J. ltal. Statist. Soc. (1999)I, pp. 25-47

A NEW APPROACH TO STOCK PRICE MODELLINGAND THE EFFICIENCY OF THEITALIAN STOCK EXCHANGE

Attilio Gardini, Giuseppe Cavaliere*, Michele CostaDepartment ofStatistical Sciences, University ofBologna, Italy

Summary

In this paper, we propose a new model of asset prices which takes account of the invest­ment strategies of three different kinds of agents: the market-makers, who operate rational­lyon the basis of the asset fundamentals, the smart buy-and-sell agents, who intervenewhen the prices reach particular levels and the non-smart buy-and-sell agents, who tradeinfrequently, mainly following psychological motivations. The different behavior of thesegroups of agents can determine temporary inefficiences on financial markets and we showthat, by considering these inefficiences, it is possible to improve forecasting of asset prices.

Keywords: Stock price dynamics, price barriers, market efficiency.

1. Introduction

One of the main references of the theory of stock market efficiency (Fama, 1970;1991) and of the quantitative analyses of stock price dynamics is the unpredicta­bility of prices, which are generally described by means of a random walk withdrift.

In our work, we start from the observation that in some cases markets canexhibit heterogeneous dynamics which can be used in order to predict future

* Addressfor correspondence: Dipartimento di Scienze Statistiche, Universita degli Studidi Bologna, Via delle Belle Arti, 41- 1-40126, Bologna, Italy. Tel. + 390512098230, fax+39051 232153, e-mail: cavalier@stat.unibo.it

This paper was presented at the XXXIX Scientific Meeting of The Italian StatisticalSociety held in Sorrento, April 1998.

We are indebted to seminar partecipants at the 9th (ECi conference «Forecasting inEconometrics», Stockholm, December 1998, and the 52nd session of the InternationalStatistical Institute, Helsinki, August 1999. We thank Bruno Sitzia and the partecipants atthe C.I.D.E. seminar, Milan, June 1999. For official purposes only we specify that section1 is by A. Gardini, sections 2 and 4 are by G. Cavaliere, sections 3, 5 and 6 are by M.Costa.

25

A. GARDINI . G. CAVALIERE· M. COSTA

values of asset prices. Such heterogeneities can be identified in the presence ofstable periods, where prices move approximately between two price barriers ­which define a price band - and periods of strong trends in prices, which movefrom one band to a new one. We call these last phases transitions.

We motivate the existence of price bands and transitions through the presenceof heterogeneities and inefficiencies in the stock market, caused by smart andnon-smart speculation, the former being based on a rational (fully informed)knowledge of the real asset value, while the latter is due to incomplete informa­tion'. Both speculations are assumed to take place infrequently, i.e. smart andnon-smart speculators do not trade continuously.

When the asset's implicit value (i.e. the expected discounted flow of futuredividends) is stable, smart speculators adopt a buy low/sell high strategy, whichstarts when the price reaches a price barrier, thus determining a stabilizing effectwhose main implication is the presence of a price band. Such a band can providesignificant information about the dynamics of the price.

On the other side, when changes in the implicit values occur, smart specula­tors can turn their investment strategy into a sequential buy-and-hold (sell-and­wait) at the upper (lower) price barrier. During this phase, the asset price movesrapidily toward its implicit value, therefore determining the transition.

The transition periods can also be characterized by the presence of non-smartspeculators, who accelerate the velocity and the size of the transition. Theirinvestment causes an overshooting of the price, which reaches unrealistic lev­els.

Within this theoretical framework, the problem of predicting stock prices canbe turned into the problem of predicting price band changes. Futhermore, ifpricebands are detected, then the inclusion of the price barriers within the informationset can provide a new tool for price forecasting.

Although the idea that financial asset prices move within specific intervals isan important point of reference for other theoretical (Balduzzi et al., 1997; Berto­la and Caballero, 1992a) and empirical studies (Donaldson and Kim, 1993; Leyand Varian, 1994; De Ceuster et al., 1997) and represents one of the main sup­ports to technical analysis (see, e.g., Corielli, 1991; Gencay, 1998; Gallo andPacini, 1998), the idea of incorporating price barriers in the prediction of futureprices is still at a very preliminary stage.

The price barrier hypothesis has great similarities with the literature relativeto exchange rate modelling in the presence of a target zone (Krugman, 1991;Froot and Obstfeld, 1991; Delgado and Dumas, 1992; Ball and Roma, 1993,

1. Therefore, our work has been strongly inspired by the literature about asymmetricinformation and heterogeneous expectation (see, e.g. Brunnermeier, 1998; Calcagno andLovo, 1998; Dow and Rahi, 1998; Kurz and Motolese, 1999).

