Society for Industrial and Applied Mathematics: SIAM Journal on Financial Mathematics: Table of Contents

Price Impact and Long-Term Profitability of Energy Storage

Roxana Dumitrescu — 2026-04-01T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 281-322, June 2026.
Abstract.We study the price impact of storage facilities in electricity markets and analyze the long-term profitability of these facilities in prospective scenarios of energy transition. To this end, we begin by characterizing the optimal operating strategy for a stylized storage system, assuming an arbitrary exogenous price process. We next determine the equilibrium price in market comprising storage systems (acting as price takers), renewable energy producers, and conventional producers with a defined supply function, facing an exogenous demand process. The price process is characterized as a solution to a fully coupled system of forward-backward stochastic differential equations, for which we establish existence and uniqueness under appropriate assumptions. We finally illustrate the impact of storage on intraday electricity prices through numerical examples and show how the revenues of storage agents may evolve in prospective energy transition scenarios from RTE, the French electricity network operator, taking into account both the increasing penetration of renewable energies and the self-cannibalization effect of growing storage capacity. We find that both the average revenues and the interquantile ranges increase as a function of time in all scenarios, highlighting higher expected profits and higher risk for storage assets.

Dual Representations for Quasiconvex Compositions with Applications to Systemic Risk Measures

Çağın Ararat — 2026-04-07T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 323-370, June 2026.
Abstract.Motivated by the problem of finding dual representations for quasiconvex systemic risk measures in financial mathematics, we study quasiconvex compositions in an abstract infinite-dimensional setting. We calculate an explicit formula for the penalty function of the composition in terms of the penalty functions of the ingredient functions. The proof makes use of a nonstandard minimax inequality (rather than equality as in the standard case) that is available in the literature. In the second part of the paper, we apply our results in concrete probabilistic settings for systemic risk measures, in particular, in the context of the Eisenberg–Noe clearing model. We also provide novel economic interpretations of the dual representations in these settings.

Kullback–Leibler Barycenter of Stochastic Processes

Sebastian Jaimungal — 2026-04-09T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 371-405, June 2026.
Abstract.We consider the problem where an agent aims to combine the views and insights of different experts’ models. Specifically, each expert proposes a diffusion process over a finite time horizon. The agent then combines the experts’ models by minimizing the weighted Kullback–Leibler divergence to each of the experts’ models. We show existence and uniqueness of the barycenter model and prove an explicit representation of the Radon–Nikodym derivative relative to the average drift model. We further allow the agent to include their own constraints, resulting in an optimal model that can be seen as a distortion of the experts’ barycenter model to incorporate the agent’s constraints. We propose two deep learning algorithms to approximate the optimal drift of the combined model, allowing for efficient simulations. The first algorithm aims at learning the optimal drift by matching the change of measure, whereas the second algorithm leverages the notion of elicitability to directly estimate the value function. The paper concludes with an extended application to combine implied volatility smile models that were estimated on different datasets.

Nonconcave Utility Maximization with Transaction Costs

Shuaijie Qian — 2026-04-15T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 406-449, June 2026.
Abstract.This paper studies a finite-horizon portfolio selection problem with nonconcave terminal utility and proportional transaction costs, in which the commonly used concavification principle for terminal value is no longer applicable. We establish a proper theoretical characterization of this problem via a two-step procedure. First, we examine the asymptotic terminal behavior of the value function, which implies that any transaction close to maturity only provides a marginal contribution to the utility. Second, we establish the theoretical foundation in terms of the discontinuous viscosity solution, incorporating the proper characterization of the terminal condition. Via extensive numerical analyses involving several types of utility functions, we find that the introduction of transaction costs into nonconcave utility maximization problems can make it optimal for investors to either hold on to a larger long position in the risky asset compared to the frictionless case or hold on to a large short position in the risky asset despite a positive risk premium.

Wasserstein Ergodicity of a Chen-Type Model with Correlated Noise

Giacomo Ascione — 2026-04-16T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 450-490, June 2026.
Abstract.In this study, we examine a Chen-type model with correlated noise, where the stochastic trend (process) is conditioned by stochastic volatility. This model is commonly referred to as the [math] model. The paper focuses on establishing the Wasserstein ergodicity of this model, a task that is not achievable through conventional means such as the Dobrushin theorem. Instead, alternative mathematical approaches are employed, including considerations of topological aspects of Wasserstein spaces and Kolmogorov equations for measures. The methodology developed in this study not only provides new insights but also extends these results to the widely used Chen-type model, which, despite its practical applications, lacks a solid mathematical foundation.

