Papers

Recent advances in LLM agents enable a new paradigm for asset pricing, which we call Agentic Empirical Asset Pricing (AEAP): systems that autonomously conduct the scientific discovery process itself. We define AEAP and identify its core building blocks. Existing evaluation practices backtest only the outputs—factors or trades—not the autonomous discovery system that produced them. We focus on factor discovery, contributing a reference architecture, a rigorous evaluation standard for discovered factors, and a method for out-of-sample backtesting the discovery system. As a concrete instance of that architecture, we evaluate SEADS against five re-implemented baselines on two US equity panels using this standard: no single metric ranks the systems consistently, motivating evaluation on multiple axes at once. A separate rolling re-execution then asks the complementary question of whether the discovery process itself, not one static output, is reliable. We also report negative findings and limitations that surface further evaluation pitfalls for future AEAP systems.

We study online conformal prediction in a partially observed panel: a new cross-section of peer outcomes is observed before each target outcome, target feedback may be intermittent or absent, and neither units nor rounds need be exchangeable. We propose Weighted Temporal Quantile Adjustment (W-TQA), which combines similarity weights learned from unit histories with an adaptive target-specific miscoverage level. We prove that neither target–peer mismatch nor coverage on unrevealed rounds is identifiable, so assumptions on cross-unit similarity and on the feedback mechanism cannot be avoided. We bound the past-conditional miscoverage in terms of this mismatch, quantify the cost of learning the weights under a profile-similarity assumption, and show that the implemented procedure, which falls back on the largest peer score, attains long-run average coverage under missing-completely-at-random feedback and a feasibility condition on the calibration panel. On synthetic panels and four real panels, including a genuinely asynchronous U.S.-to-Tokyo equity panel, W-TQA attains the highest coverage on the worst-covered target units on every panel; similarity weighting contributes most when target feedback is scarce, and temporal adaptation more as feedback accumulates.

Many asset classes—including private equity, real estate, corporate, municipal, and mortgage bonds, as well as structured products—are illiquid, with prices that are observed only sporadically. Yet frequent, objective, and accurate price estimates are essential for portfolio valuation, investment decision-making, and regulatory compliance. This paper develops point and interval estimators for illiquid asset prices, offering both consistency guarantees and non-asymptotic coverage for the resulting intervals. The estimators are based on a scalable, assumption-light semiparametric conditional factor model for the quantiles of asset prices. Empirical results for mortgage-backed securities demonstrate the method’s effectiveness and represent the first treatment of illiquid MBS pricing in the literature, highlighting the bond and market characteristics that drive prices.

Using a mixture of advanced data analytics derived from the explosion of information from quant credit markets and machine learning techniques, this paper develops a systematic framework to predict ratings migration in corporate credit. By systematically identifying firms with elevated possibility of downgrade or upgrade, the model supports more proactive portfolio management and better capital allocation. These insights are especially critical for financial institution investors like insurers, who are highly sensitive to credit ratings of securities on their balance sheets. The ability to forecast rating transitions consistently and accurately will enhance insurer risk management and bottom-up credit selection, thereby ensuring a more stable and robust financial intermediation process.

We develop Monte Carlo estimators for the expectation of a function of a multivariate jump-diffusion with general state-dependent drift, diffusion and jump characteristics. We provide technical conditions on the coefficient and objective functions guaranteeing the unbiasedness and finite-moment properties of the estimators. Numerical results for a class of “parametrix” estimators illustrate the significant efficiency gains they can achieve relative to discretization-based and exact alternatives.

Risk management and other applications often require estimating the tail distribution of total default losses on a portfolio of credit-sensitive positions such as loans and corporate bonds. This paper develops, analyzes and tests an importance sampling estimator of large-loss probabilities. This is the first estimator that is implementable for any reduced-form model of correlated name-by-name event timing. Its construction relies on state-dependent intensity scaling rather than the usual exponential twisting arguments. Under technical conditions we prove the asymptotic optimality of the estimator. We also prove that the estimator provides asymptotically significant variance reduction even when optimality conditions are violated. Numerical results illustrate the performance of the estimator.

Machine learning is central to modern science, industry, and policy, yet its predictive power often comes at the cost of transparency: we rarely know which input features drive a model’s predictions. Without such understanding, researchers cannot draw reliable conclusions, practitioners cannot ensure fairness or accountability, and policymakers cannot trust or govern model-based decisions. Existing tools for assessing feature influence are limited; most lack statistical guarantees, and many require costly retraining or surrogate modeling, making them impractical for large modern models. We introduce AICO, a broadly applicable framework that turns model interpretability into an efficient statistical exercise. AICO tests whether each feature contributes to predictive performance by masking its information and measuring the resulting change. The method provides exact, finite-sample feature p-values and confidence intervals for feature importance through a simple, non-asymptotic hypothesis testing procedure. It requires no retraining, surrogate modeling, or distributional assumptions, making it feasible for large-scale algorithms. In both controlled experiments and real applications, from credit scoring to mortgage-behavior prediction, AICO identifies variables that contribute to model behavior, providing a scalable and statistically principled path toward transparent and trustworthy machine learning.