26

STOCK PRICE MODELLING

1994). In this context, international agreements require the exchange rate to liewithin a declared target zone by means of Central Banks interventions. As longas the target zone is realistic - in terms of the distance between the regulatedexchange rate and its theoretical market value' - intervention policy is sustain­able and the band can be controlled with a limited amount of reserves. On theother side, when the target zone is not realistic any longer, its defence (whichforces the exchange rate system into disequilibrium) cannot be feasible, andrealignments of central parity are usually necessary'. In the framework of stockprice modelling, smart speculation can be interpreted as the source of interven­tion in the financial market, which causes the price to have reduced fluctuationsinside two «flexible» price levels; moreover, when the price band is unrealistic,smart speculation can push the stock price to move rapidly to a new equilibriumlevel. For these similarities, the statistical tools for target zone modelling, whichare developed in a continuous time setting", playa lead role in our stock pricemodel.

The paper is organized as follows. In the next paragraph we formalize a styl­ized model for stock prices which allows for the presence of heterogeneous agentsand price bands; in paragraph 3 we tum our attention to the Italian Stock Market,detecting, on the basis of a descriptive analysis, the presence of stable price peri­ods and of transitions. Then, in paragraph 4 we analyze the problem ofestimatingthe parameters which drive the stock price dynamics. In paragraph 5 it is shownhow to nest price barrier information into the forecasting process, and the pro­posed methodologies are applied to the Italian Stock Exchange index. Some con­cluding remarks are reported in paragraph 6.

2. A model for stock prices

In this paragraph we formalize a model for stock price determination, which isbased on the investment strategies of three different classes of economic agents:the market makers, the smart speculators and the non-smart agents. Market mak­ers are fully informed agents which trade in continuous-time in order to maxi­mize their consumption utility; on the other side, smart and non-smart agents

2. In theoretical works, the implied exchange rate value is usually determined by com­bining the Quantity Theory of Money and the Relative Purchasing Power Parity.3. See, e.g., Bertola and Caballero (1992a, 1992b), Bertola and Svensson (1993), Bertola(1994); recent developments in target zone exchange rate modelling in the presence ofstochastic realignments can be found in Christiensen et at. (1997).4. See Svensson (1992) for a survey of target zone literature. Bertola (1994) provides anexcellent review of continuous-time exchange rate modelling.

27

A. GARDINI • G. CAVALIERE' M. COSTA

trade infrequently. However, smart agents are assumed to be informed and ableto assess the stock value.

We start by assuming that between times t and t + dt, the stock pays anexogeneous dividendf,·dt. Then, we assume that there is a market maker, whopurchases or sells the asset in order to determine its optimal consumption flowc('), which is achieved by maximizing the value function

where p is a discount factor and E, (-) denotes expectation conditional on theinformation available up to time t.

Under the hypothesis that the market maker is risk neutral and that he can onlyhold stocks and bonds, it can be shown (see Balduzzi et al., 1997) that the stockprice must satisfy the relation

Therefore, price determination is related to the dividend process and to the ex­pected instant variation of the price. An equivalent reparametrization of this equa­tion is given by

(1)

where a = IIp andf, = j,*lp is the dividend, adjusted for the discount factor p.By assuming that prices satisfy the trasversality condition

1· E (-p(T-r) ) =0im t e PTT--7~

which rules out speculative bubbles and the related arbitrage opportunities,the model can be solved by adding more specialized assumptions on thedividend dynamics. While in general it is enough to require, for instance,that in continuous time dividends follow a Lipschitz-continuous' diffusionprocess, we go further by assuming that dividends evolve as a Brownianmotion with drift

dj,* = pudBr + pud:

28

(2)

STOCK PRICE MODELLING

where B is a standard Brownian motion, pa is the instantaneous volatility andPJ.L is the instantaneous drift. In the following, in order to simplify the presentationof the results we impose, without loss of generality, the restriction J.L = O. In thiscase, it is easy to show that the solution to equation (1) is

p, =1" (3)

which states that the observable price process follows a Brownian motion andthat the relation P if) between the adjusted dividend flow and the asset price issimply P if) = f

We finally note that in this framework the price Ptis equal to the implicitstockvalue Pt', defined as the expected sum of future dividends discounted at the ratep:

(4)

where the integral and the expectation operators can be exchanged for the Lip­schitz continuity of the fundamental process. In other words, by defining theprice spread

.s, = P,-P,

as the difference between the market price and the implicit value, the previousassumptions ensure that s, = 0, all t.

