Factoring Uncertainties into Climate Policy: Hedging Emission Pathway
Abstract
At this time, scientists are calling for mitigation policy to limit global warming within 2 0C, a
temperature target that should prevent major irreversible changes in the climatic system.
However, economists express concern that such an ambitious environmental goal could be
prohibitively expensive. Current efforts to curb carbon emissions will generate some short and
mid term costs while pushing benefits sometime into the distant future. The expected value
approach to the cost-benefit analysis of climate policy dominates literature. With uncertainties,
the approach relies on the aggregated estimation of various outcomes of climate policy, weighted
and averaged by probabilities. The variance, skewness, and kurtosis are important characteristics
of uncertainties and risk associated with selected climate policy, but can be easily lost in
aggregation. Alternatively, the real option analysis (“ROA”) explicitly accounts for the expected
value of underlying assets but also considers the shape of distribution and therefore factors in the
variance of the expected value (as well as crucial moments of distribution). In this paper, we
explain application of the real options methodology as an alternative to the conventional costbenefit analysis of climate policy. Due to the irreversibility of negative transformations in the
climatic system, flexibility has economic value that could be estimated as an option value of
climate policy. The fat tail distribution of damage attributed to global climate change suggests
that flexibility in terms of future climate policy could be more valuable than flexibility in terms
of irreversible investment into sunk costs needed to ensure flexibility of climate policy.
We illustrate the real option analysis of climate policy based on numerical experiments with
different option pricing formulas and discuss the advantages and disadvantages of these formulas
on climate policy. We formulate some policy recommendations based on quantitative analysis of
relative uncertainties of cost and benefits of climate policy.
Key words: Climate policy; uncertainty; real options
1. Introduction
The expected value approach to the cost-benefit analysis of climate policy dominates literature.
With uncertainties, the approach relies on the aggregated estimation of various outcomes of
climate policy, weighted and averaged by probabilities. The variance, skewness, and kurtosis are
important characteristics of uncertainties but can be easily lost in aggregation. Alternatively, the
real option analysis explicitly accounts for the expected value of underlying assets and considers
the shape of distribution factoring in the variance of the expected value (as well as crucial
moments of distribution).
Irreversibility is an important characteristic of climate policy. Selection of a particulate emission
target may lead to irreversible consequences, regarding both climate and the economy (e.g. sunk
cost). Therefore, climate policy should focus on a dynamic balance between two irreversible
decisions to maximize flexibility and avoid irrecoverable damages and costs.
Page 1 of 23
The real options methodology emerges as a relevant analytical framework. The decision on
climate policy could be formulated as a deferral option. Acceptance of a policy target means that
policy makers may select a particular state of environment that prevents socio-economic damage
associated with a status quo policy, i.e. defer further “depletion” of climate asset. So,
implementation of an interim climate policy could be described as the purchase of an option on a
climate asset. The value of this climate asset equates to averted damage to the environment. This
value, however, is unknown. Policy makers assume a distribution that captures the magnitude of
uncertainties.
In the view of scientists and some politicians, limiting a temperature increase within 20C above
the pre-industrial era should prevent irreversible changes in the climatic system. But most
economists view such a policy as excessively expensive. If decision makers reject this policy,
they save on the mitigation sunk cost, but abandon the climate asset whose value may potentially
be higher than the mitigation cost required to prevent degradation of the climate asset. Policy
makers may decide temporarily to maintain flexibility until they learn more about climate change
and the cost of mitigation policy and then make a final decision on how to use the climate asset.
The mitigation sunk cost for the “learning period” is exactly the price to keep this asset in
possession.
In the context of irreversibility, flexibility has economic value. The fat tail distribution of
damage attributed to global climate change suggests that flexibility in terms of correction of
emission pathways could be more valuable than flexibility in terms of avoidance of sunk costs
needed to ensure flexibility of climate policy.
Obviously, peaks and variances are the most critical parameters that describe the avoided
damage and mitigation costs. The expected value approach uses probabilities directly as
coefficients to calculate a weighted outcome from uncertain parameters, not accounting for
variance. Like many others, we believe that the expected value, or expected utility, is not an
ideal valuation instrument for climate policy due to the fat tail nature of the distribution of
negative climate outcomes. There is a need for an instrument that considers both "peaks" and
"tails”.
An option pricing formula provides policy makers with a single number that is a “one variable
image” of a multivariable object. But a scalarization process based on option pricing models
preserves important information on underlying uncertainties. The only condition for application
of option pricing models is finite variance. Damage still could be significant but is bounded. In
the next section, we present ROA methodology as an alternative to the conventional aggregation
technique offered by the expected value approach in detail and explain how it could be applied to
climate policy valuation.
In this paper, we propose an application of the real options analysis to formulate rules to select
an interim climate policy (emission target) and estimate the economic value of the future
flexibility. The interim climate policy may be corrected in the future in response to new
knowledge that would hopefully reduce uncertainties.
Page 2 of 23
2. Conventional approach and state of debates on uncertainties and climate policy
Despite abundant literature on the economics of climate change, it is hard to disagree with
William Nordhaus that the key questions regarding climate change policy - "how much, how
fast, and how closely" - remain unanswered (Nordhaus 2007). On one hand, publications like
the Stern Review (Stern 2006) and AR4 (IPCC 2007) (see also Stern 2009) urge immediate
actions to drastically curb carbon emissions, appealing to newly available information in the field
of climate science. On the other hand, the application of conventional economic analysis
(Nordhaus 2007; 2008) show that such an urgent and sharp GHG reduction does not warrant the
cost.
