Robust Pricing of Worst-Of Autocallable Options

• The paper presents a robust, model-independent framework for pricing worst-of autocallable options using multi-marginal martingale optimal transport.
• It leverages entropic regularization and sequential martingale embedding to efficiently handle high-dimensional, path-dependent optimization problems.
• Practical insights include detailed recommendations for discretization, state augmentation, and convergence monitoring in exotic derivative pricing.

Robust pricing of worst-of autocallable options concerns the computation of model-independent super-replication price bounds for structured products whose payoff depends on the minimum-performing asset in a basket and incorporates early redemption features. The structured multi-marginal martingale optimal transport (MOT) framework enables such calculations by exploiting information from liquid option-implied marginal distributions and martingale constraints, without committing to a specific dynamics for the underlying assets. Recent advances leverage entropic regularization and sequential martingale structures to make these high-dimensional, path-dependent, and challenging optimization problems tractable, particularly for exotic derivatives such as worst-of autocallables (Engström et al., 2024).

1. General Primal Formulation

The robust pricing problem is set in discrete time $t = 0,\,1,\,\ldots,\,T$, where $S_t\in\mathbb{R}^d$ is the vector of underlying asset spot prices. Marginals $\mu_t = \operatorname{Law}(S_t)$ are inferred from option market data at selected times $\mathcal{T} \subset \{0,1,\ldots,T\}$. The robust super-replication price seeks the highest expected payoff over all martingale measures $\pi$ matching these marginals. This is formulated as a multi-marginal MOT problem: $$\sup_{\pi} \, \mathbb{E}_\pi\bigl[\phi(S_0,\ldots,S_T)\bigr]$$ subject to: $$\pi\circ S_t^{-1} = \mu_t\quad(t\in\mathcal{T}), \qquad \mathbb{E}_\pi[S_t\mid S_{t-1},\ldots,S_0] = S_{t-1} \quad (t=1,\ldots,T).$$ The path-dependent payoff $\phi$ encodes the autocallable’s structure. The constraints ensure consistency with observed market data and absence of arbitrage through the martingale property.

2. Entropic Regularization for Numerical Feasibility

Direct solution of the MOT problem is computationally intractable for high dimension or many time steps. Entropic regularization introduces a penalization term $\mathrm{KL}(\pi\Vert\otimes_{t=0}^T m_t)$, where $m_t$ is a (typically uniform) reference measure, and $\varepsilon>0$ is the regularization strength. The regularized objective becomes: $$\sup_\pi \left\{ \mathbb{E}_\pi[\phi(S)] - \varepsilon\, \mathrm{KL}\left(\pi \,\Vert\, \bigotimes_{t=0}^T m_t\right) \right\}$$ subject to the same constraints. On a discretized grid ($n$ points per time step), this results in a convex optimization over a transport tensor $Q$, with linear marginal and martingale constraints, while $\varepsilon$ controls the bias-variance trade-off in the approximation.

3. Sequential Martingale and Markovian Embedding

For payoff functionals $\phi$ with limited path-dependence—such as those only entangling consecutive $(S_{t-1},S_t)$ or using a compact state augmentation $X_t$—the full history-dependent martingale constraint can be replaced by a reduced, Markovian one: $\mathbb{E}[S_t | S_{t-1}, X_{t-1}] = S_{t-1}$ The Markovian embedding constructs an extended state $\text{state}_t = (S_t, X_t)$, reducing the original high-dimensional problem to a chain of conditional measures linked only at adjacent time steps. This dynamic decomposition allows the joint law $\pi$ to factor via transition kernels: $$\pi(d\,\text{state}_0,\ldots,d\,\text{state}_T) = \mu_0(d\,\text{state}_0) \prod_{t=1}^T \kappa_t(\text{state}_{t-1}, d\,\text{state}_t)$$ with the martingale condition applying only pairwise between consecutive states.