The second class of operators, whom we call the smart agents, trade infre­quently in the market. We do not model the utility maximization problems ofthese agents, whose asset demand is here assumed to be exogenouslydetermined and «taken» by the market maker; instead, we assume that theiroptimal strategy is to enter the market when prices reach particular levels, orbarriers, determined on the basis of the stock price historical maxima andminima.

Suppose that the smart agent follows a sell high-buy low strategy, by sellingthe asset when it is close to the upper barrier, set to the highest historical price,and by buying when the asset price is close to the lowest level. By indicating withPt and Ptthe historical minimum and maximum value ofPtin the period [t/» t], i.e.Pt= mint05~ {Pr} and p, = maxto5~ {Pr}, we assume that smart speculation startsonly ifPt= Pt(selling strategy) or Pt = Pt(buying strategy). In a stochastic setting(see Balduzzi et al., 1997), we can assume that if such price levels are reachedthe speculation starts with a time-varying probability JIt - for the smart buying ­and ~ - for the smart selling.

29

A. GARDINI • G. CAVALIERE' M. COSTA

Smart agents' demand inflates or deflates the stock price. Instead ofmodellingthis price component in the price domain, without loss of generality we assumethat the dividend is influenced by the amount dU, in the presence of sellingactivity, and of dL, in the presence of buying activity: dU, and dL, represent thechange of the market value due to speculation. This assumption derives from thefact that assuming barriers for the price process implies the existence of barriersfor the fundamentals: ft $.ft s]., where the midpoint of the band is indicated as c,= (ft +I,) /2, such that p (ft) = p, and p if,) = p;

From the previous assumptions it follows that the stochastic differential equa­tion which rules the fundamental is

df, = adb, + 1J,dL, - it,dU,

which, by integrating from toto time t, implies that

(5)

where Land U can be interpreted as two regulator processes in the sense of Har­rison (1985).

Firstly we consider the case it, = 1J, = 1, so that smart operators intervene withprobability one. The previous equations describe an economic problem whichhas been initially examined by Krugman (1991) and then by several authors (see,e.g., Froot and Obstfeld, 1991). These authors prove that the relation p if) be­tween the adjusted dividend flow and the asset price must satisfy the equation

2

p(f) =f + p"(f) ~

whose general solution has the form

p if) =f + C (e,1.(f-cJ - e-.l.(f-C»)

with

(6)

the constant of integration C = - (l/A) (/I]-cJ e-.l.I]-cJrl is determined by imposingthe conditions

30

STOCK PRICE MODELLING

p (j) = p

p' (j) =°The relation between fundamentals and price is illustrated in Figure 1: smart spec­ulation has a stabilizing effect, as the stock price is lower (when tt = 1) than thevalue which can be determined in the absence of speculation (re =0, dashed line).

Consider, now, the intermediate case, in which tt,and ITt can take any value inthe real interval [0, 1]. By assuming, without loss of generality, that tt,= ITt =x;the following relation holds:

Proposition 1. The relation between the asset price and the adjusted dividendflow is given by

with

./

(8)

'"0'"u.~

.x:U

2'"

'"oI

-1.0 -0.5 0.0

fundamental

0.5 1.0 1.5

Fig. 1 - Relation between fundamentals and stock price in the case of sell highlbuy lowspeculation.

31

A. GARDINI . G. CAVALIERE' M. COSTA

and Ais given in equation (7).

Proof See Appendix.

In Figure I the relation between the asset price and the adjusted expected divi­dend flow for tt =1/2 shows how the presence of speculation causes a stabiliza­tion of the asset price, which is proportional to the probability of buying/sellingat the boundaries.

From normal to transition periods. From the previous results it arises that thepresence of «smart» speculation has a stabilizing effect on the stock price. Inparticular, the price stabilization is caused by two factors; firstly, selling at themaximum and buying at the minimum reduces the excursion of price; secondly,the market maker, by «discounting» the future expected speculation, automati­cally adjusts the current price, thus anticipating future movements due to smartagents.

