An analytical framework to price long‐dated climate‐exposed assets

Abstract

This paper uses a tractable stochastic integrated‐assessment model to analyze the influence of climate change on asset returns across time and maturity. Quasi‐analytical, or recursive, formulas allow to price various long‐dated assets, including fixed‐income products, derivatives, and equities. We find that climate risks will increasingly drive down long‐term risk‐free yields, reducing them by about 30 basis points by the end of the century. This decline reflects weaker growth and increased uncertainty, leading to a rise in precautionary savings. We illustrate the concept of climate risk premiums by examining model‐implied prices of long‐term assets vulnerable to sea level rise or temperatures. Climate risk premiums are particularly sensitive to damage assumptions.

Introduction

Climate change introduces a pervasive and multifaceted set of challenges that impact virtually every sector of the global economy. It poses a profound risk to long-term assets, threatening the value and sustainability of investments over extended periods. Increasing environmental volatility and the fast-evolving awareness of investors regarding climate risks necessitate a reevaluation of how long-term assets are priced.

We develop a modeling framework aimed at determining equilibrium prices for long-dated assets whose payoffs are exposed to climate risk.¹ Whereas it captures complex interactions between climate and macroeconomic variables, our sea-level-augmented Integrated Assessment Model (IAM) is solved instantaneously and offers quasi-analytical pricing formulas for various financial instruments, including bonds and derivatives.²

The tractability of the model relies on the characteristics of affine processes, which are widely utilized in the asset pricing literature (see, e.g., Duffie (1996), Duffie, Filipovic, and Schachermayer (2003), Piazzesi (2010)). These processes have been introduced into IAMs by Traeger (2015, 2021, 2023). It is important to emphasize that the affine nature of the dynamics in our state vector does not restrict the magnitude of risk within the model; damages, in particular, can be arbitrarily large within the current framework. Tractability is also key at the calibration stage. Indeed, the availability of computationally efficient solutions for first- and second-order moments of state variables (such as temperature and sea levels in 2100) enables us to align the model with the current evaluation of the climate trajectory and its linked economic consequences, both in terms of expectations and uncertainty. Let us take three examples. First, our model takes into account nonindustrial emissions of carbon into the atmosphere due to permafrost thaw; we parameterize a dedicated process to match the current expectation (consensus) and variance (uncertainty) regarding these specific emissions, for a given horizon (2100). Second, we make the model consistent with the 2100 expectations of sea level rise (SLR) and damages conditional on temperature anomalies above preindustrial levels (namely +2°C and +4°C). Third, we constrain the model to match a targeted level of uncertainty associated with temperature in 2100.

We find that climate risks will exert downward pressure on long-term real yields over the coming decades: in our base calibration, climate risks lead to a decrease in the 10-year rate of around 30 basis points. About a third of this effect is attributed to risk premiums, reflecting higher precautionary savings; the remaining portion is due to lower expected growth. Besides, simulations suggest that if inflation is positively affected by climate-related damage, there will be relatively less downward pressure on nominal interest rates, due to higher inflation risk premiums.

We use our model as a laboratory to examine climate risk premiums, which are price components influenced by agents' risk aversion and the inability to diversify climate risks. A model recognizes risk premiums when risk-neutral (or risk-adjusted) distributions diverge from their physical counterparts. Intuitively, risk-neutral distributions assign greater weight to unfavorable states of the world compared to physical distributions. All stochastic IAMs with risk-averse agents implicitly feature risk-neutral distributions across variables and time horizons. But while they can be challenging to exhibit in standard IAMs, our affine framework allows for straightforward derivation of both the physical and risk-neutral distributions. We find that the risk-neutral temperature distributions are right-shifted relative to their physical counterparts, indicating a positive temperature risk premium. Our results show that this premium—the difference between the averages of the physical and risk-neutral distributions—would be larger without exogenous technological uncertainty and smaller without damage or permafrost-related uncertainty. Indeed, two forces push the temperature risk premium in opposite directions: (−) everything else equal, an increase in the uncertainty about exogenous technological progress augments the correlation between temperature and consumption since (net of damages) consumption positively depends on capital, whose production generates carbon emissions; this tends to generate a negative correlation between temperatures and marginal utility of consumption (or the stochastic discount factor, SDF); (+) higher temperatures imply larger damage intensity, which contributes positively to the covariance between temperature and the SDF. Although the channels are not exactly the same, this is reminiscent of the two opposite forces that are at play in the insurance component of the social cost of carbon (Dietz, Gollier, and Kessler (2018), Lemoine (2021), Giglio, Kelly, and Stroebel (2021)): (−) a scaling effect arises under conventional damage specifications, where climate-induced losses increase in consumption because damages affect consumption multiplicatively; in that context, emission reductions increase future consumption by a larger amount when the future marginal utility of consumption is high, generating a negative correlation between consumption benefits associated with lower carbon concentrations and the SDF (hence a negative risk premium); (+) the damage-reduction effect is more intuitive—reducing damage intensity avoids consumption losses, generating a positive correlation between benefits and the SDF.

A concrete asset-pricing implication of temperature risk premiums is that, according to the model, investors would be willing to accept an additional return—similar to an insurance premium—in exchange for entering into a swap that pays the realized temperature at maturity. This excess return would precisely coincide with the difference between the means of the risk-neutral and physical distributions (the temperature risk premium discussed in the previous paragraph). More broadly, the pricing of climate derivatives, such as this temperature swap, serves as a natural illustration of climate risk premiums.³ Consider, for instance, a (hypothetical) digital temperature option that yields a unit payoff if future temperatures $T_{t+h}$ reach a certain threshold value $T_K$, say, and zero otherwise. If $T_K$ is high, this option's price incorporates a risk premium reflecting agents' aversion to extreme climate risks; this specific risk premium can be expressed as the difference between the risk-adjusted probability that $T_{t+h}>T_K$ and its physical counterpart. As an example, the physical and risk-neutral probabilities of having a temperature anomaly larger than 4°C in 2100 are respectively of 11% and 13% in our benchmark calibration.

Pricing long-dated assets amounts to computing (risk-adjusted) expectations of functions of the state vector at long horizons. For nonaffine IAMs, the analytical computation of such expectations is usually out of reach and necessitates Monte Carlo simulations. While the computational time required to calculate the price of one long-term asset using simulations is not necessarily prohibitive, it can become so if the price needs to be calculated many times, for example, as part of a sensitivity analysis, or because one wishes to calibrate the model to match certain prices. Maybe more importantly, relying on simulations to price long-dated assets may be misleading in contexts where these prices may not exist (i.e., are infinite), which may happen in the context of IAM featuring catastrophic events (Weitzman (2009)). Unlike pricing approaches based on simulations, those benefiting from analytical or computationally-efficient solutions make it possible to directly detect situations characterized by infinite prices. Importantly, this type of issue is not limited to pricing. It may also arise at the solution stage, where the computation of conditional expectations is needed to determine the value function—as is typically the case with Epstein–Zin or other recursive preferences. Related problems may occur if the solution approach is not based on Monte Carlo simulations but on grid-based techniques: when a model has fat-tailed distributions, grid-based solutions may depend heavily on the choices of grid endpoints (even if the probability of draws beyond these endpoints is extremely low).⁴

The remainder of this paper is organized as follows. Section 2 reviews the literature to which this paper is linked. Section 3 outlines our modeling framework. Section 4 presents pricing formulas and illustrates them in the context of numerical examples. Section 5 focuses on the Social Cost of Carbon (SCC). Section 6 concludes. Appendices provide details regarding some modeling ingredients and the calibration. Additional proofs and results are gathered in the Supplemental Appendix (Chikhani and Renne (2025)).

Literature review

Our research relates to the literature investigating the pricing of climate risks. A large share of the theoretical literature is concerned with the computation of the Social Cost of Carbon, defined as the marginal value of emission reductions (e.g., Weitzman (2013)). Uncertainty and aversion to ambiguity are found to have profound implications on the SCC calculations (e.g., Cai and Lontzek (2019), Barnett, Brock, and Hansen (2020), Van den Bremer and Van der Ploeg (2021)). While the present paper focuses on the pricing of various financial instruments, we also examine the SCC resulting from our model. Our baseline estimate of $270 per ton of CO₂ is broadly consistent with several recent studies, notably those entailing a strong focus on uncertainty (e.g., Daniel, Litterman, and Wagner (2019), Lemoine (2021), Traeger (2021), Environmental Protection Agency (2023)).

The empirical literature on the impact of climate risks on asset pricing is rapidly expanding. Several articles analyze the relative value of green or ESG bonds, revealing small premiums compared to similar ordinary bonds (Baker, Bergstresser, Serafeim, and Wurgler (2018), International Monetary Fund (2019), Andersson, Bolton, and Samama (2019), Larcker and Watts (2020), Pástor, Stambaugh, and Taylor (2022)). Given that investors have only recently begun considering climate risks, estimates of climate risk premiums from short historical samples should be viewed cautiously (Giglio, Kelly, and Stroebel (2021), Pástor, Stambaugh, and Taylor (2022)). EGKL+ (2020) suggest dynamically hedging associated risks using stocks-based factor-mimicking portfolios. Other studies investigate bond market evidence of climate risk premiums (e.g., Huynh and Xia (2021), Painter (2020)). Some studies utilize climate projections to simulate effects on asset prices; for instance, KABK+ (2023) and Zenios (2022) simulate the effect of climate change on sovereign credit ratings, aligning with predictions from Mallucci (2022)'s structural model. Bernstein, Gustafson, and Lewis (2019) and Baldauf, Garlappi, and Yannelis (2020) document the adverse impact of sea level rise on residential real estate prices. Goldsmith-Pinkham, Gustafson, Lewis, and Schwert (2023) emphasize the significance of uncertainty in the transmission of climate risks into current prices.

The present paper is particularly close to those studies that investigate asset pricing in the context of stochastic integrated assessment models (IAMs). Some of these studies utilize models that simplify the climate block of the Dynamic Integrated Climate-Economy model (DICE) developed by Nordhaus (1992) (e.g., Bansal, Ochoa, and Kiku (2016), Bansal, Kiku, and Ochoa (2019), Karydas and Xepapadeas (2022)). Other studies employ standard DICE-related IAMs and look for feasible numerical pricing solutions (e.g., Daniel, Litterman, and Wagner (2019), Barnett, Brock, and Hansen (2020)). We manage to combine DICE-type equations and simple recursive pricing solutions. For that, we build on Traeger (2015, 2021) who demonstrates how stochastic integrated assessment models can be solved analytically when affine stochastic processes govern the uncertainties. We expand on this analysis by demonstrating how this framework enables the rapid pricing and analysis of long-dated financial instruments.

Model

Like any integrated assessment model (IAM), our model aims to capture the relationship between the climate and the economy. Figure 1 synthesizes these interactions: a representative agent uses capital to produce a good that she can consume or invest in the productive capital; the good can also be used by a technology that mitigates the carbon emissions resulting from the production process. Carbon emissions affect temperatures. High temperatures damage productive capital.

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Climate block

As in the DICE2016 model of Nordhaus (2017), we focus on two global temperatures: lower ocean ($T_{\mathrm{LO}}$) and atmosphere ($T_{\mathrm{AT}}$), expressed as anomalies from a baseline period.⁵ The dynamics of these two temperatures depend on radiative forcings due to greenhouse gases (F): Earth receives radiant energy from the Sun and emits some energy back into space; at equilibrium, Earth should absorb as much radiant energy as it radiates out of our atmosphere; the difference between the two is radiative forcings. When the latter goes up, absorption increases, and the Earth warms.⁶ Specifically, we have 1${\mathbb{E}}_{t-1}(T_{\mathrm{AT},t})=T_{\mathrm{AT},t-1}+{\xi }_1\mathrm{\Delta }t(F_{t-1}-\frac{{\displaystyle \tau }}{{\displaystyle \nu }}{T}_{\mathrm{AT},t-1}-{\xi }_2[T_{\mathrm{AT},t-1}-T_{\mathrm{LO},t-1}]),$ where ${\mathbb{E}}_{t-1}$ denotes the expectation based on the information available on date $t-1$, which we denote by ${\mathcal{I}}_{t-1}$ (i.e., ${\mathbb{E}}_{t-1}(T_{\mathrm{AT},t})=\mathbb{E}(T_{\mathrm{AT},t}|{\mathcal{I}}_{t-1})$). While (1) is considered to be deterministic in standard IAMs (in the sense that $\mathbb{V}a{r}_{t-1}(T_{\mathrm{AT},t})=0$), we capture the uncertainty surrounding the temperature's law of motion by assuming that $T_{\mathrm{AT},t}$ follows: 2$T_{\mathrm{AT},t}|{\mathcal{I}}_{t-1}\sim i.i.d.{\gamma }_0({\lambda }_{T,t},{\mu }_T),\text{with }{\lambda }_{T,t}={\mathbb{E}}_{t-1}(T_{\mathrm{AT},t})/{\mu }_T,$ where ${\mathbb{E}}_{t-1}(T_{\mathrm{AT},t})$ is given in (1), and where ${\gamma }_0$ is the gamma-zero distribution introduced by Monfort, Pegoraro, Renne, and Roussellet (2017). Appendix A outlines the properties of this distribution. Note that (2) is consistent with (1) since the mean of a ${\gamma }_0(\lambda ,\mu )$ distribution is λμ. Another ${\gamma }_0$ property implies that $\mathbb{V}a{r}_{t-1}(T_{\mathrm{AT},t})=2{\mu }_T{\mathbb{E}}_{t-1}(T_{\mathrm{AT},t})$. Hence, ${\mu }_T$ characterizes the uncertainty associated with the temperature dynamics. (In particular, we are back to the deterministic case if ${\mu }_T\approx 0$.) This original use of the ${\gamma }_0$ distribution is discussed in more detail in Appendix A.2.⁷