Over the past decade, blockchains and cryptocurrencies have taken a central stage in financial technology (FinTech) innovation. In 2020–2021, as the academic finance and management community began actively investigating this domain, we issued a call for papers for a special issue to encourage interdisciplinary research in this emerging area. This section of Management Science presents the first systematic collection of knowledge, both theoretical and empirical, focusing on blockchain economics, crypto assets, decentralized finance, and Web3 ecosystems. We describe the editorial protocol employed for this special issue (now included in this volume as a special section), summarize what we learn about the field, and introduce the 15 articles included in the special section. We also offer several observations to highlight foundational issues in the new field and to guide future research in this exciting new area at the intersection of technology and finance.

Stochastic point process models of event timing are common in many areas, including finance, insurance, and reliability. Monte Carlo simulation is often used to perform computations for these models. The standard sampling algorithm, which is based on a time-change argument, is widely applicable but generates biased simulation estimators. This article develops and analyzes a change of probability measure that can reduce or even eliminate the bias without restricting the scope of the algorithm. A result of independent interest offers new conditions that guarantee the existence of a broad class of point process martingales inducing changes of measure. Numerical results illustrate our approach.

We develop a pivotal test to assess the statistical significance of the feature variables in a single-layer feed-forward neural network regression model. We propose a gradient-based test statistic and study its asymptotics using non-parametric techniques. Under technical conditions, the limiting distribution is given by a mixture of chi-square distributions. The tests enable one to discern the impact of individual variables on the prediction of a neural network. The test statistic can be used to rank variables according to their influence. Simulation results illustrate the computational efficiency and the performance of the test. An empirical application to house price valuation highlights the behavior of the test using actual data.

We examine the behavior of mortgage borrowers over several economic cycles using an unprecedented dataset of origination and monthly performance records for over 120 million mortgages originated across the United States between 1995 and 2014. Our deep learning model of multi-period mortgage delinquency, foreclosure, and prepayment risk uncovers the highly nonlinear influence on borrower behavior of an exceptionally broad range of loan-specific and macroeconomic variables down to the zip-code level. In particular, most variables strongly interact. Prepayments involve the greatest nonlinear effects among all events. We demonstrate the significant implications of the nonlinearities for risk management, investment management, and mortgage-backed securities.

This paper formulates and analyzes a discretization scheme for a jump-diffusion process with general state-dependent drift, volatility, jump intensity, and jump size. The jump times of the process are constructed as time-changed Poisson arrival times, and the Euler method is used to generate the process between the jump epochs. Under conditions on the coefficient functions specifying the process, the scheme is proved to converge weakly with order one for functions with polynomial growth. The use of higher-order methods between the jumps does generally not improve the weak order of convergence. Numerical experiments illustrate the results.

We treat the parameter estimation problem for mean‐field models of large interacting financial systems such as the banking system and a pool of assets held by an institution or backing a security. We develop an asymptotic inference approach that addresses the scale and complexity of such systems. Harnessing the weak convergence results developed for mean‐field financial systems in the literature, we construct an approximate likelihood for large systems. The approximate likelihood has a conditionally Gaussian structure, enabling us to design an efficient numerical method for its evaluation. We provide a representation of the corresponding approximate estimator in terms of a weighted least‐squares estimator, and use it to analyze the large‐system and large‐sample behavior of the estimator. Numerical results for a mean‐field model of systemic financial risk highlight the efficiency and accuracy of our estimator.

This paper develops an unbiased Monte Carlo approximation to the transition density of a jump-diffusion process with state-dependent drift, volatility, jump intensity, and jump magnitude. The approximation is used to construct a likelihood estimator of the parameters of a jump-diffusion observed at fixed time intervals that need not be short. The estimator is asymptotically unbiased for any sample size. It has the same large-sample asymptotic properties as the true but uncomputable likelihood estimator. Numerical results illustrate its properties.

We study the sources of corporate default clustering in the United States. We reject the hypothesis that firms’ default times are correlated only because their conditional default rates depend on observable and latent systematic factors. By contrast, we find strong evidence that contagion, through which the default by one firm has a direct impact on the health of other firms, is a significant clustering source. The amount of clustering that cannot be explained by contagion and firms’ exposure to observable and latent systematic factors is insignificant. Our results have important implications for the pricing and management of correlated default risk.

Point processes are widely used in finance and economics to model the timing of defaults, market transactions, unemployment spells, births, and a range of other events. We develop and analyze likelihood estimators for the parameters of a marked point process and incompletely observed explanatory factors that influence the arrival intensity and mark distribution. We establish an approximation to the likelihood and analyze the convergence and large-sample properties of the associated estimators. Numerical results illustrate the behavior of our estimators.

Financial institutions, government-sponsored enterprises, and asset-backed security investors are often exposed to delinquency and prepayment risk from large numbers of loans. Examples include mortgages, credit cards, and auto, student, and business loans. Because of the size of the pools, the measurement of these exposures is computationally expensive. This paper develops and tests efficient numerical methods for the analysis of large pools of loans as well as asset-backed securities backed by such pools. For a broad class of statistical and machine learning models of loan-level risk in discrete time, we prove a law of large numbers and a central limit theorem for the pool-level risk. The asymptotics are then used to construct computationally efficient approximations for a large pool. The approximations aggregate the full loan-level dynamics, making it possible to take advantage of the detailed loan-level data often available in practice. We furthermore prove a convergence rate for the approximation and a uniform integrability result. The latter allows the approximation to be applied to a wide class of pool-level risk statistics. To demonstrate the effectiveness of our approach, we implement it on a data set of over 25 million mortgages. The results show the accuracy and speed of the approximation in comparison to brute-force simulation of a pool, for a variety of pools with different risk profiles.

