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.
I am PI of the focus platform Quantitative analysis of stochastic and rough systems within the Weierstrass Institute.
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 and Benjamin Stemper) 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). We find that a surprisingly simple model using a stochastic volatility component involving a fractional Brownian motion allows for great fit of model prices with market option prices using only three parameters. The model is non-Markovian, which leads to significant numerical problems. Together with Peter Friz, I was PI of a DFG project on "Rough stochastic volatility and related topics" (BA 5484/1-1).
A second research interest is the numerical approximation of partial differential equations with random coefficients using stochastic representations based on stochastic (ordinary) differential equations and regression in the spacial variable, in collaboration with Martin Eigel and John Schoenmakers. I also want to study these techniques for partial differential equations driven by random or deterministic rough paths. The advantage of this method is that it enables us to use well known techniques on numerical simulation of diffusion processes and on regression to numerically approximate a much more complicated object. This research is supported by the Research Unit FOR 2402 funded by the DFG.
I am also interested in Monte Carlo algorithms for various more complicated problems. In the past, I have worked on reflected diffusions and on establishing heuristic, efficient and reliable criteria for the choice of the number of samples in general Monte Carlo procedures.
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. However, 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, in a project supported by the DFG via the Berlin Mathematics Research Center MATH+, project EF1-5/EF1-13.
I am part of the DFG International Research Training Group IRTG 2544 Stochastic Analysis in Interaction, supervising the PhD student Simon Breneis jointly together with Terry Lyons as co-supervisor.
Publications and preprints
- Christian Bayer, Josef Teichmann: The proof of Tchakaloff's Theorem (pdf), Proc. Amer. Math. Soc. 134, 3035-3040, 2006.
- Christian Bayer, Josef Teichmann: Cubature on Wiener space in infinite dimension (pdf), Proceedings of the Royal Society A, 464(2097), 2493-2516, 2008.
- Christian Bayer, Anders Szepessy, Raul Tempone: Adaptive weak approximation of reflected and stopped diffusions (pdf), Monte Carlo Methods and Applications 16, 1-67, 2010.
- Christian Bayer, Peter Friz, Ronnie Loeffen: Semi-closed form cubature and applications to financial diffusion models (pdf), Quantitative Finance 13(5), 769-782, 2013.
- Christian Bayer, Peter Friz: Cubature on Wiener space: Pathwise convergence (pdf), Applied Mathematics and Optimization 67(2):261-278, 2013.
- Christian Bayer, Klaus Wälde: The dynamics of distributions in continuous time stochastic models (pdf), preprint, 2013.
- Christian Bayer, Jim Gatheral, Morten Karlsmark: Fast Ninomiya-Victoir calibration of the Double-Mean-Reverting model (pdf), Quantitative Finance 13(11), 1813-1829, 2013.
- Christian Bayer, John Schoenmakers: Simulation of forward-reverse stochastic representations for conditional diffusions (pdf), Annals of Applied Probability 24(5), 1994-2032, 2014.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Christian Bayer, Alvaro Moraes, Raul Tempone, Pedro Vilanova: An Efficient Forward-Reverse Expectation-Maximization Algorithm for Statistical Inference in Stochastic Reaction Networks (pdf), Stochastic Analysis and Applications, 34(2), 193-231, 2016.
- 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.
- Christian Bayer, Peter Friz, Jim Gatheral: Pricing under rough volatility (pdf), Quantitative Finance, 16(6), 887-904, 2016.
- 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.
- Christian Bayer, Ulrich Horst, Jinniao Qiu: A functional limit theorem for limit order books with state dependent price dynamics(pdf), preprint, Annals of Applied Probability, 27(5), 2753-2806, 2017.
- 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.
- 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.
- 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.
- Christian Bayer, Markus Siebenmorgen, Raul Tempone: Smoothing the payoff for efficient computation of basket option prices (pdf), Quantitative Finance, 18(3), 491-505, 2018.
- Christian Bayer, Juho Häppölä, Raul Tempone: Implied Stopping Rules for American Basket Options from Markovian Projection (pdf), Quantitative Finance, 19(3), 371-390, 2019
- 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.
