Papers

Unlike the classical Heston model, Fourier pricing under the rough Heston model requires solving a fractional Riccati equation at every quadrature point. Since the required resolution varies with model parameters and quadrature point, a single uniform time discretization can be inefficient. We develop single- and multilevel Gauss-Laguerre quadrature methods that balance the time discretization and Fourier quadrature errors. Both methods scale the laguerre weight to the estimated Fourier integrand decay. The single-level method allocates a prescribed tolerance between the two errors. The multilevel method splits the integrand into a level-zero term and level differences, selecting quadrature points separately at each level. Suppose that the Fourier integrand discretization error is $O(\Delta t^p)$, that evaluating the characteristic function once costs $O(\Delta t^{-\beta})$, and that the algebraic Gauss-Laguerre quadrature error is $O(N^{-s_{SL}/2})$, where $s_{{SL}}$ is the smoothness index. Under this estimate and assumptions on the regularity and decay of level differences, we prove that the proposed single-level method requires $O(\epsilon^{-(\beta/p+2/s_{{SL}})})$ computational work to achieve accuracy $\epsilon$, whereas the proposed multilevel method requires $O(\epsilon^{-\beta/p})$ computational work. We also study root-exponential Gauss-Laguerre error models for practical multilevel quadrature allocation. Numerical experiments support the observed fractional Riccati and Fourier integrand convergence rates and root-exponential quadrature behavior, and show substantial reductions in quadrature cost from the proposed scaling. The multilevel method provides clear computational savings over the single-level method. We further benchmark the multilevel fractional Riccati method against the BL2 Markovian approximation and report lower total CPU time in the tested configurations.

Generative models for implied volatility surfaces must produce outputs that satisfy static no-arbitrage constraints. We study these constraints in latent space. For a fixed generator, we assign each latent code a scalar margin determined by the no-arbitrage conditions of the generated surface. The codes with nonnegative margin form the admissible latent set. We establish conditions under which strictly admissible codes remain admissible under small perturbations and the boundary of the admissible set is characterized by zero margin. For regular boundary components, we formulate a level-set equation whose local dynamics are directed toward the zero-margin set. The analysis treats the generator as a map from latent variables to surfaces and is therefore not restricted to a particular architecture. It applies to variational autoencoders, generative adversarial networks, and other generative models with a deterministic realization map. Numerical tests recover known boundaries in analytic examples. Experiments with a variational autoencoder trained on Heston surfaces show that similar reconstruction errors can correspond to different admissible regions and that the latent prior may be concentrated inside such a region. The computed boundary can also be used to modify latent codes that generate violating surfaces.

Quantitative trading is moving from isolated predictive models toward agentic workflows that combine reasoning, tool use, memory, and feedback. This survey reviews agentic quantitative trading across five stages: factor mining, signal discovery, portfolio construction, order execution, and risk management. We further examine agentic quant trading systems through architecture, coordination, and adaptation, while comparing benchmarks across strategy construction, offline trading, live market evaluation, and reliability assessment. Our review finds that current systems remain concentrated on signal discovery, while complete integration with portfolio construction, execution, and risk control is still uncommon. Multi-agent systems also rely heavily on aggregation despite increasingly diverse workflow structures. Benchmark evidence further shows that strong model or forecasting capability does not reliably translate into trading performance under live market conditions and reliability controls. We conclude with future directions for more complete trading workflows, stronger coordination, and evaluation matched to the capability being assessed.

We propose a neural calibration method to construct a recombining binomial tree directly from a set of given option prices. Rather than estimating a continuous option pricing function or a local volatility surface as an intermediate object, a neural network is used to deform a benchmark lattice. This leads to a discrete pricing model which is guaranteed to be arbitrage-free, complete, easy to interpret, and can be used directly for pricing and to find replicating trading strategies. Calibration is formulated as a penalized optimization problem that combines a repricing error with an admissibility penalty, and an optional spatial regularization term based on implied local volatilities. Numerical experiments on synthetic and SPX market data show that the proposed approach yields accurate repricing and is very competitive when compared to recently proposed other neural calibration methods. It preserves the computational advantages of lattice-based valuation and hedging. In particular, the calibrated tree can be reused to price contracts that allow early exercise, and could even be calibrated directly with American option prices.

