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

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