Stochastic Capital Theory
Many problems in capital theory — particularly ‘Austrian’ capital theory —take the following form: an asset has an intrinsic value X(t) at time t. If he takes a particular action at time T, then the asset’s owner gets X(T) at T. In anticipation of future usage we shall call the action taken at T stopping and refer to T as a stopping time. This set-up raises two natural, and related, questions. When should the intrinsic process be stopped? What is the present value of the asset? The standard examples are when to drink the wine whose quality at t is given by X(t) or when to cut down the tree which contains lumber with a value of X(t). If the discount rate is r then these questions may be simply answered. The optimal stopping time T* maximizes e -rT X(T) and the present value of the tree is its discounted valueTo distinguish from intrinsic value, call this latter quantity the market value of the asset.
KeywordsInterest Rate Discount Rate Free Boundary Local Increase Geometric Brownian Motion
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© George R. Feiwel 1989