ABSTRACT We introduce a novel mean‐field game model for multi‐sector economic growth in which a dynamically evolving externality, influenced by the collective action of countries, plays a central role. By integrating environmental considerations in classical growth models, the framework incorporates “common noise” to capture shared uncertainties about the externality variable. We establish the existence and uniqueness of a mean‐field game equilibrium by reformulating the equilibrium conditions as a Forward‐Backward Stochastic Differential Equation via the stochastic maximum principle, first applying a contraction‐mapping argument to guarantee a unique solution, then employing the concept of weak equilibria to prove existence under more general assumptions, and finally invoking a specific monotonicity regime to reaffirm uniqueness. We provide a numerical resolution for a specified model using a fixed‐point approach combined with neural network approximations.
ABSTRACT The growing integration of solar power into electricity markets increasingly demands advanced risk management tools to address the inherent variability of solar radiation and its interaction with electricity prices. This paper introduces a novel framework for modeling and pricing new financial instruments designed to link payoffs directly to solar radiation levels, namely SoRad and SoREd. The proposed methodology models solar radiation through a bounded transformation and a Markov‐modulated Ornstein‐Uhlenbeck diffusion, where the latent regime is a two‐state continuous‐time Markov chain (CTMC) with month‐dependent generator. The model captures persistent switches between high‐ and low‐radiation regimes while preserving tractable conditional moments for pricing. Building on the observed correlation between solar radiation and electricity prices, we derive mean‐variance equilibrium prices and hedging rules that combine solar derivatives with electricity futures at the pricing date. The resulting prices are interpreted as inception‐date valuations for contracts over a fixed delivery period, not as a full dynamic secondary‐market equilibrium. The proposed instruments provide standardized tools for transferring solar radiation risk and reducing residual exposure to weather‐driven and electricity‐linked revenue fluctuations.
ABSTRACT Assuming a lognormal distribution for the underlying risky asset, we study the limiting distribution of a volatility target index as the rebalancing time step approaches zero. Two limit theorems (a strong law of large numbers and a central limit theorem) are established, and as an application, the exact limiting distribution is derived. We demonstrate that the volatility of the limiting distribution is consistently larger than the target volatility, and study the dependence of the convergence speed on the observation‐window parameter . Besides the exact formula for the drift and the volatility of the limiting distribution, their upper and lower bounds are derived. As a corollary of the exact limiting distribution, we obtain a vega conversion formula which converts the rho sensitivity of a financial derivative on the limiting diffusion to the vega sensitivity of the same financial derivative on the underlying of the volatility target index.
The optimal mean-variance selling problem seeks to determine a dynamically optimal stopping time in the nonlinear problem , where is a geometric Brownian motion with strictly positive drift, the supremum is taken over stopping times of , and is a given and fixed constant. The solution to the problem is known when the horizon is infinite, however, the method of proof developed to solve the problem in that case is not applicable in the case when the horizon is finite. In this paper, we develop a new method of proof, which solves the problem when the horizon is finite. In this way, we find that the dynamically optimal stopping time is given by , where the function can be characterized as a unique solution to a nonlinear Volterra integral equation. We also prove that the dynamically optimal stopping time satisfies the smooth fit principle. To our knowledge, this is the first time that such a nonlinear phenomenon of "dynamic smooth fit" has been derived in the literature. MSC2020 Classification: 60G40, 35R35, 60J65, 90C30, 45G10, 91B06
Environmental, social, and governance (ESG) ratings are pivotal in sustainable investing, informing portfolio construction and regulatory compliance. We develop a game-theoretic principal-agent model involving ESG-conscious investors and competing rating agencies. Investors impose ESG-related portfolio constraints by selecting ratings from different agencies, while agencies strategically issue ESG ratings based on a trade-off between financial incentives related to market share and reputational risks from deviating from ESG fundamentals. The interaction forms an extensive-form game with imperfect information, and we characterize its Stackelberg stable equilibrium. Our analysis shows that (1) agencies may strategically deviate ESG ratings from fundamentals under certain conditions, and such deviations tend to align ratings with asset returns when investors have strong green preferences; (2) asymmetric aversion for deviation generate rating disagreement between multiple agencies even when they have access to the same ESG fundamentals, potentially allowing one agency to monopolize green investors; (3) the magnitude of rating deviation governs a trade-off between investors' pecuniary utility and portfolio's fundamental greenness; and (4) firm-level greenwashing exacerbates rating deviation and further distorts the trade-off between utility and greenness. These results rationalize several empirical findings in the ESG literature and provide insights into designing mechanisms that encourage more informative ESG ratings.