We introduce ISCOS, a cross-entropy importance-sampling calibration method for rare credit-portfolio losses. We derive Gaussian and Gaussian–inverse-Gamma proposals and analyse the propagation of finite-COS approximation errors to the fitted parameters. Numerical experiments for Gaussian and Student t-copula credit portfolios show the efficiency of this method.

The factor HJM stochastic volatility model introduced by Sepp and Rakhmonov (2025) obtains tractable swaption pricing by freezing the nonlinear swap-rate loading along a deterministic expected-state path. This removes the dependence of conditional swap-rate variance on the current yield-curve state. We introduce a first-order Taylor correction to the loading that adds no calibration parameters. Conditional on retaining the frozen annuity-measure drift, we show that the first variation of the swap-rate transform is affine in the centered rate states and reduces to one-dimensional equations in volatility. For quadratic-drift lognormal stochastic volatility, these equations yield a finite-dimensional ODE representation and a direct log-volatility formulation. Calibrations to independently generated nonlinear-model prices show substantially lower stochastic volatility parameter bias and held-out pricing error, with only modest changes in local calibration identifiability.

Classical option-hedging methods like Black-Scholes delta assume constant, free rebalancing, which real markets don't allow. Deep hedging trains a neural network to handle these frictions directly, and prior work reports strong results. But those comparisons usually pit deep hedging against a frictionless classical baseline on simulated price data. That's not a fair fight, and it leaves open whether the advantage is real. We test this using five years of actual BTC options data from Deribit (2020-2024), comparing Black-Scholes delta, Leland's cost-adjusted hedge, and the Whalley-Wilmott no-trade band against three deep hedging setups: an LSTM and a feedforward network, each trained with a CVaR loss and, in some runs, a penalty for trading too often. All six strategies face the same 5 basis point transaction cost. On 11,546 test episodes from September 2023 to December 2024, Whalley-Wilmott cuts transaction costs significantly versus hourly rebalancing, saving $1.79 per episode against plain BS delta (95% CI [-2.21, -1.39], p < 0.0001) by trading about eight times less often. Its P&L and tail-risk numbers are better too, though not quite significant at this sample size. None of the three deep hedging models beat any classical benchmark on any metric, and all kept trading almost every hour regardless of penalty weight, a twenty-fold range barely moved the needle. A calmer validation period shows Whalley-Wilmott's P&L edge shrinks or disappears there while its cost edge holds, so the result depends on market regime. We think the likely causes are a small training set and the lack of any built-in mechanism for sitting still in the architectures tested. Not a flashy result, but an honest one: a real-data check on a claim mostly supported by simulations so far.

We develop a market-informed valuation framework for guaranteed minimum maturity benefit (GMMB) riders with rational surrender under the Heston stochastic-local volatility (SLV) model. The guarantee is written on the fee-deducted account value and is considered both in its terminal-only form and in the presence of early surrender rights. The Heston SLV specification combines stochastic volatility with a leverage function calibrated to a prescribed local-volatility surface. The leverage surface is obtained through a forward Markovian-projection equation so that, at the model level, the SLV dynamics are constrained to the same one-dimensional marginals as the corresponding local-volatility (LV) model. The latter is used only as a one-factor benchmark, allowing us to isolate the effect of stochastic volatility on continuation values and surrender decisions while preserving the same option-calibrated local-volatility target. We derive the associated backward pricing equations and propose a hybrid tree/finite-difference algorithm for the SLV model with a calibrated leverage function. Synthetic experiments and a market-informed case study show that SLV and LV valuations are numerically close for terminal-only guarantees, as expected from the common marginal target, whereas materially larger differences can arise once surrender is allowed. These differences are reflected in guarantee values, fair insurance fees and volatility-dependent surrender regions. The results indicate that matching one-date marginals implied by vanilla-option prices does not eliminate model risk for insurance liabilities whose value depends on conditional continuation dynamics and endogenous surrender decisions.

