Journal of Financial Economics 18(1987)277-311.North-Holland CONSTRAINTS ON SHORT-SELLING AND ASSET PRICE ADJUSTMENT TO PRIVATE INFORMATION* Douglas W.DIAMOND University of Chicago.Chicago.IL 60637.USA Robert E.VERRECCHIA University of Pennsylvania,Philadelphia,PA 19104.USA Received September 1985,final version received May 1986 This paper models effects of short-sale constraints on the speed of adjustment (to private information)of security prices.Constraints eliminate some informative trades,but do not bias prices upward.Prohibiting traders from shorting reduces the adjustment speed of prices to private information,especially to bad news.Non-prohibitive costs can have the reverse effect,but this is unlikely.Implications are developed about return distributions on information announcement dates.Periods of inactive trade are shown to impart a downward bias to measured returns.An unexpected increase in the short-interest of a stock is shown to be bad news. 1.Introduction This paper models the effects of constraints on short-sales on the distribu- tion and speed of adjustment (to private information)of security prices.A very simple rational expectations model of trade with bid and ask prices posted by a specialist is used to clarify the informational effects of these constraints.The model yields results concerning the effects of constraints on short-sales on the distribution of security prices,the absolute speed of adjust- ment of prices to private information,and the relative speed of adjustment to (private)good,versus bad,news.This,in turn,has implications for the 'informational efficiency'of security prices that are subject to constraints on short-selling.When combined with the notation that introducing traded put and call options can reduce the cost of establishing what is effectively a short position,these implications have empirical content.In particular,the model .We are grateful for useful comments from the referee.Haim Mendelson,Jennifer Conrad. Kenneth French,Robert Holthausen,Richard Leftwich,Robert Vishny and workshop par- ticipants at Carnegie-Mellon.Chicago,Columbia,Duke.UCLA,University of Pennsylvania and Yale.Diamond is grateful to acknowledge financial support from a Batterymarch Fellowship and the Center for Research in Security Prices at the University of Chicago.Verrecchia is grateful to acknowledge financial support from the Arthur Young Foundation. 0304-405X/87/$3.501987,Elsevier Science Publishers B.V.(North-Holland)
Journal of Financial Economics 18 (1987) 277-311. North-Holland CONSTRAINTS ON SHORT-SELLING AND ASSET PRICE ADJUSTMENT TO PRIVATE INFORMATION* Douglas W. DIAMOND Universiry of Chicago, Chicago, IL 60637, USA Robert E. VERRECCHIA Unlversir) o/Pennsylvania, Philadelphia, PA 19104, USA Received September 1985. final version received May 1986 This paper models effects of short-sale constraints on the speed of adjustment (to private information) of security prices. Constraints eliminate some informative trades, but do not bias prices upward. Prohibiting traders from shorting reduces the adjustment speed of prices to private information, especially to bad news. Non-prohibitive costs can have the reverse effect, but this is unlikely. Implications are developed about return distributions on information announcement dates. Periods of inactive trade are shown to impart a downward bias to measured returns. An unexpected increase in the short-interest of a stock is shown to be bad news. 1. Introduction This paper models the effects of constraints on short-sales on the distribution and speed of adjustment (to private information) of security prices. A very simple rational expectations model of trade with bid and ask prices posted by a specialist is used to clarify the informational effects of these constraints. The model yields results concerning the effects of constraints on short-sales on the distribution of security prices, the absolute speed of adjustment of prices to private information, and the relative speed of adjustment to (private) good, versus bad, news. This, in turn, has implications for the ‘informational efficiency’ of security prices that are subject to constraints on short-selling. When combined with the notation that introducing traded put and call options can reduce the cost of establishing what is effectively a short position, these implications have empirical content. In particular, the model ‘We are grateful for useful comments from the referee, Haim Mendelson. Jennifer Conrad, Kenneth French, Robert Holthausen, Richard Leftwich, Robert Vishny and workshop participants at Carnegie-Mellon, Chicago, Columbia, Duke, UCLA, University of Pennsylvania and Yale. Diamond is grateful to acknowledge financial support from a Batterymarch Fellowship and the Center for Research in Security Prices at the University of Chicago. Verrecchia is grateful to acknowledge financial support from the Arthur Young Foundation. 0304-405X/87/$3.500 1987, Elsevier Science Publishers B.V. (North-Holland)
278 D.W.Diamond and R.E.Verrecchia,Price adjustment to private information predicts how introducing these options influences the magnitude of price adjustments to public information,such as earnings announcements.A second set of empirical implications is contained in a characterization of the impact on prices of the announcement each month of the short-interest in a stock:we show that an unexpected increase in the short-interest is bad news.We analyze the relation between these announcement effects and the speed of adjustment to private information,producing joint empirical predictions.A final im- portant implication of short-constraints is identified:the last transaction price is an upward biased measure of the value of a stock during periods when no trade is observed. Existing studies of short-sales constraints stress that it is pessimists who would want to sell short [e.g.,Miller (1977),Figlewski (1981)].This approach concludes that constraining pessimists without constraining optimists imparts an upward bias to stock prices.An analogy with voting on a referendum illustrates this point.In an 'unconstrained'vote,voters may choose yes or no, and the motion passes if subtracting no votes from yes votes yields a positive number.If the voters were constrained to choose between voting yes or abstaining and the election rule were unchanged,the results would be biased in favor of the yes voters.Changing the election rule at the same time as the voting constraint could remove the bias.One example of a new rule is to require a fixed number of yes votes for the referendum to pass.This