Dr. Hans Buehler
Quant of the Year 2022

hans@quantitative-research.de (SSRN, Arxiv, LinkedIn)


  • 1998-2001 Co-founder codex design software, Berlin
  • 2001-June 2008: Global Head of Equity Derivatives Quantitative Research, Deutsche Bank, London; Intern to Director (2006)
  • June 2008: APAC head of Equities Quantitative Research, JP Morgan
  • 2010: Global Head of Equities, Sales, and Securities Services Quantitative Research (Electronic Trading & Risk Management, Financing, Prime, Derivatives, x-asset Sales, fund services,..), JP Morgan, London; Managing Director
  • Sep 2018 in addition: Global Head Equities Analytics, Automation and Optimization (AAA) at JP Morgan, London; Managing Director, driving the transformation of the Equities business with data and analytics, from clearing, prime through cash to derivatives.
  • Summer 2022: visiting Professor TU Munich with Blanka Horvath
  • From July 2022: Deputy CEO, XTX Markets.
  • From June 2023: Co-CEO, XTX Markets.

Quant Finance 2.0 - Learning To Trade

My primary focus until 2022 was the use of modern quant finance, data-driven, and AI methods for financial applications in markets with a strong focus on reinforcement learning for execution, market making, derivatives risk management and pricing/quoting to drive client business. The use of big data and cloud compute technology allows pushing forward the barrier from analytics, automation to optimization accross the Equities and markets businesses.
Here is my view on record: Deep Hedging and where we are. Brief summary of all our papers on this area on http://deep-hedging.com.

My next step is to work with the leading team at XTX markets.


  • Equity Hybrid Derivatives
    (with M.Overhaus, A.Bermudez, A.Ferraris, C.Jordinson, A.Lamnouar)
    The fourth book of the Deutsche Bank GME Quantitative Products Analytics team (formerly Global Quantitiative Research) covers a wide range equity modelling issues in general - such as dividend handling, variance swaps, local volatility, CPPIs - and hybrid risk from rates and credit markets.
    Wiley, 2006
  • Volatility Markets
    Revised and update version of my PhD thesis in print, incorporating new results presented since the publication of the thesis itself, in particular on the subject of "fitted models". A particlar section of "Fitted Heston" goes beyond the material presented in "Equity Hybrid Derivatives".
    VDM Verlag Dr. Muller, 2009

Public Commentary


Published Papers

  • Deep Hedging: Learning to Remove the Drift
    (with P.Murray, M.Pakkanen, B.Wood)
    We use our Deep Hedging algorithm to construct equivalent measures which are free of "Statistical Arbitrage" in the sense that there is trading strategy which will make money under utility-based objective functions, i.e. we ``remove the drift". We apply this to otherwise pure ML based simulators of option markets.
    An earlier version with partial results can be found here.

