Irreversible exit decisions under mean-reverting uncertainty

Abstract

Although many economic variables of interest exhibit a tendency to revert to long-run levels, mean reverting processes are rarely used in investment and disinvestment models in the literature. Previous work by Sarkar (J Econ Dyn Control 28(2):377–396, 2003), that focuses on irreversible entry decisions, showed that mean reversion has three effects on investment: (a) the “variance effect” (mean reversion reduces the long-run uncertainty and thus brings closer the critical investment level), (b) the “realized price effect” (the lower variance resulting from mean reversion makes it less likely to reach extreme high or low price levels, thereby reducing the likelihood of reaching the investment trigger) and (c) the “risk discounting effect” (mean reversion lowers the required rate of return, which affects both the project value and the value of the real option to invest). Metcalf and Hassett (J Econ Dyn Control 19(8):1471–1488, 1995) and Sarkar (J Econ Dyn Control 28(2):377–396, 2003) showed that (a) and (b) work in opposite directions, essentially canceling each other out, however the effect of (c) depends on parameter values, making the overall effect (a–c) of mean reversion on entry decisions ambiguous and parameter-dependent. In this paper, we show that as far as irreversible exit decisions are concerned, the effect of mean reversion is negative: Mean reversion unambiguously lowers the rate of irreversible disinvestment/exit for reasonable parameter values, since the mean reversion in this case only affects the value of the real option to exit and not the value resulting from (real) option exercise.

Appendix A

In order to solve (8) subject to (9), write the homogenous part of Eq. (8) in the following form

$$\begin{aligned} V_{H}^{\prime \prime }\left( P\right) +\left[ \frac{2\kappa \theta }{\sigma ^{2} P}-\frac{2\left( \kappa +\lambda \rho \sigma \right) }{\sigma ^{2}}\right] V_{H}^{\prime }\left( P\right) -\frac{2r}{\sigma ^{2}P}V_{H}\left( P\right) =0. \end{aligned}$$

(22)

with $V_{H}$ representing the homogenous part.

From (Abramowitz and Stegun (1972), eq. 13.1.35), the general confluent differential equation

$$\begin{aligned}&w^{\prime \prime }\left( z\right) +\left[ \frac{2A}{z}+2f^{\prime }\left( z\right) +\frac{b h^{\prime }\left( z\right) }{h\left( z\right) }-h^{\prime }\left( z\right) -\frac{h^{\prime \prime }\left( z\right) }{h^{\prime }\left( z\right) }\right] w^{\prime }\left( z\right) \nonumber \\&\quad + \left[ \left( \frac{b h^{\prime }\left( z\right) }{h\left( z\right) }-h^{\prime }\left( z\right) -\frac{h^{\prime \prime }\left( z\right) }{h^{\prime }\left( z\right) }\right) \left( \frac{A}{z}+f^{\prime }\left( z\right) \right) +\frac{A\left( A-1\right) }{z^{2}}+\frac{2Af^{\prime }\left( z\right) }{z}\right. \nonumber \\&\quad +\left. f^{\prime \prime }\left( z\right) +\left[ f^{\prime }\left( z\right) \right] ^{2}-\frac{a\left[ h^{\prime }\left( z\right) \right] ^{2}}{h\left( z\right) }\right] w\left( z\right) =0 \end{aligned}$$

(23)

has general solution of the form

$$\begin{aligned} w\left( z\right) =C z^{-A}e^{-f\left( z\right) }M\left( a,b,h\left( z\right) \right) +D z^{-A}e^{-f\left( z\right) }U\left( a,b,h\left( z\right) \right) \end{aligned}$$

(24)

with $C,D$ arbitrary constants, $M\left( a,b,h\left( z\right) \right)$ Kummer’s confluent hypergeometric function, and $U\left( a,b,h\left( z\right) \right)$ Tricomi’s confluent hypergeometric function given by

$$\begin{aligned} U\left( a,b,h\left( z\right) \right)&= \frac{\pi }{\sin {b \pi }}\left[ \frac{M\left( a,b,h\left( z\right) \right) }{\Gamma \left( 1+a-b\right) \Gamma \left( b\right) }\right. \nonumber \\&\left. -\left[ h\left( z\right) \right] ^{1-b} \frac{M\left( 1+a-b,2-b,h\left( z\right) \right) }{\Gamma \left( a\right) \Gamma \left( 2-b\right) }\right] , \end{aligned}$$

(25)

where $\Gamma \left( .\right)$ is the Gamma function.

Set $A=-\gamma$, $f\left( z\right) =0$ and $h\left( z\right) =\frac{2 \kappa \theta }{\sigma ^{2} z}$ in (23) to get after several algebraic computations

$$\begin{aligned}&z w^{\prime \prime }\left( z\right) +\left[ 2-b-2\gamma +\frac{2\kappa \theta }{\sigma ^{2} z}\right] w^{\prime }\left( z\right) \nonumber \\&\quad +\left[ \frac{b\gamma -2\gamma +\gamma \left( \gamma +1\right) }{z}-\frac{2\kappa \theta \left( \gamma +a\right) }{\sigma ^{2} z^{2}}\right] w\left( z\right) =0 \end{aligned}$$

(26)

with general solution

$$\begin{aligned} w\left( z\right) =C z^{\gamma }M\left( a,b,\frac{2\kappa \theta }{\sigma ^{2} z}\right) +D z^{\gamma }U\left( a,b,\frac{2\kappa \theta }{\sigma ^{2} z}\right) \end{aligned}$$

