David B. Colwell

Martingale representation and hedging policies

The integrand, when a martingale under an equivalent measure is represented as a stochastic integral, is determined by elementary methods in the Markov situation. Applications to hedging portfolios in finance are described.

A structural model for credit risk with switching processes and synchronous jumps

This paper studies a switching regime version of Mertons structural model for the pricing of default risk. The default event depends on the total value of the firms asset modeled by a switching Lévy process. The novelty of this approach is to consider that firms asset jumps synchronously with a change in the regime. After a discussion of dynamics under the risk neutral measure, two models are presented. In the first one, the default happens at bond maturity, when the firms value falls below a predetermined barrier. In the second version, the firm can enter bankruptcy at multiple predetermined discrete times. The use of a Markov chain to model switches in hidden external factors makes it possible to capture the effects of changes in trends and volatilities exhibited by default probabilities. With synchronous jumps, the firms asset and state processes are no longer uncorrelated. Finally, some econometric evidence that switching Lévy processes, with synchronous jumps, fit well historical time series is provided.

Component Structure of Credit Default Swap Spreads and Their Determinants

In this paper, we decompose credit default swap (CDS) spreads into a transitory component and a persistent component and test how these components are affected by the theoretical explanatory variables. We use three benchmark iTraxx Europe indices of two different maturities (5 and 10 years) and extract the components in the framework of Schwartz and Smith (2000). We then regress these components against proxies for several commonly used explanatory variables. We find significant but differing impacts of these explanatory variables on the extracted components. For example, equity volatility seems to have a larger influence on the transitory component, suggesting that its effect may be mostly short-lived, while our proxy for illiquidity has a bigger impact on the persistent component, which suggests that its effect is more enduring. Surprisingly, our proxy for the credit rating premium is not even significant in explaining two of the 10-year indices, but has a large effect on the persistent component. Finally, the slope of the yield curve has impacts with opposite signs on the two components and thus helps address the conflicting results reported in earlier studies without such a component framework. These results indicate that a two factor formulation, similar to Hull and White (1994) interest rate model, may be needed to model CDS options.

A Class of Stochastic Volatility HJM Interest Rate Models

This paper considers a class of stochastic volatility HJM term structure models with explicit finite dimensional realisations. The resulting bond market is arbitrage free but incomplete resulting in a non-unique martingale measure. Nevertheless, the market price of risk is partially determined by the forward rate drift and volatility. Numerical simulation for bond and bond option prices are included to illustrate the effect of stochastic volatility on these prices.

Variance Minimizing Strategies for Stochastic Processes with Applications to Tracking Stock Indices

This paper extends the notion of variance optimal hedging of contingent claims under the incomplete market setting to the hedging of entire processes, and applies the results to the problem of tracking stock indices. Sufficient conditions under which this is possible are given, along with the corresponding variance optimal strategy in feedback form as given in Schweizer (1996) and Pham, Rheinlander, and Schweizer (1998) for contingent claims. The performances of tracking error variance minimizing, locally risk minimizing, and variance minimizing strategies in tracking stock indices are investigated using both simulated and historical market data.

A Structural Model for Credit Risk with Markov Modulated Lévy Processes and Synchronous Jumps

This paper presents a switching regime version of the Mertons structural model for the pricing of default risk. The default event depends on the total value of the firms asset modeled by a Markov modulated Levy process. The novelty of our approach is to consider that firms asset jumps synchronously with a change in the regime. After a discussion of dynamics under the risk neutral measure, we present two models. In the first one, the default occurs at bond maturity if the firms value falls below a predetermined barrier. In the second version, the company can bankrupt at multiple predetermined discrete times. The use of a Markov chain to model switches in hidden external factors makes it possible to capture the effects of changes in trends and volatilities exhibited by default probabilities. Finally, with synchronous jumps, the firms asset and state processes are no longer uncorrelated.

Regime dependent causality : equity and credit markets

We apply a Markov switching model to investigate the possibility of an asymmetric causal relationship between the volatility process inferred from the iTraxx CDS options market and the implied volatility from the stock index options market. We find strong evidence that the stock market leads the CDS market and the effect of the implied stock market volatility is more significant during the volatile regime. We also find that a large jump in the stock return, up or down, may indeed be followed by a regime shift.