We derive comparison results for Markov processes with respect to stochastic orderings induced by function classes. Our main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators implies ordering of the processes. Unlike in previous work no boundedness assumptions on the function classes are needed anymore. We also present an integral version of the comparison result which does not need the local comparability assumption of the generators. The method of proof is also used to derive comparison results for time-discrete Markov processes.

In this paper, we determine the lowest cost strategy for a given payoff in Lévy markets where the pricing is based on the Esscher martingale measure. In particular, we consider Lévy models where prices are driven by a normal inverse Gaussian (NIG)-or a variance Gamma (VG)-process. Explicit solutions for cost-efficient strategies are derived for a variety of vanilla options, spreads, and forwards. Applications to real financial market data show that the cost savings associated with these strategies can be quite substantial. The empirical findings are supplemented by a result that relates the magnitude of these savings to the strength of the market trend. Moreover, we consider the problem of hedging efficient claims, derive explicit formulas for the deltas of efficient calls and puts and apply the results to German stock market data. Using the time-varying payoff profile of efficient options, we further develop alternative delta hedging strategies for vanilla calls and puts. We find that the latter can provide a more accurate way of replicating the final payoff compared to their classical counterparts. Int. J. Theor. Appl. Finan. 2014.17. Downloaded from www.worldscientific.com by INDIANA UNIVERSITY @ BLOOMINGTON on 02/04/15. For personal use only. 1450041-2 Int. J. Theor. Appl. Finan. 2014.17. Downloaded from www.worldscientific.com by INDIANA UNIVERSITY @ BLOOMINGTON on 02/04/15. For personal use only. 1450041-3 Int. J. Theor. Appl. Finan. 2014.17. Downloaded from www.worldscientific.com by INDIANA UNIVERSITY @ BLOOMINGTON on 02/04/15. For personal use only. E. A. von Hammerstein et al.provided that the expectation exists. Note that here and in the following the expectation E[·] = E P [·] is always calculated w.r.t. the real-world measure P if not stated otherwise. Definition 2.1 (Cost-efficient and most-expensive strategies).(a) A strategy (or payoff) X T ∼ G is called cost-efficient w.r.t. the payoffdistribution G if any other strategy X T that generates the same payoffdistribution G costs at least as much, that is,the payoffdistribution G if any other strategy X T that generates the same payoffdistribution G costs at most as much, that is,3) 1450041-6 Int. J. Theor. Appl. Finan. 2014.17. Downloaded from www.worldscientific.com by INDIANA UNIVERSITY @ BLOOMINGTON on 02/04/15. For personal use only. Optimality of Payoffs in Lévy ModelsRemark 2.1. Note that within this framework one cannot assume the state price density to be square integrable in general. The interval (a, b) on which the mgf M L1 is well defined and finite, can be fairly small. Thus, it might happen that a solution θ ∈ (a, b − 1) of (2.7) exists, but 2θ ∈ (a, b), implying that E[e 2θLt ] is infinite and hence Zθ t is not square integrable for any t > 0. Now we can reformulate Theorem 2.1 in terms of the driving Lévy process instead of the state price process.

In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.

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