OVERCONFIDENCE 1199 this effect,which vanishes when A=0,to focus on the extra volatility component caused by the option value. VII.Properties of Equilibria for Small Trading Costs In this section,we discuss several of the characteristics of the cquilibrium for small trading costs,including the volume of trade and the magni- tudes of the bubble and of the extra volatility component.We also provide some comparative statics and show how parameter changes co- move price,volatility,and turnover. A. Trading Volume It is a property of Brownian motion that if it hits the origin at l,it will hit the origin at an infinite number of times in any nonempty interval [t+At).In our limit case of c =0,this implies an infinite amount of trade in any nonempty interval that contains a single trade.When the cost of trade c 0,in any time interval,turnover is either zero or infinity,and the unconditional average volume in any time interval is infinity.The expected time between trades depends continuously on c so it is possible to calibrate the model to obtain any average daily volume.However,a serious calibration would require accounting for other sources of trading,such as shocks to liquidity,and should match several moments of volume,volatility,and prices. B.Magnitude of the Bubble When c=0,a trade occurs each time traders'fundamental beliefs "cross."Nonetheless,the bubble at this trading point is strictly positive since 1h(0) b= 2(r+)h'(0) Owners do not expect to sell the asset at a price above their own val- uation,but the option has a positive value.This result may seem coun- terintuitive.To clarify it,it is worthwhile to examine the value of the option when trades occur whenever the absolute value of the differences in fundamental valuations equals an e>0.An asset owner in group A (B)expects to sell the asset when agents in group B (A)have a fun- damental valuation that exceeds the fundamental valuation of agents The unconditional probability that it is zero depends on the volatility and mean reversion of the process of the difference of opinions and on the length of the interval. As the length of the interval goes to infinity,the probability of no trade goes to zero. Reproduced with permission of the copyright owner.Further reproduction prohibited without permission.Reproduced with permission of the copyright owner. Further reproduction prohibited without permission