Lower-ocean temperature evolves according to 3$T_{\mathrm{LO},t}=T_{\mathrm{LO},t-1}+{\xi }_3\mathrm{\Delta }t(T_{\mathrm{AT},t-1}-T_{\mathrm{LO},t-1}),$ and radiative forcings follow: 4$F_t=[\tau {\log }_2(m_{0,t})+\frac{{\displaystyle \tau }}{{\displaystyle \log (2)m_{0,t}}}(\frac{{\displaystyle M_{\mathrm{AT},t}}}{{\displaystyle M_{\mathrm{PI}}}}-m_{0,t}\left)\right]+F_{\mathrm{EX},t},$ where $F_{\mathrm{EX},t}$ is the (exogenous) part of radiative forcings that is due to non-CO₂ greenhouse gases, $M_{\mathrm{AT},t}$ is the mass of carbon in the atmosphere, and $M_{\mathrm{PI}}$ is its pre-industrial level. The standard expression for the term in curly brackets is $\tau {\log }_2(M_{\mathrm{AT},t}/M_{\mathrm{PI}})$. In (4), for each date t, this expression is linearized at $m_{0,t}:=M_{\mathrm{AT},t}^{\mathrm{RCP}}/M_{\mathrm{PI}}$, where $M_{\mathrm{AT},t}^{\mathrm{RCP}}$ is set to the average of the RCP4.5 and RCP6.0 values for that date.⁸ Appendix B demonstrates that the influence of this linearization on temperature trajectories is minimal for carbon concentration pathways remaining close to the RCP4.5 and RCP6.0 and that this relative insensitivity persists for pathways that are not significantly divergent from these reference scenarios: For instance, considering pathways where the quantity of carbon in the atmosphere is ultimately 500 GtC higher than that in the RCP6.0 scenario by 2300, the expected atmospheric temperature is 5.8°C when using (4), compared to 5.5°C with the nonlinear specification. In the case of an overshoot of the RCP6.0 scenario by 1000 GtC, the linearized model predicts an end temperature of 7.1°C, while the nonlinear specification results in a temperature of 6.2°C. These differences falls within a ±1 standard confidence interval that reflects the uncertainty regarding the temperature process, as captured by the ${\gamma }_0$ distribution of (2).⁹

The atmospheric carbon mass $M_{\mathrm{AT},t}$ appearing in (4) constitutes one of the three reservoirs used to synthesize the carbon cycle (atmosphere, AT, upper ocean, UP, and lower ocean, LO). This cycle determines the journey of all carbon atoms on earth: a loop between atmosphere, land, and ocean. The carbon cycle is modeled as follows: 5$\begin{array}{cc}\hfill \left[\begin{array}{c}\hfill M_{\mathrm{AT},t}\hfill \\ \hfill M_{\mathit{UP},t}\hfill \\ \hfill M_{\mathrm{LO},t}\hfill \end{array}\right]=& {\left[\begin{array}{ccc}\hfill 1-{\phi }_{12}\hfill & \hfill {\phi }_{12}\frac{{\displaystyle \mathit{mateq}}}{{\displaystyle \mathit{mueq}}}\hfill & \hfill 0\hfill \\ \hfill {\phi }_{12}\hfill & \hfill 1-{\phi }_{12}\frac{{\displaystyle \mathit{mateq}}}{{\displaystyle \mathit{mueq}}}-{\phi }_{23}\hfill & \hfill {\phi }_{23}\frac{{\displaystyle \mathit{mueq}}}{{\displaystyle \mathit{mleq}}}\hfill \\ \hfill 0\hfill & \hfill {\phi }_{23}\hfill & \hfill 1-{\phi }_{23}\frac{{\displaystyle \mathit{mueq}}}{{\displaystyle \mathit{mleq}}}\hfill \end{array}\right]}^{\mathrm{\Delta }t}\left[\begin{array}{c}\hfill M_{\mathrm{AT},t-1}\hfill \\ \hfill M_{\mathit{UP},t-1}\hfill \\ \hfill M_{\mathrm{LO},t-1}\hfill \end{array}\right]\hfill \\ \hfill & +\frac{{\displaystyle \mathrm{\Delta }t}}{{\displaystyle 3.666}}\left[\begin{array}{c}\hfill {\mathcal{E}}_{t-1}\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right].\hfill \end{array}$ In the previous equation, ${\mathcal{E}}_t$ is the carbon dioxide (CO₂) emitted into the atmosphere, on average, for each year of period t (one period lasts $\mathrm{\Delta }t=5$ years), converted into carbon masses by applying the conversion rate 1/3.666. Total emissions are the sum of three components: 6${\mathcal{E}}_t={\mathcal{E}}_{\mathrm{Land},t}+{\mathcal{E}}_{\mathrm{Ind},t}+N_t,$ where ${\mathcal{E}}_{\mathrm{Ind},t}$ are industrial emissions due to human activity, assumed to grow at the same pace as capital (see equation (20) in Section 3.2), ${\mathcal{E}}_{\mathrm{Land},t}$ represents an exogenous component including, in particular, emissions due to deforestation, and $N_t$ accounts for carbon releases that are endogenous but do not directly stem from the production process. The latter variable intends to capture positive feedback effects documented in the literature on tipping points (LHKH+ (2008))—such positive feedback effects underlie arrows (b)↔(e) in Figure 1. The world's climatic system has an unknown number of positive and negative feedback loops. While negative feedback loops decrease the pressure on global warming by absorbing greenhouse gases from the atmosphere, positive feedback loops amplify the positive imbalances in radiative forcings by creating a vicious circle strengthening global warming. Examples of positive feedback loops include the release of hundreds of gigatons of methane trapped in the permafrost and in ice lattices in the shallow ocean, the acidification of oceans, the melting of the Greenland ice sheet, or the weakening of the Atlantic thermohaline circulation.

Our framework focuses on feedback loops involving emissions of nonproduction-based greenhouse gases ($N_t$ in equation (6)), foremost among which are those linked to permafrost (e.g., SACE+ (2022)). These emissions are assumed to be drawn from a second ${\gamma }_0$ distribution (as in equation (2)): 7$N_t|{\mathcal{I}}_{t-1}\sim {\gamma }_0({\lambda }_{N,t},{\mu }_N),\text{with }{\lambda }_{N,t}={\kappa }_N^{t-1}(a^{(N)}+b^{(N)}{T}_{\mathrm{AT},t-1})/{\mu }_N.$ We impose that $0\le {\kappa }_N<1$, which ensures that $N_t$ goes to zero in the long run and that ${\lim }_{t\to \mathrm{\infty }}{N}_1+\mathrm{\cdots }{N}_t$ does not diverge, consistently with the fact that the carbon trapped in permafrost is not in infinite supply. As for (2), ${\mu }_N$ characterizes the uncertainty associated with $N_t$. While our modeling of this feedback loop accommodates sudden and large effects, with highly persistent adverse effects, it does not feature an explicit threshold as in, for example, Lemoine and Traeger (2014, 2016), Bansal, Ochoa, and Kiku (2016), or Taconet, Guivarch, and Pottier (2021).

We augment the model with global mean sea level, $H_t$. It is assumed to evolve as follows: 8$H_t-H_{t-1}|{\mathcal{I}}_{t-1}\sim i.i.d.{\gamma }_0({\lambda }_{H,t},{\mu }_H),\text{with }{\lambda }_{H,t}=(a^{(\mathrm{\Delta }H)}+b^{(\mathrm{\Delta }H)}{T}_{\mathrm{AT},t-1})/{\mu }_H.$

This specification is a stochastic extension of semiempirical approaches à la Rahmstorf (2007), which aim to exploit the link between global sea level and global temperature in past observational data for projecting the future.¹⁰ While the dynamic relationship between sea level and temperature is usually expressed in a deterministic fashion, our ${\gamma }_0$-based specification introduces uncertainty in the process. The amount of uncertainty underlying the process is captured by ${\mu }_H$; if the latter is infinitely small, (8) boils down to the deterministic specification used, for example, by Rahmstorf (2007), that is, $H_t=H_{t-1}+a^{(\mathrm{\Delta }H)}+b^{(\mathrm{\Delta }H)}{T}_{\mathrm{AT},t-1}$.

Economic block

We consider a production economy where capital quality can be damaged by climatic disasters. On each period, agents allocate production ($Y_t$) between consumption ($C_t$), investment in productive capital (${\mathrm{Inv}}_t$), and investment in low carbon emissions technologies (${\mathrm{\Psi }}_t$):¹¹ 9$Y_t=C_t+{\mathrm{Inv}}_t+{\mathrm{\Psi }}_t.$ Production $Y_t$ results from AK technology: 10$Y_t=A_t{K}_t,\text{with }{A}_t=Ā+{\sigma }_A{\eta }_{A,t},$ where $A_t$ and $K_t$ are productivity and productive capital, respectively. Ā is the average productivity, and ${\eta }_{A,t}\sim i.i.d.\mathcal{N}(0,1)$ is a productivity shock.¹² The dynamics of productive capital are governed by 11$K_t^{*}=(1-\mathrm{dep})K_{t-1}+{\mathrm{Inv}}_{t-1},$ 12$K_t=\exp (-D_t-b_{\mathrm{SK}}\mathrm{\Delta }{H}_t)K_t^{*},$ where dep is the depreciation rate, $D_t$ (≥0) represents climatic disasters, and $b_{\mathrm{SK}}\mathrm{\Delta }{H}_t$ corresponds to destruction of capital due to sea level rise (SLR)—one of the costliest dangers from climate change (e.g., Hauer, Evans, and Mishra (2016), DKKKN+ (2021)). Productive capital $K_t$ is smaller than planned capital $K_t^{*}$ when climatic disasters occur ($D_t>0$) and when sea level rises ($\mathrm{\Delta }{H}_t>0$). That is, damages here take the form of capital depreciation (as in Gourio (2012), Moore and Diaz (2015), Gomes, Grotteria, and Wachter (2019)). Climatic disasters are drawn from a ${\gamma }_0$ distribution (Monfort, Pegoraro, Renne, and Roussellet (2017)): 13$D_t|{\mathcal{I}}_{t-1}\sim i.i.d.{\gamma }_0({\lambda }_{D,t},{\mu }_D),\text{with }{\lambda }_{D,t}=(a^{(D)}+b^{(D)}{T}_{\mathrm{AT},t-1})/{\mu }_D.$ According to the properties of this distribution (Appendix A.3), the probability of having $D_t>0$ is equal to $1-\exp (-{\lambda }_{D,t})$. When ${\lambda }_{D,t}$ is small, this probability is close to ${\lambda }_{D,t}$, which therefore represents the disaster intensity.

Increasing industrial emissions pushes temperatures upwards (equations (2) to (5)), which increases disaster intensity (equation (13)) and, in turn, affects production (equations (10) to (12)). This adverse mechanism prompts agents to invest in low-carbon technologies. Investing in these technologies reduces the emissions from producing one unit of goods by a factor ${\mu }_t$, called mitigation rate (with $0\le {\mu }_t\le 1$). Increasing ${\mu }_t$ is costly. As in the DICE model, we posit that associated abatement costs (${\mathrm{\Psi }}_t$ in equation (9)) are given by 14${\mathrm{\Psi }}_t={\mathrm{\Lambda }}_t{Y}_t,\text{with }{\mathrm{\Lambda }}_t={\theta }_{1,t}{\mu }_t^{{\theta }_2},$ where ${\theta }_{1,t}$ deterministically declines over time to account for reductions in the price of backstop technology and the carbon intensity of production.¹³ With ${\theta }_2>1$, (14) implies that, on a given date, the marginal cost of mitigation increases with the mitigation rate.

Our representative agent faces the following trade-off: investing in mitigation reduces long-term risks—making disasters less likely—but comes at the cost of lowering immediate consumption. Her decision regarding how to allocate output between mitigation (${\mathrm{\Psi }}_t$), investment in productive capital (${\mathrm{Inv}}_t$), and consumption ($C_t$) depends on their preferences, which we take of the Epstein–Zin–Weil type, with a unit elasticity of intertemporal substitution (EIS).^14,15 Specifically, the time-t utility of a consumption stream $(C_t)$ is then recursively defined by (e.g., Piazzesi and Schneider (2007)): 15$u_t=(1-\delta )c_t+\frac{{\displaystyle \delta }}{{\displaystyle 1-\gamma }}\log \left({\mathbb{E}}_t\exp \right[(1-\gamma )u_{t+1}\left]\right),$ where $c_t$ denotes the logarithm of the agent's consumption level $C_t$, δ is the time discount factor (i.e., $-\log (\delta )$ is the pure rate of time preference), and γ is the risk aversion parameter.

To get instant results, we simplify the agent's optimization problem. While, on each date, the representative agent optimally determines how to allocate post-mitigation output $(1-{\mathrm{\Lambda }}_t)Y_t$ between consumption $C_t$ and investment ${\mathrm{Inv}}_t$ (see equation (9)), she decides the mitigation rate path (${\mu }_t$) only once and for all on the initial date, and further commits to that parametric path. Formally, she solves 16$\begin{array}{cc}\hfill & \underset{\{{\theta }_a,{\theta }_b\},C_0,{\mathrm{Inv}}_0}{\text{max}}{u}_0[{\mu }_1({\theta }_a,{\theta }_b),\mathrm{\dots },{\mu }_k({\theta }_a,{\theta }_b),\mathrm{\dots }]\text{for }t=0,\hfill \\ \hfill & \underset{C_t,{\mathrm{Inv}}_t}{\text{max}}{u}_t\text{for }t>0,\hfill \end{array}$ where 17${\mu }_t({\theta }_a,{\theta }_b)=\min \left[\exp \right(-|{\theta }_a|+|{\theta }_b|\times t);1],t>0.$ The choice of the parametric time function (17) is based on the usual shapes of emission control rates obtained in standard IAMs. Importantly, as demonstrated in Proposition 3, our framework provides a computationally efficient solution for $u_t$, and in particular for $u_0$, which facilitates the numerical optimization that underlies (16).

Supplemental Appendix E presents the results of exercises designed to assess the impact of our assumptions regarding the mitigation path. A first exercise demonstrates that if agents were allowed to optimize the entire sequence of future ${\mu }_t$ ex ante, rather than being constrained by the parametric function, the resulting sequence would closely resemble the optimized parametric one. In a second exercise, the agent has the opportunity to revise the ${\mu }_t$ trajectory in future periods (but she always thinks it is the last time she can do so). The results suggest that the updates would moderately differ from the initial decision at $t=0$, even in scenarios where temperatures and carbon concentrations are 2 standard deviations above date-0 expectations. Importantly, we also find that in the latter scenarios, the trajectory of SCCs and climate risk premiums would only be slightly modified compared to our baseline case.

Deriving the state vector's dynamics

Closing the model

At that stage, closing the model essentially involves establishing (a) the law of motion for consumption and (b) the link between industrial emissions and consumption.

