Alfonsi, Aurélien (2006) Credit risk modeling. Calibration and discretization of financial models. PhD thesis, ENPC.

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Abstract

The first part of this thesis deals with credit risk. After a short introduction (in French) to this market and its modelling, we present a reduced-form model for the default time called SSRD (Shifted Square-Root Diffusion). One interesting feature of this model is that it can be automatically calibrated to the Credit Default Swap prices that come from the market. Moreover, it leaves the possibility to have dependence between the default intensity and the interest short rate. Then, we present a new family of copula functions named "periodic copulas because their construction relies on periodic functions. Copula functions are used in the credit risk area to model dependence between different default times. Periodic copulas allow to explore a large range of dependence and can also be extended to the multivariate case. Furthermore, they can be easy sampled.
Then, we focus on the discretization schemes for the stochastic differential equation of Cox-Ingersoll-Ross which is used in the SSRD model. We present several discretization schemes of both the implicit and explicit types. We study their strong and weak convergence. We also examine numerically their behaviour and compare them to the schemes already proposed by Deelstra and Delbaen [27] and Diop [28]. Gathering all the results obtained, we recommend, in the standard case, the use of one of our explicit schemes.
In the last, part we turn to the equity market. It is well known that it is possible, in a local volatility function model, to grab from the European option prices a volatility function which is consistent with. We consider here the analogous problem for American options: can we find a volatility function that explains ail the American option prices observed ? To tackle this problem, we consider perpetual American options and time-homogeneous local volatility functions. We put in evidence a relation named Call-Put Duality that allows to interpret a Put price as a Call price where strike value and spot value have been interchanged. Thanks to this duality result, we design a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices for a fixed spot price.

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