee...
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