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Does the “uptick rule” stabilize the stock market?

Insights from Adaptive Rational Equilibrium

Dynamics

F. Dercole∗*and D. Radi†**

*Department of Electronics and Information, Politecnico of Milan, Italy

**Department of Management, Economics and Quantitative Methods, University of Bergamo

Abstract

This paper investigates the effects of the “uptick rule” (a short selling regulation

1Introduction

Short selling is the practice of selling financial instruments that have been borrowed, typi-

cally from a broker-dealer or an institutional investor, with the intent to buy the same class

of financial instruments in a future period and return them back at the maturity of the loan.

By short selling, investors open a so-called “short position”, that is technically equiv-

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asset will rise in value.

Short selling is considered the father of the modern derivatives and, as such, it has a

doublefunction: itcanbeusedasaninsurancedevice, byhedgingtheriskoflongpositions

in related stocks thus allowing risky financial operations, or for speculative purposes, to

profit from an expected downward price movement. Moreover, financial speculators can

sell short stocks in an effort to drive down the related price by creating an imbalance of

sell-side interest, the so called “bear raid” action. This feedback may lead to the market

collapse, and has indeed been observed during the financial crises of 1937 and 2007, see,

e.g. [Misra et al. 2012].

Many national authorities have developed different kinds of short selling restrictions

to avoid the negative effect of this financial practice1. Most of the regulations are based on

“price tests”, i.e. short selling is allowed or restricted depending on some tests based on

recent price movements. The best known and most widely applied of such regulations is

the so-called “uptick rule”, or rule 10a-1, imposed in 1938 by the U.S. Securities and Ex-

change Commission2(hereafter SEC) to protect investors and was in force until 2007. This

rule regulated short selling into all U.S. stock markets and in the Toronto Stock Exchange.

Other financial markets, like the London Stock Exchange and the Tokyo Stock Exchange,

have different or no short selling restrictions (for a summary of short sale regulations in

approximately 50 different countries see [Bris et al. 2007]).

The uptick rule originally stated that short sales are allowed only on an uptick, i.e. at

1It is worth mentioning that short sale restrictions are nearly as old as organized exchanges. The first

short selling regulation was enacted in 1610 in the Amsterdam stock exchange. For a review of the history

of short sale restrictions, see Short History of the Bear, Edward Chancellor, October 29, 2001, copyright

David W. Tice and Co.

2The rule was originally introduced under the Securities Exchange Act of 1934.

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a price higher than the last reported transaction price. The rule was later relaxed to allow

short sales to take place on a zero-plus-tick as well, i.e. at a price that is equal to the last

sale price but only if the most recent price movement has been positive. Conversely, short

sales are not permitted on minus- or zero-minus-ticks, subject to narrow exceptions3.

In adopting the uptick rule, the SEC sought to achieve three objectives4:

(i) allowing relatively unrestricted short selling in an advancing market;

(ii) preventing short selling at successively lower prices, thus eliminating short selling

as a tool for driving the market down; and

(iii) preventing short sellers from accelerating a declining market by exhausting all re-

maining bids at one price level, causing successively lower prices to be established

by long sellers.

The last two objectives have been partially confirmed by the empirical analysis (see,

e.g. [Alexander and Peterson, 1999] and reference therein). Instead, the regulation does

not seem to be effective in producing the first desired effect. The observed number of

executed short sales is indeed lower under uptick rule than in the unconstrained case,

during phases with an upward market trend, see again [Alexander and Peterson, 1999].

This is due to the asynchrony between placement and execution of a short-sell order, since

the rising of the price in between these two operations can make the trade not feasible

under the uptick rule.

3In the Canadian stock markets, the tick test was introduced under rule 3.1 of UMIR (Universal Market

Integrity Rules). It prevents short sales at a price that is less than the last sale price of the security.

4Quoted from the Securities Exchange Act Release No. 13091 (December 21, 1976), 41 FR 56530 (1976

Release).

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Moreover, empirical evidence provides uniform support of the idea that short selling

restrictions often cause share prices to rise. From a theoretical point of view, there is

no clear argument for explaining this mispricing effect of the uptick rule. According to

[Miller, 1977] this is due to a reduction in stock supply owing to the short sale restriction.

