Managing Risks of Supply-Chain Disruptions: Dual ... - CiteSeerX

Let’s consider a portfolio comprising the project and n units of a short position in X. This costs[ F( x,t)− nX ] dollars and the total return is:1 2 2[ π ( x,t)+ Ft ( x,t)+ αxFx( x,t)+ σ x Fxx(x,t)− nX ( D(x,t)+ A(x,t)]dt + [ bFx( x,t)− nBX ] dz2bFxTo have a riskless portfolio, we must choose n so that n = .BXAnd the expected return of the portfolio must equal the riskless return r, so that:1 b(x,t)b 2 ( x,t)F ( , ) + { ( , ) − [ ][ ( x , t ) − r ]} F ( x , t ) − rF ( x , t ) + F ( x , t ) + ( x , t ) =xxx t a x tµXxtπ 02B(x,t)Using limit conditions, this equation can be used to obtain the value of the project.b) Dynamic programmingContingent claim analysis requires the existence of a traded asset that tracks exactly the uncertaintyof our real asset. This is not always possible, and another method is therefore required. In thatrespect, dynamic programming is quite relevant. This method relies on traditional NPV calculationand, as we can imagine from what we saw previously, it deals with the discount rate in a mannerthat is less rigorous than with contingent claim analysis. However it presents some advantages andis in fact closely related to contingent claim analysis – those 2 methods lead to the same results inmany cases.The basis of Dynamic programming is to split the whole process of decision in two parts: first, theimmediate decision and the remaining decisions.Let’s begin with a discrete time approach. Let’s assume that the current state is xt. We will denoteF x ) the value of the cash flows that will occur in the future if the firm acts optimally from thet( tcurrent state x t. At each period, the firm has to take a decision that is comprised in the setThis will lead to an immediate profit π x , u ) . At t, the firms choose the action that willt(t tmaximize the sum of its immediate payoff and the discounted expected value of the optimalremaining decision process. The result will be the value F x ) .t( tUt.41