Published June 19, 2026 | Version v1
Working paper Open

The Projected-Generator Method for Path-Dependent Derivative Pricing

Authors/Creators

  • 1. Independent Researcher

Description

The projected-generator method is a deterministic, simulation-free engine for pricing path-dependent derivatives — American and Bermudan exercise, barriers, autocallables, coupon memory, and structured products. Instead of Monte Carlo paths, regression continuation estimates, or high-dimensional PDE grids, it projects the model's infinitesimal generator onto a finite basis and propagates continuation values by the resulting matrix semigroup; path dependence is carried by finite event-state transitions around the same propagation primitive.

The same finite expansion yields Greeks without bump-and-reprice, and the projection residual gives an explicit error budget: pricing is exact when the finite space is invariant under the generator, and for analytic payoffs under spectral bases the error decays exponentially. The closure and error-certificate results are formally verified in Lean 4 (with Python verification scripts).

Key results: 7 theorems / propositions. Maturity: Draft. Target venue: Mathematical Finance / SIAM Journal on Financial Mathematics. Part of The Latent research program.

The method is strongest for low-to-moderate-dimensional models with a tractable generator; high-dimensional baskets, rough volatility without a controlled lift, and severe discontinuities remain better served by Monte Carlo, finite differences, or adapted bases.

Notes

Topic: fin_projected_generator_method. Source: topics/fin_projected_generator_method/paper.md. Status: Draft. Contains 7 formal results. Formal verification: Lean 4 (with Python verification scripts). Related topics: rough_volatility, universal.

Files

Nagy_2026_the_projected_generator_method_for_path_dependent_derivative_pricing.pdf

Files (320.6 kB)

Additional details

References

  • Ackerer, D. and Filipović, D. (2020). Option pricing with orthogonal
  • polynomial expansions. Mathematical Finance, 30(1), 47–84.
  • Ahn, K., Cheng, X., Daneshmand, H. and Sra, S. (2023). Transformers learn to
  • implement preconditioned gradient descent for in-context learning. *Advances
  • in Neural Information Processing Systems (NeurIPS)*, 36.
  • Bertsekas, D. P. (2012). *Dynamic Programming and Optimal Control, Vol. II:
  • Approximate Dynamic Programming*. 4th ed. Athena Scientific.
  • Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate
  • Liabilities. Journal of Political Economy, 81(3), 637–654.
  • Brown, D. B., Smith, J. E. and Sun, P. (2010). Information relaxations and
  • duality in stochastic dynamic programs. Operations Research, 58(4),
  • Broadie, M. and Glasserman, P. (2004). A stochastic mesh method for
  • pricing high-dimensional American options. *Journal of Computational
  • Finance*, 7(4), 35–72.
  • Boyd, J. P. (2001). Chebyshev and Fourier Spectral Methods. Dover Publications.
  • Cuchiero, C., Keller-Ressel, M. and Teichmann, J. (2012). Polynomial
  • processes and their applications to mathematical finance. *Finance and
  • Stochastics*, 16(4), 711–740.
  • Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes.
  • Chapman & Hall/CRC.
  • Fang, F. and Oosterlee, C. W. (2008). A novel pricing method for European
  • options based on Fourier-cosine series expansions. *SIAM Journal on
  • Scientific Computing*, 31(2), 826–848.
  • Feynman, R. P. (1948). Space-time approach to non-relativistic quantum
  • mechanics. Reviews of Modern Physics, 20(2), 367–387.
  • Filipović, D. and Larsson, M. (2016). Polynomial diffusions and
  • applications in finance. Finance and Stochastics, 20(4), 931–972.
  • Galerkin, B. G. (1915). Rods and plates. Series in some problems of elastic
  • equilibrium of rods and plates. Vestnik Inzhenerov i Tekhnikov, 19, 897-908.
  • Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering.
  • Kac, M. (1949). On distributions of certain Wiener functionals.
  • Transactions of the AMS, 65(1), 1–13.
  • Kolmogorov, A. N. (1931). On analytical methods in probability theory.
  • Mathematische Annalen, 104(1), 415–458.
  • Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation. 2nd ed.
  • Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by
  • simulation: a simple least-squares approach. Review of Financial Studies,
  • 14(1), 113–147.
  • Merton, R. C. (1976). Option pricing when underlying stock returns are
  • discontinuous. Journal of Financial Economics, 3(1-2), 125–144.
  • Nagy, T. (2026). The Latent: Finite Sufficient Representations of Smooth
  • Systems. Zenodo. DOI: 10.5281/zenodo.19101209.
  • Øksendal, B. (2003). *Stochastic Differential Equations: An Introduction with
  • Applications*. 6th ed. Springer.
  • Reed, M. and Simon, B. (1978). Methods of Modern Mathematical Physics.
  • Academic Press.
  • Trefethen, L. N. (2013). Approximation Theory and Approximation Practice.
  • von Oswald, J., Niklasson, E., Randazzo, E., Sacramento, J., Mordvintsev, A.,
  • Zhmoginov, A. and Vladymyrov, M. (2023). Transformers learn in-context by
  • gradient descent. International Conference on Machine Learning (ICML),
  • 35151–35174.

Subjects

Quantitative finance
http://id.loc.gov/authorities/subjects/sh2003001188
Stochastic processes
http://id.loc.gov/authorities/subjects/sh85128175