More generally, theoretical models with heterogeneous agents and differences in trading

strategies support the idea that share values become overvalued under short selling re-

strictions due to the fact that “pessimistic” and “bear” traders (expecting negative price

movements) are ruled out of the market (see, e.g. [Harrison and Kreps, 1978]). In con-

trast, theoretical models based on the assumption that all agents have rational expectations

suggest that short selling restrictions do not change on the average the stock prices (see,

e.g. [Diamond and Verrecchia, 1987]).

However, given the complexity of the phenomena, and the impossibility of isolating

the effects of a regulation from other concomitant changes in the economic scenario, the

effectiveness of the uptick rule in meeting the three above objectives, and its possible side

effects on shares’ prices, are still far from being completely clarified.

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adaptability based on past performances, have shown to endogenously sustain price fluc-

tuations. Asset pricing models of A.R.E.D. (hereafter referred to simply as ARED asset

pricing models) are discrete-time dynamical systems based on the empirical evidence that

investors with different trading strategies coexist in the financial market (see, e.g. [Taylor

and Allen, 1992]). These simple models provide a theoretical justification for many “styl-

ized facts” observed in the real financial time series, such as, financial bubbles and volatil-

ity clustering (see [Gaunersdorfer, 2001], and [Gaunersdorfer et al. 2008]). Stochastic

models based on the same assumptions are even used to study exchange rate volatility and

the implication of some specific financial policies (see, e.g. [Westerhoff, 2001]).

We extend, in particular, the deterministic model introduced in [Brock and Hommes,

1998], where, in the simplest case, agents choose between two predictors of future prices

of a risky asset, i.e. a fundamental predictor and a non-fundamental predictor. Agents

that adopt the fundamental predictor are called fundametalists, while agents that adopt the

non-fundamental predictor are called noise traders or non-fundamental traders. Funda-

mentalists believe that the price of a financial asset is determined by its fundamental value

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fundamental traders predict the fundamental price. As long as the sensitivity of traders in

switching to the best performing predictor is relatively low, the fundamental equilibrium

is stable, but the fundamental stability is typically lost at higher intensities of the traders’

choice across the predictors, making room for financial bubbles.

It is worth to remember that [Brock and Hommes, 1998] investigated the peculiar case

of zero supply of outside shares. Under this assumption each bought share is sold short.

Wethereforeconsiderapositivesupplyofoutsideshare, thatisessentialtoensurefinancial

transactions when short selling is forbidden5. Moreover, we pair the fundamental predictor

with first a technical linear predictor and then with a technical nonlinear predictor and

compare the results obtained with and without the uptick rule.

Aslinearpredictor, weconsiderthechartistpredictorintroducedin[BrockandHommes,

1998]. This facilitates the comparison of our results with those in [Brock and Hommes,

1998]andrelatedpapers. Asnonlinearpredictor, weintroduceanewpredictor, Smoothed

Price Rate Of Change or S-ROC predictor, that extrapolates future prices by applying the

rate of change averaged on past prices with a confidence mechanism smoothing out ex-

treme unrealistic rates (for an overview of this class of predictors, see [Elder, 1993]).

For what concerns the implementation of the regulation, we implement the uptick rule

as it was in its original formulation, i.e. short selling is allowed only on an uptick. Note,

however, that in an artificial asset pricing model a zero-tick is possible only at equilibrium,

so that allowing or forbidding short sales on zero-plus-ticks makes basically no change in

the observed price dynamics. In fact, with a positive supply of shares, traders take long

5We consider a positive supply of outside shares for the asset pricing model under Walrasian market

clearing at each period. A similar model under the market maker scenario has been considered by [Hommes

et al. 2005].

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positions at the fundamental equilibrium, so only the non-fundamental equilibria at which

one type of trader is prohibited to go short are affected by the rule behavior on zero-plus-

ticks (moreover, such equilibria are irrelevant to study the global price dynamics, as will

be explained in Section 2.3).