In most integrated assessment models (“IAMs”), damage appears as insignificant losses of GDP
(for further discussion, see Golub et al 2009). Climate policy imposes costs now, but does not
return enough to justify a strict emission target like stabilization of temperature increase within
20C. An optimal policy could be formulated as “do something, but not much”. DICE-2007
computes an optimal trajectory with temperatures going up about 3.50C. Optimal carbon tax is
about $7.5/tCO2 (see Nordhaus 2008). Adding catastrophic events, we have “infinite damage”
(Weitzman 2007), and then even drastic and immediate reduction is insufficient and geoengineering becomes the only way to tackle climate change.
Application of either methodology reaches a similar conclusion: rejection of near-term interim
temperature target. Cost benefit analysis based on expected value maximization suggests
introduction of a modest carbon tax. Consideration of distributions with infinite variance
suggests that even immediate conversion to carbon-free economy does not eliminate unbounded
damage. Then geoengineering (that we can interpret as adaptation to inevitable climate change)
appears as the dominant policy and mitigation plays a secondary role. If, however, damage is
bounded, ROA offers an optimal mitigation policy that balances risks of uncontrollable costs and
fat tail unrecoverable damage that is subject to adaptation. Geoengineering would be an
important but not dominant element of a climate hedging portfolio.
In the literature there are several attempts to correct the expected value methodology. A full
review is beyond the scope of this paper; however, some relevant work must be mentioned. As
noted in the literature, the conventional approach does not account for the presence of thresholds
in the concentration-response function and the risk of global catastrophic climate events that,
though characterized by a low probability of occurrence, would lead to significant economic
damage. Tol (2003) and Weitzman (2007) present a strong case highlighting the major
limitations of cost benefit analysis based on the expected value calculation in the context of
climate policy.
Stern attempts to solve the problem by applying an exogenous social discount rate (0.1-0.15%)1,
but this artificially low discount rate coan not be explain in Ramsey framework that constitutes
foundation for most of IAMs. Yet risk-adjusted discount rate could be an instrument to account
for negative impact of climate induced productivity shocks on economy.
1
See for example Stern 2006
Page 3 of 23
The nature of productivity shocks is critical too. Gollier and Weitzman 2009 and Gollier 2010
and consider a hilly stylized model with irreversible investment decision should be taken now
when future return on capital is a subject to unknown productivity shocks and proposed
application of risk adjusted discount rate that depend on magnitude of permanent shocks.
As extension of Gollier and Weitzman (2009) approach Golub and Keohane (2010) consider
endogenous productivity shocks and distinct between i.i.d. and permanent shocks. Climate
change is a perfect example of endogenous productivity shocks. Accumulation of greenhouse
gases in the atmosphere leads to irreversible transformation of environment and result in
permanent (or at least long-term) negative productivity shocks. Two different growth alternatives
(or two different economies) that have different impact on environment and as result are subjects
to shocks of a different kind. “Clean economy” is subject to i.i.d. and “Dirty economy” is subject
to both i.i.d and permanent shocks induced by irrevercible changes in climatic system.
Magnitude of permanent shocks is a function of accumulated pollution. Then there is a feedback
between current climate policy and future productivity shocks. In Ramsey framework risks
attributed to these shocks could be reflected in adjusted discount rate.
2.1.Uncertainty is at the heart of the climate change debate
There are several unknown parameters that constrain environmental policy. We do not know:
•
•
•
•
Magnitude of climate response to increased GHG concentrations;
Cost of climate policy to mitigate GHG emissions;
Adaptation potential and magnitude of adaptation cost;
Business response to various policy instruments.
Technological developments also are a significant random component. We cannot say for sure
what the cost of environmental policy would be or where the optimal balance between mitigation
and adaptation is. Yet through careful risk management, a decision should be made regarding
these uncertainties.
Underlying uncertainties could be summarized as in Figure 1:
Page 4 of 23
Figure 1. Sources of uncertainties in climate change estimations
What we do know:
• The temperature increase could be significant but bounded;
• Even catastrophic damage is a finite number;
• Mitigation cost could be significant but truncated by existence of back-stop
technology;
• Learning narrows uncertainties over time.
Consensus is summarized in IPCC reports. Based on IPCC reports and various peer reviewed
publications, we can quantify these uncertainties and assign particular distributions to underlying
parameters. These publications represent the current state of knowledge about climate change,
mitigation and adaptation cost. Then we can run Monte-Carlo simulations and compute
combined uncertainties. Monte-Carlo simulations simply combine uncertainties and do not apply
additional assumptions. One important requirement remains for future application of ROA: each
underlying uncertain parameter should be described by distribution with finite variance. MonteCarlo simulation to calculate implied volatility creates the basis for forward-looking analysis of
current climate policy without aggregation or losses of primary information that are available
now.
2.2.Peaks, tails and variance
While a number of scientists and politicians call for a mitigation policy to limit global warming
within 20C (see for example Stern Review 2006, IPCC AR4, Stern 2009, etc), a temperature
target that should prevent major irreversible changes in the climatic system, economists express
concern that such an ambitious environmental goal could be very expensive.
Applying conventional methodology that focuses on the expected value approach (also known in
the literature as expected utility) may not be the best way to understand benefits of climate
policy, where there is an immediate economic burden but distant positive effect in the future.
The expected value approach averages out various climate outcomes; therefore it may not be an
Page 5 of 23
appropriate tool for accurate analysis of irreversible processes with outcomes described by fat
tailed distribution.
Figure 2 presents results of Monte-Carlo simulations for a particular numerical example we
selected to illustrate the methodology. A detailed description of this example is provided in
Annex 1 (for more detailed description see Anda, Golub, Strukova 2009).
