q-fin.PR

I propose a cap-axis integral diagnostic for factor-model evaluation. Low-dimensional factor models can improve the maximum-Sharpe frontier while leaving zero-alpha violations on economically fixed subspaces. The diagnostic studies one such subspace by lifting pricing errors into a bridge-alpha curve along the market-capitalization rank axis. Under an aggregate-market gate, a zero curve is equivalent to pricing the market's internal cap-rank subspace. In 1967-2024 CRSP data, q5's daily negative bridge attenuates under lead-lag correction, while Fama-French and Carhart bridges are more visible monthly. Across 154 factors, the cap-axis norm is distinct from Sharpe gain and size exposure.

This paper proposes a novel asset pricing framework that augments large language model (LLM) embeddings of annual report disclosures with supply chain knowledge graph (KG) propagation. Using FinBERT embeddings of 10-K MD&A sections for 255 S&P 500 firms over 2011-2025, two sets of return predictors are constructed: direct LLM embeddings and network-augmented embeddings, where firm-level signals propagate through inter-firm linkages. Fama-MacBeth cross-sectional regressions reveal that the network-augmented factor (net_pc_5) carries significant return predictability with a Newey-West t-statistic of -2.64, even after controlling for momentum, volatility, and firm size. A long-short portfolio sorted on net_pc_5 achieves an annualized Sharpe ratio of 0.86 and a Fama-French five-factor alpha of 7.27% per year (t = 2.30). The predictive power survives out-of-sample tests, placebo experiments, sector-neutralization, and subsample analysis. The findings suggest that inter-firm network structure contains pricing-relevant information beyond firm-level textual disclosures.

A flexible forward (FF) is a customized FX hedging instrument that guarantees a fixed exchange rate while letting the holder choose the delivery date within a pre-agreed window. It is therefore an American-style option on timing, and its valuation must respect the volatility skew of the underlying currency pair. We price FF contracts (and, more generally, American options) under a time-inhomogeneous Heston model which captures the forward-skew term structure while preserving analytical tractability through a recursive (matrix) Riccati solution for the joint characteristic function. Extending the integral-equation (decomposition) approach to time-dependent coefficients, we derive a Volterra equation characterizing the early-exercise surface. The expectation in the decomposition formula is evaluated by two complementary spectral methods: a double cosine (COS) expansion of the transition density, and a damped-Sinc (DSINC) local-basis scheme that is more accurate and stays robust when a low Feller ratio or large vol-of-vol induces Gibbs oscillations in the COS series. Benchmarked against a penalty-iteration MCS-ADI finite-difference solver, both methods price a contract in about 1-2 seconds, roughly an order of magnitude faster than the finest finite-difference grid, while DSINC improves median accuracy over COS by about a factor of twelve. The experiments also show that the early-exercise surface is a substantially nonlinear function of the variance, contrary to the linear-in-variance approximation common in earlier work.

In this paper, we present a numerical method for option pricing and the computation of Greeks under stochastic volatility Bachelier-type models, based on elementary linear algebra. The method allows option prices and Greeks to be computed for infinitely many strikes (within a range of convergence) by evaluating only a finite number of expectations, independent of the number of strikes. For the SABR model, we derive an explicit range of convergence. Numerical examples are provided for both the SABR and the rough Bergomi models.

Commodity futures can be represented hierarchically, with underlying assets at the upper level and individual futures contracts at the lower level. Entities at each level can be connected by edges reflecting inherent correlations, with cross-level edges capturing contract-to-underlying asset connections. Building on our observations of these structures, we propose a hierarchical graph learning approach for calendar spread (CS) strategies in commodity futures markets, addressing two significant gaps in the machine-learning literature: (i) the absence of learning-based methods for CS strategies in futures markets, and (ii) the lack of consideration of maturity-dependent interrelationships across commodity futures. We first establish the efficacy of CS strategies by analytically showing that CS strategies can possess higher risk-adjusted returns, measured by the information ratio, and lower risk, measured by variance and delta, than long-only strategies. We then introduce a method to convert learning-based predictions into CS positions. Next, we develop a hierarchical graph learning method that predicts futures price movements by utilizing the maturity-dependent interrelationships, thereby yielding a CS trading algorithm. Empirical results on commodity futures markets traded on the Chicago Mercantile Exchange Group demonstrate that our method outperforms benchmark models in both prediction and trading performance. We find that maturity-dependent interrelationships across commodity futures are instrumental in prediction and that CS trading based on hierarchical graph learning is effective for statistical arbitrage.

In this paper, we consider pricing a Bermudan swaption with a small number of exercise dates. We begin with the case of two exercise dates. In this limit, we show that the Bermudan price decomposes into the sum of short-dated European swaptions, setting an upper bound, minus a correction term. This correction is expressed as an integral involving a forward volatility agreement type payoff with start at the first exercise date, and it can be evaluated in closed form. The magnitude of the correction is smaller when variance is front loaded and larger when it is back-loaded. We extend to three-exercise Bermudans via backward induction under rolling forward measures. A key feature is boundary linearity enabling further analytic steps. The exercise boundary of options splits into a strike-dependent term and a variance term; together they determine optimal exercise. The linear term is negative, supressing the exponentials in subsequent steps and aiding analytic calculations. This boundary linearity extends to multiple exercise dates and yields pricing formulas with the same decomposition, showing how optionality accumulates across exercise dates. We conclude that the Bermudan can be reconstructed by adding, at each exercise date, the initial short swaption with an increasingly higher strike and subtracting the integrated payoffs of all forward-starting receiver swaptions starting at that date. The corresponding double and higher-order integrals decrease rapidly and, in the presence of only a few exercise dates, can be safely neglected without materially impacting the valuation. The general case is discussed at the end.

Intelligent transportation systems increasingly rely on decentralized mechanisms to allocate limited resources such as freight capacity, warehouse availability, charging infrastructure, and network bandwidth. Efficient allocation requires pricing mechanisms that adapt dynamically to demand while preserving system stability. This paper investigates weighted constant-function market makers as a decentralized resource allocation mechanism for intelligent transportation systems, adapting the weighted invariant from Balancer-type automated market makers to model a generalized formulation over multiple tokenized resources. The standard literature documents exactly four resource allocation operations: proportional contribution, proportional withdrawal, single-resource contribution, and single-resource withdrawal, each obtained via separate derivations. This paper presents a single closed-form formula that unifies all four cases and extends them to two previously undocumented operations: partial-proportional contributions and fully non-proportional operations. The unified formula reveals that the conservation invariant and the allocation formula are structurally identical; the invariant itself is the general allocation formula. We prove two swap-decomposition theorems showing that, in a fee-less environment, any non-proportional operation is equivalent to an internal rebalancing swap combined with a proportional operation. Both theorems generalize previous propositions from single-resource to arbitrary multi-resource operations. The proposed framework provides a mathematically grounded mechanism for decentralized market-based coordination in transportation networks.

