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Research Notes

This page records the modeling perspective behind the package and clarifies the distinction between exact and regularized computations.

Problem Statement

Given two marginals mu1 and mu2, the objective is to find couplings of S1 and S2 such that:

  • S1 ~ mu1
  • S2 ~ mu2
  • E[S2 | S1] = S1

and then optimize a payoff under this martingale restriction.

This is a robust-pricing problem in which only the marginals and the martingale structure are fixed. The admissible price range is generated by all couplings consistent with these constraints.

Role Of The Martingale Constraint

Without the martingale condition, the problem reduces to a standard transport problem. The martingale restriction imposes preservation of conditional mean and substantially narrows the set of admissible couplings. This is the central structural constraint in the project.

Exact Discrete Formulation

The exact solver uses a discrete linear program. On grids x_i and y_j, with coupling weights pi_ij, the problem is:

maximize or minimize   sum_{i,j} pi_ij c(x_i, y_j)
subject to             sum_j pi_ij = mu_i
                       sum_i pi_ij = nu_j
                       sum_j pi_ij (y_j - x_i) = 0    for each i
                       pi_ij >= 0

A longer discussion of the discrete construction is provided in Discrete Formulation.

Entropic Regularization

The regularized solver is not intended to replace the LP benchmark. Its purpose is to provide a smooth approximation path across values of eps.

The interpretive distinction is:

  • the LP value is the discrete benchmark
  • the entropic value is an approximation path
  • the regularized objective includes entropy and is not identical to the raw expected payoff

This distinction is maintained explicitly throughout the package.

Several recurring patterns are visible in the shipped examples:

  • The reference |S2 - S1| case produces a nontrivial interval, approximately [0.6087, 0.9975].
  • The call-on-spread and put-on-spread examples have the same width in the current discretization but different interval levels.
  • The centered straddle produces one of the widest intervals in the gallery.
  • The quadratic spread nearly collapses to a single value in the current discrete setting.
  • Increasing the variance of the second marginal widens the robust interval in the absolute-spread and put-on-spread wide examples.
  • A broad spread straddle on the original support pair remains highly nontrivial despite its symmetry.

Conditions For Confidence

Results are most reliable when the following conditions hold simultaneously:

  • convex order indicates martingale feasibility
  • the exact LP solves cleanly
  • marginal and martingale errors are small
  • the regularized path moves toward the LP benchmark as eps decreases
  • the behavior is stable under nearby discretizations

Scope

The package is designed as a compact workbench for two-step discrete MOT experiments. It is not intended as:

  • a continuous-time MOT library
  • a general derivatives system
  • a universal framework for all transport problems

The emphasis is on precise, reproducible experiments in a deliberately limited setting.

Consequence Of The Package Rewrite

The move from notebook to package separated three layers that were previously intertwined:

  • mathematics
  • numerics
  • presentation

This separation improves reproducibility and makes it easier to distinguish exact statements from approximation results and implementation details.