تاثیر محدودیت های فروش استقراضی بر ثبات بازار مالی در مدل عوامل ناهمگن

Recent turmoil on global financial markets has led to a discussion on which policy measures should or could be taken to stabilize financial markets. One such a measure that resurfaced is the imposition of short-selling constraints. It is conjectured that these short-selling constraints reduce speculative trading and thereby have the potential to stabilize volatile financial markets. The purpose of the current paper is to investigate this conjecture in a standard asset pricing model with heterogeneous beliefs. We model short-selling constraints by imposing trading costs for selling an asset short. We find that the local stability properties of the fundamental rational expectations equilibrium do not change when trading costs for short-selling are introduced. However, when the asset is overvalued, costs for short-selling increase mispricing and price volatility.

The practice of short-selling – borrowing a financial instrument from another investor to sell it immediately and close the position in the future by buying and returning the instrument – is widespread in financial markets. In fact, short-selling is the mirror image of a “long position”, where an investor buys an asset which did not belong to him before. While a long position can be thought of as a bet on the increase of the assets' value (with dividend yield and opportunity costs taken into account), short-selling allows investors to bet on a fall in stock prices. Some people have argued that such betting may increase volatility of financial markets and even lead to the incidence of crashes. A proposed policy would then restrict short-selling. In this paper we investigate consequences of such a restriction in a heterogeneous agents model of a financial market and show that it may increase mispricing as well as price volatility. The historical account of Galbraith (1954) provides evidence that short sales were common during the market crash of 1929. As short-sellers were often blamed for the crash, the Securities and Exchange Commission (SEC) introduced the so-called “uptick rule” in 1938, which prohibited the selling short “on a downtick”, i.e., at prices lower than the previous transaction price. Curiously enough, the uptick rule was removed on 6 July, 2007, right before the market crash of 2008–2009 began. Fig. 1 shows the evolution of the S&P500 index and indicates the end of the uptick rule period by the dotted vertical line in the left part of the figure. Since its removal, calls to restore the uptick rule have been recurrent. The dates of the statements by different practitioners, authority experts, congressmen and senators for restoring the uptick rule are indicated in Fig. 1 as well. These calls did not remain unanswered and in the fall of 2008 – at the peak of the credit crisis – the SEC temporarily prohibited short-selling in 799 different financial companies. The SEC's chairman, Christopher Cox, argued that: “The emergency order temporarily banning short selling of financial stocks will restore equilibrium to markets.”1 The period for which the short-selling ban was imposed is indicated by two vertical lines in the right part of the figure. Even more stringent policies have been adopted in other countries, see Beber and Pagano (2013) for an overview. It is not clear, however, whether such a ban on short-selling has actually been helpful in stabilizing financial markets. According to Boehmer et al. (2009) the price for the banned stocks sharply increased when the ban was announced, but gradually decreased during the ban period. The whole S&P500 index continued to fall during the short-sell ban as well as afterwards, see Fig. 1. Full-size image (35 K) Fig. 1. S&P 500 and the uptick rule. The end of the “uptick rule”, the period between 19 September 2008 and 2 October 2008, when the short-sales ban for 799 financial stocks was in place, and the major calls for reinstatement of the rule are shown. Figure options The traditional academic view on constraints on short-selling is that they may lead to overpricing of the asset.2Miller (1977), for example, argues that the equilibrium price between demand and supply for a risky asset reflects an average view among heterogeneous investors about the asset's value. The investors with the most pessimistic view on the future price of the asset may sell the asset short at the equilibrium price. Therefore, the constraints on short-selling effectively restrict the supply of shares, leading to a higher equilibrium price level than would emerge in the absence of constraints. In the more sophisticated, dynamic model of Harrison and Kreps (1978) risk-neutral investors have different expectations about the dividends of a certain asset and perfect foresight about beliefs of the other investors. In the absence of short-selling constraints, investors with different opinions take infinitely large, opposite positions. When the constraints are imposed, the price reflects the beliefs of the most optimistic investors, and due to speculative motives the actual price may even be higher. However, since these first contributions other models have been developed that predict no mispricing or even underpricing as a consequence of short-selling constraints. Diamond and Verrecchia (1987) argue that since the constraints are common knowledge, financial market participants should take them into account both in their behavior as well as in their beliefs about the behavior of the other market participants. In their model of asymmetric information (based on Glosten and Milgrom, 1985) short-selling prevents some investors from desired trading. Even if not all private information is fully incorporated into the order flow, the fully rational and risk-neutral market-maker will take the existence of short-selling constraints into account, and will set bid and ask prices at the correct level. Bai et al. (2006) show that this result might change when rational traders are risk-averse. In this case uninformed traders will ask a premium for their higher perceived risk (because the short-selling constraints slow down price recovery), which leads to lower prices. But the model of Bai et al. (2006) may also result in the opposite prediction, as a consequence of smaller supply. Similarly, in a general equilibrium economy considered by Gallmeyer and Hollifield (2008) the short-sell constraints can lead either to overpricing or to underpricing depending on the intertemporal elasticity of substitution of the optimists. Notice that many of these results are obtained by assuming that investors are unboundedly rational. Laboratory experiments with paid human