# Efficient Pricing Barrier Options and CDS in Lévy Models with Stochastic Interest Rate

29 Pages Posted: 2 Jan 2015 Last revised: 6 Dec 2015

See all articles by Svetlana Boyarchenko

## Svetlana Boyarchenko

University of Texas at Austin - Department of Economics

## Sergei Levendorskii

Calico Science Consulting

Date Written: January 13, 2015

### Abstract

Recently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of L'evy models, the Heston model and other affine models. Similar deformations were used in one-factor L'evy models to price options with barrier and lookback features and CDSs. In the present paper, we generalize this approach to models of structural default, where the dynamics of assets follows an exponential L'evy process $X_t$, and the interest rate $r_t$ is stochastic. Assuming that $X_t$ and $r_t$ are independent, and the infinitesimal generator of the pricing semigroup in the model for the short rate, is (block)-diagonalizable, we develop a variation of the pricing procedure for L'evy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 sec suffice to calculate prices of 8 options of same maturity in a two-factor model with the error tolerance $5\cdot 10^{-5}$ sec. and less; in a three-factor model, accuracy of order 0.001-0.005 is achieved in about 0.2 sec. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of L'evy models with the stochastic interest rate driven by 1-2 (possibly, 3) factors, which allows for fast calculations. This class can satisfy the current requirements by regulators for banks to have sufficiently sophisticated credit risk models.

Keywords: Stochastic interest rate, quadratic term structure models, L'evy processes, Wiener-Hopf factorization, barrier options, credit default swaps, parabolic inverse Laplace transform, parabolic inverse Fourier transform, quanto CDS

JEL Classification: G13

Suggested Citation

Boyarchenko, Svetlana I. and Levendorskii, Sergei Z., Efficient Pricing Barrier Options and CDS in Lévy Models with Stochastic Interest Rate (January 13, 2015). Available at SSRN: https://ssrn.com/abstract=2544271 or http://dx.doi.org/10.2139/ssrn.2544271