Highlights
The jumpdiffusion and stochastic volatility Libor models developed in the preceding years, were adapted, extended, and casted into a multicurve framework. The resulting multicurve Libor models turned out to be well suited for calibration to a whole range of usual Libor options, such as Caps, Floors, und Swaptions. Moreover, by incorporating a displacement feature, the nowadays typical occurrence of negative rates can be matched with these models as well. For the calibration to certain liquid markets, for example the S&P 500 Index, a breakthrough was achieved by leaving the usual semimartingale paradigm for modeling stochastic volatility. In fact, several data studies from time series and calibration reveal that the roughness of volatility is actually higher than feasible within the class of ordinary diffusion processes.Publications
Monographs

D. Belomestny, J. Schoenmakers, Advanced SimulationBased Methods for Optimal Stopping and Control: With Applications in Finance, Macmillan Publishers Ltd., London, 2018, 364 pages, (Monograph Published), DOI 10.1057/9781137033512 .

CH. Bayer, J.G.M. Schoenmakers, Option Pricing in Affine Generalized Merton Models, in: Advanced Modelling in Mathematical Finance  In Honour of Ernst Eberlein, J. Kallsen, A. Papapantoleon , eds., Springer Proceedings in Mathematics & Statistics, Springer International Publishing Switzerland, Cham, 2016, pp. 219239, (Chapter Published).
Abstract
In this article we consider affine generalizations of the Merton jump diffusion model Merton (1976) and the respective pricing of European options. On the one hand, the Brownian motion part in the Merton model may be generalized to a logHeston model, and on the other hand, the jump part may be generalized to an affine process with possibly state dependent jumps. While the characteristic function of the logHeston component is known in closed form, the characteristic function of the second component may be unknown explicitly. For the latter component we propose an approximation procedure based on the method introduced in Belomestny, Kampen, Schoenmakers (2009). We conclude with some numerical examples. 
J.G.M. Schoenmakers, Chapter 12: Coupling Local Currency Libor Models to FX Libor Models, in: Recent Developments in Computational Finance, Th. Gerstner, P. Kloeden, eds., 14 of Interdisciplinary Mathematical Sciences, World Scientific Publishers, Singapore, 2013, pp. 429444, (Chapter Published).

J.G.M. Schoenmakers, Robust Libor Modelling and Pricing of Derivative Products, Chapman & Hall CRC Press, 2005, 202 pages, (Monograph Published).
Articles in Refereed Journals

CH. Bayer, M. Fukasawa, S. Nakahara, Short communication: On the weak convergence rate in the discretization of rough volatility models, SIAM Journal on Financial Mathematics, ISSN 1945497X, 13 (2022), pp. SC66SC73, DOI 10.1137/22M1482871 .

M. Redmann, Ch. Bayer, P. Goyal, Lowdimensional approximations of highdimensional asset price models, SIAM Journal on Financial Mathematics, ISSN 1945497X, 12 (2021), pp. 128, DOI 10.1137/20M1325666 .
Abstract
We consider highdimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have nonzero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out. 
P. Pigato, Extreme atthemoney skew in a local volatility model, Finance and Stochastics, 23 (2019), pp. 827859, DOI 10.1007/s00780019004062 .