Factor-model performance depends not only on the model but also on how test assets are constructed. We form characteristic-unsorted random portfolios from a broad CRSP universe and vary stock selection, initial weighting, holding, and rebalancing. Rankings shift materially: buy-and-hold favors FF5 and FF6, whereas daily constant-weighting favors FF3, the most stable model across designs. Although q5 attains the highest maximum Sharpe ratio in factor-spanning tests, it leaves comparatively large and construction-sensitive pricing errors on random portfolios. These results reflect construction-specific weighting of each model's pricing-error vector. Test-asset construction, including dynamic weight management, is therefore a design choice in model evaluation.

Short-dated index options make scheduled macro-announcement risk visible in market prices, but visibility does not imply identification: a flexible no-event surface fitted to event-spanning quotes can absorb event premia, while a jump calibrated without event-spanning quotes is unidentified. To separate the continuous surface from the scheduled jump, we model Federal Open Market Committee (FOMC) decisions, Consumer Price Index (CPI) releases, and nonfarm payroll (NFP) reports as deterministic-time jumps in risk-neutral option pricing and propose a non-spanning identification protocol. Non-spanning expiries identify the no-event volatility surface, event-spanning training quotes calibrate the scheduled jump, and held-out event-spanning quotes are used only for pricing evaluation. On PM-settled S\&P 500 index (SPX) options from May 2022 to August 2025, Gaussian and two-component mixture jumps improve held-out event-spanning pricing, with the clearest gains in robust median pricing errors and in event-volatility option combinations (straddles and strangles) rather than directional risk reversals. A contaminated-surface stress test confirms the identification concern: allowing event-spanning training quotes into the no-event surface fit produces strong held-out performance by absorbing event premia rather than identifying scheduled jump risk. An amortized mixture density network (MDN) benchmark shows limited cross-event transfer: pure leave-one-event-out amortization reduces implied-volatility errors but not mean dollar or mean spread-normalized pricing errors, while the scale-calibrated variant restores Gaussian-level performance yet remains below event-specific mixture calibration. Scheduled-jump identification is strongest for CPI and FOMC and weaker for NFP.

We consider the role of a continuum operational time u and its mapping to calendar time t and how these relate to event time for option pricing problems. We derive option-pricing equations from an operational-time Markov lattice rather than from a calendar-time diffusion. The primitive model is a nearest-neighbour log-price lattice with state- and time-dependent transition probabilities. Its Chapman-Kolmogorov decomposition yields discrete forward and backward equations, which converge under local finite-variance scaling to the usual continuum adjoint pair. In price variables, the backward equation gives a generalized European pricing PDE and reduces to Black-Scholes-Merton under the risk-neutral drift restriction and constant volatility. Interpreted as a reaction-boundary model for limit-order-book mid-prices, the construction identifies local volatility with an activity-rescaled risk-neutral bid-ask reaction-boundary variance. The framework separates the operational kernel, calendar-time projection, and pricing-measure choice, to clarify how unspanned clock, jump, or renewal risks can lead to incomplete-market pricing.

This paper presents an open-economy macroeconomic equilibrium model for Proof-of-Stake (PoS) networks with fee-burn mechanics (EIP-1559) that formalizes the strategic interplay between a Kelly-optimizing rational institutional investor and a utility-driven retail consumer. We analyze network dynamics across two behavioral regimes. In The Unbounded Accumulation Model, the consumer purely accumulates tokens, creating an exclusive buy-side pressure that interacts with institutional portfolio rebalancing to fuel an ever-expanding speculative bubble and generate compounding excess returns for investors. Conversely, in The Utility-Consumption Model, the consumer dynamically buys and sells tokens to balance crypto wealth against real-world fiat consumption. Within this framework, we derive an explicit steady-state equilibrium price for ETH, demonstrating how token valuation anchors to a stable fundamental baseline that scales directly with network adoption while completely dissolving the institutional yield premium. Our numerical simulations show that while exogenous traditional finance (TradFi) shocks propagate through portfolio rebalancing to drive high token price volatility, network inflation remains highly stable. Furthermore, we prove that network security is insulated from institutional monopoly by counter-cyclical consumer behavior. Our findings reveal that institutional excess wealth creation in PoS ecosystems is not native to the staking protocol itself, but is strictly driven by the leveraged extraction of the retail consumer's continuous demand for transactional utility.

We present a study of the leading-order asymptotics for VIX option prices in Bergomi models in the short-maturity and small volatility-of-volatility regimes. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered for one-factor, two-factor Bergomi and $N$-factor models. The leading-order asymptotics are obtained in closed-form, which are translated into predictions for the small-maturity asymptotics of the VIX implied volatility. Numerical illustrations are provided to illustrate the efficiency of the closed-form asymptotic formulas.

Q-variance (so-called) posits a statistical relationship $\mathbf{E}(\sigma^2 | z) = \sigma_0^2 + \tfrac{1}{2}z^2$ between an asset's volatility $\sigma^2$, as observed in a time interval $T$, and its (suitably scaled) return $z$ in the same interval. We here show that this relationship is {\em exactly equivalent} to to positing an Inverse Gamma probability distribution for $\sigma^2$ itself. We then show that such a distribution is exactly generated by a multiplicative Langevin process with an arbitrary, settable coherence time $\tau_c$, so that very nearly the same Q-variance relationship will hold for all $T \ll \tau_c$.

We study Bermudan option pricing under the Gatheral Double Mean-Reverting (GDMR) stochastic volatility model. The model features a variance process together with a stochastic long-run mean variance process and allows Constant Elasticity of Variance (CEV)-type exponents in the diffusion coefficients. This model is attractive since it provides a flexible specification for volatility dynamics. However, the pricing of early-exercise derivatives under the GDMR model remains largely unexplored in the literature. To address this challenge, we adapt a Hybrid Least-Squares Monte Carlo-Partial Differential Equation (LSMC-PDE) framework to the GDMR model and provide a detailed model-specific implementation. Conditioning on simulated variance paths, the pricing problem reduces to a one-dimensional problem in the asset price, which is solved by a Fourier-based approach, while the remaining dependence on the variance variables is approximated by least-squares regression. Our numerical experiments demonstrate that the Hybrid LSMC-PDE approach yields accurate pricing estimates and often lower pricing errors than plain LSMC, particularly for low and moderate numbers of simulation paths, showing the benefit of using the model structure in early-exercise option pricing.

