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Market Viability and Completeness for Multinomial Models

Nahuel I. Arca

Abstract

In this paper we aim to study viability and completeness in finite markets. In order to do that, we characterize the set of equivalent martingale measures of two-period markets as convex combinations of a finite number of martingale measures. We provide an algorithm for finding such measures, that can be applied in other problems of convex geometry, and represents the starting point for a study of such characterizations of convex sets' intersections. We apply these results to the study of a discrete-time version of the Korn-Kreer-Lenssen model, and give an example of the limitations of using discrete-time models to understand continuous-time ones.

Market Viability and Completeness for Multinomial Models

Abstract

In this paper we aim to study viability and completeness in finite markets. In order to do that, we characterize the set of equivalent martingale measures of two-period markets as convex combinations of a finite number of martingale measures. We provide an algorithm for finding such measures, that can be applied in other problems of convex geometry, and represents the starting point for a study of such characterizations of convex sets' intersections. We apply these results to the study of a discrete-time version of the Korn-Kreer-Lenssen model, and give an example of the limitations of using discrete-time models to understand continuous-time ones.

Paper Structure

This paper contains 8 sections, 10 theorems, 76 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Theory
  3. Classification of martingale measures
  4. More trading dates
  5. Concrete examples
  6. Application
  7. The Korn–Kreer–Lenssen model
  8. Conclusions

Key Result

Proposition 1

The market is arbitrage-free if and only if there exists $q\in\mathbb{R}_{>0}^b$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Binomial model with 3 trading dates.
  • Figure 2: Trinomial tree model with 3 trading dates and $ud=1$.
  • Figure 3: Information tree of the binomial model with 3 trading dates.
  • Figure 4: Information tree of the trinomial tree model with 3 trading dates.
  • Figure 5: The generators are $p^1$ and $p^2$.

Theorems & Definitions (25)

  • Proposition 1
  • Proposition 2
  • Theorem 3: Krein-Milman
  • Definition 1
  • Proposition 4
  • Remark
  • Remark
  • Example 1
  • Proposition 5
  • proof
  • ...and 15 more