Full grid solution for multi-asset options pricing with tensor networks

TL;DR

This work presents a deterministic, full-grid approach to pricing and hedging high-dimensional multi-asset options by employing quantized tensor trains (QTT). It introduces two TT-based solvers: a time-stepping QTT method for European and American options with dense Greeks, and a space–time QTT method that solves the entire space–time system for European options, both leveraging TT-Cross and ALS to maintain polynomial-in-d ranks. The authors demonstrate accurate full-grid prices and Greeks for 3–5 assets on a laptop, with favorable trade-offs between space–time and time-stepping methods depending on the dimension, and discuss extensions to richer market models and potential for rapid re-pricing within a fixed grid. The results indicate QTT as a compelling deterministic alternative to sparse grids and Monte Carlo in genuinely high-dimensional Black–Scholes settings, enabling fast re-pricing, calibration, and risk monitoring on commodity hardware. These contributions offer practical impact for hedging and model validation by delivering full solution surfaces and dense sensitivities without stochastic noise.

Abstract

Pricing multi-asset options via the Black-Scholes PDE is limited by the curse of dimensionality: classical full-grid solvers scale exponentially in the number of underlyings and are effectively restricted to three assets. Practitioners typically rely on Monte Carlo methods for computing complex instrument involving multiple correlated underlyings. We show that quantized tensor trains (QTT) turn the d-asset Black-Scholes PDE into a tractable high-dimensional problem on a personal computer. We construct QTT representations of the operator, payoffs, and boundary conditions with ranks that scale polynomially in d and polylogarithmically in the grid size, and build two solvers: a time-stepping algorithm for European and American options and a space-time algorithm for European options. We compute full-grid prices and Greeks for correlated basket and max-min options in three to five dimensions with high accuracy. The methods introduced can comfortably be pushed to full-grid solutions on 10-15 underlyings, with further algorithmic optimization and more compute power.

Figures (6)

• Figure 1: Schematic overview of the QTT time-stepping pipeline for the $d$-asset Black--Scholes PDE. The grey box starts from Eq. \ref{['eq:multi_asset_BS']}, applies a backward Euler discretization in time and centered finite differences in the log–prices, and yields a linear system at each time level that is never formed explicitly. The red box builds a matrix-product-operator (MPO) representation of the discrete differential operator. The blue box constructs the QTT representation of the payoff. Using an analytical construction for the "blue terms" and TT-Cross to evaluate the $\max$ function. The green box imposes the boundary conditions at each time step: each boundary is encoded in QTT form using the same procedure as for the payoff. Finally, the purple box solves the resulting linear system in the QTT framework and, in the American case, applies the early-exercise condition and a rank reduction of the solution (lines 8--9 of Alg. \ref{['alg:qtt_ts_main']}).
• Figure 2: Rank convergence of the right-hand side approximation for the $d=3$ basket put. MSE versus capped TT-rank $\chi$. Increasing the number of cores per dimension $c$ improves accuracy at fixed $\chi$.
• Figure 3: QTT time-stepping American option
• Figure 4: QTT Time-Stepping Solver for the 2D European Black--Scholes Equation
• Figure 5: Dense vs. QTT payoff and pointwise squared error for the 3-asset worst-of put on a $(256)^{\times 3}$ grid (slice at fixed $S_3$). The QTT payoff is built via TT-Cross for $\min(S_1,S_2,S_3)$ and $\max(\cdot,0)$, then we truncate the rank at $12$ and $24$. As expected the error concentrates near the exercise kink but can be reduced by increasing the truncation rank.

Theorems & Definitions (1)

• Lemma 1: kazeev2012