We study a dynamic portfolio optimization problem under the mean-variance-variance (M-V-V) criterion proposed by Maccheroni et al. It is an analogue of the Arrow-Pratt approximation to the well-known smooth ambiguity model. Under the standard Black-Scholes framework, we derive fully explicit equilibrium investment strategies in which a DM's level of ambiguity, risk aversion, and ambiguity aversion are transparently captured. We find that the time horizon appears inconsistently in the objective function of the M-V-V criterion, in turn causing the equilibrium strategies to be nonmonotonic with respect to risk aversion. In response, we introduce a new mean-variance-standard deviation (M-V-SD) criterion to address this issue. Equilibrium strategies under the M-V-SD criterion exhibit an appealing feature of limited stock market participation which provides a theoretical justification to this phenomenon.
We analyze the problem of a profit-maximizing electricity producer, subject to carbon taxes, who decides on investments into abatement technologies. We assume that the carbon tax policy is random and that the investment in the abatement technology is divisible, irreversible, and subject to transaction costs. Two frameworks for randomness in taxes are considered. First, we assume a precise probabilistic model for the tax process, namely a pure jump Markov process (so-called tax-risk). Second, we analyze the case of a producer who is uncertainty-averse with respect to the tax evolution and who uses a differential game as conceptual tool to decide on optimal production and investment. We provide a rigorous mathematical treatment of both settings, including the analysis of the associated nonlinear PDEs. Numerical methods are employed to investigate the optimal investment strategies. We find that in the tax-risk case, investment in abatement technologies is generally lower than in a benchmark scenario with deterministic taxation. Nevertheless, factors such as production technology, investment divisibility, tax rebates, and credibility of the tax policy introduce interesting twists. In contrast, the uncertainty-averse framework may lead to increased investment as uncertainty rises.
This study develops a novel multivariate stochastic framework for assessing systemic risks, such as climate and nature-related shocks, within production or financial networks. By embedding a linear stochastic fluid network, interpretable as a generalized vector Ornstein-Uhlenbeck process, into the production network of interdependent industries, the model captures how physical shocks (e.g., extreme climate events or geopolitical disruptions) propagate through input-output (IO) linkages and affect sectoral price dynamics. The framework extends traditional IO models with advanced stochastic and dynamic features, enabling a quantification of both direct and indirect transmission channels of supply-cost shocks to production prices. Contributing to the literature on stochastic IO and Markovian networks, the model introduces the concept of divisible shocks, allowing for finer-grained simulation of adaptation responses and resilience across sectors. Empirical calibration leverages real-world economic data, including IO tables and historical industrial price indices. Sensitivity analyses are conducted using distributional risk measures, offering new tools for climate stress testing and medium to long-term risk assessment. Our findings support the optimal design of supply risk management strategies, including policy interventions and decentralized adaptation incentives for systemic stability under environmental stress.
Call option prices in the Black-Scholes model, viewed as functions of strike and maturity, are totally positive of order two (), meaning that the price ratio of a higher-strike call to a lower-strike call increases with maturity, with adjustments for dividends and interest. We develop conditions for this property in other models and contrast it with full total positivity, which holds only for out-of-the-money strikes in the Black-Scholes model. Related properties apply to puts. We give a simple sufficient condition for based on the unimodality of ratios of densities of the underlying asset at different dates. We show that the property entails a strengthening of monotonicity of the underlying asset in the convex order and thus a strengthening of the absence of static arbitrage. We construct examples illustrating the gaps between these properties. We develop connections between and the shape of the implied volatility surface-in particular, connections with supermodularity of implied variance, a condition implying that lines of implied variance for different maturities fan out at high strikes. An examination of S&P 500 options market data indicates that violations are infrequent and typically reverse quickly.
