Christian Bayer - Personal Homepage

I am working within the research group Stochastic Algorithms and Nonparametric Statistics at the Weierstraß Institute for Applied Analysis and Stochastics.

Research Interests

My main research interests are financial mathematics and stochastic numerics.

In finance, one of my major current research project (with Jim Gatheral, Peter Friz) is about modeling stock indices like the S & P 500 index (SPX) consistently with respect to the implied volatility surface, and the volatility index (VIX). A specific issue is the behaviour of the implied volatility of options for very short maturities, which is largely believed to exhibit explosion in the form of a power law as maturity goes to zero. This type of behaviour is inconsistent with standard diffusive models of volatility, but can be easily replicated when a stochastic volatility process of fractional Brownian motion type is chosen. As this leads to volatility processes, which are rougher than Brownian motion, the name rough volatility models was established. Our work on rough volatility models is very fruitful, because they lead to severe challenges both on the theoretical and on the numerical side, due to lack of Markov and semi-martingale propertie, as well as the highly irregular path behaviour.

An important research problem in computational finance is numerical approximation of stochastic optimal control problems, in particular optimal stopping. I am collaborating with Denis Belomestny, Raul Tempone, John Schoenmakers, and others on these topics. This strand of research is supported by the DFG via the Berlin Mathematics Research Center MATH+, project AA4-2.

The theory of rough paths has many applications in the field of machine learning. I am, in particular, interested in applications to stochastic optimal control. We use the path signature to derive efficient numerical approximation methods for stochastic optimal control problems, when the state process is not a Markov process. These problems are partially motivated by rough volatility models. Methods from rough path analysis can also be used for theoretical analysis of the properties of deep neural networks. I collaborate with Peter Friz on this subject.

I am part of the DFG International Research Training Group IRTG 2544 Stochastic Analysis in Interaction, the DFG CRC/TRR 388 Rough Analysis, Stochastic Dynamics and Related Fields. I am also a member of Math+, the Berlin cluster of excellence.


I am teaching a course on Computational Finance at TU Berlin.


  1. Peter Bank, Christian Bayer, Paul Hager, Tobias Nauen, Sebastian Riedel: Stochastic control with signatures (pdf), preprint, 2024.
  2. Christian Bayer, Luca Pelizzari, John Schoenmakers: Primal and dual optimal stopping with signatures (pdf), preprint, 2023.
  3. Christian Bayer, Simon Breneis: Efficient option pricing in the rough Heston model using weak simulation schemes (pdf), preprint, 2023.
  4. Christian Bayer, Simon Breneis: Weak Markovian approximations of rough Heston (pdf), preprint, 2023.
  5. Peter Bank, Christian Bayer, Peter Friz, Luca Pelizzari: Rough PDEs for local stochastic volatility models (pdf), preprint, 2023.
  6. Christian Bayer, Simon Breneis, Terry Lyons: An adaptive algorithm for rough differential equations (pdf), preprint, 2023.
  7. Christian Bayer, Chiheb Ben Hammouda, Antonis Papapantoleon, Michael Samet, Raul Tempone: Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Levy Models (pdf), preprint, 2022.
  8. Christian Bayer, Benjamin Stemper: Deep calibration of rough stochastic volatility models (pdf), preprint 2018.


