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.
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.
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 CIR3 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.
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 \lambda . 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 Le'\vy 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 \lambda \rightarrow \infty 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.
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.
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 \gamma \%. The asymptotic analysis is conducted as \gamma \rightarrow 0 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.
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-takesall contest.
Local stochastic volatility refers to a popular model class in applied mathematical finance that allows for "calibration-on-the-fly", typically via a particle method, derived from a formal McKean-Vlasov equation. Well-posedness of this limit is a well-known problem in the field; the general case is largely open, despite recent progress in Markovian situations. Our take is to start with a well-defined Euler approximation to the formal McKean-Vlasov equation, followed by a newly established half-step-scheme, allowing for good approximations of conditional expectations. In a sense, we do Euler first, particle second in contrast to previous works that start with the particle approximation. We show weak order one for the Euler discretization, plus error terms that account for the said approximation. The case of particle approximation is discussed in detail and the error rate is given in dependence of all parameters used.
We provide an extension of the unbiased simulation method for SDEs developed in Henry-Labordere et al. [Ann Appl Probab. 27:6 (2017) 1-37] to a class of path-dependent dynamics, pertaining for Asian options. In our setting, both the payoff and the SDE's 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 bounded variations. In particular, this applies to the numerical resolution of the class of path-dependent PDEs whose regularity, in the sens of Dupire, is studied in Bouchard and Tan [Ann. I.H.P., to appear].
In this paper we use Malliavin Calculus techniques in order to obtain expressions for the short-time behavior of the at-the-money implied volatility (ATM-IV) level and skew for a jump-diffusion stock price. The diffusion part is assumed to be the stochastic volatility Bachelier model and the jumps are modeled by a pure-jump Lévy process with drift so that the stock price is a martingale. Regarding the level, we show that the short-time behavior of the ATM-IV level is the same for all pure-jump Lévy processes and, regarding the skew, we give conditions on the law of the jumps for the skew to exist. We also give several numerical examples of stochastic volatilities and Lévy processes that confirm the theoretical results found in the paper.
This paper investigates systemic risk measures for stochastic financial networks of explicitly modelled bilateral liabilities. We extend the notion of systemic risk measures from Biagini, Fouque, Fritelli and Meyer-Brandis (2019) to graph structured data. In particular, we focus on an aggregation function that is 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 study numerical methods for the approximation of systemic risk and optimal random allocations. We propose to use permutation equivariant architectures of neural networks like graph neural networks (GNNs) and a class that we name (extended) permutation equivariant neural networks ((X)PENNs). We compare their performance to several benchmark allocations. 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.
This paper proposes a new approach using the stochastic projected gradient method and Malliavin calculus for optimal reinsurance and investment strategies. Unlike traditional methodologies, we aim to optimize static investment and reinsurance strategies by directly minimizing the ruin probability. Furthermore, we provide a convergence analysis of the stochastic projected gradient method for general constrained optimization problems whose objective function has Hölder continuous gradient. Numerical experiments show the effectiveness of our proposed method.
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 reproduces stylized facts of real financial market data.
This paper studies an optimal consumption problem with both relaxed benchmark tracking and consumption drawdown constraint, leading to a stochastic control problem with dynamic 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 smoothfit principle and the super-contact 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 estimations. Some numerical examples and financial insights are also presented.
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.
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.
Abstract. In this paper we investigate a Merton-type portfolio optimization problem with a minimum comfortable consumption constraint, utilizing a stochastic control approach. By translating the Hamilton–Jacobi–Bellman (HJB) equations into second-order ordinary differential equations through a novel method, we precisely characterize the set of candidate value functions. We then identify the optimal consumption rate, investment strategy, and the value function explicitly by extending the recent stochastic perturbation method presented in [M. Herdegen et al., Math. Finance, 31 (2021), pp. 1218–1239]. This approach can be applied to derive explicit solutions for other portfolio choice problems under constraints, with detailed studies of the corresponding HJB equations. In addition, we have extended the model when inflation is considered. We also discuss some applications, such as retirement funds, pension funds, endowment portfolios and the AK model for economic growth.
We analyse the regret arising from learning the price sensitivity parameter κ of liquidity takers in the ergodic version of the Avellaneda-Stoikov market making model. We show that a learning algorithm based on a maximum-likelihood estimator for the parameter achieves the regret upper bound of order ln^2 T in expectation. To obtain the result we need two key ingredients. The first is the twice differentiability of the ergodic constant under the misspecified parameter in the Hamilton-Jacobi-Bellman (HJB) equation with respect to κ, which leads to a second–order performance gap. The second is the learning rate of the regularised maximum-likelihood estimator which is obtained from concentration inequalities for Bernoulli signals. Numerical experiments confirm the convergence and the robustness of the proposed algorithm.
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.
This paper studies a loss-averse version of the multiplicative habit formation preference and the corresponding optimal investment and consumption strategies over an infinite horizon. The agent's consumption preference is depicted by a general S-shaped utility function of her consumption-to-habit ratio. By considering the concave envelope of the S-shaped utility and the associated dual value function, we provide a thorough analysis of the HJB equation for the concavified problem via studying a related nonlinear free boundary problem. Based on established properties of the solution to this free boundary problem, we obtain the optimal consumption and investment policies in feedback form. Some new and technical verification arguments are developed to cope with generality of the utility function. The equivalence between the original problem and the concavified problem readily follows from the structure of the feedback controls. We also discuss some quantitative properties of the optimal policies, complemented by illustrative numerical examples and their financial implications.
