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The McCormick martingale optimal transport

Erhan Bayraktar, Bingyan Han, Dominykas Norgilas

TL;DR

This study extendsMartingale optimal transport by incorporating causality constraints among assets, inspired by the nonanticipativity condition of stochastic processes, and proposes McCormick relaxations to ease the bicausal formulation.

Abstract

Martingale optimal transport (MOT) often yields broad price bounds for options, constraining their practical applicability. In this study, we extend MOT by incorporating causality constraints among assets, inspired by the nonanticipativity condition of stochastic processes. This, however, introduces a computationally challenging bilinear program. To tackle this issue, we propose McCormick relaxations to ease the bicausal formulation and refer to it as McCormick MOT. The primal attainment and strong duality of McCormick MOT are established under standard assumptions. Empirically, we apply McCormick MOT to basket and digital options. With natural bounds on probability masses, the average price reduction for basket options is approximately 1.08% to 3.90%. When tighter probability bounds are available, the reduction increases to 12.26%, compared to the classic MOT, which also incorporates tighter bounds. For most dates considered, there are basket options with suitable payoffs, where the price reduction exceeds 10.00%. For digital options, McCormick MOT results in an average price reduction of over 20.00%, with the best case exceeding 99.00%.

The McCormick martingale optimal transport

TL;DR

This study extendsMartingale optimal transport by incorporating causality constraints among assets, inspired by the nonanticipativity condition of stochastic processes, and proposes McCormick relaxations to ease the bicausal formulation.

Abstract

Martingale optimal transport (MOT) often yields broad price bounds for options, constraining their practical applicability. In this study, we extend MOT by incorporating causality constraints among assets, inspired by the nonanticipativity condition of stochastic processes. This, however, introduces a computationally challenging bilinear program. To tackle this issue, we propose McCormick relaxations to ease the bicausal formulation and refer to it as McCormick MOT. The primal attainment and strong duality of McCormick MOT are established under standard assumptions. Empirically, we apply McCormick MOT to basket and digital options. With natural bounds on probability masses, the average price reduction for basket options is approximately 1.08% to 3.90%. When tighter probability bounds are available, the reduction increases to 12.26%, compared to the classic MOT, which also incorporates tighter bounds. For most dates considered, there are basket options with suitable payoffs, where the price reduction exceeds 10.00%. For digital options, McCormick MOT results in an average price reduction of over 20.00%, with the best case exceeding 99.00%.
Paper Structure (22 sections, 8 theorems, 73 equations, 10 figures, 8 tables)

This paper contains 22 sections, 8 theorems, 73 equations, 10 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Motivation and implication of causality
  4. Nonanticipativity condition
  5. Non-convexity
  6. Bicausal MOT in the finite discrete case
  7. McCormick relaxation
  8. The finite discrete case
  9. The absolutely continuous case with capacity constraints
  10. The strong duality
  11. Numerical study
  12. An illustrating example
  13. A calibration method of risk-neutral densities
  14. Basket options
  15. Equal weights
  16. ...and 7 more sections

Key Result

Proposition 3.1

Suppose $\pi \in \Pi(\bar{\mu}, \bar{\nu})$ and the following martingale condition holds with the filtration generated by each asset: If $\pi$ is bicausal, then eq:mart_ind implies that the martingale condition eq:mart under the filtration generated by $(X, Y)$ also holds.

Figures (10)

  • Figure 1: GILD and GSK as underlying assets
  • Figure 2: JPM and MS as underlying assets
  • Figure 4: Minimum ratio achievable for a given weight $w$ across 247 dates.
  • Figure 5: Minimum ratio achievable on a given date across 100 weight parameters.
  • Figure 6: Average ratios across all dates with fixed weight parameters.
  • ...and 5 more figures

Theorems & Definitions (18)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • Proposition 4.2: aspremont2003relaxationsvandenberghe1996semidefinite
  • Definition 5.1
  • Remark 5.2
  • Remark 5.3
  • Definition 5.4
  • Theorem 5.5
  • Lemma 5.6
  • ...and 8 more