In the context described above, the optimal consumption growth path satisfies¹⁶ 18$\mathrm{\Delta }{c}_t\approx {\mu }_{c,t}+{\sigma }_{c,t}{\eta }_{A,t}-D_t-b_{\mathrm{SK}}\mathrm{\Delta }{H}_t,$ where ${\mu }_{c,t}$ and ${\sigma }_{c,t}$ are deterministic functions of the model parameters, 19$\{\begin{array}{c}{\mu }_{c,t}=\log \delta +\log [(1-{\mathrm{\Lambda }}_t)Ā+(1-\mathrm{dep})],\hfill \\ {\sigma }_{c,t}=\frac{{\displaystyle (1-{\mathrm{\Lambda }}_t)}}{{\displaystyle (1-{\mathrm{\Lambda }}_t)Ā+(1-\mathrm{dep})}}{\sigma }_A.\hfill \end{array}$ Equation (18) shows that climate-related capital depreciation (i.e., $D_t+b_{\mathrm{SK}}\mathrm{\Delta }{H}_t$ in equation (12)) translates into an equivalent decrease on consumption growth. That is, our framework eventually entails growth-type damages, as in Moore and Diaz (2015), Bansal, Ochoa, and Kiku (2016), Bansal, Kiku, and Ochoa (2019), Lemoine (2021), and Hambel, Kraft, and Schwartz (2021).¹⁷

Define carbon intensity (${\sigma }_t$) as the emission resulting from the production of one unit of good in the absence of mitigation. Assuming that industrial emissions would grow at the same pace as (planned) capital if the mitigation rate ${\mu }_t$ and the carbon intensity ${\sigma }_t$ were constant, we obtain the following approximate process for ${\mathcal{E}}_{\mathrm{Ind},t}$: 20${\mathcal{E}}_{\mathrm{Ind},t}={\lambda }_t(1+{\tilde{y}}_{t-1}),$ with ${\tilde{y}}_t={\sum }_{i=1}^t{\sigma }_{c,i}{\eta }_{A,i}$ and ${\lambda }_t={\sigma }_t(1-{\mu }_t)q_0\exp [{\sum }_{i=1}^{t-1}({\mu }_{c,i}+{\sigma }_{c,i}^2/2)]$, where $q_0$ is the date-0 production.¹⁸

Notice that one could also employ the present framework without being specific about the production function (as, e.g., in Bansal, Kiku, and Ochoa (2019), Daniel, Litterman, and Wagner (2019), Lemoine (2021)). More precisely, one could combine the climate block of Section 3.1 with consumption dynamics entailing the effects of climate damages (potentially including mitigation costs, as in Daniel, Litterman, and Wagner (2019)), together with a link between consumption growth and industrial emissions.

Conditional Laplace transform of the state vector

Proposition 1 presents the conditional Laplace transform of the state vector (denoted by $X_t$) derived from the previous assumptions. The conditional Laplace transform of a vector characterizes its dynamics.

$1$

Proposition

State vector dynamics. The model described by equations (1) to (20) is such that if $X_t={[Z_t^{\prime }{W}_t^{\prime }]}^{\prime }$, with $Z_t={[{\tilde{y}}_t{\mathcal{E}}_t{\mathcal{E}}_{\mathrm{Ind},t}{F}_t{M}_{\mathrm{AT},t}{M}_{\mathit{UP},t}{M}_{\mathrm{LO},t}{T}_{\mathrm{LO},t}{H}_t]}^{\prime },\text{and}{W}_t={[{\eta }_{A,t}{D}_t{N}_t{T}_{\mathrm{AT},t}\mathrm{\Delta }{H}_t]}^{\prime },$ then the conditional log Laplace transform of $X_t$ is given by 21${\psi }_t(u):=\log {\mathbb{E}}_t\left(\exp \right(u^{\prime }{X}_{t+1}\left)\right)={\alpha }_t(u)+{\beta }_t{(u)}^{\prime }{X}_t,$ with 22$\{\begin{array}{c}{\alpha }_t(u)=u_Z^{\prime }{\omega }_{0,t+1}+{\alpha }_{W,t}(u_W+{\omega }_{t+1}^{\prime }{u}_Z),\hfill \\ {\beta }_t(u)=\left[\begin{array}{c}\hfill A_{1,t+1}^{\prime }{u}_Z\hfill \\ \hfill 0_{n_W\times 1}\hfill \end{array}\right]+{\beta }_{W,t}(u_W+{\omega }_{t+1}^{\prime }{u}_Z),\hfill \end{array}$ where $n_W$ is the dimension of $W_t$ and $u={[u_Z^{\prime },u_W^{\prime }]}^{\prime }$. Matrices $A_{1,t}$, ${\omega }_{0,t}$, and ${\omega }_t$ deterministically depend on time; they are such that $Z_t=A_{1,t}{Z}_{t-1}+{\omega }_{0,t}+{\omega }_t{W}_t$, and are detailed in Appendix C. Functions ${\alpha }_{W,t}$ and ${\beta }_{W,t}$ characterize the Laplace transform of $W_t$. Specifically, for all t, we have 23${\psi }_{W,t}(u_W):=\log {\mathbb{E}}_t\left(\exp \right[u_W^{\prime }{W}_{t+1}\left]\right)={\alpha }_W(u_W)+{\beta }_W{(u_W)}^{\prime }{X}_t,$ with $u_W={[u_{\eta },u_D,u_N,u_T,u_H]}^{\prime }$ and $\{\begin{array}{c}{\alpha }_{W,t}(u_W)=\frac{{\displaystyle u_{\eta }^2}}{{\displaystyle 2}}+\frac{{\displaystyle u_D{\mu }_D}}{{\displaystyle 1-u_D{\mu }_D}}{\ell }_0^{(D)}+\frac{{\displaystyle u_N{\mu }_N}}{{\displaystyle 1-u_N{\mu }_N}}{\ell }_{0,t+1}^{(N)}+\frac{{\displaystyle u_T{\mu }_T}}{{\displaystyle 1-u_T{\mu }_T}}{\ell }_0^{(T_{\mathrm{AT}})}\hfill \\ \phantom{{\alpha }_{W,t}(u_W)=}+\frac{{\displaystyle u_H{\mu }_H}}{{\displaystyle 1-u_H{\mu }_H}}{\ell }_0^{(\mathrm{\Delta }H)},\hfill \\ {\beta }_{W,t}(u_W)=\frac{{\displaystyle u_D{\mu }_D}}{{\displaystyle 1-u_D{\mu }_D}}{\ell }_1^{(D)}+\frac{{\displaystyle u_N{\mu }_N}}{{\displaystyle 1-u_N{\mu }_N}}{\ell }_{1,t+1}^{(N)}+\frac{{\displaystyle u_T{\mu }_T}}{{\displaystyle 1-u_T{\mu }_T}}{\ell }_1^{(T_{\mathrm{AT}})}\hfill \\ \phantom{{\beta }_{W,t}(u_W)=}+\frac{{\displaystyle u_H{\mu }_H}}{{\displaystyle 1-u_H{\mu }_H}}{\ell }_1^{(\mathrm{\Delta }H)},\hfill \end{array}$ where scalars ${\ell }_{0,t}^{(w)}$ and vectors ${\ell }_{1,t}^{(w)}$ ($w\in \{D,N,T_{\mathrm{AT}},\mathrm{\Delta }H\}$) are such that $w_t|{\mathcal{I}}_{t-1}\sim {\gamma }_0({\ell }_{0,t}^{(w)}+{{\ell }_{1,t}^{(w)}}^{\prime }{X}_{t-1},{\mu }_w)$ (see Table 6 in Appendix C for detailed specifications of the ${\ell }_0$s and ${\ell }_1$s.).

The previous proposition shows that the dynamics of the state vector is affine, in the sense that the conditional Laplace transform of $X_{t+1}$ is exponential affine in $X_t$. The properties of affine processes imply in particular that $X_t$'s law of motion admits the following vector autoregressive representation: 24$X_t={\mu }_{X,t}+{\mathrm{\Phi }}_{X,t}{X}_{t-1}+{\mathrm{\Sigma }}_{X,t}(X_{t-1}){\epsilon }_t,$ where (i) ${\mu }_{X,t}$, ${\mathrm{\Phi }}_{X,t}$, and function ${\mathrm{\Sigma }}_{X,t}$ deterministically depend on the model parameters, (ii) ${\mathrm{\Sigma }}_{X,t}(X_{t-1}){\mathrm{\Sigma }}_{X,t}^{\prime }(X_{t-1})$ is affine in $X_{t-1}$, and (iii) ${\epsilon }_t$ is a martingale difference sequence satisfying $\mathbb{V}a{r}_t({\epsilon }_{t+1})=\mathbf{\text{Id}}$.¹⁹ From (24), one can derive recursive formulas to compute ${\mathbb{E}}_t(X_{t+h})$ and $\mathbb{V}a{r}_t(X_{t+h})$, both of which are affine in $X_t$ (see Supplemental Appendix A).

Proposition 1 is complemented by the following one, which demonstrates that recursive formulas can be employed to derive the conditional multihorizon Laplace transform of $X_t$.

$2$

Proposition

Multihorizon log Laplace transform of $X_t={[Z_t^{\prime }{W}_t^{\prime }]}^{\prime }$. We have 25$\begin{array}{rcl}{\psi }_t^{(h)}(u_1,\mathrm{\dots },u_h)& :=& \log {\mathbb{E}}_t\left[\exp \right(u_1^{\prime }{X}_{t+1}+\mathrm{\cdots }+u_h^{\prime }{X}_{t+h}\left)\right]\\ & =& {\psi }_{0,t}^{(h)}(u_1,\mathrm{\dots },u_h)+{\psi }_{1,t}^{(h)}{(u_1,\mathrm{\dots },u_h)}^{\prime }{X}_t,\end{array}$ where, for all t and for $h>1$, 26$\{\begin{array}{c}{\psi }_{0,t}^{(h)}(u_1,\mathrm{\dots },u_h)={\psi }_{0,t+1}^{(h-1)}(u_2,\mathrm{\dots },u_h)+{\alpha }_t[u_1+{\psi }_{1,t+1}^{(h-1)}(u_2,\mathrm{\dots },u_h)],\hfill \\ {\psi }_{1,t}^{(h)}(u_1,\mathrm{\dots },u_h)={\beta }_t[u_1+{\psi }_{1,t+1}^{(h-1)}(u_2,\mathrm{\dots },u_h)],\hfill \end{array}$ with ${\psi }_{0,t}^{(1)}(u)={\alpha }_t(u)$ and ${\psi }_{1,t}^{(1)}(u)={\beta }_t(u)$ for all t. Functions ${\alpha }_t$ and ${\beta }_t$ are defined in (22).

Knowing the multihorizon Laplace transform allows in particular to determine the conditional distribution of any linear combination of future values of $X_t$ by inverse Fourier transform.²⁰ This can be used to derive the model-implied distributions of the variables involved in the state vector (as done in Figures 4 and 5). It is also particularly useful to calculate the prices of long-dated assets, as demonstrated in Section 4.

Steady-state time-invariant affine representation

We assume that the deterministic processes involved in the model will have converged for $t\ge t_{\mathrm{\infty }}$, with $t_{\mathrm{\infty }}=100$ (i.e., in 500 years). Our model implies indeed that emissions ${\mathcal{E}}_t$, defined as the sum of ${\mathcal{E}}_{\mathrm{Ind},t}$, ${\mathcal{E}}_{\mathrm{Land},t}$ and $N_t$ (equation (6)), converge to zero.²¹ (Note that, despite the cessation of new emissions, climate-related damages will still occur due to the persistence of elevated atmospheric temperatures.) Hence, for $t\ge t_{\mathrm{\infty }}$, functions ${\alpha }_t$ and ${\beta }_t$ in (21) do not depend on time anymore. The model then becomes stationary. Formally, for $t\ge t_{\mathrm{\infty }}$, the conditional log Laplace transform of $X_t$ reads 27${\psi }_t(u)=\log {\mathbb{E}}_t\left[\exp \right(u^{\prime }{X}_{t+1}\left)\right]=\alpha (u)+\beta {(u)}^{\prime }{X}_t.$ This implies in particular that we have ${\mu }_{X,t}={\mu }_X$, ${\mathrm{\Phi }}_{X,t}={\mathrm{\Phi }}_X$, and ${\mathrm{\Sigma }}_{X,t}(•)={\mathrm{\Sigma }}_X(•)$ in (24) for $t\ge t_{\mathrm{\infty }}$. That is, for $t\ge t_{\mathrm{\infty }}$, the process followed by the state vector takes the form of a semistrong vector autoregressive model (Monfort, Pegoraro, Renne, and Roussellet (2017)).

Calibration

Numerous parameters are borrowed from existing literature, in particular from Barrage and Nordhaus (2023), and from Folini, Friedl, Kübler, and Scheidegger (2025) (see Table 7 in Appendix D). Importantly, the fact that the parameters underlying the temperature and carbon cycle dynamics (equations (2) to (5)) are taken from the CDICE model of Folini, Friedl, Kübler, and Scheidegger (2025) makes our model consistent with a relatively fast reaction of temperatures to changes in carbon concentrations—a feature called for by Dietz and Venmans (2019) and Dietz, van der Ploeg, Rezai, and Venmans (2021), building on JRFP+ (2013). Figure 2 displays the model-implied temperature response resulting from a pulse of CO₂ emissions. The two black lines illustrate the results from the present study (labeled CR): the solid line excludes permafrost-related emissions, while the line marked with crosses includes them. When permafrost-related emissions are incorporated into the model—our baseline specification—the temperature response is higher, as increased emissions result in elevated temperatures, which consequently increase the likelihood of releasing carbon currently stored in permafrost.

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Several parameters in our framework lack direct equivalents in prior studies; this includes the parameters that govern the positive feedback loops ($N_t$, equation (7)), economic damages ($D_t$, equation (13)), and sea level rise ($H_t$, equation (8)). We calibrate these parameters to align with specific “targets” (or conditional moments) that aim to reflect the current understanding of climate science and for which our framework provides analytical formulas. Table 1 reports these parameters together with the targeted moments. While some targets are expressed in terms of conditional expectations; others are presented as conditional variances. The latter aims to align our model with the uncertainties recognized by climate scientists. We refer to Appendix D and Supplemental Appendix C for computational details regarding the calibration process.

Table 1 Model-specific parameters.

