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

therefore its price is easily known. If the asset is not traded, the current market is sufficiently richto permit to replicate it with a portfolio of traded assets. Then the value of the investment project isequal to the total value of the replicating portfolio to prevent arbitrage profits.Let’s analyze further the case where the asset is not traded. Suppose that the profit flow of theproject depends on a variable x that follows a geometric Brownian movement:dx = a( x,t)dt + b(x,t)dz = α xdt + σxdz(11)where α is the drift parameter, σ the variance parameter, dz the increment of a Wiener Process.To value the project we just need to be able to find some other asset that is traded and “tracks orspans” perfectly the uncertainty in x. Let’s denote X the price of this “spanning asset”. Thestochastic process of X must verify:dX = A( x,t)Xdt + B(x,t)Xdz(12)where dz is the same increment of Wiener Process as the one in the stochastic fluctuations of x.If we assume that the replicating asset pays a dividend at rate D ( x,t), its generates the total return:[ D ( x,t)+ A(x,t)]dt + B(x,t)dzwhere D( x,t)+ A(x,t)= µ ( x,t)is the required expected return.XNow, let’s note F(x,t) the value of the project . The project yields a random capital gain that wecalculate using Ito’s formula:1 2 2dF = [ Ft ( x,t)+ α xFx( x,t)+ σ x Fxx(x,t)]dt + σxFx( x,t)dz(13)2The project also pays a dividend to the owner that is the profit flow π ( x,t), and therefore the totalreturn per dollar invested is:1 2π ( x,t)+ Ft( x,t)+ αxFx( x,t)+ σ x2F(x,t)2Fxx( x,t)σxFx( x,t)dt + dzF(x,t)40