We analyze the optimal execution problem of a portfolio manager trading multiple assets. In addition to the liquidity and risk of each individual asset, we consider cross asset interactions in these two dimensions, which substantially enriches the nature of the problem. Focusing on the market microstructure, we develop a tractable order book model to capture liquidity supply/demand dynamics in a multiasset setting, which allows us to formulate and solve the optimal portfolio execution problem. We find that cross asset risk and liquidity considerations are of critical importance in constructing the optimal execution policy. We show that even when the goal is to trade a single asset, its optimal execution may involve transitory trades in other assets. In general, optimally managing the risk of the portfolio during the execution process affects the time synchronization of trading in different assets. Moreover, links in the liquidity across assets lead to complex patterns in the optimal execution policy. In particular, we highlight cases where aggregate costs can be reduced by temporarily “overshooting” one’s target portfolio.

We consider the problem of optimally selecting a large portfolio of risky loans, such as mortgages, credit cards, auto loans, student loans, or business loans. Examples include loan portfolios held by financial institutions and fixed-income investors as well as pools of loans backing mortgage- and asset-backed securities. The size of these portfolios can range from the thousands to even hundreds of thousands. Optimal portfolio selection requires the solution of a high-dimensional nonlinear integer program and is extremely computationally challenging. For larger portfolios, this optimization problem is intractable. We propose an approximate optimization approach that yields an asymptotically optimal portfolio for a broad class of data-driven models of loan delinquency and prepayment. We prove that the asymptotically optimal portfolio converges to the optimal portfolio as the portfolio size grows large. Numerical case studies using actual loan data demonstrate its computational efficiency. The asymptotically optimal portfolio’s computational cost does not increase with the size of the portfolio. It is typically many orders of magnitude faster than nonlinear integer program solvers while also being highly accurate even for moderate-sized portfolios.

We provide a simple and easy to use goodness-of-fit test for the misspecification of the volatility function in diffusion models. The test uses power variations constructed as functionals of discretely observed diffusion processes. We introduce an orthogonality condition which stabilizes the limit law in the presence of parameter estimation and avoids the necessity for a bootstrap procedure that reduces performance and leads to complications associated with the structure of the diffusion process. The test has good finite sample performance as we demonstrate in numerical simulations.

We establish a central limit theorem and a large deviations principle for affine point processes, which are stochastic models of correlated event timing widely used in finance and economics. These limit results generate closed-form approximations to the distribution of an affine point process. They also facilitate the construction of an asymptotically optimal importance sampling estimator of tail probabilities. Numerical tests illustrate our results.

We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogenous portfolios. The density of the limiting measure is shown to solve a non-linear stochastic PDE, and certain moments of the limiting measure are shown to satisfy an infinite system of SDEs. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustrate the accuracy of the approximation, and highlight its computational advantages over a direct Monte Carlo simulation of the original stochastic system.

Using an extensive data set on corporate bond defaults in the US from 1866 to 2010, we study the macroeconomic effects of bond market crises and contrast them with those resulting from banking crises. During the past 150 years, the US has experienced many severe corporate default crises in which 20–50% of all corporate bonds defaulted. Although the total par amount of corporate bonds has at times rivaled the amount of bank loans outstanding, we find that corporate default crises have far fewer real effects than do banking crises. These results provide empirical support for current theories that emphasize the unique role that banks and the credit and collateral channels play in amplifying macroeconomic shocks.

We analyze the fluctuation of the loss from default around its large portfolio limit in a class of reduced-form models of correlated firm-by-firm default timing. We prove a weak convergence result for the fluctuation process and use it for developing a conditionally Gaussian approximation to the loss distribution. Numerical results illustrate the accuracy and computational efficiency of the approximation.

This paper formulates and solves the selection problem for a portfolio of credit swaps. The problem is cast as a goal program that entails a constrained optimization of preference-weighted moments of the portfolio value at the investment horizon. The portfolio value takes account of the exact timing of protection premium and default loss payments, as well as any mark-to-market profits and losses realized at the horizon. The constraints address collateral and solvency requirements, initial capital, position limits, and other trading constraints that credit swap investors often face in practice. The multimoment formulation accommodates the complex distribution of the portfolio value, which is a nested expectation under risk-neutral and actual probabilities. It also generates computational tractability. Numerical results illustrate the features of optimal portfolios. In particular, we find that credit swap investment constraints can have a significant impact on optimal portfolios, even for simple investment objectives. Our problem formulation and solution approach extend to corporate bond portfolios and mixed portfolios of corporate bonds and credit derivatives.

This paper develops a method for the exact simulation of a skeleton, a hitting time, and other functionals of a one-dimensional jump diffusion with state-dependent drift, volatility, jump intensity, and jump size. The method requires the drift function to be C¹, the volatility function to be C², and the jump intensity function to be locally bounded. No further structure is imposed on these functions. The method leads to unbiased simulation estimators of security prices, transition densities, hitting probabilities, and other quantities. Numerical results illustrate its features.