Parameter	Notation	Equation	Value	Unit/Note
Production function
Target: avg mean growth in the absence of climate risks	8.0	%/period (5yrs)
Target: Std dev. of growth in the absence of climate risks	3.0	%/period (5yrs)
Average TFP	Ā	(10)	0.41
Standard deviation of TFP shock	σA	(10)	3.42	%
Damage specification (excluding SLR-based damages)
Target: Expected 2100 fractional damage conditional on $T_{\mathrm{AT}}=2$°C	5.0	%
Target: Expected 2100 fractional damage conditional on $T_{\mathrm{AT}}=4$°C	10.0	%
Target: Std dev. of 2100 fractional damage conditional on $T_{\mathrm{AT}}=4$°C	7.5	%
Uncertainty parameter	μD	(13)	0.0352
Constant term in damage specification	a(D)	(13)	−0.0020
Slope coefficient in damage specification	b(D)	(13)	0.0034
Permafrost-related emissions
Target: Expected 2100 release conditional on $T_{\mathrm{AT}}=2$°C	205.4	GtCO2
Target: Expected 2100 release conditional on $T_{\mathrm{AT}}=4$°C	370.4	GtCO2
Target: Std dev. of 2100 release conditional on $T_{\mathrm{AT}}=4$°C	191.7	GtCO2
Target: Expected release conditional on $T_{\mathrm{AT},\mathrm{\infty }}=4$°C	748.1	GtCO2
Uncertainty parameter	μN	(7)	49.6
Constant term in carbon-release specification	a(N)	(7)	−5.7
Slope coefficient in carbon-release specification	b(N)	(7)	15.0
Decay rate of releases	κN	(7)	0.96
Sea-level rise (SLR)
Target: Expected 2100 SLR conditional on $T_{\mathrm{AT}}=2$°C	0.45	meter
Target: Expected 2100 SLR conditional on $T_{\mathrm{AT}}=4$°C	0.93	meter
Target: Std dev. of 2100 SLR conditional on $T_{\mathrm{AT}}=4$°C	0.36	meter
Uncertainty parameter in SLR specif.	μH	(8)	0.0715
Constant in SLR specif.	a(ΔH)	(8)	−0.0260
Coefficient on TAT in SLR specif.	b(ΔH)	(8)	0.0295
Uncertainty about temperature process
Target: Std dev. of 2100 atmospheric temperature	0.75	°C
Uncertainty parameter in TAT specification	μT	(2)	0.0647
Mitigation process
Based on agents' optimization on initial date
Coef. on imposed function form of μt	θa,opt	(17)	−1.288
Coef. on imposed function form of μt	θb,opt	(17)	0.064

Figure 3 illustrates how our calibrated damage function compares to other specifications used in the literature. It depicts damages, expressed as a fraction of the output (net of damages), as a function of atmospheric temperature in 2100.²² The black solid and dashed lines represent expected damages for the present model, with the solid line including sea-level-related damages and the dashed line excluding them. The two red squares indicate the targeted conditional means, which pertain to damages excluding those associated with sea-level rise.²³

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Figures 4 and 5 display, in blue, the resulting model-implied trajectories of atmospheric temperature and global mean sea level, respectively. The graphs also show, in the form of blue areas, the confidence intervals for both variables up to 2100; these intervals reflect the model-implied uncertainty regarding the variables in question (as of the initial date). We stress that the curves and confidence intervals traced on this figure are obtained by quasi-analytical formulas (i.e., based on a recursive algorithm), and not by simulation. The conditional means are of the form ${\alpha }^{\prime }{\mathbb{E}}_t(X_{t+h})$, where ${\mathbb{E}}_t(X_{t+h})$ directly stems from the VAR representation of $X_t$ (equation (24)). The confidence intervals are based on the conditional distribution, whose computation is based on inverse Fourier transforms (see footnote 20).

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Pricing

This section presents formulas for pricing various financial instruments whose payoffs are directly or indirectly affected by climate. Section 4.1 determines the stochastic discount factor, which serves as a key component of our pricing formulas. Section 4.2 provides formulas for pricing generic payoffs. Sections 4.3 and 4.4 demonstrate how to apply these formulas in the contexts of bonds (real and nominal) and climate derivatives, respectively. Supplemental Appendix F further shows how these formulas can be combined with a Black and Scholes (1973)-Merton (1974) structural approach to explore the influence of climate risks on credit and equity.

Solving for the value function and the stochastic discount factor

Knowing the SDF at any horizon allows to price long-dated assets. In general settings, deriving long-dated SDF can be computationally demanding and often necessitates sophisticated numerical tools, particularly when recursive preferences are involved (e.g., Daniel, Litterman, and Wagner (2019)). By contrast, the present model offers straightforward recursive formulas to compute the SDF at any date. The SDF turns out to be exponential affine in $X_t$, which opens the door to the machinery employed in the no-arbitrage term-structure modeling literature (e.g., Duffie, Filipovic, and Schachermayer (2003), Piazzesi (2010)).

Before determining the SDF (Proposition 4), we need to solve for the value function (Proposition 3). These two propositions essentially proceed under the assumptions that (a) the consumption process is affine in the state variable $X_t$, (b) the latter follows an affine process (with parameters that deterministically depend on time), and (c) agents feature Epstein–Zin preferences with unit EIS. Hence, this approach can be used in a wider context than the present one.

$3$

Proposition

Consider a general consumption growth process of the form: 28$\mathrm{\Delta }{c}_t={\mu }_{c,0,t}+{\mu }_{c,1,t}^{\prime }{X}_t,$ where ${\mu }_{c,0,t}$ and ${\mu }_{c,1,t}$ are deterministic functions of time, and are such that ${\mu }_{c,0,t}={\mu }_{c,0}$ and ${\mu }_{c,1,t}={\mu }_{c,1}$ for $t\ge t_{\mathrm{\infty }}$. Assume further that agents' preferences are of the Epstein–Zin type with unit EIS (equation (15)) and that the state vector dynamics is defined by (21) for $t<t_{\mathrm{\infty }}$, and by (27) after $t_{\mathrm{\infty }}$. In that context, a solution to the value function is of the form: 29$u_t=c_t+{\mu }_{u,0,t}+{\mu }_{u,1,t}^{\prime }{X}_t,$ where ${\mu }_{u,0,t}$ and ${\mu }_{u,1,t}$ do not depend on time for $t\ge t_{\mathrm{\infty }}$, and can be computed in a backward fashion for $t<t_{\mathrm{\infty }}$. More precisely, we have the following.

• Step 1:For $t\ge t_{\mathrm{\infty }}$, ${\mu }_{u,0,t}={\mu }_{u,0}$ and ${\mu }_{u,1,t}={\mu }_{u,1}$, where ${\mu }_{u,1}$ solves 30${\mu }_{u,1}=\frac{{\displaystyle \delta }}{{\displaystyle 1-\gamma }}\beta \{(1-\gamma )({\mu }_{u,1}+{\mu }_{c,1})\},$ and ${\mu }_{u,0}$ is given by 31${\mu }_{u,0}=\frac{{\displaystyle \delta }}{{\displaystyle 1-\delta }}{\mu }_{c,0}+\frac{{\displaystyle \delta }}{{\displaystyle 1-\delta }}\frac{{\displaystyle 1}}{{\displaystyle 1-\gamma }}\alpha \{(1-\gamma )({\mu }_{u,1}+{\mu }_{c,1})\}.$
• Step 2:For $t<t_{\mathrm{\infty }}$, $\{{\mu }_{u,0,t},{\mu }_{u,1,t}\}$ is deduced from $\{{\mu }_{u,0,t+1},{\mu }_{u,1,t+1}\}$ via 32$\{\begin{array}{c}{\mu }_{u,0,t}=\delta ({\mu }_{u,0,t+1}+{\mu }_{c,0,t+1})+\frac{{\displaystyle \delta }}{{\displaystyle 1-\gamma }}{\alpha }_t\{(1-\gamma )({\mu }_{u,1,t+1}+{\mu }_{c,1,t+1})\},\hfill \\ {\mu }_{u,1,t}=\frac{{\displaystyle \delta }}{{\displaystyle 1-\gamma }}{\beta }_t\{(1-\gamma )({\mu }_{u,1,t+1}+{\mu }_{c,1,t+1})\}.\hfill \end{array}$

The initial step involves solving the nonlinear vector equation (30). This is done by iterating on (30), starting from an arbitrary initial value for the vector ${\mu }_{u,1}$.²⁴ Once convergence is reached (typically within a few dozen iterations), ${\mu }_{u,0}$ is computed using (31).

The first column of Table 2 presents the scalar ${\mu }_{u,0}$ and the vector ${\mu }_{u,1}$ that characterize the stationary value function (for $t\ge t_{\mathrm{\infty }}$), in the context of baseline calibration discussed in Section 3.4. The second column gives $\{{\mu }_{u,0,t},{\mu }_{u,1,t}\}$ for $t=0$, or year 2020, obtained after the backward recursions (32). Observing that the value function is given by $U_t=\exp (u_t)=C_t\exp ({\mu }_{u,0,t}+{\mu }_{u,1,t}^{\prime }{X}_t)$, it comes that the entries of ${\mu }_{u,1,t}$ are the elasticities of the value function with respect to the state variables. This implies in particular that the coefficient of ${\mu }_{u,1,t}$ associated to $M_{\mathrm{AT}}$ is critical to determine the Social Cost of Carbon (see Section 5).²⁵

Table 2 Intertemporal utility solution for t ≥ t_∞ and t = 0.

	Utility (t ≥ t∞)	Utility (Ini. Period)	rt (t ≥ t∞)	rt (Ini. Period)
μu,0,t/μr,0,t	1.562	1.555	2.567 × 10−02	2.548 × 10−02
${\tilde{y}}_t$	−	−5.166 × 10−02	−	−
${\mathcal{E}}_t$	−1.450 × 10−04	−2.377 × 10−04	−	−
Ft	−2.986 × 10−03	−4.912 × 10−03	−	−
MAT,t	−1.010 × 10−04	−1.626 × 10−04	−	−
MUP,t	−8.597 × 10−05	−1.305 × 10−04	−	−
MLO,t	−2.169 × 10−05	−2.463 × 10−05	−	−
TAT,t	−4.575 × 10−03	−7.752 × 10−03	−1.271 × 10−03	−1.271 × 10−03
TLO,t	−2.694 × 10−02	−3.685 × 10−02	−	−

Knowing the value function, we can proceed with the derivation of the SDF, for any date t.

$4$

Proposition

Under the assumptions of Proposition 3, the SDF is given by ${\mathcal{M}}_{t,t+1}=\exp [-r_t+{\mathrm{\Pi }}_{t+1}^{\prime }{X}_{t+1}\underset{=-{\psi }_t({\mathrm{\Pi }}_{t+1}),\text{ see Proposition 1}}{\underbrace{-{\alpha }_t({\mathrm{\Pi }}_{t+1})-{\beta }_t{({\mathrm{\Pi }}_{t+1})}^{\prime }{X}_t}}],$ where $r_t:=-\log {\mathbb{E}}_t({\mathcal{M}}_{t,t+1})={\mu }_{r,0,t+1}+{\mu }_{r,1,t+1}^{\prime }{X}_t$ is the risk-free rate, and with 33$\{\begin{array}{c}{\mathrm{\Pi }}_{t+1}=(1-\gamma ){\mu }_{u,1,t+1}-\gamma {\mu }_{c,1,t+1},\hfill \\ {\mu }_{r,0,t+1}=-\log \delta +{\mu }_{c,0,t+1}+{\alpha }_t\{(1-\gamma )({\mu }_{u,1,t+1}+{\mu }_{c,1,t+1})\}-{\alpha }_t({\mathrm{\Pi }}_{t+1}),\hfill \\ {\mu }_{r,1,t+1}={\beta }_t\{(1-\gamma )({\mu }_{u,1,t+1}+{\mu }_{c,1,t+1})\}-{\beta }_t({\mathrm{\Pi }}_{t+1}),\hfill \end{array}$ where ${\mu }_{u,1,t+1}$ characterizes the value function (see Proposition 3).

The resulting SDF is exponential affine in the state vector, as is usual in the affine term structure literature (e.g., Piazzesi (2010)). A specificity is however here that the short-term rate $r_t$ specification (i.e., ${\mu }_{r,0,t}$ and ${\mu }_{r,1,t}$) and the prices of risk ${\mathrm{\Pi }}_t$ deterministically depend on time.

Let us highlight that while (33) defines the stochastic discount factor (SDF) based on our assumptions about agents' preferences and the production function, the pricing formulas provided below remain valid under any alternative specifications of ${\mu }_{r,0,t}$, ${\mu }_{r,1,t}$, and ${\mathrm{\Pi }}_t$. That is, one could use the pricing formulas of Section 4.2 after specifying $r_t$ to align with a desired term structure of interest.²⁶ Alternatively, one can adjust ${\mu }_{r,0,t}$, ${\mu }_{r,1,t}$, and ${\mathrm{\Pi }}_t$ so that ${\mathcal{M}}_{t,t+1}=\exp (\log (\delta )-\gamma \mathrm{\Delta }{c}_{t+1})$—which implicitly amounts to assuming that agents possess time-separable CRRA preferences, as demonstrated in Supplemental Appendix D. This adjustment would imply that the short-term rate is given by the Ramsey formula, aligning the model with the approach proposed in Chapter 6 of National Academies of Sciences, Engineering, and Medicine (2017) and implemented by REPR+ (2022), and the Environmental Protection Agency (2023), among others.

Generic pricing formulas

The fact that the SDF is exponential affine in the state vector, which itself follows an affine process, ensures the fast pricing of various generic assets. To start with, consider a basic asset, whose only payoff is $\exp ({\omega }^{\prime }{X}_{t+h})$, settled on date $t+h$.

