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.     

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.     

A martingale representation theorem and valuation of defaultable securities

We consider a financial framework with two levels of information: the public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. As the random time cannot be seen before its occurrence, the progressive enlargement of filtration seems tailor‐fit to model the larger flow of information that incorporates both the public flow and the information about the random time. In this context, our interest focuses on the following challenges: (a) How to single out the various risks coming from the financial assets, the random time, and their correlations? (b) How these risks interplay and lead to the formation of any risk in the larger flow of information? It is clear that understanding how risks build‐up and interact, when one enlarges the flow of information, is vital for an efficient risk management and derivatives' evaluation in those informational markets. Our answers to these challenges are full and complete no matter what the model for the random time is and no matter how the random time is related to the public flow. In fact, we introduce “pure default” risks, and quantify and classify these risks afterward. Then we elaborate our martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into orthogonal local martingales (i.e., local martingales whose product remains a local martingale). This constitutes our first principal contribution, while our second contribution consists in evaluating various defaultable securities according to the recovery policy, within our financial setting that encompasses any default model, using a martingale “basis.” Our pricing formulas explain the impact of various recovery policies on securities and determine the types of pure default risk they entail.     

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.     

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.     

Implications of marketing capability and research and development intensity on firm default risk

Marketing capability and research and development intensity are firm resources used to increase firm performance and reduce investor risk. This study aims to link marketing capability and research and development intensity, and their interaction to firm default risk. This study is the first to examine marketing capability and research and development intensity regarding their influences on firm default vulnerability and to demonstrate how marketing capability may strengthen research and development intensities’ power on risk reduction. The results reveal a U-shaped relationship between research and development intensity and firm default risk, while marketing capability’s impact is unidirectional. Further, marketing capability strongly moderates the relationship between research and development intensity and firm default risk. For low marketing capability firms, the U-shaped pattern is more significant. For high marketing capability firms, the pattern is not salient and the risk reduction power of research and development intensity is stronger. This research provides useful implications for marketing theories, as well as business practice.     

Valuing Seller‐Defaultable Options

This study analyzes seller‐defaultable options that allow option writers to have a free‐will right to default, along with some prespecified default mechanisms. We analytically and numerically examine the pricing, hedging, defaulting, and profitability of the seller‐defaultable options, considering three possible scenarios for seller default. Analyzing the essential implications of seller‐defaultable options, we show that the option price is positively correlated with the default fine, underlying asset price, and volatility. The seller‐defaultable option's Greeks appear more complicated than those of the plain vanilla options. The likelihood of sellers defaulting increases with the underlying asset price, interest rate, volatility, and maturity time. Subject to the default mechanism, the buyers’ trading involves a trade‐off between profits and costs. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:129–157, 2013     

Corporate security prices in structural credit risk models with incomplete information

The paper studies derivative asset analysis in structural credit risk models where the asset value of the firm is not fully observable. It is shown that in order to determine the price dynamics of traded securities, one needs to solve a stochastic filtering problem for the asset value. We transform this problem to a filtering problem for a stopped diffusion process and apply results from the filtering literature to this problem. In this way, we obtain an stochastic partial differential equation characterization for the filter density. Moreover, we characterize the default intensity under incomplete information and determine the price dynamics of traded securities. Armed with these results, we study derivative assets in our setup: We explain how the model can be applied to the pricing of options on traded assets and we discuss dynamic hedging and model calibration. The paper closes with a small simulation study.     

DEFAULTABLE OPTIONS IN A MARKOVIAN INTENSITY MODEL OF CREDIT RISK

This paper is a follow‐up to “Valuation and Hedging of Defaultable Game Options in a Hazard Process Model” by the same authors. In the present paper we give user friendly assumptions ensuring that the general conditions in the previous paper are satisfied. We also give a systematic procedure to construct suitable intensity models of credit risk, and, in the Markovian case, we provide a variational inequality approach to the pre‐default pricing problem. We finally illustrate our results on a study of defaultable convertible bonds.     

LOCAL RISK MINIMIZATION FOR DEFAULTABLE MARKETS

We study the local risk minimization approach for defaultable markets in a general setting where the asset price dynamics and the default time may influence each other. We find the Föllmer-Schweizer decomposition in this general setting and compute it explicitly in two particular cases, when default time depends on the risky asset's behavior and when only a dependence of discounted asset price on default time is occurring.     

资本资产定价模型的扩展及其在保险中的应用

传统的资本资产定价模型是在一系列过于严格化、理想化的条件下建立起来的。针对现实资本市场情况,通过对资本资产定价模型的应用条件的部分修改,如增加保险公司存在违约风险、交易费用和税收的条件,并且讨论交易费用分别为固定值和保费的函数时的情形以及税收分为固定值和变量的情形,对保费定价问题进行模型扩展。理论推导结果显示,在存在违约风险情况下,保险公司所收保费应该更低;承保费用越少,所需保费就越少;存在税负条件下的公平保费与税收水平有关。     

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.     

