Hysteresis effects under CIR interest rates

Most decision making research in real options focuses on revenue uncertainty assuming discount rates remain constant. However, for many decisions revenue or cost streams are relatively static and investment is driven by interest rate uncertainty, for example the decision to invest in durable machinery and equipment. Using interest rate models from Cox et al. (1985b), we generalize the work of Ingersoll and Ross (1992) in two ways. Firstly, we include real options on perpetuities (in addition to zero coupon cash flows). Secondly, we incorporate abandonment or disinvestment as well as investment options, and thus model interest rate hysteresis (parallel to revenue uncertainty in Dixit (1989a)). Under stochastic interest rates, economic hysteresis is found to be significant, even for small sunk costs.     

Efficient rare-event simulation for perpetuities

We consider perpetuities of the form

D=_B1exp(_Y1)+_B2exp(_Y1+_Y2)+?,$D=B_{1}exp\left(Y_1\right)+B_{2}exp(Y_1+Y_2)+?\text{,}$

where the _Yj$Y_j$’s and _Bj$B_j$’s might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the _Yj$Y_j$’s satisfy the so-called Cramér condition with associated root _θ∗∈(0,∞)${\theta }_{\ast }\in (0,\infty )$ and that the tails of the _Bj$B_j$’s are appropriately behaved so that D$D$ is regularly varying with index _θ∗${\theta }_{\ast }$. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the _Yj$Y_j$’s according to _θ∗${\theta }_{\ast }$ fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.     

Matrix refinement type equations

Taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the Fourier transform of an L¹-solution of the following random matrix refinement type equation

f(x)=_∫Ω|detL(ω)|C(ω)f(L(ω)x-M(ω))P(dω),