Papers

We study the quadratic tracking problem of a general stochastic target process with absolutely continuous controls, with and without terminal constraint. We derive explicit, non-asymptotic upper bounds in terms of a Besov-type modulus of the target. These bounds yield sharp explicit rates that specialize to the square-root order for semimartingale targets. We then apply these results to a generalized Obizhaeva–Wang execution model with random terminal inventory. We first develop a Hilbert-space approach to characterize its optimal strategy, which includes jumps. To avoid such trading spikes, one regularizes the problem by a quadratic trading-rate penalty with coefficient $\varepsilon$. We then show that the regularized optimal execution cost—and therefore the excess price impact cost of the regularized optimal strategy—converges at the sharp rate $O(\sqrt{\varepsilon})$. Since the regularized optimal strategy is not available in closed form, we further construct a nearly optimal strategy which is readily implementable and shares the same approximation rate.

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).

We consider the distribution-constrained optimal stopping problem introduced by Bayraktar and Miller (Mathematical Finance, 2019) and Beiglbock et al. (PTRF, 2018). Motivated by the multi-marginal Skorokhod embedding problem (SEP) and applications in financial mathematics, we generalize the Root-type solution to the multi-marginal case. The key difficulty is that the associated stopping barriers need not be ordered, so the problem in general cannot be reduced to a sequence of one-marginal problems. First, we give a probabilistic characterization in terms of sequential optimal stopping for an auxiliary time-reversed process, in the same spirit as Cox et al. (PTRF, 2019). Then, we prove the optimality of this construction by martingale arguments for a general class of reward functions, including multi-marginal versions of all examples in Beiglbock et al. (PTRF, 2018) as special cases.

We model equity markets using geometric Brownian particles entering and exiting at rank-dependent intensities. In the many-firm limit, the capital distribution converges to the solution of a reaction-diffusion equation with reaction term built from the intensities. Calibrated on CRSP data, the reaction term is bistable, and the long-run distribution is a traveling wave: we prove existence, uniqueness, and, for constant coefficients, exponential relaxation. Turnover, not drift, stabilizes the calibrated market. With measured volatility, the wave tracks the empirical capital distribution in every decade, determines the capitalization growth of diversity-weighted portfolios, and places the market just inside the boundary of the diverse phase. Turnover reclaims most of what rebalancing gains.

We develop a PDE-based methodology for pricing and hedging European contingent claims in general one-dimensional diffusion markets characterized solely by their scale function and speed measure, possibly without a classical SDE representation, and with constant interest rate. We derive a hedging equation whose solution generates a self-financing hedging strategy and provide sufficient conditions on scale, speed, and interest rate, under which this strategy achieves the minimal hedging capital, expressed through the no free lunch with vanishing risk (NFLVR) condition. We further prove necessary and sufficient conditions for NFLVR and characterize the class of equivalent local martingale measures through an auxiliary diffusion whose scale and speed characteristics are determined by those of the real-world diffusion and by the interest rate. When the NFLVR condition fails, the framework may produce multiple hedging equations corresponding to non-minimal strategies, whose associated prices can exceed the minimal hedging capital. We illustrate both the effectiveness and limitations of the approach through numerical experiments involving diffusion models with irregular features.

This work complements our previous paper, which studies borrower-side strategies in decentralized lending markets, by focusing on lender-side capital allocation. We consider a lender who seeks to allocate a fixed budget across multiple markets sharing the same supplied asset. Accounting for the impact of supplied capital on lending rates, we derive closed-form solutions under three interest-rate models: linear, kinked, and adaptive (Morpho's AdaptiveCurveIRM). Backtests are conducted first on USDC and then on WETH Morpho lending markets on Ethereum. We also show that, under the kinked rate model, an allocation that brings a market exactly to the kink is never optimal on the lender side, whereas it can be optimal on the borrower side. This asymmetry may create tension between lenders and borrowers around the kink and thereby exacerbate rate volatility.

