John Hull

The Relationship between Credit Default Swap Spreads, Bond Yields, and Credit Rating Announcements

A company’s credit default swap spread is the cost per annum for protection against a default by the company. In this paper we analyze data on credit default swap spreads collected by a credit derivatives broker. We first examine the relationship between credit default spreads and bond yields and reach conclusions on the benchmark risk-free rate used by participants in the credit derivatives market. We then carry out a series of tests to explore the extent to which credit rating announcements by Moody’s are anticipated by participants in the credit default swap market.

Valuation of a CDO and an n th to Default CDS Without Monte Carlo Simulation

Many of the new credit derivative products are based on default experience for a portfolio of financial instruments. These include collateralized debt obligations (CDOs) and similar tranched credit products, and “n-th to default swaps.” Devising good default risk models for single-name credits has been challenging enough, but applying them to credit portfolios introduces much greater complexity, because of the critical importance of correlation. The most common valuation technology is Monte Carlo simulation, but with many bonds, each of which is subject to both correlated and idiosyncratic risk factors, the simulation is time-consuming and limited in scope. In this article, Hull and White offer two straightforward approximation techniques for evaluating default risk within the industry-standard “copula“ model that eliminate simulation of the idiosyncratic risks. Their approach greatly accelerates the solution while still allowing a large degree of flexibility in the choice of factor correlation structure and probability distributions. For example, Student-t distributed shocks that have fatter tails than the normal are easily accommodated.

NUMERICAL PROCEDURES FOR IMPLEMENTING TERM STRUCTURE MODELS I: SINGLE-FACTOR MODELS

This article presents a new approach for constructing no-arbitrage models ofthe term structure in terms ofthe process followed by the short rate, r. The approach, which makes use oftrinomial trees, is relatively simple and computationally much more ejkient than previously proposed procedures. The advantages ofthe new approach are particularly noticeable when hedge statistics such as delta, gamma, and Vega are computed. The procedure is appropriate for models where there is some function x of the short rate r that follows a meanreverting arithmetic process. I t can be used for the Ho-Lee model, the HullWhite model, and the Black-Karasinski model. Also, it is a tool that can be usedfor developing a wide range ofnew models. The key element of the procedure is that it produces a tree that is symmetrical about the expected value ofx. A forward induction procedure is used to find the positions of the central nodes at the end ofeach time step. In the w e of the Ho-Lee and Hull-White models, this forward induction procedure is entirely analytic. In the w e of other models, it is necessary to use the Newton-Raphson or other iterative search procedure at each time step, but only a small number of iterations are required. We illustrate the procedure using numerical examples and explain how the models can be calibrated to market data on interest rate option prices.

One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities

This paper compares different approaches to developing arbitrage-free models of the term structure. It presents a numerical procedure that can be used to construct a wide range of one-factor models of the short rate that are both Markov and consistent with the initial term structure of interest rates.

Valuing Credit Default Swaps I: No Counterparty Default Risk

One of the fastest growing areas of both derivatives trading and research right now is in contracts based on credit risk. The credit default swap is a standard instrument, offering the possibility of hedging against default by the issuer of an underlying bond. Several existing valuation methodologies differ in their assumptions about the payoff in case of a credit event. In this article, Hull and White present an approach based on the realistic assumption that the amount bondholders will claim in a default is based on the difference between the bond&’s post-default market value and its face value. An important contribution of this article is to use the term structure of risk-neutral implied default probabilities obtained from market prices for a set of bonds of the same issuer. The dependence of swap values on assumed recovery rates and the shape of the yield curve are explored.

Valuing Credit Default Swaps Ii: Modeling Default Correlations

“In the Fall 2000, Journal of Derivatives, Hull and White presented a model for pricing credit default swaps based on the realistic assumption that in a default the bondholders will claim the difference between the bond&s post-default market value and its face value. An important feature of the approach is the use of market prices for a set of bonds from the same issuer to obtain a term structure of risk-neutral implied default probabilities. This article extends the model significantly to allow for the existence of multiple correlated default risks. Correlations are important either when the swap is subject to counterparty credit risk, or when there are multiple underlyings with correlated risks, as in a basket default swap.”

Valuing Derivative Securities Using the Explicit Finite Difference Method

This paper suggests a modification to the explicit finite difference method for valuing derivative securities. The modification ensures that, as smaller time intervals are considered, the calculated values of the derivative security converge to the solution of the underlying differential equation. It can be used to value any derivative security dependent on a single state variable and can be extended to deal with many derivative security pricing problems where there are several state variables. The paper illustrates the approach by using it to value bonds and bond options under two different interest rate processes.

The impact of default risk on the prices of options and other derivative securities

Abstract This paper presents a model for valuing derivative securities when there is default risk. The holder of a security is assumed to recover a proportion of its no-default value in the event of a default by the counterparty. Both the probability of default and the size of the proportional recovery are random. The paper shows how data on bonds issued by the counterparty can be used to provide information about model parameters.

INCORPORATING VOLATILITY UPDATING INTO THE HISTORICAL SIMULATION METHOD FOR VALUE AT RISK

This paper proposes a procedure for using a GARCH or exponentially weighted moving average model in conjunction with historical simulation when computing value at risk. It involves adjusting historical data on each market variable to reflect the difference between the historical volatility of the market variable and its current volatility. We compare the approach using nine years of daily data on 12 exchange rates and 5 stock indices with the historical simulation approach and show that it is a substantial improvement.

VALUE AT RISK WHEN DAILY CHANGES IN MARKET VARIABLES ARE NOT NORMALLY DISTRIBUTED

This paper proposes a new model for calculating VaR where the user is free to choose any probability distributions for daily changes in the market variables and parameters of the probability distributions are subject to updating schemes such as GARCH. Transformations of the probability distributions are assumed to be multivariate normal. The model is appealing in that the calculation of VaR is relatively straightforward and can make use of the RiskMetrics or a similar database. We test a version of the model using nine years of daily data on 12 different exchange rates. When the first half of the data is used to estimate the model’s parameters we find that it provides a good prediction of the distribution of daily changes in the second half of the data.