Bitcoin inverse options, traded on the Deribit exchange and settled in the underlying cryptocurrency rather than in fiat currency, combine extreme and genuinely rough volatility dynamics with a non-linear, currency-dependent payoff structure. This paper develops and empirically validates a pricing and calibration framework for these instruments based on the rough Bergomi (rBergomi) model of Bayer, Friz and Gatheral (2016). We adapt the rBergomi dynamics to the inverse payoff max(S_T - K, 0)/S_T, and implement and compare three computational pipelines that differ in the simulation scheme for the driving fractional Brownian motion (coarse-grid Cholesky vs. the Hybrid Scheme of Bennedsen et al., 2017) and in the Monte Carlo pricing estimator (plain log-Euler vs. the Mixed Estimator of McCrickerd and Pakkanen, 2018). The model is calibrated to thirty implied volatility surfaces extracted from Deribit trade data between May 2022 and March 2025, spanning seven major market-stress events and nine baseline regimes stratified by volatility level. The Hybrid and Mixed pipeline is simultaneously the most accurate (mean unweighted RMSE 22.83 percentage points, versus 41.76 pp for the Cholesky and Euler benchmark) and the fastest (17 seconds per snapshot, a 20-fold speed-up). The calibrated Hurst exponent is consistently close to the lower bound of the search space (H approximately equal to 0.01–0.06 in most regimes), confirming that Bitcoin's volatility is genuinely rough, and calibration error scales approximately linearly with the level of at-the-money implied volatility (Pearson r = 0.89).

This paper develops a unified framework for assessing systemic risk and identifying contagion channels in the global banking system using a Temporal Heterogeneous Multiplex Graph Neural Network. We construct a harmonised quarterly panel combining bank fundamentals, CDS spreads, and macroeconomic indicators, and represent these data as dynamic multiplex networks linking banks through financial similarity and liquidity co-movement, augmented with country-level macroeconomic relationships. The model integrates graph convolutional layers with recurrent GRU dynamics and incorporates a learnable fusion gate to capture time-varying reliance on alternative contagion channels. Empirical results show that the framework outperforms conventional econometric, machine learning, and graph-based benchmarks for short-term changes in CDS spreads. Beyond forecasting, we provide an interpretable framework to quantify bank-level systemic importance via stress testing, assess country-level spillovers under macroeconomic shocks, and uncover transmission pathways through edge perturbation analysis. Robustness tests confirm the stability of both predictive accuracy and systemic risk rankings.

Algorithmic trading now represents a market exceeding $20 billion, where even marginal gains in signal robustness can translate into economically significant returns. Existing evaluations of equity prediction models do not explicitly target regime robustness during hyperparameter selection. Five model classes are trained on daily observations from approximately 300 large-cap US equities over eleven years, with Bayesian optimisation configured to target trading performance across three statistically different market regimes. Regime-robust hyperparameter selection is associated with out-of-sample generalisation, as signal precision remains above the random baseline across all four quarters of the test period, and portfolio performance slowly degrades under simulated input noise before collapsing beyond a defined threshold. No individual tabular deep learning architecture outperforms gradient-boosted trees, but combining XGBoost and TabNet using rank aggregation produces a Hybrid ensemble with an annualised return of 51.26%, a Sharpe ratio of 2.44, and a statistically significant CAPM alpha of 0.423 (p = 0.011). A near-zero beta indicates this outperformance is driven by stock selection, not market exposure. Alternative data plays a secondary role once technical and fundamental features are accounted for, as well as contributing more strongly on the short side than the long, and varies by model class. An interactive application makes these results explorable in real time, with live data integration the remaining step toward practical deployment.

The computation of Black-Scholes implied volatility is a fundamental task in quantitative finance, underpinning option valuation, model calibration and risk management. Although implied volatility is routinely used in practice, the inversion of the Black-Scholes pricing formula remains a challenging numerical problem, particularly in asymptotic regimes corresponding to extreme option prices, strikes or maturities, where the inverse map becomes highly sensitive to perturbations of the price. In this paper, we introduce a new family of asymptotically-informed neural-network architectures for implied-volatility computation. Exploiting the distinct behaviours of the Black-Scholes pricing function in different volatility regimes, we propose a family of architectures that learn a trainable partition of the price-log-moneyness domain through a system of gating functions and combines specialised local approximations of the implied-volatility function within each region. Extensive numerical experiments demonstrate that the proposed models consistently outperform standard feed-forward neural networks across a wide range of parameter domains, often by several orders of magnitude in relative accuracy while maintaining excellent generalisation properties. Furthermore, the neural-network outputs provide highly accurate initial guesses for a third-order Householder scheme, allowing near machine-precision implied-volatility computations after only two refinement iterations.