paper analyzes the ways that market forces change the 'election rules'in a security market when short-sale constraints are imposed.Previous work assumes that these 'rules'are unchanged.We show that unchanged 'rules'are inconsistent with common knowledge that short-selling is constrained(since no one would argue that this constraint is a secret),when the differences in votes(security trades)is due to information differences rather than to differences in tastes. Rational expectation formation changes the election rules and removes any upward bias to prices,but there remain important implications of short-con- straints that we identify. The model is structured to examine the observable effects of constraints on short-selling.Our approach is to assume that not all traders face the same cost of short-selling a stock (although our model can analyze situations where all face the same cost).Some traders and market makers can sell short at no cost and immediately obtain the sale proceeds for reinvestment,others cannot sell short at all,and a third group can sell short but cannot immediately receive the sale proceeds. We model a market with a competitive market maker who sets bid and ask prices at each instant of time.The basic structure of the model is based on Glosten and Milgrom (1985),though the logic goes back to Bagehot (1971) 1Jarrow (1980)shows that the 'bias'can be cither positive,negative or zero.None of these papers have investors with rational expectations
278 D. W. Dtamond and R. E. Verrecchia. Price adjustment to pricate tnformation predicts how introducing these options influences the magnitude of price adjustments to public information, such as earnings announcements. A second set of empirical implications is contained in a characterization of the impact on prices of the announcement each month of the short-interest in a stock: we show that an unexpected increase in the short-interest is bad news. We analyze the relation between these announcement effects and the speed of adjustment to private information, producing joint empirical predictions. A final important implication of short-constraints is identified: the last transaction price is an upward biased measure of the value of a stock during periods when no trade is observed. Existing studies of short-sales constraints stress that it is pessimists who would want to sell short [e.g., Miller (1977) Figlewski (1981)]. This approach concludes that constraining pessimists without constraining optimists imparts an upward bias to stock prices.’ An analogy with voting on a referendum illustrates this point. In an ‘unconstrained’ vote, voters may choose yes or no, and the motion passes if subtracting no votes from yes votes yields a positive number. If the voters were constrained to choose between voting yes or abstaining and the election rule were unchanged, the results would be biased in favor of the yes voters. Changing the election rule at the same time as the voting constraint could remove the bias. One example of a new rule is to require a fixed number of yes votes for the referendum to pass. This paper analyzes the ways that market forces change the ‘election rules’ in a security market when short-sale constraints are imposed. Previous work assumes that these ‘rules’ are unchanged. We show that unchanged ‘rules’ are inconsistent with common knowledge that short-selling is constrained (since no one would argue that this constraint is a secret), when the differences in votes (security trades) is due to information differences rather than to differences in tastes. Rational expectation formation changes the election rules and removes any upward bias to prices, but there remain important implications of short-constraints that we identify. The model is structured to examine the observable effects of constraints on short-selling. Our approach is to assume that not all traders face the same cost of short-selling a stock (although our model can analyze situations where all face the same cost). Some traders and market makers can sell short at no cost and immediately obtain the sale proceeds for reinvestment, others cannot sell short at all, and a third group can sell short but cannot immediately receive the sale proceeds, We model a market with a competitive market maker who sets bid and ask prices at each instant of tune. The basic structure of the model is based on Glosten and Milgrom (1985), though the logic goes back to Bagehot (1971) ‘Jarrow (1980) shows that the ‘bias’ can be either positive, negative or zero. None of these papers have investors with rational expectations
D.W.Diamond and R.E.Verrecchia.Price adjustment to pricate information 279 and Copeland and Galai (1983).The information structure is the simplest other than perfect information:there are informed traders who observe identical private information and uninformed traders who observe only public information.The competitive,risk-neutral market maker does not observe the private information,but does observe all trades as they take place.Potential competition implies that a risk-neutral market maker will earn a zero expected profit on each transaction.He sets a bid-ask spread such that,on average,his losses from transacting with informed traders are equal to his profits from transacting with uninformed traders.This requires that each bid and ask price be set equal to the conditional expectation of the value of the asset given all past trades,and given the information of the current trade(e.g.,a buy at the ask,or a sale or short-sale at the bid).Changing the constraints on short-sell- ing affects the information content of observed transactions.Rational market makers and investors take this into account when formulating their demand and pricing decisions. Imposing a cost on short-selling obviously makes it less attractive,and one expects that those willing to pay the cost are the ones with the greatest anticipated benefits from selling short.This implies that imposing a cost on short-selling both reduces the number of short-sales and influences the mix of relatively informed and relatively uninformed traders who remain in the pool of short-sellers.To examine the implications of both effects,we specify two types of short-selling costs,each of which has only one of the two effects.In practice,most costs would have both effects(we discuss the empirical implica- tions of this in section 5). The first effect arises from the prohibition,or elimination,of short-sales.We refer to this as the short-prohibition effect.Here,we assume there exists a cost that prevents investors who want to short