This paper develops a valuation framework for guaranteed lifetime withdrawal benefit (GLWB) contracts with long-term care (LTC) features when the reference fund follows exponential Levy dynamics and the short rate follows the Hull-White model. The contract combines financial guarantees, longevity protection, health-contingent LTC payments, and surrender optionality, requiring the joint treatment of jump risk, stochastic discounting, and disability risk. The numerical method couples a recombining Hull-White trinomial tree with an implicit-explicit (IMEX) finite difference scheme. The framework incorporates a seven-state health model, annual fees, LTC payments, guaranteed withdrawals, and bang-bang policyholder actions, and is benchmarked against Monte Carlo simulation. Numerical results show that the hybrid tree-IMEX method delivers stable long-maturity prices consistent with simulation benchmarks. They also show that Levy equity dynamics and stochastic interest rates have a material impact on fair fees and surrender incentives, and affect the decomposition of contract value. The findings highlight the importance of modelling financial tail risk and interest-rate risk jointly when pricing long-term insurance guarantees with LTC-contingent benefits.

We develop a partial integro-differential equation (PIDE) framework for option pricing under joint stochastic volatility and jump dynamics, and evaluate its empirical content using the S&P500 index option contracts across three maturities. The framework is derived from the infinitesimal generator of an affine L\'evy-type process and implemented via finite-difference discretization with FFT-based treatment of the nonlocal jump operator. Calibration via GMM reveals that stochastic volatility accounts for the dominant share of pricing improvement, where relative to Black-Scholes, the Heston specification reduces implied-volatility RMSE by 39%. Jump augmentation via either Merton or CGMY specifications yields marginal improvements concentrated at short maturities and in the deep out-of-the-money region. The calibrated CGMY activity index supports a compound-Poisson structure, consistent with high-frequency evidence on S&P500 index returns.

This paper estimates the carry embedded in listed IBIT options and compares it with the carry embedded in matched CME bitcoin futures. Put-call parity recovers an implied forward on the ETF; BlackRock's daily holdings file maps each ETF share into bitcoin units; and CME futures prices and BRRNY, a U.S. close bitcoin reference rate, provide the corresponding futures-market carry. The difference in carry implied by these two products is consistent with frictions that limit cross-margining between spot bitcoin or ETF exposure and CME futures. In the selected-strike IBIT sample of 386 date-bucket observations, the mean wedge is 2.58 percent and the median wedge is 2.52 percent, both measured in annual percentage points. The result is consistent with segmented collateral and margin systems limiting arbitrage between regulated bitcoin-exposure venues.

This study develops an integrated stochastic modeling framework for pricing short and medium-maturity equity options and assessing interest-rate risk using the Heston (1993), Bates (1996), and CIR (1985) models. We calibrate the Heston model using both the Lewis (2001) Fourier inversion and the Carr-Madan (1999) FFT approach, finding near-identical parameter sets, which is consistent with the calibration stability reported in recent studies such as Agazzotti et al. (2025). Extending the model to Bates shows that jump intensities converge to values effectively equal to zero for 60-day maturities, echoing empirical findings that jumps contribute marginally to short-term smile fitting. We further compare our calibration approach with the joint volatility-surface and variance-term-structure framework proposed by Yoo (2025), confirming that standard Heston/Bates calibration remains robust for the maturities considered. Finally, we calibrate the CIR short-rate model to the Euribor term structure, generating positive and economically consistent forward-rate scenarios in line with recent stochastic-rate option-pricing research by Jeon and Kim (2025). Overall, our results show that continuous stochastic volatility dominates near-term pricing dynamics, while stochastic interest rates materially influence valuations beyond one year.

In general, the pricing of variable annuities with guarantees can be done by solving the corresponding optimal stochastic control problem if the contract withdrawal strategy is assumed to be optimal. This is typically solved as a dynamic programming problem using deterministic grid methods, which become computationally infeasible for more than a few state variables. In such situations, one needs to rely on simulation methods. The least-squares Monte Carlo (LSMC) method has become a popular simulation method for solving optimal stochastic control problems in quantitative finance over the last decades. In principle, the LSMC, originally developed for pricing Bermudan options, cannot be used directly for pricing variable annuities without simplifying assumptions because the underlying state variables are affected by the control decisions. This paper presents modifications of the LSMC algorithm that makes the pricing of general variable annuities feasible. For numerical illustrations, the pricing of variable annuities with guaranteed minimum withdrawal benefit under optimal withdrawal strategies is obtained with and without stochastic interest rates, using either polynomial regression or neural network regression in the LSMC algorithm. We found that the classical polynomial LSMC can give very accurate prices, at the cost of manual feature engineering, and with a standard deviation of the estimator that increases greatly when interest rates are made stochastic. By contrast, neural network LSMC gives slightly less accurate prices, requires more training time, but does not require manual feature engineering, and making interest rates stochastic makes no visible difference to its accuracy, suggesting a more stable and robust pricing performance of deep LSMC for higher-dimensional pricing problems.

We study risk-neutral density extraction from short-dated option chains. As expiry approaches, option premia decline and bid--ask spreads can be large relative to prices, making mid quotes particularly uninformative. Stale or asynchronous quotes may also generate potential static arbitrages, rendering standard procedures infeasible or unstable. We develop a model-free pipeline that treats bid-ask quotes as the primitive market constraint. The pipeline consists of two steps. First, a procedure called ``Arbitrage Removal Iterative Executable Strategy'' (ARIES) filters executable static arbitrage at quoted bid and ask prices under market-depth constraints. Second, the ``Smooth Entropic Density EXtraction'' (SEDEx) then recovers the density through a criterion leveraging smoothness and entropy under bid-ask constraints. We test the pipeline on synthetic Heston panels and short-dated SPX option data, sampled from a few hours to one week before expiry. Computation is fast and returns robust densities across various market conditions, including scheduled macroeconomic announcements. As an empirical application, we use the recovered densities to construct short dated implied-volatility smiles.