This paper develops the first end-to-end application of cross-sectional learning-to-rank to the S&P 500 weekly options (SPXW) zero-day-to-expiration surface, integrated with margin-aware position sizing, an abstention rule driven by model uncertainty, and a strict out-of-time integrity check. A LightGBM LambdaRank ranker scores a daily nine-strategy cross-section composed of eight delta-targeted short-put positions and a SKIP candidate, trained against a path-aware Sortino-on-bars label computed at one-minute resolution. The framework is evaluated under index-option margin requirements, a tiered fee schedule, and bid-to-mid execution assumptions across a four-window walk-forward over 2021-2024 and a strictly held-out 2025 out-of-time slice. Seven sizing methods produce out-of-time annualized Sharpe ratios between 4.31 and 5.76, with the headline method reaching a Probabilistic Sharpe Ratio of 0.964 and a sample-period maximum drawdown of -2.28%, on a single hold-out year against a walk-forward range of 1.90 to 3.11. Out of time, every method exceeds three passive benchmarks (CBOE PUT, CBOE WPUT, SPX buy-and-hold) by at least 3.84 in Sharpe ratio and five internal selection baselines by at least 3.69. A two-by-two ablation of the confidence gate against the tail-risk features places 5.05 of the 5.59 out-of-time Sharpe gap over the CBOE PUT with the ranker and the selection layer, the two risk controls adding 0.54 between them. On walk-forward, where the gate binds, neither control comes close to the headline alone and their interaction supplies most of the result. A fifteen-group feature ablation shows that removing the multiplicative regime interactions collapses walk-forward statistical confidence.

Physical crash probabilities recovered from option prices depend on a pricing kernel and on a risk-neutral distribution that finitely many bid and ask quotes do not identify. For a power utility investor, we characterize the pairs of physical crash probability and expected loss below the crash threshold that the quotes admit; the boundary of this set is the physical crash frontier. Both coordinates are ratios of moments, yet when the index is bounded above the closed set is convex, and under an interior regularity condition second-order cone programs compute it exactly at the benchmark risk aversion of two. In a decade of weekly S&P 500 cross sections, the quotes beyond the two puts nearest a 10 percent decline shrink the range of admissible crash probabilities by about 80 percent, yet its upper end remains two to three times its lower end. Under strict quote slack and the interior-mass condition stated below, removing the cap without further tail control drives the lower bounds on crash probability and unconditional shortfall to zero for risk aversion above one. A vanishing probability far in the right tail inflates the normalizing moment while every quote remains inside its spread. In this setting, a positive floor requires additional tail information, supplied here by the support cap.

A companion paper \cite{ItkinDF2026} introduced the Diagonal Frog (DF) positivity-preserving schemes for anisotropic Fokker–Planck equations, advancing each directional substep by a Krylov-computed matrix exponential, which dominates the cost. Replacing that exponential by a rational map $r(\gamma L)$ reduces the substep to a banded solve, but the positivity argument no longer applies. We prove that for eventually exponentially positive generators the entrywise sign of $r(\gamma L)$ at large steps is decided by a single number, the value $r(\infty)$ taken on infinitely stiff modes. Nonnegativity holds above a computable threshold when $0\le r(\infty)<1$, and at most on a bounded interval, empty or vanishingly narrow in all our tests, when $r(\infty)<0$. The criterion rejects the Crank–Nicolson (trapezoidal) method, where $r(\infty)=-1$, and selects the subdiagonal Padé$(0,2)$ method, which is second order, L-stable and provably positive above an explicit threshold. The resulting DF-ADI scheme costs $O(N)$ per step, keeps the implicit factorized mixed derivative unchanged, is second order in space and time, and conserves discrete mass exactly. In the strong cross-diffusion regime, however, the directional factors demand a step larger than the mixed derivative permits, so the composite second-order scheme is only empirically positive there, and the criterion serves to discriminate the well-behaved multiplicative factors from the stabilizing-correction schemes rather than to guarantee positivity. Against the Krylov exponential it runs ten to thirty-two times faster at matched accuracy in our tests with the gain growing with the mesh. We extend the construction to the backward Kolmogorov equation and to jump-diffusion models.