V. Krätschmer, M. Ladkau, R.J.A. Laeven, J.G.M. Schoenmakers, M. Stadje, Optimal stopping under uncertainty in drift and jump intensity, Mathematics of Operations Research, 43 (2018), pp. 11771209, DOI 10.1287/moor.2017.0899 .
Abstract
This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general timeconsistent ambiguity averse preferences and general payoff processes driven by jumpdiffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem %represented by the Snell envelope using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backwardforward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples. 
D. Belomestny, H. Mai, J.G.M. Schoenmakers, Generalized PostWidder inversion formula with application to statistics, Journal of Mathematical Analysis and Applications, 455 (2017), pp. 89104.
Abstract
In this work we derive an inversion formula for the Laplace transform of a density observed on a curve in the complex domain, which generalizes the well known PostWidder formula. We establish convergence of our inversion method and derive the corresponding convergence rates for the case of a Laplace transform of a smooth density. As an application we consider the problem of statistical inference for variancemean mixture models. We construct a nonparametric estimator for the mixing density based on the generalized PostWidder formula, derive bounds for its root mean square error and give a brief numerical example. 
Z. Grbac, A. Papapantoleon, J.G.M. Schoenmakers, D. Skovmand, Affine LIBOR models with multiple curves: Theory, examples and calibration, SIAM Journal on Financial Mathematics, ISSN 1945497X, 6 (2015), pp. 9841025.
Abstract
We introduce a multiple curve LIBOR framework that combines tractable dynamics and semianalytic pricing formulas with positive interest rates and basis spreads. The dynamics of OIS and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows to derive Fourier pricing formulas for caps, swaptions and basis swaptions. A model specification with dependent LIBOR rates is developed, that allows for an efficient and accurate calibration to a system of caplet prices. 
CH. Bayer, J. Gatheral, M. Karlsmark, Fast NinomiyaVictoir calibration of the doublemeanreverting model, Quantitative Finance, 13 (2014), pp. 18131829.

M. Ladkau, J.G.M. Schoenmakers, J. Zhang, Libor model with expirywise stochastic volatility and displacement, International Journal of Portfolio Analysis and Management, 1 (2013), pp. 224249.
Abstract
We develop a multifactor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own squareroot stochastic volatility process. The main advantage of this approach is that, maturitywise, each squareroot process can be calibrated to the corresponding cap(let)volastrike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modeling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in volastrikematurity panels are typically observed. 
S. Balder, A. Mahayni, J.G.M. Schoenmakers, Primaldual linear Monte Carlo algorithm for multiple stopping  An application to flexible caps, Quantitative Finance, 13 (2013), pp. 10031013.
Abstract
In this paper we consider the valuation of Bermudan callable derivatives with multiple exercise rights. We present in this context a new primaldual linear Monte Carlo algorithm that allows for efficient simulation of lower and upper price bounds without using nested simulations (hence the terminology). The algorithm is essentially an extension of a primaldual Monte Carlo algorithm for standard Bermudan options proposed in Schoenmakers et al (2011), to the case of multiple exercise rights. In particular, the algorithm constructs upwardly a system of dual martingales to be plugged into the dual representation of Schoenmakers (2010). At each level the respective martingale is constructed via a backward regression procedure starting at the last exercise date. The thus constructed martingales are finally used to compute an upper price bound. At the same time, the algorithm also provides approximate continuation functions which may be used to construct a price lower bound. The algorithm is applied to the pricing of flexible caps in a Hull White (1990) model setup. The simple model choice allows for comparison of the computed price bounds with the exact price which is obtained by means of a trinomial tree implementation. As a result, we obtain tight price bounds for the considered application. Moreover, the algorithm is generically designed for multidimensional problems and is tractable to implement. 
A. Papapantoleon, J.G.M. Schoenmakers, D. Skovmand, Efficient and accurate logLévy approximations to Lévy driven LIBOR models, Journal of Computational Finance, 15 (2012), pp. 344.
Abstract
The LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a Lévydriven LIBOR model and aim at developing accurate and efficient logLévy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps and swaptions show that the approximations perform very well. In addition, we also consider the logLévy approximation of annuities, which offers good approximations for high volatility regimes. 
P. Friz, S. Gerhold, A. Gulisashvili, S. Sturm, On refined volatility smile expansion in the Heston model, Quantitative Finance, 11 (2011), pp. 11511164.

D. Belomestny, J.G.M. Schoenmakers, A jumpdiffusion Libor model and its robust calibration, Quantitative Finance, 11 (2011), pp. 529546.

