Trajectory based models. Evaluation of minmax pricing bounds

Discrete, Non Probabilistic Market Models. Arbitrage and Pricing Intervals

“…We emphasize that S W (s 0 , d, c, Q), as characterized above, does not need to be, in general, the set of all trajectories S satisfying the listed constraints in Definition (11); specific examples are described in [12]. Comparing with (4.2), we see that we have allowed F to be an independent variable W .…”

“…Not imposing (4.3) allows to incorporate 0-neutral nodes which are arbitrage nodes (see Definition 10 and related comments afterwards.) An analysis of these considerations in the context of the example is outside the scope of the paper, details are given in [12]. A natural discretization leading to an implementation of S W (s 0 , c, d, Q) is obtained by introducing real numbers δ , β > 0.…”

“…Remark 4. Under some assumptions on the market M , such as conditionally 0-neutral, it can be proven that the conditional bounds V k (S, Z, M ) and/or V k (S, Z, M ) are finite when Z is an upper or lower minimax function, for reasons of space details are provided elsewhere ( [12]).…”

“…In the present paper we describe a general discrete example of a class of trajectory sets extending substantially the model in [11], in particular, the example incorporates trajectory dependent volatility. A computational and more detailed analysis is developed in the companion paper [12]. Section 10 also introduces trajectory sets associated to continuous-time martingale processes.…”

“…The emphasis of the paper is to establish basic theoretical properties that follow from the proposed framework. A detailed computational analysis of related examples is available in [12]; we expect to make clear that the present setting is quite flexible, several numerical examples and processing of market data can be found in [24].…”