This is the situation during normal periods, but in the presence of a strongpositive (negative) drift in the fundamental process and when the distance be­tween the market price and the implicit price is high, smart agents do not playastabilizing role and, like any fully informed operator, intervene by buying (sell­ing) the asset instead of selling (buying). The model previously introduced cantake this possibility into account. In fact, we can generalize the model by assum­ing that if the upper price level is reached then the smart speculators sell with atime-varying probability ftit while they buy with probability 1;,. In the same way,when the lower price level is reached, smart speculators buy with probability 7]1'

and sell with probability 7]2"

We can nest this hypothesis in our model by assuming again that speculationmodifies the fundamental process which, at the upper barrier, is lowered (in­creased) by the amount dU, in the presence of selling (buying) activity, andat the lower barrier, is raised (decreased) by the value dl; in the presence ofbuying (selling) activity. In terms of stochastic differential equation, thiscase can be represented through the following representation of the dividendprocess

(9)

which, by integrating from to to time t, implies that

and the processes L and U are the usual regulators of Harrison.

32

STOCK PRICE MODELLING

Let us consider only the case when the upper barrier is hit: the behavior of thedividend process is determined by the sign of the difference (it j , - ftzJ If it isgreater than zero, i.e. itlt > ftz" the selling activity is stronger than the buyingactivity, thus determining a price fall; on the other side, if it j , < it:l" then purchaseswill be strong enough to determine an increase in the price.

In the following, by setting without loss of generality J.1 = 0 and ~t= 7Jj' =1rj ,

j =1, 2, we simply generalize Proposition 1.

Proposition 2. The relation between the asset price and the adjusted dividendflow is given by

with

c= 1r j-1rz)1,(eA(]-C) +e-A(]-C»)

and A is given in equation (7).

'"0QlU.;:a..x:o.s"' '"c:i

I

-1.0 -0.5 0.0

fundamental

0.5 1.0 1.5

Fig. 2 - Relation between fundamentals and stock price with sell highlbuy low specula­tion (ll} < llz) and with buy-and-hold (sell-and-wait) speculation (ll}> Tlz).

33

A. GARDIN! . G. CAVALIERE· M. COSTA

Proof See Appendix.

This result is illustrated in Figure 2. If 1r] > 1r2 we have the previous case ofstabilization, but if 1r] < 1r2 then the price is inflated with respect to its theoreticallevel, thus producing an overshooting effect. In the latter case, the price can becharacterized by a strong growth (decrease), which is partially due to the repeat­ed purchases driven by buying (selling) orders.

Non-smart speculation and price overshooting. There exists another class ofoperators which act in the financial markets and can cause temporary disequi­libria and overshooting of the asset prices with respect to their reaivalues: thenon-smart operators. We can assume that these agents, who generally follow abuy-and-hold strategy, do not buy or sell by following some special rules but,driven by psychological motivations, usually enter the market by purchasing theasset after a strong price growth, and by selling the asset after a strong pricedecrease. This behavior is justified by the presence of asymmetric information:the non-smart operators are not able to evaluate the asset price correctly (throughrelation (4» and, consequently, they select their investment strategies just throughan extrapolation of the short-term price trend. Clearly, this behavior causes pos­itive overshooting in periods of price growth and negative overshooting in peri­ods of price decrease, thus being another source of price volatility.

Our stylized model can be generalized by including this possibility too; inparticular, by assuming that at the upper price barrier non-smart agents purchasewith probability tt31 and, at the lower price barrier, they sell with probability 1]31.

Again, we assume that the implied variations in the fundamentals are dU, anddl.; and consequently the fundamental process (9) can be expressed as

By setting 1;1 =1]jl = ~, j = 1, 2, 3, Proposition 2 can be generalized into thefollowing relation between fundamentals and prices

with

which produces an overshooting near the barrier, which is proportional to 1r]-.le;. - 1r3•

When only smart speculators enter the market there is a stabilizing effect ifthey follow a buy-low/sell-high strategy; on the other side, when they turn their

34

STOCK PRICE MODELLING

strategy to a buy-and-hold (or sell-and-wait) there may be overshooting phenom­ena, which can become even stronger in the presence of non-smart speculation.

3. Price bands and the Italian Stock Exchange

The analysis of stock price dynamics on the Italian stock market shows that twodifferent situations can be found in a number of cases: on one side, there arenormal periods, characterized by bounded variations of prices, on the other side,there are periods of transition, characterized by strong movements of stock prices.The former may correspond to the case of smart speculation with stabilizing ef­fects, the latter may correspond to the smart speculation without stabilizing ef­fect plus the intervention of non-smart agents.

Figure 3 reports the daily values of the COMIT index (in logarithmic form) ofthe Italian stock market' from January 1970 to September 1998. During this pe­riod, it is possible to detect three phases with a strong growth of the stock index- the first between 1980 and 1981, the second between 1985 and 1986, the thirdfrom 1997 until mid-1998.

