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 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. 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