from so doing.This eliminates short-sales by informed and uninformed traders alike.Examples include legal or contractual prohibitions of shorting by certain institutional investors and corporate insiders,the inability to borrow stock to short,and (in the short run) the 'no short-sale on a down-tick'rule,which prohibits short-sales at prices below the last differing price. The second effect arises from the restriction of short-sales through the imposition of additional costs.We refer to this as the short-restriction effect.If sale proceeds cannot be reinvested,or there is an additional cost of borrowing securities to short,only investors who have strong beliefs that a significant price decline will soon occur will choose to short.Thus,the restriction of short-sales due to costs changes the composition of the remaining pool of short-sellers.In contrast to the prohibition of short-sales,a restriction drives relatively uninformed traders out of the pool of shorts more so than it drives out relatively informed traders.We specify a cost that drives out only the uninformed traders.Observed changes in the costs of establishing short-posi- tions probably contain elements of both effects,driving out some informed
D. W. Diamond and R. E. Verrecchla. Pnce adjurment to pricate information 279 and Copeland and Galai (1983). The information structure is the simplest other than perfect information: there are informed traders who observe identical private information and uninformed traders who observe only public information. The competitive, risk-neutral market maker does not observe the private information, but does observe all trades as they take place. Potential competition implies that a risk-neutral market maker will earn a zero expected profit on each transaction. He sets a bid-ask spread such that, on average, his losses from transacting with informed traders are equal to his profits from transacting with uninformed traders. This requires that each bid and ask price be set equal to the conditional expectation of the value of the asset given all past trades, and given the information of the current trade (e.g., a buy at the ask, or a sale or short-sale at the bid). Changing the constraints on short-selling affects the information content of observed ,transactions. Rational market makers and investors take this into account when formulating their demand and pricing decisions. Imposing a cost on short-selling obviously makes it less attractive, and one expects that those willing to pay the cost are the ones with the greatest anticipated benefits from selling short. This implies that imposing a cost on short-selling both reduces the number of short-sales and influences the mix of relatively informed and relatively uninformed traders who remain in the pool of short-sellers. To examine the implications of both effects, we specify two types of short-selling costs, each of which has only one of the two effects. In practice, most costs would have both effects (we discuss the empirical implications of this in section 5). The first effect arises from the prohibition, or elimination, of short-sales. We refer to this as the shorr-prohibition effect. Here, we assume there exists a cost that prevents investors who want to short from so doing. This eliminates short-sales by informed and uninformed traders alike. Examples include legal or contractual prohibitions of shorting by certain institutional investors and corporate insiders, the inability to borrow stock to short, and (in the short run) the ‘no short-sale on a down-tick’ rule, which prohibits short-sales at prices below the last differing price. The second effect arises from the restriction of short-sales through the imposition of additional costs. We refer to this as the short-restriction effect. If sale proceeds cannot be reinvested, or there is an additional cost of borrowing securities to short, only investors who have strong beliefs that a significant price decline will soon occur will choose to short. Thus, the restriction of short-sales due to costs changes the composition of the remaining pool of short-sellers. In contrast to the prohibition of short-sales, a restriction drives relatively uninformed traders out of the pool of shorts more so than it drives out relatively informed traders. We specify a cost that drives out only the uninformed traders. Observed changes in the costs of establishing short-positions probably contain elements of both effects, driving out some informed
280 D.W.Diamond and R.E.Verrecchia,Price adjustment to pricate information and some relatively uninformed traders.Therefore,our strategy is to identify the implications of each effect,and identify predictions we can make without directly knowing which one dominates. The balance of the paper proceeds as follows.Section 2 develops the model. Section 3 examines the effect of prohibiting short-selling on the speed of adjustment of prices to private information,on the magnitude of price adjustments to announcements of public information,and on the bid-ask spread.Section 4 presents analogous results on the effect of restricting the receipt and reinvestment of the proceeds of a short-sale,rather than prohibit- ing such sales.Section 5 presents empirical implications of the model's results on informational efficiency,on short-interest announcements,and on the implications for measuring returns after periods of inactive trade.Section 6 concludes the paper. 2.The model The basic structure of the model is based on Glosten and Milgrom(1985): market makers are risk-neutral,face no inventory costs or constraints,and earn zero expected profits from each trade.Traders are also risk-neutral and are either informed or uninformed.There is an infinite number of each type of trader.Informed traders know (privately)the true liquidating value of the risky.asset,while uninformed traders make an inference about its value based on all public information.The prior distribution of the risky asset's value is Bernoulli:its liquidating value is one with probability one-half,and zero with probability one-half.The liquidating value is paid in the distant future,but we abstract from discounting in determining market prices.Apart from short-sale constraints,an informed trader buys the asset if it is underpriced and sells if it is overpriced.A share is underpriced if its ask price is less than the trader's conditional expectation of the liquidating value,and overpriced if its bid price is above the trader's conditional expectation. An informed trader makes a particular trade on the basis of his information and the current price of the stock.If the market maker traded only with informed traders,he would lose money because informed traders would buy when the price was too low and sell only when the price was too high.Absent a motive for trade other than speculative profit,there would exist no prices that allow the specialist to break even and the market would break down. Therefore,we introduce another motive to trade by considering the role of 'liquidity trading'.Liquidity trading occurs for reasons exogenous to our model,and involves the need to buy or sell at a particular time.The reasons might include immediate consumption needs,tax planning,and alternative outside investment opportunities.With liquidity trading,voluntary trade is possible because the specialist can earn enough profit on non-informational trades to offset losses from transactions with informed traders