We develop a dynamic point process model of correlated default timing in a portfolio of firms, and analyze typical default profiles in the limit as the size of the pool grows. In our model, a firm defaults at a stochastic intensity that is influenced by an idiosyncratic risk process, a systematic risk process common to all firms, and past defaults. We prove a law of large numbers for the default rate in the pool, which describes the “typical” behavior of defaults.

This paper develops a formula for a transform of a vector point process with totally inaccessible arrivals. The transform is expressed in terms of a Laplace transform under an equivalent probability measure of the point process compensator. The Laplace transform of the compensator can be calculated explicitly for a wide range of model specifications, because it is analogous to the value of a simple security. The transform formula extends the computational tractability offered by extant security pricing models to a point process and its applications, which include valuation and risk management problems arising in single‐name and portfolio credit risk.

We provide a sequential Monte Carlo method for estimating rare-event probabilities in dynamic, intensity-based point process models of portfolio credit risk. The method is based on a change of measure and involves a resampling mechanism. We propose resampling weights that lead, under technical conditions, to a logarithmically efficient simulation estimator of the probability of large portfolio losses. A numerical analysis illustrates the features of the method and contrasts it with other rare-event schemes recently developed for portfolio credit risk, including an interacting particle scheme and an importance sampling scheme.

Dynamic, intensity-based point process models are widely used to measure and price the correlated default risk in portfolios of credit-sensitive assets such as loans and corporate bonds. Monte Carlo simulation is an important tool for performing computations in these models. This paper develops, analyzes, and evaluates two simulation algorithms for intensity-based point process models. The algorithms extend the conventional thinning scheme to the case where the event intensity is unbounded, a feature common to many standard model formulations. Numerical results illustrate the performance of the algorithms for a familiar top-down model and a novel bottom-up model of correlated default risk.

Point processes with stochastic arrival intensities are ubiquitous in many areas, including finance, insurance, reliability, health care, and queuing. They can be simulated from a Poisson process by time scaling with the cumulative intensity. The paths of the cumulative intensity are often generated with a discretization method. However, discretization introduces bias into the simulation results. The magnitude of the bias is difficult to quantify. This paper develops a sampling method that eliminates the need to discretize the cumulative intensity. The method is based on a projection argument and leads to unbiased simulation estimators. It is exemplified for a point process whose intensity is a function of a jump-diffusion process and the point process itself. In this setting, the method facilitates the exact sampling of both the point process and the driving jump-diffusion process. Numerical experiments demonstrate the effectiveness of the method.

We study corporate bond default rates using an extensive new data set spanning the 1866–2008 period. We find that the corporate bond market has repeatedly suffered clustered default events much worse than those experienced during the Great Depression. For example, during the railroad crisis of 1873–1875, total defaults amounted to 36% of the par value of the entire corporate bond market. Using a regime-switching model, we examine the extent to which default rates can be forecast by financial and macroeconomic variables. We find that stock returns, stock return volatility, and changes in GDP are strong predictors of default rates. Surprisingly, however, credit spreads are not. Over the long term, credit spreads are roughly twice as large as default losses, resulting in an average credit risk premium of about 80 basis points. We also find that credit spreads do not adjust in response to realized default rates.

This paper develops dynamic measures of the systemic risk of the financial sector as a whole. It defines systemic risk as the conditional probability of failure of a sufficiently large fraction of the total population of financial institutions. This definition recognizes that the cause of systemic distress is the correlated failure of institutions to meet obligations to creditors, customers, and trading partners. The likelihood estimators of the failure probability are based on a dynamic hazard model of correlated failure timing that captures the influence on failure timing of time-varying macroeconomic and sector-specific risk factors, and of spillover effects. Tests indicate that our measures provide accurate out-of-sample forecasts of the term structure of systemic risk in the United States for the period from 1998 to 2009.

A multiname credit derivative is a security that is tied to an underlying portfolio of corporate bonds and has payoffs that depend on the loss due to default in the portfolio. The value of a multiname derivative depends on the distribution of portfolio loss at multiple horizons. Intensity-based models of the loss point process that are specified without reference to the portfolio constituents determine this distribution in terms of few economically meaningful parameters and lead to computationally tractable derivatives valuation problems. However, these models are silent about the portfolio constituent risks. They cannot be used to address applications that are based on the relationship between portfolio and component risks, for example, constituent risk hedging. This paper develops a method that extends these models to the constituents. We use random thinning to decompose the portfolio intensity into a sum of constituent intensities. We show that a thinning process, which allocates the portfolio intensity to constituents, uniquely exists, and is a probabilistic model for the next-to-default. We derive a formula for the constituent default probability in terms of the thinning process and the portfolio intensity, and develop a semi-analytical transform approach to evaluate it. The formula leads to a calibration scheme for the thinning processes and an estimation scheme for constituent hedge sensitivities. An empirical analysis for September 2008 shows that the constituent hedges generated by our method outperform the hedges prescribed by the Gaussian copula model, which is widely used in practice.

Using data on corporate default experience in the U.S. and market rates of CDX index and tranche swaps of various maturities, we estimate reduced-form models of correlated default timing in the CDX High Yield and Investment Grade portfolios under actual and risk-neutral probabilities. The striking contrast between the estimated processes followed by the actual and risk-neutral arrival intensities of defaults, and between the parameters governing the actual and risk-neutral dynamics of the risk-neutral intensities, indicates the presence of substantial default risk premia in CDX swap market rates. The effects of risk premia on swap rates covary strongly across maturities, and depend on general stock market volatility and several measures of credit spreads.

