Structural Credit Modeling Under Stochastic Volatility

Abstract

This paper presents a structural credit model with underlying stochastic volatility, a CIR process, combining the Black/Cox framework with the Heston Model. We allow to calibrate a Heston Model for a non-observable process as underlying of the Black/Cox Model. A closed-form solution for the price of a down-and-out call option on the assets with the debt as barrier and strike price is derived using the concept of optional sampling. Furthermore, estimators are derived with the Method of Moments for Hidden Markov Chains. As an application in Statistical Finance, the default probabilities of Merrill Lynch during the financial crisis are examined.

Multidimensional Structural Credit Modeling under Stochastic Volatility

“…In [1] a stochastic volatility model is presented to describe the process of a company's assets and the equity of that company is modeled as a barrier call option on its assets. This model will now be generalized to a multi-dimensional framework allowing for correlation between the asset processes of different companies.…”

“…The model defined by (1) and 2is a so-called stochastic volatility model and also referred to as "Heston Model" due to the fact that it had been introduced by [4]. This model setup is further examined in the one-dimensional case in [1]. The variance process was first used by [6] to model interest rates.…”

“…In this paper we combine a structural credit model with the Heston model of stochastic volatility in a multidimensional framework. In [1] the value of the equity of a company is modeled as a down-and-out barrier call option on the company's assets with its liabilities as strike price, following the motivation of structural credit models, for example, by [2] or [3]. As markets have shown, in particular during the financial crisis, volatility is not constant over time.…”

“…In [1] the default risk of Merrill Lynch was examined utilizing the structural credit model with stochastic volatility. As recent events have shown defaults are unlikely to incur independently.…”

“…This paper derives an estimator for the correlation parameter of the assets based on the method of moments which, together with estimators already presented in [1] for the volatility parameters, allows for a full set of closed-form estimators of the model parameters. These estimators are consistent and quick to be found numerically as shown in this paper which is highly beneficial to practitioners in part due to the negligent literature on consistent estimators for continuous time process with stochastic covariance.…”