Solving the dual Russian option problem by using change-of-measure arguments

We apply the change-of-measure arguments of Shepp and Shiryaev [38]to study the dual Russian option pricing problem proposed by Shepp and Shiryaev [39] as an optimal stopping problem for a one-dimensional diffusion process with reflection. We recall the solution to the associated free-boundary problem and give a solution to the resulting onedimensional optimal stopping problem by using the martingale approach of Beibel and Lerche [6] and [7]

American Step-Up and Step-Down Default Swaps under Levy Models

This paper studies the valuation of a class of default swaps with the embedded option to switch to a different premium and notional principal anytime prior to a credit event. These are early exercisable contracts that give the protection buyer or seller the right to step-up, step-down, or cancel the swap position. The pricing problem is formulated under a structural credit risk model based on Levy processes. This leads to the analytic and numerical studies of several optimal stopping problems subject to early termination due to default. In a general spectrally negative Levy model, we rigorously derive the optimal exercise strategy. This allows for instant computation of the credit spread under various specifications. Numerical examples are provided to examine the impacts of default risk and contractual features on the credit spread and exercise strategy.Comment: 35 pages, 5 figure

Optimal Stopping in Levy Models, for Non-Monotone Discontinuous Payoffs

We give short proofs of general theorems about optimal entry and exit problems in Levy models, when payoff streams may have discontinuities and be non-monotone. As applications, we consider exit and entry problems in the theory of real options, and an entry problem with an embedded option to exit

ADAPTIVE WEAK APPROXIMATION OF DIFFUSIONS WITH JUMPS

This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler–Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows and discrete dual backward functions. Combined with proper estimation of the statistical error, they lead to efficient and accurate computation of global error estimates, extending the results by A. Szepessy, R. Tempone, and G. E. Zouraris [Comm. Pure Appl. Math., 54 (2001), pp. 1169–1214]. Adaptive algorithms for either deterministic or trajectory-dependent time stepping are proposed. Numerical examples show the performance of the proposed error approximations and the adaptive schemes