Collateralized debt obligations, which are securities with payoffs that are tied to the cash flows in a portfolio of defaultable assets such as corporate bonds, play a significant role in the financial crisis that has spread throughout the world. Insufficient capital provisioning due to flawed and overly optimistic risk assessments is at the center of the problem. This paper develops stochastic methods to measure the risk of positions in collateralized debt obligations and related instruments tied to an underlying portfolio of defaultable assets. It proposes an adaptive point process model of portfolio default timing, a maximum likelihood method for estimating point process models that is based on an acceptance/rejection resampling scheme, and statistical tests for model validation. To illustrate these tools, they are used to estimate the distribution of the profit or loss generated by positions in multiple tranches of a collateralized debt obligation that references the CDX High Yield portfolio and the risk capital required to support these positions.

Correlated default risk plays a significant role in financial markets. Dynamic intensity-based models, in which a firm default is governed by a stochastic intensity process, are widely used to model correlated default risk. The computations in these models can be performed by Monte Carlo simulation. The standard simulation method, which requires the discretization of the intensity process, leads to biased simulation estimators. The magnitude of the bias is often hard to quantify. This paper develops an exact simulation method for intensity-based models that leads to unbiased estimators of credit portfolio loss distributions, risk measures, and derivatives prices. In a first step, we construct a Markov chain that matches the marginal distribution of the point process describing the binary default state of each firm. This construction reduces the original estimation problem to one involving a Markov chain expectation. In a second step, we estimate the Markov chain expectation using a simple acceptance/rejection scheme that facilitates exact sampling. To address rare event situations, the acceptance/rejection scheme is embedded in an overarching selection/mutation scheme, in which a selection mechanism adaptively forces the chain into the regime of interest. Numerical experiments demonstrate the effectiveness of the method for a self-exciting model of correlated default risk.

This paper analyzes a family of multivariate point process models of correlated event timing whose arrival intensity is driven by an affine jump diffusion. The components of an affine point process are self- and cross-exciting and facilitate the description of complex event dependence structures. ODEs characterize the transform of an affine point process and the probability distribution of an integer-valued affine point process. The moments of an affine point process take a closed form. This guarantees a high degree of computational tractability in applications. We illustrate this in the context of portfolio credit risk, where the correlation of corporate defaults is the main issue. We consider the valuation of securities exposed to correlated default risk and demonstrate the significance of our results through market calibration experiments. We show that a simple model variant can capture the default clustering implied by index and tranche market prices during September 2008, a month that witnessed significant volatility.

A credit investor such as a bank granting loans to firms or an asset manager buying corporate bonds is exposed to correlated corporate default risk. A multiname credit derivative is a financial security that allows the investor to transfer this risk to the credit market. In this paper, we study the valuation and risk analysis of multiname derivatives. To capture the complex economic phenomena that drive the pricing of these securities, we introduce a time-changed birth process as a probabilistic model of correlated event timing. The self-exciting property of a time-changed birth process captures the feedback from events that is often observed in credit markets. The stochastic variation of arrival rates between events captures the exposure of firms to common economic risk factors. We derive a closed-form expression for the distribution of a time-changed birth process, and develop analytically tractable pricing relations for a range of multiname derivatives valuation problems. We illustrate our results by calibrating a tranche forward and option pricer to market rates of index and tranche swaps.

In a traditional structural model of default it is implicitly assumed that the information used to calibrate and run the model is publicly available. In reality, model inputs and parameters are unobservable. In this article we analyze the role of information in structural models, which we specify through a model definition of the default time and a model filtration. The model definition relates the default of a firm to its assets and liabilities. The model filtration describes the information of investors relative to the model definition. It parameterizes a family of default models for a given default time. An important situation is when the default is not observable with respect to the model filtration. Examples include models with incomplete information about firm assets and models with incomplete information about the liability-dependent barrier that triggers default. Here the default time is typically totally inaccessible, as in the intensity-based, reduced-form models of default. In this case the model admits generalized reduced-form security pricing formulae in terms of the trend, which is the cumulative intensity. The trend can be explicitly characterized through the conditional default probability given the model filtration. If the trend is absolutely continuous with respect to the Lebesgue measure, then its density is the intensity and our formulae simplify to the classical intensity-based formulae. Not every incomplete information model admits an intensity. A model in which investors cannot observe the default barrier is a first example.

Credit contagion refers to the propagation of economic distress from one firm to another. This article proposes a reduced-form model for these contagion phenomena, assuming they are due to the local interaction of firms in a business partner network. We study aggregate credit losses on large portfolios of financial positions contracted with firms subject to credit contagion. In particular, we provide an explicit Gaussian approximation of the distribution of portfolio losses. This enables us to quantify the relation between the volatility of losses and the determinants of credit contagion. We find that contagion processes have typically a second order effect on portfolio losses. They induce additional fluctuations of losses around their averages, whose size depends on the denseness of the business partner network.