Figure 2. Costs and benefits of climate policy
Figure 2 highlights an important aspect of climate policy analysis: the tradeoff between expected
values on one hand and tail and variance on the other. In this particular example, expected cost
of the policy “outweighs” expected benefits. Conventional cost-benefit analysis will reject this
policy. However, the presence of a fat tail in the benefits distribution suggests potential high
damages if the policy is rejected. With relatively low yet significant probability, the damage
may reach double-digit figures. There is a 10% probability that the irreversible damage process
results in costs of more than 5.7% of the gross world product (“GWP”), while there is a 90%
probability that the cost of a policy is less than 4.4% of the GWP. Therefore, the choice is
between higher costs versus higher risk. The expected value approach masks this trade-off.
In this paper, we illustrate the application of a “real option” analysis to climate policy
assessment. Numerical experiments with simplified version of the DICE 2007 model
demonstrate the positive value of flexibility afforded by an interim climate policy aimed at
avoiding irreversible changes. We found that the economic value of flexibility exceeded the net
present value (“NPV”) of the costs of an interim policy. While this requires a more substantial
analysis, we hope to provide a framework within which this can be done.
3. Irreversibility and value of waiting
Page 6 of 23
When outcome or cost of policy is uncertain and policy decision is irreversible, there could be a
positive value of postponing decision until major uncertainties are resolved. Flexibility has an
option value that is higher when uncertainties are higher. For illustration we adopt a simple
approach presented in Dixit (1992) and Pindyck (2004) that is easy to understand and provide
some illustrations.
Let capital investment K yields annualized revenue R . The simplest decision rule is to invest
when project has a positive expected return;
R  rK  M
(1),
where r stands for risk free discount rate and M denotes brake even point for investment. In
other words, M is a minimum level of return for investments to pay back. The investment rule
(1) holds in deterministic case only. If benefits are uncertain, then decision rule (1) transforms
into (2):
B  H  rK  M
(2)
Waiting for a certain amount of time allows an investor to avoid downside risk (Dixit 1992), but
it doesn’t make sense to wait any longer if net revenue reaches sufficiently high level H . If
however, policy maker is forced to make a decision before observed revenue reaches level H ,
then lost flexibility value could be considered as a penalty for adoption of that policy.
In Ramsey model discount rate is equal to marginal product of capital. If marginal product of
capital is subject to independent identically distributed (i.i.d.) random shocks, then discount rate
equals to the expected productivity.
Benefits of climate policy depend on uncertain repose of climatic system to anthropogenic
emissions and uncertain response of socioeconomic system to changing climate. Cost of climate
policy is uncertain too. Unlike in the case described above, decision maker does not observe
process in time and does not have a flexibility to implement climate policy when it is clear that
avoided damage is large enough to recover sunk cost of mitigation policy. It is overdue to act
when there is a 100% certainty about the damage attributed to climate change. Therefore a
decision maker should pick a climate policy (say, a concentration target) now, substantially
before the moment when major uncertainties regarding cost and benefits of climate policy are
fully resolved.
In the beginning a decision maker has a climate asset A. Long-term expected value of this asset
is D that is a function of the selected concentration target. Implementation of climate policy
requires cost Z . If the policy is discontinued, then decision maker irreversibly loses the asset A,
but has some savings on climate policy cost. Let’s call these savings an asset B. Application of
the Dixit rule allows to formulate conditions when it is optimal to discontinue climate policy
taking into account value of waiting attributed to uncertainties that determine actual value of
assets A and B.
Page 7 of 23
Asset A has an expected value D , it could be exchanged for an asset B that has an expected
value Z . Investment decision to continue to posses A results in expected cost Z  D , penalty
 D and “premium”  Z since holder of the asset A postpones decision to exchange the asset A.
Or vise versa, investment decision to switch to the asset B yields instant benefits Z  D , but
implies cost equal to the lost net value of waiting  Z  D .
In terms of a stylized analysis presented in this section, selection of a climate policy is selection
of a long-term stabilization GHG concentration target Q . Selecting particular target a policy
maker loses flexibility that has a value equal to the value of waiting presented in figure 3 by
curves  D (Q) and  Z (Q) . According to Dixit-Pindyck formula this value is a convex function:
 (Q)   BD (Q) 
(3),
where B is a constant, and D (Q ) stands for economic benefits of stabilization target Q . Power
coefficient  is a function of volatility  and discount  :

1
8 
1  1  2 
2
 
(4)

 0 . It implies that value to postpone climate policy is higher if

volatility of climate benefits are lower and, vise versa, the value of waiting and defer climate
policy is lower, if uncertainties associated with future damage are higher. For instance in the
numerical example presented above,  D  1.04 and  Z  1.46 .
Note that   1
and
As presented in figure 3, we assumed D  Z then:
Z  D  H D  H Z
It implies that in order to commit climate policy a decision maker should observe relatively
lower expected return on climate asset. It also implies that in the neighborhood of D  , net
penalty for abandoning a climate asset is positive. These results hold for marginal cost and
benefits of climate policy too.
Page 8 of 23

Z
D
HD
R
HZ
Z D
Figure 3. Value to wait and climate policy (R denotes annualized revenue).
Next we consider more general case when D  Z . In the numerical example presented in
section 2, D  Z If a decision maker chooses to save on mitigation cost, he looses the value of
waiting, but saves on abatement cost simultaneously loosing the value of climate asset. In the
numerical example from section 2 the difference between the savings on cost and the lost
benefits is 14. Net savings in this case is the difference between the expected values of cost and
benefits minus penalty that equals to the lost value of waiting. The decision maker picks a
climate policy minimizing net losses.