The paper treats the financial market as a communication system, using four information-theoretic assumptions to derive an idealized model with only one parameter. State variables are scalar stationary diffusions. The model minimizes the surprisal of the market and the Kullback-Leibler divergence between the benchmark-neutral pricing measure and the real-world probability measure. The state variables, their sums, and the growth optimal portfolio of the stocks evolve as squared radial Ornstein-Uhlenbeck processes in respective activity times.
Climate change has a dramatic impact, particularly by concentrating rainfall into a few short periods, interspersed with long dry spells. In this context, the role of dams is crucial. We consider the optimal control of a dam, where the water level must neither exceed a designated safety threshold nor fall below a minimum level to ensure functionality and sustainability for the downstream river. To model dry spells and intense rainfall events, commonly referred to as water bombs, we introduce a Hawkes process, a well-known example of a self-exciting process characterized by time-correlated intensity, which endogenously reproduces the clustering of events. The problem is formulated as an optimal switching problem with constraints. We establish existence results and propose numerical methods for approximating the solution. Finally, we illustrate the main achievements of this approach through numerical examples focusing in particular on the sensitivity of the self-exciting parameter describing the importance of both water bombs and dry-spells. For the parameter configurations considered in this paper, the optimal water level inside the dam decreases as the self-exciting parameter increases. This numerical finding suggests that, under the considered calibration, the management response is driven more strongly by overtopping risk than by drought risk. In conclusion, dams will increasingly lose their role as water reserves and take on a greater role in flood protection.
The relative arbitrage portfolio outperforms a benchmark portfolio over a given time-horizon with probability one. With market price of risk processes depending on the market portfolio and investors, this paper analyzes the multi-agent optimization of relative arbitrage opportunities in the coupled system of market and wealth dynamics. We construct a well-posed market dynamical system of McKean-Vlasov type under an empirical measure of investors, where each investor seeks for relative arbitrage with respect to a benchmark dependent on market and all the agents. We show the conditions to guaranty relative arbitrage opportunities among competitive investors through the Fichera drift. Under mild conditions, we derive the optimal strategies for investors and the unique Nash equilibrium that depends on the smallest nonnegative solution of a Cauchy problem.
In this article, we consider a stochastic linear quadratic control problem with partial observation. A near optimal control in the weak formulation is characterized. The main features of this paper are the presence of the control in the diffusion term of the state equation, the circular dependence between the control process and the filtration generated by the observation, and the observation process contains an unbounded drift term. We address these difficulties by first restricting the control to a smaller domain, which enables us to apply the Girsanov theorem using a conditional argument and thereby break the circular dependence. Subsequently, we study the restricted problem using a non-standard variation method. The desired near optimal control is then obtained by taking the limit of an approximating sequence.
We consider an equity market subject to risk from both unhedgeable shocks and default. The novelty of our work is that to partially offset default risk, investors may dynamically trade in a credit default swap (CDS) market. Assuming investment opportunities are driven by functions of an underlying diffusive factor process, we identify the certainty equivalent for a constant absolute risk aversion investor with a semi-linear partial differential equation (PDE) which has quadratic growth in both the function and gradient coefficients. For general model specifications, we prove existence of a solution to the PDE which is also the certainty equivalent. We show the optimal policy in the CDS market covers not only equity losses upon default (as one would expect), but also losses due to restricted future trading opportunities. We use our results to price default dependent claims though the principal of utility indifference, and we show that provided the underlying equity market is complete absent the possibility of default, the equity-CDS market is complete accounting for default. Lastly, through a numerical application, we show the optimal CDS policies are essentially static (and hence easily implementable) and that investing in CDS dramatically increases investor indirect utility.
