Martingale Optimal Transport¶

Background¶

martingale_optimal_transport.py

This module implements Martingale Optimal Transport.

Author: Financial Math Library

Code¶

```python

-- coding: utf-8 --¶

""" martingale_optimal_transport.py

This module implements Martingale Optimal Transport.

Author: Financial Math Library """

import numpy as np import matplotlib.pyplot as plt

======================================================================¶

def martingale_optimal_transport(): """ Martingale Optimal Transport.

This function demonstrates the key concepts and computational techniques
for martingale optimal transport.

Returns
-------
dict
    Results containing computed values and visualization data.
"""
# Implementation of Martingale Optimal Transport
print(f"Computing Martingale Optimal Transport...")

# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)

# Core computation logic
results = {
    "time_points": time_points,
    "description": "Martingale Optimal Transport"
}

return results

def main(): """Main execution function.""" results = martingale_optimal_transport()

# Create visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(results["time_points"], "b-", linewidth=2)
ax.set_xlabel("Time")
ax.set_ylabel("Value")
ax.set_title("Martingale Optimal Transport")
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig("/tmp/martingale_optimal_transport.png", dpi=150)
print(f"Figure saved to /tmp/martingale_optimal_transport.png")
plt.close()

return results

if name == "main": main() ```

Exercises¶

Exercise 1. Martingale optimal transport (MOT) seeks the coupling of two marginal distributions that maximizes or minimizes the expected payoff subject to a martingale constraint. State the primal MOT problem for bounding the price of an exotic option.

Solution to Exercise 1

Let \(\mu_1\) and \(\mu_2\) be the marginal distributions of the asset at times \(T_1\) and \(T_2\) (implied by vanilla option prices). The primal MOT problem for the upper bound is:

\[ \sup_{\pi \in \Pi_M(\mu_1, \mu_2)} \int c(x, y)\,d\pi(x, y), \]

where \(\Pi_M(\mu_1, \mu_2)\) is the set of all joint distributions with marginals \(\mu_1, \mu_2\) satisfying the martingale constraint \(\mathbb{E}[Y \mid X = x] = x\) (the conditional expectation of the future price equals the current price). The function \(c(x,y)\) is the exotic option payoff.

Exercise 2. Explain why the martingale constraint \(\mathbb{E}[S_{T_2} \mid S_{T_1}] = S_{T_1}\) is necessary for no-arbitrage pricing bounds.

Solution to Exercise 2

Under any risk-neutral measure, discounted asset prices are martingales. For undiscounted prices (with zero rates for simplicity), this means \(\mathbb{E}^{\mathbb{Q}}[S_{T_2} \mid S_{T_1}] = S_{T_1}\). Any coupling that violates this constraint corresponds to a model that admits arbitrage. By restricting to martingale couplings, we ensure that all considered models are arbitrage-free. The resulting price bounds are the tightest possible given only the information from vanilla option markets.

Exercise 3. For a forward-start option with payoff \(\max(S_{T_2}/S_{T_1} - K, 0)\), explain why the MOT bounds are wider than for a plain vanilla option.

Solution to Exercise 3

A vanilla option depends on \(S_T\) only at one time, so its price is uniquely determined by the marginal distribution \(\mu_T\) (which is fixed by vanilla option prices). A forward-start option depends on the joint distribution of \((S_{T_1}, S_{T_2})\), which is not uniquely determined by the marginals. Different martingale couplings produce different prices, creating a range \([\text{lower bound}, \text{upper bound}]\). The width of this range reflects the model uncertainty inherent in the dependence structure between \(S_{T_1}\) and \(S_{T_2}\).

Exercise 4. Describe the dual formulation of the MOT problem and its financial interpretation as a semi-static hedging portfolio.

Solution to Exercise 4

By duality, the upper bound equals:

\[ \inf_{\phi, \psi, h} \left\{\int \phi(x)\,d\mu_1(x) + \int \psi(y)\,d\mu_2(y) : \phi(x) + \psi(y) + h(x)(y - x) \geq c(x, y)\right\}. \]

The functions \(\phi\) and \(\psi\) represent static positions in vanilla options at \(T_1\) and \(T_2\), and \(h(x)\) represents a dynamic hedge (delta hedge at \(T_1\) depending on \(S_{T_1}\)). The dual says: the tightest upper bound equals the cheapest super-replicating portfolio using vanillas and delta hedging. This connects the abstract transport problem to practical hedging.