280 D. W. Diamond and R. E. Verrecchia, Price arijustment to pncate information and some relatively uninformed traders. Therefore, our strategy is to identify the implications of each effect, and identify predictions we can make without directly knowing which one dominates. The balance of the paper proceeds as follows. Section 2 develops the model. Section 3 examines the effect of prohibiting short-selling on the speed of adjustment of prices to private information, on the magnitude of price adjustments to announcements of public information. and on the bid-ask spread. Section 4 presents analogous results on the effect of restricting the receipt and reinvestment of the proceeds of a short-sale, rather than prohibiting such sales. Section 5 presents empirical implications of the model’s results on informational efficiency, on short-interest announcements, and on the implications for measuring returns after periods of inactive trade. Section 6 concludes the paper. 2. The model The basic structure of the model is based on Glosten and Milgrom (1985): market makers are risk-neutral, face no inventory costs or constraints, and earn zero expected profits from each trade. Traders are also risk-neutral and are either informed or uninformed. There is an infinite number of each type of trader. Informed traders know (privately) the true liquidating value of the risky. asset, while uninformed traders make an inference about its value based on all public information. The prior distribution of the risky asset’s value is Bernoulli: its liquidating value is one with probability one-half, and zero with probability one-half. The liquidating value is paid in the distant future, but we abstract from discounting in determining market prices. Apart from short-sale constraints, an informed trader buys the asset if it is underpriced and sells if it is overpriced. A share is underpriced if its ask price is less than the trader’s conditional expectation of the liquidating value, and overpriced if its bid price is above the trader’s conditional expectation. An informed trader makes a particular trade on the basis of his information and the current price of the stock. If the market maker traded only with informed traders, he would lose money because informed traders would buy when the price was too low and sell only when the price was too high. Absent a motive for trade other than speculative profit, there would exist no prices that allow the specialist to break even and the market would break down. Therefore, we introduce another motive to trade by considering the role, of ‘liquidity trading’. Liquidity trading occurs for reasons exogenous to our model, and involves the need to buy or sell at a particular time. The reasons might include immediate consumption needs, tax planning, and alternative outside investment opportunities. With liquidity trading, voluntary trade is possible because the specialist can earn enough profit on non-informational trades to offset losses from transactions with informed traders
D.W.Diamond and R.E.Verrecchia,Price adjustment to private information 281 Formally,we model liquidity trading as a shock to an individual's time preference.All traders discount future consumption by the factor p,so the present utility value of consumption,Cr,on the date of the liquidating dividend,is p x Cr.We assume that absent a shock,p equals one.Unin- formed traders (but no one else)receive one of two possible shocks:either p =0,which implies that one sells the asset to satisfy consumption today,or p=+oo,which implies that one buys the asset as a means of deferring consumption indefinitely.Modeling preference shocks as we do is a'reduced form'for many possibilities.We use extreme values purely for simplicity of interpretation.The Glosten-Milgrom (1985)model is consistent with more general shocks and types of private information.Some motive for trade other than speculative profit is necessary to construct a model of trade by unin- formed individuals:unless they have some potential gains from trade,they will be unwilling to pay the bid-ask spread.Nothing of substance depends on the assumption that only uninformed traders are subject to liquidity shocks. To offer the simplest setting for the role of information and liquidity shocks toward generating observable trades,we abstract from the possible variations in the size of trades established by traders.Specifically,a trader is allowed to buy a single share,sell a single share,short-sell a single share,or do nothing. For example,if a trader is informed and observes that the stock is underpriced at the ask price,he then buys one share and holds it (because under our assumptions,it will never subsequently become overpriced at the bid).Simi- larly,if a trader receives a liquidity shock and has a desire to invest (i.e.. p=+co),he buys one share.If a trader who already owns the stock finds it overpriced at the bid price or receives a liquidity shock and must sell (i.e., p=0).he sells one share. A trader's willingness to short-sell is influenced by the cost associated with this transaction.We assume a simple cost function that is independent of a trader's level of information or type of liquidity shock.The cost associated with selling short falls into one of three categories:no-cost,proceeds-restric- tions,and short-prohibitions.The no-cost scenario allows full reinvestment or consumption of short-sale proceeds,implying that a short-sale generates funds on its initiation date.The proceeds-restrictions scenario delays receipt of proceeds.In this circumstance a short-sale generates no funds today but does allow one to profit if the price falls.Finally,short-prohibitions eliminate any opportunity to short-sell,either because an individual trader is prohibited from engaging in this activity,or the cost is so high that no trader would avail himself of the opportunity regardless of what he knows.We represent that fraction of the population which encounters no cost associated with short-sell- ing by c,that fraction which faces proceeds-restrictions by c2,and that fraction which is essentially prohibited from this activity by c3.All traders, independent of whether they are informed or uninformed,fall into one of these three categories,i.e.,c+c2+c3=1 at all times.Under the assumption