$5$

Proposition

The date-t price of an asset providing the payoff $\exp ({\omega }^{\prime }{X}_{t+h})$ on date $t+h$ is 34${\phi }_t^{(h)}(\omega ):=\exp ({\phi }_{0,t}^{(h)}(\omega )+{\phi }_{1,t}^{(h)}{(\omega )}^{\prime }{X}_t),$ with 35$\{\begin{array}{c}{\phi }_{0,t}^{(h)}(\omega )=-{\mu }_{r,0,t+1}-{\alpha }_t({\mathrm{\Pi }}_{t+1})-\mathrm{\cdots }-{\mu }_{r,0,t+h}-{\alpha }_{t+h-1}({\mathrm{\Pi }}_{t+h})\hfill \\ \phantom{{\phi }_{0,t}^{(h)}(\omega )=}+{\psi }_{0,t}^{(h)}(u_1,\mathrm{\dots },u_h)\hfill \\ {\phi }_{1,t}^{(h)}(\omega )=-{\mu }_{r,1,t+1}-{\beta }_t({\mathrm{\Pi }}_{t+1})+{\psi }_{1,t}^{(h)}(u_1,\mathrm{\dots },u_h),\hfill \end{array}$ where functions ${\alpha }_t$ and ${\beta }_t$ characterize the Laplace transform of $X_t$ (see Proposition 1), ${\psi }_{0,t}^{(h)}$ and ${\psi }_{1,t}^{(h)}$ are defined in Proposition 2, ${\mu }_{r,0,t}$, ${\mu }_{,r1,t}$, and ${\mathrm{\Pi }}_t$ are defined in Proposition 4, and where 36$u_k=\{\begin{array}{cc}-{\mu }_{r,1,t+1+k}-{\beta }_{t+k}({\mathrm{\Pi }}_{t+1+k})+{\mathrm{\Pi }}_{t+k}\hfill & \text{for }k=1,\mathrm{\dots },h-1,\hfill \\ {\mathrm{\Pi }}_{t+k}+\omega \hfill & \text{for }k=h.\hfill \end{array}$

A direct implication of the previous proposition is that the date-t price of an asset providing the payoff ${\omega }^{\prime }{X}_{t+h}$ on date $t+h$ is²⁷ 37${\tilde{\phi }}_t^{(h)}(\omega )=\underset{\epsilon \to 0}{\lim }\frac{{\displaystyle {\phi }_t^{(h)}(\epsilon \omega )-{\phi }_t^{(h)}(0)}}{{\displaystyle \epsilon }}.$

Moreover, using the Fourier-transform approach of Duffie, Pan, and Singleton (2000), one can price instruments providing option-like payoffs, of the form $\exp ({\omega }^{\prime }{X}_{t+h}){\mathbb{1}}_{\{a^{\prime }{X}_{t+h}<b\}}$: 38$\begin{array}{rcl}{\hat{\phi }}_t^{(h)}(\omega ,a,b)& :=& {\mathbb{E}}_t\left({\mathcal{M}}_{t,t+h}\exp \right({\omega }^{\prime }{X}_{t+h}\left){\mathbb{1}}_{\{a^{\prime }{X}_{t+h}<b\}}\right)\\ & =& \frac{{\displaystyle {\phi }_t^{(h)}(\omega )}}{{\displaystyle 2}}-\frac{{\displaystyle 1}}{{\displaystyle \pi }}{\int }_0^{\mathrm{\infty }}\frac{{\displaystyle \text{Im}[{\phi }_t^{(h)}(\omega +iax)\exp (-ibx)]}}{{\displaystyle x}}dx,\end{array}$ where $\text{Im}(x)$ denotes the imaginary part of x and where function ${\phi }_t^{(h)}$ is defined in (34). It is important to note that the integral appearing in (38) is one-dimensional, irrespective of the dimension of $X_t$. As a result, Monte Carlo simulations are not required to evaluate this integral, as it is easily evaluated using Riemann sums.

Finally, the date-t price of an asset providing $({\omega }^{\prime }{X}_{t+h}){\mathbb{1}}_{\{a^{\prime }{X}_{t+h}<b\}}$ on date $t+h$ is²⁸ 39$\begin{array}{rl}{\overline{\phi }}_t^{(h)}(\omega ,a,b)& :={\mathbb{E}}_t\left({\mathcal{M}}_{t,t+h}\right({\omega }^{\prime }{X}_{t+h}\left){\mathbb{1}}_{\{a^{\prime }{X}_{t+h}<b\}}\right)\\ & =\underset{\epsilon \to 0}{\lim }\frac{{\displaystyle {\hat{\phi }}_t^{(h)}(\epsilon \omega ,a,b)-{\hat{\phi }}_t^{(h)}(0,a,b)}}{{\displaystyle \epsilon }},\end{array}$ where ${\hat{\phi }}_t^{(h)}(\omega ,a,b)$ is given by (38).

The following subsections show how the previous formulas can be used to price different types of assets, directly exposed to climate risks (e.g., climate derivatives), or indirectly. The latter category includes for instance treasury bonds. While these bonds are risk-free in a credit-risk sense, their pricing is affected by climate risks through economic growth, to which the SDF is linked. We start with this asset class.

Real and nominal bonds

As long as the model is not augmented with an inflation process (which will be done below), we can only price real, or inflation-linked, bonds. The SDF given in Proposition 4 is indeed real—it allows to price payoffs expressed in units of goods. In particular, ${\mathbb{E}}_t({\mathcal{M}}_{t,t+h})$ is the price of a bond providing a payoff of one unit of good on date $t+h$. (Note that ${\mathcal{M}}_{t,t+h}={\mathcal{M}}_{t,t+1}\times \mathrm{\cdots }\times {\mathcal{M}}_{t+h-1,t+h}$.) In other words, ${\mathbb{E}}_t({\mathcal{M}}_{t,t+h})$ is the price of a zero-coupon inflation-linked bond of maturity h. Equation (34) gives 40${\mathcal{B}}_{t,h}:={\mathbb{E}}_t({\mathcal{M}}_{t,t+h})={\phi }_t^{(h)}(0).$ Denote by $r_{t,h}$ the associated yield-to-maturity ($r_{t,1}$ being equal to the short-term real rate $r_t$, introduced in Proposition 4; we have 41$r_{t,h}:=-\frac{{\displaystyle 1}}{{\displaystyle h}}\log {\mathcal{B}}_{t,h}=-\frac{{\displaystyle 1}}{{\displaystyle h}}{\phi }_{0,t}^{(h)}(0)-\frac{{\displaystyle 1}}{{\displaystyle h}}{\phi }_{1,t}^{(h)}{(0)}^{\prime }{X}_t.$ This expression is reminiscent of the solutions of the so-called market models (Brace, Gatarek, and Musiela (1997), Miltersen, Sandmann, and Sondermann (1997)), in the sense that the resulting interest rates linearly depend on the state vector, but in a way that deterministically depends on time (since functions ${\phi }_{0,t}^{(h)}$ and ${\phi }_{1,t}^{(h)}$ are time varying, but deterministic).

Panel (a) of Figure 6 displays the model-implied term structure of real rates (black solid line). It is downward sloping, starts at less than 2.4% and settles below 2.1% for ultralong maturities, a level that is consistent with the upper bound of 2.6% determined by GMRS+ (2021) to price very-long-term payoffs. These properties of the risk-free term structure of real rates are broadly in line with Gollier and Hammitt (2014). As explained for instance in Gollier and Weitzman (2010), downward-sloping term structure of real interest rates are mechanically obtained in structural frameworks when growth shocks are persistent; this is typically the case in IAMs, as temperature increases have persistent adverse effects on growth. Panel (a) also compares our yield curve with that resulting from different frameworks (Bansal, Kiku, and Ochoa (2019), Lemoine (2021), Bauer and Rudebusch (2023)); all these yield curves are decreasing and show values of similar orders of magnitude.

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While the term structures shown in panel (a) of Figure 6 are those associated with the initial period ($t=0$), equation (41) can be used to determine the model-implied yields for any value of the state vector $X_t$, at any date t, and for any maturity h, exploiting that panel (b) of Figure 6 shows the expected trajectories of the 10-year rate until the end of the century. Specifically, panel (b) displays ${\mathbb{E}}_0(r_{t,h})$ for dates t going from the initial date to 2100, and for $h=2$ (one period is 5 years). The black line with circles indicates that the 10-year yield is anticipated to decline over time, with the decrease projected to reach 30 basis points (bps) by the year 2100. Two mechanisms account for this: first, consumption growth is expected to decrease over time, under the effect of rising temperatures and associated damages (expected growth channel); second, growth uncertainty will increase over time, accentuating precautionary effects (growth risk channel). To get a sense of the relative importance of these two effects, we compute the same interest rates under a version of the model where we remove climate-related risks (setting ${\mu }_D={\mu }_H={\mu }_N={\mu }_T=0$). The resulting term structures are displayed in grey in panel (b) of Figure 6 (triangles). The decrease in long-term rates, of about 20 bps over the next 80 years, is smaller than in the baseline base (30 bps).²⁹ The difference can be attributed to the climate-induced growth risk channel.

Let us now turn to nominal bonds. As mentioned above, for that purpose, we need to introduce a price index. We can add such a variable among the state variables of $X_t$, allowing for some effects of the other state variables on inflation, defined as the change in the log price index. Formally, if inflation (${\pi }_t$) can be expressed as an affine deterministic function of $X_t$, that is if ${\pi }_t={\mu }_{\pi ,0,t}+{\mu }_{\pi ,1}^{\prime }{X}_t,$ then we can augment $X_t$ with the log price index ($\log {\mathrm{PI}}_t$). The latter then satisfies 42$\log {\mathrm{PI}}_t=\log {\mathrm{PI}}_{t-1}+{\pi }_t=\log {\mathrm{PI}}_{t-1}+{\mu }_{\pi ,0,t}+{\mu }_{\pi ,1}^{\prime }{X}_t.$ The price-index-augmented state vector $X_t$ remains an affine process.³⁰

Consider the payoff of a nominal zero-coupon bond of maturity h, issued on date t. At maturity, its real payoff is ${\mathrm{PI}}_t/{\mathrm{PI}}_{t+h}=\exp ({\mu }_{\mathrm{PI}}^{\prime }(X_t-X_{t+h}))$, where ${\mu }_{\mathrm{PI}}$ is a selection vector, that is, such that $\log ({\mathrm{PI}}_t)={\mu }_{\mathrm{PI}}^{\prime }{X}_t$. Using (34), the date-t price of a zero-coupon nominal bond of maturity h is ${\mathcal{B}}_{t,h}^{\mathrm{\$}}:=\exp ({\phi }_{0,t}^{(h)}(-{\mu }_{\mathrm{PI}})+{[{\mu }_{\mathrm{PI}}+{\phi }_{1,t}^{(h)}(-{\mu }_{\mathrm{PI}})]}^{\prime }{X}_t).$ Denoting by $i_{t,h}$ the yield-to-maturity of a maturity-h nominal zero-coupon bond, we get 43$i_{t,h}:=-\frac{{\displaystyle 1}}{{\displaystyle h}}\log {\mathcal{B}}_{t,h}^{\mathrm{\$}}=-\frac{{\displaystyle 1}}{{\displaystyle h}}{\phi }_{0,t}^{(h)}(-{\mu }_{\mathrm{PI}})-\frac{{\displaystyle 1}}{{\displaystyle h}}{[{\mu }_{\mathrm{PI}}+{\phi }_{1,t}^{(h)}(-{\mu }_{\mathrm{PI}})]}^{\prime }{X}_t.$

To fix ideas, we can for instance have ${\pi }_t={\mu }_{\pi ,0,t}+{\mu }_{\pi ,D}{D}_t$, where the parameter ${\mu }_{\pi ,D}\ge 0$ determines the influence of climate-related damages ($D_t$) on inflation. Such a specification, in which damages have opposing effects on inflation and production, aligns with the interpretation of climate-related damages as supply shocks (see, e.g., Batten (2018), Andersson, Baccianti, and Morgan (2020), Chavleishvili and Moench (2024)).³¹ Several mechanisms may account for inflationary climate-related disasters. Supply chain disruptions and damages to infrastructure can lead to shortages, driving up prices. Additionally, reduced agricultural yields can raise food prices, while higher production costs—due to increased insurance rates and energy price volatility—are expected to result in businesses passing these expenses onto consumers.

Panel (a) of Figure 7 shows nominal yield curves (solid lines) obtained for different values of ${\mu }_{\pi ,D}$, namely 0.1, 0.5, and 5. The intermediate value corresponds to a scenario where a 1% decrease in output due to climate-related damages leads to a 0.5 percentage point rise in inflation.³² For each value of ${\mu }_{\pi ,D}$, we determine the deterministic sequence of ${\mu }_{\pi ,0,t}$ such that expected annualized inflation is 2% at any horizon.³³ Panel (a) shows that the higher the sensitivity of inflation to damages (captured by ${\mu }_{\pi ,D}$), the less downward-sloping the nominal yield curve. This is because the break-even inflation rate (BEIR)—the difference between nominal and real yields (Panels b to d)—is steeper for larger values of ${\mu }_{\pi ,D}$. The BEIR comprises two components: (i) expected inflation, which we have set to 2% by construction for the three values of ${\mu }_{\pi ,D}$ and for any horizon, and (ii) inflation risk premium, that is, the difference between the colored line and the black dashed line. The fact that supply shocks imply upward-sloping inflation risk premiums is a well-known result of the macro-finance literature (see, e.g., Rudebusch and Swanson (2012), Gurkaynak and Wright (2012)): supply shocks generate a negative correlation between consumption and inflation, which implies that nominal bonds tend to depreciate during recessions; as a result, risk-averse investors are willing to hold these bonds only if, on average, they provide an excess return with respect to risk-free (i.e., real) bonds. That is, in this case, the BEIR is larger than expected inflation. In the present case, this difference can be seen as a climate-induced inflation risk premium.

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Climate derivatives and risk premiums

A straightforward application of our framework is the pricing of climate derivatives, or climate linkers. Because of their nature, the valuation of these instruments provides a natural illustration of the notion of climate risk premium and, more generally, of the risk-adjusted—or risk-neutral—distributions of climate variables. Although our discussion focuses on temperature derivatives, we could equally have centered it on sea-level derivatives, or any other climate variable included in the state vector.

We start with the definition of a temperature-indexed swap. A swap is a basic derivative product that materializes the agreement between two counterparties to exchange cash flows at predetermined dates. The first type of cash flow—the fixed leg of the swap—is determined at the time the swap is negotiated (t, say) and paid by the protection buyer to the protection seller. In the context of a temperature-indexed swap, this cash flow would be $T_{\mathrm{AT},t,h}^{S}N$, where N denotes the notional amount, expressed in dollars per degree Celsius, and $T_{\mathrm{AT},t,h}^S$ is the negotiated swap temperature, h being the swap maturity. The second type of cash flow—the floating leg of the swap—depends on an index that is observed at maturity, the temperature $T_{\mathrm{AT},t+h}$ in the present case. This second cash flow, namely $T_{\mathrm{AT},t+h}N$, is paid by the protection seller to the protection buyer on date $t+h$. In the following discussion, we focus on a swap with a notional value of one unit ($N=1$$ per degree Celsius). This sometimes leads us to treat degrees Celsius as a price unit, as the fixed and floating payoffs of such a basic swap correspond to $T_{\mathrm{AT},t,h}^S$ and $T_{\mathrm{AT},t}$, respectively, making them homogeneous with temperatures. (By the same token, for a sea-level swap, one would use meters to express the values of the legs of a swap with a unit notional value.)