74 78 82 86 90 94 98

Fig. 3 - Italian Stock Exchange Index (log-form) and frequency histogram.

5. Source: Datastream.

35

A. GARDINI . G. CAVALIERE' M. COSTA

The same property can be observed by examining the right side of the figure,where the frequency histogram of the index is reported: a big percentage of theobservations seem to belong to three clusters, which are formed by the observa­tions which belong to the three periods with limited price movements. Betweenthese periods, low frequencies belonging to the three transition periods can benoticed: in such cases, the stock index grows fast and does not frequently take thesame values. Therefore, the Italian stock index seems to be «switching» betweennormal and transition phases, instead of having a homogeneous behavior overthe whole sample.

As assumed by our model, the hypothesis of the presence of a band whichlimits the dynamics of stock prices has the same meaning as assuming that agents'expectations are influenced by the previous maximum and minimum prices. There­fore the agents' buying and selling strategies and, consequently, supply and de­mand functions, are affected by the existence of particular price «critical val­ues», in this work interpreted as the limits of the band, which - if reached - caninduce agents to expect a reversion of the index trend: this is what we call thenormal phase. By referring to this concept, in this paragraph we analyze the sta­tistical properties of the price bands which can be detected by examining thehistorical values of the price index.

In order to choose the band it is necessary to define the width of the band aswell as a rule for determining its upper and lower edges. Financial literature doesnot suggest any particular solution to these problems (Balduzzi et al., 1997; Leyand Varian, 1994) and as a first attempt we propose to refer to the maxima and theminima of the price index. The first relative maximum determines the upper edgeof the band, while the lower edge is determined by subtracting the width of theband from the upper edge. The band is shifted upward (downward) if a newmaximum is reached (if the lower edge is exceeded). Therefore, from the se­quences of the maxima and the minima of the index it is possible to establish theedges of the band.

A ±40% band (see Figure 4) allows to detect three different normal periods(from April 1970 to August 1980; from June 1981 to May 1985; from May 1986to June 1997), which are separated by two transitions characterized by a fastgrowth of the market index.

Together with this price analysis, which allows to discriminate between trans­tions and normal periods, further useful information can be obtained by examin­ing the index daily returns" (Figure 5): whole sample statistics are reported inTable 1, while Table 2 shows the same statistics computed for the transitions andthe normal periods.

6. We consider the continuously compounded return, i.e. r,= (P, - P,-I) . 100 with P, = In(P,), Returns are not annualized.

36

STOCK PRICE MODELLING

o ll----f~--...,

'"oI

""oI II-__~--'-l!.-\!,

I 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00

Fig. 4 - Italian Stock Exchange Index (log-form) and 40% price band.

coI

'"'I

""I~ ",",",-,~......." ~.Iu.J...~ ~ w..u..~ ..L...1.J .....JI 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98. 00

Fig. 5 - Daily returns of the Italian Stock Market index.

37

A. GARDINI • G. CAVALIERE· M. COSTA

Table IItalian Stock Exchange: descriptive statistics of the daily percentage returns of the

market index, 1970-1998, whole sample

whole sample1.70-9.98

number of observationsmean

asymmetrydaily autocorr.

weekly autocorr.monthly autocorr.

73890.028

-0.4580.1220.0060.008

standard deviationkurtosis

daily ARCHweekly ARCHmonthly ARCH

1.32612.1770.2150.0910.056

Table 2Italian Stock Exchange: descriptive statistics of the daily percentage returns ofthe

market index, 1970-1998

I period I transition II period II transition III period III transition4.70-8.80 8.80-6.81 6.81-5.85 5.85-5.86 5.86-6.97 6.97-4.98

no.ofobs. 2647 204 1015 263 2859 206

mean -0.010 0.428 -0.001 0.435 -0.005 0.354st. dey. 1.238 1.952 1.505 1.404 1.215 1.502

asymmetry 0.305 -0.351 -0.890 -0.089 -1.017 -0.725

kurtosis 12.052 5.958 13.603 6.193 14.390 8.625daily autocorr, 0.076 0.093 0.166 0.070 0.154 -0.061

weekly autocorr; 0.019 -0.071 0.001 -0.048 0.000 -0.003monthly autocorr, 0.095 0.237 0.042 ·0.095 -0.038 -0.060

daily ARCH 0.281 0.070 0.183 0.Q75 0.180 0.431weekly ARCH 0.030 0.112 0.142 0.036 0.069 -0.057

monthly ARCH 0.037 0.198 0.094 0.167 0.030 -0.073

It can be immediately noticed the presence of three long normal periods withzero average return, while the three transitions are characterized by a high aver­age return (more than 0.4 during the first and the second transition, and above 0.3during the last transition). High values of the average return strictly depend onthe short length of the transition periods: around 11 months for the first transi­tion, 12 for the second, 10 for the third. When the first and the last transitions areentered, an increment of the volatility is observed, while it is not observed when