D. W. Diamond and R. E. Verrecchia. Puce adjustment roprioate information 281 Formally, we model liquidity trading as a shock to an individual’s time preference. All traders discount future consumption by the factor p, so the present utility value of consumption, Cr. on the date of the liquidating dividend, is p X Cr. We assume that absent a shock, p equals one. Uninformed traders (but no one else) receive one of two possible shocks: either p = 0, which implies that one sells the asset to satisfy consumption today, or p = + cc, which implies that one buys the asset as a means of deferring consumption indefinitely. Modeling preference shocks as we do is a ‘reduced form’ for many possibilities. We use extreme values purely for simplicity of interpretation. The Glosten-Milgrom (1985) model is consistent with more general shocks and types of private information. Some motive for trade other than speculative profit is necessary to construct a model of trade by uninformed individuals: unless they have some potential gains from trade, they will be unwilling to pay the bid-ask spread. Nothing of substance depends on the assumption that only uninformed traders are subject to liquidity shocks. To offer the simplest setting for the role of information and liquidity shocks toward generating observable trades, we abstract from the possible variations in the size of trades established by traders. Specifically, a trader is allowed to buy a single share, sell a single share, short-sell a single share, or do nothing. For example, if a trader is informed and observes that the stock is underpriced at the ask price, he then buys one share and holds it (because under our assumptions, it will never subsequently become overpriced at the bid). Similarly, if a trader receives a liquidity shock and has a desire to invest (i.e., p = + co), he buys one share. If a trader who already owns the stock finds it overpriced at the bid price or receives a liquidity shock and must sell (i.e., p = 0), he sells one share. A trader’s willingness to short-sell is influenced by the cost associated with this transaction. We assume a simple cost function that is independent of a trader’s level of information or type of liquidity shock. The cost associated with selling short falls into one of three categories: no-cost, proceeds-restrictions, and short-prohibitions. The no-cost scenario allows full reinvestment or consumption of short-sale proceeds, implying that a short-sale generates funds on its initiation date. The proceeds-restrictions scenario delays receipt of proceeds. In this circumstance a short-sale generates no funds today but does allow one to profit if the price falls. Finally, short-prohibitions eliminate any opportunity to short-sell, either because an individual trader is prohibited from engaging in this activity, or the cost is so high that no trader would avail himself of the opportunity regardless of what he knows. We represent that fraction of the population which encounters no cost associated with short-selling by ct, that fraction which faces proceeds-restrictions by c2, and that fraction which is essentially prohibited from this activity by cs. All traders, independent of whether they are informed or uninformed, fall into one of these three categories, i.e., ci + c2 + cj = 1 at all times. Under the assumption
282 D.W.Diamond and R.E.Verrecchia,Price adjustment to private information Table 1 A summary of the types of traders who sell short,assuming that they lack stock in their portfolios to sell directly.where the liquidity preference shock p0 implies a low valuation of claims to future consumption Informed with Uninformed Cost bad news with p=0 Other types C:No cost Yes Yes No C2:Deferred receipt of proceeds Yes No No C:Prohibitive costs No No No that the distribution of traders across cost functions does not depend on their type,c,i=1,2,3,represents the probability that a randomly selected trader faces cost i. The implications of differential short-selling costs are as follows.Those who are in the first category face no cost,and therefore short-sell whenever they do not own the stock and need to consume (i.e.,p=0),or have bad news.Those in the second category encounter a proceeds-restriction when selling short: namely,an inability to consume or reinvest the proceeds.If a trader is in this category and informed with bad news,he shorts a stock if he does not otherwise own a share (in which case he will simply sell).Because p=1 for informed traders and interest rates are zero,the lack of proceeds does not deter them from shorting.If a trader is uninformed and needs to consume immediately (i.e.,p=0)he does not short (even if he does not own a share) because the transaction raises no immediate proceeds.Thus,restrictions drive out uninformed short-sellers,while allowing informed traders to short if the occasion arises.Those who are in the third category are prohibited from short-selling because of its cost.This prohibition applies to both informed and uninformed traders.Consequently,it does not influence the proportion of short-sales that are informed as all traders facing this cost are constrained. Table 1 provides a summary of which traders short,assuming their portfolio contains no stock. Our economy operates as follows (refer to fig.1 for illustration of its operation,and table 2 for a summary of the notation).Before trade begins, nature moves to choose either 0 or 1 as the value of the risky asset:we refer to this choice as the true state-of-nature.After nature's move,time is divided into T discrete intervals,with arbitrary length between them.At each interval,there is a probability g that a single trader potentially wants to trade(depending on the costs of trading)and 1-g that no trader has a reason to entertain trading (in which case no-trade is observed).A trader who potentially wants to trade is a random draw from the (infinite)population of all traders.He is either an informed trader with probability a or an uninformed trader with probability
