OPTIMAL INVESTMENT IN CREDIT DERIVATIVES PORTFOLIO UNDER CONTAGION RISK

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.     

ARBITRAGE‐FREE BILATERAL COUNTERPARTY RISK VALUATION UNDER COLLATERALIZATION AND APPLICATION TO CREDIT DEFAULT SWAPS

We develop an arbitrage‐free valuation framework for bilateral counterparty risk, where collateral is included with possible rehypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e., the difference between the on‐default exposure and the predefault collateral account. We then specialize our analysis to credit default swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of rehypothecation, collateral margining frequency, and default correlation‐induced contagion is illustrated with numerical examples.     

TIME‐CHANGED MARKOV PROCESSES IN UNIFIED CREDIT‐EQUITY MODELING

This paper develops a novel class of hybrid credit‐equity models with state‐dependent jumps, local‐stochastic volatility, and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time‐changed Markov diffusion process with state‐dependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with state‐dependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has local‐stochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has state‐dependent jumps, local‐stochastic volatility, and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time‐changed process is expressed in closed form as a single integral in the complex plane. If the payoff is square integrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jump‐to‐default extended constant elasticity of variance model of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, local‐stochastic volatility, and default intensity. These models can be used to jointly price equity and credit derivatives.     

BOUNDING WRONG‐WAY RISK IN CVA CALCULATION

A credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk to estimate a counterparty's risk of default together with the market value of exposure to the counterparty at default. Wrong‐way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong‐way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, a linear program computes the worst‐case CVA. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure by constraining the deviation of the joint model from a baseline reference model. Measuring the deviation through relative entropy leads to a tractable convex optimization problem that can be solved through the iterative proportional fitting procedure. Here, too, we prove convergence of the resulting estimate of the penalized worst‐case CVA and the joint distribution that attains it. We consider extensions with additional constraints and illustrate the method with examples.     

RECOVERING PORTFOLIO DEFAULT INTENSITIES IMPLIED BY CDO QUOTES

We propose a stable nonparametric algorithm for the calibration of “top‐down” pricing models for portfolio credit derivatives: given a set of observations of market spreads for collateralized debt obligation (CDO) tranches, we construct a risk‐neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk‐neutral loss process “closest” to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton–Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct a default intensity process which leads to tranche spreads consistent with the observations. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the previous number of defaults, and offer quantitative evidence for contagion effects in the (risk‐neutral) loss process.     

Dividend policy and capital structure of a defaultable firm

Default risk significantly affects the corporate policies of a firm. We develop a model in which a limited liability entity subject to default at an exponential random time jointly sets its dividend policy and capital structure to maximize the expected lifetime utility from consumption of risk‐averse equity investors. We give a complete characterization of the solution to the singular stochastic control problem. The optimal policy involves paying dividends to keep the ratio of firm's equity value to investors' wealth below a critical threshold. Dividend payout acts as a precautionary channel to transfer wealth from the firm to investors for mitigation of losses in the event of default. Higher the default risk, more aggressively the firm leverages and pays dividends.     

Model risk in credit risk

We provide sharp analytical upper and lower bounds for value‐at‐risk (VaR) and sharp bounds for expected shortfall (ES) of portfolios of any dimension subject to default risk. To do so, the main methodological contribution of the paper consists in analytically finding the convex hull generators for the class of exchangeable Bernoulli variables with given mean and for the class of exchangeable Bernoulli variables with given mean and correlation in any dimension. Using these analytical results, we first describe all possible dependence structures for default, in the class of finite sequences of exchangeable Bernoulli random variables. We then measure how model risk affects VaR and ES.     

DEFAULT AND SYSTEMIC RISK IN EQUILIBRIUM

We develop a finite horizon continuous time market model, where risk‐averse investors maximize utility from terminal wealth by dynamically investing in a risk‐free money market account, a stock, and a defaultable bond, whose prices are determined via equilibrium. We analyze the endogenous interaction arising between the stock and the defaultable bond via the interplay between equilibrium behavior of investors, risk preferences and cyclicality properties of the default intensity. We find that the equilibrium price of the stock experiences a jump at default, despite that the default event has no causal impact on the underlying economic fundamentals. We characterize the direction of the jump in terms of a relation between investor preferences and the cyclicality properties of the default intensity. We conduct a similar analysis for the market price of risk and for the investor wealth process, and determine how heterogeneity of preferences affects the exposure to default carried by different investors.     

HAZARD PROCESSES AND MARTINGALE HAZARD PROCESSES

In this paper, we build a bridge between different reduced‐form approaches to pricing defaultable claims. In particular, we show how the well‐known formulas by Duffie, Schroder, and Skiadas and by Elliott, Jeanblanc, and Yor are related. Moreover, in the spirit of Collin Dufresne, Hugonnier, and Goldstein, we propose a simple pricing formula under an equivalent change of measure. Two processes will play a central role: the hazard process and the martingale hazard process attached to a default time. The crucial step is to understand the difference between them, which has been an open question in the literature so far. We show that pseudo‐stopping times appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when τ is an honest time, providing an explicit expression for the difference. Eventually we provide a solution to another open problem: we show that if τ is an arbitrary random (default) time such that its Azéma's supermartingale is continuous, then τ avoids stopping times.     

RATING BASED LÉVY LIBOR MODEL

In this paper, we consider modeling of credit risk within the Libor market models. We extend the classical definition of the default‐free forward Libor rate and develop the rating based Libor market model to cover defaultable bonds with credit ratings. As driving processes for the dynamics of the default‐free and the predefault term structure of Libor rates, time‐inhomogeneous Lévy processes are used. Credit migration is modeled by a conditional Markov chain, whose properties are preserved under different forward Libor measures. Conditions for absence of arbitrage in the model are derived and valuation formulae for some common credit derivatives in this setup are presented.     

