Energy Price Jumps, Fat Tails and Climate Policy

Abstract

Many authors who have analyzed key energy prices, such as crude oil and natural gas, have found that these prices exhibit “fat tails”—the feature that large percentage changes occur far more often than would be predicted by a conventional model. These fat tails can arise either because of time-varying volatility or because of rapid, unexpected changes—also known as jumps. Addressing global climate change is likely to require broad-based deployment of new infrastructure. This new infrastructure is likely to be both costly to build and difficult to reverse—suggesting the deployment of new infrastructure is an example of “investment under uncertainty” [1]. In this context, a key concept is the “option value of waiting,” i.e., the potential gain in value that arises from waiting to learn more about the evolution of some key underlying stochastic ingredient, such as a commodity price or the cost of a carbon permit. We argue that this option value of waiting is likely to be increased by the presence of jumps. Assuming there is some urgency in undertaking these investments, the increase in option value of waiting is worrisome and motivates the deployment of a policy intervention that reduces this option value.

Appendix

In this Appendix, we provide the mathematical details underlying our numerical simulations. We start by fleshing out the problem under geometric Brownian motion (GBM) process. As noted in eq. (4), the optimal value function satisfies

$$\begin{aligned} \rho F(X)= \frac{1}{dt}E[\text {d}(F)]. \end{aligned}$$

(12)

We proceed by expanding the expression on the right-hand side, which gives rise to¹⁵

$$\begin{aligned} \frac{1}{dt}E[\text {d}(F)] = \alpha X F^{\, \prime }(X) + \frac{1}{2} \sigma ^2 X^2 F^{\, \prime \prime }(X). \end{aligned}$$

As a result, the fundamental equation of optimality reduces to the second-order differential equation

$$\begin{aligned} \frac{1}{2} \sigma ^2 X^2 F^{\, \prime \prime }(X) + \alpha X F^{\, \prime }(X) - \rho F(X) = 0; \end{aligned}$$

(13)

the generic solution of this differential equation is the power function

$$\begin{aligned} F(X) = a X^\beta . \end{aligned}$$

Inserting this form into the fundamental equation of optimality, taking derivatives as appropriate, and combining terms leads to the characteristic equation

$$\begin{aligned} \frac{\sigma ^2}{2} \beta (\beta -1)+ \alpha \beta - \rho = 0, \end{aligned}$$

(14)

which the parameter $\beta$ must satisfy. There are two roots, which we denote as $\beta _1$ and $\beta _2$; it is easy to see that these roots satisfy $\beta _1 < 0$ and $\beta _2 > 1$. The complete solution to the fundamental equation of optimality is then

$$\begin{aligned} F(X)= a_1 X^{\beta _1} + a_2 X^{\beta _2}. \end{aligned}$$

The solution must also satisfy a boundary condition, namely that the optimal value becomes negligible as the spot price becomes very small (since it will almost surely not be optimal to invest under those conditions, the option to wait is essentially worthless). Since $\beta _1 < 0$, the first term describing F would be unbounded if slope parameter $a_1$ were non-zero. Hence, the requirement that F tends to zero as X tend to zero implies $a_1 = 0$. Accordingly, the value function can be simplified to

$$\begin{aligned} F(X)= a X^{\beta }, \end{aligned}$$

(15)

where we have substituted a for $a_2$ and $\beta$ for $\beta _2$ to reduce notational clutter.

In addition, the solution must satisfy two other conditions, the “value-matching condition” and the “smooth-pasting condition” [1]. At $X^*$, the point at which it is optimal to build, the value-matching condition requires that the value of building equals the value of waiting:

$$\begin{aligned} F(X^*) = X^* - K. \end{aligned}$$

(16)

Because the value function F(X) can be interpreted as the value of an option to invest in the future [1], this value must equal the difference between the value upon investing when the value of investing immediately, $X^*$, and the sunk cost of investing, K. The smooth-pasting condition requires that the slope of the optimal value function equals 1, the slope of the function reflecting the value of waiting [1]. Accordingly,

$$\begin{aligned} a(X^*)^\beta = X^* - K; \end{aligned}$$

(17)

$$\begin{aligned} a \beta (X^*)^{(\beta -1)} = 1. \end{aligned}$$

(18)

Multiplying both sides of eq. (18) by $X^*/\beta$, substituting into eq. (17) and rearranging then yields eq. (7) in the text.

Now suppose the value X evolves according to the mixed jump-diffusion process

$$\begin{aligned} dX = \tilde{\alpha } X dt + \sigma X dz + (Y-1)dq, \end{aligned}$$

where as above dz is an increment of a standard Weiner process; here, we interpret Y as the magnitude of a jump if it occurs, and dq as an increment of a Poisson process (which captures the probability that a jump occurs). Because the term capturing jumps will add something to the expected value, the mean term ($\alpha$ in the GBM representation) must be adjusted. Denoting the arrival rate under the Poisson process by $\lambda$, and the mean value of a jump (should one occur) by $\theta$, the drift term is [41]

$$\begin{aligned} \frac{1}{dt}E[\text {d}(F)]{X} = \alpha^{+} = \alpha + \lambda \theta . \end{aligned}$$

In this setting, the equation governing optimal value function is more complicated than that of eq. (13). Here, the equation governing the optimal value function is determined by the interaction between jump size, Y, and continuation value, V:

$$\begin{aligned} \frac{1}{2} \sigma ^2 X^2 F^{\, \prime \prime }(X) + \tilde{\alpha } X F^{\, \prime }(X) + \lambda \int _0^{\infty } F(XY) G(Y) dY = 0, \end{aligned}$$

(19)

where G(J) is the probability density function governing jump size. As with the GBM problem, the solution here is governed by the value-matching condition (5), as well as a smooth-pasting condition. This equation cannot be solved analytically, and so we use numerical methods as discussed in Sect. 4.