As for any swap, the swap temperature negotiated by the two counterparties on date t (namely $T_{\mathrm{AT},t,h}^S$) has to be such that the values of the two legs are equal on date t. Formally, we have ${\mathbb{E}}_t({\mathcal{M}}_{t,t+h}{T}_{\mathrm{AT},t+h})={\mathbb{E}}_t({\mathcal{M}}_{t,t+h}{T}_{\mathrm{AT},t,h}^S)$, or 44$T_{\mathrm{AT},t,h}^S={\mathbb{E}}_t\left(\frac{{\displaystyle {\mathcal{M}}_{t,t+h}}}{{\displaystyle {\mathbb{E}}_t({\mathcal{M}}_{t,t+h})}}{T}_{\mathrm{AT},t+h}\right)={\phi }_t^{(h)}({\mu }_{\mathrm{TAT}})/{\phi }_t^{(h)}(0),$ where ${\mu }_{\mathrm{TAT}}$ is the selection vector that is such that $T_{\mathrm{AT},t}={\mu }_{\mathrm{TAT}}^{\prime }{X}_t$. Equation (44) shows that $T_{\mathrm{AT},t,h}^S$ can be seen as a risk-adjusted expectation of $T_{\mathrm{AT},t+h}$. Formally, (44) rewrites $T_{\mathrm{AT},t,h}^S={\mathbb{E}}_t^{{\mathbb{Q}}^h}(T_{\mathrm{AT},t+h})$, where ${\mathbb{E}}_t^{{\mathbb{Q}}^h}$ denotes the expectation under the h-forward risk-neutral measure. The latter is defined with respect to the physical measure using the Radon–Nikodym derivative ${\mathcal{M}}_{t,t+h}/{\mathbb{E}}_t[{\mathcal{M}}_{t,t+h}]$.³⁴ Because the SDF reflects the marginal utility of consumption, it is elevated in “bad states of the world.” As a result, the Radon–Nikodym derivative is greater than 1 in bad states and less than 1 in good states. This indicates that, in comparison to the physical measure, the risk-neutral measure is biased towards unfavorable states of the world.

A third way to rewrite (44) is 45$T_{\mathrm{AT},t,h}^S={\mathbb{E}}_t[T_{\mathrm{AT},t+h}]+{\mathrm{prem}}_{t,h},\text{with }{\mathrm{prem}}_{t,h}=\frac{{\displaystyle \mathbb{C}o{v}_t[T_{\mathrm{AT},t+h},{\mathcal{M}}_{t,t+h}]}}{{\displaystyle {\mathbb{E}}_t[{\mathcal{M}}_{t,t+h}]}},$ which shows that ${\mathrm{prem}}_{t,h}$—the difference between the swap-implied temperature ($T_{\mathrm{AT},t,h}^S$) and the expected temperature (${\mathbb{E}}_t[T_{\mathrm{AT},t+h}]$)—depends on the covariance between temperatures and the SDF. In economic terms, if states of higher temperature are perceived as “bad states of the world” (states of high marginal utility, or high SDF), then the covariance term is positive, and the swap-implied temperature is above its expectation. In that case, the protection buyer is willing to lose money, on average, to be hedged against temperature risk—which is akin to paying an insurance premium.

As discussed above, Figure 4 shows the expected trajectories of temperatures under the physical measure (in blue). It also displays the risk-neutral expectations (in orange). Since $T_{\mathrm{AT},t,h}^S={\mathbb{E}}_t^{{\mathbb{Q}}^h}[T_{\mathrm{AT},t+h}]$, the solid orange line in panel (b) represents the term structure of the temperature swap as of the initial date ($t=0$). The deviation between the orange line and the blue line therefore corresponds to temperature risk premiums, denoted ${\mathrm{prem}}_{t,h}$ in (45). (Figure 5 provides similar information for sea levels; in this figure, the difference between the orange line and the blue line represents sea-level risk premiums.)

For the baseline calibration, we find a 2100 temperature risk premium of about 0.09°C; this value corresponds to the deviation between the means of the physical and risk-neutral distributions plotted on panel (c) of Figure 4, where these means are indicated by vertical solid lines. Results reported in Supplemental Appendix F document the sensitivity of the temperature risk premium to various model ingredients. We find, in particular, that this premium would be larger if there was no exogenous technological uncertainty (${\sigma }_A=0$), and lower in the absence of damage uncertainty (${\mu }_D=0$). (The premiums would then be 0.12°C and 0.06°C, respectively.) This is suggestive of two forces that push the temperature risk premium in opposite directions. The first is the capital accumulation channel: everything else equal, an increase in ${\sigma }_A$ augments the correlation between temperature and consumption since net damage consumption positively depends on capital, whose production generates carbon emissions; this tends to generate a negative correlation between temperatures and the SDF. The second is the damage channel: higher temperatures imply larger damage intensity, which works oppositely. As explained in the introduction, this is reminiscent of the two opposite forces that are at play to account for the insurance component of the SCC, namely the scaling and the damage-reduction channels (see Lemoine (2021)).

We have observed that temperature risk premiums, defined as the difference between a temperature swap and the physical expectation of temperature, align with the difference in the first-order moments of the risk-neutral and physical conditional distributions of temperature (equation (45)). Higher-order moments have an impact on different temperature risk premiums related to assets with payoffs not linearly dependent on climate variables. Consider for instance a digital option that provides a one-dollar payoff if temperature, at maturity, is larger than a given strike ($T_K$, say), and zero otherwise. The price of this option embeds a risk premium that depends on the difference between the right tails of the two distributions. Formally, the date-t price of this option is ${\mathbb{E}}_t({\mathcal{M}}_{t,t+h}{\mathbb{1}}_{\{T_{\mathrm{AT},t+h}>T_K\}})$. Supposing that the option payment is deferred until maturity, the option price is³⁵ 46$\frac{{\displaystyle {\mathbb{E}}_t({\mathcal{M}}_{t,t+h}{\mathbb{1}}_{\{T_{\mathrm{AT},t+h}>T_K\}})}}{{\displaystyle {\mathbb{E}}_t({\mathcal{M}}_{t,t+h})}}={\mathbb{E}}_t\left(\frac{{\displaystyle {\mathcal{M}}_{t,t+h}}}{{\displaystyle {\mathbb{E}}_t({\mathcal{M}}_{t,t+h})}}{\mathbb{1}}_{\{T_{\mathrm{AT},t+h}>T_K\}}\right)={\mathbb{P}}_t^{{\mathbb{Q}}_h}(T_{\mathrm{AT},t+h}>T_K).$ The previous expression shows that the option price is the size of the right-hand tail of the risk-neutral distribution. It can be decomposed as ${\mathbb{P}}_t(T_{\mathrm{AT},t+h}>T_K)+\underset{\text{risk premium}}{\underbrace{{\mathbb{P}}_t^{{\mathbb{Q}}_h}(T_{\mathrm{AT},t+h}>T_K)-{\mathbb{P}}_t(T_{\mathrm{AT},t+h}>T_K).}}$ The risk premium is positive if the risk-adjusted tail is larger than its physical counterpart, which, according to (46), corresponds to situations where there is a positive conditional correlation between high temperatures and the SDF.

Our framework offers computationally-efficient formulas to price such digital options, as well as calls and puts. Table 3 reports the payoffs and pricing formulas for the three types of options. For expository purposes, our numerical experiment focuses on digital options, whose prices can be interpreted as risk-adjusted probabilities of high temperatures (equation (46)). Figure 8 plots the prices of such digital options for $T_K=$2°C, 3°C, and 4°C, and maturities up to 2100. Consistently with what precedes, we observe that risk-adjusted probabilities are higher than their physical counterparts. For example, for the maturity year 2100, the risk-adjusted probability of exceeding $T_K=4$°C is 13%, compared to an 11% physical probability.

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Table 3 Climate options: types, payoffs, and pricing formulas.

Type	Payoff (maturity, date t + h)	Price (date t)
Digital	${\mathbb{1}}_{\{T_{t+h}>T_K\}}$	${\hat{\phi }}_t^{(h)}(0,-{\mu }_T,-T_k)$
Call	${(T_{t+h}-T_K)}^{+}={\mathbb{1}}_{\{T_{t+h}>T_K\}}(T_{t+h}-T_K)$	${\overline{\phi }}_t^{(h)}({\mu }_T,-{\mu }_T,-T_k)-T_K{\overline{\phi }}_t^{(h)}(0,-{\mu }_T,-T_k)$
Put	${(T_{t+h}-T_K)}^{-}={\mathbb{1}}_{\{T_{t+h}<T_K\}}(T_K-T_{t+h})$	$T_K{\overline{\phi }}_t^{(h)}(0,{\mu }_T,T_k)-{\overline{\phi }}_t^{(h)}({\mu }_T,{\mu }_T,T_k)$

Social cost of carbon

Definition and decomposition

Estimates of the social cost of carbon (SCC) have garnered significant attention over the past few decades, as they are a critical metric for shaping climate policy and informing benefit-cost analyses by governments and decision-makers (e.g., National Academies of Sciences, Engineering, and Medicine (2017), Environmental Protection Agency (2023)). This cost quantifies the monetary value of societal damages resulting from an additional metric tonne of carbon emissions. In the context of IAMs, it is formally defined as the reduction in current consumption that agents would be willing to accept for a one-unit decrease in $M_{\mathrm{AT},t}$. Using the value function $U_t$, this general definition reads 47${\mathit{SCC}}_t=-\frac{{\displaystyle \partial U_t}}{{\displaystyle \partial M_{\mathrm{AT},t}}}/\frac{{\displaystyle \partial U_t}}{{\displaystyle \partial C_t}}.$ In our framework, the SCC admits a simple expression since we can solve for the value function (Proposition 3). Specifically, in our framework, we have³⁶ 48${\mathit{SCC}}_t=-\frac{{\displaystyle {\mu }_{1,u,t,\mathrm{MAT}}}}{{\displaystyle 1-\delta }}{C}_t,$ where ${\mu }_{1,u,t,\mathrm{MAT}}$ is the entry of ${\mu }_{1,u,t}$ that corresponds to $M_{\mathrm{AT},t}$ (see Table 2 for values of ${\mu }_{1,u,t}$ at $t=0$ and $t=t_{\mathrm{\infty }}$). Equation (48) indicates that, at date t, agents are willing to lose a fraction ${\mu }_{1,u,t,\mathrm{MAT}}/(1-\delta )$ of their current consumption in exchange for the reduction of $M_{\mathrm{AT},t}$ by 1 GtC. With our baseline calibration, this gives a current SCC of $270 per ton of CO₂, which is higher than early estimates found in the literature, but rather in line with recent studies (see Table 4).

Table 4 SCC comparison (USD per ton of CO₂).

Study	SC-CO2	Tipping Points	Stochast. IAM	Discount.	Pref. for Present	Discount Rate
Jensen and Traeger (2014)	10–23		✓	Internal	1.5%
Lemoine and Traeger (2014)	10–15	✓	✓	Internal	1.5%
Bansal, Ochoa, and Kiku (2016)	1–28	✓	✓	Internal	1%
Bansal, Kiku, and Ochoa (2019)	45		✓	Internal	1%
Nordhaus (2017)	31			Internal	1.5%
Cai and Lontzek (2019)	10–27	✓	✓	Internal	1.5%
Daniel, Litterman, and Wagner (2019)	106–142	✓	✓	Internal	0.5%
Pindyck (2019)	67–86			External		2.6%
Barnett, Brock, and Hansen (2020)	65–112		✓	Internal	1%
Hambel, Kraft, and Schwartz (2021)	18–43		✓	Internal	1.5%
Lemoine (2021)	710		✓	Internal	1.5%
Van den Bremer and Van der Ploeg (2021)	40–66		✓	Internal	1.5%
Traeger (2021)	143–201		✓	Internal	1.4%
−	602		✓	Internal	0.7%
Traeger (2023)	30–134			Internal	1.4%
−	75–547			Internal	0.5%
−	290–2330			Internal	0.1%
Bauer and Rudebusch (2023)	101–963			External		1–2.5%
Environmental Protection Agency (2023)	117–337			Hybrid	0–0.5%	1.5–2.5%
REPR+ (2022)	59–185			Hybrid	0–0.8%	2–3%
This paper	270	✓	✓	Internal	1%

To gain insights into the potential determinants of the SCC, it is useful to consider the time-separable case. In that case, the SCC can be written as the sum of discounted future benefits resulting from the removal of one ton of carbon dioxide from the atmosphere on date t. Formally, ${\mathit{SCC}}_t=\sum _{h=1}^{\mathrm{\infty }}{\mathbb{E}}_t({\mathcal{M}}_{t,t+h}{\mathcal{P}}_{t,t+h}),$ where ${\mathcal{P}}_{t,t+h}$ represents such benefits for date $t+h$, that is, ${\mathcal{P}}_{t,t+h}=-\partial C_{t+h}/\partial M_{\mathrm{AT},t}$.³⁷ The SCC rewrites 49${\mathit{SCC}}_t=\sum _{h=1}^{\mathrm{\infty }}{\mathcal{B}}_{t,h}{\mathbb{E}}_t({\mathcal{P}}_{t,t+h})+\sum _{h=1}^{\mathrm{\infty }}\mathbb{C}o{v}_t({\mathcal{M}}_{t,t+h},{\mathcal{P}}_{t,t+h}),$ where ${\mathcal{B}}_{t,h}$ is the discount factor of maturity h, that is, the date-t price of a real zero coupon bond of maturity h (given in equation (40)). Hence, the first sum corresponds to the net present values of expected benefits. The second sum is nonzero when benefits ${\mathcal{P}}_{t,t+h}$ correlate with the stochastic discount factor ${\mathcal{M}}_{t,t+h}$; it captures SCC risk premiums. The sign of these risk premiums determines the sign of the so-called climate beta (e.g., Dietz, Gollier, and Kessler (2018), Lemoine (2021)). Everything else equal, (49) shows that higher expected damages increase the SCC. It also indicates that higher discount factors ${\mathcal{B}}_{t,h}$ or, equivalently, lower risk-free interest rates, have a positive impact on the SCC (looking at the first sum). We emphasize, however, that most model parameters simultaneously influence different components of (49). For example, alterations in the specification of damages impact ${\mathbb{E}}_t({\mathcal{P}}_{t,t+h})$, but they also result in changes to the consumption growth process. This, in turn, affects the stochastic discount factors ${\mathcal{M}}_{t,t+h}$, which subsequently influences the discount factors ${\mathcal{B}}_{t,h}$ and the components related to risk premiums.