    Risk, Feb 2022 Risk Article
  • Generating Financial Markets with Signatures
    (with B.Horvath, T.Lyons, I.Perez, B.Wood)
    We show how a rough paths-based feature map encoded by the signature of the path outperforms returns based market generation both numerically and from a theoretical point of view. Finally, we also propose a suitable performance evaluation metric for financial time series and discuss some connections of our signature-based Market Generator to deep hedging.
    Risk, June 2021
  • Deep Hedging in math finance notation, published in Quantitative Finance.
    Version usign machine learning notation.
    (with L.Gonon, J.Teichmann, B.Wood)
    We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, liquidity constraints or risk limits using modern deep reinforcement machine learning methods. We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show in Section 4 that the set of constrained trading strategies used by our algorithm is large enough to e-approximate any optimal solution. Our algorithm does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available.
    Quantitative Finance, vol 0, num 0, 2019, pages 1-21
    Accepted as poster sesssion at NeurIPS 2018, ICML 2019
  • The Heston Model (Encyclopedia of Quantiative Finance)
    (with O.Chybiryakov) A review of the Heston model and its applications.
    Encyclopedia of Quantitative Finance, Cont.R (Ed.), John Wiley & Sons Ltd, pp. 889-897 (2010)
  • Volatility Markets: Consistent Modelling, Hedging and Practical Implementation
    Published version of my dissertation, updated 2008
    Contains extended material on consistent variance curves, a proof that "smooth" diffusion markets are always complete, comments on pricing in local martingale models, fitting models to the market (general, Bergomi, Dupire, Heston), Heston-type models with semi-closed forms, algorithms to perform parameter hedging with linear programming, computation of variance, gamma and entropy swaps, expensive martingales, and the implementation of a particular four-factor variance curve model.
    Defended June 26th, 2006 (summa cum laude)
  • Recent Developments in Mathematical Finance: A Practitioner's Point of View
    (with M.Overhaus, A.Bermudez, A.Ferraris, C.Jordinson, A.Lamnouar, A.Puthu)
    An introductory text on mathematical finance which explains basic concepts and shows applications in practise, in particular pricing of options on variance. Covers the nature of hedging and a simple derivation of the idea of "delta hedging".
    DMV Jahresbericht, 2006 (first version May 2005)
  • Consistent Variance Curve Models
    Generalized term-structure market model approach to variance swaps for hedging of products on realized variance. Completeness of such models is discussed. We also apply the results to the application re-calibration of stochastic volatility models
    Finance and Stochastics, Volume 10, Number 2 / April, 2006 (first version June 2004)
  • Expensive Martingales
    Calibration of discrete transition kernels between the marginal distributions of a stock price process using weak information such as Cliquet prices. The resulting one-factor process reprices spot started options and is optimized to fit forward started options. (Generalization of Derman-Kani trees.)
    Quantitative Finance, Volume 6, Number 3 / June 2006 (first version March 2004)
  • Information-equivalence: On filtrations created by independent increments
    Two Brownian motions generate the same filtration iff they are a.s. deterministic integrals of each other (and related results).
    Seminaire de Probabilites XXXVIII, p.195, Berlin, Springer 2004
  • Zur Struktur Brownscher Filtrationen (in German)
    A Brownian motion remains extremal on its filtration after a change of measure, but it may not generate that filtration anymore (thesis is based on a paper by Prof. Schachermayer; relevant new results have been published in the paper above.)
    Diploma-Thesis, 2001 (1.0)

Working Papers

  • Deep Bellman Hedging
    (with P. Murray, B. Wood)
    We present a dynamic programming "Bellman" approach to the Deep Hedging problem of hedging a portfolio of financial instruments with other instruments, including derivatives, under market frictions. Compared to the vanilla Deep Hedging problem the approach here attempts to learn the optimal hedging strategy for "all" portfolios across all market states.
    SSRN Working paper; publication in preparation, version 1.0 June 30th 2022
  • Multi-Asset Spot and Option Market Simulation
    (with M.Wiese, B. Wood, A. Pachoud. R. Korn. P. Murrat, L. Bai)
    We construct realistic spot and equity option market simulators for a single underlying on the basis of normalizing flows. We address the high-dimensionality of market observed call prices through an arbitrage-free autoencoder that approximates efficient low-dimensional representations of the prices while maintaining no static arbitrage in the reconstructed surface. Given a multi-asset universe, we leverage the conditional invertibility property of normalizing flows and introduce a scalable method to calibrate the joint distribution of a set of independent simulators while preserving the dynamics of each simulator. Empirical results highlight the goodness of the calibrated simulators and their fidelity.
    Arxiv Working paper; publication in preparation, version 1.0 December 2021
  • A Data-Driven Market Simulator for Small Environments
    (with B.Horvath, Terry Lyons, Immanol P. Arribas, Ben Wood)
    We present a generative model based on paths signatures which is tuned towards small-data environments commonly found in finance, and discuss various success metrics for time series simulation.
    Arxiv Working paper, version 1.0 June 2020
  • Statistical Hedging: Motivating the Use of Convex Risk Measures for Hedging Portfolios of Derivatives Over One Time Step in the Presence of General Transaction Cost. A Summary for Derivative Quants
    This note presents an extension of the generalized Markoviz-type "mean-variance" portfolio optimization approach over one period to portfolios of derivatives. Most notably, we show that once "writing off" parts of the portfolio is allowed, we naturally arrive at using "cash-invariant monotone hulls" a'la Filipovic/Kupper to construct sensible measures of risk. In particular, we show that the resulting risk-adjusted implementation cost function for hedging a portfolio is bounded (by the best and worst possible outcome), monotone decreasing (better portfolios are cheaper) and convex (diversification works) - note that the classic mean-variance framework fails to satisfy the first two properties when considered over non-symmetric returns such as those arising from working with derivatives.
    This note summarizes results presented at Global Derivatives 2013 and 2014 and provides a more generalized view on the problem at hand.
    This work contains little original contributions; its aim to motivate the use of convex risk measures and their construction via cash-invariant monotone hulls from a practitioner's point of view.

