CyberSec Research

We propose a framework for transfer learning of discount curves across different fixed-income product classes. Motivated by challenges in estimating discount curves from sparse or noisy data, we extend kernel ridge regression (KR) to a vector-valued setting, formulating a convex optimization problem in a vector-valued reproducing kernel Hilbert space (RKHS). Each component of the solution corresponds to the discount curve implied by a specific product class. We introduce an additional regularization term motivated by economic principles, promoting smoothness of spread curves between product classes, and show that it leads to a valid separable kernel structure. A main theoretical contribution is a decomposition of the vector-valued RKHS norm induced by separable kernels. We further provide a Gaussian process interpretation of vector-valued KR, enabling quantification of estimation uncertainty. Illustrative examples demonstrate that transfer learning significantly improves extrapolation performance and tightens confidence intervals compared to single-curve estimation.

In this paper, we study the continuous-time multi-asset mean-variance (MV) portfolio selection using a reinforcement learning (RL) algorithm, specifically the soft actor-critic (SAC) algorithm, in the time-varying financial market. A family of Gaussian portfolio selections is derived, and a policy iteration process is crafted to learn the optimal exploratory portfolio selection. We prove the convergence of the policy iteration process theoretically, based on which the SAC algorithm is developed. To improve the algorithm's stability and the learning accuracy in the multi-asset scenario, we divide the model parameters that influence the optimal portfolio selection into three parts, and learn each part progressively. Numerical studies in the simulated and real financial markets confirm the superior performance of the proposed SAC algorithm under various criteria.

We propose the resilience rate as a measure of financial resilience. It captures the rate at which a dynamic risk evaluation recovers, i.e., bounces back, after the risk-acceptance set is breached. We develop the associated stochastic calculus by establishing representation theorems of a suitable time-derivative of solutions to backward stochastic differential equations (BSDEs) with jumps, evaluated at stopping times. These results reveal that our resilience rate can be represented as an expectation of the generator of the BSDE. We also introduce resilience-acceptance sets and study their properties in relation to both the resilience rate and the dynamic risk measure. We illustrate our results in several examples.

We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary stochastic maximum principle tailored to Epstein-Zin utility and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for the equilibrium investment and consumption policies.

Option pricing often requires solving partial differential equations (PDEs). Although deep learning-based PDE solvers have recently emerged as quick solutions to this problem, their empirical and quantitative accuracy remain not well understood, hindering their real-world applicability. In this research, our aim is to offer actionable insights into the utility of deep PDE solvers for practical option pricing implementation. Through comparative experiments in both the Black--Scholes and the Heston model, we assess the empirical performance of two neural network algorithms to solve PDEs: the Deep Galerkin Method and the Time Deep Gradient Flow method (TDGF). We determine their empirical convergence rates and training time as functions of (i) the number of sampling stages, (ii) the number of samples, (iii) the number of layers, and (iv) the number of nodes per layer. For the TDGF, we also consider the order of the discretization scheme and the number of time steps.

Although modern blockchains almost universally produce blocks at fixed intervals, existing models still lack an analytical formula for the loss-versus-rebalancing (LVR) incurred by Automated Market Makers (AMMs) liquidity providers in this setting. Leveraging tools from random walk theory, we derive the following closed-form approximation for the per block per unit of liquidity expected LVR under constant block time: \[ \overline{\mathrm{ARB}}= \frac{\,\sigma_b^{2}} {\,2+\sqrt{2\pi}\,\gamma/(|\zeta(1/2)|\,\sigma_b)\,}+O\!\bigl(e^{-\mathrm{const}\tfrac{\gamma}{\sigma_b}}\bigr)\;\approx\; \frac{\sigma_b^{2}}{\,2 + 1.7164\,\gamma/\sigma_b}, \] where $\sigma_b$ is the intra-block asset volatility, $\gamma$ the AMM spread and $\zeta$ the Riemann Zeta function. Our large Monte Carlo simulations show that this formula is in fact quasi-exact across practical parameter ranges. Extending our analysis to arbitrary block-time distributions as well, we demonstrate both that--under every admissible inter-block law--the probability that a block carries an arbitrage trade converges to a universal limit, and that only constant block spacing attains the asymptotically minimal LVR. This shows that constant block intervals provide the best possible protection against arbitrage for liquidity providers.

We propose a reinforcement learning (RL) framework under a broad class of risk objectives, characterized by convex scoring functions. This class covers many common risk measures, such as variance, Expected Shortfall, entropic Value-at-Risk, and mean-risk utility. To resolve the time-inconsistency issue, we consider an augmented state space and an auxiliary variable and recast the problem as a two-state optimization problem. We propose a customized Actor-Critic algorithm and establish some theoretical approximation guarantees. A key theoretical contribution is that our results do not require the Markov decision process to be continuous. Additionally, we propose an auxiliary variable sampling method inspired by the alternating minimization algorithm, which is convergent under certain conditions. We validate our approach in simulation experiments with a financial application in statistical arbitrage trading, demonstrating the effectiveness of the algorithm.

The paper addresses a singular control problem, where controls are processes of finite variation, with a focus on portfolio investment under transaction costs. We establish the existence of an optimal strategy in the class of randomized strategies for a range of goal functionals, including cumulative prospect theory preferences, subject to the nonnegativity constraint on the portfolio wealth. Unlike the existing approaches, we employ a metrizable Meyer-Zheng topology. Additionally, we investigate the applicability of the Skorokhod convergence for processes, adapted to a specific filtration. The presented framework provides a clear distinction between constraints imposed on the market model and on the goal functional.