We study long-range correlated Wigner-type matrices built from row-independent stationary Gaussian sequences. For exponentially decaying (AR(1)) correlations, the bulk spectral density deforms from the semicircle law via an explicit combinatorial "hub" mechanism, yet we verify the flatness and decay hypotheses of the matrix-Dyson-equation framework (MDE), with numerical evidence supporting Tracy-Widom edge universality for every fixed $\rho<1$ of the exponential decay correlations; the degenerate limit $\rho\to1^-$ reduces to a symmetrized Volterra operator, connecting to the singular-value cascade identified in a companion BBP analysis. For power-law correlations $dt\sim t^{-\gamma}$, we identify $\gamma_c=1/2$ as the critical point for divergence of the bulk fourth-moment, while $\gamma=1$ marks the breakdown of the flatness condition governing the MDE edge analysis. We prove the fourth-moment transition exactly and find numerically that the self-consistent edge varies smoothly across $\gamma=1$, with no evidence of a kink or discontinuity.

W-shaped smiles appear in near-expiry options around binary events such as earnings, and have been associated with bimodal risk-neutral densities. The three-parameter eSSVI slice cannot produce them. This paper defines WSVI, a parametric family for implied volatility that admits negative at-the-forward curvature and bimodal implied densities, and develops its static no-arbitrage structure. The construction factorizes total variance into a level and a dimensionless shape of normalized log-moneyness. The shape extends the per-slice eSSVI form with bounded one-sided basis terms, which add flexibility in the interior while leaving the leading-order wing behavior controlled by the affine and quadratic components. We characterize the family's exact domain and write the butterfly, vertical spread, and calendar conditions directly in shape coordinates.

Complex systems are often organised into hierarchies whose internal interactions are stronger or faster than interactions across levels. When markets are treated as genuinely multilevel systems it becomes natural to represent them as systems with hierarchical causality. Here we show that reflexivity can be formulated within such a discrete hierarchical causal system; but one in which a higher-level actor state restricts the lower-level transition kernels that remain admissible. Then event dynamics can be separated from calendar embeddings: a set-valued actor-conditioned correspondence can be used to define the admissible family of event kernels, while joint state and waiting-time laws can be used to determine compatible timing to then natural demonstrate reflexivity. A selected event-state law need not determine a unique calendar embedding. Locally, uniqueness of the joint event or timing specification requires uniqueness of both the admissible event kernel and its compatible timing law. Reflexivity is thus the endogenous closure of a hierarchical constraint loop, while timing and projection can generate calendar-time memory or causal ambiguity even for Markov event dynamics.

Implied volatility surfaces summarise the option market and are central to many financial applications. Forecasting their future evolution requires modelling two-dimensional geometry, temporal dependence, and predictive uncertainty while preserving economic admissibility. We propose a conditional latent diffusion framework for generating joint 30-step trajectories of implied volatility surfaces and underlying returns. An arbitrage-aware autoencoder learns a low-dimensional surface representation, while the diffusion model captures the conditional joint evolution. Evaluated on SPX surfaces, the framework generates realistic probabilistic multi-step scenarios while also outperforming the persistence benchmark in point forecasting.

We study discrete-time asset pricing with bid-ask spreads and model uncertainty. The family of probability measures enters the no-arbitrage condition through the union of its supports. In the single-period setting, we establish fundamental theorems of asset pricing with and without short-sale constraints. In the unconstrained market, no arbitrage is equivalent to the existence of a full-support martingale consistent price system. Under short-sale constraints, the martingale condition is replaced by a supermartingale condition. We then extend these results to a finite multi-period tree. The initial information is allowed to be nontrivial, so the initial trading cost and valuation bounds may depend on the initial state, and the corresponding inequalities are formulated conditionally. Finally, for a family of pricing measures, we introduce lower and upper robust supermartingale consistent price systems. We show that no arbitrage implies the existence of a lower system, while the existence of an upper system is sufficient for no arbitrage. A two-state example shows that the lower condition alone is not sufficient.