We present ThiopheneIV, a Black-Scholes implied-volatility solver with a monotone core and explicit production guards. The solver starts from the simple Choi-Huh-Su L3 lower-bound seed and applies three Euler-Chebyshev steps on a lower branch and three Halley steps on the remaining upper branch. We prove that, in exact arithmetic, the seed lies below the root and both maps increase monotonically without overshooting. We also detail the practical challenges encountered for a double-precision implementation: parity normalisation, microscopic Bachelier-limit handling, saturated price treatment, and an optional J\"ackel-Newton polish. Across standard grids, market-like data, high-volatility cases, and adversarial corners, ThiopheneIV agrees closely with multiprecision Black reference prices at low latency. We provide detailed comparisons with recent solvers, including J\"ackel's Let's Be Rational. The broader lesson is that a convergence proof gives a clean core, but robust production inversion still depends on boundary handling and on the pricing objective one chooses to match.

This paper studies empirical deep hedging for S&P 500 index options under a local downside-shortfall reward. It moves beyond performance comparison by asking what the learned hedge does, when it fails, and whether it can be made auditable. TD3 agents are compared with a daily-updated Black-Scholes delta hedge on the same option episodes. In walk-forward tests from 2015 to 2023, the agents usually learn a systematic delta haircut relative to Black-Scholes. The correction is explained by spot-implied-volatility co-movement and often improves accumulated reward and terminal downside variance, but it is regime-fragile: 2022 exposes losses in adverse daily states, while 2023 shows that underhedging can raise ordinary variance when option P&L is spot-dominated and the volatility channel is unusually weak. Symbolic regression distills the neural policies into compact formulas that can be traded out of sample; these formulas preserve much of the reward, downside-variance, and CVaR advantage over Black-Scholes, and sometimes sharpen it, but inherit the same fragility in difficult regimes.

We present two explicit rational formulae for Bachelier, or normal, implied volatility. The formulae take the option price, forward, strike, and expiry as inputs and return the implied normal volatility without iteration. They follow the branch structure of LFK-4, but use the simpler near-the-money variable given by the absolute forward-strike difference divided by the tail time value, avoiding a logarithm and a small-argument Taylor branch in that region. LFK-2026 is the accuracy-oriented formula and approximates reciprocal absolute standardized moneyness directly in the far tail. LFK-2026C keeps the same shifted out-of-the-money rational tail approximation, but splits the near-the-money branch two low degree rationals. In double precision tests both remain close to machine accuracy, while LFK-2026C is the faster scalar implementation on the current benchmark mix.

Monte-Carlo valuation engines can generate pathwise sensitivities of a derivative value with respect to a high-dimensional vector of model primitives. Hedge ratios with respect to market instruments are then linked to these primitive sensitivities by a pathwise linear relation. Solving this relation independently on every simulated path may be expensive, unstable, and unnecessarily high-dimensional. This paper studies reduced stochastic hedge ratios of the form $\phi_j^r=\sum_{q=1}^r\xi_j^qX_q$, where the number of solution basis functions is much smaller than the number of Monte-Carlo paths. The hedge-instrument sensitivity tensor is not replaced by its own basis expansion; it is retained through empirical averages over the simulated paths. The basis ansatz alone does not determine the coefficients, so two coefficient criteria are distinguished. The first minimizes the full empirical pathwise residual $\sum_\ell\|A_\ell\phi_\ell^r-b_\ell\|_2^2$. The second is a projected moment equation requiring $\langle A\phi^r-b,Y_s\rangle_N=0$ for selected test functions. The special case $Y_s=X_s$ is the usual Galerkin choice; different test functions give a Petrov--Galerkin formulation. The criteria coincide in special cases but differ when the hedge-instrument sensitivities are path-dependent. The paper gives the tensor and matrix forms of both reductions, discusses regularization and conditioning, and records implementation considerations. The constructions are motivated by sensitivity-based margin valuation adjustment and replication-consistent liquidity forecasting, where pathwise primitive sensitivities have to be converted into hedge ratios with respect to market instruments.

This paper develops a deep learning-based framework for pricing convertible bonds with path-dependent contractual features, namely downward conversion price reset and issuer call clauses under rolling-window trigger rules, which are widespread in the convertible bond market. We formulate the valuation problem as a path-dependent partial differential equation (PPDE), which explicitly captures the dependence of the convertible bond value on the historical path of the underlying asset and the dynamic evolution of the conversion price. We derive consistent PPDE formulations for three canonical underlying dynamics: geometric Brownian motion (GBM), constant elasticity of variance (CEV) and Heston stochastic volatility. We then construct a discrete-time dynamic programming scheme in which conditional expectations are approximated by neural networks, which remains tractable in such high-dimensional path-dependent setting. Empirical tests on China CITIC Bank Convertible Bond show that our framework produces stable and accurate prices and sensitivity patterns across all model specifications. Three key economic insights emerge: 1. Contractual features dominate underlying dynamics in determining convertible bond values. 2. The call provision decreases convertible bonds prices by truncating upside gains. 3. Counterintuitively, despite improving conversion terms, the downward reset provision further decreases the price of convertible bonds by lowering the effective call threshold and making early redemption more likely. The proposed PPDE-deep learning approach provides an efficient, flexible tool for pricing convertible bonds with complex path-dependent structures.

Paper 1 of this research programme develops a resolution-aware risk-design framework for the simplest event-linked perpetual: a contract whose underlying tracks a single binary prediction-market probability through resolution. The instrument class is broader. Variants span conditional probabilities P(A|B), spreads p^A - p^B, weighted baskets sum w_i p^(i), derivatives on variance or entropy of the probability process, contracts on liquidity itself, perpetual-on-expiring-event roll structures, and funding-only derivatives with no settlement. Each variant inherits some framework components from the single-market binary case and requires its own design adaptations. This paper develops a formal taxonomy of seven pure-form canonical variants beyond the probability-index perpetual of Paper 1, organised along four orthogonal design axes: underlying geometry, temporal structure, settlement structure, and venue composition. The list is not exhaustive; combinations are not treated separately. For each variant we provide a precise payoff definition; an inheritance map identifying which Paper 1 components carry over, are modified, or fail; variant-specific design constraints; microstructure properties; empirical evaluability on the PMXT v2 archive; and limitations. Notable findings: the conditional variant admits a candidate non-portability proposition (denominator instability as the conditioning event becomes improbable); the spread variant requires a three-channel decomposition of resolution risk; the volatility/entropy variant avoids random binary terminal-collapse but introduces estimator-convention and entropy-decay issues; the basket variant requires multi-period jump-aware margin whose aggregation is correlation-dependent. The paper is theoretical primarily; it specifies how demonstrative time series can be constructed and provides evaluability criteria to guide future work.