Deep hedging is a data-driven approach to learn hedging strategies. It relies on synthetic price paths generator, as real market data is often limited for training. Existing approaches primarily evaluate such generators based on realism, i.e., how well they capture statistical properties of real markets, but the relationship between realism and hedging performance remains unclear. In this work, we introduce a decision-centric perspective on synthetic data for deep hedging based on the notion of compatibility. Compatibility measures the extent to which strategies trained on synthetic scenarios remain effective in the true market. We theoretically show that 1) hedging performance decomposes into learning error and a compatibility gap, and 2) realism and compatibility can diverge. Empirically, we find that hedging performance is governed not by realism alone, but by the alignment between the generator and the hedger, together with task structure. Taken together, this work provides a principled basis for designing synthetic data in finance aligned with decision tasks.

The Marketron model of \cite{HalperinItkin2025Mark} and its option pricing extension in \cite{HalperinItkinMarketron2} suffer from structural non-identifiability: an eighteen-parameter space traps solvers in suboptimal local minima and renders economic quantities unmeasurable. By removing exact scaling gauges and sign symmetries, freezing non-financial parameters by explicit criteria, and adiabatically eliminating the fast hidden signal, we derive a robust nine-parameter reduced model. A Gauss-Newton Hessian with empty null space and a manifold-boundary analysis confirm that the reduced core carries no exact symmetry and admits no further reduction. A diffusive correlation between flow and return innovations captures the short-maturity skew. A staged calibration from the physical measure to the risk-neutral measure, illustrated on SPX options, fits the whole surface with a single parameter set. The same reduction turns the wedge between the physical and pricing values of the flow block into a well-defined market price of flow risk rather than a ridge artifact, identifiable here for the first time, though a single surface constrains its level only weakly. Finally, our analysis reveals that in the Marketron model the log-price obeys a generalized Langevin equation with a closed-form, state-modulated memory kernel, and that the memory variable itself is the exact Markovian lift of this kernel. This mapping also yields a testable condition, the equality of the signal and memory relaxation rates, which on the SPX surface come out well separated, though both weakly identified, placing the fitted market tentatively in the driven, non-equilibrium regime and turning the active-matter reading from an analogy into a falsifiable constraint.

We introduce Deep-MKV-TS, a path-dependent McKean-Vlasov framework for financial scenario generation. The stochastic dynamics are chosen by matching selected path and volatility features of generated scenarios to those observed in the data. Starting from an interpretable reference model, Deep-MKV-TS preserves the reference drift and adjusts its volatility, while a regularization penalty limits unnecessary departures from the calibrated dynamics. We solve the resulting control problem using a neural, sample-based implementation of the stochastic maximum principle. We validate the method against an exactly computable oracle. On Heston and Heston-mixture models, Deep-MKV-TS substantially reduces path-dependent and volatility-related deficiencies of the reference model. In delayed-volatility experiments, the correction remains effective as the forecasting horizon increases, while direct training becomes less reliable. On held-out intraday equity-index futures, the corrected model improves conditional forecasts relative to the reference and reaches a level of performance comparable to flexible generative and historical baselines. The resulting scenarios also support greater exposure than the reference under a fixed drawdown-risk target. These results show that path-dependent McKean-Vlasov control can enrich an interpretable reference model without replacing it.

Automated market makers (AMMs) are a cornerstone of decentralised finance (DeFi). Constant product markets with concentrated liquidity, such as UniswapV3, are now a well-established design. In these markets, liquidity providers (LPs) face a sequential decision problem: they must decide when to rebalance their positions and which price ranges to allocate capital to as market conditions evolve. We formulate dynamic liquidity provision as a stochastic impulse control problem and use reinforcement learning (RL) to solve it, focusing on providing interpretable solutions. We show that learned policies exhibit rich state-dependent behaviour, allocating liquidity according to mispricing, rebalancing costs, uncertainty, inventory exposure, and heterogeneous risk preferences. These behaviours help compress the left tail of the Profit and Loss (PnL) distribution and avoid catastrophic outcomes under high uncertainty. Finally, we benchmark the RL agents against baseline and sophisticated agents from the AMM microstructure literature and analyse their performance.