We model aggregate credit losses on large portfolios of financial positions contracted with firms subject to both cyclical default correlation and direct default contagion processes. Cyclical correlation is due to the dependence of firms on common economic factors. Credit contagion phenomena are associated with the local interaction of firms with their business partners. We provide an explicit normal approximation of the distribution of total portfolio losses. We quantify the relation between the variability of global economic fundamentals, strength of local firm interaction, and the fluctuation of losses. We find that cyclical oscillations in fundamentals dominate average portfolio losses, while local interaction cause additional fluctuations of losses around their average. The strength of the contagion-induced loss variability and hence the degree of extreme loss risk depends on the complexity of the business partner network, a relation that was recently confirmed by empirical studies.

We propose a multi-firm first-passage credit model in which investors have incomplete information. In this model, investors observe neither a firm's value nor its default barrier. The model accounts for the short term risk inherent in default events, the market-wide impact of defaults on security prices due to counterparty relations among firms, and the cyclical default correlation effects observed in credit markets. We explicitly calculate the pricing trend and the arrival intensity of the kth-to-default. These results furnish (1) tractable reduced form formulae for arrival probabilities of sequential dependent defaults and prices of multi-name credit derivatives and (2) an algorithm for the simulation of sequential unpredictable default times.

The recent accounting scandals at Enron, WorldCom, and Tyco were related to the (intentional) non-disclosure of a part of the firm's liabilities. We provide a structural model of correlated multi-firm default, in which public bond investors are uncertain about the liability-dependent barrier at which individual firms default. In lack of complete information, investors form prior beliefs on the barriers, which they update with the default status information of firms arriving over time. Whenever a firm suddenly defaults, investors learn about the default threshold of closely associated business partner firms. Due to the unpredictable nature of defaults in our model, this updating leads to "contagious" jumps in credit spreads of business partner firms, which correspond to re-assessments of these firms' health by investors. We characterize joint default probabilities and the default dependence structure as assessed by imperfectly informed investors, where we emphasize the modeling of dependence with copulas. A case study based on Brownian asset dynamics illustrates our results.

We give an empirical assessment of I², a structural credit model based on incomplete information. In this model, investors cannot observe a firm's default barrier. As a consequence, I² exhibits both the economic appeal of a structural model and the tractable pricing formulae and empirical plausibility of a reduced form model. We compare default probability and credit spread forecasts generated by I² and the well-known structural models of Merton (1974) and Black & Cox (1976). We find that I² reacts more quickly to new information and, unlike the other two models, it forecasts positive short term credit spreads.

Many prediction problems across science and engineering, especially in finance and economics, involve large cross-sections of individual time series, where each unit (e.g., a loan, stock, or customer) is driven by unit-level features and latent cross-sectional dynamics. While sequence models have advanced per-unit temporal prediction, capturing cross-sectional effects often still relies on hand-crafted summary features. We propose Set-Sequence, a model that learns cross-sectional structure directly, enhancing expressivity and eliminating manual feature engineering. At each time step, a permutation-invariant Set module summarizes the unit set; a Sequence module then models each unit’s dynamics conditioned on both its features and the learned summary. The architecture accommodates unaligned series, supports varying numbers of units at inference, integrates with standard sequence backbones (e.g., Transformers), and scales linearly in cross-sectional size. Across a synthetic contagion task and two large-scale real-world applications—equity portfolio optimization and loan risk prediction—Set-Sequence significantly outperforms strong baselines, delivering higher Sharpe ratios, improved AUCs, and interpretable cross-sectional summaries.

As high-quality public web corpora become increasingly exhausted, clean long-context documents have become a scarce and expensive source of training data for large language models (LLMs). Existing long-context corpora are often proprietary and costly to acquire, synthetically generated, or concentrated in narrow domains such as programming. We introduce the Stanford EDGAR Filings Dataset (SEFD), an open reconstruction of SEC filings into layout-faithful MultiMarkdown for financial language modeling and evaluation. SEFD makes audited financial statements, risk disclosures, ownership reports, accounting notes, and market-moving event filings usable as long-context pretraining data and as a basis for financial reasoning, forecasting, compliance, and document understanding. The resulting corpus is token-efficient, model-ready, and has less than 0.1% overlap with Common Crawl-derived corpora. We release SEFD-v1, a 152B-token initial public snapshot, and provide corpus-level analyses of a larger 18.5M-filing archive estimated at 550B tokens. We further introduce two SEFD-derived benchmarks: EDGAR-Forecast, which evaluates filing-grounded numerical forecasting after model knowledge cutoffs, and EDGAR-OCR, which evaluates transcription of complex financial tables.

We develop a simple and computationally efficient significance test for the features of a predictive model. Our forward-selection approach applies to any model specification, learning task and variable type. The test is non-asymptotic, straightforward to implement, and does not require model refitting. It identifies the statistically significant features as well as feature interactions of any order in a hierarchical manner, and generates a model-free notion of feature importance. This testing procedure can be used for model and variable selection. Experimental and empirical results illustrate its performance.

Many applications from the financial industry successfully leverage clustering algorithms to reveal meaningful patterns among a vast amount of unstructured financial data. However, these algorithms suffer from a lack of interpretability that is required both at a business and regulatory level. In order to overcome this issue, we propose a novel two-steps method to explain clusters. A classifier is first trained to predict the clusters labels, then the Single Feature Introduction Test (SFIT) method is run on the model to identify the statistically significant features that characterise each cluster. We describe a real wealth management compliance use-case that highlights the necessity of such an interpretable clustering method. We illustrate the performance of our method using simulated data and through an experiment on financial ratios of U.S. companies.

