Economic reasonable interpretation and realistic features
+ The model incorporates default contagion. Effects are twofold. On the one hand, the general setup changes due to conditional copulas and locked processes. On the other hand, a random default change is instantaneously triggered. Thus, defaults directly affect intensities. + Default and non-default information is used simultaneously and go into the model as one
relevant filtration. As such a separation of information is not observable in markets, this is a realistic feature from an economic point of view.
+ The discrete-time setup for the intensity reduces complexity and facilitates implementation. As side effect, several defaults can possibly occur in one interval.
In parts, the above mentioned advantages are mutually dependent. However, they also cause drawbacks:
- Truncation of tails lead to rescaled probabilities and hence to a reduced volatility. This could not be desired or realistic, respectively.
- Due to the special model features (truncation of margins, differentiation of copulas in case of defaults), analytical in sense of closed-form or unique solutions do not exist in general. They depend on the applied distributions (i.e., copulas and margins). It may happen that requirements on the intensity (e.g., validity of pricing formulas, martingale properties) can hardly or not at all be fulfilled under certain conditions.
- Model construction bases on the fact that default and non-default information are simultane- ously exploited. Hence, the main pro of the general information-based setup (i.e., modeling and evaluating default-free and defaulted variables separately) is suspended.
The last aspect leads to a sort of “certain” default prediction which in turn triggers the default change. Nevertheless, this framework is comparable to the general setup. In our model, the perspective on time is extended or - in other words - more granular:
• In the general model, the intensity is evaluated first. Second, defaults are simulated after- words on the whole time horizon.
• In our model, intensities and defaults are simultaneously analyzed on one interval. This evaluation produces default times. Interpreting this interval as complete time horizon, this matches the common model up to this point. Now, these default times effect intensities through default changes and conditional copulas. Thus, it can be regarded as extension of the general setup by a component of default time. In particular, this allows to model contagion effects on dependence structures of intensities and is not incorporated in the generic intensity-based framework.
12.2 Implementation - Pros and Cons
Concerning numerical implementation, our model provides following advantages:
+ Due to the inverse modeling approach, only copula-distributed random variables are used. + Truncation of marginal distributions can easily be implemented. A particular evaluation of
conditional margins is not required.
+ Implementation is possible and provides reasonable outcomes. Model effects are observable. + Calibration has to be performed only once. A recalibration due to defaults is not necessary
as their effects are considered.
Numerical drawbacks are mainly provoked by the special model construction. Partly, these cons depend on specifications and can therefore be diminished.
- Numerical differentiation of copulas is complex - even though closed-from solutions exits, e.g., for the Gaussian copula.
- The model is path-dependent. Sequent simulations outputs rely on all former realizations. - For default prediction and identification, all defaulted positions must be memorized. - Due to the previous facts, implementation is computationally intensive.
- Calibration is only valid on the predetermined time grid. A sort of time-scaling as for the Brownian motion for instance is not possible. Moreover, default changes is most likely difficult to fit due to scarce data.
Summarized, we come to the conclusion that implementation of the developed model is possible. For the given specifications and under the fixed assumptions, fit tests provide good results. Model peculiarities are reflected in numerical outcomes. The model works out as whole period valuation, it outperforms a benchmark approach in backtesting from a long-term view. Short-term results are worse. The high complexity is attended by
• higher computing times and
• a difficult numerical implementation.
These disadvantages overthrow the gain on performance. Therefore, an application asDP predic- tion for a portfolio perspective is probably not workable in practice.
12.3 Outlook and Literature
For future research, the exact numerical interpretation and understanding of model details can be focused. Here, an in-depth analysis of a one-dimensional problem may reveal impacts of the marginal truncation. Moreover, an additional benchmark approach with the independence copula may offer dues to the effects of conditional copulas.
As further problem, analytical and numerical outcomes may be investigated with respect to their dependence of the time grid variableN, i.e., the number of predetermined random changes. As possible central aspect for future research, it could be analyzed if our model can be calibrated for pricing problems and if it is of practical use for this scope of application. Concerning calibration, requirements on the intensity restrict the set of applicable distributions. Moreover, the underlying probability measure may not be unique and / or techniques of change of numeraires must be considered. Therefore, model fitting is exceptionally challenging. If calibration is reasonable, the model could be benchmarked with a standard intensity model specified by mean-reversion processes, for instance.
Concerning literature for copulas, the reader is referred to the monographs mentioned in Section 6.3, especially [12], [15], [37], [49] and [52].
For a general introduction to the world of credit risk, “(An Introduction to) Credit Risk Modeling“ by Bluhm et al. [7] provides a comprehensive survey. Detailed descriptions of credit risk models and applications are found in
• Bielecki and Rutkowski’s ”Credit Risk: Modeling, Valuation and Hedging” [4],
• Brigo and Mercurio’s ”Interest Rate Models - Theory and Practice” [9] or
• Sch¨onbucher’s ”Credit Derivatives Pricing Models” [58], for example.
For the mathematical treatment of probability measures in combination with Cox process, ”Con- vergence of probabilities” [5] by Billingsley and ”Doubly Stochastic Poisson Processes” [32] by Grandell are useful.
Part III
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