Herramientas de programación.NET 22

The form that a good multi-firm credit model should take is clearly the subject of great debate, but broadly speaking it seems reasonable to require that it should

1. Be intuitive. The model should be based on economic fundamentals and interested parties should be able to understand how the model relates to the event of company default. This is the primary advantage of the structural model over most other approaches.

2. Capture real-world default dynamics. Default events should be dynamic and driven by macroeconomic and market-wide influences as well as idiosyn- cratic factors. Defaults should be at times predictable, but more often, unex- pected.

3. Incorporate a realistic dependence structure. Both positive and negative

credit dependence should be possible, and it should be asymmetric3_{. The model}

should allow default propagation in the form of a type of credit contagion, with the possibility of default clustering triggered by either cyclical or common factors.

4. Be implementable. The model should not be so complicated and incompre- hensible that it is impossible to understand or use, either from an analytical or a numerical standpoint.

5. Enable calibration. Models which highlight the sensitivity of prices to the dependence specification and default dynamics are good, but at the end of the day, the aim is to price multi-name credit derivatives. The model should be

consistent with market CDS prices and enable the valuation of kth_-to-default

baskets and CDO tranches.

Taken together, the above represent the holy grail of credit modelling and the route to riches. As a starting point, the aim of this thesis is to extend the first passage struc- tural model to account for a more realistic dependence structure, one that is dynamic, asymmetric with respect to default risk, incorporates both macro driving influences and contagion and can result in periods of default clustering. There are many further attributes that would be attractive, for example removing the predictable nature of

3_{Symmetrical credit dependence is a frequent modelling assumption, but is unrealistic. The}

bankruptcy of Microsoft, for example, would have a huge impact on a local computer supplier but Microsoft would be unlikely to notice if the latter went out of business.

defaults at the individual level. However, these are for the future. By necessity, there is a balance to be struck between extending beyond a naive asset correlation structure and obtaining an easily calibratable model. Our focus is on the former, and so we leave questions of calibration to one side. By proposing new ways of incorporating a realistic dependence structure, our aim is to highlight the importance of credit de- pendence assumptions and to develop a framework that has the potential to form the basis for an implementable approach to multi-asset credit modelling.

In Chapters 4 and 5, we consider an analytical approach, providing formulæ and results for the valuation of bonds in the presence of default contagion and two-firm CDS baskets. Prohibited from extending this to larger baskets by the intractability of the mathematics in higher dimensions, we turn to numerics in Chapter 6, enabling us to incorporate a much richer dependence structure.

Chapter 4

Calculating Joint Survival

Probabilities

In this chapter we consider the joint survival probability for two correlated Brownian motions with constant default barriers. By a simple change of variables, this case can be used to consider the more general formulation of correlated geometric Brownian motions with exponential default barriers; the results obtained in this chapter are applied to the pricing of corporate bonds and credit default swaps in Chapter 5. This is the first work to date using a two-dimensional structural model with default contagion to price corporate bonds. Elements of the underlying framework, however, are considered in two previous papers. In the first, Hua et al. (1998) derive the gen- eral formula for the joint survival probability and explore its use in the valuation of double lookback options. As outlined in Chapter 3, Zhou (2001a) considers the de- fault correlation that arises between two companies modelled as correlated Brownian motions. Results are primarily given for the simpler, constant-leverage case, with formulæ derived solely for this case.

This chapter is in four sections. Section 4.1 formulates the problem, describing the general case of interest and its transformation into the simpler case of two correlated Brownian motions with constant default barriers that is considered in the rest of this chapter. The joint survival probability is derived, and simple cases are discussed. Section 4.2 considers the numerical evaluation of the joint survival probability, whilst Section 4.3 illustrates the results and analyses the impact of different parameter values; conclusions are in Section 4.4.

explain the general framework, derive the main formulæ and consider their imple- mentation. These are then used as the basis for pricing credit instruments in Chapter 5. Much of the detail is in the appendices for ease of exposition.