We develop and analyze Monte Carlo simulation estimators for path integrals of a multivariate diffusion with a general state-dependent drift and volatility. We prove that our estimators are unbiased and have finite variance by extending the regularity conditions of the parametrix method. The performance of our estimators is illustrated on numerical examples that highlight some applied problems for which our estimators apply.

We develop and analyze an unbiased Monte Carlo estimator for a functional of a one-dimensional jump-diffusion process with a state-dependent drift, volatility, jump intensity and jump size. The approach combines a change of measure to sample the jumps with the parametrix method to simulate the diffusions. Under regularity conditions on the coefficient functions as well as the functional, we prove the unbiasedness and the finite variance property of the estimator. Numerical experiments illustrate the performance of the scheme.

Although neural networks can achieve very high predictive performance on various different tasks such as image recognition or natural language processing, they are often considered as opaque "black boxes". The difficulty of interpreting the predictions of a neural network often prevents its use in fields where explainability is important, such as the financial industry where regulators and auditors often insist on this aspect. In this paper, we present a way to assess the relative input features importance of a neural network based on the sensitivity of the model output with respect to its input. This method has the advantage of being fast to compute, it can provide both global and local levels of explanations and is applicable for many types of neural network architectures. We illustrate the performance of this method on both synthetic and real data and compare it with other interpretation techniques. This method is implemented into an open-source Python package that allows its users to easily generate and visualize explanations for their neural networks.

We consider a continuous-time, inhomogeneous Markov chain M taking values in {0,1}ⁿ. Processes of this type arise in finance as models of correlated default timing in a portfolio of firms, in reliability as models of failure timing in a system of interdependent components, and in many other areas. We develop a logarithmically efficient importance sampling scheme for estimating the tail of the distribution of the total transition count of M at a fixed time horizon.

In this paper we study rare event simulation for the tail probability of an affine point process (Jₜ)ₜ≥₀ that generalizes the Hawkes process. By constructing a suitable exponential martingale, we are able to construct an importance sampling algorithm that is logarithmically efficient in the Gartner-Ellis asymptotic regime.

Point processes with stochastic intensities are ubiquitous in many application areas, including finance, insurance, reliability and queuing. They can be simulated from standard Poisson arrivals by time-scaling with the cumulative intensity, whose path is typically generated with a discretization method. However, discretization introduces bias into the simulation results. This paper proposes a method for the exact simulation of point processes with stochastic intensities. The method leads to unbiased estimators. It is illustrated for a point process whose intensity follows an affine jump-diffusion process.

A credit derivative is a path dependent contingent claim on the aggregate loss in a portfolio of credit sensitive securities. We estimate the value of a credit derivative by Monte Carlo simulation of the affine point process that models the loss. We consider two algorithms that exploit the direct specification of the loss process in terms of an intensity. One algorithm is based on the simulation of intensity paths. Here discretization introduces bias into the results. The other algorithm facilitates exact simulation of default times and generates an unbiased estimator of the derivative price. We implement the algorithms to value index and tranche swaps, and we calibrate the loss process to quotes on the CDX North America High Yield index.

The rise in the complexity and globalization of financial services has contributed to stronger interconnections or linkages. While more extensive linkages contribute to economic growth by smoothing credit allocation and allowing greater risk diversification, they also increase the potential for disruptions to spread swiftly across markets and borders. In addition, financial complexity has enabled risk transfers that were not fully recognized by financial regulators or by institutions themselves, complicating the assessment of counterparty risk, risk management, and policy responses. Thus the importance of assessing the systemic implications of financial linkages. This chapter illustrates the type of methodologies that can provide some prospective metrics to facilitate discussions on systemic linkages and, specifically, the “too-connected-to-fail” problem, thereby contributing to enhanced systemically focused surveillance and regulation.

Though these methodologies have limitations, together they represent a set of valuable surveillance tools and can form the basis for policies to address the too-connected-to-fail problem, specifically in two areas:

Perimeter of regulation. To maintain an effective perimeter of prudential regulation without stifling innovation, the tools provided in the chapter could help address questions such as whether to limit an institution’s exposures, the desirability of capital surcharges based on systemic linkages, and the merits of additional liquidity regulations.

Information gaps. The chapter also discusses the importance of filling existing information gaps on cross-market, cross-currency, and cross-country linkages to refine analyses of systemic linkages. Closing information gaps would require improved data collection procedures and impose additional demands on financial institutions, but would be a far better alternative to waiting until a crisis ensues to obtain information as events unfold.

Credit risk is the distribution of financial loss due to a broken financial agreement, for example failure to pay interest or principal on a loan or bond. It pervades virtually all financial transactions, and therefore plays a significant role in financial markets. A credit derivative is a security that allows investors to transfer credit risk to other investors who are willing to take it. By facilitating the allocation of risk, these instruments have an important economic function. Yet they have hit the headlines recently. This paper gives an overview of credit derivatives. It discusses the mechanics of standard contracts, describes their application, and outlines the mathematical challenges associated with their analysis.

Risk management is an important component of the investment process. It requires quantitative measures of risk that provide a metric for the comparison of financial positions. In this expository note we give an overview of risk measures. In particular, we contrast different risk measures with respect to their sensitivity to potentially large losses due to market wide shocks. The industry standard value at risk exhibits many deficiencies. It does not account for the size of the losses and may penalize diversification. We compare value at risk to alternative risk measures including average value at risk and the less well known but superior utility-based shortfall risk.

