Martingale OT and Options Pricing

Цели урока

• Formulate martingale OT and derive its connection to model-free option pricing via the dual semi-static hedging problem
• Prove strong duality: the option upper bound equals the cost of the super-replicating portfolio
• Apply entropic MOT for numerical solution while preserving the martingale constraint

Предварительные знания

• Kantorovich OT duality and Kantorovich potentials
• Martingale measures and risk-neutral pricing (Black-Scholes framework)
• Convex order and stochastic dominance
• Linear programming: feasibility and duality

• Multi-Marginal OT
• Causal OT

After the 2008 financial crisis, regulators required banks to price exotic derivatives without model assumptions. Martingale OT provides exactly this: model-free price bounds derived only from observed vanilla option prices, with no assumptions about volatility dynamics.

• Goldman Sachs and Societe Generale use MOT-based bounds for regulatory capital calculations on path-dependent derivatives
• Volatility surface calibration: checking that implied vol surfaces satisfy the convex order condition (no-arbitrage) is an MOT feasibility problem
• Robust portfolio optimization: worst-case portfolio returns over all martingale measures calibrated to market prices
• Cryptocurrency derivatives: MOT bounds are especially important in crypto markets where model assumptions break down frequently

From Skorokhod Embedding to Model-Free Finance

The Skorokhod embedding problem (1961) asks how to stop a Brownian motion to match a target distribution - this is an early form of martingale transport. Model-free finance was pioneered by Dupire (1994) and Derman-Kani (1994) for single-period options. The multi-period model-free bounds problem remained open until Beiglbock, Henry-Labordere, and Penkner (2013) formulated it as martingale OT. They proved strong duality and characterized extremal couplings. Beiglbock and Juillet (2016) then discovered the left-curtain coupling as the canonical solution for the 1D problem. Today the field bridges pure OT mathematics, stochastic analysis, and quantitative finance.

Martingale OT: Transport Under No-Arbitrage

Martingale optimal transport adds a financial no-arbitrage constraint to the Kantorovich problem: the coupling must be a martingale measure. This connects OT theory to model-free derivatives pricing and robust finance.

The martingale constraint requires that nu is a dilation of mu (mu is stochastically dominated by nu in convex order). This is the Strassen theorem: pi_M is non-empty iff mu <=_cx nu.

Model-Free Derivative Bounds via MOT Duality

The dual of MOT is a semi-static hedging problem. The hedging portfolio consists of static vanilla option positions (calibrated to market) plus a dynamic delta hedge. Model-free bounds are the cheapest super-replicating portfolios of this form.

In practice, market quotes for vanilla options at multiple strikes directly encode the marginal distributions via the Breeden-Litzenberger formula: nu(K) = d^2 C / dK^2. MOT takes this as input and requires no model assumption on the dynamics.

Numerical Methods for MOT

MOT on paper is an LP with a martingale constraint. In practice: how to solve it for real markets with continuous distributions? Three approaches: discretization (standard), Sinkhorn with a soft martingale penalty, neural-network dual variables. Each is a tradeoff between accuracy and scale.

Discretization of MOT on a grid requires careful verification: the martingale constraint E[Y|X]=X must hold for discrete transition probabilities. The discretization error E[Y|X=x_i] = x_i is violated on a coarse grid.

Finance as a Laboratory for Transport Theory

Martingale OT shows that optimal transport theory is not just a mathematical tool for ML - it is the correct language for model-free financial mathematics. The duality between primal martingale couplings and dual semi-static hedges mirrors the Kantorovich duality. The left-curtain coupling mirrors the Brenier map. Financial markets have inadvertently been doing OT for decades; the theory just caught up.

• Optimal Transport — Related topic

Итоги

• Martingale OT adds constraint E[Y|X=x]=x (no-arbitrage) to the Kantorovich coupling problem
• The dual of MOT is a semi-static hedging problem: cheapest super-replicating portfolio of vanilla options plus a delta hedge
• Strong duality holds: option upper bound = minimum super-hedge cost
• Entropic regularization enables Sinkhorn-based numerical solution while keeping the martingale constraint exact

Вопросы для размышления

• The martingale constraint requires E[Y|X=x]=x. What happens to the MOT problem if this constraint is relaxed to E[Y|X=x] in [x-delta, x+delta]? Does the feasible set enlarge and do the price bounds widen?
• In finance, the convex order condition mu <=_cx nu must hold for Pi_M to be non-empty. Empirically, are option-implied marginal distributions always in convex order across maturities? What happens when they are not?
• The left-curtain coupling maximizes expected payoff for functions increasing in Y. What coupling minimizes the same class of payoffs, and what is its geometric interpretation?

Связанные уроки

• ot-26-multi-marginal — Martingale OT is MOT with additional martingale constraints on the coupling
• ot-01-monge — MOT builds on the Monge-Kantorovich primal-dual framework
• ot-28 — Causal OT generalizes martingale OT from the martingale constraint to general causality constraints
• ot-02-kantorovich