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Computation of Robust Option Prices via Structured Multi-Marginal Martingale Optimal Transport

Linn Engström, Sigrid Källblad, Johan Karlsson

TL;DR

This work addresses robust, model-independent pricing bounds for exotic options by formulating the problem as a multi-marginal martingale optimal transport (MOT) with given market marginals. The authors exploit a structured payoff class to reformulate the MOT on an extended, Markovian state space and then discretize it into a tensor-based linear program, enabling entropic regularisation and efficient projections. A coordinate dual ascent algorithm leverages graph-structured costs to update dual variables efficiently, achieving strong duality and scalability to many marginals. The methodology is demonstrated on lookback, digital, and Asian options, and shows accurate bounds and convergence behavior as the number of marginals grows, illustrating its practical impact for robust pricing and risk management. Overall, the paper provides a scalable, structure-exploiting framework for robust option pricing under market-implied marginals, with potential to inform model risk and hedging strategies in complex path-dependent payoffs.

Abstract

We introduce an efficient computational framework for solving a class of multi-marginal martingale optimal transport problems, which includes many robust pricing problems of large financial interest. Such problems are typically computationally challenging due to the martingale constraint, however, by extending the state space we can identify them with problems that exhibit a certain sequential martingale structure. Our method exploits such structures in combination with entropic regularisation, enabling fast computation of optimal solutions and allowing us to solve problems with a large number of marginals. We demonstrate the method by using it for computing robust price bounds for different options, such as lookback options and Asian options.

Computation of Robust Option Prices via Structured Multi-Marginal Martingale Optimal Transport

TL;DR

This work addresses robust, model-independent pricing bounds for exotic options by formulating the problem as a multi-marginal martingale optimal transport (MOT) with given market marginals. The authors exploit a structured payoff class to reformulate the MOT on an extended, Markovian state space and then discretize it into a tensor-based linear program, enabling entropic regularisation and efficient projections. A coordinate dual ascent algorithm leverages graph-structured costs to update dual variables efficiently, achieving strong duality and scalability to many marginals. The methodology is demonstrated on lookback, digital, and Asian options, and shows accurate bounds and convergence behavior as the number of marginals grows, illustrating its practical impact for robust pricing and risk management. Overall, the paper provides a scalable, structure-exploiting framework for robust option pricing under market-implied marginals, with potential to inform model risk and hedging strategies in complex path-dependent payoffs.

Abstract

We introduce an efficient computational framework for solving a class of multi-marginal martingale optimal transport problems, which includes many robust pricing problems of large financial interest. Such problems are typically computationally challenging due to the martingale constraint, however, by extending the state space we can identify them with problems that exhibit a certain sequential martingale structure. Our method exploits such structures in combination with entropic regularisation, enabling fast computation of optimal solutions and allowing us to solve problems with a large number of marginals. We demonstrate the method by using it for computing robust price bounds for different options, such as lookback options and Asian options.
Paper Structure (19 sections, 10 theorems, 100 equations, 5 figures, 2 algorithms)

This paper contains 19 sections, 10 theorems, 100 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and background
  3. Problem formulation for structured payoff functions
  4. Financial examples
  5. Structured multi-marginal optimal transport
  6. Reformulation as a structured OT problem
  7. Markovian reformulation of the problem
  8. Formulation as a linear programming problem
  9. Solving the structured problem via regularisation
  10. Strong duality for the regularised problem
  11. Exploiting the structure for computing the projections
  12. Summary of the full algorithm
  13. Computational results
  14. Monotone transport
  15. Late and early transports
  16. ...and 4 more sections

Key Result

Proposition 2.9

Let the elements of the tensors $\mathbf{K}$ and $\mathbf{U}$ be of the form for matrices $K_t \in \mathbb{R}^{n \times n}$, $t \in [T] \backslash 0$, and vectors $u_t \in \mathbb{R}^n$, $t \in [T]$. Then,

Figures (5)

  • Figure 1: Optimal solutions to the upper bound bi-marginal MOT and OT problems corresponding to the payoff of a variance swap when the given marginals are uniform, as in \ref{['ex:sol_monotone']}. The computed solution of the MOT problem is displayed in \ref{['fig:sol_monotone_mtg']}; it exhibits the shape of the left-monotone transport. For comparison, the computed solution of the corresponding OT problem is displayed in \ref{['fig:sol_monotone_nonmtg']}; it displays the shape of the anti-monotone coupling.
  • Figure 2: Marginal distributions corresponding to the optimal solutions of the upper- and lower bound MOTs from \ref{['ex:sol_lateearly']}. The given initial marginal $\mu_0$ is displayed in light blue bars, while the given terminal marginal $\mu_T$ is shown in light red bars. The marginals corresponding to the computed transport plan are shown for $t \in \{0, 10, 20, 30, 40, 50\}$ with solid lines. \ref{['fig:sol_late']} corresponds to the lower bound martingale transport plan while \ref{['fig:sol_early']} corresponds to the upper bound martingale transport plan.
  • Figure 3: The given terminal marginal used in \ref{['ex:sol_maxofmax']} is displayed in \ref{['fig:sol_AY_marginal']}. It is centered in $0.5$ and its support is contained within $[0,1]$. The corresponding distribution of the maximum of the maximum, solved approximately for an increasing number of marginals $T+1$, is shown in \ref{['fig:sol_maxofmax_cdf']}. The corresponding distribution for the continuous-time case is included for reference (bold red). We note that the discrete-time computed solutions approach the continuous-time solution as the number of time steps used increases, and that the cumulative distribution function decreases. The corresponding problem values are shown as a function of the number of marginals $T+1$ in \ref{['fig:sol_maxofmax_price']}; the problem value for the continuous-time case is included for comparison (bold red). The maximum element of the computed residuals, obtained prior to updating the corresponding dual variable, for the marginal constraints (dashed line) and for some of the martingale constraints (bold line) are displayed in a linlog scale in \ref{['fig:sol_maxofmax_residuals']}, along with the threshold level used in Newton's method (black dashed line). Note the high level of accuracy used in the computations.
  • Figure 4: The computed upper bound on the probability that the digital option with barrier $B$ is not activated as a function of the barrier level $B$, as given in \ref{['ex:sol_mot_dig_ay']}. The cumulative distribution function of the maximum of the corresponding Azéma--Yor martingale is displayed in bold red for reference. The grey vertical lines illustrate the support of the price process contained within $[0.5, 0.92]$. Note that the resolution of the computed approximations follows this grid.
  • Figure 5: The marginals of the optimal couplings corresponding to the lower bound MOT problem of \ref{['ex:sol_asian_straddle']}, subject to marginal constraints on the marginals belonging to $\mathcal{T}$. Marginals subject to constraints are marked with a green background.

Theorems & Definitions (38)

  • Remark 2.1
  • Example 2.2: Lookback and barrier options
  • Example 2.3: Asian options
  • Example 2.4: Barrier options
  • Example 2.5: Variance swaps
  • Example 2.6: Parisian options
  • Example 2.7: Dual expiry options
  • Example 2.8: Cliquet options
  • Proposition 2.9: ElvanderHaaslerJakobssonKarlsson2020, Proposition 2
  • Theorem 3.1
  • ...and 28 more