Abstract
This paper studies the explicit calculation of the set of superhedging (and underhedging) portfolios where one asset is used to superhedge another in a discrete time setting. A general operational framework is proposed and trajectory models are defined based on a class of investors characterized by how they operate on financial data leading to potential portfolio rebalances. Trajectory market models will be specified by a trajectory set and a set of portfolios. Beginning with observing charts in an operationally prescribed manner, our trajectory sets will be constructed by moving forward recursively, while our superhedging portfolios are computed through a backwards recursion process involving a convex hull algorithm. The models proposed in this thesis allow for an arbitrary number of stocks and arbitrary choice of numeraire. Although price bounds, V 0 (X0, X2 ,M) ≤ V 0(X0, X2 ,M), will never yield a market misprice, our models will allow an investor to determine the amount of risk associated with an initial investment v.
Highlights
The theory of asset pricing has been studied extensively throughout the academic and financial literature, with a large emphasis being on the study of options pricing
We limit ourselves to d = 2 in this paper, the generalized notation goes to show that trajectory market models are not limited as we provide the framework for an arbtirary number of assets
The most basic model will be limited to the coordinates (Xi1, Xi2, i), which will act as the base model, as it contains the least amount of infomation for the investor
Summary
X∗(x, T , i) and X∗(x, T , i) for i ≥ 0 represent the maximum and minimum ratio of normed vector changes that occurs at the i’th δ-movement within the charts x(t), respectively This constraint will limit the amount our trajectory asset values may fluctuate since an initial portfolio rebalancing (i = 0). W ∗(x, T , ρ) and W∗(x, T , ρ) for ρ ∈ [0, T ] represent the maximum and minimum amount of accumulated variation between historical portfolio rebalancing times This is used to limit the amount that model asset values X1, X2 can vary after time ρ has elapsed.
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