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A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options
A. Galichon, P. Henry-Labordère, N. Touzi
TL;DR
The paper studies robust superhedging under volatility uncertainty with dynamic underlying trading and static trading of European calls across all strikes. It replaces the classical SEP route with a continuous-martingale optimal-transport duality, recovers known lookback bounds including Azéma–Yor optimality, and supports numerical approximation.
Problem
Robust superhedging is studied when the underlying is dynamically traded and all-strike European calls with common maturity are statically traded, while the process distribution remains unknown.
Method
The paper directly solves the robust superhedging problem through a Kantorovich-dual formulation as continuous martingale optimal transport, producing the hedging strategy and worst-case model.
Results
The formulation recovers the known robust upper bound for a class of lookback options and provides a new proof that the Azéma–Yor SEP solution attains it.
Takeaways & Limitations
Unlike the SEP approach, the framework imposes no time-change-invariance restriction on exotic payoffs and is suitable for numerical approximation.
Takeaways & Limitations
Explicit robust superhedging solutions should not generally be expected, and the stated duality framework relies on payoff regularity and integrability assumptions that may require extension.
Abstract
from arXiv · showhide
We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset, and statically trade European call options for all possible strikes with some given maturity. This problem is classically approached by means of the Skorohod Embedding Problem (SEP). Instead, we provide a dual formulation which converts the superhedging problem into a continuous martingale optimal transportation problem. We then show that this formulation allows us to recover previously known results about lookback options. In particular, our methodology induces a new proof of the optimality of Azéma-Yor solution of the SEP for a certain class of lookback options. Unlike the SEP technique, our approach applies to a large class of exotics and is suitable for numerical approximation techniques.
1. Introduction.
The paper studies robust superhedging with dynamic underlying-asset trading and static calls across all strikes, replacing the classical SEP route with a stochastic-control and continuous-martingale optimal-transport formulation. It recovers known lookback bounds while extending applicability beyond time-change-invariant exotics and supporting numerical approximation.
- Problem: The market combines unrestricted dynamic trading of underlying assets with static European calls of common maturity and all possible strikes.Under linearity and continuity, vanilla derivatives at maturity can be replicated from call prices, although the joint process distribution remains unknown.
- Existing approach: The classical SEP approach applies only to derivatives whose payoffs are invariant under time change.The paper avoids the time-change step rather than imposing that restriction on the exotic payoff.
- Contribution: The paper formulates robust superhedging through Kantorovich duality, relating it to stochastic control and continuous-martingale optimal transportation.The dual solution also provides an optimal hedging strategy and worst-case model.
- Applications: For lookback derivatives, the dual formulation recovers the known robust bound and gives a new presentation of Azéma–Yor optimality for a class of lookback options.It also recovers the robust superhedging cost for the forward lookback option.
- Implications: The optimal-transport formulation complements SEP by emphasizing the superhedging problem and producing the optimal semi-static superhedging strategy.Related work uses this strategy to extend Azéma–Yor results to multiple intermediate marginals.
- Applications: The approach is suitable for numerical approximation, including finite-difference or Monte Carlo treatment of the fixed-multiplier stochastic-control problem followed by optimization over the multiplier.The authors note that robust superhedging generally should not be expected to admit an explicit solution.
2. Model-free bounds of derivatives securities.
The section formulates model-free superhedging under martingale and volatility uncertainty, incorporating static trading in all-strike European options. Its dual formulation connects robust bounds to stochastic control and optimal transportation, while retaining access to hedging strategies and numerical approximation.
- 2.1. The probabilistic framework.: The market model uses uniformly integrable martingale measures for the underlying price process and defines the model-free superhedging bound as the minimal initial capital for a robust hedge.The framework includes admissible dynamic portfolios and a value-process interpretation of the no-arbitrage upper bound.
- 2.3. Dual formulation of the super-hedging bound.: Theorem 2.1 provides a dual representation of the robust superhedging problem for uniformly continuous claims under an integrability condition.When the bound is finite, the theorem also yields a superhedging portfolio and nondecreasing predictable processes.
