The mean comparison theorem cannot be extended to the Poisson case

Abstract

In this paper, we show that the Mean Comparison Theorem which is valid for Brownian motion, cannot be extended to Poisson process. A counter example in the Poisson case, for which the Mean Comparison Theorem does not hold, is provided.

Comparison results for path-dependent options

“…Henderson (2000) establishes stochastic comparison of the supremum of a stochastic volatility model and the supremum of a time-homogeneous diffusion model, making use of a comparison result for diffusions in Hajek (1985). Večeř and Xu (2004) give a counterexample to show that the result of Hajek (1985) does in general not apply to Poisson models.…”

European and Asian Greeks for Exponential Lévy Processes

“…Dufresne (2005) and finally given in general form for some general classes of semimartingale models. For a detailed overview of this development, we refer to Vecer (2002) and Vecer and Xu (2004), see also Albrecher (2004) and Albeverio and Lütkebohmert (2005).…”

“…The main methods developed in this context are based on the inversion of the Laplace transform (extending Geman and Yor (1993)), its connection to the fast Fourier transform, on the analytical expansion method of Linetsky (2004) and the (integro-)differential equation method of Rogers and Shi (1995), Vecer and Xu (2004) and Vecer (2014). Based on a change of measure technique as in Shreve and Vecer (2000) it was shown in Vecer (2002) and Vecer and Xu (2004) that the path dependency in the formulation of the Asian option pricing problem can be simplified to the case of European options with modified payoff functions where the underlying asset is driven by a special semimartingale process leading to an integro-differential equation where the stock price is driven by a process with independent increments. This reduction is given for generalized Asian options of the form which includes for K 1 = 0 the fixed strike option and for K 2 = 0 the floating strike option.…”