Dynamite dynamics

By Jesper Andreasen

INTRODUCTION

In the context of stochastic models, explosion means that the variable that is modelled can go to infinity with positive probability. An example of this is the following model:
 

$$x\left( t \right) = \frac{1}{{a - W\left( t \right)}}$$

where a is a positive constant and W is a Brownian motion with W(0) = 0. Here x will go to infinity as W approaches a, and the process x is thus only well defined up to the first passage time of W to the level a.

In most finance models the stochastic dynamics of the modelled variables are restricted to prevent explosions, as we are rarely interested in prices or rates that are infinite. But in this chapter we actually make use of explosive processes to give us more realistic dynamics of credit default swap (CDS) spreads.

The case for this is based on the observation that defaults can be anticipated as well as unanticipated. If a default is anticipated it is well known in the market that the company has financial problems and the actual default will tend to happen after a period of steep increases in credit spreads and corresponding declines in bond and equity prices. Default can also be a complete surprise to the financial markets,

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