While the decomposition (49) is useful to fix ideas, it is not applicable with Epstein–Zin preferences, where the SCC cannot be expressed in time-separated terms, as noted for instance by Jensen and Traeger (2014). In the Epstein–Zin case, we can nevertheless consider an ad hoc SCC risk premium defined as the difference between the SCC and the sum of future expected benefits, that is, ${\mathit{SCC}}_t-{\sum }_{h=1}^{\mathrm{\infty }}{\mathcal{B}}_{t,h}{\mathbb{E}}_t({\mathcal{P}}_{t,t+h})$.³⁸ In our baseline scenario, this risk premium is negative, with the SCC accounting for about 70% of the sum of future expected benefits.³⁹

SCC sensitivity

Table 5 presents SCC estimates under different model calibrations. Overall, the findings reveal that the damage specification, the coefficient of risk aversion, the rate of pure time preference, and climate sensitivity are key determinants of the SCC.

Table 5 Social cost of carbon, in US $ per ton of CO₂.

A. Epstein–Zin Preferences
Specification	$\gamma =7$	$\gamma =2$	$\gamma =10$
Baseline	270		223		313
Large rate of pref. for present (−log(δ)/Δt = 1.5%)	114	−57.7%	95	−57.3%	131	−58.0%
Small rate of pref. for present (−log(δ)/Δt = 0.5%)	1289	+376.6%	1039	+366.6%	1541	+392.9%
Large climate sensitivity (ν = 4)	324	+19.7%	265	+19.0%	376	+20.5%
Small climate sensitivity (ν = 2)	174	−35.8%	144	−35.2%	199	−36.3%
Deterministic model	209	−22.7%	209	−6.1%	209	−33.1%
No exog. techno. uncertainty (σA = 0)	270	+0.0%	223	+0.0%	313	+0.0%
No uncert. on damage specif. (μD = 0)	213	−21.4%	210	−5.9%	215	−31.4%
No uncert. on permafrost specif. (μN = 0)	266	−1.7%	222	−0.2%	303	−3.1%
No uncert. on sea-level specif. (μH = 0)	270	−0.1%	223	−0.0%	312	−0.1%
No uncert. on temperature specif. (μT = 0)	270	−0.3%	223	−0.0%	311	−0.5%
Small damages (twice lower)	112	−58.5%	104	−53.4%	118	−62.2%
No permaf. releases (a(N) = b(N) = 0)	199	−26.3%	167	−25.1%	227	−27.4%
No sea-level-specific damages (a(H) = b(H) = 0)	239	−11.6%	193	−13.5%	280	−10.4%
B. Time-Separable CRRA Preferences
Specification	$\gamma =1.5$	$\gamma =1.001$	$\gamma =2$
Baseline	73		215		35
Large rate of pref. for present (−log(δ)/Δt = 1.5%)	49	−32.7%	92	−57.2%	28	−20.3%
Small rate of pref. for present (−log(δ)/Δt = 0.5%)	118	+61.3%	853	+297.4%	45	+28.3%
Large climate sensitivity (ν = 4)	87	+19.3%	255	+18.8%	41	+18.4%
Small climate sensitivity (ν = 2)	47	−35.0%	139	−35.1%	23	−33.7%
Deterministic model	69	−5.3%	208	−2.9%	32	−7.6%
No exog. techno. uncertainty (σA = 0)	73	−0.3%	215	+0.0%	35	−0.8%
No uncert. on damage specif. (μD = 0)	69	−4.9%	208	−2.9%	33	−6.7%
No uncert. on permafrost specif. (μN = 0)	73	−0.1%	215	−0.0%	35	−0.2%
No uncert. on sea-level specif. (μH = 0)	73	−0.0%	215	−0.0%	35	−0.0%
No uncert. on temperature specif. (μT = 0)	73	−0.0%	215	−0.0%	35	−0.0%
Small damages (twice lower)	32	−55.7%	102	−52.5%	15	−56.9%
No permaf. releases (a(N) = b(N) = 0)	58	−20.6%	161	−24.9%	29	−16.8%
No sea-level-specific damages (a(H) = b(H) = 0)	62	−14.8%	185	−13.9%	30	−14.9%

Panel A reports results based on Epstein–Zin preferences—our baseline specification. The first line underscores the significant impact of the rate of time preference on the SCC, a topic widely discussed in the literature (e.g., Nordhaus (2017), Traeger (2021)). The estimate increases more than fourfold when this rate drops from 1% to 0.5% The following lines indicate that the SCC also depends heavily on climate sensitivity (ν in equation (1)). Additionally, the SCC decreases when risk is removed from the model, making it deterministic (${\mu }_T={\mu }_N={\mu }_H={\mu }_D={\sigma }_A=0$). This reduction varies with the coefficient of risk aversion γ: it is of 23% when $\gamma =7$, of 6% when $\gamma =2$, and of 33% when $\gamma =10$. Further lines indicate that uncertainty about damages accounts for most of the difference between the deterministic and baseline SCCs. Halving average damages reduces the SCC by 58% in our baseline case ($\gamma =7$). Eliminating permafrost-related emissions and sea-level-specific damages decreases the SCC by 26% and 12%, respectively.

Panel B of Table 5 presents the results of a similar exercise conducted with time-separable CRRA preferences. For comparable levels of risk aversion, SCCs are lower under CRRA than with Epstein–Zin preferences (see the columns for $\gamma =2$, which appear in both panels). This is primarily because discount rates are higher with CRRA preferences. Specifically, with CRRA preferences and in the deterministic case, the short-term real rate is given by a Ramsey-type equation, that is, $r=-\log (\delta )+\gamma \mathrm{\Delta }c$, where γ is the coefficient of risk aversion but also the inverse of the elasticity of intertemporal substitution (EIS). Thus, increasing γ leads to higher discount rates, and consequently, lower SCCs (see equation (49)). In this context, using γ larger than 2 or 3 implies large risk-free rates and, as a result, low SCCs. (This effect persists in the stochastic framework as well.) By breaking the inverse relationship between the EIS and risk aversion, Epstein–Zin preferences enable us to take into account higher coefficients of relative risk aversion without significantly increasing interest rates (Cai and Lontzek (2019), Daniel, Litterman, and Wagner (2019)). In the latter case, discount rates are less sensitive to γ, and higher levels of risk aversion generally result in higher SCCs (see panel A of Table 5).

Concluding remarks

This paper proposes a framework aiming to investigate the effects of climate risks on the pricing of long-dated assets. Our integrated assessment model (IAM) is solved instantaneously and offers quasi-analytical (recursive) pricing formulas for various assets. Tractability results from the fact that conditional on the path of some deterministic exogenous variables, the state vector follows an affine process. The model accommodates climate-related capital destruction as well as adverse feedback loops reflecting, for example, permafrost-related carbon releases.

Equipped with this model, we examine climate risk premiums. These premiums arise from differences between the physical and risk-neutral distributions, which our framework allows us to calculate. We define, in particular, the temperature risk premium as the deviation between the mean of the risk-neutral and physical distributions of the atmospheric temperature for a given horizon. This premium would stand as the difference between the price of a swap indexed to the future temperature and the physical expectation of this temperature. We argue that the sign of the temperature risk-premium is ambiguous, as it would depend on the relative importance of two forces driving it in opposite directions: (i) everything else equal, temperature is higher in scenarios where exogenous productivity shocks have led a rapid accumulation of capital, whose production implies carbon emissions; this capital accumulation channel contributes to the positiveness of the correlation between consumption and temperatures, and therefore to the negativeness of the temperature risk premium; but (ii) higher temperatures increase the intensity of climate-related damages; this damage channel has opposite effects. (These two opposite forces are reminiscent, but not equivalent, to those shaping the sign of the climate beta discussed, e.g., in Dietz, Gollier, and Kessler, 2018, and Lemoine, 2021.) The second effect dominates in our baseline calibration, where the temperature risk premium is about 0.1°C at horizon 2100.

The affine nature of our state vector process paves the way for various equity and bond pricing methodologies built upon the properties of such processes. One could, for instance, price equity by employing the Bansal and Yaron (2004) approach, which consists of specifying a process for the dividend growth rate and deducing an approximate expression for the stock return using Campbell and Shiller (1988)'s linearization. One can also price defaultable bonds by specifying default intensities that linearly depend on the climate variables included in the state vector, such as temperatures of sea levels (Duffie and Singleton (1999), Duffie (2005)). In Supplemental Appendix F, we show how to combine the present framework with a more structural approach (based on Black and Scholes (1973), Merton (1974)) to explore the influence of climate risks on credit and equity.

The present framework is flexible and could be enriched in several dimensions. As demonstrated by Traeger (2021), tractability is maintained in affine Integrated Assessment Models (IAMs) by linearly integrating any affine process into the state variables.⁴⁰ For instance, as shown in Supplemental Appendixes A and B, one can incorporate additional persistent Gaussian exogenous processes without compromising tractability.⁴¹ Such an extension could be used, for instance, to incorporate business-cycle components within the economic block of the model, to introduce exogenous long-term risks into the productivity growth process (as seen in, e.g., Bansal and Yaron (2004), Bansal and Shaliastovich (2013)), or to add sources of uncertainty to the system, similar to the temperature shocks described by Jensen and Traeger (2021), or the Brownian process used by Barnett, Brock, and Hansen (2020). These avenues for exploration remain open for future research.

A

Appendix

The use of the ${\gamma }_0$ distribution

A.1

Presentation of the ${\gamma }_0$ distribution

In (2), (7), (8), and (13), randomness is captured by means of the ${\gamma }_0$ distribution, introduced by Monfort, Pegoraro, Renne, and Roussellet (2017). We have $X\sim {\gamma }_0(\lambda ,\mu )$ if and only if there exists a Poisson variable $Z\sim \mathcal{P}(\lambda )$ such that the conditional distribution of X given Z is a Dirac mass at zero if $Z=0$, and a gamma distribution featuring a shape parameter Z and a scale parameter μ otherwise (i.e., if $Z>0$). In other words, this distribution arises as a Poisson mixture of gamma distributions.

It is easily seen that $X=0$ if and only if $Z=0$. As a consequence, 50$\mathbb{P}(X=0)=\exp (-\lambda ).$

Monfort, Pegoraro, Renne, and Roussellet (2017) show that the ${\gamma }_0$ distribution admits a simple Laplace transform: 51$\mathbb{E}(\exp [uX])=\exp (\lambda u\mu /(1-u\mu )).$ Importantly, the fact that this Laplace transform is exponential affine in λ implies that it can be used as a basic block to construct affine processes. (These processes can be multivariate, as is the case in the present paper.) For the same reason, all the cumulants of X are affine in λ. In particular, we extensively exploit the analytical expressions of the first two (mean and variance) in the present paper: 52$\mathbb{E}(X)=\mu \lambda ,\text{and}\mathbb{V}ar(X)=2\mu \mathbb{E}(X)=2{\mu }^2\lambda .$

A.2

Modeling functional uncertainty with the ${\gamma }_0$ distribution

In (2), (7), (8), and (13), the ${\gamma }_0$ distribution is used to model uncertainty regarding the effect of some variables on others. Consider two random variables $x_t$ and $y_t$. Suppose we are uncertain about how $x_t$ depends on $y_t$, although we expect an average linear relationship: $x=a+by$. We propose to model this situation in the following way: $x_t|y_t\sim {\gamma }_0((a+b{y}_t)/\mu ,\mu ).$ Owing to the properties of the ${\gamma }_0$ distribution, this implies $\mathbb{E}(x_t|y_t)=a+b{y}_t,\text{and}\mathbb{V}ar(x_t|y_t)=2\mu \mathbb{E}(x_t|y_t).$ Hence, this modeling encompasses the case where $x_t$ deterministically depends on $y_t$. Indeed, when μ is infinitely small, $x_t=a+b{y}_t$. When μ is large, $x_t$ may be far from $a+b{y}_t$.

This is illustrated by the following figure (Figure 9). We consider a given trajectory of $y_t$ (in black). We set $a=0$ anf $b=1$, so that $\mathbb{E}(x_t|y_t)=y_t$. (In a more general setting, as is the case in our model, the conditioning variable $y_t$ can itself be stochastic; it can also depend on the lags of $x_t$; uncertainty is then compounded.) The graphs display 80% confidence bands, which widen as μ increases. Panel (a) (small μ) is very close to the deterministic case, where $x_t=a+b{y}_t$.

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A.3

${\gamma }_0$ distribution and disasters

Variable $D_t$, that appears in the capital accumulation process (12), captures climate-related damages. Specification (13) is flexible regarding the “disastrous” nature of the damages, which can be measured by the relative fractions of draws that are strictly positive (the others being zero). The ${\gamma }_0$ distribution features a Dirac mass at zero, but the importance of this Dirac mass depends on the specification. Consider the distribution ${\gamma }_0(y/\mu ,\mu )$ (whose mean is y, and variance is $2\mu y$); the Dirac mass at zero is equal to $\exp (-y/\mu )$. Hence, if $y/\mu$ is large, the Dirac mass is infinitesimal (which is the case in equations (2), (7), and (8)). For our baseline calibration (Table 1), we have $D_{t+1}|T_{\mathrm{AT},t}\sim {\gamma }_0([-0.0020+0.0034T_{\mathrm{AT},t}]/0.0352,0.0352),$ which implies that 87% (resp., 79%) of the draws are equal to zero when $T_{\mathrm{AT}}=2$°C (resp., $T_{\mathrm{AT}}=3$°C). Note that when it comes to valuing long-term assets, it is the cumulative damage that counts, that is, ${\sum }_{i=1}^h{D}_{t+i}$. Conditionally on a given sequence of temperatures, and when damages are serially independent (which we assume), we have $\sum _{i=1}^h{D}_{t+i}|T_{\mathrm{AT},t:t+h}\sim {\gamma }_0([-h0.0020+0.0034(T_{\mathrm{AT},t}+\mathrm{\cdots }+T_{\mathrm{AT},t+h-1})]/0.0352,0.0352).$ The Dirac mass of this distribution is much lower, reducing the “disastrous” nature of damages for long horizons. (By the central limit theorem, this sum converges to a Gaussian distribution.) Considering an average temperature of $T_{\mathrm{AT}}=2$°C (resp., 3°C) over the next 16 periods (until 2100), we obtain a probability to have zero damages of 11% by then (resp., 2%). These features are illustrated by Figure 10, for an average temperature $T_{\mathrm{AT}}$ of 2°C.

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B

Appendix

From carbon concentrations to radiative forcing

The linearization of the standard relationship between $F_t$ and $M_{\mathrm{AT},t}$ (equation (4)) is important for the tractability of the model. This Appendix documents the effects of this simplifying assumption on the temperature trajectory.