    SSRN Working paper, version 0.931, April 2017
  • Discrete Local Volatility for Large Time Steps (short version) see also the extended Version with many details, but no advanced applications.
    We construct a state-and-time discrete martingale which is calibrated globally to a set of given input option prices which may exhibit arbitrage. We also provide a method to take small steps, fully consistent with the transition kernels of the large steps.
    The model's robustness vs. arbitrage violations in the input surface makes our approach particularly suited for computations in stressed scenarios. Indeed, our method of finding a globally closest arbitrage-free surface under constraints on implied and local volatility is useful in its own right.
    We demonstrate the power of our approach by showing its application to affine dividends calibrated to option prices given by proportional dividends, availability of Likelihood Greeks, and to mean-reverting assets such as VIX. We also comment on how to introduce jumps into our processes.
    The material discussed here was also presented at Global Derivatives 2016.
    SSRN Working paper. This is the first proper version of the "short" paper after our presentation at GD'16. In particular, it discusses the incorporation of jumps, Likelihood Greeks, and - indeed - modelling VIX with a Discrete Local Volatility process.
  • Volatility and Dividends II - Consistent Cash Dividends
    We discuss a time-homogeneous equity stock price modelling approach with a consistent dividend process such that at any point, conditional on the state variables of the model, short-term implied dividends are "cash-like" (constant) and long-term dividends are "proportional".
    Our approach is based on a general representation for dividend paying stocks where we prove that the stock price process is the sum of an "inner" process plus the sum of all future appropriately discounted dividends under risk-neutral measure.
    This note summarizes results presented in 2012 at Global Derivatives. We discuss dividend dynamics in the proposed approach; calibration to dividend options and the equity implied volatility surface are only touched upon as it can be acccomplished\ by standard methods.
    This note summarizes results presented at Global Derivatives 2012.
    SSRN Working paper, draft Version 1.00 (missing graphs), April 2012, August 25, September 9 2015
  • Stochastic Proportional Dividends
    (with A.S.Dhouibi and D.Sluys)
    Motivated by recently increased interest in trading derivatives on dividends, we present a simple, yet efficient equity stock price model with discrete stochastic proportional dividends.
    The model has a closed form for European option pricing and can therefore be calibrated efficiently to vanilla options on the equity. It can also be simulated efficiently with Monte-Carlo and has fast analytics to aid the pricing of derivatives on dividends. While its efficiency makes the model very appealing, it has the twin drawbacks that dividends in this model can become negative, and that it does not price in any skew on either dividends or the stock price.
    We present the model and also discuss various extensions to stochastic interest rates, local volatility and jumps.