In numerical example presented in previous section, the expected benefits of climate policy are
even below the expected cost. If climatic change and damages attributed to climatic change a
fully reversible, then the best strategy is to wait, see what happens, and then adopt the policy Q ,
as soon as there is enough evidence that benefits of climate policy are equal or higher than H D .
But climatic change and economic damage attributed to climatic change is irreversible.
Therefore the target Q may not be feasible any more.
Rejection of climate policy Q yields some savings on mitigation cost Z on expense of future
damage D . H Z is a threshold. The expected savings on abatement cost should reach it in order
to reject a climate policy.  Z (Q) is the value of waiting and postponing a decision about
dropping stabilization target Q and saving on abatement cost. In other words,  is a penalty for
an early adoption of a climate policy. For lower concentration target this penalty should be added
to the negative expected benefits (difference between avoided damage and mitigation cost). But
if policy is relaxed and emission target increased, then penalty    Z (Q) is even higher. The
Page 9 of 23
cost of inaction would be D   Z (Q) /  , while cost of climate policy would be Z   D (Q) /  .
Selection of a long-term climate policy implies significant reduction of flexibility and loss of
value of flexibility.
For a given concentration target Q we can estimate net annualized benefits of adoption that
target:
D (Q)  Z (Q)  
Z
(Q)   D (Q)
(5)
And from (5):
D (Q)  Z (Q)  B( Z Z  D  D )
(6)
For instance, in numerical example from section 2,  D  1.04 and  Z  1.46 . Recall
D (Q)  Z (Q)  14 for Q  450 ppm , if B  1 , then annualized value of an option to prevent
climate change by keeping the target at 450 ppm level equals 2.1.
Further analysis of (5) requires closed-form expression for B . Let 20C limit on global
temperature increase is an environmental target. Then 450 ppm could be considered as a starting
brake-even point for cost benefit analysis of climate policy. Then
Q  450  ( D (Q)  Z (Q))   B( z Z  d D )  0
and B 
Z D
,
( z  d D )
Z
where z  Z ; d  D .
In our numerical example B  0.2
Starting with an initial target, policy makers observe dynamics of an expected damage and
expected policy cost and correct (if possible) concentration target Q , solving (5) for Q and
plugging in new expected values for benefits and cost of climate policy. Although initial
decisions could be mostly irreversible, there could be a slight flexibility on a side of climatic
system and greater flexibility on a side of an abatement target.
In Dixit (1992), value of waiting is interpreted as an adjustment coefficient for a discount factor.
As we pointed in section 2, it is a plausible approach to estimate risk adjusted cost and benefits
of climate policy that is coherent with Gollier (2010) and Gollier, Weitzman (2009) analysis.
D
 would be applied to savings in
Relatively higher adjusted discount coefficient
D 1
Page 10 of 23
abatement cost attributed to untightening of emission target. Relatively lower discount rate
Z
 would be applied to avoided damage.
Z 1
Note that this adjustment of discount rate does not require any artificial steps like in Stern
(2006), and follows from application of expected value approach. This approach is consistent
with the Ramsey framework. The result holds for irreversible shocks only, and therefore it is
applicable to irreversible losses.
From (4) it is easy to see that relative adjustments of discount are a function of variance.
Although this method is better than “raw” expected value approaches since it takes into account
first and second moments of distribution. Risk adjusted discount rate may be a good tool to use
in IAMs and for other practical applications that require simplified numerical analysis. It is not
good enough to capture distribution shape and tail. Advanced option pricing formulas we discuss
in the next section take into account third and fourth moments.
4. Real option alternative
The ROA methodology provides a way to quantify relative uncertainties and takes into
account both "peaks” and “tails". For instance, a higher variance yields a higher option
value. If the abatement cost variance is kept well below the variance of climate
outcomes, we are "going long" on the relatively cheap option of new technologies to
cover our "short" position on a relatively expensive climate option. In other words,
under this methodology, we would undertake emission reductions now in order to keep
the option of stabilizing emissions at safe levels. We would keep this option open until
new information on climate change and its impact becomes available. To use this
methodology, we need better variance (as well as skewness and kurtosis) data from
both sides of the trade, meaning positive feedback loops on the climate side and
narrowing technology cost ranges on the abatement side.
The ROA methodology was developed relatively recently, and it is one of the fastest growing
areas of investment analysis (see for example, Copeland et al. 2000; Copeland and Antikarov
2003, Mun 2005, Trigeorgis 1996, Schwartz and Trigeorgis 2004). Recently, this methodology
was applied to environmental policy analysis, focusing on option analysis into abatement
technologies. It is the first time we apply ROA to evaluate climate policy (see Golub at al 2008,
Anda, Golub, Strukova 009), although there were examples of applying the event tree approach
(see Webster 2008).
4.1. Testing new tool
As we noticed in Figure 2, the underlying asset – avoided damage - exhibits a more significant
variance than the cost needed to prevent degradation, i.e. irreversible losses of this asset. A high
relative variance of damages over costs favors a more stringent emission target. Average return
on this investment does not cover the cost of climate policy, yet the option value is positive. In
Page 11 of 23
this numerical example, the positive option value of climate policy outweighs negative net
expected value. High relative skew and kurtosis of damages over costs indicates further benefits
of more stringent targets.
150
100
95
80
93 95
93
Benefit
52
50
23
NPV
0
-50
Cost
Mean
-15
Median
-39
SD
Figure 4. Expected value and uncertainties
Figure 4 illustrates potential problems with valuation of climate policy based on expected value
of cost and benefits and helps to understand how ROA brings different perspectives into
assessment of climate policy. In the example, the mean value of net benefits of 450 ppm
concentration target is negative. Median analysis of net benefits suggests that the difference
between cost and benefit of that climate policy is even more significant. The standard deviation
of climate benefit that captures uncertainties of climate outcome, however, is about four times
greater than standard deviation of the cost. Moreover, SD of the net benefits is six times higher
than the absolute value of net benefits, which are negative in the example. The standard
deviation or volatility is the simplest indicator of riskiness of climate policy. In two-standard
deviation range, present value of climate benefits almost reaches 270 trillion while cost of the
policy is just slightly higher than 140 trillion.