Structured launch signals on Product Hunt contain statistically significant predictive information for Series A funding outcomes. We construct PHBench from 67,292 featured Product Hunt posts spanning 2019-2025, linked to Crunchbase funding records via deterministic domain matching, identifying 528 verified Series A raises within 18 months of launch (positive rate: 0.78%). Our best-performing model, a three-component ensemble (ENS_avg, ENS_ISO, XGB) selected by validation F0.5, achieves F0.5 = 0.097 and AP = 0.037 (95% CI: 0.024-0.072; 4.7x lift over random) on the private held-out test set (103 positives). A paired bootstrap confirms a statistically credible advantage over the logistic regression baseline (AP delta: +0.013, 95% CI: [0.004, 0.039], p < 0.001; F0.5 delta: +0.056, 95% CI: [0.006, 0.122], p = 0.016). Validation-set metrics (F0.5 = 0.284, AP = 0.126) reflect best-of-144 selection bias on 53 positives and are reported for benchmark reproducibility only. We further evaluate three zero-shot Gemini models (Gemini 2.5 Flash, Gemini 3 Flash, and Gemini 3.1 Pro) in an anonymized numerical setting. The best LLM achieves AP = 0.034 (Gemini 3 Flash), below the LR baseline AP of 0.044. Notably, the most capable Gemini variant (Gemini 3.1 Pro, AP = 0.023) performs worst -- an unexpected pattern that warrants further investigation across providers and prompting strategies. Both ML and LLM models show the same temporal performance decay tracking the 2020-2021 funding boom and subsequent contraction, confirming the dataset captures genuine market structure rather than noise. PHBench provides a reproducible framework comprising public training, validation, and blind test splits; 61 engineered features; a five-metric evaluation harness; and a public leaderboard at https://phbench.com. All code, baseline models, and anonymized dataset splits are publicly available.

We present fast-vollib, an open-source Python library that provides high-performance European option pricing, implied volatility (IV) computation, and Greeks under the Black-76, Black-Scholes, and Black-Scholes-Merton models. The library is designed as a drop-in alternative to the de-facto-standard py_vollib and py_vollib_vectorized packages, with pluggable PyTorch and JAX execution backends, a CUDA fused-kernel Triton contribution for batched IV workloads, and a compatibility-first public API. In addition to a vectorized Halley-method IV solver, fast-vollib ships an experimental, fully-vectorized implementation of J\"ackel's "Let's Be Rational" (LBR) algorithm with NumPy/Numba, torch.compile, JAX, and Triton single-pass GPU kernels for batched option chains. This note announces the library and describes its public API surface, with source, documentation, and packaging artifacts available at: GitHub (https://github.com/raeidsaqur/fast-vollib), Docs (https://raeidsaqur.github.io/fast-vollib/), PyPI (https://pypi.org/project/fast-vollib/).

We introduce the Local Occupied Volatility (LOV) model that sits between Dupire's local volatility and fully path-dependent dynamics. By design, the LOV model ensures automatic calibration to European vanilla options, while offering the flexibility to capture stylized facts of volatility or fit additional instruments. This is achieved by tuning the occupation sensitivity function that quantifies the effect of path-dependent shocks on volatility. We validate the model through the joint American-European calibration of options chain on non-dividend paying stocks.

Persistent shifts in term-structure dynamics undermine the stability of single-regime models in long samples. We develop an arbitrage-free regime-switching generalized CIR (RS-GCIR) model that jointly prices the Chinese government bond (CGB) curve and corporate bond curves. To capture the systematic transmission from interest-rate conditions to credit spreads, we structure the model into two blocks and price corporate bonds conditional on the prevailing rate regime. The rate block features a two-state RS-GCIR short-rate process estimated from CGB zero-coupon curves, while the credit block embeds CIR-type credit factors in an intensity-based framework for rating migration and default. We implement a block-recursive Unscented Kalman Filter (UKF) procedure--filtering the rate block first and the credit block next--using weekly data from 2014--2025, a period that begins with the onset of China's modern corporate default cycle. We identify two persistent rate regimes with distinct level--volatility profiles. Relative to single-regime benchmarks, regime switching improves joint curve fit, delivers economically interpretable filtered regime probabilities, and sharpens the decomposition of corporate yields into discounting and credit compensation.

Black-Scholes implied volatility is a quantile. The insight follows from the normalized option price being a probability on the variance scale, with the inverse Gaussian distribution providing the link. It enables analytically exact and explicit formulas for implied volatility in terms of existing quantile functions, with volatility on the left-hand side and only observable option inputs on the right-hand side. The result is not another approximation or asymptotic expansion. Instead, it rewrites the price-to-volatility map itself as a distributional transform. The representation gives implied volatility a first-passage-time interpretation, identifies variance as the natural coordinate of inversion, and reorganizes Greeks and no-arbitrage restrictions in the same variance-quantile coordinates. Numerically, the formula achieves machine precision faster than a state-of-the-art solver in the benchmark considered. The paper therefore provides a new coordinate system for computing, interpreting, and decomposing one of the central quantities in option markets.

We demonstrate that machine learning methods provide a powerful framework for modelling conditional asymmetric risk. Using a large cross-section of US stocks and a comprehensive set of firm characteristics, we show that allowing for nonlinearities significantly increases the out-of-sample performance across a wide range of asymmetric beta measures and forecasting horizons. Trading frictions, followed by characteristics related to intangibles, momentum and growth, emerge as the most important drivers of future risk dynamics. Reconstructing CAPM beta from forecasts of asymmetric beta components indicates that a more granular decomposition of systematic risk yields a more accurate representation of market beta. We also find that incorporating conditional beta forecasts into discounted cash flow models that account for the term structure of betas enhances equity valuation accuracy. Finally, we show that the statistical outperformance of conditional betas translates into economically significant benefits for market-neutral portfolio investors.

