Copulas and Dependence models in Credit Risk: Diffusions versus Jumps

The most common approach for default dependence modelling is at present copula functions. Within this framework, the paper examines factor copulas, which are the industry standard, together with their latest development, namely the incorporation of sudden jumps to default instead of a pure diffusive behavior. The impact of jumps on default dependence - through factor copulas - has not been fully explored yet. Our novel contribution consists in showing that modelling default arrival through a pure jump asset process does matter, even when the copula choice is thestandard, factor one, and the correlation is calibrated so as to match the diffusive and non diffusive case. An example from the credit derivative market is discussed.credit risk, correlated defaults, structural models, Lévy processes, copula functions, factor copula

Copula-Based Default Dependence Modelling: Where Do We Stand?

Copula functions have proven to be extremely useful in describing joint default and survival probabilities in credit risk applications. We overview the state of the art and point out some open modelling issues. We discuss first joint default modelling in diffusion based structural models, then in intensity based ones, focusing on the possibility - and the dynamic inconsistency - of re-mapping a model of the second type into one of the first. For both types of models, we discuss calibration issues under the risk neutral measure, using the factor copula device. The survey leads us to focus on a non-diffusive structural model, which can be re-mapped in a dynamic consistent intensity-based one, and which can be calibrated under a risk neutral measure without assuming equicorrelation.default dependence, copula functions, risk neutral versus historical dependence

Calibrating risk-neutral default correlation.

The implementation of credit risk models has largely relied on the use of historical default dependence, as proxied by the correlation of equity returns. However, as is well known, credit derivative pricing requires risk-neutral dependence. Using the copula methodology, we infer risk neutral dependence from CDS data. We also provide a market application and explore its impact on the fees of some credit derivatives.

Demographic risk transfer: is it worth for annuity providers? ICER Working Paper

Longevity risk transfer is a popular choice for annuity providers such as pension funds. This paper formalizes the trade-off between the cost and the risk relief of such choice, when the annuity provider uses value-at-risk to assess risk. Using first-order approximations we show that, if the transfer is fairly priced and the aim of the fund is to maximize returns, the funds' alternatives can be represented in the plane expected return-VaR. We build a risk-return frontier, along which the optimal transfer choices of the fund are located and calibrated it to the 2010 UK annuity and bond market

A Generalized Normal Mean Variance Mixture for Return Processes in Finance

Time-changed Brownian motions are extensively applied as mathematical models for asset returns in Finance. Time change is interpreted as a switch to trade-related business time, different from calendar time. Time-changed Brownian motions can be generated by infinite divisible normal mixtures. The standard multivariate normal mean variance mixtures assume a common mixing variable. This corresponds to a multidimensional return process with a unique change of time for all assets under exam. The economic counterpart is uniqueness of trade or business time, which is not in line with empirical evidence. In this paper we propose a new multivariate definition of normal mean-variance mixtures with a flexible dependence structure, based on the economic intuition of both a common and an idiosyncratic component of business time. We analyze both the distribution and the related process. We use the above construction to introduce a multivariate generalized hyperbolic process with generalized hyperbolic margins. We conclude with a stock market example to show the ease of calibration of the model.multivariate normal mean variance mixtures, multivariate generalized hyperbolic distributions, Levy processes, multivariate subordinators

Multivariate Option Pricing with Copulas.

In this paper we suggest the adoption of copula functions in order to price multivariate contingent claims. Copulas enable us to imbed the marginal distributions extracted from vertical spreads in the options markets in a multivariate pricing kernel. We prove that such kernel is a copula function, and that its super -replication strategy is represented by the Fréchet bounds. As applications, we provide prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, we provide no-arbitrage pricing bounds, as well as the values consistent with independence of the underlying assets. As a final reference value, we use a copula function calibrated on historical data.option pricing; basket options; copula functions; non-normal returns

Credit risk in pure jump structural models

Structural models of credit risk are known to present vanishing spreads at very short maturities. This shortcoming, which is due to the diffusive behavior assumed for asset values, can be circumvented by considering discontinuities of the jump type in their evolution over time. In particular, assuming a pure jump process. Moreover, when applied to market data diffusion-based structural models tend to produce too low spreads, even over longer horizons. In this paper we show that a jump process of the Variance-Gamma type for the asset value can also circumvent this practical shortcoming. We calibrate a terminal-default jump structural model to single-name data for the CDX NA IG and CDX NA HY components. We show that the VG model provides not only smaller errors, but also a better qualitative fit than other diffusive structural models. Indeed, it avoids both the spread underprediction of the classical Merton model and the excessive overpredictions of other well known diffusive models, as recently explored by Eom, Helwege, Huang (2004) or Demchuk and Gibson (2005).

Intercorporate guarantees, leverage and taxes

This paper characterizes optimal intercorporate guarantees, under the classical trade-off between bankruptcy costs and taxation. Conditional guarantees, allowing the guarantor - or Holding company - to maintain limited liability vis-a-vis the beneficiary - or Subsidiary - maximize joint value. They indeed achieve the highest tax savings net of default costs. We provide conditions ensuring that - at the optimum - guarantees increase total debt, which bears mostly on the Subsidiary. This difference in optimal leverage between Holding company and Subsidiary explains why optimal conditional guarantees (i) generate value independently of cash flow correlation (ii) are unilateral rather than mutual, at least for moderate default costs (iii) dominate the unconditional ones, that are embedded in mergers, at least when firms have high cash-flow correlation. We also endogenize the choice of the guarantor, showing that it has higher proportional bankruptcy costs, lower tax rates and bigger size.debt; taxes; bankruptcy costs; limited liability; capital structure; subsidiary; groups; mergers

Ownership links, leverage and credit risk

This paper explores the relationship between optimal leverage and credit risk under ownership links. It develops a structural model of a parent and a subsidiary, which issues debt in its own name under a guarantee by the parent. We find that zero leverage can be optimal for the guarantor, while leverage close to one can be optimal for the guaranteed company, as this optimally exploits the tax shield of debt while minimizing default costs. As far as credit risk is considered, their joint default probability is lower than that of stand alone units, despite their higher debt capacity. Higher group optimal leverage and lower default probability increase value with respect to conglomerate mergers and stand alone arrangements. Default probability, spreads and loss given default of the subsidiary are higher than for a stand alone with similar size and volatility. We also study the situation when the subsidiary is constrained to a debt equal to the optimal stand alone level. Only in this case group credit risk depends on the ownership share. Consistently with intuition, our unconstrained model rationalizes the capital structure typical of private equity; the constrained model instead is able to explain observed features of public business groups and more regulated environments.credit risk; default risk; structural models; optimal leverage; zero leverage; ownership structure; parent-subsidiary

A note on stochastic survival probabilities and their calibration.

In this note we use doubly stochastic processes (or Cox processes) in order to model the evolution of the stochastic force of mortality of an individual aged x. These processes have been widely used in the credit risk literature in modelling the default arrival, and in this context have proved to be quite flexible and useful. We investigate the applicability of these processes in describing the individual's mortality, and provide a calibration to the Italian case. Results from the calibration are twofold. Firstly, the stochastic intensities seem to better capture the development of medicine and long term care which is under our daily observation. Secondly, when pricing insurance products such as life annuities, we observe a remarkable premium increase, although the expected residual lifetime is essentially unchanged.