Paper: arXiv 2609.39256

Authors: Mathias Beiglböck, Manuel Hasenbichler, Gudmund Pammer

Abstract

European option smiles determine the risk-neutral marginal laws of an asset, but not their intertemporal coupling, which is decisive for many applications. The Bass martingale construction selects, among all calibrated martingales, the one closest to Bachelier dynamics; it permits fast calibration at discrete maturities and recovers the Dupire local-volatility (LV) model as the maturity grid is refined. This article develops a modular calibration overlay for existing stochastic and path-dependent volatility models. We recursively construct a martingale that matches all prescribed marginals exactly while remaining as close as possible, in an adapted Knothe-Rosenblatt sense, to the reference dynamics. As with the Bass LV model, each calibration step is amenable to an efficient Martingale Sinkhorn algorithm. We develop the theoretical foundations, numerical implementation and consider convergence to the SLV model. We also benchmark the method for Heston and Bergomi finite-factor path-dependent volatility dynamics and develop the multi-asset extension.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical and novel approach to calibrating stochastic local volatility models, building on advanced concepts like optimal transport and martingale Sinkhorn algorithms. While strong on theoretical foundations and numerical implementation, the empirical rigor is demonstrated through benchmarking rather than extensive backtesting. The modular design and multi-asset extension enhance its practical applicability.

Research Flowchart

  flowchart TD
    A[Research Goal/Question: Calibrate SLV Models Efficiently] --> B{Key Methodology: Stochastic Knothe-Rosenblatt}
    B --> C[Inputs: European Option Smiles, Existing SLV/Path-Dependent Volatility Models (Heston, Bergomi)]
    C --> D{Computational Process: Recursive Martingale Construction & Martingale Sinkhorn Algorithm}
    D --> E[Outcomes: Fast Calibration, Exact Marginal Matching, Convergence to SLV, Multi-asset Extension]

Cited References (93)