Equilibrium with Default and Endogenous Collateral

We study a two‐period general equilibrium model with incomplete asset markets and default. We make collateral endogenous by allowing each seller of assets to fix the level of collateral. Sellers are required to provide collateral whose first‐period value, per unit of asset, exceeds the asset price by an arbitrarily small amount. Moreover, borrowers are also required to be fully covered by the purchase, in the first period, of state‐by‐state default insurance. These insurance contracts are offered by lenders. The insurance cost or revenue is a linear charge and plays the role of a spread penalizing borrowers who will incur in default and benefiting lenders who will suffer default. Under these assumptions, equilibrium always exists.     

The risk of betting on risk: Conditional variance and correlation of bank credit default swaps

Credit default swaps (CDS) have been used to speculate on the default risk of the reference entity. The risk of CDS can be measured by their second moments. We apply a Glosten, Jagannathan, and Runkle (GJR)-t model for the conditional variance and a Dynamic Conditional Correlation (DCC)-t model for the conditional correlation. Based on the CDS of six large US banks from 2002 to 2018, we find that CDS conditional variance is asymmetric and leptokurtic. A positive innovation actually increases CDS conditional variance more than a negative innovation does. CDS conditional correlations have stayed elevated since the financial crisis, in contrast to the decreasing stock conditional correlations.     

Institutional quality and sovereign credit default swap spreads

We examine how the quality of political, legal, and regulatory institutions impacts sovereign risk premia. An improvement in institutional quality significantly lowers a country's sovereign credit default swap (CDS) spread, even after controlling for domestic and global macroeconomic factors. The incremental effect of institutional quality may also be economically important in explaining the variations in the level of sovereign CDS spreads. The basic results are robust to alternative model specifications, samples, control variables, measures of institutional quality, estimation methods, and controls for endogeneity. Overall, the evidence suggests that institutional quality may play a significant role in explaining sovereign CDS spreads.     

Assessing credit quality from the equity market: can a structural approach forecast credit ratings?

We investigate the empirical performance of default probability prediction based on Merton's (1974) structural credit risk model. More specifically, we study if distance‐to‐default is a sufficient statistic for the equity market information concerning the credit quality of the debt‐issuing firm. We show that a simple reduced form model outperforms the Merton (1974) model for both in‐sample fitting and out‐of‐sample predictability for credit ratings, and that both can be greatly improved by including the firm's equity value as an additional variable. Moreover, the empirical performance of this hybrid model is very similar to that of the simple reduced form model. As a result, we conclude that distant‐to‐default alone does not adequately capture the firm's credit quality information from the equity market. Copyright © 2007 ASAC. Published by John Wiley & Sons, Ltd.     

MODELING SOVEREIGN RISKS: FROM A HYBRID MODEL TO THE GENERALIZED DENSITY APPROACH

Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model that combines an accessible part taking into account the evolution of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed‐form formulas for the probability that the default occurs at critical political dates in a Markovian setting. Moreover, we introduce a generalized density framework for the hybrid default time and deduce the compensator process of default. Finally, we apply the hybrid model and the generalized density to the valuation of sovereign bonds and explain the significant jumps in long‐term government bond yields during the sovereign crisis.     

CORRELATED DEFAULTS IN INTENSITY-BASED MODELS

This paper presents an intensity-based model of correlated defaults with application to the valuation of defaultable securities. The model assumes that the intensities of the default times are driven by common factors as well as other defaults in the system. A recursive procedure called the "total hazard construction" is used to generate default times with a broad class of correlation structures. This approach is compared to standard reduced-form models based on conditional independence as well as alternative approaches involving copula functions. Examples are given for the pricing of defaultable bonds and credit default swaps of the regular and basket type.     

EXPECTATIONS OF FUNCTIONS OF STOCHASTIC TIME WITH APPLICATION TO CREDIT RISK MODELING

We develop two novel approaches to solving for the Laplace transform of a time‐changed stochastic process. We discard the standard assumption that the background process () is Lévy. Maintaining the assumption that the business clock () and the background process are independent, we develop two different series solutions for the Laplace transform of the time‐changed process . In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the standard class of default intensity models of credit risk, and show that stochastic time‐change has a very large effect on the pricing of deep out‐of‐the‐money options on credit default swaps.     

Credit portfolio selection with decaying contagion intensities

We develop a fixed‐income portfolio framework capturing the exponential decay of contagious intensities between successive default events. We show that the value function of the control problem is the classical solution to a recursive system of second‐order uniformly parabolic Hamilton–Jacobi–Bellman partial differential equations. We analyze the interplay between risk premia, decay of default intensities, and their volatilities. Our comparative statics analysis finds that the investor chooses to go long only if he is capturing enough risk premia. If the default intensities deteriorate faster, the investor increases the size of his position if he goes short, or reduces the size of his position if he goes long.     

信贷资产证券化的违约风险分析

识别和控制资产证券化过程中的风险是金融监管的客观需要。以信贷资产支持证券为例探讨资产证券化的特有风险—违约风险,并运用KMV模型测度个案违约风险,在此基础上提出一些控制违约风险的策略。     

PRICING SWAPTIONS UNDER MULTIFACTOR GAUSSIAN HJM MODELS

Several approximations have been proposed in the literature for the pricing of European‐style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin‐Dufresne and Goldstein is able to characterize the order of the approximation error. Under a multifactor HJM Gaussian framework, this paper proposes a new approximation for European‐style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the conditioning approach initiated by Curran and Rogers and Shi. All the proposed pricing bounds will arise as a simple by‐product of the Nielsen and Sandmann setup, and will be shown to provide a better accuracy–efficiency trade‐off than all the approximations already proposed in the literature.