Smiling jumps

Local jump intensity models, in which the volatility of a Lévy process is made spot-dependent, are difficult to parameterise and calculate. However, they are an important ingredient of credit barrier models, in which the firm value is modelled as a geometric Lévy process and default as the first passage time below a barrier. Richard Martin shows how to apply the local jump intensity to the Merton framework, gives examples on a variety of credits, and explains why some classes of issuers have more pronounced local volatility than others

Volatility skews have been the subject of much interest in the past couple of decades. Effort has been directed down three routes (Lipton, 2002a and 2002b): models with jumps (Lipton, 2002c); local volatility, where the volatility is a deterministic function of spot price and time (Dupire, 1994); and stochastic volatility, where the volatility is a separate variable but correlated to the spot price, and the state space becomes one dimension higher (Heston, 1993). The main emphasis has