  1. Christian Bayer, Chiheb Ben Hammouda, Raul Tempone: Numerical smoothing and hierarchical approximations for efficient option pricing and density estimation (pdf), SIAM Journal on Scientific Computing, to appear, 2023.
  2. Christian Bayer, Denis Belomestny, Oleg Butkovsky, John Schoenmakers: RKHS regularization of singular local stochastic volatility McKean-Vlasov models (pdf), Finance & Stochastic, to appear, 2023.
  3. Christian Bayer, Peter Friz, Nikolas Tapia: Stability of Deep Neural Networks via discrete rough paths (pdf), SIAM J. on Mathematics of Data Science 5, 2023.
  4. Christian Bayer, Paul Hager, Sebastian Riedel, John Schoenmakers: Optimal stopping with signatures (pdf), Annals of Applied Probability 33 (1), 238-273, 2023.
  5. Christian Bayer, Martin Eigel, Leon Sallandt, Philipp Trunschke: Pricing high-dimensional Bermudan options with hierarchical tensor formats (pdf), SIAM Journal on Financial Mathematics, 14(2), 383-406, 2023.
  6. Christian Bayer, Eric Hall, Raul Tempone: Weak error rates for option pricing under linear rough volatility (pdf), International J. Theoretical and Applied Finance, 2250029, 2023.
  7. Christian Bayer, Simon Breneis: Markovian approximations of stochastic Volterra equations with the fractional kernel (pdf), Quantitative Finance, 23(1), 53-70, 2023.
  8. Christian Bayer, Masaaki Fukasawa, Shonosuke Nakahara: On the weak convergence rate in the discretization of rough volatility models (pdf) SIAM Journal on Financial Mathematics 13(2), 2022, available online.
  9. Christian Bayer, Chiheb Ben Hammouda, Raul Tempone: Numerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing (pdf), Quantitative finance, published online, 2022.
  10. Christian Bayer, Denis Belomestny, Paolo Pigato, Paul Hager, John Schoenmakers, Vladimir Spokoiny: Reinforced optimal control (pdf), Communications in Mathematical Sciences 20(7), 1951-1978, 2022.
  11. Christian Bayer, Jinniao Qiu, Yao Yao: Pricing Options Under Rough Volatility with Backward SPDEs (pdf), SIAM Journal on Financial Mathematics 13 (1), 179-212, 2022.
  12. Christian Bayer, Fabian Harang, Paolo Pigato: Log-modulated rough stochastic volatility models (pdf), SIAM Journal on Financial Mathematics 12 (3), 1257-1284, 2021.
  13. Christian Bayer, Denis Belomestny, Paul Hager, Paolo Pigato, John Schoenmakers: Randomized optimal stopping algorithms and their convergence analysis (pdf), SIAM Journal on Financial Mathematics 12 (3), 1201-1225, 2021.
  14. Martin Redmann, Christian Bayer, Pawan Goyal: Low-dimensional approximations of high-dimensional asset price models (pdf), SIAM Journal on Financial Mathematics 12 (1), 1-28, 2021.
  15. Christian Bayer, Martin Redmann, John Schoenmakers: Dynamic programming for optimal stopping via pseudo-regression (pdf), Quantitative Finance 21 (1), 29-44, 2021.
  16. Christian Bayer, Chiheb Ben Hammouda, Raul Tempone: Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model (pdf), Quantitative Finance 20(9), 1457-1473, 2020.
  17. Christian Bayer, Raul Tempone, Sören Wolfers: Pricing American Options by Exercise Rate Optimization (pdf), Quantitative Finance 20 (11), 1749-1760, 2020.
  18. Christian Bayer, Denis Belomestny, Martin Redmann, Sebastian Riedel, John Schoenmakers: Solving linear parabolic rough partial differential equations (pdf), Journal of Mathematical Analysis and Applications, p.124236, 2020.
  19. Christian Bayer, Peter Friz, Paul Gassiat, Jörg Martin, Benjamin Stemper: A regularity structure for rough volatility (pdf), Mathematical Finance 30(3), 782-832, 2020.
  20. Christian Bayer, Peter Friz, Archil Gulisashvili, Blanka Horvath, Benjamin Stemper: Short-time near-the-money skew in rough fractional volatility models (pdf), Quantitative Finance, 19(5), 779-798, 2019.
  21. Christian Bayer, Juho Häppölä, Raul Tempone: Implied Stopping Rules for American Basket Options from Markovian Projection (pdf), Quantitative Finance, 19(3), 371-390, 2019
  22. Christian Bayer, Markus Siebenmorgen, Raul Tempone: Smoothing the payoff for efficient computation of basket option prices (pdf), Quantitative Finance, 18(3), 491-505, 2018.
  23. Christian Bayer, Hilmar Mai, John Schoenmakers: Forward-reverse EM algorithm for Markov chains: Convergence and numerical analysis (pdf), Advances in Applied Probability 50(2):621-644, 2018.