(27)

Observe that if we represent $P$ with $z$ and $V_{H}\left( P\right)$ with $w\left( z\right)$, differential Eq. (22) and (26) would coincide if and only if

$$\begin{aligned} 2-b-2\gamma&= -\frac{2\left( \kappa +\lambda \rho \sigma \right) }{\sigma ^{2}} \nonumber \\ b\gamma -2\gamma +\gamma \left( \gamma +1\right)&= -\frac{2r}{\sigma ^{2}} \\ 2\kappa \theta \left( \gamma +a\right)&= 0 \nonumber \end{aligned}$$

(28)

Solving the system of Eqs. (28) for $a,b,\gamma$ yields two sets of solutions:

$$\begin{aligned} \gamma _{1,2}&= \frac{2\left( \kappa +\lambda \rho \sigma \right) +\sigma ^{2}\pm \sqrt{8r\sigma ^{2}+\left( -2\kappa -2\lambda \rho \sigma -\sigma ^{2}\right) }}{2\sigma ^{2}} \nonumber \\ b_{1,2}&= 2-2\gamma _{1,2}+\frac{2\left( \kappa +\lambda \rho \sigma \right) }{\sigma ^{2}} \\ a&= -\gamma _{1,2} \nonumber \end{aligned}$$

(29)

The boundary condition (9) rules out the set that depend on the positive root, say $\gamma _{1}$, and thus the solution of Eq. (22) is

$$\begin{aligned} V_{H}\left( P\right) =\left[ C M\left( -\gamma ,b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) +D U\left( -\gamma ,b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) \right] P^{\gamma }, \end{aligned}$$

(30)

with $\gamma \equiv \gamma _{2}=\frac{2\left( \kappa +\lambda \rho \sigma \right) +\sigma ^{2}-\sqrt{8r\sigma ^{2}+\left( -2\kappa -2\lambda \rho \sigma -\sigma ^{2}\right) }}{2\sigma ^{2}}$.

A solution to the inhomogeneous part of (8) is obviously

$$\begin{aligned} V_{I}\left( P\right) =\frac{P}{r+\kappa +\lambda \rho \sigma }+\frac{\kappa \theta }{r\left( \kappa +\lambda \rho \sigma \right) }-\frac{\kappa \theta }{\left( r+\kappa +\lambda \rho \sigma \right) \left( \kappa +\lambda \rho \sigma \right) }-\frac{c}{r}\qquad \end{aligned}$$

(31)

[see the condition in (9)]. Combining the solutions of the homogenous and inhomogeneous part, the general solution of (8) becomes

$$\begin{aligned} V\left( P\right)&= V_{H}\left( P\right) +V_{I}\left( P\right) =\left[ C M\left( -\gamma ,b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) +D U\left( -\gamma ,b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) \right] P^{\gamma }\nonumber \\&+\frac{P}{r+\kappa +\lambda \rho \sigma }+\frac{\kappa \theta }{r\left( \kappa +\lambda \rho \sigma \right) }-\frac{\kappa \theta }{\left( r+\kappa +\lambda \rho \sigma \right) \left( \kappa +\lambda \rho \sigma \right) }-\frac{c}{r}\quad \quad \end{aligned}$$

(32)

Equation (10) in the text results from Eq. (32) by doing the following: First, substitute the definition of Tricomi’s confluent hypergeometric function from (25) in (32), collect similar terms and simplify to get

$$\begin{aligned} V\left( P\right)&= \left[ BM\left( -\gamma ,b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) \right. \nonumber \\&\left. +L\left( \frac{2 \kappa \theta }{\sigma ^{2} P}\right) ^{1-b}M\left( 1-\gamma -b,2-b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) \right] P^{\gamma }\nonumber \\&+\frac{P}{r+\kappa +\lambda \rho \sigma }+\frac{\kappa \theta }{r\left( \kappa +\lambda \rho \sigma \right) }-\frac{\kappa \theta }{\left( r+\kappa +\lambda \rho \sigma \right) \left( \kappa +\lambda \rho \sigma \right) }-\frac{c}{r},\quad \quad \quad \end{aligned}$$

(33)

with $B,L$ constants that involve $C,D$ and the $\Gamma \left( .\right)$ function. To ensure the boundary condition (9), it must be that

$$\begin{aligned}&\lim _{P \rightarrow +\infty }\left[ BM\left( -\gamma ,b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) \right. \\&\qquad \qquad \left. +L\left( \frac{2 \kappa \theta }{\sigma ^{2} P}\right) ^{1-b}M\left( 1-\gamma -b,2-b,\frac{2\kappa \theta }{\sigma ^{2} P}\right) \right] P^{\gamma }=0. \end{aligned}$$

As $P\rightarrow +\infty$, $\frac{2\kappa \theta }{\sigma ^{2}P}\rightarrow 0$, $M\left( .,.,\frac{2\kappa \theta }{\sigma ^{2}P}\right) \rightarrow 1$ and thus for

$$\begin{aligned} \lim _{P \rightarrow +\infty }\left[ B+L\left( \frac{2 \kappa \theta }{\sigma ^{2} P}\right) ^{1-b}\right] P^{\gamma }=0 \end{aligned}$$

to hold, it must be that $L=0$, given that $\gamma <0$ and $1-b<0$ from (29).