The main goal of this study is to develop a general theoretical pricing framework that will capture some practically relevant properties, such as: the prices are not homogeneous in number of shares traded; the underlying securities bear transaction costs; the securities pay dividends; the dividends may be different for a long or short position. To achieve this goal, we use sub-scale invariant Dynamic Acceptability Indices (DAIs) as the main tool in developing the pricing methodology, and consequently, we present a representation of proposed prices in terms of a class of Backward Stochastic Difference Equations and g-Expectations. Besides the above mentioned properties, we also prove that: considered market models do not admit arbitrage; bid and ask prices do shrink the super hedging pricing interval; the prices are time consistent in some appropriate sense; if the drivers are linear we recover the classical martingale pricing theory. Finally, we provide some practical examples.
This is a joint work with Tomasz R. Bielecki and Tao Chen.
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