subjects show that short-selling constraints may lead to considerable mispricing, with the important reservation that relaxing the constraints reduces mispricing, but does not eliminate it completely, see, e.g., Haruvy and Noussair (2006). Empirical research on short-sell bans tends to support the view that banned securities are overpriced. Jones and Lamont (2002) study data on the costs of short-selling between 1926 and 1933 and find that those assets which were expensive to sell short subsequently earned lower returns. Similarly, Chang et al. (2007) examine the effect of revisions in the list of securities which cannot be sold short at the Hong Kong stock exchange. They find that inclusion of a stock to the list leads to an abnormal negative return, while exclusion from the list is associated with an abnormal positive return. These results imply that a short-selling ban leads to overpricing. Lamont and Thaler (2003) discuss 3Com/Palm and other examples of clear mispricing, where an arbitrage opportunity is obviously present. They attribute the failure to correct mispricing to the high cost of selling short. On the other hand, recent analysis of Beber and Pagano (2013) shows that, with the exception of the US, there is no evidence of overpricing of the banned stocks during the recent wave of short-selling constraints. They compute the cumulative abnormal return (with respect to the market) after the day the ban is introduced for the stocks in the countries where the ban was imposed. Comparison of the cumulative returns of the stocks subject to a ban with the remaining stocks shows that the effect of ban on the stock price was positive during the first 30 days after it was introduced, but changed sign afterwards, even if the restrictions were not yet relaxed. These results indicate that the ban leads to a price increase for the banned stocks but only in the short run (which is also consistent with the US data). Most of the models discussed above are static in nature and assume that investors are fully rational. This assumption of full rationality has been challenged on theoretical as well as empirical grounds. Theoretically, it can be argued that to actually compute rational beliefs, agents would need to know the precise structure and laws of motion for the economy, even though this structure depends on other agents' beliefs, see, e.g., Evans and Honkapohja (2001). Empirically, some important market regularities, such as recurrent periods of speculative bubbles and crashes, fat tails of the return distribution, excess volatility, long memory and volatility clustering are difficult to explain with models with fully rational investors. Moreover, there is an abundance of experimental evidence that suggests that theoretical models with fully rational agents do not even provide accurate descriptions of the behavior of relatively simple laboratory markets (see, e.g., Smith et al., 1988, Lei et al., 2001, Hommes et al., 2005b and Anufriev and Hommes, 2012). An alternative approach is to consider models of behavioral finance (see, e.g., Shleifer, 2000 and Barberis and Thaler, 2003 for reviews) or, closely related, heterogeneous agents models (HAMs, see Hommes, 2006 and LeBaron, 2006 for reviews). In HAMs, for example, traders choose between different heuristics or rules of thumb when making an investment decision. Typically, heuristics that turned out to be more successful in the (recent) past will be used by more traders. Such models are also successful empirically (by reproducing many of the empirical regularities discussed above, see Lux, 2009), and therefore become an increasingly accepted alternative to the traditional models with a fully rational, representative agent. In this paper we investigate the impact of introducing short-selling constraints in such a heterogeneous agents model. We take the well-known and widely used asset pricing model with heterogeneous beliefs from Brock and Hommes (1998) as our benchmark model. Traders in this model have to decide every period how much to buy or sell of an inelastically supplied risky asset, and they base their decision on one of a number of behavioral prediction strategies (e.g., a fundamentalist or a trend following / chartist prediction strategy). As new data become available agents not only update their forecasts but they also switch from one prediction strategy to another depending on the past performance of those strategies. Such a low-dimensional heterogeneous agents model is able to generate the type of dynamics typically observed in financial markets, in particular when traders are sensitive to differences in profitability between prediction strategies. We investigate the impact of imposing short-selling constraints in this framework, by analytical as well as numerical methods. Specifically we assume that traders need to pay additional ‘trading costs' when they take a short position. We find that the imposition of these costs for short-selling does not affect the local stability properties of the fundamental steady state, that is, the financial market is neither stabilized nor destabilized due to these costs. However, if the price dynamics are volatile to begin with, the costs for short-selling affect the global dynamics and may lead to even more volatile price dynamics. The intuition for this result is that the implicit constraint on short-selling reduces the potential of the financial market to quickly correct mispricing. Limited liquidity in the market in each time period contributes to this effect. Furthermore, the temporary mispricing gets reinforced by the population dynamics. A recent, independent study by Dercole and Radi (2012) gives results that are qualitatively similar to ours. They also study the effect of imposing short-selling constraints in the Brock and Hommes (1998) framework, but their model differs from ours in two respects. First, their analysis is restricted to the case of a full ban on short-selling, whereas our model allows for a wide range of intermediate settings, with the full ban as a limiting case. Second, the ban in their model is imposed only in periods in which the asset price decreases. They therefore focus on the effects the up-tick rule, mentioned above, has on price stability, whereas we are interested in the more general effects of trading costs of short-selling on market dynamics. The rest of the paper is organized as follows. In the next section we extend the Brock–Hommes model to the case of positive outside supply and costly short-selling. The dynamics of the model for the typical and familiar case of fundamentalists versus chartists are studied in Section 3. Section 4 concludes the paper.