38

STOCK PRICE MODELLING

the second transition starts. By referring to the distribution shape, normal periodsseem to becharacterized by a higher kurtosis; a significant daily autocorrelationis found during two normal phases but not during transitions. Finally, the analy­sis of the autocorrelation structure of the squared returns, i.e. the analysis of ARCHeffects, does not detect any particular difference between normal periods andtransitions. Thus, the high average return with respect to the average return dur­ing normal periods seems to be the most characterizing elements of transitionperiods.

From this descriptive analysis, a normal/transition «switching» representationfor index dynamics seems to hold for the Italian index? Now, by considering thenormal periods, when it is possible to assume that prices are influenced by targetbands, we evaluate how their inclusion affects the inferential procedures.

4. Modelling issues in the presence of price bands

Traditionally, the unpredictability of prices and returns in financial markets isrelated to Fama's (1970, 1991) market efficiency hypothesis, which, under someparticular conditions (see Campbell et al., 1997), can be represented through theunit root process for the log-prices P,

P, = J1 + Pt-J + E, (10)

where J1 and e, are the expected price variation and a zero-mean error term re­spectively. When e, is independent and identically distributed, equation (10) iden­tifies a random walk with drift. In the literature, the market efficiency hypothesisis analyzed by referring to a number of different methodologies, which focus onthe evaluation of the existence of autocorrelation in stock returns and time-vary­ing first and second moments.

In our context, however, the analysis of the previous paragraph suggests thatthe investigation of stock prices predictability must be performed by referring toa statistical model which includes a band for stock prices among all Sources ofvariation.

The model described in section 2 identifies a very general model, characterizedby a stochastic structure for the dividend process (whose parameters are theexpected variation J1 and the instantaneous volatility 0"), a discount factor p = I/o;

7. Researchers are paying an increasing attention to the presence of switching dynamicsin the conditional mean of asset prices, both in theoretical and empirical studies; seeSchaller and Van Norden (1997), Timmermann (1998), inter alia.

39

A. GARDIN! • G. CAVALIERE' M. COSTA

and the stochastic processes {gI" g 2" g 3" ttl" tt2t, tt3t}, which determine thespeculative behavior of the smart and non-smart operators. In the absence oftransitions the model turns to the special case described in Proposition 1, and thefollowing restrictions hold: g2t = g3t = 1Czt = tt3t = 0 while ttlt and 'Jlt are non­negative processes. In order to have a flexible specification, in the following weimpose the coefficient a to be equal to zero, so we have exact equality betweenfundamentals and the stock price. This allows to describe the price dynamicsthrough the stochastic differential equation (5).

As, at time t, the position of the upper and the lower price barriers are assumedto be known, we can avoid the estimation of the probability of band realignmentsttlt and g It' thus concentrating on the price position within the price band. Inrelation to the stochastic differential equation (5), the price generation process is- in continuous time - a Brownian motion with drift J1 and instantaneous vari­ance o', reflected between the upper and the lower price barriers (Karatzas andShreve, 1988; Cox and Miller, 1965), i.e. Ptand Pt.

Given that such barriers are known, the reflected Brownian motion is com­pletely characterized by its instantaneous drift J1 and variance ~. Anyway, thecorresponding transition density function in the presence of a time-varying bandis not known. However, by assuming that the price barriers are invariant" on thereal set (t -1, t], i.e. Pt-8 = Ptand Pr-8 = p,for all 8 E [0, 1), the transition densityof the process from time t - 1 to time t Is equal to the transition density functionof a reflected Brownian motion, reflected within a fixed band [Pt, Pt], which isgiven by (Svensson, 1991; de long, 1994) -

(11)

where

at =(Pt- Pt) / 1T:

h = 2J1/(j2

Ak =o' (k%; + h2/4) / 2

gk(x) =2k cos (k (x - Pt) / at) + h . at sin (k (x - Pt) / at)- -

8. This assumption involves an approximation error which goes to zero as the distancebetween two subsequent observations decreases.