D. Belomestny, A. Kolodko, J.G.M. Schoenmakers, Pricing CMS spreads in the Libor market model, International Journal of Theoretical and Applied Finance, 13 (2010), pp. 4562.
Abstract
We present two approximation methods for pricing of CMS spread options in Libor market models. Both approaches are based on approximating the underlying swap rates with lognormal processes under suitable measures. The first method is derived straightforwardly from the Libor market model. The second one uses a convexity adjustment technique under a linear swap model assumption. A numerical study demonstrates that both methods provide satisfactory approximations of spread option prices and can be used for calibration of a Libor market model to the CMS spread option market. 
D. Belomestny, Spectral estimation of the fractional order of a Lévy process, The Annals of Statistics, 38 (2010), pp. 317351.

P. Friz, S. Benaim, Regular variation and smile asymptotics, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 19 (2009), pp. 112.

D. Belomestny, S. Mathew, J.G.M. Schoenmakers, Multiple stochastic volatility extension of the Libor market model and its implementation, Monte Carlo Methods and Applications, 15 (2009), pp. 285310.
Abstract
In this paper we propose a Libor model with a highdimensional specially structured system of driving CIR volatility processes. A stable calibration procedure which takes into account a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy to implement. 
D. Belomestny, G.N. Milstein, V. Spokoiny, Regression methods in pricing American and Bermudan options using consumption processes, Quantitative Finance, 9 (2009), pp. 315327.
Abstract
Here we develop methods for efficient pricing multidimensional discretetime American and Bermudan options by using regression based algorithms together with a new approach towards constructing upper bounds for the price of the option. Applying sample space with payoffs at the optimal stopping times, we propose sequential estimates for continuation values, values of the consumption process, and stopping times on the sample paths. The approach admits constructing both low and upper bounds for the price by Monte Carlo simulations. The methods are illustrated by pricing Bermudan swaptions and snowballs in the Libor market model. 
R. Krämer, P. Mathé, Modulus of continuity of Nemytskiĭ operators with application to a problem of option pricing, Journal of Inverse and IllPosed Problems, 16 (2008), pp. 435461.

O. Reiss, J.G.M. Schoenmakers, M. Schweizer, From structural assumptions to a link between assets and interest rates, Journal of Economic Dynamics & Control, 31 (2007), pp. 593612.
Abstract
We derive a link between assets and interest rates in a standard multiasset diffusion economy from two structural assumptions ? one on the volatility and one on the short rate function. Our main result is economically intuitive and testable from data since it only involves empirically observable quantities. A preliminary study illustrates how this could be done. 
J.G.M. Schoenmakers, B. Coffey, Systematic generation of parametric correlation structures for the LIBOR market model, International Journal of Theoretical and Applied Finance, 6 (2003), pp. 507519.
Abstract
We present a conceptual approach of deriving parsimonious correlation structures suitable for implementation in the LIBOR market model. By imposing additional constraints on a known ratio correlation structure, motivated by economically sensible assumptions concerning forward LIBOR correlations, we yield a semiparametric framework of nondegenerate correlation structures with realistic properties. Within this framework we derive systematically low parametric structures with, in principal, any desired number of parameters. As illustrated, such structures may be used for smoothing a matrix of historically estimated LIBOR return correlations. In combination with a suitably parametrized deterministic LIBOR volatility norm we so obtain a parsimonious multifactor market model which allows for joint calibration to caps and swaptions. See Schoenmakers [2002] for a stable full implied calibration procedure based on the correlation structures developed in this paper.
Contributions to Collected Editions

J.G.M. Schoenmakers, SHOWCASE 17  Expirywise Heston LIBOR model, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 314315.

P. Friz, M. KellerRessel, Moment explosions in financial models, in: Encyclopedia of Quantitative Finance, R. Cont, ed., Wiley, Chichester, 2010, pp. 12471253.

