Paper: arXiv 2307.00807

Authors: Tongseok Lim

Abstract

The Black-Scholes-Merton model is a mathematical model for the dynamics of a financial market that includes derivative investment instruments, and its formula provides a theoretical price estimate of European-style options. The model’s fundamental idea is to eliminate risk by hedging the option by purchasing and selling the underlying asset in a specific way, that is, to replicate the payoff of the option with a portfolio (which continuously trades the underlying) whose value at each time can be verified. One of the most crucial, yet restrictive, assumptions for this task is that the market follows a geometric Brownian motion, which has been relaxed and generalized in various ways. The concept of robust finance revolves around developing models that account for uncertainties and variations in financial markets. Martingale Optimal Transport, which is an adaptation of the Optimal Transport theory to the robust financial framework, is one of the most prominent directions. In this paper, we consider market models with arbitrarily many underlying assets whose values are observed over arbitrarily many time periods, and demonstrates the existence of a portfolio sub- or super-hedging a general path-dependent derivative security in terms of trading European options and underlyings, as well as the portfolio replicating the derivative payoff when the market model yields the extremal price of the derivative given marginal distributions of the underlyings. In mathematical terms, this paper resolves the question of dual attainment for the multi-period vectorial martingale optimal transport problem.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper tackles a sophisticated theoretical question in mathematical finance by resolving dual attainment for multi-period vectorial martingale optimal transport. While highly advanced mathematically and conceptually novel, it is purely theoretical with virtually no empirical backtesting or real-world data application.

Research Flowchart

  flowchart TD
    A["Research Goal: Resolve dual attainment in multi-period, multi-asset Martingale Optimal Transport (MOT)"] --> B["Inputs: Observed marginal distributions of underlyings & option payoff structure"]
    B --> C["Methodology: Extend MOT framework to vector-valued, multi-period settings"]
    C --> D["Process: Construct static trades in European options & dynamic trades in underlyings"]
    D --> E["Outcome 1: Prove existence of optimal sub-/super-hedging portfolios for path-dependent derivatives"]
    D --> F["Outcome 2: Establish exact derivative replication under extreme market models"]