Cross-Currency Basis Swaps Referencing Backward-Looking Rates

Yining Ding — 2026-04-23T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 491-544, June 2026.
Abstract.The financial industry has undergone a significant transition from the London Interbank Offered Rate (LIBOR) to Risk Free Rates (RFRs) such as the Secured Overnight Financing Rate (SOFR) in the U.S. and the Cash Rate (AONIA) in Australia, as primary benchmark rates for borrowing costs. The paper examines the pricing and hedging method for financial products in a cross-currency framework with the special emphasis on the Compound SOFR vs Average AONIA cross-currency basis swap (CCBS) where both reference rates are backward-looking and the swap is collateralized. While the SOFR and AONIA are used as particular instances of RFRs in a cross-currency basis swap, the proposed approach is able to handle backward-looking rates for any two currencies. We give explicit pricing and hedging results for a constant notional cross-currency basis swap with either domestic or foreign collateralization using interest rate futures and currency futures as hedging instruments within an arbitrage-free cross-currency multicurve setting.

Collateralized Networks with Two Interacting Channels of Fire Sales

Raymond Ka-Kay Pang — 2026-04-30T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 545-564, June 2026.
Abstract.We develop a model for financial contagion in collateralized networks in which two channels of fire sales interact. We consider a financial market with multiple assets that can be used for both investment purposes and to satisfy collateral requirements. In our model, a fire sale can be triggered both before default, when illiquid assets are sold to satisfy payment obligations, and after default, when collateral of defaulted institutions is sold. We investigate contagion that arises from the overlap in assets used for investment purposes and as collateral. In particular, we illustrate how fire sales triggered prior to default can reduce the effectiveness of collateralization after a default. Our results highlight the importance of using high-quality assets as collateral to improve financial stability.

Computing Systemic Risk Measures with Graph Neural Networks

Lukas Gonon — 2026-05-21T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 565-619, June 2026.
Abstract.This paper investigates systemic risk measures for stochastic financial networks of explicitly modeled bilateral liabilities. We extend the notion of systemic risk measures based on random allocations from Biagini, Fouque, Fritelli, and Meyer-Brandis (2019) to graph structured data. In particular, we focus on aggregation functions that are derived from a market clearing algorithm proposed by Eisenberg and Noe (2001). In this setting, we show the existence of an optimal random allocation that distributes the overall minimal bailout capital and secures the network. We then study numerical methods for the approximation of systemic risk and optimal random allocations. Our proposition is to use permutation-equivariant architectures of neural networks such as graph neural networks (GNNs) and a class that we name (extended) permutation-equivariant neural networks ((X)PENNs). The performance of these architectures is benchmarked against several alternative allocation methods. The main feature of GNNs and (X)PENNs is that they are permutation-equivariant with respect to the underlying graph data. In numerical experiments, we find evidence that these permutation-equivariant methods are superior to other approaches.

Global Convergence of Deep Galerkin and PINN Methods for Solving Partial Differential Equations

Deqing Jiang — 2026-05-29T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page 620-645, June 2026.
Abstract.Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. This is in particular a fundamental challenge for the solution of financial models, which are often inherently high-dimensional. Option pricing, hedging, mean-field financial models, order book models, and dynamic portfolio investment can all require the solution of high-dimensional PDEs. A variety of deep learning methods have recently been developed to try and solve high-dimensional PDEs by approximating the solution using a neural network. These deep learning methods have been widely applied to high-dimensional PDEs in financial engineering. In this paper, we prove global convergence for one of the commonly used deep learning algorithms for solving PDEs, the deep Galerkin method (DGM). DGM trains a neural network approximator to solve the PDE using stochastic gradient descent. We prove that, as the number of hidden units in the single-layer network goes to infinity (i.e., in the “wide network limit”), the trained neural network converges to the solution of an infinite-dimensional linear ordinary differential equation (ODE). The PDE residual of the limiting approximator converges to zero as the training time [math]. Under mild assumptions, this convergence also implies that the neural network approximator converges to the solution of the PDE. A closely related class of deep learning methods for PDEs is physics-informed neural networks (PINNs), which have been widely used in a variety of fields (including financial mathematics but also physics and engineering). Using the same mathematical techniques, we can prove a similar global convergence result for the PINN neural network approximators. Both proofs require analyzing a kernel function in the limit ODE governing the evolution of the limit neural network approximator. A key technical challenge is that the kernel function, which is a composition of the PDE operator and the neural tangent kernel operator, lacks a spectral gap, therefore requiring a careful analysis of its properties.