4. Sinkhorn-Type Algorithm

After regularization, dual variables separate into static marginal Lagrange multipliers $\{\lambda_t\}_{t\in\mathcal{T}}$ and dynamic martingale multipliers $\{\gamma_t\}_{t=0}^{T-1}$. The dual maximization involves: $$\max_{\lambda, \gamma} \sum_{t\in\mathcal{T}} \lambda_t^\top m_t - \varepsilon \left\langle K, U^\lambda \odot G^\gamma \right\rangle$$ with $$K_{i_{t-1},i_t}^{(t)} = \exp(-C_t(i_{t-1},i_t)/\varepsilon)$$, $$U^\lambda = \prod_{t\in \mathcal{T}} \exp(\lambda_t/\varepsilon)$$, $$G^\gamma_{t-1,t} = \exp(\gamma_{t-1}(i_{t-1})\Delta_t(i_{t-1},i_t)/\varepsilon)$$. The numerical solution uses coordinate-ascent in $(\lambda,\gamma)$:

1. Initialize $\lambda_t \gets 0$, $\gamma_t \gets 0$.
2. Repeat until convergence:
   • Marginal update for each $t\in \mathcal{T}$: $$u_t \gets m_t / P_t(K \odot U^\lambda \odot G^\gamma)$$.
   • Martingale update for each $t=0,\ldots,T-1$: solve $$[P_{t,t+1}(K \odot U^\lambda \odot G^\gamma) \odot \Delta_{t+1}]\,1 = 0$$ for $\gamma_t$.

Efficient forward/backward recursions (complexity $O(T n^2)$ per sweep) exploit the chain structure. Under mild conditions, coordinate-ascent converges linearly. A few hundred iterations typically suffice for residuals in the range $10^{-6}$–$10^{-8}$.

5. Specialization to Robust Worst-Of Autocallable Payoffs

A worst-of autocallable typically references a basket of $d$ assets and contains two components:

• Observation: At each $t=1,\ldots,T-1$, if $\min_{1\leq i \leq d} S^i_t \geq B_{\mathrm{obs}}$, the structure is automatically redeemed at coupon $c_t$.
• Survival to Maturity: If not called, at $T$ the payoff is $F(\min_{i} S^i_T)$ (commonly $F(x) = \max(x-K,0)$).

To encode the autocallable mechanism, an extended state variable $X_t$ is introduced: $$X_0 = 1, \quad X_t = \begin{cases} 0, & \text{if redeemed earlier} \ X_{t-1}\mathbf{1}_{\{\min_i S^i_t \ge B_{\mathrm{obs}}\}}, & 1\leq t \leq T-1 \ X_{t-1} F(\min_i S^i_T), & t = T \end{cases}$$ The total payoff functional is: $$\phi(S, X) = \sum_{t=1}^{T-1} c_t X_{t-1} \mathbf{1}_{\{\min_i S^i_t \ge B_{\mathrm{obs}}\}} + X_{T-1} F(\min_i S^i_T)$$ This $\phi$ conforms to the pairwise-decoupled form required by the entropic-Sinkhorn algorithm.

6. Implementation Recommendations

Practical implementation requires careful discretization and tuning:

• Marginal discretization: Implied densities $\mu_t$ are extracted from call smiles and discretized on grids of 50–300 points.
• Extended grid: For each $X_t$, use indicator states (e.g., $\{0,1\}$) or 1–20 quantiles of the relevant minimum.
• Regularization parameter $\varepsilon$: Start with $\varepsilon \sim 10^{-2}$ and reduce until residuals stall or MC error $O(\varepsilon\log n)$ emerges.
• Computational cost: Complexity per iteration scales as $O(n^2T)$ (with $n\sim 100$, $T\sim 20$ typical).
• Error monitoring: Track $\max$ marginal violation $\|P_t(Q) - m_t\|_\infty$ and martingale residual $\|(P_{t,t+1}(Q)\odot\Delta_{t+1})1\|_\infty$, terminating when both fall below $10^{-6}$ or chosen tolerance.

An experienced practitioner can generalize this methodology to other path-dependent exotics via suitable state augmentation $X_t$ and corresponding definition of pairwise payoff blocks $\phi_t$ (Engström et al., 2024).

7. Context and Extensions

The structured MOT framework with entropic regularization represents a significant advance in robust, model-independent pricing for complex exotics. The practical recipes for worst-of autocallables highlight the tractability of path-dependent, high-dimensional robust pricing within this paradigm. The approach is flexible, as state augmentation and pairwise payoff decompositions extend naturally to a range of other exotics, contingent on the ability to obtain market-implied marginals and efficient discretization. The numerical performance enables routine robust price computations for problem sizes ($n\sim100$, $T\sim 20$) in line with realistic derivative specifications (Engström et al., 2024).