Option-implied rates are often treated as frictionless because completed boxes deliver riskless payoffs. I show that this interpretation requires option- and benchmark-side implementation wedges to offset. Using SPX and RUT options from 2012-2025, I find a 2-3 bp unannualized increase in the option funding basis when maturity first crosses December 31. The effect behaves as a fixed price wedge, strengthens in the mid-2010s, survives alternative contracts and benchmarks, appears independently in government-bond CIP, and is reproduced in an independently constructed option panel. Put-call parity can identify discount rates precisely without establishing their economic purity.

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.

Hedge ratios, factor models and diversified portfolios all rest on an estimate of which firms move together. That estimate is not stable: firms migrate between the groupings the market treats as coherent, and when enough migrate the organizing axes of the cross-section turn. We measure the rate of that turning as the mean squared sine of the principal angles between subdominant eigenspaces of consecutive twelve-month S&P 500 correlation matrices. A typical month rewrites a fifth of the structure and carries four-fifths forward. That rate is priced: it couples to the aggregate variance risk premium at t = 5.40, no level measure correlates above 0.32, and the implied-correlation surface spans at most 6.7 percent of it. Only the persistent component is priced - the premium compensates the pace of revision, not the distance traveled. The mechanism is prepayment: implied variance rises on impact, volatility follows two to three quarters later (simulated-null p < 0.03 at h = 1-9), and the premium converges as it arrives. Three pre-registered boundaries hold: no timing alpha, no crash protection, and a downside version inseparable from intensity. The premium is, in part, rent on exposure held over a map still being redrawn.

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.

We consider the question of the optimal timing of the sale of an asset with stochastic dynamics. Our analysis is based on the method of the distribution builder introduced by Sharpe, Goldstein and Blythe [SGB00] for the purpose of optimal portfolio selection. Instead of specifying a utility function or risk aversion coefficient, this tool directly elicits the target distribution of the investor. We show how the problem of an optimal asset sale is in this setting linked to the problem of finding a Skorokhod embedding of a distribution into a diffusion process. In the case where the asset process follows a geometric Brownian motion and a specific family of distributions is targeted, one can observe a risk-return tradeoff.

This paper develops a revolving-door model in which a regulator designs policy signals that affect a regulated firm's capital structure while holding an implicit career option to leave public service for the regulated firm or its advisers. The model has three linked mechanisms. First, regulatory signals alter the firm's pricing, leverage, and risk choices by changing the tradeoff between consumer surplus and firm profit. Second, because the regulator's outside payoff is convex in regulated-firm value, the career option rises with firm value, leverage, and signal-induced volatility. Third, when signal-induced volatility raises the option vega of firm equity, the same mechanism can increase tail risk and reduce distance to default. A proof-of-concept calibration illustrates leverage-vega warning regions and shows how a regulator-induced signal can move a firm from a go-leverage region into an early-warning or crash-risk region. The calibration includes maturity comparisons, sensitivity grids over compensation convexity and initial leverage, a bank-like parameter anchor, stochastic signal simulations, and a stopping-horizon diagnostic. Evidence from federal payroll data on post-employment restrictions supports the career-option primitive by showing that limits on revolving-door opportunities affect post-agency pay. The exercises also organize historical regulatory failures: Lehman Brothers as a regulatory-vacuum case, Silicon Valley Bank as a distorted-signal and weak-supervision case, and Signature Bank as a converted-signal case involving post-public-service incentives, lighter oversight, rapid growth, uninsured-deposit exposure, and tail risk. The model does not predict collapse dates; it identifies conditions under which regulatory forbearance, weak debt penalties, and career-option incentives can delay oversight until firms cross a leverage-vega warning threshold.