Barrier derivatives depend on extrema and first-passage events and are therefore highly sensitive to volatility dynamics -- especially to the instantaneous return-volatility correlation $\rho$, often called ``leverage''. This sensitivity makes accurate and fast pricing under realistic stochastic-volatility specifications difficult: two-dimensional PDE solvers are expensive inside calibration loops, while Monte Carlo methods converge slowly when barrier hits are rare and discretely monitored. In equity markets in particular, the pronounced implied-volatility skew motivates factoring in a negative return-volatility correlation. We study a class of continuous-path stochastic-clock volatility models in which the log-price is represented as a Brownian motion run on a random increasing clock. In the baseline independent-clock case (\rho=0), a broad family of barrier-relevant objects-maximum distributions, survival probabilities, and killed joint laws-reduces to one-dimensional quantities determined by the Laplace transform of the terminal clock. This yields transform-only pricing formulas for single- and double-barrier contracts that are fast and numerically stable once the clock transform is available, notably for affine and quadratic clocks. To incorporate leverage without forfeiting tractability, we develop a systematic small-\rho expansion around the \rho=0 backbone. The expansion produces a hierarchy of forced problems whose forcing terms are semi-analytic and computable from baseline barrier objects. We provide two implementable leverage-correction routes\,: forced PDEs and a Duhamel-type Monte Carlo representation, and we show how Pad{\'e} acceleration can extend practical accuracy to equity-like correlations. Calibration then proceeds by\,: (i) fitting clock parameters from vanillas using only one-dimensional transforms, (ii) precomputing the \rho=0 barrier backbone once, and (iii) iterating on \rho (and any remaining parameters) using the fast semi-analytic corrections-optionally Pad{\'e}-accelerated-inside a standard least-squares loop.

We study cash-flow forecasting for derivatives used in liquidity management and clarify its relation to risk-neutral valuation and replication. While it is well known that expectations under different measures (e.g., $\mathbb{P}$ vs. $\mathbb{Q}$) can yield different undiscounted cash-flows, further inconsistencies arise when payment times are stochastic. We show that using discounting sensitivities (funding-curve hedge ratios) instead of "expected cash-flows" aligns forecasting with the self-financing replication strategy and avoids measure-mixing/aggregation issues. We then illustrate how a standard valuation model delivers pathwise funding requirements and propose a simple liquidity valuation adjustment to capture settlement lags and related timing frictions. The note provides implementation hints (American Monte Carlo with adjoint differentiation) and clarifies when "expected cash-flows" are informative and when sensitivities should be used instead.

The Nelson-Siegel-Svensson (NSS) interest rate curve model yields a separable nonlinear least-squares problem whose inner linear block is often ill-conditioned because the basis functions become nearly collinear. We analyze this instability via an exact orthogonal reparametrization of the design matrix. A thin QR decomposition produces orthogonal linear parameters for which, conditional on the nonlinear parameters, the Fisher information matrix is diagonal. We also derive a finite-horizon analytical orthogonalization: on $[0,T]$, the $4\times 4$ continuous Gram matrix has closed-form entries involving exponentials, logarithms, and the exponential integral $E_1$, yielding an explicit horizon-dependent orthogonal NSS basis. Together with Jacobian-rank and profile-likelihood arguments, this representation clarifies the degenerate manifold $\lambda_1=\lambda_2$, where the Svensson extension loses two degrees of freedom. Orthogonalization leaves the least-squares fit and uncertainty of the original linear parameters unchanged, but isolates the conditioning structure. When the decay parameters are estimated jointly, the full first-order covariance in orthogonal coordinates admits an explicit Schur-complement form. The approach also yields a scalar identifiability diagnostic through the QR element $R_{44}$ and separates model reduction from numerical instability. Synthetic experiments confirm that orthogonal parametrization eliminates correlations among the linear parameters and keeps their conditional uncertainty uniform. A daily U.S. Treasury study on a reduced fixed 9-tenor grid from 1981 to 2026 shows smoother orthogonal parameter series than classical NSS parameters while the moving QR basis remains nearly constant.

Sparsity or complexity? In modern high-dimensional asset pricing, these are often viewed as competing principles: richer feature spaces appear to favor complexity, while economic intuition has long favored parsimony. We show that this tension is misplaced. We distinguish capacity sparsity-the dimensionality of the candidate feature space-from factor sparsity-the parsimonious structure of priced risks-and argue that the two are complements: expanding capacity enables the discovery of factor sparsity. Revisiting the benchmark empirical design of Didisheim et al. (2025) and pushing it to higher complexity regimes, we show that nonlinear feature expansions combined with basis pursuit yield portfolios whose out-of-sample performance dominates ridgeless benchmarks beyond a critical complexity threshold. The evidence shows that the gains from complexity arise not from retaining more factors, but from enlarging the space from which a sparse structure of priced risks can be identified. The virtue of complexity in asset pricing operates through factor sparsity.

Using only the characteristic function, we derive short-time at-the-money (ATM) call-price asymptotics for the exponential CGMY model with activity parameter $Y\in(1,2)$. The Lipton--Lewis formula expresses the normalized ATM call price, denoted $c(t,0)$, in terms of the characteristic exponent, which, upon rescaling at the rate $t^{-1/Y}$ from the $Y$-stable domain of attraction, yields $c(t,0) = d_{1} t^{1/Y} + d_{2} t + o(t)$ as $t\downarrow 0$. The first-order coefficient $d_{1}$ is the known stable limit from the domain of attraction of a symmetric $Y$-stable law, and $d_{2}$ is given by an explicit integral involving the characteristic exponent and the limiting stable exponent. We then extract closed-form higher-order coefficients by keeping the full Lipton--Lewis integrand intact and introducing a dynamic cutoff that partitions the domain into inner, core, and tail regions, establishing the expansion with controlled remainder. All coefficients are verified numerically against existing closed-form expressions where available.

Large language models (LLMs) are increasingly deployed in quantitative finance for stock price forecasting. This review synthesizes recent applications of LLMs in this domain, including extracting sentiment from financial news and social media, analyzing financial reports and earnings-call transcripts, tokenizing or symbolizing stock price series, and constructing multi-agent trading systems. Particular attention is paid to practical pitfalls that are often understated in the literature, such as fragility in sentiment analysis, dataset and horizon design, performance evaluation metrics, data leakage, illiquidity premia, and limits of stock price predictability. Organized from a hedge-fund perspective, the review is intended to guide both academic researchers and hedge fund managers in integrating LLMs into real-world trading pipelines and in stress-testing their robustness under realistic market frictions.

Corporate bond factor research faces a replication crisis. The crisis stems from two sources that inflate reported factor premia: transaction prices whose measurement error enters both sorting signals and return denominators, creating a correlated errors-in-variables bias, and asymmetric ex-post return filtering that embeds future information into factor construction. Applying our framework to a 'factor zoo' of 108 signals across nine thematic clusters, we show that the majority of previously documented factors do not produce statistically significant bond CAPM alphas after correction. We provide an open source framework via Open Bond Asset Pricing, including error-corrected TRACE data, bias corrected factors, and software for reproducible research.