From the mathematical point of view, the uptick rule makes the asset pricing model a

piecewise-smooth dynamical system6, namely a system in which different mathematical

rules can be applied to determine the next price, and the rule to be applied depends on

the current state of the system, that is, on the fact that trader types are interested in going

short and whether short selling is allowed or not. Non-smooth dynamical systems are

certainly more problematic to analyze, both analytically and numerically (though non-

smooth dynamics is a very active topic in current research, see [di Bernardo et al. 2008],

and[Colomboetal.,2012], andreferencestherein)sowewilllimittheanalyticaltreatment

to stationary solutions.

Piecewise-smooth dynamical systems have already been used as models in finance.

[Tramontanaetal.,2011]proposedaone-dimensionalpiecewise-linearassetpricingmodel,

where traders adopt different buying and selling strategies in response to different market

movements. Other examples can be found in [Tramontana et al. 2010a], [Tramontana

et al. 2010b], and [Tramontana and Westerhoff, 2013]. Two ARED piecewise-smooth

systems modeling short selling restrictions have been also proposed. Modifying the model

in [Brock and Hommes, 1998], [Anufriev and Tuinstra, 2009] restricted short selling by

allowing limited short positions at each trading period, whereas [Dercole and Cecchetto,

2010] investigated the complete ban on short selling. Thus both contributions implement

6To be precise, the model is a piecewise-continuous dynamical system. However, the class of piecewise-

smooth dynamical systems contains the class of piecewise-continuous dynamical systems.

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short selling restrictions that are not based on price tests.

The results of our theoretical analyses are in line with the empirical evidence. The sale

price that is established in our model when one trader type is prohibited from going short

is indeed systematically higher than the unconstrained price. Thus, constrained downward

movements below the fundamental value are less pronounced, whereas constrained up-

ward movements above the fundamental value can be larger. We provide a more complete

explanation for this effect, suggesting that it is due to the combination of two mecha-

nisms: on one side, the short selling restriction reduces the possibility for pessimistic or

bear traders to bet on downward movements below the fundamental value, avoiding ex-

cessive underpricing, but at the same time, when prices are above the fundamental value,

the restriction reduces the possibility for fundamentalists to drive down the prices back to

the fundamental value by opening short positions. This is in agreement with the last two

goals established by the SEC (see above). The first stated objective of the uptick is always

realized in our model, since the market clearing is assumed to be synchronous among all

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set pricing model, summarizing from [Brock and Hommes, 1998] and setting the notation

and most of the modeling equations that will be used in next Sections. Section 2.2 is

also preliminary and recaps the concept of fundamental equilibrium, including its stability

analysis and some new results. Section 2.3 formulates the piecewise-smooth model con-

strained by the uptick rule, and discusses the existence and stability of fundamental and

non-fundamental equilibria. So far, no explicit price predictors is introduced, whereas Sec-

tion 2.4 presents the price predictors for which the unconstrained and constrained models

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a constraint on it can be imposed. In this generalized version of the original model, we

introduce a negative demand constraint according to the uptick rule, in order to study the

effects of this regulation on price fluctuations.

2.1The unconstrained ARED asset pricing model

Consider a financial market where traders invest either in a single risky asset, supplied in

S shares7of (ex-dividend) price ptat period t, or in a risk free asset perfectly elastically

supplied at gross return R (where R = 1 + r, with r ∈ (0,1)). The risky asset pays

random dividend ˜ ytin period t, where the divided process ˜ ytis IID (Identically Indepen-

dently Distributed) with Et[˜ yt+1] = ¯ y constant. Thus, denoting by Wh,tthe economic

wealth of a generic trader of type h at the beginning of period t, and by zh,tthe number of

shares held by the trader in period t, we have the following wealth equation (or individual

intertemporal budget constraint):

˜ Wh,t+1= R(Wh,t−ptzh,t)+ ˜ pt+1zh,t+ ˜ yt+1zh,t= RWh,t+(˜ pt+1+ ˜ yt+1−Rpt)zh,t, (1)

where tilde denotes random variables, Wh,t− ptzh,tis the amount of money invested in

the risk free asset in period t and˜Rt+1= ˜ pt+1+ ˜ yt+1− Rptis the excess return per share

realized at the end of the period.

Let Eh,t,Vh,tdenote the beliefs of investor of type h about the conditional expecta-

7S is in fact the supply of traded assets in each period. Obviously when short selling is allowed assets

are borrowed outside the pool of this S shares making the total supply higher than S.