An Examination of the Systematic Risks in the Multi-Name Credit and Equity Markets

Abstract

The focus of this dissertation is to examine the systematic risks in the multi-name credit and equity markets during the recent financial crisis. We first consider a hybrid intensity-based model that can price both credit and equity instruments, and here, we analyze the multi-name credit spreads and the option-implied volatility skews. Secondly, we develop a top-down utility-indifference model for valuing credit and equity instruments, and in this case, we compare the implied risk aversions of investors in the two markets.

In the first part, we present an intensity-based common factor model that can be used to link the credit market to the equity market. In particular, we use the hybrid intensity model to price single-name credit instruments such as credit default swaps (CDSs), multi-name credit derivatives such as collateralized debt obligations (CDOs), and equity index options such as calls and puts on the S\&P 500. The CDS prices have analytical expressions; the CDO prices have to be computed numerically using a recursion algorithm and Fourier transform methods; and the equity index option prices have semi-analytical expressions that are computed using numerical integration. Once we have the expressions for the model prices of these instruments, we then calibrate the model parameters to fit the market data. We study two problems, the ``forward'' and ``backward'' problems: in the former, we start from equity index options and then compute the CDO tranche spreads, while in the latter, we fit the parameters to the CDO tranche spreads and then back out the equity index option prices and implied volatilities. In both cases, we analyze the systematic risks inherent in the credit and equity markets by examining the tranche spreads and implied volatility skews from the model and the market. We find that based on our hybrid model, the systematic risks in the two markets were similar from 2004 to 2007, while the credit market incorporated far greater systematic risk than the equity market during the financial crisis from 2008 to 2010.

In the second part, we consider a top-down utility-indifference model that incorporates the investor's risk aversion for valuing both multi-name credit derivatives and equity index options. In particular, we assume that the default loss process is a self-exciting counting process in which the intensity is mean-reverting with feedback from defaults, while the stock index process has stochastic variance which also contains feedback from defaults. Now, for the optimal control problem, the investor can invest in the equity/credit derivative, the money market and the stock index. The indifference prices then arise from the Merton and tranche holder's value functions, which are the solutions to systems of multi-variable Hamilton-Jacobi-Bellman PDEs. Here, the differential equations can be solved recursively using either finite differences or trinomial trees. From our numerical tests, we find that the investors' implied risk aversions for CDOs were increasing with seniority, whereas the risk aversions for equity put options increased as moneyness levels decreased. In addition, we find that over the 16-month period from June 2009 to September 2010, the largest risk aversions from the credit market far exceeded those from the equity market, indicating greater systematic risk in the credit market during the crisis, consistent with the conclusions from the first part of the thesis.

Lastly, we also discuss two extensions to the top-down model, first where we allow for correlation between the equity, variance, and intensity processes and secondly, where we impose that the stock index drops by proportional amounts at times of default.