At the same time, the numbers are indicative rather than definitive. It is not likely that we could
determine an optimal climate policy (optimal concentration or emission target) from this
analysis. Indications given by option tools are well outside the ranges for conventional NPV
results. To the degree that the policy objective is hedging catastrophic climate risk, the
indications given by option based tools should be considered. Most likely, ROA is a new way of
thinking about climate policy and understanding and communicating the benefits and costs while
implicitly including the value of risk.
Risk management in climate policy requires forward looking expert judgments regarding future
negative impacts on climate and the cost of climate policy. Econometric analysis focused on
identification and extrapolation of existing trends does not help since historically observed
Page 12 of 23
climate change is not a good predictor. Underestimation of risks caused by safeguard policies
built on extrapolations of historical observation led to the financial crisis of 2008-2009.
Historical data was not a helpful indicator. Risk management should rely on forward-looking
implied volatility analysis. Weak correlation between random variables leads to the emergence
of a fat tail. Presence of a weak correlation among returns on different securities was ignored and
the fat tail was overlooked. Non linearity also leads to a fat tail emergence.
4.2.Option pricing and tail quantification
Direct calculations based on Monte-Carlo simulation give the best estimations of option value.
However, depending on the number of underlying uncertainties, Monte-Carlo simulation can be
time-consuming. Monte-Carlo simulation embedded into optimization procedure (i.e.
maximization of an option value) may create significant computational complications. Therefore
it would be helpful to apply a closed-form solution for option value. The well known BlackScholes formula is easy to apply but in cases of fat tails, it significantly underestimates option
value (see Figure 5 and Table 3).
100
80
60
40
20
0
85
76
61
32
21
11
24
3640
Damage
Cost
Policy Value
Black
Gram Edgeworth
Scholes Charlier Binomial
Figure 5. Selecting option pricing formula
The ideal model for option pricing is computable distribution of an underlying asset with
filtration (continues analogy of event tree). However if one is looking for a closed-form solution
needed in integrated assessment models, one should apply different option pricing formulas.
Black Scholes is the most famous option pricing formula that offers closed form solution for an
option, however Black-Scholes assumes normal distribution of return and therefore log-normal
distribution of the underlying asset. For a climate asset, this analysis understates value since it
does not have enough flexibility to factor in skewness and kurtosis. Gram Charlier and
Edgeworth Binomial option models offer an alternative to Black-Scholes. These models better
Page 13 of 23
capture shapes of distributions and most importantly allow for quantifying a fat tail, but still
present an approximation that could understate or overstate the value of ROA on a climate asset.
In section 3 we discussed valuation of flexibility method based on the Dixit-Pindyck approach.
Its application to climate policy suggests that the value of climate asset should be discounted less
then savings on abatement cost. This result is coherent with the Gollier-Weitzman interpretation
of risk adjusted discount rate. Both methodologies take into account first and second moments of
benefits and cost distributions. ROA allows accounting for skewness and kurtosis, and therefore
for better capturing the shape of distributions and tails.
5. Climate hedging policy: real options analysis of interim emission target
5.1.Integrated assessment modeling and uncertainty analysis
Integrated assessment models (“IAM”) are useful instrument of policy analysis. With regard to
simplification and arbitral assumptions on important but not yet observed characteristics (like
damage), IAMs should be treated more like a way of thinking about climate policy, rather than a
way to compute an optimal emission pathway, carbon tax, etc. Scenario analysis is one way to
address uncertainties (see for example Nordhaus 2008). In this case, the model solves in
deterministic forward-looking mode. Each solution was attributed to a particular set of
realization of uncertainty parameters. A user of this model should select the most plausible
scenario. Stochastic dynamic optimization is another way to address uncertainties. A recursive
solution allows accommodating adjustments in response to realization of sequential shocks.
ROA permits application of a simplified forward looking optimization procedure. First, MonteCarlo simulations are conducted to calculate expected value and combined uncertainties of cost
and benefit of an optimal climate policy. Option value could be presented as a function of
expected value and volatility. If the distribution of benefits or costs demonstrates a fat tail, a
more sophisticated pricing formula should be applied. Next, the option value of benefits should
be added to the criteria and the option value of sunk costs subtracted. Then the optimization
procedure should be repeated. If the option value of climate benefits is greater than the option
value from the recalculation, the solution should be a lower emission trajectory. Experiments
with DICE 2007 demonstrate that volatility is a constant. In this case, a simple formula for
option pricing could be applied: ROV  0.4v , where v denotes volatility and  stands for
expected value of the underlying asset. If v  v  , then Monte-Carlo simulation should be
repeated and the optimal solution recalculated.
In a sense, ROA is an extension of deterministic optimization with perfect foresight. In contrast
to optimization of expected value of climate policy, we optimize risk-adjusted value.
Adjustment is equal to the difference between option value of climate benefits and option value
of abatement sunk cost.
5.2. Uncertainties, risks and learning process
Thousands of papers in the field of climate science and economics of climate change have been
published every year in peer reviewed journals and books. IPCC critically reviews these
Page 14 of 23
publications, extracts and summarizes the most robust results in regularly published IPCC
reports. The IPCC process creates a background for quantification of uncertainties like we use in
our numerical example. Over time, it also ensures learning and policy adjustment.