282 D. W. Diamond and R. E. Verrecchia, Pnce adjustment to prtcate information Table 1 A summary of the types of traders who sell short, assuming that they lack stock in their portfolios to sell directly, where the liquidity preference shock p = 0 implies a low valuation of claims to future consumption. cost Informed with bad news Uninformed with p=O Other types c,: No cost c2: Deferred receipt of proceeds cj: Prohibitive costs Yes Yes No Yes No No No No No that the distribution of traders across cost functions does not depend on their type, ci, i = 1,2,3, represents the probability that a randomly selected trader faces cost i. The implications of differential short-selling costs are as follows. Those who are in the first category face no cost, and therefore short-sell whenever they do not own the stock and need to consume (i.e., p = 0), or have bad news. Those in the second category encounter a proceeds-restriction when selling short: namely, an inability to consume or reinvest the proceeds. If a trader is in this category and informed with bad news, he shorts a stock if he does not otherwise own a share (in which case he will simply sell). Because p = 1 for informed traders and interest rates are zero, the lack of proceeds does not deter them from shorting. If a trader is uninformed and needs to consume immediately (i.e., p = 0) he does not short (even if he does not own a share) because the transaction raises no immediate proceeds. Thus, restrictions drive out uninformed short-sellers, while allowing informed traders to short if the occasion arises. Those who are in the third category are prohibited from short-selling because of its cost. This prohibition applies to both informed and uninformed traders. Consequently, it does not influence the proportion of short-sales that are informed as all traders facing this cost are constrained. Table 1 provides a summary of which traders short, assuming their portfolio contains no stock. Our economy operates as follows (refer to fig. 1 for illustration of its operation, and table 2 for a summary of the notation). Before trade begins, nature moves to choose either 0 or 1 as the value of the risky asset: we refer to this choice as the true state-of-nature. After nature’s move, time is divided into T discrete intervals, with arbitrary length between them. At each interval, there is a probability g that a single trader potentially wants to trade (depending on the costs of trading) and 1 - g that no trader has a reason to entertain trading (in which case no-trade is observed). A trader who potentially wants to trade is a random draw from the (infinite) population of all traders. He is either an informed trader with probability a or an uninformed trader with probability
D.W Dia ond and R E Verrecch ia Price adiustment to private information 283 ape/L-ON BPR/1-ON sAng :oo=d S SilRS g lassv aul 104 an1 EA e SM 3n12N
D. W. Diamond and R. E. Verrecchia, Pnce ndjusment to pnoare informalion 283
284 D.W.Diamond and R.E.Verrecchia,Price adjustment to pricate information Table 2 A summary of notation used in fig.1 and throughout the paper. Variable Definition Value of the asset,either one or zero. 8 Probability that one trader potentially wants to trade (for either liquidity or information based motives). a Probability that a given trader is informed.This also represents the fraction of traders who are informed among those who actively participate in the market. h Probability that a trader already owns the stock.This also represents the fraction of traders who already own the stock independent of their typc. Probability that a trader faces cost i of short-selling.This also represents the fraction of traders who face this cost independent of whether they are informed or uninformed. p The liquidity preference shock that affects uninformed traders.It assumes the value zero.or(positive)infinity with equal probability.If p=0,the trader wants to sell.If p=+oo.the trader wants to buy. The probability of observing action A when the value of the asset is v. The price or conditional expectation associated with an action of type A at time t. 1-a.If an informed trader's private information is 'good news'(i.e.,=1), then he buys a single share because the price is never greater than one.If an informed trader's private information is'bad news'(i.e.,v=0)and he already owns shares of the asset (this occurs with probability h),he sells one share because price is never less than zero;if he has bad news and owns no shares (this occurs with probability 1-h),he shorts a single share if he faces no costs or proceeds-restrictions on short-selling (with probabilities c and c2,respec- tively).The oniy circumstances in which an informed trader with bad news does nothing (i.e,no-trade)is when he owns no shares (with probability 1-h)and encounters shorts-prohibitions(with probability c3).2 An uninformed trader participates in the market if he has experienced a liquidity shock (otherwise he has no reason to trade against better informed traders and pay the bid-ask spread).Independent of the true state-of-nature (known only to the informed),a randomly selected uninformed trader wants to buy (with probability one-half)or sell(with probability one-half)a single share for liquidity reasons.However,while he can always buy,and he can 2The exogenous probabilities g.a,and h lie in the open interval (0.1)
284 D. W. Diamond and R.E. Verrecchla. Pnce adjustment to prwate mformanon Table 2 A summary of notation used in fig. 1 and throughout the paper. Variable Definition ” g Value of the asset. either one or zero. Probability that one trader potentially wants to trade (for either liquidity or information based motives). a Probability that a given trader is informed. This also represents the fraction of traders who are informed among those who actively participate in the market. h Probability that a trader already owns the stock. This also represents the fraction of traders who already own the stock independent of their type. c, Probability that a trader faces cost i of short-selling. This also represents the fraction of traders who face this cost independent of whether they are informed or uninformed. P The liquidity preference shock that affects uninformed traders. It assumes the value zero. or (positive) infinity with equal probability. If p= 0, the trader wants to sell. If p = + a~, the trader wants to buy. A 41, P, The probability of observing action A when the value of the asset is o. The price or conditional expectation associated with an action of type A at time t. 1 - a. If an informed trader’s private information is ‘good news’ (i.e., u = l), then he buys a single share because the price is never greater than one. If an informed trader’s private information is ‘bad news’ (i.e., u = 0) and he already owns shares of the asset (this occurs with probability h), he sells one share because price is never less than zero; if he has bad news and owns no shares (this occurs with probability 1 - h), he shorts a single share if he faces no costs or proceeds-restrictions on short-selling (with probabilities ct and cz, respectively). The o&y circumstances in which an informed trader with bad news does nothing (i.e., no-trade) is when he owns no shares (with probability 1 - h) and encounters shorts-prohibitions (with probability cs).’ An uninformed trader participates in the market if he has experienced a liquidity shock (otherwise he has no reason to trade against better informed traders and pay the bid-ask spread). Independent of the true state-of-nature (known only to the informed), a randomly selected uninformed trader wants to buy (with probability one-half) or sell (with probability one-half) a single share for liquidity reasons. However, while he can always buy, and he can 2The exogenous probabilities g, a, and h lie in the open interval (0,l)
D.W.Diamond and R.E.Verrecchia,Price adjustment to private information 285 always sell if he owns shares of the asset(which occurs with probability h),his decision to short depends upon the costs associated with this transaction.If he wants to sell (which occurs with probability one-half)and owns none of the risky asset (which occurs with probability 1-h),he will short if he is a trader who faces no costs (with probability c1),and does not short if he faces proceeds-restrictions or short-prohibitions (with probabilities c2 and c3,re- spectively).In the latter events he does nothing,and no-trade is observed. The tree diagram in fig.1 illustrates the calculation of the probability of each type of observed action,conditional upon the true state-of-nature.There are four actions available to each trader:buy,sell,or short a single share,or do not trade.When no-trade occurs,neither the market maker nor other traders can distinguish whether this arises because no trader wants to trade,or a trader chooses not to trade because of short-selling costs.In addition,when a sale occurs,neither the market maker,nor other traders,can distinguish whether the share sold is one owned by the seller,or is a short-sale.As a result,there are two possible partitions of the action space:the set of actions taken and the set of actions observed.The set of actions taken includes buy, sell,short,and no-trade,while the set of actions obserued is restricted to buy, 'sell-or-short',and no-trade.Let v represent the true state-of-nature (i.e.,v=0 or v=1),and qo represent the probability of observing action A4 conditional on state v.The conditional probabilities of the possible observable actions are given in table 3. The market maker posts a bid price at which he is willing to buy one share (in response to a'sell-or-short'order),or an ask price at which he is willing to sell one share (in response to a buy order).At time t,the bid price is Ps and the ask price is p8.Free entry into market making is assumed.This,along with risk-neutrality and no inventory constraint implies that the expected profit from each trade is zero.The bid price at time t is the conditional expectation of the value of the asset given previous public information and the fact that the current transaction is a 'sell-or-short'order.The ask price at Table 3 Conditional probabilities of actions directly observed.where g is the probability that some trader potentially wants to grade,a is the probability a trader is informed,h is the probability a trader owns the stock,and c is the probability that a trader faces cost i of short-selling. Actions Conditional probabilities Conditional probabilities directly when state-of-nature is when state-of-nature is observed v=1() D=0(96) Buy 8(1+a) g1-a) Sell-or-short g(1-a(h+[1-h]c) g(1+a(h+[1-h]c1)+ga(1-h)c2 No-trade 1-g+g(1-h(1-a(cz+c3) 1-8+8(1-h(1-a(c2+c3)+2ac3】
D. W. Diamond and R E. Verrecchla, Pnce adjwtment to pricate mformatlon 285 always sell if he owns shares of the asset (which occurs with probability h), his decision to short depends upon the costs associated with this transaction. If he wants to sell (which occurs with probability one-half) and owns none of the risky asset (which occurs with probability 1 - h), he will short if he is a trader who faces no costs (with probability c,), and does not short if he faces proceeds-restrictions or short-prohibitions (with probabilities c2 and cj, respectively). In the latter events he does nothing, and no-trade is observed. The tree diagram in fig. 1 illustrates the calculation of the probability of each type of observed action, conditional upon the true state-of-nature. There are four actions available to each trader: buy, sell, or short a single share, or do not trade. When no-trade occurs, neither the market maker nor other traders can distinguish whether this arises because no trader wants to trade, or a trader chooses not to trade because of short-selling costs. In addition, when a sale occurs, neither the market maker, nor other traders, can distinguish whether the share sold is one owned by the seller, or is a short-sale. As a result, there are two possible partitions of the action space: the set of actions taken and the set of actions observed. The set of actions taken includes buy, sell, short, and no-trade, while the set of actions observed is restricted to buy, ‘sell-or-short’, and no-trade. Let u represent the true state-of-nature (i.e., u = 0 or u = l), and qz represent the probability of observing action A conditional on state u. The conditional probabilities of the possible observable actions are given in table 3. The market maker posts a bid price at which he is willing to buy one share (in response to a ‘sell-or-short’ order), or an ask price at which he is willing to sell one share (in response to a buy order). At time 1, the bid price is Pts and the ask price is P, ‘. Free entry into market making is assumed. This, along with risk-neutrality and no inventory constraint implies that the expected profit from each trade is zero. The bid price at time I is the conditional expectation of the value of the asset given previous public information and the fact that the current transaction is a ‘sell-or-short’ order. The ask price at Table 3 Conditional probabilities of actions directly observed. where g is the probability that some trader potentially wants to grade, a is the probability a trader is informed, h is the probability a trader owns the stock, and c, is the probability that a trader faces cost i of short-selling. Actions directly observed Buy Sell-or-short No-trade Conditional probabilities Conditional probabilities when state-of-nature is when state-of-nature is u-1(4?) o=O(& ig(l + a) ig(1 -a) ig(l - a)(h + [l - h]c,) ig(l + a)(h + [l - h]c,) + ga(l - h)c, 1 - g + $g(l - h)(l - a)(cr + c3) 1 - g + ig(l - h)[(l - a)(q + c3) + Zac,]