Panel (a) of Figure 11 compares (4) with its standard nonlinear version for three different dates (2070, 2100, and 2150). The right end of the segments ends on the value of RCP8.5 for the corresponding date, as higher values are unlikely. (For the three dates, the $M_{\mathrm{AT},t}$ values for RCP4.5, RCP6.0, and RCP8.5 are indicated on the x-axis of the plot.) The figure also shows the distributions of $M_{\mathrm{AT},t}$ for the three dates, as derived from the present model and computed using the inverse Fourier transform. It suggests that the approximation is reasonable for the range of values deemed plausible by the model.

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Panels (b) to (e) of Figure 11 illustrate the effects of the linearization on the temperature trajectory for specific trajectories of carbon concentrations $M_{\mathrm{AT}}$. The black dotted line and the grey line correspond to the present model; while the former uses the linearized version of the model (i.e., equation (4)), the latter uses the standard nonlinear equation (in the latter case, the specification coincides with Folini, Friedl, Kübler, and Scheidegger (2025)). In both cases, the same linear equations govern the dynamics of $T_{\mathrm{AT}}$ and $T_{\mathrm{LO}}$. Panel (b) shows that the linearization has a negligible impact on the RCP6.0 (this is also the case for the RCP4.5). To check whether the approximation remains reasonable for higher trajectories of $M_{\mathrm{AT},t}$, we consider, in panels (c) and (d), scenarios where we linearly augment the RCP6.0 trajectories of $M_{\mathrm{AT},t}$ with emissions summing up to 500 and 1000 GtC by 2300, respectively. (Note that the “RCP6.0+1000 GtC trajectory” attains carbon concentration levels that are already particularly high, for instance, the associated level for 2150 is beyond the 99th percentile of the distribution of $M_{\mathrm{AT},t}$ resulting from our baseline model.) Panels (b) and (c) show that the linearized version of the model then overshoots the nonlinear one, but this difference remains contained in a ±1 standard deviation interval that reflects the temperature uncertainty captured by (2). The discrepancy between the linearized and nonlinearized model is more significant for RCP8.5 scenarios (see panel (e)). The ACE model delivers temperature responses that are closer to the CDICE ones for this scenario. Panel (a) however suggests that the probabilities of attaining such elevated values of $M_{\mathrm{AT},t}$ are negligible in the context of our model.

Unreported results—available on request—show that, for $M_{\mathrm{AT},t}$ scenarios generated by our baseline model, the distributions of temperatures resulting from the linearized and non-linearized versions of the model are small. For 2300, for instance, the resulting average temperature is 3.92°C for the linearized version and 3.88°C for the nonlinearized one; we respectively obtain standard deviations of 0.724 and 0.720.

C

Appendix

Matrices $A_{1,t}$, ${\omega }_{0,t}$, and ${\omega }_t$ (in Proposition 1)

The state vector is $X_t={[Z_t^{\prime }{W}_t^{\prime }]}^{\prime }$, with $Z_t={[{\tilde{y}}_t{\mathcal{E}}_t{F}_t{M}_{\mathrm{AT},t}{M}_{\mathit{UP},t}{M}_{\mathrm{LO},t}{T}_{\mathrm{AT},t}{T}_{\mathrm{LO},t}{H}_t]}^{\prime }$, and $W_t={[{\eta }_t{D}_t{N}_t{T}_{\mathrm{AT},t}\mathrm{\Delta }{H}_t]}^{\prime }$. The conditional dynamics of $Z_t$ is given by 53$Z_t=A_{1,t}{Z}_{t-1}+{\omega }_{0,t}+{\omega }_t{W}_t.$ Matrices $A_{1,t}$, ${\omega }_{0,t}$, and ${\omega }_t$ are given by $A_{1,t}={(A_0^{*})}^{\mathrm{-1}}{A}_{1,t}^{*}$, ${\omega }_{0,t}={(A_0^{*})}^{\mathrm{-1}}{\omega }_{0,t}^{*}$, and ${\omega }_t={(A_0^{*})}^{\mathrm{-1}}{\omega }_t^{*}$, where $A_0^{*}$, $A_{1,t}^{*}$, ${\omega }_{0,t}^{*}$, and ${\omega }_t^{*}$ are such that 54$A_0^{*}{Z}_t=A_{1,t}^{*}{Z}_{t-1}+{\omega }_{0,t}^{*}+{\omega }_t^{*}{W}_t,$ with $A_0^{*}{=}_{9\times 9}-\frac{\tau }{\log (2)M_{\mathrm{PI}}(m_{0,t})}{e}_3{e}_4^{\prime }$ ($e_i$ denoting the ith column of ${\mathbf{\text{Id}}}_{9\times 9}$), and where where ${\sigma }_{c,t}=\frac{(1-{\mathrm{\Lambda }}_t)}{(1-{\mathrm{\Lambda }}_t)Ā+(1-\mathrm{dep})}{\sigma }_A$ (equation (19)). As in the DICE model, the abatement cost ${\mathrm{\Lambda }}_t$ (expressed as a fraction of GDP) is given by ${\mathrm{\Lambda }}_t={\theta }_{1,t}{\mu }_t^{{\theta }_2}$, where ${\mu }_t$ is the mitigation rate and with 55${\theta }_{1,t}=\frac{{\displaystyle {\mathit{BP}}_t{\sigma }_t}}{{\displaystyle 1000\times {\theta }_2}},\text{with }{\mathit{BP}}_t=\mathit{pback}{(1-\mathit{gback})}^{t-1},$ where the carbon intensity ${\sigma }_t$ is given by 56${\sigma }_t={\sigma }_{t-1}(1+g_{\sigma ,t}),\text{with }{g}_{\sigma ,t}=g_{\sigma ,t-1}{(1+{\delta }_{\sigma })}^{\mathrm{\Delta }t}.$ Besides, following the DICE specification, exogenous radiative forcings ($F_{\mathrm{EX},t}$) and exogenous land emissions (${\mathcal{E}}_{\mathrm{Land},t}$) are respectively given by 57$F_{\mathrm{EX},t}=\{\begin{array}{cc}{\varphi }_0+\frac{{\displaystyle 1}}{{\displaystyle 16}}({\varphi }_1-{\varphi }_0)(t-1)\hfill & \text{if }t\le 16\text{(}t=16\text{ corresponds to 2100)},\hfill \\ {\varphi }_1\hfill & \text{if }t>16,\hfill \end{array}$ 58${\mathcal{E}}_{\mathrm{Land},t}={\epsilon }_0{(1-{\rho }_{\mathrm{land}})}^{t-1},\text{with }0<{\rho }_{\mathrm{land}}<1.$

Table 6 details the specifications of the ${\lambda }_t$ variables appearing in (2), (7), (8), (13), as well as $\mathrm{\Delta }{c}_t$ (28).

Table 6 Detailed specifications of λ_T,t (2), λ_N,t (7), λ_H,t (8), λ_D,t (13), and Δc_t (28).

	${\tilde{y}}_t$	${\mathcal{E}}_t$	Ft	MAT,t	MUP,t	MLO,t	TAT,t	TLO,t	Ht	ηt	Dt	Nt	TAT,t	ΔHt
${\ell }_{1,t+1}^{(D)}$	−	−	−	−	−	−	$\frac{b^{(D)}}{{\mu }_D}$	−	−	−	−	−	−	−
${\ell }_{1,t+1}^{(T_{\mathrm{AT}})}$	−	−	$\frac{{\xi }_1\mathrm{\Delta }t}{{\mu }_T}$	−	−	−	$\frac{1-{\xi }_1\mathrm{\Delta }t(\frac{\tau }{\nu }+{\xi }_2)}{{\mu }_T}$	$\frac{{\xi }_1{\xi }_2\mathrm{\Delta }t}{{\mu }_T}$	−	−	−	−	−	−
${\ell }_{1,t+1}^{(N)}$	−	−	−	−	−	−	$\frac{{\kappa }_N^t{b}^{(N)}}{{\mu }_N}$	−	−	−	−	−	−	−
${\ell }_{1,t+1}^{(\mathrm{\Delta }H)}$	−	−	−	−	−	−	$\frac{b^{(\mathrm{\Delta }H)}}{{\mu }_H}$	−	−	−	−	−	−	−
μc,1,t	−	−	−	−	−	−	−	−	−	σc,t	−1	−	−	−bSK

D

Appendix

Calibration

While numerous parameters of the model are directly taken from existing literature (see Table 7), several parameters of our framework have no direct equivalent in previous studies; this is notably the case of the parameters governing the positive feedback loops ($N_t$, equation (7)), economic damages ($D_t$, equation (13)), or sea level rise ($H_t$, equation (8)). We calibrate these parameters to match “targets” (or conditional moments) that reflect the current state of knowledge among climate scientists. The targets and results are reported in Table 1. Here are details on this calibration step:

Table 7 Calibrated parameters.

Parameter	Notation	Equation	Value	Unit/Note	Reference
Time step	Δt		5	years
Risk aversion	γ	(15)	7.0
Discount factor (annual)	$1-{\delta }^{\frac{1}{\mathrm{\Delta }t}}$	(15)	1.00	percent per year
Capital depreciation rate	dep	(12)	6	percent per year
Decline rate of decarbonization	δσ	(56)	−4.00	percent per period	DICE2023
Industrial emissions in 2020	e0	(σ0)	37.60	GtCO2 per year	DICE2023
Initial world gross output in 2020	q0	(20)	135.70	trillions of 2019 USD	DICE2023
Emission control rate in 2020	μ0	(17)	5.00	percent	DICE2023
Carbon intensity in 2020	σ0	(56)	$\frac{e_0}{q_0(1-{\mu }_0)}=0.29$	kgCO2 per 1 thousand USD of 2020 prod.	DICE2023
Initial growth of carbon intensity σ1	gσ,1	(56)	−1.50	percent per period	DICE2023
Backstop price	pback	(55)	695	2019 USD	DICE2023
Initial cost decline backstop cost	gback	(55)	5.00	percent per period	DICE2023
Exponent control cost function	θ2	(14)+(55)	2.60		DICE2023
Carbon emissions from land in 2020	ε0	(58)	5.90	GtCO2 per year	DICE2016
Decline rate in land emissions	ρ	(58)	10.00	percent per period	DICE2016
2020 forcings of non-CO2 GHG	ϕ0	(57)	0.52	W/m2	DICE2023
2100 forcings of non-CO2 GHG	ϕ1	(57)	0.80	W/m2	DICE2023
Equilibrium concentration in atmosphere	mateq	(5)	607	GtC	CDICE
Equilibrium concentration in upper strata	mueq	(5)	489	GtC	CDICE
Equilibrium concentration in lower strata	mleq	(5)	1281	GtC	CDICE
Transfer coef. from atm. to up. ocean	φ12	(5)	0.05300		CDICE
Transfer coef. from up. to lo. ocean	φ23	(5)	0.00820		CDICE
Climate eq. coefficient for upper level	ξ1	(2)	0.13700	per period	CDICE
Transfer coef. upper to lower stratum	ξ2	(2)	0.73000	per period	CDICE
Transfer coef. for lower stratum	ξ3	(3)	0.00689	per period	CDICE
Forcings of equilibrium CO2 doubling	τ	(4)+(2)	3.45	W/m2	CDICE
Equilibrium temperature impact	ν	(2)	3.25	°C per doubling CO2	CDICE
Destruction of capital due to rising SL	bSK	(18)	0.019		Diaz (2016)

(N). The four parameters characterizing $N_t$'s dynamics (${\mu }_N$, $a^{(N)}$, $b^{(N)}$, and ${\kappa }_N$) are calibrated to match the following conditional moments: (a) the expectation of the 2100 permafrost-related carbon release ($N_1+\mathrm{\cdots }+N_h$, period h corresponding to 2100) conditional on $T_{\mathrm{AT},h}=2$°C, (b) its expectation, but conditional on $T_{\mathrm{AT},h}=4$°C, (c) its standard deviation conditional on $T_{\mathrm{AT},h}=4$°C, and (d) the expectation of total permafrost-related carbon release conditional on a long-run temperature value ($T_{\mathrm{AT},\mathrm{\infty }}$). The targets are based on Table 1 in MacDougall and Knutti (2016), assuming that RCP2.6 (resp., RCP8.5) corresponds to a 2100 temperature anomaly of 2°C (resp., 4°C); our 4th target is based on their 2300 result (RCP6.0). Notice that targets (a) and (b) are also broadly in line with SACE+ (2022).

(D). The damage specification involves three parameters (${\mu }_D$, $a^{(D)}$, and $b^{(D)}$) that are calibrated to match (a) the expectation of 2100 cumulated damages conditional on $T_{\mathrm{AT},h}=2$°C, (b) their expectation conditional on $T_{\mathrm{AT},h}=4$°C, and (c) their standard deviation conditional on $T_{\mathrm{AT},h}=4$°C. The first two damage targets are consistent with KMNP+ (2021) and Howard and Sterner (2017); the relatively large standard deviation of damages is aimed at capturing the significant uncertainty surrounding the damage-function estimates, and to allow for estimates such as those of Burke, Hsiang, and Miguel (2015) (about −25% at 4°C, see their Figure 5.d) to remain in the plausible model-implied range.⁴² This specification does not covers sea-level-related damages; see footnote 43.

(H). The three parameters of the SLR specification (${\mu }_H$, $a^{(\mathrm{\Delta }H)}$, and $b^{(\mathrm{\Delta }H)}$) are calibrated to match (a) the expectation of the 2100 sea level conditional on $T_{\mathrm{AT},h}=2$°C, (b) its expectation conditional on $T_{\mathrm{AT},h}=4$°C, and (c) its standard deviation conditional on $T_{\mathrm{AT},h}=4$°C. The targets are based on HKCL+ (2020).

Table 1 shows the numerical values of the targeted moments and the associated parameters (target sources are given in the caption of the table). Figure 1, displayed in Supplemental Appendix C, proposes a graphical representation of the previous three steps of the calibration.

Moreover, we calibrate Ā and ${\sigma }_A$ appearing in (10) so that the average and the standard deviation of pre-damage growth match those computed on per capita real GDP growth over the last 40 years. Another parameter to calibrate is $b_{\mathrm{SK}}$, which determines SLR-related damages (in equation (12)). We set it to 0.019, based on Diaz (2016).⁴³

The last step of our calibration approach pertains to ${\mu }_T$, which characterizes the uncertainty in temperature dynamics (see equation (2)). Once all the other parameters have been calibrated, we determine the value of this parameter to obtain a conditional standard deviation of atmospheric temperature in 2100 of 0.75°C, which broadly reflects the temperature uncertainty reported in the IPCC AR6 report.⁴⁴

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