- Christian Bayer, Peter Friz, Paul Gassiat, Jörg Martin, Benjamin Stemper: A regularity structure for rough volatility (pdf), Mathematical Finance 30(3), 782-832, 2020.
- 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.
- Christian Bayer, Benjamin Stemper: Deep calibration of rough stochastic volatility models (pdf), preprint 2018.
- Christian Bayer, Martin Redmann, John Schoenmakers: Dynamic programming for optimal stopping via pseudo-regression (pdf), Quantitative Finance, available online, 2020.
- Christian Bayer, Raul Tempone, Sören Wolfers: Pricing American Options by Exercise Rate Optimization (pdf), Quantitative Finance, available online, 2020.
- 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.
- Christian Bayer, Chiheb Ben Hammouda, Raul Tempone: Numerical smoothing and hierarchical approximations for efficient option pricing and density estimation (pdf), preprint, 2020.
- Martin Redmann, Christian Bayer, Pawan Goyal: Low-dimensional approximations of high-dimensional asset price models (pdf), preprint, 2020.
- Christian Bayer, Denis Belomestny, Paul Hager, Paolo Pigato, John Schoenmakers: Randomized optimal stopping algorithms and their convergence analysis (pdf), preprint, 2020.
- Christian Bayer, Jinniao Qiu, Yao Yao: Pricing Options Under Rough Volatility with Backward SPDEs (pdf), preprint, 2020.
- Christian Bayer, Fabian Harang, Paolo Pigato: Log-modulated rough stochastic volatility models (pdf) , preprint, 2020.
- Christian Bayer, Denis Belomestny, Paolo Pigato, Paul Hager, John Schoenmakers, Vladimir Spokoiny: Reinforced optimal control (pdf), preprint, 2020.
Selected presentations
- Discretization of SDEs: Euler Methods and Beyond (pdf), 2006.
- Calculation of the Greeks Using Cubature Malliavin Calculus (pdf), 2006.
- Weak adaptive approximation of reflected diffusions (pdf), 2008.
- Hypo-elliptic simulated annealing (pdf), 2009.
- Cubature on Wiener space for Heath-Jarrow-Morton interest rate models (pdf), 2009.
- Some applications of cubature on Wiener space (pdf), 2011.
- Cubature and splitting schemes for stochastic differential equations (pdf), 2012.
- Existence, uniqueness and stability of invariant distributions in continuous-time stochastic models (pdf), 2012.
- Simulation of conditional diffusions via forward-reverse stochastic representations (pdf),2013.
- Asymptotics beats Monte Carlo: The case of correlated local vol baskets (pdf), 2013.
- Pricing under rough volatility (pdf), 2015.
- Minicourse on rough path analysis (pdf), 2015.
- SDE based regression for random PDEs (pdf), 2016.
- A regularity structure for rough volatility ( pdf), 2017.
- Rough volatility models (pdf), 2018.
- Short dated option pricing under rough volatility (pdf), 2018.
- Smoothing the payoff for efficient computation of basket option prices ( pdf), 2018.
- A regularity structure for rough volatility (pdf ), 2019.
- Pricing American Options by Exercise Rate Optimization (pdf), 2020.
Lecture notes and other short manuscrips
- Brownian Motion and Ito Calculus (pdf), Lecture notes for a short course given at the WK summer camp 2006.
- The Geometry of Iterated Stratonovich Integrals (pdf), notes 2006.
- Computational Finance (pdf), Lecture notes for a course given at TU Berlin in 2010 to 2014. (2011 and 2012 given by Antonis Papapantoleon).
- Advanced probability theory (pdf), Lecture notes for a course given at University of Vienna, 2011.
- Stochastik I (pdf), Lecture notes (partial) for a course given at HU Berlin, 2015.
My theses
- Diploma thesis: Cubature on Wiener space extended to higher order operators (pdf); supervisor: Josef Teichman
- PhD thesis: Selected topics in numerics of stochastic differential equations (pdf); supervisor: Josef Teichmann