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# The methodology which has been used is the analysis of the logical steps made by mathematical theories

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The present treatise constitutes the work for a master degree thesis of the author as a completion of her study curruculum in the analytic-probabilistic field.

The aim of the thesis is to provide mathematical models for the evaluation of financial bonds with a fixed interest rate, however liable

to insolvency. The gap between safe state bonds (government securities), which for the Euro-area are the German 'bunds' and all the others, express the idea that a higher interest may compensate the risk of buying a bond that is more likely to be exposed to insolvency.

If the idea is clear, what remains is the actual interest of Rating Agencies or traders, to evaluate how much this risk may be worth, not only on the basis of statistical data but also on the evolutions' trends of these data.

The methodology which has been used is the analysis of the logical steps made by mathematical theories. Two major approaches as a solution to the problem of determining default probabilities are structural models and reduced-form models, which are based on an intensity parameter.

While the former are founded on an endogenous description of a possibility of

insolvency, the latter consider as throughly exogenous, therefore unpredictable,the event of a bankruptcy.

From this study undertaken, one can emphasize the inadequacy of the first type model to reach the prefigured aim. The reason of this is twofold. On one hand, they make use of a whole description of the issuing cash flow, whereas, as we know, the banks or credit institutions can keep part of its budget and assets hidden.

On the other hand, these models do not contemplate the possibility of failure because of external factors.

Models based on intensity have been elaborated exploiting some stochastic differential equations, for example CIR-short rate models. That allowed the

evaluation of insolvency probability, as well as the pricing of the defaultable claims of "call"

options exposed to the risk of a lack of covering or security.

One hybrid model has been studied too, the one of Duffie and Lando (2001). It can be seen the way they exploit intuitions coming from optimal stochastic equations and problems, as well as Black- Scholes did with Feynman-Kac equations.

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