Recent studies document strong empirical support for multifactor models that aim to explain the cross-sectional variation in corporate bond expected excess returns. We revisit these findings and provide evidence that common factor pricing in corporate bonds is exceedingly difficult to establish. Based on portfolio- and bond-level analyses, we demonstrate that previously proposed bond risk factors, with traded liquidity as the only marginal exception, do not have any incremental explanatory power over the corporate bond market factor. Consequently, this implies that the bond CAPM is not dominated by either traded- or nontraded-factor models in pairwise and multiple model comparison tests.

We analyze 18 quadrillion models for the joint pricing of corporate bond and stock returns. Strikingly, we find that equity and nontradable factors alone suffice to explain corporate bond risk premia once their Treasury term structure risk is accounted for, rendering the extensive bond factor literature largely redundant for this purpose. While only a handful of factors, behavioral and nontradable, are likely robust sources of priced risk, the true latent stochastic discount factor is dense in the space of observable factors. Consequently, a Bayesian Model Averaging Stochastic Discount Factor explains risk premia better than all low-dimensional models, in- and out-of-sample, by optimally aggregating dozens of factors that serve as noisy proxies for common underlying risks, yielding an out-of-sample Sharpe ratio of 1.5 to 1.8. This SDF, as well as its conditional mean and volatility, are persistent, track the business cycle and times of heightened economic uncertainty, and predict future asset returns.

We examine whether model-based spot volatility estimators extracted from traded options data enhance the predictive power of the Heterogeneous Autoregressive (HAR) model for realized volatility. Specifically, we infer spot volatility under the rough stochastic volatility model via an iterative two-step approach following Andersen et al. (2015a) and adopt a deep learning surrogate to accelerate model estimation from large-scale options panels. Benchmarked against traditional stochastic volatility models (Heston, Bates, SVCJ) and the VIX index, our results demonstrate that the augmented HAR-RV-RHeston model improves daily realized volatility forecasting accuracy and sustains superior performance across horizons up to one month.

The multidimensional Uncertain Volatility Model leads to robust option pricing problems under joint volatility and correlation uncertainty. Their numerical resolution quickly becomes challenging because the associated stochastic control problem is high-dimensional. We propose a backward actor-critic stochastic policy gradient scheme tailored to this setting. The method combines a discrete dynamic programming principle with Proximal Policy Optimization and shallow neural-network approximations of both the value function and the control policy. A key ingredient is the policy parameterization: continuous controls are represented through a squashed Gaussian policy built on a C-vine representation of correlation matrices, which enforces positive semidefiniteness by construction. Numerical experiments on a range of multidimensional derivatives show that the method yields accurate prices, remains computationally efficient, and compares favorably with existing Monte Carlo and machine-learning-based benchmarks for robust pricing in the Uncertain Volatility Model.

We develop at-the-money call-price and implied volatility asymptotic expansions in time to maturity for a class of asset-price models whose log returns follow a L\'evy process. Under mild assumptions placing the driving L\'evy process in the small-time domain of attraction of an $\alpha$-stable law with $\alpha \in (1,2)$, we give first-order at-the-money call-price and implied volatility asymptotics. A key observation is that both the stable domain of attraction and the finiteness of the centering constant $\bar{\mu}$ are preserved under the share measure transformation, so that all of the distributional input needed for the call-price expansion can be read off from the regular variation of the L\'evy measure near the origin. When the L\'evy process has no Brownian component, new rates of convergence of the form $t^{1/\alpha} \ell(t)$ where $\ell$ is a slowly varying function are obtained. We provide an example of an exponential L\'evy model exhibiting this behavior, with $\ell$ not asymptotically constant, yielding a convergence rate of $(t / \log(1/t))^{1/\alpha}$. In the case of a L\'evyprocess with Brownian component, we show that the jump contribution is always lower order, so that the leading $\sqrt{t}$ behavior of the at-the-money call price is universal and driven entirely by the Gaussian part of the characteristic triplet.

This paper develops an autonomous framework for systematic factor investing via agentic AI. Rather than relying on sequential manual prompts, our approach operationalizes the model as a self-directed engine that endogenously formulates interpretable trading signals. To mitigate data snooping biases, this closed-loop system imposes strict empirical discipline through out-of-sample validation and economic rationale requirements. Applying this methodology to the U.S. equity market, we document that long-short portfolios formed on the simple linear combination of signals deliver an annualized Sharpe ratio of 3.11 and a return of 59.53%. Finally, our empirics demonstrate that self-evolving AI offers a scalable and interpretable paradigm.

This paper develops a computational framework for Multi-Period Martingale Optimal Transport (MMOT), addressing convergence rates, algorithmic efficiency, and financial calibration. Our contributions include: (1) Theoretical analysis: We establish discrete convergence rates of $O(\sqrt{\Delta t} \log(1/\Delta t))$ via Donsker's principle and linear algorithmic convergence of $(1-\kappa)^{2/3}$; (2) Algorithmic improvements: We introduce incremental updates ($O(M^2)$ complexity) and adaptive sparse grids; (3) Numerical implementation: A hybrid neural-projection solver is proposed, combining transformer-based warm-starting with Newton-Raphson projection. Once trained, the pure neural solver achieves a $1{,}597\times$ online inference speedup ($4.7$s $\to 2.9$ms) suitable for real-time applications, while the hybrid solver ensures martingale constraints to $10^{-6}$ precision. Validated on 12,000 synthetic instances (GBM, Merton, Heston) and 120 real market scenarios.

We introduce the Consensus-Bottleneck Asset Pricing Model (CB-APM), which embeds aggregate analyst consensus as a structural bottleneck, treating professional beliefs as a sufficient statistic for the market's high-dimensional information set. Unlike post-hoc explainability approaches, CB-APM achieves interpretability-by-design: the bottleneck constraint functions as an endogenous regularizer that simultaneously improves out-of-sample predictive accuracy and anchors inference to economically interpretable drivers. Portfolios sorted on CB-APM forecasts exhibit a strong monotonic return gradient, robust across macroeconomic regimes. Pricing diagnostics further reveal that the learned consensus encodes priced variation not spanned by canonical factor models, identifying belief-driven risk heterogeneity that standard linear frameworks systematically miss.