We propose a novel valuation framework for contingent convertible (CoCo) bonds based on the issuing bank's Common Equity Tier 1 (CET1) ratio, which is widely acknowledged as an indicator of a bank's solvency. Our approach develops a bivariate jump-diffusion model that captures the dynamic relationship linking the CET1 ratios, share prices, and CoCo bond prices, incorporating both continuous market movements and correlated jump risk. The model advances existing literature through three key innovations: (1) a hybrid mechanism for modeling regulatory discretion in trigger decisions, (2) a class of power conversion schemes that generalizes traditional approaches while maintaining analytical tractability, and (3) a method to overcome the temporal discrepancy between high-frequency market data and low-frequency regulatory reporting. We derive semi-closed form formulas for both write-down and equity-convertible CoCo bonds and validate our model through five case studies spanning from 2009 to 2023, including an in-depth analysis of the 2023 Credit Suisse collapse. The results demonstrate a significant improvement in pricing and hedging performance while highlighting the model's data-adaptive nature that enables short-term predictions.

We measure volatility roughness across asset classes using a common data infrastructure and pipeline. Our data covers 3,926 United States equities, 34 CME futures roots, rates, FX, and commodities, and options on 44 underlyings over 2010-2025. Realized volatility is rough everywhere. The class-median Hurst estimate ranges from $0.05$ (livestock) through $0.07-0.10$ (rates, FX, agriculture, energy, metals) to $0.13$ (single stocks) and $0.20$ (equity indices). The option-implied measure identifies $H$ only where the leverage effect produces a clean skew term structure. For the equity indices, implied estimates of $0.21-0.28$ are just above realized volatility $H$, while for rates and FX the ATM skew regression fails with an R-squared near zero even though realized volatility remains rough. We also show a mean-reversion contamination formula for the second-moment estimator of the roughness for the stationary fractional Ornstein-Uhlenbeck process. The local slope of the increment second moment deviates from $2H$ by $(1-H)\Gamma(2H+1)(\kappa\Delta)^{2-2H}$ for all $H\in(0,1)$. A correction framework for the second moment, when the log realized volatility measure has additive noise, raised the $H$ estimate slightly but nowhere near the Brownian diffusion framework. Finally, a failure taxonomy discusses where rough-volatility methods apply and where they fail, suggesting alternative paths to further explore the rough-volatility paradigm.

We study optimal investment for insurers managing participating (profit-sharing) contracts under probability distortion and probability benchmark (aspiration) constraints. The problem combines three theoretical complexities: (i) nonconcave effective utilities induced by embedded guarantees and surplus-sharing rules, (ii) probability weighting capturing behavioral aspects of long-horizon decisions, and (iii) aspiration-type constraints formalizing solvency requirements. Using quantile formulations and concavification techniques, we derive explicit closed-form solutions for optimal terminal wealth and trading strategies in both complete and incomplete Black-Scholes markets. Our utility class accommodates the piecewise hyperbolic absolute risk aversion (PHARA) family and covers nonconcavities arising naturally in insurance contexts. The framework reveals how probability distortion weakens lock-in behavior and induces time inconsistency: under inverse S-shaped distortions, insurers overestimate upside probabilities and increase risky investment relative to undistorted benchmarks. Asymptotic analysis and numerical illustrations demonstrate regime switches in optimal policies driven by regulatory thresholds and capital constraints. Our results extend the hope-fear-aspirations framework of He and Zhou (2016) and provide practical insights for managing insurance balance sheets under behavioral preferences and solvency constraints.