    SSRN Working paper, draft Version 1.013 December 2010 (first version January 2010, based on work from 2006 with C.Jordinson)
  • Volatility and Dividends - Volatility Modelling with Cash Dividends and simple Credit Risk
    This article discusses incorporating cash dividends and simple credit risk into equity derivatives risk management. It is shown that the only consistent way is via a simple affine transformation of the ``pure" local martingale of the form S(t) = {F(t) - D(t)} X(t) + D(t) up to default.
    Implementation and is discusseed for: plain Europeans, American options, Barriers and finally variance swaps and related derivatives. Risk management for volatilty hedging and variance swaps in general is discussed in detail. To our best knowledge, this paper is the only one discussing the incorporation of cash dividends into variance swap pricing.

    The aim of the article is to present results discussed in Equity Hybrid Derivatives in a more intuitive way (in the book all results have been derived rigourously). It is a reference summary on volatility and dividend modelling for equity derivatives. The updated version 1.2 contains two additional proofs compared to 1.00 from March 2009.
    SSRN Working paper, version 1.3 October 2010 (first version March 2007)
  • Delta Hedging Works: On Market Completeness for Diffusion Processes
    This article provides new criteria for the completeness of markets driven by diffusion processes. In particular, we show that if the coefficients of the SDE are C1 almost surely, the the market of payoffs measurable with respect to the market process is complete, regardless of the non-negativity of the instantaneous covariance matrix.
    Our approach is in marked contrast wto the classic requirement that the volatility matrix of the SDE is invertible in order to retrieve the background driving motion which is much stronger and often violated in practice due to differing trading times for underlyings in different time zones. It is also not a very natural approach since a period of zero volatility "in one direction" should not impede replicability in another risk factor.

    SSRN Working paper, version 1.1 October 3rd, 2009 (first version March 2006)