P. Friz, Implied volatility: Large strike asymptotics, in: Encyclopedia of Quantitative Finance, R. Cont, ed., Wiley, Chichester, 2010, pp. 909913.
Preprints, Reports, Technical Reports

CH. Bayer, Ch. Ben Hammouda, A. Papapantoleon, M. Samet, R. Tempone, Optimal damping with hierarchical adaptive quadrature for efficient Fourier pricing of multiasset options in Lévy models, Preprint no. 2968, WIAS, Berlin, 2022.
Abstract, PDF (1686 kByte)
Efficient pricing of multiasset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourierbased methods become competitive compared to alternative techniques because the integrand in the frequency space has often higher regularity than in the physical space. However, when designing a numerical quadrature method for most of these Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of the damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of the high dimensionality. To address these challenges, we propose an efficient numerical method for pricing European multiasset options based on two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of damping parameters based on a proposed heuristic optimization rule. Second, we use sparsification and dimensionadaptivity techniques to accelerate the convergence of the quadrature in high dimensions. Our extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some Lévy models demonstrates the advantages of adaptivity and our damping rule on the numerical complexity of the quadrature methods. Moreover, our approach achieves substantial computational gains compared to the Monte Carlo method. 
CH. Bayer, E. Hall, R.F. Tempone, Weak error rates for option pricing under linear rough volatility, Preprint no. 2916, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2916 .
Abstract, PDF (1685 kByte)
In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887904, 2016], seek to fit observed market data based on the observation that the logrealized variance behaves like a fractional Brownian motion with small Hurst parameter, H < 1/2, over reasonable timescales. Both time series of asset prices and optionderived price data indicate that H often takes values close to 0.1 or less, i.e., rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the nonMarkovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only H. We prove rate H + 1/2 for the weak convergence of the Euler method for the rough SteinStein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by secondorder polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses TalayTubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion.
Talks, Poster

N. Tapia, Transport and continuity equations with (very) rough noise, MiniWorkshop ``Regularization by Noise: Theoretical Foundations, Numerical Methods and Applications driven by Levy Noise'', February 13  19, 2022, Mathematisches Forschungsinstitut Oberwolfach, February 18, 2022.

CH. Bayer, RKHS regularization of singular local stochastic volatility McKeanVlasov models (online talk), MiniWorkshop ``Regularization by Noise: Theoretical Foundations, Numerical Methods and Applications driven by Levy Noise'', February 13  20, 2022, Mathematisches Forschungsinstitut Oberwolfach, February 14, 2022.

CH. Bayer, Calibration of rough volatility models by deep learning, Rough Workshop 2019, September 4  6, 2019, Technische Universität Wien, Financial and Actuarial Mathematics, Austria.

CH. Bayer, Deep calibration of rough volatility models, New Directions in Stochastic Analysis: Rough Paths, SPDEs and Related Topics, WIAS und TU Berlin, March 18, 2019.

CH. Bayer, Deep calibration of rough volatility models, SIAM Conference on Financial Mathematics & Engineering, June 4  7, 2019, Society for Industrial and Applied Mathematics, Toronto, Ontario, Canada, June 7, 2019.

CH. Bayer, Learning rough volatility, Algebraic and Analytic Perspectives in the Theory of Rough Paths and Signatures, November 14  15, 2019, University of Oslo, Department of Mathematics, Norway, November 14, 2019.

M. Ladkau, A new multifactor stochastic volatility model with displacement, First BerlinSingapore Workshop on Quantitative Finance and Financial Risk, May 21  24, 2014, WIASBerlin und HumboldtUniversität zu Berlin, May 22, 2014.

J.G.M. Schoenmakers, Affine LIBOR models with multiple curves: Theory, examples and calibration, 11th German Probability and Statistics Days 2014, March 5  7, 2014, Universität Ulm, March 6, 2014.

M. Ladkau, A new multifactor stochastic volatility model with displacement, PreMoLab Workshop on: Advances in predictive modeling and optimization, May 16  17, 2013, WIASBerlin, May 16, 2013.

CH. Bayer, Asymptotics beats Monte Carlo: The case of correlated local volatility baskets, Stochastic Methods in Finance and Physics, July 15  19, 2013, University of Crete, Department of Applied Mathematics, Heraklion, Greece, July 19, 2013.

