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Monotone Mean-Variance Portfolio Selection in Semimartingale Markets: Martingale Method

Yuchen Li, Zongxia Liang, Shunzhi Pang

Abstract

We use the martingale method to discuss the relationship between mean-variance (MV) and monotone mean-variance (MMV) portfolio selections. We propose a unified framework to discuss the relationship in general financial markets without any specific setting or completeness requirement. We apply this framework to a semimartingale market and find that MV and MMV are consistent if and only if the variance-optimal signed martingale measure keeps non-negative. Further, we provide an example to show the application of our result.

Monotone Mean-Variance Portfolio Selection in Semimartingale Markets: Martingale Method

Abstract

We use the martingale method to discuss the relationship between mean-variance (MV) and monotone mean-variance (MMV) portfolio selections. We propose a unified framework to discuss the relationship in general financial markets without any specific setting or completeness requirement. We apply this framework to a semimartingale market and find that MV and MMV are consistent if and only if the variance-optimal signed martingale measure keeps non-negative. Further, we provide an example to show the application of our result.
Paper Structure (12 sections, 7 theorems, 61 equations)

This paper contains 12 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Complete Market
  3. MV Problem in Complete Market
  4. MMV Problem in Complete Market
  5. General Market
  6. General Framework
  7. Semimartingale Market
  8. MV Problem in Semimartingale Market
  9. MMV Problem in Semimartingale Market
  10. Example
  11. Conclusion
  12. Acknowledgments

Key Result

Theorem 1

In any complete market (with Assumption assump complete), solutions to MV Problem prob com MV 1 and MMV Problem prob com MMV 1 are the same.

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 4 more