Numerical valuation of basket credit derivatives in structural jump-diffusion models

Abstract

We consider a model where each company's asset value follows a jump-diffusion process, and is connected with other companies via global factors. Motivated by ideas in Bush et al. (2011), where the joint density of asset values is evolved in a large basket approximation, we develop an algorithm for the efficient estimation of CDO index and tranche spreads consistent with underlying CDSs, through a finite difference simulation of the resulting SPDE. We verify the validity of this approximation numerically by comparison to results obtained by direct Monte Carlo simulation of the basket constituents. A calibration exercise assesses the flexibility of the model and its extensions to match CDO spreads from pre-crisis and crisis periods.

Multilevel Monte Carlo methods

“…This work has been extended by Bujok and Reisinger (2012) to include a systemic jump-diffusion term. They also switch from continuous monitoring of default, which gives the boundary condition for all , to discrete monitoring which introduces the boundary condition for for each discrete default date .…”

Stochastic Finite Differences and Multilevel Monte Carlo for a Class of SPDEs in Finance

“…This, however, puts the model precisely in the realm of the large basket approximation (2). See [Bujok & Reisinger(2012)] for a numerical study of this large basket approximation. The loss functional is thereby approximated by…”

Fast mean-reversion asymptotics for large portfolios of stochastic volatility models

“…under the fast mean-reverting volatility setting to the loss under an appropriate constant volatility setting. 4 , the function h is bounded, and g has the positive recurrence property, in which case we have…”

“…, c 1,n } of values of the vector C 1 . In the special case when asset prices are modelled as simple constant volatility models, the numerics (see Giles and Reisinger [14] or Bujok and Reisinger [4] for jump-diffusion models) have a significantly smaller computational cost, which motivates the investigation of the existence of accurate approximations using a constant volatility setting in the general case. We also note that one-dimensional SPDEs describing large portfolio limits in constant volatility envi-ronments have been found to have a unique regular solution (see Bush et al [5] or Hambly and Ledger [18] for a loss-dependent correlation model), an important component of the numerical analysis and a counterpoint to the fact that we have been unable to establish uniqueness of solutions to the two-dimensional SPDE arising in the CIR volatility case [15].…”