TRIVARIATE SUPPORT OF FLAT‐VOLATILITY FORWARD LIBOR RATES

This paper investigates the multivariate support of forward Libor rates in the one‐factor, constant volatilities Libor market model. The comparatively simple bivariate case was solved in Jamshidian (2008) in connection to the recent finding by Davis and Mataix‐Pastor (2007) of positive probability of negative Libor rates in the swap market model. The approach here builds on Jamshidian (2008) but becomes really effective only in the trivariate case, and there particularly for a special “flat‐volatility” case, leading to an analytic solution. The main idea is a certain recursion in the Libor market model by means of which the calculation of the support is reduced to a calculus of variation problem (with bounds on the slope).     

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.     

Multiple curve Lévy forward price model allowing for negative interest rates

In this paper, we develop a framework for discretely compounding interest rates that is based on the forward price process approach. This approach has a number of advantages, in particular in the current market environment. Compared to the classical as well as the Lévy Libor market model, it allows in a natural way for negative interest rates and has superb calibration properties even in the presence of extremely low rates. Moreover, the measure changes along the tenor structure are significantly simplified. These properties make it an excellent base for a postcrisis multiple curve setup. Two variants for multiple curve constructions based on the multiplicative spreads are discussed. Time‐inhomogeneous Lévy processes are used as driving processes. An explicit formula for the valuation of caps is derived using Fourier transform techniques. Relying on the valuation formula, we calibrate the two model variants to market data.     

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.     

Economic determinants of default risks and their impacts on credit derivative pricing

This study constructs a credit derivative pricing model using economic fundamentals to evaluate CDX indices and quantify the relationship between credit conditions and the economic environment. Instead of selecting specific economic variables, numerous economic and financial variables have been condensed into a few explanatory factors to summarize the noisy economic system. The impacts on default intensity processes are then examined based on no‐arbitrage pricing constraints. The approximated results show that economic factors indicated credit problems even before the recent subprime mortgage crisis, and economic fundamentals strongly influenced credit conditions. Testing of out‐of‐sample data shows that credit evolution can be identified by dynamic explanatory factors. Consequently, the factor‐based pricing model can either facilitate the evaluation of default probabilities or manage default risks more effectively by quantifying the relationship between economic environment and credit conditions. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Affine multiple yield curve models

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or Heath–Jarrow–Morton modeling, can be consolidated. We model a numéraire process and multiplicative spreads between Libor rates and simply compounded overnight indexed swap rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multicurve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed‐form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.     

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.     

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.     

Multiple Ratings Model of Defaultable Term Structure

A new approach to modeling credit risk, to valuation of defaultable debt and to pricing of credit derivatives is developed. Our approach, based on the Heath, Jarrow, and Morton (1992) methodology, uses the available information about the credit spreads combined with the available information about the recovery rates to model the intensities of credit migrations between various credit ratings classes. This results in a conditionally Markovian model of credit risk. We then combine our model of credit risk with a model of interest rate risk in order to derive an arbitrage‐free model of defaultable bonds. As expected, the market price processes of interest rate risk and credit risk provide a natural connection between the actual and the martingale probabilities.     

Estimation of physical intensity models for default risk

The estimation of physical intensity processes in the context of default risk is investigated here. Using data from Moody's Corporate Bond Default Database, a term structure of default probabilities for different rating classes is constructed each year from 1970 to 2001. Two specifications used for modeling the dynamics of the (risk‐neutral) intensity process in the bond‐pricing literature are then examined empirically: the Ornstein–Uhlenbeck and square‐root cases. The results reveal that the Ornstein–Uhlenbeck case is not an adequate modeling alternative with a rejection of this specification in five out of seven credit classes and nonsignificant mean reverting behavior for all credit classes. The square‐root case obtains better results with four credit classes out of seven for which this specification cannot be rejected and significant mean reversion parameters in many cases. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 29:95–113, 2009     

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.     

When and Why Usury Should be Prohibited

Usury ceilings seem indefensible. Their opponents insist these caps harm the consumers they are intended to help. Low ceilings are said to prevent the least advantaged agents from accessing legal credit and drive them into the black market, where prices are higher and collection methods are harsher. But in this paper, I challenge these arguments and show that the benefits of interest-rate limitations in the most expensive credit markets clearly outweigh the costs. The test case is payday lending. Deregulated pricing in this market produces negative externalities that justify usury restrictions. Unless prices are capped, the more solvent majority of borrowers is compelled to cross-subsidize the least solvent debtors, who have a high rate of default. Rationing the riskiest debtors out of this market by means of a moderate usury cap puts an end to this unfairness and produces fewer bad consequences than the advocates of deregulated pricing recognize. I argue that only an extreme principle like maximizing the minimum could justify a free market in payday credit.     