The discreet character of climate policy allows for accommodating new knowledge and new
information into climate regulation periodically in response to new IPCC findings.
Internationally, UNFCCC (1992) creates foundations for modern policy. Key elements of the
convention were an environmental goal to avoid dangerous interference with the climatic system;
foundation of economic mechanism; and common but differentiated responsibility. It took five
years to negotiate the Kyoto Protocol and make a first attempt to implement the idea of
differentiated responsibility and establish a “pilot phase” of a global “cap and trade” system. It
took 15 years to build consensus regarding the 20C threshold as a long-term environmental
target. It may take several years to get the major developing countries to commit to absolute
emission targets comparable to this environmental goal. Over that time, new knowledge will be
accumulated and periodically integrated into the policy making process. The discreet
mechanism of global climate policy formation should ensure sufficient flexibility. Option
analysis offers dynamic hedging of climate policy and provides guidance for preserving
flexibility in selecting interim targets.
5.3.Option value of interim target
Recognition of the 20C threshold as an environmental target does not mean its automatic
translation into a quantitative target on emission. Figure 6 suggests a wide range of uncertainties
of climate outcomes for particular emission targets.
A stringent concentration target reduces the range of negative outcomes, but it does not
guarantee a particular result formulated in terms of the global temperature increase.
While traditional models oriented on cost benefit analysis (expected value optimization in case
of uncertainties) aim to provide “precise” recommendations like level of emission cap or level of
emission tax, ROA offers an algorithm for climate hedging policy. We call that algorithm
“climate risk management” (“CRM”). The ideas of CRM are presented climate policy evalustion
as a bottom up process. On the top of the pyramid is a scalar indicator that is assigned to a
particular sequence of emission targets, but in contrast to expected value this indicator captures
more information and allows comparison between relative risks driven by the shape of
distribution and spread of underlying uncertainties. The bottom of the pyramid represents
theoretical foundations behind that aggregated number. The middle part of the pyramid
represents the institutional mechanism of climate policy implementation. Regulations should
take into account institutional barriers, including behavioral aspects like a firm’s risk aversion
and financial constraints. Climate policy based on CRM offers a uniform metric to compare risks
and tools to minimize integrated risk.
Application of the real options methodology provides a new perspective for price versus quantity
debates. In this paper, we put aside the differences in incentives created by price and quantity in
cases of risk neutral and risk averse firms (see Golub et al 2008) and just discuss carbon tax
versus emission cap from the point of view of a regulator aiming for a particular temperature
Page 15 of 23
target (e.g. 20C increase above pre-industrial era). Carbon tax provides assurance regarding
maximum level of marginal cost of climate policy, while it leaves climate outcome uncertain. An
emission cap provides assurance regarding total emissions, but it leaves room for uncertainties.
Mitigation policy “slims” but does not truncate a fat tail. Figure 9 and Table 1 illustrate this
point. Therefore, even aggressive climate policy could result in significant damage (even if
damage distribution has a finite variance). Thus, a climate hedging portfolio should have
additional instruments to truncate the tail. Adaptation and R&D into geoengineering should be
incorporated into a policy portfolio.
Figure 6. Policy choice and and outcome’s uncertanties.
Page 16 of 23
Figure 6 illustrates that even selection of emission targets, formulated as ppm concentration, leaves room
for significant uncertainties.
Table 1. Sliming a fat tail
Statistics
Trials
Mean
Median
Mode
Standard Deviation
Variance
Skewness
Kurtosis
Percentiles
96%
97%
98%
99%
99.5%
Damage 350 ppm target Damage 400 ppm target Damage 450 ppm target Damage 500 ppm target Damage 550 ppm target Damage 600 ppm target
100000
100000
100000
100000
100000
100000
0.57
1.52
2.73
4.11
5.60
7.17
0.42
1.10
1.98
2.98
4.06
5.20
------------0.54
1.43
2.58
3.88
5.29
6.76
0.29
2.06
6.64
15.05
27.96
45.75
3.54
3.54
3.54
3.54
3.54
3.54
29.52
29.52
29.52
29.52
29.52
29.52
Damage 350 ppm target Damage 400 ppm target Damage 450 ppm target Damage 500 ppm target Damage 550 ppm target Damage 600 ppm target
1.69
4.47
8.04
12.11
16.50
21.11
1.87
4.97
8.93
13.45
18.33
23.45
2.15
5.70
10.24
15.41
21.01
26.87
2.68
7.10
12.75
19.20
26.17
33.47
3.26
8.64
15.53
23.39
31.88
40.77
A major factor underlying uncertainty in this case is climate sensitivity. Many recent
publications explains why climate sensitivity is not predictable and confirms the log-normal
shape of the climate sensitivity probability distribution function (“PDF”) (see for example, Roe
and Baker 2007). Since economic damage is a function of an increase in temperatures, the
distribution of climate sensitivities that links particulate concentration and temperature rise
predetermines the distribution of economic damage. Higher concentrations lead to higher
expected damage but also, more importantly, to higher variance of temperature increase (see
Figure 2), and therefore higher variance of economic damage.
Roe and Baker’s paper, and the associated commentary, has an important policy implication.
The premise, that as feedback approaches 1, climate sensitivity approaches infinity, leads
logically to measuring the amount of carbon consistent with 2°C rather than sensitivity. The
policy extension of Roe and Baker’s framework is to evaluate policy choices with an a priori
assumption of a 2°C goal. The application of a real options methodology allows us to estimate
the benefits of such a policy. Suppose we start with an initial assumption about emissions
between now and 2050 consistent with 2°C. We can then attach an aggregate cost to that
abatement and assume, initially, that the aggregate benefit derived is equal to that cost. The cost
buys us an option on 2°C. The value of that option is zero if the volatility of costs and benefits
are equal. If, however, benefit volatility exceeds cost volatility, then the option is valuable. In
this case, we are inherently under spending – favoring a quantity target. If cost volatility exceeds
benefit volatility, then the option has a negative value – favoring a price target to avoid over
spending.
