OVERCONFIDENCE 1201 mental process.(2)As i,or ip increases,o.is unchanged and p increases since 0<<1.Therefore,b and g(x),for x<k',increase.The bubble increases with the amount of information in the signals and the dividend flow.(3)As increases,o,increases and p decreases.Thus an increase in o has offsetting effects on the size of the bubble.However,numerical exercises indicate that the size of the bubble always increases with o. C.Magnitude of the Extra Volatility Component As the difference of opinions x approaches the trading point,the vol- atility of the option value approaches 20j h(h*) r+入()+h'(-k) We have the following lemma. LEMMA 4.If c is small,n(k')decreases with the interest rate r and the degree of mean reversion A and increases with the overconfidence pa- rameter o and the fundamental volatility o This lemma implies that an increase in the volatility of fundamentals has an additional effect on price volatility at trading points,through an increase in the volatility of the option component. D.Price,Volatility,and Turnover Our model provides a link between asset prices,price volatility,and share turnover.Since these are endogenous variables,their relationship will typically depend on which exogenous variable is shifted.In this subsection,we illustrate this link using numerical examples with a small trading cost. Figure I shows the effect of changes in on the equilibrium when there is a small transaction cost on the trading barrier k',expected duration between trades,the bubble at the trading point b,and the volatility of the bubble at the trading point,(k).The expected duration between trades is measured in years.The terms k',n(k),and b are measured in multiples of the fundamental volatility o/(r+A).Recall that,as o increases,the volatility parameter o,in the difference of beliefs increases,whereas the mean reversion parameter p decreases.As a result, the resale option becomes more valuable to the asset owner,the bubble and the extra volatility component become larger,and the optimal trad- ing barrier becomes higher.The duration between trades is determined Since the bubble is generated through an option value,it is natural to normalize it by the volatility of the underlying fundamental value,i.e.,the price volatility that would prevail if fundamentals were observable. Reproduced with permission of the copyright owner.Further reproduction prohibited without permission.Reproduced with permission of the copyright owner. Further reproduction prohibited without permission