  1. A nonlinear partial integro-differential equation from mathematical finance (2010) — link
  2. Existence of $q$-Bass martingales in the semidiscrete setting (2026) — link
  3. Fixed Points for the $q$-Bass Martingale: Existence, Stability, and Convergence (2026) — link
  4. Calibration of the Bass local volatility model — → 2311.14567
  5. Neural optimal transport with general cost functionals (2024)
  6. Calibrating volatility surfaces via relative-entropy minimization (1997)
  7. Skorokhod imbedding via stochastic integrals (1983) — link
  8. Jumps and stochastic volatility: exchange rate processes implicit in D eutsche M ark options (1996)
and 85 more
  1. Adapted W asserstein distances and stability in mathematical finance (2020) — link
  2. Bridging classical and martingale Schr$\backslash$" odinger bridges — → 2604.01299
  3. Martingale B enamou- B renier: A probabilistic perspective (2020)
  4. Causal transport in discrete time and applications (2017)
  5. Existence, duality, and cyclical monotonicity for weak transport costs (2019)
  6. Existence, duality, and cyclical monotonicity for weak transport costs (2019)
  7. Existence of B ass martingales and the martingale B enamou– B renier problem in $\mathbb R ^d$ (2023) — link
  8. Existence of B ass Martingales and the Martingale B enamou– B renier Problem in $\mathbb R ^d$ (2026) — link
  9. Geometric martingale Benamou–Brenier transport and geometric Bass martingales — → 2406.04016
  10. The $L^2$ gradient flow of the Bass functional in martingale optimal transport (2025)
  11. The B ass functional of martingale transport (2025)
  12. Stochastic Volatility Modeling (2016)
  13. Stability of Weak Martingale Optimal Transport in $\mathbb R ^d$ (2026)
  14. Model-independent bounds for option prices—a mass transport approach (2013)
  15. Stability of the weak martingale optimal transport problem (2023)
  16. On a Problem of Optimal Transport under Marginal Martingale Constraints (2016)
  17. Complete Duality for Martingale Optimal Transport on the Line (2017) — link
  18. The K nothe- R osenblatt distance and its induced topology (2023) — link
  19. Change of numeraire for weak martingale transport — → 2406.07523
  20. The Fundamental Theorem of Weak Optimal Transport (2025) — link
  21. From B achelier to D upire via optimal transport (2022)
  22. Prices of State-contingent Claims Implicit in Option Prices (1978)
  23. From K nothe’s transport to B renier’s map and a continuation method for optimal transport (2010)
  24. Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization (2025) — link
  25. Convergence of S inkhorn’s algorithm for entropic martingale optimal transport problem (2026)
  26. Bass construction with multi-marginals: Lightspeed computation in a new local volatility model (2021)
  27. Calibration of a hybrid local-stochastic volatility stochastic rates model with a control variate particle method (2019)
  28. Irreducible convex paving for decomposition of multidimensional martingale transport plans (2019) — link
  29. Martingale optimal transport and robust hedging in continuous time (2014)
  30. Pricing with a smile (1994)
  31. Pricing autocallables under local-stochastic volatility (2022)
  32. The Volatility Surface: A Practitioner’s Guide (2006)
  33. Pricing and Calibration in the 4-Factor Path-Dependent Volatility Model — → 2406.02319
  34. A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options (2014)
  35. Stochastic optimization for large-scale optimal transport (2016)
  36. Structure of optimal martingale transport plans in general dimensions (2019) — link
  37. Kantorovich duality for general transport costs and applications (2017)
  38. Dispersion-constrained martingale S chr "o dinger problems and the exact joint S&P 500/ VIX smile calibration puzzle (2024)
  39. Nonlinear option pricing (2013)
  40. Volatility is (mostly) path-dependent (2023) — link
  41. Path dependent optimal transport and model calibration on exotic derivatives (2021)
  42. Local volatility calibration by optimal transport (2019)
  43. Calibration of local-stochastic volatility models by optimal transport (2022)
  44. Computational methods for martingale optimal transport problems (2019)
  45. The smile calibration problem solved (2011)
  46. Mimicking the one-dimensional marginal distributions of processes having an It ^o differential (1986) — link
  47. The Martingale S inkhorn Algorithm — → 2310.13797
  48. Managing smile risk (2002)
  49. Duality and Complementary Slackness for Weak Martingale Optimal Transport (2026)
  50. A weak transport approach to the S chr "o dinger– B ass bridge (2026) — link
  51. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  52. Robust hedging of the lookback option (1998)
  53. Option pricing with linear market impact and nonlinear B lack- S choles equations (2018) — link
  54. Robust bounds for forward start options (2012)
  55. A B enamou– B renier formulation of martingale optimal transport (2019) — link
  56. Pricing exotics under the smile (1999)
  57. An extension of martingale transport and stability in robust finance — → 2304.09551
  58. Existence of a calibrated regime switching local volatility model (2020)
  59. Contributions to the theory of convex bodies. (1957)
  60. Neural optimal transport (2022) — link
  61. Inverting the M arkovian projection, with an application to local stochastic volatility models (2020)
  62. Inverting the Markovian projection, with an application to local stochastic volatility models (2020)
  63. The vol smile problem (2002)
  64. Bid–Ask Martingale Optimal Transport — → 2603.24605
  65. Fitting martingales to given marginals (2008) — link
  66. Limits of One-Dimensional Diffusions (2009)
  67. Optimal transport mapping via input convex neural networks (2020)
  68. Existence and Uniqueness of Monotone Measure-Preserving Maps (1995)
  69. The Measure Preserving Martingale Sinkhorn Algorithm (2023) — link
  70. $q$- Bass Martingales: Beyond the Gaussian Reference (2026)
  71. Density and gradient estimates for non degenerate B rownian SDE s with unbounded measurable drift (2021) — link
  72. Martingale S chr "o dinger bridges and optimal semistatic portfolios (2023)
  73. Structure of martingale transports in finite dimensions (2017) — link
  74. Weak Optimal Transport: When Is the Dual Potential Convex? (2025) — link
  75. Calibrating and pricing with embedded local volatility models (2007)
  76. Remarks on a multivariate transformation (1952)
  77. Variational Analysis (1998)
  78. The decomposition of stretched Brownian motion into Bass martingales (2026)
  79. Optimal transportation under controlled stochastic dynamics (2013)
  80. $q$-Bass martingales (2024) — link
  81. Markov-Functional Models with Local Drift — → 2411.15053
  82. Supplementary material to ``Stochastic Knothe–Rosenblatt: Light-speed Calibration of Stochastic Local Volatility Models’’ (2026) — link
  83. Stochastic Knothe–Rosenblatt : Light-speed Calibration of Stochastic Local Volatility Models (2026)
  84. $q_x$-Bass Martingales: Optimality, Existence and Convergence (2026) — link
  85. Mimicking Volatilities under Marginal Constraints via $q_x$-Bass Martingales: A Parallel Computing Approach (2026) — link

9 of 93 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.