CONTINUOUSLY MONITORED BARRIER OPTIONS UNDER MARKOV PROCESSES

In this paper, we present an algorithm for pricing barrier options in one‐dimensional Markov models. The approach rests on the construction of an approximating continuous‐time Markov chain that closely follows the dynamics of the given Markov model. We illustrate the method by implementing it for a range of models, including a local Lévy process and a local volatility jump‐diffusion. We also provide a convergence proof and error estimates for this algorithm.     

EFFICIENT PRICING OF BARRIER OPTIONS AND CREDIT DEFAULT SWAPS 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 Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Lévy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.