Short Communication: Martingale Expansion for Stochastic Volatility

Masaaki Fukasawa — 2026-04-13T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 2, Page SC1-SC12, June 2026.
Abstract.The martingale expansion provides a refined approximation to the marginal distributions of martingales beyond the normal approximation implied by the martingale central limit theorem. We develop a martingale expansion framework specifically suited to continuous stochastic volatility models. Our approach accommodates both small volatility-of-volatility and fast mean-reversion models, yielding first-order perturbation expansions under essentially minimal conditions.

Uniswap V3: Impermanent Loss Modeling and Swap Fees Asymptotic Analysis

Mnacho Echenim — 2026-01-07T08:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 1-40, March 2026.
Abstract.Automated Market Makers have emerged quite recently, and Uniswap is one of the most widely used platforms (it covers 60% of the total value locked on Ethereum blockchain at the time of writing this article). This protocol is challenging from a quantitative point of view because it allows participants to choose where they wish to concentrate liquidity. There has been an increasing number of research papers on Uniswap v3 but often, these articles use heuristics or approximations that can be far from reality: for instance, the liquidity in the pool is sometimes assumed to be constant over time, which contradicts the mechanism of the protocol. The objectives of this work are fourfold. First, we revisit Uniswap v3’s principles in detail (starting from the open source code) to build an unambiguous knowledge base. Second, we analyze the Impermanent Loss of a liquidity provider by detailing its evolution, with no assumption on the swap trades or liquidity events that occur over the time period. Third, we introduce the notion of a liquidity curve. For each curve, we can construct a payoff at a given maturity, net of fees. Conversely, we show how any concave payoff can be synthetized by an initial liquidity curve and some tokens outside the pool; this paves the way for using Uniswap v3 to create options. Fourth, we analyze the asymptotic behavior of collected fees without any simplifying hypothesis (like a constant liquidity), given the mild assumption that the pool price coincides with a latent price (general Ito process) every time the latter changes by [math]. The asymptotic analysis is conducted as [math] within the arbitrage model by [Angeris et al., 2021]. The value of the collected fees then coincides with an integral of call and put prices. Our derivations are supported by graphical illustrations and experiments.

Portfolio Selection in Contests

Yumin Lu — 2026-01-08T08:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 41-77, March 2026.
Abstract.In an investment contest with incomplete information, a finite number of agents dynamically trade assets with idiosyncratic risk and are rewarded based on the relative ranking of their terminal portfolio values. We explicitly characterize a symmetric Nash equilibrium of the contest and rigorously verify its uniqueness. The connection between the reward structure and the agents’ portfolio strategies is examined. A top-heavy payout rule results in an equilibrium portfolio return distribution with high positive skewness, which suffers from a large likelihood of poor performance. Risky asset holding increases when competition intensifies in a winner-takes-all contest.

Optimal Consumption under Relaxed Benchmark Tracking and Consumption Drawdown Constraint

Lijun Bo — 2026-02-20T08:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 78-117, March 2026.
Abstract.This paper studies an optimal consumption problem with both relaxed benchmark tracking and consumption drawdown constraint, leading to a stochastic control problem with dynamical state-control constraints. In our relaxed tracking formulation, it is assumed that the fund manager can strategically inject capital to the fund account such that the total capital process always outperforms the benchmark process, which is described by a geometric Brownian motion. We first transform the original regular-singular control problem with state-control constraints into an equivalent regular control problem with a reflected state process and consumption drawdown constraint. By utilizing the dual transform and the optimal consumption behavior, we then turn to study the linear dual PDE with both Neumann boundary condition and free boundary condition in a piecewise manner across different regions. Using the smooth-fit principle and the supercontact condition, we derive the closed-form solution of the dual PDE and obtain the optimal investment and consumption in feedback form. We then prove the verification theorem on optimality by some novel arguments with the aid of an auxiliary reflected dual process and some technical estimates. Some numerical examples and financial insights are also presented.