  24. Felix Anker, Christian Bayer, Martin Eigel, Johannes Neumann, John Schoenmakers: A fully adaptive interpolated stochastic sampling method for linear random PDEs (pdf), International Journal for Uncertainty Quantification, 7(3), 189-205, 2017.
  25. Felix Anker, Christian Bayer, Martin Eigel, Marcel Ladkau, Johannes Neumann, John Schoenmakers: SDE based regression for linear random PDEs (pdf), SIAM Journal on Scientific Computing 39(3), A1168 - A1200, 2017.
  26. Christian Bayer, Ulrich Horst, Jinniao Qiu: A functional limit theorem for limit order books with state dependent price dynamics(pdf), Annals of Applied Probability, 27(5), 2753-2806, 2017.
  27. Christian Bayer, Peter Friz, Sebastian Riedel, John Schoenmakers: From rough-path estimates to multi-level Monte Carlo (pdf), SIAM J. Numer. Anal., 54(3), 1449-1483, 2016.
  28. Christian Bayer, Peter Friz, Jim Gatheral: Pricing under rough volatility (pdf), Quantitative Finance, 16(6), 887-904, 2016.
  29. Christian Bayer, John Schoenmakers: Option pricing in affine generalized Merton models (pdf), Appeared in: Advanced Modelling in Mathematical Finance. Springer Proceedings in Mathematics & Statistics, vol 189, 2016.
  30. Christian Bayer, Alvaro Moraes, Raul Tempone, Pedro Vilanova: An Efficient Forward-Reverse Expectation-Maximization Algorithm for Statistical Inference in Stochastic Reaction Networks‚Äč (pdf), Stochastic Analysis and Applications, 34(2), 193-231, 2016.
  31. Christian Bayer, Håkon Hoel, Ashraful Kadir, Petr Plechac, Mattias Sandberg, Anders Szepessy: Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces (pdf), Appl Math Res Express, 2015.
  32. Christian Bayer, Peter Friz, Peter Laurence: On the probability density function of baskets (pdf), appeared in: Large Deviations and Asymptotic Methods in Finance, Springer, 2015.
  33. Christian Bayer, Peter Laurence: Small-time asymptotics for at-the-money implied volatility in amulti-dimensional local volatility model (pdf), appeared in: Large Deviations and Asymptotic Methods in Finance, Springer, 2015.
  34. Christian Bayer, Bezirgen Veliyev: Utility maximization in a binomial model with transaction costs: A duality approach based on the shadow price process (pdf), Int. J. Theor. Appl. Finan. 17, 1450022, 2014.
  35. Christian Bayer, Håkon Hoel, Erik von Schwerin, Raul Tempone: On nonasymptotic optimal stopping criteria in Monte Carlo simulations (pdf), SIAM Journal on Scientific Computing, 36(2), A869-A885, 2014.
  36. Christian Bayer, Peter Laurence: Asymptotics beats Monte Carlo: The case of correlated local vol baskets (pdf, [available at Wiley]), Communications on Pure and Applied Mathematics, 67(10), 1618-1657, 2014.
  37. Christian Bayer, John Schoenmakers: Simulation of forward-reverse stochastic representations for conditional diffusions (pdf), Annals of Applied Probability 24(5), 1994-2032, 2014.
  38. Christian Bayer, Jim Gatheral, Morten Karlsmark: Fast Ninomiya-Victoir calibration of the Double-Mean-Reverting model (pdf), Quantitative Finance 13(11), 1813-1829, 2013.
  39. Christian Bayer, Alan D Rendall , Klaus Wälde: The invariant distribution of wealth and employment status in a small open economy with precautionary savings (pdf), Journal of Mathematical Economics 85, 17-37, 2019.
  40. Christian Bayer, Peter Friz: Cubature on Wiener space: Pathwise convergence (pdf), Applied Mathematics and Optimization 67(2):261-278, 2013.
  41. Christian Bayer, Peter Friz, Ronnie Loeffen: Semi-closed form cubature and applications to financial diffusion models (pdf), Quantitative Finance 13(5), 769-782, 2013.
  42. Christian Bayer, Anders Szepessy, Raul Tempone: Adaptive weak approximation of reflected and stopped diffusions (pdf), Monte Carlo Methods and Applications 16, 1-67, 2010.
  43. Christian Bayer, Josef Teichmann: Cubature on Wiener space in infinite dimension (pdf), Proceedings of the Royal Society A, 464(2097), 2493-2516, 2008.
  44. Christian Bayer, Josef Teichmann: The proof of Tchakaloff's Theorem (pdf), Proc. Amer. Math. Soc. 134, 3035-3040, 2006.

Selected presentations

Lecture notes and other short manuscrips

My theses