In this work, we introduce amortizing perpetual options (AmPOs), a fungible variant of continuous-installment options suitable for exchange-based trading. Traditional installment options lapse when holders cease their payments, destroying fungibility across units of notional. AmPOs replace explicit installment payments and the need for lapsing logic with an implicit payment scheme via the decay of the claimable notional. This amortization ensures all units evolve identically, preserving fungibility. We demonstrate that AmPO valuation can be reduced to an equivalent vanilla perpetual American option on a dividend-paying asset. This enables analytical expressions for the exercise boundaries and risk-neutral valuations for calls and puts. These formulas and relations allow us to derive the Greeks and study comparative statics with respect to the amortization rate. Illustrative numerical case studies demonstrate how the amortization rate shapes option behavior and reveal the resulting tradeoffs in the effective volatility sensitivity.

Differential ML (Huge and Savine 2020) is a technique for training neural networks to provide fast approximations to complex simulation-based models for derivatives pricing and risk management. It uses price sensitivities calculated through pathwise adjoint differentiation to reduce pricing and hedging errors. However, for options with discontinuous payoffs, such as digital or barrier options, the pathwise sensitivities are biased, and incorporating them into the loss function can magnify errors. We consider alternative methods for estimating sensitivities and find that they can substantially reduce test errors in prices and in their sensitivities. Using differential labels calculated through the likelihood ratio method expands the scope of Differential ML to discontinuous payoffs. A hybrid method incorporates gamma estimates as well as delta estimates, providing further regularization.

Environmental, Social, and Governance (ESG) data provides non-financial insights into corporations. In this study, we aim to identify relevant ESG raw variables to assess financial risk, measured by logarithmic volatility of return. We propose a framework specifically designed for ESG datasets characterized by a hierarchical data structure and a significantly larger number of variables than observations. We show that raw variables selected by the proposed framework are significantly more relevant to financial risk than aggregated ESG scores. Furthermore, these selected risk variables provide additional insights beyond the traditional financial factors. We validate the robustness of this framework using out-of-sample data. We illustrate our framework using company data from various sectors of the US economy. We further identify the specific ESG risk variables relevant to large and small companies within each sector.

Traditional models for pricing catastrophe (CAT) bonds struggle to capture the complex, relational data inherent in these instruments. This paper introduces CATNet, a novel framework that applies a geometric deep learning architecture, the Relational Graph Convolutional Network (R-GCN), to model the CAT bond primary market as a graph, leveraging its underlying network structure for spread prediction. Our analysis reveals that the CAT bond market exhibits the characteristics of a scale-free network, a structure dominated by a few highly connected and influential hubs. CATNet demonstrates higher predictive performance, significantly outperforming strong Random Forest and XGBoost benchmarks. Interpretability analysis confirms that the network's topological properties are not mere statistical artifacts; they are quantitative proxies for long-held industry intuition regarding issuer reputation, underwriter influence, and peril concentration. This research provides evidence that network connectivity is a key determinant of price, offering a new paradigm for risk assessment and proving that graph-based models can deliver both state-of-the-art accuracy and deeper, quantifiable market insights.

This paper studies the pricing of contingent claims of American style, using indifference pricing by fully dynamic convex risk measures. We provide a general definition of risk-indifference prices for buyers and sellers in continuous time, in a setting where buyer and seller have potentially different information, and show that these definitions are consistent with no-arbitrage principles. Specifying to stochastic volatility models, we characterize indifference prices via solutions of Backward Stochastic Differential Equations reflected at Backward Stochastic Differential Equations and show that this characterization provides a basis for the implementation of numerical methods using deep learning.

We present a generative approach to price options and extract risk-neutral densities from the market. Specifically, we model the underlying log-returns on the time-to-maturity continuum as a generative model from standard normal. Neural nets are used to represent the term structures of the location, the scale, and the higher-order moments. We impose stringent conditions on the learning process to ensure no arbitrage. This model allows for the efficient generation of samples to price options across strikes and maturities. We have validated the effectiveness of this approach by benchmarking it against a comprehensive set of baseline models. Experiments show that the extracted risk-neutral densities accommodate a diverse range of shapes. Its accuracy significantly outperforms the extensive set of baseline models--including three parametric models and nine stochastic process models--in terms of accuracy and stability. The success of this approach is attributed to its capacity to offer flexible term structures for risk-neutral skewness and kurtosis.

We study how trading fees and continuous-time arbitrage affect the profitability of liquidity providers (LPs) in Geometric Mean Market Makers (G3Ms). We use stochastic reflected diffusion processes to analyze the dynamics of a G3M model under the arbitrage-driven market. Our research focuses on calculating LP wealth and extends the findings of Tassy and White related to the constant product market maker (Uniswap v2) to a wider range of G3Ms, including Balancer. This allows us to calculate the long-term expected logarithmic growth of LP wealth, offering new insights into the complex dynamics of AMMs and their implications for LPs in decentralized finance.

This study proposes a fast exact simulation scheme for the Ornstein-Uhlenbeck driven stochastic volatility model. With the Karhunen-Lo\`eve expansions, the stochastic volatility path (Ornstein-Uhlenbeck process) is expressed as a sine series, and the time integrals of volatility and variance are analytically derived as infinite series of independent normal random variables. The new method is several hundred times faster than the existing method using numerical transform inversion. The simulation variance is further reduced with conditional simulation and the control variate.

L\'evy-driven Ornstein-Uhlenbeck (OU) processes represent an intriguing class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. However, in the current state of play, Monte Carlo simulations of these processes are not straightforward for two main reasons: i) algorithms are available only for some specific processes within this class; ii) they are often computationally expensive. In this paper, we introduce a new simulation technique designed to address both challenges. It relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all L\'evy-driven OU processes. Moreover, leveraging FFT, the proposed methodology ensures fast and accurate simulations, providing a solid basis for the widespread adoption of these processes in the energy sector. Lastly, the algorithm allows explicit control of the numerical error. We apply the technique to the pricing of energy derivatives, comparing the results with the existing benchmarks. Our findings indicate that the proposed methodology is at least one order of magnitude faster than the existing algorithms, while maintaining an equivalent level of accuracy.

We derive a backward and forward nonlinear PDEs that govern the implied volatility of a contingent claim whenever the latter is well-defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. We also discuss suitable initial and boundary conditions for those PDEs. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach.

We propose a new model for electricity pricing based on the price cap principle. The particularity of the model is that the asset price is an exponential functional of a jump L\'evy process. This model can capture both mean reversion and jumps which are observed in electricity market. It is shown that the value of an European option of this asset is the unique viscosity solution of a partial integro-differential equation (PIDE). A numerical approximation of this solution by the finite differences method is provided. The consistency, stability and convergence results of the scheme are given. Numerical simulations are performed under a smooth initial condition.