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