- 2.4. Calibration adjusted no-arbitrage bound.: Static trading in European calls and puts across all strikes identifies a terminal marginal distribution µ and improves the corresponding no-arbitrage upper bound.A payoff λ(XT) has price µ(λ) and can be replicated using options across strikes.
- 2.3. Dual formulation of the super-hedging bound.: The dual formulation gives access to an optimal vanilla profile, a worst-case model, and the dynamic hedge obtained from the residual claim.The static component is λ*, while the dynamic component is obtained by representing ξ − λ*(XT).
- 2.5. Connection with optimal transportation theory.: For fixed λ, the dual problem is a singular stochastic control problem that can be approximated by finite differences or Monte Carlo methods, followed by optimization over λ.The numerical procedure therefore combines an inner control approximation with an additional multiplier-optimization stage.
- 2.5. Connection with optimal transportation theory.: The calibration-adjusted lower-bound problem is an optimal transportation problem over martingale models with fixed initial and terminal marginals, although direct equality with sub-hedging cost is not obvious.The paper notes that the relevant value function is convex in µ, but lower semicontinuity is not immediate without a uniform quadratic-variation bound.
3. Application to lookback derivatives.
The paper applies its stochastic-control and optimal-transport formulation to lookback derivatives, recovering the Azéma–Yor bound through a new dual approach. The analysis converts the problem to optimal stopping and characterizes the relevant obstacle using the barycenter function.
- The lookback application seeks to reproduce the known Azéma–Yor upper bound from the paper’s dual formulation.
- The robust superhedging problem is converted into an infinite-horizon optimal stopping problem.
- For admissible convex λ, the dynamic value function uλ is independent of time and can be represented through the associated stopping problem.
- The resulting obstacle construction and verification argument establish the optimal bound and recover the Azéma–Yor solution for lookback options.
- The candidate value function is constructed from g and λ, with a tangent continuation region determined by a free-boundary function ψ.
- Because λ may be nonsmooth, the free-boundary ODE is treated in a relaxed sense using the measure-valued second derivative of convex λ.
4. Forward start lookback options.
The forward-start lookback extension considers two maturities with call-price marginals ordered in convex order. Dynamic programming reduces the problem to the earlier lookback analysis, yielding an obstacle identified with Hobson’s solution.
- The forward-start lookback problem uses call prices at two maturities, with the later marginal dominating the earlier one in convex order.
- Its model-free superhedging cost combines dynamic trading in the underlying with static positions in calls from both maturities.
- The paper recovers Hobson’s previously known result using the stochastic-control approach.
- Dynamic programming reduces the value function for the two-maturity problem to the previously studied one-maturity problem at the first maturity.
- Pointwise minimization produces an optimal obstacle characterized through the functions c1, c2, and the associated free boundary.
- The resulting free boundary is recognized as Hobson’s solution, induces a Skorohod embedding, and gives the optimal upper bound.
5. Proof of the duality result.
The duality proof establishes the superhedging lower bound through supermartingale arguments and proves the converse under uniformly continuous payoffs. A dynamic value process is constructed, shown to be a supermartingale, and decomposed into hedging and increasing components.
- Any admissible superhedging wealth process is a supermartingale under every model, implying its initial capital dominates the expected payoff.
- The converse inequality requires the payoff ξ to belong to the uniformly continuous class UC(ΩX0).
- The proof uses regular conditional probability distributions, shifted canonical spaces, concatenation, and conditional model constructions in the nondominated setting.
- The dynamic value process V is defined by taking the supremum of conditional expected payoffs over admissible continuation models.
- V has a right-continuous adapted version and is a supermartingale under every admissible probability measure.
- A Doob–Meyer decomposition and aggregation argument yield a universal dynamic strategy and an increasing process that support the converse superhedging inequality.
- Uniform continuity is used essentially for measurability of the dynamic value process, and extensions require relaxing or replacing this condition.