For any given GHG emission target, costs of such a policy are uncertain. Regulators are
challenged by another decision-making problem with uncertainty: introducing climate policy
now, they face uncertain investment cost, but by postponing policy through inaction, regulators
also postpone irreversible investment decisions. Hence regulators have another real option:
whether or not to postpone irreversible socio-economic costs of climate mitigation policy
resulting from investments to meet GHG emission limits. Keeping this option open, regulators
could implement climate policy later, when there is better knowledge about carbon emission
Page 17 of 23
reduction costs. The key question any regulator should ask is, which option is worth more, the
option to postpone the costs of policy to a later date, or maintaining the option to avoid damage
from climate change at a later date?
The next 15-20 years are critical determinants of long-term climate policy. Aggressive emission
reduction may generate unreasonable and unrecoverable sunk costs, while insufficient policy
may lead to irreversible changes in climatic system and significant economic damage in the
future. Selection of interim policy that prevents irreversible decisions regarding stabilization
targets in the future could be the solution to the problem.2
6. Conclusions
The raison d’être of ROA in climate policy is to reflect the relative uncertainty of costs and
benefits. Admittedly, the policy value of ROA depends on the predictability of relative
volatility, as well as relative skewness and kurtosis. Since these parameters are not highly
predictable, the scenario analysis common to a conventional marginal cost equals marginal
benefit (MC=MB) approach is still needed.
Scenario analysis using ROA, however, is
sufficiently robust to be additive to conventional analysis, particularly when cost and benefit
distributions are dissimilar.
To some degree, climate policy has a dynamic of “you get what you measure”. If we measure
policy on the parallel parameters of tail skimming and technology learning, and use ROA as a
tool to inform this process, we can better match a policy to its objectives. Borrowing from
financial economics, one way to state the objective of policy is to hedge the fat tail risk of
climate change. To accomplish this we need sufficient technology, learning for rapid
deployment of abatement technology when the hedge ratio, informed by scientific data on
climate, changes. Thus, part of the benefit to using ROA as a policy tool is its close alignment
with the objectives of policy.
In the context of a high degree of uncertainty, the preservation of flexibility may be more
important than cost control in the short run. With regard to the option value of climate policy, the
steep slope of the marginal cost over the next 20 to 30 years does not mean that this policy
cannot be justified just because the marginal benefits are relatively flat. Higher variance in
benefits “compensates for a lower slope.”
Taking into account the discreet character of climate negotiations and the continual nature of
knowledge accumulation, a ROA methodology helps to explain the value of a hedging strategy
based on the adoption of an interim emission target more rigid relative to the target that could be
justified based on the expected value approach. First of all, the estimation of an option value is a
call for politicians to adopt tighter emission targets in order to prevent radical, and therefore
expensive, corrections of the emission trajectory in the future, assuming the climatic system
turns out to be more sensitive to GHG accumulation in the atmosphere and that the socioeconomic system appears less resilient to the climate change.
2
Appendix B present in simplified form optimal condition for interim emission target.
Page 18 of 23
A real option approach helps balance any future upward or downward corrections in the climate
policy and minimize possible distortions in the carbon market. An upward correction of the
emissions target may flatten the market while downward corrections may disobey rights of
allowance holders (governments may be forced to withdraw or re-call previously distributed
carbon allowances or change allocation plans or introduce additional policies and measures).
Understanding the option value of the climate policy will help to minimize these costs. Any
discreet corrections of the emissions target based on forward-looking assessments of the option
value will reduce the magnitude of corrections and therefore reduce any disincentives and market
distortions. The alternative is to wait until all uncertainty is resolved and then act. But as we
mentioned before, the necessary corrections may be enormous and, in all likelihood, not feasible
due to the inertia of the economic system and the irreversibility of climate change.
In the context of irreversibility, flexibility has economic value that cannot be captured by the
expected value approach. The fat tail distribution of damage attributed to global climate change
suggests that flexibility in terms of future climate policy, could be more valuable than flexibility
in terms of irreversible investment into sunk costs needed to ensure flexibility of climate policy.
Uncertainties in climate sensitivity should play an essential role in the design of climate policy
and particularly in the targets and tools selected. Current efforts to curb carbon emissions would
generate some short term and mid term costs while benefits would be harvested only in the
distant future. Because of this time lag, the net present values of expected benefits of climate
policy are not sufficient to justify an expensive mitigation policy.
One important conclusion can be drawn regarding the role of uncertainties. Those that criticize
immediate actions for preventing climate change often refer to the uncertainties of the emissiondamage chain as a reason to postpone a policy that yields uncertain benefits. Real option analysis
interprets these uncertainties in opposite ways. The more uncertain the irreversible outcome, the
higher value of flexibility and the more reason a policy maker has to prevent it.
Page 19 of 23
References
Anda, A. Golub, A. Strukova, E. Economics of climate change under uncertainty: Benefits of
flexibility. Energy policy 37 (2009) 1345–1355
Baker, M. and Roe, G. 2007. Why is Climate Sensitivity So Unpredictable? Science. Vol. 318,
No. 5850. pp. 629-632.
Copeland T., Antikarov V. 2003. Real Options. A Practitioner’s Guide. London: Texere.
Copeland, T., et al. 2000. Valuation: Measuring and Managing the Value of Companies, Third
Edition. McKinsey & Company Inc.
