Article Detail

A General Structural Approach For Credit Modeling Under Stochastic Volatility

Journal 32: Applied Finance

Marcos Escobar, Tim Friederich, Luis Seco, Rudi Zagst

This paper assumes a structural credit model with underlying stochastic volatility combining the Black/Cox approach with the Heston model. We model the equity of a company as a barrier call option on its assets. The assets are assumed to follow a stochastic volatility process; this implies an equity model with most documented stylized facts incorporated. We derive the price of this option under a general framework where the barrier and strike are different from each other, allowing for richer financial applications. The expression for the probability of default under this framework is also provided. As the calibration of this model gets much more complex, we present an iterative fitting algorithm with which we are able to nicely estimate the parameters of the model, and we show via simulation the consistency of the estimator. We also study the sensitivity of the model parameters to the difference between the barrier and strike price.

When we aim to characterize the performance of a stock, the stock price is mainly the result of the behavior of the company’s assets and its liabilities. Yet, the evolution of assets and liabilities is usually not reported on a daily basis. For this reason, we modeled a company’s equity as a barrier call option on the assets, with the liabilities as barrier and strike price. The value of the equity is therefore given by the price of the down-and-out call option [Escobar et al. (2010)]. In this manner, we can calculate the asset price time series from the given equity price time series by inverting the option pricing formula, in particular, the asset price volatility becomes the volatility implied from the option price. This model is mainly based on the foundations of structural credit models laid by Merton (1973) and Black and Cox (1976). We combined this interpretation of the equity price as a call option with the Heston model [see Heston (1993)] by modeling the asset of the company as a stochastic volatility process. We also derived estimators for the calibration of the model, inspired by the work of Genon-Catalot et al. (2000).

In Escobar et al. (2010), we assumed the knock-out barrier and strike price to be equal and found the option pricing formula to be of a very simple form with a straightforward inverse formula, which allows us to calculate the asset price if the value of the option is known. In this paper, we do not make this assumption anymore and derive the price of the equity in the form of a general barrier call option. A strike price D+K with K>0 can be economically interpreted in various ways. For example, the additional costs of fulfilling the option contract at maturity. These costs are not involved when selling the option before maturity. Such costs are likely for over-the-counter (OTC) options and common for customized options. As a practical example, for OTC options on commodities or other physical goods, the transportation or storage costs incurred when the option is exercised might affect the price someone is willing to pay for such an option.

An alternative explanation of D and K is to assume that total liabilities consist of the debt D and an additional debt K granted to the cash account. The credit limit on the cash account does not require any backing by assets but is granted for the liquidity of daily business cash flows on the company’s cash account. However, the company only defaults if the assets fall below D. In other words, the debt on the cash account is not considered for determining the case of a default, but only the true debt D. Another interpretation comes from taking D+K1 as the actual debt with K1 < K so that K1 and T are the maximum amount and time that equity holders (EH) and debt holders (DH) are willing to wait to avoid bankruptcy. Default would mean a total loss for EH as they are the first to be left out, so allowing the asset to wander in [D, D+K1] is beneficial to them. On the other hand, DH are usually willing to accept this risk for some extra return K - K1 on the money lent.


Reaching credit after economic failure is indeed a harsh job, but not an impossible one if you are aware of smart tips to satisfy lender's requirements. After reading the suggestions in this site, you will easily understand how to get a credit line after bankruptcy.

Mega Search

Leave a comment

Comments are moderated and will be posted if they are on-topic and not abusive. For more information, please see our Comments FAQ
This question is for testing whether you are a human visitor and to prevent automated spam submissions.