MODELING THE RECOVERY RATE IN A REDUCED FORM MODEL

This paper provides a model for the recovery rate process in a reduced form model. After default, a firm continues to operate, and the recovery rate is determined by the value of the firm's assets relative to its liabilities. The debt recovers a different magnitude depending upon whether or not the firm enters insolvency and bankruptcy. Although this recovery rate process is similar to that used in a structural model, the reduced form approach is maintained by utilizing information reduction in the sense of Guo, Jarrow, and Zeng. Our model is able to provide analytic expressions for a firm's default intensity, bankruptcy intensity, and zero-coupon bond prices both before and after default.     

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.     

The effect of default options on choice—Evidence from online product configurators

Many firms use product configurators to enable customers to specify their desired products online. In such systems, defaults are pre-specified for levels of product features by the manufacturer or dealer. For example, when configuring a racing bike online, a default is predefined (e.g., the Shimano Ultegra model) for all required features (e.g., the gearshift levers). Such defaults, which may even adapt to previous choices, ensure that a functional and fully defined product emerges at the end of the configuration process. However, when designing sales systems, companies often fail to realize that these defaults also affect customer decision-making. We demonstrate the effect by a study that makes use of a fully simulated racing bike configurator. We find the following results: Moving the default of one feature (e.g. wheels) from the lowest to the highest level results in an increase in sales. In addition, the feature level defined as the default also acts as a reference point by increasing the sales of levels near to it. In order to maximize sales, the default should be set at the level of a feature that is between the medium and the highest price level. To conclude we discuss how manufacturers and dealers subtly yet powerfully influence the decision-making process with their sales systems.     

The determinants of households’ repayment difficulties on mortgage loans: evidence from Italian microdata

This article investigates the main determinants of households’ repayment difficulties on mortgage loans in Italy. We contribute to the empirical literature on household financial vulnerability by assessing the joint impact of socio‐demographic factors, loan characteristics and institutional variables on the likelihood of mortgage insolvency and on the intensity of arrears. Using data from the Italian component of the 2008 European Union‐Statistics on Income and Living Conditions (EU‐SILC) survey, we firstly identify which types of households are more vulnerable to unexpected adverse events that may trigger repayment difficulties. Specifically, households whose head is young, unemployed or immigrant show a higher probability of arrears and emerge as those suffering more from the adverse economic conditions connected to the crisis. Moreover, household repayment behaviour is affected by mortgage characteristics and, in particular, having modified contract terms significantly increases current arrears probability. Finally, regional institutional and credit market factors mainly impact on the conditional intensity of arrears. This evidence suggests that, although repayment difficulties mainly arise from a genuine inability to repay, households are less likely to pay on time when institutions are less effective at punishing default, confirming the existence of some strategic behaviour.     

Defaults in bank loans to SMEs during the financial crisis

We investigate the role of (business) collateral and (personal) guarantees alongside small and medium enterprise (SME), lending bank and loan characteristics, macroeconomic conditions, sectors, and geographic locations while controlling for unobserved time effects in predicting default at the peak of the financial crisis. First, we find a positive relation between collateral and default, and a negative relation between guarantees and default. Second, we find a negative relation between the joint influence of collateral and high credit score, and a positive relation between the joint influence of collateral and low credit score and default. We also find a negative relation between the joint influence of guarantees and high credit score. These findings are relevant for SME policies aimed at facilitating access to credit, reducing the cost of borrowing, and decreasing default; risk management of banks; and the application of theories of financial economics in the context of a financial crisis.     

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.     

A simple method for extracting the probability of default from American put option prices

We present a novel method for extracting the risk-neutral probability of default (PD) of a firm from American put option prices. Building on the idea of a default corridor proposed by Carr and Wu, we derive a parsimonious closed-form formula for American put option prices from which the PD can be inferred. The method is easy to implement. Our empirical results based on seven large US firms for the period 2002–2010 show that, in some cases, our option-implied PD can provide a more accurate estimate of default probability than the estimates implied from credit default swaps.     

RETHINKING DYNAMIC CAPITAL STRUCTURE MODELS WITH ROLL‐OVER DEBT

Dynamic capital structure models with roll‐over debt rely on widely accepted arguments that have never been formalized. This paper clarifies the literature and provides a rigorous formulation of the equity holders’ decision problem within a game theory framework. We spell out the linkage between default policies in a rational expectations equilibrium and optimal stopping theory. We prove that there exists a unique equilibrium in constant barrier strategies, which coincides with that derived in the literature. Furthermore, that equilibrium is the unique equilibrium when the firm loses all its value at default time. Whether the result holds when there is a recovery at default remains a conjecture.