In this paper we have analyzed the quantitative consequences of imposing short-selling constraints for asset-pricing dynamics in a model with heterogeneous beliefs. Most of the existing literature points out that short-selling restrictions may lead to systematic overvaluation of the security. The intuition for that was provided by Miller, 1977 who shows, in a two-period setting, that a diversity of expectations among investors leads to overpricing. Our model formalizes this intuition and extends it to a dynamic setting. In our model the demand of myopic investors depends on their expectations of the future price. Expectations are heterogeneous and agents are allowed to switch between different forecasting rules over time. As is well known, the dynamical behavior of asset prices in such a model is qualitatively for higher values of the intensity of choice parameter ββ. For low values of this parameter prices converge to their fundamental steady state. For high values of the intensity of choice the model may exhibit price oscillations with excess volatility. We introduce short-selling constraints in this environment by imposing trading costs for selling the risky asset short. Since trading costs do not have to be paid at the fundamental steady state, where all traders hold a positive amount of the risky asset, existence and local stability of that steady state are unaffected by these costs. However, typically non-fundamental steady states also exist in this environment, and their existence and local stability depends crucially on trading costs. In particular, when there are trading costs for short-selling, these non-fundamental steady states emerge for a wider range of parameters of the underlying model, they may correspond to a much larger degree of mispricing – in particular when the asset is overvalued – and they lose stability for lower values of the intensity of choice parameter. Introducing trading costs for short-selling may therefore very well increase mispricing and price volatility, a feature which is quite robust and confirmed by our numerical simulations. To study the effect of short-selling constraints, we have deliberately chosen a model with heterogeneous expectations, capable of generating the patterns of bubbles and crashes that financial authorities aim to prevent or mitigate by restricting short-selling.13 Some features of this model may influence our findings. Consequently, in future research we would like to extend the model in several directions. First, the myopic agents of the model do not take the short-sell constraints into account while forming their expectations. While we believe that such an assumption is quite reasonable in the framework of boundedly rational agents, it would be also interesting to analyze the model with some fraction of rational agents, who do take the short-sell constraints into account. Alternatively, one might look at a larger set of belief types and investigate whether the introduction of trading costs changes the (steady state) distribution of the population of traders over these belief types. Second, the constraints which we analyzed are individual, while on the real markets there are many aggregate constraints. For example, the total amount of shares available for short sales is, in reality, limited. The effect of that type of constraint can be analyzed in a large scale agent-based version of the current model. Finally, the effect of short-selling constraints is closely related to the role of margin requirements. Indeed, in a real market selling a share short requires providing some collateral to the broker. If the price of an asset rises, the investor who is short should cover his nominal losses to an extent which depends on the margin requirement. It is not surprising that among the most important questions discussed in the literature on margin requirements is their role in market volatility and the prevention of bubbles. Two opposing points of views can be found in the literature. On the one hand, Seguin and Jarrell (1993) and Hsieh and Miller (1990) argue that margin requirements are empirically irrelevant for price behavior, whereas, e.g., Garbade (1982) and Hardouvelis and Theodossiou (2002) provide theoretical arguments why an increase in margin requirements is beneficial for market stability. Again, with an agent-based extension of the model presented here we plan to analyze the joint effect of short-selling constraints and margin requirements on financial market dynamics and price volatility.