We study continuous-time multi-asset portfolio choice and consumption under smooth pointwise constraints, including state-dependent feasible sets. The method separates dynamic information acquisition from local constrained recovery. A pointwise-feasible neural actor generates reference rollouts; after training, its realized latent outputs are frozen and first- and second-order adjoints are harvested from a fixed-latent open-loop backpropagation-through-time graph. Feedback therefore generates the reference trajectory without restricting the adjoint formulation to Markov controls. Conditional on the harvested adjoints, deployment solves a local generalized Pontryagin-Hamiltonian problem: quadratic-affine portfolio blocks are recovered exactly by a quadratic program, while a log barrier approximates more general regular KKT branches. We establish local chart representations, an OL-BPTT-to-adjoint correspondence retaining orthogonal martingale residuals, and an end-to-end bound from reference value loss and numerical errors to recovered-policy and local QP-gap errors. Analytical constant- and predictable-opportunity benchmarks validate the adjoints. Common-input experiments show that recovery reduces residual PMP/KKT error left by finite-budget direct policy optimization, including under a state-dependent consumption cap and with up to 100 risky assets. Scalability concerns the constrained action block rather than dimension-free state-space complexity.

Suppose there are $N$ heterogeneous agents in a market with idiosyncratic risks but no uninsurable systematic risk factors. These agents may agree arbitrary financial contracts with one another, subject to the condition that contracts are self-enforcing under coalitions of agents in a common state. We show that, under mild conditions, this uniquely determines the limiting utility of every agent as $N$ tends to infinity. The result is an ergodic theorem: as the population grows, the number of degrees of freedom in the problem collapses, so that agents in the same state are treated identically in the limit. We exhibit an explicit, practically realisable mechanism achieving this limit using only short-dated contracts. The model can be applied either to an economy of heterogeneous agents pooling idiosyncratic risk through self-enforcing contracts or to the design of optimal insurance products such as pensions.

We study dynamic physical hedging for insurers exposed jointly to catastrophe losses and stochastic reconstruction costs. Surplus evolves as a controlled jump diffusion whose loss amplitude combines marked catastrophe severity, an exogenous mean-reverting cost factor, and endogenous mitigation. We establish well-posedness, moment and stability estimates, and a stopping-time dynamic programming principle, and prove that the value function is the unique viscosity solution of the resulting nonlocal Hamilton-Jacobi-Bellman (HJB) equation Strategic interaction is introduced through a mean field game (MFG) with reduced-form vulnerability costs, yielding a coupled backward-forward HJB-Kolmogorov system. We establish relaxed equilibrium existence, Markovian realization, and uniqueness under appropriate compactness and monotonicity conditions. Numerical experiments show that reconstruction costs and capitalization materially affect optimal hedging and that cross-sectional vulnerability alters equilibrium costs. Tail-family robustness calculations further assess the sensitivity of these conclusions to alternative catastrophe-severity specifications.

Narrow Uniswap v3 liquidity ranges resemble short dated options, and Panoptic's streaming premium echoes the short maturity concentration of Black-Scholes theta near the strike. This motivates a natural question: can implied volatility be extracted from Uniswap v3 and Panoptic using only on chain observables? A direct identification of theta with realized fee income is too strong, since fee income captures only the compensation leg of a narrow range LP position. The remaining leg, the cost of dynamically hedging the LP's negative convexity through arbitrage aligned trades, is formalized elsewhere as predictable loss or loss versus rebalancing and is not observable from fees alone. We therefore reformulate the object of interest. We derive $\sigma_{fee} = 2 \cdot feeRate \cdot \sqrt{Volume/L_{tick}}$, and interpret it as a DEX native fee implied volatility proxy: an observable, oracle free measure of fee flow relative to active liquidity. We show that recovering a structural latent volatility from pool observables alone is not identified in general, since aggregate volume mixes informed and uninformed flow while the missing hedging cost term depends on external price dynamics and arbitrage timing. What the pool data support directly is a fee implied activity index, not a Black-Scholes consistent implied volatility.

