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

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Summary

Bibliography vii

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.

Introduction
Logical constraints from adopting an operational setting and a superhedging methodology
Possibilities to Deploy the General Methodology
General, Discrete, Trajectory Based Models
Multidimensional Trajectorial Markets with Arbitrary Numeraire
No-Arbitrage and 0-Neutrality
Local No-Arbitrage, Local 0-Neutrality and Geometric Characterizations
Price Bounds for One Asset Relative to Another Asset
Operational Setting
Chart Values
Charts and Investors’ Operations
Discretization of Observed Quantities
Unfolding Chart Parameters
Observable Worst-Case Pruning Constraints
Type 0 Pruning Constraint
Type I Pruning Constraints
Type II Pruning Constraints
Trajectory Sets
Trajectory Model Specification
Trajectory Termination
Type 0 Model
Type I Model
Type II Model
Issues with Pruning of Possible Xi+1: Dealing with Arbitrage Nodes
Nested Model Values
Risk Taking in Trajectory Models
Rounded Chart Values
Calibration of δ, δ0, δup, δdown, δ1, δ2, ν0
Comments on Worst-Case Calibration and Estimation
Worst-Case Estimate for NE
Worst-Case Estimate for Pruning Constraints
Worst-Case Estimate for Stopping Number
Data Employed
Objective 1 - Constructing the Trajectory Set
Objective 2 - Superhedging Methodology
Objective 1 - Constructing our Trajectory Set
Discussion and Conclusion
Full Text
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