Optimal execution (OE) in multi-asset settings involves complex interactions across assets, particularly through shared capital constraints during portfolio rebalancing. While existing models capture cross-impact and portfolio-level dynamics, they largely overlook the role of explicit cash constraints along the execution trajectory. As a result, the feasibility of execution strategies under limited capital remains poorly understood. In this paper, we extend the classical Almgren-Chriss framework to incorporate intertemporal constraints on expected cash consumption, requiring that the expected cumulative cash spent does not exceed a prescribed budget at every trading period. We show that the resulting multi-asset OE problem can be equivalently formulated as a quadratically constrained quadratic program (QCQP), and further establish that it admits a convex representation under mild conditions. This provides a tractable framework for analyzing execution strategies under dynamic capital constraints. Through controlled synthetic experiments, we show that the proposed cash constraints qualitatively alter OE schedules toward cash-feasible sell-first executions as the constraints become tighter. Furthermore, evaluations in an out-of-sample agent-based market simulator demonstrate that our method substantially reduces peak cash drawdown while maintaining implementation shortfall comparable to existing execution strategies. Our results highlight the importance of explicitly modeling financial feasibility in multi-asset execution and provide a foundation for bridging theoretical OE models with practical capital constraints.

We develop simulation-based policy iteration for continuous-time portfolio choice with predictable returns and convex constraints. Each outer step re-evaluates a fixed-latent open-loop backpropagation-through-time (OL-BPTT) adjoint after deployment and solves the constrained update. Shifted-adjoint cancellation controls the adjoint–HJB Hamiltonian-gradient discrepancy by the policy-improvement residual. For CRRA portfolios, exact HJB policy iteration identifies the optimal reduced value factor, while population OL-BPTT iteration converges globally when the adjoint update is directionally improving and approximate stationarity is asymptotically HJB-compatible. A theorem-matched occupation audit yields maximal $95\%$ upper endpoints of $0.066$ for the primitive directional ratio and $0.074$ for a stronger norm-relative ratio, both against the half-step threshold $0.75$. In the high-precision $50$–$50$ occupancy/broad-anchor design of a three-factor, fifty-asset benchmark, current-policy re-evaluation outperforms matched pooled refinement under the on-policy and broad evaluation laws.

This paper considers European multi-asset option pricing under Lévy and affine characteristic-function models. The main obstruction is the curse of dimensionality: direct multidimensional COS pricing forms tensor-product coefficient arrays whose size grows exponentially with the number of assets. We study and extend COS-TT-CHF, a low-rank construction that uses TT-cross to compress sampled characteristic-function tensors into tensor-train COS coefficients for arithmetic basket and min/max option pricing. Once built, the compressed representation gives fast post-setup strike-grid and selected component Delta/Vega calculations. The numerical study compares with adaptive-quadrature Fourier benchmarks, direct COS, a tensor-Fourier min-option benchmark, and quasi-Monte Carlo (QMC) references based on randomized Sobol points. The reported timings show a low-dimensional crossover against direct COS as the benchmark moves from $d=2$ to $d=4$, favorable timings against the tensor-Fourier min-option benchmark from $d=3$ onward, and favorable timings against the QMC common-Heston reference already at $d=2$. The reported tests reach $d=30$ for GBM and $d=20$ for VG, NIG, and common-Heston benchmark families, with accuracy, rank, runtime, control-sensitivity, and component Delta/Vega diagnostics reported throughout.