2022  Learning to Trade: Data-Driven Quantitative Finance (plenary talk)
11th Bachelier World Congress
Hong Kong, June 2022
Learning to trade: From regression to reinforcement learning
Bloomberg Quant (BBQ) Seminar Series
February 2022
2021  Learning to Trade
QuantMinds International
Barcelona, December 2021
Deep Hedging: Volatility Market Simulation
Risk Global Quant Network
London, July 2021
Deep Hedging: Learning Risk-Neutral Market Dynamics
Oxford Newton Gateway to Mathematics, Unlocking Data Streams
Oxford, March 2021
Simulating spot and equity option markets using rough path signatures
Oxford Newton Gateway to Mathematics, Unlocking Data Streams
Oxford, March 2021
2020  Reinforcement Learning in Trading
Risk Quant Summit London 2020
London, March 2020
Deep Hedging and Market Simulation
Oxford Conference on Machine Learning in Finance
Oxford, September 2019
Digital Trading: Implementing Data-Driven Decision Making for Trading Businesses
SwissQuant Conference on Industrialization of Data Analytics
Zurich, May 2019
Deep Hedging: from Theory to Practice: From Greeks to Hedging under Market Frictions
Imperial Frontiers in Quantitative Finance
London, April 2019
Deep Hedging GAMeD: Generative Adversarial Market Dynamics for Hedging under Market Frictions
CFm-Imperial Quantitative Finance Seminar
London, February 2019
2018  Deep Hedging Machine-driven trading of derivatives under market frictions
Swissquote Conference 2018 on Machine Learning in Finance
Geneva, Nov 2018
Deep Statistical Hedging
QuantMinds 2018
Lisbon, May 2018
2017  Quant Finance: From Black-Scholes To Big Data
CornellTech 10th Anniversary, Panel Discussion
New York, Nov 2017
Deep Statistical Hedging: Hedging a Portfolio of Derivatives under Transaction cost and Liquidity with Convex Risk Measures
ETH Zurich Math Finance Seminar, June 7th, 2017
University of Freiburg, Seminar, June 27th 2017
Discrete Local Volatility & Applications
Global Derivatives Trading & Risk Management Conference 2017
Barcelona, May 2017
Discrete Local Volatility: Affine Dividends, Multi-Asset Pricing, Quanto, Likelihood Risk
MathFinance Conference Frankfurt
Frankfurt, April 2017
2016  Discrete Local Volatility: Pricing with a Discrete Smile
Global Derivatives Conference 2016
Budapest, May, 2016
2014  Statistical Hedging – Cost, Carry, Risk
Global Derivatives Conference 2014, Amsterdam
Amsterdam, May 2014
2013  Statistical Hedging: Application to Stochastic Local Volatility Models
Global Derivatives Conference 2013, Amsterdam
Amsterdam, May 2013
Statistical Hedging with Stochastic Local Vol
MathFinance Conference Frankfurt
Frnakfurt, March 2013
2012  Stochastic Dividend Modeling II: Consistent Cash Dividends
Global Derivatives Trading & Risk Management Conference 2012
Barcelona, April 2012
2011  Stochastic Dividend Modeling
Global Derivatives Trading & Risk Management Conference 2011
Paris, April 2011
2010  Dividend Modeling
Forschungsseminar Stochastische Analysis und Stochastik der Finanzmärkte
Humboldt University, Technical University Berlin
Berlin, December 2010
2009  Risk Management with Infinite dimensional SDEs
Workshop Computational Finance
Kyoto, August 2009
Delta-Hedging Works: Market Completeness for Factor Models on the example of Variance Curve Models
Conference on small time asymptotics, perturbation theory and heat kernel methods in mathematical finance TU Wien
Vienna, February 2009
2007  Hedging Options On Variance
Global Derivatives & Risk Management
Paris, May 2007
Quantitative Products Analytics - Deutsche Bank’s Equity Derivatives Quant Team
University Paris 6 Student Event
Paris, January 2007
2006  Options On Variance: Pricing And Hedging
IQPC Volatility Trading Conference
London, November 2006
Consistent Variance Curve Models
Bachelier World Congress 2006
Tokyo, August 2006
Consistent Variance Curve Models: Theory and Application
Imperial College Student Event
London, March 2006
Consistent Variance Curve Models: Theory and Application
ICMA Centre University of Reading
Reading, February 2006
Modeling Variance Swap Curves: Theory and Application
Petit Dejeuner de la Finance
Paris, February 2006
2005  Finanzmathematik in der Praxis
Humboldt University Berlin Student Event
Berlin, December 2005
Variance Swap Market Models
Seminar Stochastische Analysis and Stochastik der Finanzmaerkte TU Berlin / HU Berlin / MATHEON
Berlin, November 2005
Valuing and Hedging Equity Derivatives
Quant Congress Europe
London, October 2005
Consistent Variance Curve Models
Technische Universität Wien
Vienna, October 2005
Consistent Variance Curve Models
Workshop Stochastic Analysis and Applications in Finance Max Planck Institute for Mathematics in the Sciences
Leipzig April 2005
2004  Stochastic Volatility Models and Products
Risk Training Course
Hong Kong, July 2004
Construction of Martingales Under Constraints: From Implied Volatility to Pricing Exotics
Tandem-Workshop Stochastik-Numerik TU Berlin DFG Research Centre
Berlin, June 2004
Finanzmathematik in der Praxis
Humboldt University Berlin Student Event
Berlin, June 2004
Levy Models in Option Pricing: Utilising Volatility Smile Models to Optimise Pricing and Hedging Strategies
Volatility Modelling Risk Training
London, June 2004
2003  Volatilitätsmodelle in der Praxis
Humboldt University Berlin Student Event
Berlin, May 2003
2002  Applying stochastic volatility models for pricing and hedging derivatives
Volatility Forecasting and Modelling Techniques Risk Training
New York, Nov 2002 and London, Dec 2002
Quantitative Research in der Praxis
TU Berlin Student Event
Berlin, July 2002

From a long time ago