We start with the idea that open quantum systems can be used to represent financial markets by modelling events from the external environment and their impact on the market price. We show how to characterize distinct orbits of the time evolution, and look at the development of the reduced density matrix, that represents the state of the market, over long time frames. In particular we distinguish between classical and non-classical modes of time evolution. We show that whilst both tend to the same set of maximum entropy states, this occurs faster in classical systems, with a knock on effect on the resulting probability distributions. We demonstrate how non-classical modes of time-evolution can be used to incorporate factors such as illiquid trades and imperfect trading mechanisms, and distinguish between different mechanisms of non-classical time evolution.

We introduce a novel simulation scheme, iVi (integrated Volterra implicit), for integrated Volterra square-root processes and Volterra Heston models based on the Inverse Gaussian distribution. The scheme is designed to handle $L^1$ kernels with singularities by relying solely on integrated kernel quantities, and it preserves the non-decreasing property of the integrated process. We establish weak convergence of the iVi scheme by reformulating it as a stochastic Volterra equation with a measure kernel and proving a stability result for this class of equations. Numerical results demonstrate that convergence is achieved with very few time steps. Remarkably, for the rough fractional kernel, unlike existing schemes, convergence seems to improve as the Hurst index $H$ decreases and approaches $-1/2$.

We prove the existence of a Radner equilibrium in a model with population growth and analyze the effects on asset prices. A finite population of agents grows indefinitely at a Poisson rate, while receiving unspanned income and choosing between consumption and investing into an annuity with infinitely-lived exponential preferences. After establishing the existence of an equilibrium for a truncated number of agents, we prove that an equilibrium exists for the model with unlimited population growth. Our numerics show that increasing the birth rate reduces oscillations in the equilibrium annuity price, and when younger agents prioritize the present more than older agents, the equilibrium annuity price rises compared to a uniform demographic.

Concentrated liquidity automated market makers (AMMs), such as Uniswap v3, enable liquidity providers (LPs) to earn liquidity rewards by depositing tokens into liquidity pools. However, LPs often face significant financial losses driven by poorly selected liquidity provision intervals and high costs associated with frequent liquidity reallocation. To support LPs in achieving more profitable liquidity concentration, we developed a tractable stochastic optimization problem that can be used to compute optimal liquidity provision intervals for profitable liquidity provision. The developed problem accounts for the relationships between liquidity rewards, divergence loss, and reallocation costs. By formalizing optimal liquidity provision as a tractable stochastic optimization problem, we support a better understanding of the relationship between liquidity rewards, divergence loss, and reallocation costs. Moreover, the stochastic optimization problem offers a foundation for more profitable liquidity concentration.

We present closed analytical approximations for the pricing of Asian basket spread options under the Black-Scholes model. The formulae are obtained by using a stochastic Taylor expansion around a log-normal proxy model and are found to be highly accurate for Asian and spread options in practice. Unlike other approaches, they do not require any numerical integration or root solving.

Based on the Lie symmetry method, we investigate a Feynman-Kac formula for the classical geometric mean reversion process, which effectively describing the dynamics of short-term interest rates. The Lie algebra of infinitesimal symmetries and the corresponding one-parameter symmetry groups of the equation are obtained. An optimal system of invariant solutions are constructed by a derived optimal system of one-dimensional subalgebras. Because of taking into account a supply response to price rises, this equation provides for a more realistic assumption than the geometric Brownian motion in many investment scenarios.

We adapt Leland's dynamic capital structure model to the context of an insurance company selling participating life insurance contracts explaining the existence of life insurance contracts which provide both a guaranteed payment and surplus participation to the policyholders. Our derivation of the optimal participation rate reveals its pronounced sensitivity to the contract duration and the associated tax rate. Moreover, the asset substitution effect, which describes the tendency of equity holders to increase the riskiness of a company's investment decisions, decreases when adding surplus participation.

We provide a simple and straightforward approach to a continuous-time version of Cover's universal portfolio strategies within the model-free context of F\"ollmer's pathwise It\^o calculus. We establish the existence of the universal portfolio strategy and prove that its portfolio value process is the average of all values of constant rebalanced strategies. This result relies on a systematic comparison between two alternative descriptions of self-financing trading strategies within pathwise It\^o calculus. We moreover provide a comparison result for the performance and the realized volatility and variance of constant rebalanced portfolio strategies

Climate change is a major threat to the future of humanity, and its impacts are being intensified by excess man-made greenhouse gas emissions. One method governments can employ to control these emissions is to provide firms with emission limits and penalize any excess emissions above the limit. Excess emissions may also be offset by firms who choose to invest in carbon reducing and capturing projects. These projects generate offset credits which can be submitted to a regulating agency to offset a firm's excess emissions, or they can be traded with other firms. In this work, we characterize the finite-agent Nash equilibrium for offset credit markets. As computing Nash equilibria is an NP-hard problem, we utilize the modern reinforcement learning technique Nash-DQN to efficiently estimate the market's Nash equilibria. We demonstrate not only the validity of employing reinforcement learning methods applied to climate themed financial markets, but also the significant financial savings emitting firms may achieve when abiding by the Nash equilibria through numerical experiments.

In [8], easily computable scale-invariant estimator $\widehat{\mathscr{R}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its antiderivative. This paper extends this convergence result by proving that $\widehat{\mathscr{R}}^s_n$ also consistently estimates the Hurst parameter when applied to the antiderivative of $g \circ X$ for a general nonlinear function $g$. We also establish an almost sure rate of convergence in this general setting. Our result applies, in particular, to the estimation of the Hurst parameter of a wide class of rough stochastic volatility models from discrete observations of the integrated variance, including the fractional stochastic volatility model.