Financial returns are heavy-tailed, and accurate tail risk estimation is central to portfolio risk management. Modern neural generators sample by pushing a simple base distribution through a learned map, and for training stability that map is built from Lipschitz components. This is the binding constraint: a Lipschitz map of a Gaussian is sub-Gaussian, so heavier-tailed targets admit no exact match at any finite Lipschitz constant. The Monge–Ampère equation ties the Brenier map's local distortion to the density ratio $f/(g\circ T)$, so a deeper trough in the target density requires a higher-gain map and yields a higher-variance estimator. The argument needs only bounded distortion, so it covers normalizing flows, flow matching, GANs, and diffusion samplers alike. Semi-Discrete Optimal Transport (SDOT) relaxes the map's regularity rather than the source's tail class. Its power diagram gives every training observation a cell holding exactly $1/N$ of the source measure, and tail observations are reached by crossing a cell boundary rather than by stretching. Our primary experiment sweeps severity over a calibrated Merton jump-diffusion spanning kurtosis 94 to 1,679. SDOT holds tail ratios at $0.85$–$0.94$ with cross-seed standard deviations below $0.025$, while every learned generator either compresses the tails or inflates them with a variance that grows alongside. Further experiments carry the result to real S&P~500 returns and to a 21-year backtest, where SDOT gives the best risk-adjusted market-neutral strategy under CVaR optimization (Sharpe $0.70$, max drawdown $-2.60\%$, against $0.40$ for the next-best generator).

Reinforcement learning has gained increasing attention as a data-driven approach for stock trading. However, learning a policy that is both profitable and stable remains challenging due to non-stationary market behaviour and noisy reward signals. Auxiliary tasks are often used to improve representation learning and stabilize training, yet they are usually designed manually and depend heavily on prior assumptions about targets and prediction horizons. Such fixed designs may not remain suitable across changing market regimes. In this work, we propose a self-supervised framework that automatically discovers auxiliary tasks to support reinforcement learning for stock trading. The auxiliary tasks are formulated as General Value Functions so that their predictions enrich the learned state representation and assist policy optimization. The framework consists of two networks. The main network learns the trading policy along with the auxiliary predictions, while the secondary network generates the definitions of auxiliary tasks through learned cumulants and discount factors. These tasks are updated using a meta gradient mechanism that accounts for their long-term impact on trading performance and improves training stability. We evaluate the proposed approach across four major equity indices: DJI, FTSE, Sensex, and TAIEX. The empirical results demonstrate that automatically discovered auxiliary tasks lead to more robust learning and improved trading performance compared to existing baselines.

Indonesia has established a regulated carbon market supported by national registry infrastructure and the IDXCarbon exchange. Carbon units can be issued, recorded, traded, and retired within this framework. IDXCarbon currently uses a private blockchain for its trading infrastructure. This creates an opportunity to examine how Indonesian carbon credits could also be represented and traded through public blockchain infrastructure. This study proposes an architecture for tokenizing Indonesian carbon credits as real-world assets (RWAs), with particular focus on Sertifikat Pengurangan Emisi Gas Rumah Kaca (SPE-GRK). The proposed architecture retains the Sistem Registri Unit Karbon (SRUK) as the authoritative source of carbon-unit status. It introduces a public-blockchain layer for token representation and programmable transactions. The architecture is designed to support lifecycle management, token-based asset representation, public observability of token activity, interoperability, wallet-based transactions, and programmable settlement. The architecture consists of four layers: the authoritative carbon layer, the registry interoperability and tokenization layer, the public-blockchain RWA layer, and the market and application layer. Access to the tokenized carbon assets remains regulated. Token issuance and transfers are linked to participant eligibility and registry status. Retirement also remains dependent on the authoritative carbon registry. The proposed architecture provides a framework for introducing public-blockchain RWA infrastructure into Indonesia's existing carbon market while maintaining SRUK authority and existing market-integrity controls.

Using the local time-space calculus of Peskir (2005) and the method developed in Mijatovic (2010), we derive a new integral representation for the backward probability associated with the first-passage time (FPT) of a diffusion process through a time-dependent barrier. We present a complete three-step numerical algorithm: first, the problem is reduced to a Volterra-type integral equation; second, its kernel is approximated by a Markov chain; finally, the resulting equation is solved using a quadrature rule. The method is implemented for several representative examples, and its convergence properties are established. An extension to double barrier problems is carried out

