Paper: arXiv 2307.14887

Abstract

Introduced in the late 90s, the passport option gives its holder the right to trade in a market and receive any positive gain in the resulting traded account at maturity. Pricing the option amounts to solving a stochastic control problem that for $d>1$ risky assets remains an open problem. Even in a correlated Black-Scholes (BS) market with $d=2$ risky assets, no optimal trading strategy has been derived in closed form. In this paper, we derive a discrete-time solution for multi-dimensional BS markets with uncorrelated assets. Moreover, inspired by the success of deep reinforcement learning in, e.g., board games, we propose two machine learning-powered approaches to pricing general options on a portfolio value in general markets. These approaches prove to be successful for pricing the passport option in one-dimensional and multi-dimensional uncorrelated BS markets.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces complex stochastic control and multi-dimensional partial differential equations with extensive mathematical derivations, but its empirical validation is limited to simulated data without reported backtests or statistical performance metrics.

Research Flowchart

  flowchart TD
  A["Research Goal: Price<br>Multi-dim Passport Options"] --> B{"Methodology Approach"}
  B --> C["Analytical: Discrete-Time Solution<br>Uncorrelated BS Assets"]
  B --> D["ML: Deep Reinforcement Learning<br>Agent-based Pricing"]
  
  C --> E["Computational Process:<br>Stochastic Control Optimization"]
  D --> F["Computational Process:<br>Reinforcement Learning Simulation"]
  
  E --> G["Key Findings/Outcomes"]
  F --> G
  
  G --> H["✓ Closed-form price for uncorrelated assets<br>✓ ML successfully prices 1D & multi-dim options<br>✓ Framework for general markets"]