Credit risk - A structural model with jumps and correlations

As far as the field of finance is concerned, the vast majority of studies put forward by physicists has been devoted to market risk. The market risk is due to the unknown time evolution of the asset prices. In general, one is faced with a large spectrum of different risk types. One also distinguishes, for example, operational risk (due to failure of internal systems), political risk (due to political decisions that affect the capital markets) and legal risk (due to fraud and discontinued contracts). In this contribution, we address credit risk. It is due to the failure of a counterpart to make a promised payment. At present, risk managers and researchers are more familiar with market risk than with any of the other risk types and the corresponding mathematical description is highly developed. It is of considerable practical interest to improve the knowledge about and the modeling of the other risk types. In the case of credit risk, the probability that a promised payment is not made is usually small and difficult to estimate. Nevertheless the amount of money involved and thus the associated loss can be enormous and even jeopardize the existence of the financial institution which issued the credit.

Only recently, physicists started applying their spe-cific tools to credit risk [7,8,9,10,11,12]. The interesting point for practitioners and researchers, especially statistical physicists, is the highly asymmetric form of the loss distribution and the resulting peculiar features. This distinguishes credit risk from market risk, although the former clearly depends on the latter. In investment theory the standard deviation, referred to as volatility, of the relative asset price change is taken as a measure of how risky a certain investment is. If more uncertainty is incorporated in the investment, i.e. if the volatility is larger, then the demanded earnings are higher. Due to this fact, investors are traditionally risk averse. In other words, a potential loss is considered to be more punitive than a potential gain is beneficial, even if they are equally probable and large. The asymmetric character of the loss distributions makes risk measures other than the volatility also important in credit risk management.

In this study, we set up and analyze a structural model for credit risk, based on a jump-diffusion process for the risk factors. Our study is related to, but different from the work in Ref. [13]. These models are particularly appealing to physicists, because their starting point is, in physics terminology, microscopic and dynamical. This gives them a rather general character which makes them also suited for other problems in physics and in the theory of complex systems. With this contribution, we pursue two goals. First, we systematically explore the interplay between the different parameters of our structural model, particularly the role of leverage, jumps and correlations. In contrast to the existing literature, our main focus is on the full loss distribution of the credit portfolio, its moments and its tail behavior. Second, we review credit risk modeling and keep the whole presentation pedagogical, because we want to make this topic more accessible to the physics audience.

The paper is organized as follows. We review the present status of credit risk modeling in Sec. II. In Sec. III we introduce our model. We discuss a simplified version of it analytically in Sec. IV and the full model numerically in Sec. V. Summary and conclusion are given in arXiv:0707.3478v2 [q-fin.RM] 27 Sec. VI.

After defining debt instruments in Sec. II A, we discuss what one means by default and credit ratings in Secs. II B and II C, respectively. The credit risk measures are introduced in Sec. II D. The impact of correlations is discussed in Sec. II E. The rôle of the probability density function for credit losses is explained in Sec. II F. We conclude this short review by presenting the most important credit risk models in Sec. II G. Reviews in the financial literature on credit risk modeling can be found in Refs. [14,15] A. Debt instruments A debt instrument is simply a written promise to repay a debt. There is a wide range of such contracts. A debt instrument has two positions: a lending side (the creditor) and a borrowing side (the obligor). Bonds are very common. A bond is issued for a period of one year or more with the purpose of raising capital by borrowing. The government, states, cities, corporations, and

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