Limit order book (LOB) simulators are most useful to practitioners when they combine realistic market dynamics, computationally efficient sampling, controllable scenario generation, and the ability to generalize beyond the instruments seen during training—properties that existing agent-based and deep generative simulators provide only partially. We present FlowLOB, a conditional flow-matching generator of LOB trajectories, trained on multiple Hong Kong Exchange (HKEX) symbols at three sampling frequencies ($0.1$s, $1$s, $10$s) in tick-relative representation that transfers to unseen instruments. Because flow and diffusion models admit a common formulation, we train both with identical data, architecture, and budget, and sample both through the same fixed-step ODE solvers, yielding a controlled comparison of sampling efficiency and fidelity. Flow matching attains its best quality with only $10$ ODE-solver steps, whereas diffusion needs many more function evaluations to approach the same fidelity. At this efficient operating point, FlowLOB improves realism over baselines, two learned and two agent-based models, in most distributional metrics at the two finer sampling frequencies. We evaluate counterfactual controllability with a distributional test that asks whether changing a scenario condition moves the generated statistic toward the corresponding real tail regime; FlowLOB satisfies this criterion in most tested settings. Both realism and control effects transfer zero-shot on a held-out symbol. We additionally conduct ablation studies on the network architecture and the learning rate.

Generative models of limit orderbook (LOB) data have advanced rapidly, but their evaluation often focuses on stylised facts and selected market statistics. These measures provide useful diagnostics but may not capture the joint temporal and cross-level structure of order-book trajectories. We introduce LOB-ID, an embedding-based framework that adapts the Fréchet Inception Distance (FID) and Monge Inception Distance (MIND) to LOB data. To obtain domain-specific embeddings, we train the DeepLOB architecture on four months of Level-2 order-book data for five equities. We show that LOB-ID is stable across time, instruments, and embedding checkpoints, and rises monotonically under controlled distortions. We then construct a moment-matching attack against FID and a deep-book perturbation that evades statistic-based evaluation. MIND remains substantially more sensitive to both distortions. Finally, we score five generative LOB models, spanning stochastic baselines and deep learning approaches, and find that LOB-ID ranks them in line with the joint temporal and cross-level structure each captures by construction.

Diffusion generative models have rapidly emerged as powerful tools for modeling complex financial data. Their appeal is both structural and practical: they offer stable likelihood-based training, strong mode coverage, flexible conditioning, and a stochastic-differential-equation formulation that aligns naturally with the Itô calculus and stochastic control frameworks widely used in finance. This survey reviews the growing literature on diffusion-family generative models for financial applications. We organize prior work primarily by financial data type, covering time series, limit order books, tabular data, and other structured financial objects, while discussing the modeling goals and application contexts that arise within each category. To the best of our knowledge, this is the first survey dedicated specifically to diffusion-family models for financial data. For more detailed information, we have open-sourced a repository https://github.com/ZhuoHan1998/Diffusion-Models-In-Finance.

The dynamics of an implied volatility surface to movements of the underlying is classically described by separate rules: sticky strike, sticky delta and Bergomi's skew-stickiness ratio (SSR). We unify these as a transport of the total-variance surface along a velocity field, $\partial_u w = v\,\partial_k w$, on the class of static-arbitrage-free surfaces. The SSR is the zeroth-order term of a Taylor (jet) hierarchy that also governs skew and curvature dynamics, making the velocity profile $v(k)$ estimable from a time series of surfaces rather than a single scalar. We give explicit conditions on v under which the flow preserves butterfly and calendar arbitrage-freeness over a finite spot move, with an admissibility radius in closed form; sticky delta, sticky strike and the SSR are the constant-velocity transports, while local and rough volatility are not special cases and are reproduced only in their short-maturity at-the-money velocity. The transport representation restricts nothing unless v is bounded across smile extrema, and we turn this into a test. Writing the response as a transport part $v\,\partial_k w$ plus a residual source s, at a smile extremum, where the skew vanishes—bounded transport predicts no response at its speed, so any response measured there is pure source. This gives a joint estimator of velocity and source that never divides by a near-zero skew, with an arbitrage theorem for the source term. On five years of SPX surfaces across seven tenors we reject self-similar transport at every tenor and a strike-constant velocity at intermediate tenors; at the two shortest tenors, where the smile minimum lies within the quoted strike range, the response is pure transport at the money but carries a genuine source at the minimum. Out of sample, the extended model improves curvature dynamics over the scalar SSR, the regime most relevant for positions carrying vanna and volga.