40

STOCK PRICE MODELLING

with e'=(p, (12). By conditioning on the first observation Po, the (conditional)likelihood function of the model can be expressed in the usual way as L (e) =I1~=J(P,lp'-I; e) which can be maximized numerically by following the proce­

dures proposed by de Jong (1994) and Ball and Roma (1994).

5. Stock price forecasting within the band

Given the bands obtained in paragraph 3 and given the statistical model for stockprices specified in the previous paragraph. it is now possible to analyze the pre­dictability of stock prices.

In this paragraph, the predictive power of the statistical model with bands iscompared to the results which can be obtained by means of the random walk(I 0), which plays the role of a benchmark model", More specifically, in the fol­lowing the prediction abilities of the two models are evaluated by comparing 1­day, I-week, I,3,6-month and I year forecasts.

For the random walk (10), forecasts of the price at time t + Tare computed (i)by estimating the drift parameter pon the first t observations and (ii) by referringto the expression:

P'H = t; [P'H] = p, + f1 T.

For the price band model, forecasts are obtained (i) by estimating the parametervector eon the first t observations and (ii) by referring to the formula

(12)

where the r-step ahead transition density function which appears in (12) is ob­tained by substituting exp(-Ak) with exp(-AkT) in equation (11).

Prices available up to time t are used for forecasting through the random walk,while for the price band model the information set also contains informationprovided by price barriers, which are assumed to be constant from time t to time

9. Thesimple random walk(10) couldbeextended e.g.by introducing a stationary, pos­sibly dependent structure in theerrorterm. Thisgeneralized random walk canexplain theshort run autocorrelation which characterize the first difference of stock prices (i.e..thereturns; see tables 1-2) but.,.. due to short memory restrictions - it does not improveforecasts as thetimehorizon grows.

41

A. GARDIN! • G. CAVALIERE' M. COSTA

t + 't". In both cases parameters are re-estimated as far as a new observation isincluded in the sample.

In order to synthesize the distance between the predicted values Pt and theobserved values P"~ the forecast mean absolute percentage error is used:

TTl A IE=.iZ)pt-ptl.100:.il:. P,-P, .100T t=1 T t=1 P,

Results are illustrated in table 3.Of the three normal periods identified in paragraph 3, the first is 'characterized

by a number ofband lower barrier adjustments, as the market declined constantlyfrom the second half of 1973 until January 1978. Consequently, for this periodthe price band model cannot give better forecasts than the random walk.

For the second and the third period, the inclusion of a band allows to improveforecasts as far as a sufficiently long time horizon is concerned. In particular, inboth periods the band model «beats» the simple random walk for one week orlonger forecast horizons. Both results show that during the considered periods,the Italian index seems to be characterized by the presence of a mean revertingcomponent which is not caught by the simple random walk but, in the mediumrun, is caught by the price band model.

The higher forecast ability of the random walk model with respect to the bandmodel for short horizons should not be surprising, as the reflecting effect inducedby the presence of the band does not necessarily have to appear at short horizonsbut, on the other side, it grows as the time horizon increase.

Table 4 reports the forecast mean absolute percentage error, which is obtainedby considering the random walk model over the whole sample. It is interesting tonotice that, as far as 1 month or longer forecast horizons are considered, the

Table 3Forecast mean absolute percentage error

1 day

1 week

1 month

3 months

6 months

1 year

I period 4.70-8.80 II period 6,81-5.85 III period 5.86-6.97

random band random band random band

walk model walk model walk model

0.859 0.868 0.946 0.946 0.852 0.855

2.197 2.215 2.561 2.544 2.291 2.290

5.262 5.257 5.412 5.388 5.258 5.254

8.778 8.582 9.019 8.909 8.811 8.591

11.976 12.486 11.742 11.192 12.563 11.906

20.097 20.167 14.697 13.418 18.147 16.470

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STOCK PRICE MODELLING

Table 4Forecast mean absolute percentage error of the random walk - whole sample

whole sample1.70-9.98

1 day1 week1 month

0.9072.3925.668

3 months6 months

1 year

10.03914.75622.845

random walk performance is always worse with respect to the forecasts achievedin the normal periods.

Summarizing, the band model can lead to improved forecasts of the Italianmarket index: apart from period I, the present analysis supports the hypothesis ofthe presence of a mean reverting price component which is at least partially pre­dictable in the long run.

