OVERCONFIDENCE 1195 apply with the obvious changes to account for the finite horizon.How- ever,the option value g will now depend on the remaining life of the asset,introducing another dimension to the optimal exercise problem. The infinite horizon problem is stationary,greatly reducing the math- ematical difficulty. VI.Equilibrium In this section,we derive the equilibrium option value,duration between trades,and contribution of the option value to price volatility. A. Resale Option Value The value of the option q(x)should be at least as large as the gains realized from an immediate sale.The region in which the value of the option equals that of an immediate sale is the stopping region.The complement is the continuation region.In the mind of the risk-neutral asset holder,the discounted value of the option e"g(g")should be a martingale in the continuation region and a supermartingale in the stopping region.Using Ito's lemma and the evolution equation for g", we can state these conditions as 9≥，+q9-c (14) r+λ and 豆0q”-pxq-q≤0， (15) with equality if(14)holds strictly.In addition,the function g should be continuously differentiable (smooth pasting).We shall derive a smooth function g that satisfies equations (14)and (15)and then use these properties and a growth condition on g to show that in fact the function g solves (13). To construct the function g,we guess that the continuation region will be an interval (-k'),with k'>0.The variable k'is the minimum amount of difference in opinions that generates a trade.As usual,we begin by examining the second-order ordinary differential equation that g must satisfy,albeit only in the continuation region: 2u"-pxu'-ru 0. (16) The following proposition helps us construct an"explicit"solution to cquation (16). Reproduced with permission of the copyright owner.Further reproduction prohibited without permission.Reproduced with permission of the copyright owner. Further reproduction prohibited without permission