286 D.W.Diamond and R.E.Verrecchia.Price adjustment to pricate information time t is the conditional expectation of the value of the asset given previous public information and the fact that the current transaction is a buy order. The current transaction of either a sell or a buy is informative because of the possibility that the order is being placed by an informed trader.Because the market maker knows that all buys are at the ask and all sells at the bid,he can post the bid and ask prices before he knows which type of order will appear After a transaction takes place,the market maker can change the bid and ask prices;these prices may even change when no-trade occurs because one can draw an inference from no-trade,as well as buying and selling. Let P,denote the probability that the true state-of-nature is v=1,and 1-P,denote the probability that v=0.P,is the conditional expectation of the asset's value at time t given all public information.P,can also be interpreted as the transaction price of the asset at time t,when the transaction at time t is a buy or a'sell-or-short'.3 It turns out to be convenient to work with P/(1-P),which is analogous to the likelihood ratio of=1 versus =0.For example,before the very first trade at t=0,the likelihood ratio for =1 relative to v=0 is Po/(1-Po)=1,since here each state is equally likely. In general,for any observed action A,the conditional expectation of the value of the asset at time t,P,,is the solution to p,in the expression P,P-191 1-P,1-P,-196 where qa is the probability of observing action A conditional on state v. Because'no-trade'is an observable event,the conditionally expected value of the asset and consequently posted bid and ask prices in the future,may change at time t if no-trade is observed at t-1. P,is the conditional expectation (given all public information)of the value of the asset,implying that the unconditional expectation of the change in P on any date is zero (because the interest rate is zero).This is obviously a very general result that depends only on rational expectations and risk-neutrality. For example,we could assume that market makers only adjust prices every N periods or that no one observes when a no-trade interval occurs.The new values of P,would then be conditional expectations under this new informa- tion structure and would still exhibit no bias.Any transaction which occurs will be at a price equal to the conditional expectation.In periods when there is 3When there is no trade P is not a transaction price,but represents the effect of no-trade on future bid and ask prices.It is simplest to treat it like a transaction price,which is what we do until section 6.Section 6 discusses the empirical implications of observed periods of no-trade. 4In an economy with risk aversion,constrained short-selling could change the rate of resolution of uncertainty,and thus possibly the time series of risk premiums.In that case,the unbiased expectations would apply to the 'risk-adjusted'price
286 D. W. Dmmond and R. E. Verrecchla. Price adjuwnenr :a pncare informarlon time t is the conditional expectation of the value of the asset given previous public information and the fact that the current transaction is a buy order. The current transaction of either a sell or a buy is informative because of the possibility that the order is being placed by an informed trader. Because the market maker knows that all buys are at the ask and all sells at the bid, he can post the bid and ask prices before he knows which type of order will appear. After a transaction takes place, the market maker can change the bid and ask prices; these prices may even change when no-trade occurs because one can draw an inference from no-trade, as well as buying and selling. Let P, denote the probability that the true state-of-nature is u = 1, and 1 - P, denote the probability that u = 0. P, is the conditional expectation of the asset’s value at time t given all public information. P, can also be interpreted as the transaction price of the asset at time t, when the transaction at time t is a buy or a ‘sell-or-short’.3 It turns out to be convenient to work with P,/(l - P,), which is analogous to the likelihood ratio of u = 1 versus IJ = 0. For example, before the very first trade at t = 0, the likelihood ratio for u = 1 relative to u = 0 is P,,/(l - P,,) = 1, since here each state is equally likely. In general, for any observed action A, the conditional expectation of the value of the asset at time 1, P,, is the solution to P, in the expression P, c-1 4; -= 1 - P, l-P,_1 4,A7 where q,” is the probability of observing action A conditional on state u. Because ‘no-trade’ is an observable event, the conditionally expected value of the asset and consequently posted bid and ask prices in the future, may change at time t if no-trade is observed at t - 1. P, is the conditional expectation (given all public information) of the value of the asset, implying that the unconditional expectation of the change in P, on any date is zero (because the interest rate is zero). This is obviously a very general result that depends only on rational expectations and risk-neutrality.4 For example, we could assume that market makers only adjust prices every N periods or that no one observes when a no-trade interval occurs. The new values of P, would then be conditional expectations under this new information structure and would still exhibit no bias. Any transaction which occurs will be at a price equal to the conditional expectation. In periods when there is 3When there is no trade P, is not a transaction price, but represents the effect of no-trade on future bid and ask prices. It is simplest to treat it like a transaction price, which is what we do until section 6. Section 6 discusses the empirical implications of observed periods of no-trade. 41n an economy with risk aversion, constrained short-selling could change the rate of resolution of uncertainty, and thus possibly the time series of risk premiums. In that case, the unbiased expectations would apply to the ‘risk-adjusted’ price