6. Inefficiencies and predictability: some concluding remarks

In this work, asset price dynamics are investigated by referring to three differentkinds of operators: the market makers, who fix the asset price by means of itsexpected future dividend flow; the smart agents, who follow sell-high/buy-lowstrategies; the non-smart operators, who enter the market only after a strong one­directional price movement. By relying on the theory of target-zone exchangerate modeling, firstly introduced by Krugman (1991), we propose a continuous­time model of asset pricing, whose main implication is that price dynamics canbe characterized by periods where the price tends to vary between two pricebarriers, and periods which are characterized by strong price movements. Thefirst type of periods depend on the smart speculators, whose buying/selling activ­ity based on past price maxima and minima causes the price to lie inside a priceband; the second type of periods, which we call transitions, are caused by aninversion of the smart speculators' strategy, who evaluate as unrealistic the dis­tance between the market price and its theoretical fundamentals. This new policydetermines a trend in the price, which induces non-smart operators to enter themarket, causing a price overshooting with respect to the fundamentals.

In our model, the prediction of asset prices must take two aspects into account.The first concerns changes in the smart agents' investment policy: until the buy­low/sell-high strategy holds, the price should move within a price band, while, ifthis policy is abandoned, a break in the price process - followed by a strong pricegrowth (or decline) - should be observed; by anticipating this break, the forecast-

43

A. GARDIN! . G. CAVALIERE' M. COSTA

ing process can be greatly improved. The second aspect is that, when the smartspeculation dominates the market, the information about the price barriers can beused in order to improve forecasts.

In the second part of the paper we analyze the latter aspect. By consideringdaily data of the Italian stock market index from January 1970 to September1998, we detect the periods which are characterized by the presence of a priceband, and we nest such information in the price forecasting process. A restrictedversion of our economic model seems to be able to «beat» the forecasts based ona random walk with drift for sufficiently long forecast horizons. The determina­tion of the band limits by means of past price maxima and minima still representsa preliminary step and further investigations may be useful. Nevertheless ourchoice of a ±40% band does not affect the results, which are robust with respectto different band widths.

The results obtained by analyzing the asset price dynamics within the band arequite encouraging and allow to assert that the information related to the presenceof price bands can be a useful too. Furthermore, from an econometric viewpoint,if such bands can be identified, the usual inferential procedures are no longer validand the analysis must be carried on by means of statistical models with bands.According to the results of paragraph 5, the forecasts are only slightly improved,but we expect the greatest gain in forecasting by taking the transition phases intoaccount, since the presence of a trend ensures excellent forecasting results.

The empirical results have strong implications on the equity market efficiencyhypothesis. In particular, financial theories do not identify any particular pricelevel which can be interpreted in terms of bands; furthermore, the presence ofsuch limits and therefore of a predictable component of stock prices represent animportant reason at least to rethink the efficiency hypothesis. Therefore, the the­oretical and empirical results illustrated in this paper represent a starting point forthe revision of asset pricing theories and future price movements forecasting.

Appendix

Proofofproposition 1. In the case of a zero drift, within the band relation (6) holds

p if; c) = f + C (eA(f--<:) - e-A(f--<:»

If the asset price reaches the upper edge of the band, economic agents can deter­mine, with probability (1 - 1{;J), a shift of the band equal to L\, and, with probabil­ity 1{;J' they proceed through selling the asset, thus producing a downward shift ofthe implicit asset price equal to -L\.

In order to rule our arbitrage opportunities, the following condition has to hold(Bertola and Caballero, 1992a)

44

STOCK PRICE MODELLING

p(j;c) =(l-1C))p(j;C + L1) + 1C1p(j - L1;C), , ''---.,.----J

no intervention selling

By substituting expression (6) within this equation we obtain

7+ C(eJ..(J-C) - e-J..(J-C))

= (l-1C))(j + C(eJ..(J-c-c.) _ e-J..(J-c-c.)))

+1C)(j _L1+C(eJ..(J-c.-C) _e-J..(J-c.-C)))

which, solved for C, gives

By letting the shift L1 go to 0, we therefore obtain the proposition

(13)

o

Proofofproposition 2. We prove now the generalized proposition 2. Inthis case,as smart speculators can sell the asset with probability 1C) or buy the asset withprobability 1C2, 1C) + 1C2 S; 1, the no-arbitrage relation (13) becomes

p(j;c) =(l-1C) -1C2)p(j;C + L1) +1C,p(j - L1;c), , ' '---.,.----J

no intervention selling

By substituting expression (6) within this equation, solving for C and letting L1~ 0+ we obtain

and the proposition follows.

45

o

A. GARDIN! . G. CAVALIERE' M. COSTA

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