Empirical evidence on the dependence of credit default swaps and equity prices

We investigate the common practice of estimating the dependence structure between credit default swap prices on multi‐name credit instruments from the dependence structure of the equity returns of the underlying firms. We find convincing evidence that the practice is inappropriate for high‐yield instruments and that it may even be flawed for instruments containing only firms within a sector. To do this, we model individual credit ratings by univariate continuous time Markov chains, and their joint dynamics by copulas. The use of copulas allows us to incorporate our knowledge of the modeling of univariate processes, into a multivariate framework. However, our test and results are robust to the choice of copula. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:695–712, 2009     

BIVARIATE SUPPORT OF FORWARD LIBOR AND SWAP RATES

Based on a certain notion of "prolific process," we find an explicit expression for the bivariate (topological) support of the solution to a particular class of 2 × 2 stochastic differential equations that includes those of the three-period "lognormal" Libor and swap market models. This yields that in the lognormal swap market model (SMM), the support of the 1 × 1 forward Libor   L * _t   equals  [ l * _t , ∞)  for some semi-explicit  −1 ≤ l * _t ≤ 0  , sharpening a result of Davis and Mataix-Pastor (2007) that forward Libor rates (eventually) become negative with positive probability in the lognormal SMM. We classify the instances   l * _t < 0  , and explicitly calculate the threshold time at or before which   L * _t   remains positive a.s.     

Back to the future: Futures margins in a future credit default swap index futures market

The introduction of exchange‐traded credit default swap (CDS) index futures is eminent and this development in the credit market is the subject of this article. A theoretically appealing and practically implementable approach to computing accurate futures margins based on extreme value theory is suggested. The approach is then exemplified with a study of the increasingly popular iTraxx Europe CDS index market. Although this market is not organized through an exchange and is not a futures market, the empirical results together with an arbitrage argument nonetheless suggest margin levels in a future exchange‐traded CDS index futures market computed using extreme value theory to be superior to those computed using the traditional normal distribution or the actual historical distribution. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:85–104, 2007     

INCORPORATING RISK AND AMBIGUITY AVERSION INTO A HYBRID MODEL OF DEFAULT

It is well known that purely structural models of default cannot explain short‐term credit spreads, while purely intensity‐based models lead to completely unpredictable default events. Here we introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level. The distressed firm then defaults upon the next arrival of a Poisson process. To value defaultable bonds and credit default swaps (CDSs), we introduce the concept of robust indifference pricing. This paradigm incorporates both risk aversion and model uncertainty. In robust indifference pricing, the optimization problem is modified to include optimizing over a set of candidate measures, in addition to optimizing over trading strategies, subject to a measure dependent penalty. Using our model and valuation framework, we derive analytical solutions for bond yields and CDS spreads, and find that while ambiguity aversion plays a similar role to risk aversion, it also has distinct effects. In particular, ambiguity aversion allows for significant short‐term spreads.     

AN EXACT FORMULA FOR DEFAULT SWAPTIONS’ PRICING IN THE SSRJD STOCHASTIC INTENSITY MODEL

We develop and test a fast and accurate semi‐analytical formula for single‐name default swaptions in the context of a shifted square root jump diffusion (SSRJD) default intensity model. The model can be calibrated to the CDS term structure and a few default swaptions, to price and hedge other credit derivatives consistently. We show with numerical experiments that the model implies plausible volatility smiles.     

DYNAMIC DEFAULTABLE TERM STRUCTURE MODELING BEYOND THE INTENSITY PARADIGM

The two main approaches in credit risk are the structural approach pioneered by Merton and the reduced‐form framework proposed by Jarrow and Turnbull and by Artzner and Delbaen. The goal of this paper is to provide a unified view on both approaches. This is achieved by studying reduced‐form approaches under weak assumptions. In particular, we do not assume the global existence of a default intensity and allow default at fixed or predictable times, such as coupon payment dates, with positive probability. In this generalized framework, we study dynamic term structures prone to default risk following the forward‐rate approach proposed by Heath, Jarrow, and Morton. It turns out that previously considered models lead to arbitrage possibilities when default can happen at a predictable time. A suitable generalization of the forward‐rate approach contains an additional stochastic integral with atoms at predictable times and necessary and sufficient conditions for an appropriate no‐arbitrage condition are given. For efficient implementations, we develop a new class of affine models that do not satisfy the standard assumption of stochastic continuity. The chosen approach is intimately related to the theory of enlargement of filtrations, for which we provide an example by means of filtering theory where the Azéma supermartingale contains upward and downward jumps, both at predictable and totally inaccessible stopping times.     

Counter‐Credit‐Risk Yield Spreads: A Puzzle in China's Corporate Bond Market

In this paper, using China's risk‐free and corporate zero yields together with aggregate credit risk measures and various control variables from 2006 to 2013, we document a puzzle of counter‐credit‐risk corporate yield spreads. We interpret this puzzle as a symptom of the immaturity of China's credit bond market, which reveals a distorted pricing mechanism latent in the fundamental of this market. We also find interesting results about relationships between corporate yield spreads and interest rates and risk premia and the stock index, and these results are somewhat attributed to this puzzle.     

CREDIT SPREADS, OPTIMAL CAPITAL STRUCTURE, AND IMPLIED VOLATILITY WITH ENDOGENOUS DEFAULT AND JUMP RISK

We propose a two-sided jump model for credit risk by extending the Leland–Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) Jumps and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, this may not be true in presence of endogenous default and jumps. Pricing formulae of credit default swaps and equity default swaps are also given. In terms of mathematical contribution, we give a proof of a version of the "smooth fitting" principle under the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.