A Unified Framework for Pricing Credit and Equity Derivatives

We propose a model which can be jointly calibrated to the corporate bond term structure and equity option volatility surface of the same company. Our purpose is to obtain explicit bond and equity option pricing formulas that can be calibrated to find a risk neutral model that matches a set of observed market prices. This risk neutral model can then be used to price more exotic, illiquid or over-the-counter derivatives. We observe that the model implied credit default swap (CDS) spread matches the market CDS spread and that our model produces a very desirable CDS spread term structure. This is observation is worth noticing since without calibrating any parameter to the CDS spread data, it is matched by the CDS spread that our model generates using the available information from the equity options and corporate bond markets. We also observe that our model matches the equity option implied volatility surface well since we properly account for the default risk premium in the implied volatility surface. We demonstrate the importance of accounting for the default risk and stochastic interest rate in equity option pricing by comparing our results to Fouque, Papanicolaou, Sircar and Solna (2003), which only accounts for stochastic volatility.Comment: Keywords: Credit Default Swap, Defaultable Bond, Defaultable Stock, Equity Options, Stochastic Interest Rate, Implied Volatility, Multiscale Perturbation Metho

A Tree Implementation of a Credit Spread Model for Credit Derivatives

In this paper we present a tree model for defaultable bond prices which can be used for the pricing of credit derivatives. The model is based upon the two-factor Hull-White (1994) model for default-free interest rates, where one of the factors is taken to be the credit spread of the defaultable bond prices. As opposed to the tree model of Jarrow and Turnbull (1992), the dynamics of default-free interest rates and credit spreads in this model can have any desired degree of correlation, and the model can be fitted to any given term structures of default-free and defaultable bond prices, and to the term structures of the respective volatilities. Furthermore the model can accommodate several alternative models of default recovery, including the fractional recovery model of Duffie and Singleton (1994) and recovery in terms of equivalent default-free bonds (see e.g. Lando (1998)). Although based on a Gaussian setup, the approach can easily be extended to non-Gaussian processes that avoid negative interest-rates or credit spreads.credit derivatives; credit risk; implementation; Hull-White model

Credit Derivatives Pricing with a Smile-Extended Jump Stochastic Intensity Model

We present a two-factor stochastic default intensity and interest rate model for pricing single-name default swaptions. The specific positive square root processes considered fall in the relatively tractable class of affine jump diffusions while allowing for inclusion of stochastic volatility and jumps in default swap spreads. The parameters of the short rate dynamics are first calibrated to the interest rates markets, before calibrating separately the default intensity model to credit derivatives market data. A few variants of the model are calibrated in turn to market data, and different calibration procedures are compared. Numerical experiments show that the calibrated model can generate plausible volatility smiles. Hence, the model can be calibrated to a default swap term structure and few default swaptions, and the calibrated parameters can be used to value consistently other default swaptions (different strikes and maturities, or more complex structures) on the same credit reference name.Credit derivatives, credit default, swap, credit default swaption, jump-diffusion, stochastic intensity, doubly stochastic poisson process, cox process

Correlation Between Intensity and Recovery in Credit Risk Models

We start by presenting a reduced-form multiple default type of model and derive abstract results on the influence of a state variable X on credit spreads, when both the intensity and the loss quota distribution are driven by X. The aim is to apply the results to a concrete real life situation, namely, to the influence of macroeconomic risks on credit spreads term structures. There has been increasing support in the empirical literature that both the probability of default (PD) and the loss given default (LGD) are correlated and driven by macroeconomic variables. Paradoxically, there has been very little effort from the theoretical literature to develop credit risk models that would include this possibility. A possible justification has to do with the increase in complexity this leads to, even for the "treatable" default intensity models. The goal of this paper is to develop the theoretical framework needed to handle this situation and, through numerical simulation, understand the impact on credit risk term structures of the macroeconomic risks. In the proposed model the state of the economy is modeled trough the dynamics of a market index, that enters directly on the functional form of both the intensity of default and the distribution of the loss quota given default. Given this setup, we are able to make periods of economic depression, periods of higher default intensity as well as periods where low recovery is more likely, producing a business cycle effect. Furthermore, we allow for the possibility of an index volatility that depends negatively on the index level and show that, when we include this realistic feature, the impacts on the credit spread term structure are emphasized.Credit risk; sistematic risk; intensity models; recovery; credit spreads

Term Structure Dynamics in Theory and Reality

This paper is a critical survey of models designed for pricing fixed income securities and their associated term structures of market yields. Our primary focus is on the interplay between the theoretical specification of dynamic term structure models and their empirical fit to historical changes in the shapes of yield curves. We begin by over viewing the dynamic term structure models that have been fit to treasury or swap yield curves and in which the risk factors follow diffusions, jump-diffusion, or have \switching regimes." Then the goodness-of- ts of these models are assessed relative to their abilities to: (i) match linear projections of changes in yields onto the slope of the yield curve; (ii) match the persistence of conditional volatilities, and the shapes of term structures of unconditional volatilities, of yields; and (iii) to reliably price caps, swaptions, and other fixed-income derivatives. For the case of defaultable securities we explore the relative ts to historical yield spreads

Term Structure Dynamics in Theory and Reality

This paper is a critical survey of models designed for pricing fixed income securities and their associated term structures of market yields. Our primary focus is on the interplay between the theoretical specification of dynamic term structure models and their empirical fit to historical changes in the shapes of yield curves. We begin by overviewing the dynamic term structure models that have been fit to treasury or swap yield curves and in which the risk factors follow diffusions, jump-diffusion, or have \switching regimes." Then the goodness-of- ts of these models are assessed relative to their abilities to: (i) match linear projections of changes in yields onto the slope of the yield curve; (ii) match the persistence of conditional volatilities, and the shapes of term structures of unconditional volatilities, of yields; and (iii) to reliably price caps, swaptions, and other fixed-income derivatives. For the case of defaultable securities we explore the relative fits to historical yield spreads

Term Structure Dynamics in Theory and Reality

This paper is a critical survey of models designed for pricing fixed income securities and their associated term structures of market yields. Our primary focus is on the interplay between the theoretical specification of dynamic term structure models and their empirical fit to historical changes in the shapes of yield curves. We begin by over viewing the dynamic term structure models that have been fit to treasury or swap yield curves and in which the risk factors follow diffusions, jump-diffusion, or have \switching regimes." Then the goodness-of- ts of these models are assessed relative to their abilities to: (i) match linear projections of changes in yields onto the slope of the yield curve; (ii) match the persistence of conditional volatilities, and the shapes of term structures of unconditional volatilities, of yields; and (iii) to reliably price caps, swaptions, and other fixed-income derivatives. For the case of defaultable securities we explore the relative ts to historical yield spreads