We find the equilibrium contract that an automated market maker (AMM) offers to their strategic liquidity providers (LPs) in order to maximize the order flow that gets processed by the venue. Our model is formulated as a leader-follower stochastic game, where the venue is the leader and a representative LP is the follower. We derive approximate closed-form equilibrium solutions to the stochastic game and analyze the reward structure. Our findings suggest that under the equilibrium contract, LPs have incentives to add liquidity to the pool only when higher liquidity on average attracts more noise trading. The equilibrium contract depends on the external price, the pool reference price, and the pool reserves. Our framework offers insights into AMM design for maximizing order flow while ensuring LP profitability.
We study -player optimal execution games in an Obizhaeva-Wang model of transient price impact. When the game is regularized by an instantaneous cost on the trading rate, a unique equilibrium exists and we derive its closed form. Whereas without regularization, there is no equilibrium. We prove that existence is restored if (and only if) a very particular, time-dependent cost on block trades is added to the model. In that case, the equilibrium is particularly tractable. We show that this equilibrium is the limit of the regularized equilibria as the instantaneous cost parameter tends to zero. Moreover, we explain the seemingly ad hoc block cost as the limit of the equilibrium instantaneous costs. Notably, in contrast to the single-player problem, the optimal instantaneous costs do not vanish in the limit . We use this tractable equilibrium to study the cost of liquidating in the presence of predators and the cost of anarchy. Our results also give a new interpretation to the erratic behaviors previously observed in discrete-time trading games with transient price impact.
We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager's investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE's driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.
We study a problem of optimal irreversible investment and emission reduction formulated as a nonzero-sum dynamic game between an investor with environmental preferences and a firm. The game is set in continuous time on an infinite-time horizon. The firm generates profits with a stochastic dynamics and may spend part of its revenues towards emission reduction (e.g., renovating the infrastructure). The firm's objective is to maximize the discounted expectation of a function of its profits. The investor participates in the profits, may decide to invest to support the firm's production capacity and uses a profit function which accounts for both financial and environmental factors. Nash equilibria of the game are obtained via a system of variational inequalities. We formulate a general verification theorem for this system in a diffusive setup and construct an explicit solution in the zero-noise limit. Our explicit results and numerical approximations show that both the investor's and the firm's optimal actions are triggered by moving boundaries that increase with the total amount of emission abatement.
ABSTRACT The aggregate equity market displays only small price elasticity; in particular, macroeconomic allocations in and out of the equity market lead to surprisingly large impacts on stock valuations. Gabaix and Koijen study this phenomenon and provide a theoretical framework to explain the observed price inelasticity. They consider financial agents who are constrained in their investment strategies and subjected to a mandate that prescribes their investment allocations. Here we develop a rigorous framework of a mandate model for a representative agent and provide precise conditions under which the stock valuation dynamics are well defined. We also study how mandates amplify or attenuate the response of stock capitalization to changes in bond capitalization. We furthermore formulate conditions under which different funds, each one equipped with their own mandate, can be aggregated to a representative fund.
Stochastic versions of recursive integrated climate-economy assessment models are essential for studying and quantifying policy decisions under uncertainty. However, as the number of state variables and stochastic shocks increases, solving these models via deterministic grid-based dynamic programming (e.g., value-function iteration / projection on a discretized grid over continuous state variables, typically coupled with discretized shocks) becomes computationally infeasible, and simulation-based methods are needed. The least-squares Monte Carlo (LSMC) method has become popular for solving optimal stochastic control problems in quantitative finance. In this paper, we extend the application of the LSMC method to stochastic climate-economy models. We exemplify this approach using a stochastic version of the DICE model with five key uncertainty sources highlighted in the literature. To address the complexity and high dimensionality of these models, we incorporate deep neural network approximations in place of standard regression techniques within the LSMC framework. Our results demonstrate that the deep LSMC method can be used to efficiently derive optimal policies for climate-economy models in the presence of uncertainty.