Dynamic reduced form models of portfolio credit risk can be distinguished by the way in which the intensity of the default process is specified. In a bottom up model, the portfolio intensity is an aggregate of the constituent intensities. In a top down model, the portfolio intensity is specified without reference to the constituents. This expository article contrasts these modeling approaches. It emphasizes the role of the information filtration as a modeling tool.

Credit risk is the distribution of financial losses due to unexpected changes in the credit quality of a counterparty in a financial agreement. We review the structural, reduced form and incomplete information approaches to estimating joint default probabilities and prices of credit sensitive securities.

Extreme value statistics provides a practical, flexible, mathematically elegant framework in which to develop financial risk management tools that are consistent with empirical data. In this introductory survey, we discuss some of the basic tools including power law distributions, the peaks over thresholds estimation procedure and point processes.

The Modigliani-Miller theorem describes conditions under which the value of a firm is independent of its leverage ratio. It is one of the cornerstones of finance. A history of this result along with a modern perspective on its derivation is given in Rubinstein (2003), Journal of Investment Management 1(2). We extend this history by examining the relationship between the Modigliani-Miller theorem and quantitative models of credit risk. In the first part of the paper, we sort out the role of the Modigliani-Miller theorem and Merton's classical structural model. This material may be familiar to some readers. Subsequently, we explore the relationship between the Modigliani-Miller theorem and I², which is a hybrid structural-reduced form model based on incomplete information, Goldberg (2004), Risk 17(1), 515-518. The I² model is not consistent with the Modigliani-Miller theorem. It provides a new way to measure the deviation of real markets from the idealized markets in which the Modigliani-Miller theorem holds.

A thorough understanding of the joint default behavior of credit-risky securities is essential for credit risk measurement as well as the valuation of multi-name credit derivatives and Collateralized Debt Obligations. In this paper we study a simple and tractable intensity-based model for correlated defaults, in which unpredictable default arrival times are jointly exponentially distributed. Since all relevant results are given in closed-form, the model can be easily implemented. The efficient simulation of dependent default times for pricing and risk management purposes is straightforward as well. Parameter calibration relies on market data as well as data and figures provided by rating agencies and credit risk management solutions.

Did private mortgage securitization contribute causally to the recent subprime lending boom, or was it simply a slightly more convenient way to finance lending that would have occurred anyway? We exploit a natural experiment in which the S&P rating agency increased costs of securitizing specific types of mortgages from certain states. We find significant reductions in total originations of affected mortgage types following the change, suggesting that securitization played a causal role in fueling mortgage expansion. We also, however, find suggestive evidence for an increase in mortgages held on books of financial institutions that partially offset the restriction of securitization.

Banks often seek to reduce the default risk exposure associated with their corporate loan portfolios by entering into credit derivative positions. They can, for example, buy default protection on selected borrowers, or diversify the portfolio by selling protection on other names. The design of suitable credit derivative positions and the estimation of risk capital reserves supporting a portfolio of loans and credit derivatives requires the computation of the probability distribution of the future portfolio value. The distribution of portfolio value takes a complicated form because it takes account of losses from default, interest income, protection and default risk premia, and the mark-to-market volatility of the derivative positions. This paper develops an analytical approximation for this distribution. The approximation is based on a small-time expansion of a transform of the portfolio value. It applies to many standard intensity-based models of firm-by-firm default timing. Numerical results illustrate the approximation for the value at risk of a portfolio of loans with hedging positions in credit swaps.

Risk-averse investors in credit sensitive securities such as equity and bonds require compensation for bearing exposure to non-diversifiable corporate default risk. One component of this compensation is an event premium for the abrupt changes in security prices that occur at default. While empirical research points to the significance of event premia in corporate bond and credit swap markets, the economic nature of the event premium is not fully understood. This paper uses a structural model of corporate default risk to show that informational asymmetries can induce an event premium. If public investors are unable to observe the threshold asset value at which firm management liquidates the firm, then they face instantaneous default risk as they cannot discern the firm's distance to default. Investors are taken by surprise when the firm reaches the default threshold, causing a sudden downward jump in the prices of securities issued by or referenced on the firm. The resulting event premium is governed by the degree of investors' aversion to the randomness in the location of the unobserved default threshold. Firm management has an incentive to improve the threshold transparency in order to reduce the credit premium required by investors, and therefore the cost to the firm of equity and debt financing.

Meyer (1971) showed that any point process whose compensator has continuous paths that increase to ∞ can be time-scaled to a standard Poisson process. In this article we consider the converse to this result. We construct a time change with continuous paths increasing to ∞ that transforms a standard Poisson process into a general point process with totally inaccessible arrivals and compensator given by the time change. The time change generates path-dependent or self-affecting point processes whose dynamics depend on the information generated by the arrivals of the process as well as other observable information describing the state of the random environment. The classical Hawkes and doubly stochastic processes are special cases. Time-changed Hawkes processes are shown to combine the best features of these classical families in a flexible and tractable way. We conclude by introducing time-change techniques to multi-name credit modeling. We describe the economy-wide default process as a time-changed Poisson process, whose arrivals are temporally clustered due to contagion and depend on the economic environment. Random thinning generates sub-models for individual firms and portfolios of firms.