Dixit A., Investment and hysteresis. 1992. Journal of economic perspectives. V. 6, # 1.
Golier, C., Weitzman, M. 2009. How should the distant future be discounted when discount rates
are uncertain?
Golier., C. 2010. Expected net present value, expected net future value and the Ramsey rule.
Journal of environmental economics and management.. V. 59 # 2.
Golub, A. Markandya A. 2008. Modeling environment-improving technological innovations
under uncertainties. Routledge. UK.
IPCC, 2007. Summary of the report for policy makers. Climate Change. Cambridge University
Press, Cambridge, United Kingdom and New York, NY, USA.
Mun, Jonathan. 2005. Real Options Analysis: Tools and Techniques for Valuing Strategic
Investment and Decisions, 2nd Edition. Wiley Finance.
Nordhaus, William D. 2007. The Stern Review on the Economics of Climate
Change. Journal of Economic Literature.
Nordhaus, William D. 2008. A Question of Balance: Weighing the Options on Global Warming
Policies. Yale University Press.
Page 20 of 23
Pindyck, R. 2004. Irreversibility, Uncertanty and investment. In Schwartz and Trigeorgis (2004)
Rouah, F. and Vainberg, G. 2007. Option Pricing Models & Volatility Using Excel-VBA.
Wiley Finance.
Schwartz, E and Trigeorgis, L. 2004. Real Options and Investment under Uncertainty: Classical
Readings and Recent Contributions. MIT Press.
Stern, Nicholas. 2006. The Economics of Climate Change: The Stern Review. Cambridge, UK:
Cambridge University Press, online at http://www.hmtreasury.gov.uk/independent_reviews/stern_review_economics_climate_change/stern_review
_report.cfm.
Stern, N.2009. The global deal.
Tol, Richard S.J. 2003. Is the Uncertainty About Climate Change Too Large For
Expected Cost-Benefit Analysis? Climatic Change 56: 265-289. Kluwer Academic
Publishers.
Trigeorgis, L. 1996. Real options: managerial flexibility and strategy in resource
allocation. The MIT Press.
Webster, M, Jakobovits, L, Norton J. 2008. Learning about Climate Change and Implications for
Near-term Policy. Climatic Change.
Weitzman, Martin L. 2007. Role of Uncertainty in the Economics of Catastrophic Climate
Change. AEI-Brookings Joint Center Working Paper No. 07-11.
Page 21 of 23
Appendix A: Numerical example
For calibration of the model we use the parameters presented in Table 1. According to the
literature S is log-normally distributed with the natural logarithm mean equal to 1.09 and ln
SD=0.4. A log-normal distribution for a was applied with the mean equal to 0.5 and SD = 0.25.3
The cost estimates are based on the IPCC AR4 Summary for policy makers table SPM.7 from
IPCC (2007) p. 21. The calibration of Initial GDP in 2010, annual GDP growth and BAU
concentration is consistent with the IPCC B2 scenario (IPCC 2000).
Table 1: Parameters of the Numerical Example
Gross World Product (gross
world product) in 2010,
Trillion USD 2006
Annual growth rate
Discount rate
BAU scenario
Damage function (% GWP)
Mitigation cost 450 ppm
pathway (% GWP)
Climate sensitivity
52
2%
3%
GHG concentration reaches about 700 ppm in 2100
D  0.5T 2 where T is temperature rise above pre
industrial era
2% of GWP in 2030; 4% in 2050 and 5% in 2100 for
Monte-Carlo simulation we assumed means are equal to
numbers above and SD=50%
Median = 3; logarithm mean equal to 1.09 and ln
SD=0.4
Then we apply the Monte-Carlo simulations and calculate the combined uncertainty mean and
median value, standard deviation and confidence intervals for benefit, cost and net benefits of
climate policy. Table 2 presents the simulation results.
Table 2: Simulation Results
Avoided
damage
(Trillion
USD 2006)
Sunk cost
(Trillion
USD 2006)
Mean*
Median*
80%
confidence
interval
(CI)*
81
52
16-171
Volatility
(standard
deviation
as % of
mean)
120%
95
91
63-134
30%
3
Skewness Kurtosis
4.5
54
0.82
4.4
For the damage function we applied different distributions but the results did not change significantly.
As for the mean value of the coefficient “a” we select a moderately conservative value slightly lower than
the same coefficient in DICE-99 and slightly higher than in DICE – 2007 (see Nordhaus 2008 p 127).
Page 22 of 23
Net present
-14
-39
From -47 to
value of
37
climate
policy
(Trillion
USD)
*)All numbers discounted back to 2010.
**) Standard deviation
97 **
Table 3. Option value and fat tail
Benefits of climate policy
Black-Sholes
32.45
Gram-Charlier
216 (85*)
Edgeworth Binomial Tree
76
4.51
49
Sunk cost
21.19
61
36
*)We ignored kurtosis
As we can see, factoring a fat tail into the option analysis demonstrates a higher value of climate
policy. Thus, the relative variance, skewness, and kurtosis of cost and benefits determine the
relative value of the option of climate benefits and sunk mitigation costs. The calculations
presented in Table 4 call for a more sophisticated option pricing mechanism than Black-Sholes to
better address PDF’s shape. However, giving available information on mitigation costs and
damaging further sophistications of numerical analysis is unlikely to bring adequate added value.
Moreover, with regard to the negotiation and policy making process, any attempts to find truly
optimal numerical solutions are unlikely to pay back. The policy process is discreet and suboptimal. At the end of the day, emission targets are a political solution. This policy context
makes for an easier application of the option theory. The policy framework presents a limited
number of emission targets and a discreet timeline of international negotiations, with a mandate
to revisit environmental strategy at certain points in the future.
Page 23 of 23