An investor may be optimistic about aggregate endowment growth at some times and pessimistic at others. The weight placed on her forecast in bond valuation can therefore vary across maturities. We study whether this maturity dependence disappears at the long end of the yield curve. In a two-investor Arrow–Debreu economy, physical extinction follows when total disagreement grows without bound. We find that bond valuation depends on net disagreement. If net disagreement has no limit, neither the long-forward measure nor the long bond exists. Bond yields can converge along the same belief path. The long yield alone therefore cannot reveal whether valuation weights and bond returns converge across maturities. We also obtain finite-maturity error bounds for bond prices and forward densities.

This article presents with DYSANOS the first generative market model for smooth SANOS option surfaces for all strikes and expiries which are free of static arbitrage. Our model is designed to generate entire paths of daily spot and option prices for years in the future. We present a robust and useful if somewhat simplistic baseline in the form of an AR(1) model. We discuss model setup, data pipeline, and training and investigate market reconstruction, stability, and tail behavior. We illustrate model performance on 891 Option Metrics IvyDB S&P Index surfaces from 2022-01-03 through to 2025-08-29. We also demonstrate how to construct numerically a risk-neutral density. As part of this we develop a new test for zero conditional means under a given measure. We show that for a set of 100,000 simulated paths a trading universe of 48 options and spot is numerically free of dynamic arbitrage.

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.

We examine the shapes attainable by the forward and yield curve in the Hull-White model with Svensson-parameterized initial yield curves. For Nelson-Siegel-parameterized initial yield curves, we provide a complete classification of all attainable shapes and partition the parameter space and the state space according to these shapes. Our analysis relies on the theory of Tchebycheff systems and the envelope method. For Bliss- and Svensson-parameterized initial yield curves, we derive lower and upper bounds on the number of attainable local extrema. We examine the asymptotic behavior of the model and demonstrate that the yield curve attains only the shapes normal, inverse and humped with positive probability as $t\rightarrow\infty.$

We calibrate credit default swaps and index tranches with elastically stopped Lévy processes: each firm defaults when the running supremum of a latent, spectrally positive distress process crosses an independent exponential barrier. This yields a Cox construction with totally inaccessible default times, while retaining the interpretability and explicit formulas of a structural approach. Adding a single common compound Poisson jump factor to every firm's latent driver gives a parsimonious multi-credit model with simultaneous defaults, which is priced by an exact Wiener–Hopf Monte Carlo scheme. Its tractability rests on a single-name result we prove: a finite partial-fraction formula for the Laplace transform of the default probability under phase-type jumps. On daily CDX North American High-Yield and Investment-Grade panels, our drivers attain the lowest out-of-sample errors in a six-model field and reproduce the inverted spread curves of names heading into default, which a Lévy subordinator provably does not. At the index level, the two-parameter dependence structure closes $73\%$ to $89\%$ of the tranche pricing gap left by independent marginals with the dependence parameters frozen, and up to $95\%$ once re-marked to tranche quotes; our framework dominates a single-factor Gaussian copula and the affine intensity benchmark of Duffie–Gârleanu on both indices.

The integration of weather-dependent renewable generation increases the volatility of residual demand and raises the value of dispatchable low-carbon flexibility. This paper studies the optimal operation of a load-following nuclear power plant owned by a producer that must balance stochastic residual demand while accounting for ramping limits and costly changes in operating regimes. Nuclear output can be increased, decreased, or kept constant, and the production decision is formulated as a finite-horizon optimal switching problem. We analyze both a closed-economy benchmark, where excess production cannot be sold and shortages require costly back-up generation, and an open-economy setting, where the producer can trade electricity at prices driven by aggregate market residual demand. The value functions are characterized as viscosity solutions of a system of Hamilton-Jacobi-Bellman quasi-variational inequalities, and optimal policies are computed using a monotone semi-Lagrangian scheme. The numerical results show how shortage costs, switching costs, ramping capability, and market access shape optimal nuclear load following. The analysis highlights the economic value of controllable low-carbon capacity in renewable-intensive systems and provides implications for flexibility remuneration, balancing-market design, and interconnection policy.