The McCormick Martingale Optimal Transport

Erhan Bayraktar — 2026-02-25T08:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 118-153, March 2026.
Abstract.Martingale optimal transport (MOT) often yields broad price bounds for options, constraining their practical applicability. In this study, we extend MOT by incorporating causality constraints among assets, inspired by the nonanticipativity condition of stochastic processes. This, however, introduces a computationally challenging bilinear program. To tackle this issue, we propose McCormick relaxations to ease the bicausal formulation and refer to it as McCormick MOT. The primal attainment and strong duality of McCormick MOT are established under standard assumptions. Empirically, we apply McCormick MOT to basket and digital options. With natural bounds on probability masses, the average price reduction for basket options is approximately 1.08% to 3.90%. When tighter probability bounds are available, the reduction increases to 12.26%, compared to the classic MOT, which also incorporates tighter bounds. For most dates considered, there are basket options with suitable payoffs, where the price reduction exceeds 10.00%. For digital options, McCormick MOT results in an average price reduction of over 20.00%, with the best case exceeding 99.00%.

Optimal Reinsurance Design under the Moment-Based Premium Principle: A Representative Reinsurer’s Perspective

Tim J. Boonen — 2026-03-02T08:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 154-186, March 2026.
Abstract.This paper investigates the optimal reinsurance problem between one insurer and multiple reinsurers, where each reinsurer prices the contract based on the first two moments of the ceded loss, and the insurer aims to minimize a distortion risk measure. We provide a representative reinsurer’s perspective to solve the problem; the representative reinsurer’s premium principle admits an analytical form and possesses the properties of monotonicity and convexity. This allows us to use a convex programming approach to numerically solve the main problem. If all the reinsurers apply the same safety loading factor for the first moment of the ceded loss in their premium principles, the representative reinsurer’s premium principle also relies only on the first two moments of the ceded loss. This significantly reduces the complexity of the original problem, allowing us to use a quadratic programming approach to find the solution.

Perpetual American Options in a Jump-Diffusion Model with Random Inspection

Michael V. Boutsikas — 2026-03-02T08:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 187-222, March 2026.
Abstract.We investigate the problem of pricing perpetual American put and call options under the assumption that the option can be exercised only at random inspection times which are formulated by a Poisson process with constant intensity [math]. More specifically, we are interested in the expected payoff of the option, the optimal exercise policy, the probability of exercising, and the distribution of the time until exercise, under the real-world and the risk-neutral measure of the market. The main results are valid when the log-price of the underlying asset follows a Lévy jump-diffusion process (with two-sided jumps). Initially we present key identities to express the option’s expected payoff in terms of the distribution of the undershoot/overshoot for the put/call scenarios. We further study in more detail the case of pure diffusion (Brownian motion) and the case of Brownian motion with double-exponential jumps by offering explicit formulae for the quantities of interest. By assuming that [math] we also confirm well-established results that apply to previously studied continuous inspection models. We also include several numerical examples in order to illustrate the impact of the inspection intensity on pricing outcomes for both put and call options.

Unbiased Simulation of Asian Options

Bruno Bouchard — 2026-03-10T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 223-244, March 2026.
Abstract.We provide an extension of the unbiased simulation method for stochastic differential equations (SDEs) developed by Henry-Labordère et al. [Ann Appl Probab., 27 (2017), pp. 1–37] to a class of path-dependent dynamics, pertaining to Asian options. In our setting, both the payoff and the SDE coefficients depend on the (weighted) average of the process or, more precisely, on the integral of the solution to the SDE against a continuous function with finite variation. In particular, this applies to the numerical resolution of the class of path-dependent PDEs whose regularity, in the sense of Dupire, is studied by Bouchard and Tan [Ann. Inst. Henri Poincaré Probab. Stat., to appear].

Time-Causal VAE: Robust Financial Time Series Generator

Beatrice Acciaio — 2026-03-12T07:00:00Z

SIAM Journal on Financial Mathematics, Volume 17, Issue 1, Page 245-279, March 2026.
Abstract.We build a time-causal variational autoencoder (TC-VAE) for robust generation of financial time series data. Our approach imposes a causality constraint on the encoder and decoder networks, ensuring a causal transport from the real market time series to the fake generated time series. Specifically, we prove that the TC-VAE loss provides an upper bound on the causal Wasserstein distance between market distributions and generated distributions. Consequently, the TC-VAE loss controls the discrepancy between optimal values of various dynamic stochastic optimization problems under real and generated distributions. To further enhance the model’s ability to approximate the latent representation of the real market distribution, we integrate a RealNVP prior into the TC-VAE framework. Finally, extensive numerical experiments show that TC-VAE achieves promising results on both synthetic and real market data. This is done by comparing real and generated distributions according to various statistical distances, demonstrating the effectiveness of the generated data for downstream financial optimization tasks as well as showcasing that the generated data reproduce stylized facts of real financial market data.