CH. Bayer, Asymptotics can beat Monte Carlo, 20th Annual Global Derivatives & Risk Management, April 16  18, 2013, The International Centre for Business Information (ICBI), Amsterdam, Netherlands, April 18, 2013.

M. Ladkau, A new multifactor stochastic volatility model with displacement, International Workshop on Numerical Algorithms in Computational Finance, July 20  22, 2011, Goethe Universität Frankfurt, Goethe Center for Scientific Computing (GCSC), July 21, 2011.

P. Friz, On refined density and smile expansion in the Heston model, Workshop ``Stochastic Analysis in Finance and Insurance'', January 23  29, 2012, Mathematisches Forschungsinstitut Oberwolfach, January 29, 2012.

J.G.M. Schoenmakers, Advanced Libor modeling, Postbank Bonn, February 25, 2010.

J.G.M. Schoenmakers, Holomorphic transforms with application to affine processes, 5th General Conference in Advanced Mathematical Methods in Finance, May 4  8, 2010, University of Ljubljana, Faculty of Mathematics and Physics, Slovenia, May 6, 2010.

P. Friz, From numerical aspects of stochastic financial models to the foundations of stochastic differential equations (and back), Annual Meeting of the Deutsche MathematikerVereinigung and 17th Congress of the Österreichische Mathematische Gesellschaft, Section ``Financial and Actuarial Mathematics'', September 20  25, 2009, Technische Universität Graz, Austria, September 25, 2009.

D. Belomestny, Estimation of the jump activity of a Lévy process from low frequency data, Haindorf Seminar 2009, February 12  15, 2009, HumboldtUniversität zu Berlin, CASE  Center for Applied Statistics and Economics, Hejnice, Czech Republic, February 12, 2009.

D. Belomestny, Spectral estimation of the fractional order of a Lévy process, Workshop ``Statistical Inference for Lévy Processes with Applications to Finance'', July 15  17, 2009, EURANDOM, Eindhoven, Netherlands, July 16, 2009.

J.G.M. Schoenmakers, Statistical and numerical methods for evaluation for financial derivates and risk, Center Days 2009 (DFG Research Center scshape Matheon), March 30  April 1, 2009, Technische Universität Berlin, March 31, 2009.

P. Mathé, On nonstability of some inverse problem in option pricing, Workshop on Inverse and Partial Information Problems: Methodology and Applications, October 27  31, 2008, Austrian Academy of Sciences, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, October 30, 2008.

J.G.M. Schoenmakers, Robust Libor modelling and calibration, International Multidisciplinary Workshop on Stochastic Modeling, June 25  29, 2007, Sevilla, Spain, June 29, 2007.

J.G.M. Schoenmakers, A jumpdiffusion Libor model and its robust calibration, 4th World Congress of the Bachelier Finance Society, August 17  20, 2006, National Center of Sciences, Hitotsubashi University, ICS, Tokyo, Japan, August 20, 2006.

J.G.M. Schoenmakers, Interest rate modelling: Practical calibration and implementation techniques, June 15  16, 2006, Risk, London, UK.

J.G.M. Schoenmakers, Interest rate modelling  Practical calibration and implementation techniques, Incisive Media Events, Hong Kong, China, December 8, 2004.

J.G.M. Schoenmakers, Robust calibration of LIBOR market models, Petit Dejeuner de la Finance, November 4  5, 2003, Paris, November 5, 2003.

J.G.M. Schoenmakers, Accuracy and stability of LIBOR model calibration via parametric correlation structures and approximative swaption pricing, Risk Conference 2002, April 23  24, 2002, Paris, France, April 23, 2002.

J.G.M. Schoenmakers, Calibration of LIBOR models to caps and swaptions: A way around intrinsic instabilities via parsimonious structures and a collateral market criterion, Johann Wolfgang GoetheUniversität, MathFinance Institute, Frankfurt am Main, November 7, 2002.

J.G.M. Schoenmakers, Calibration of LIBOR models to caps and swaptions: A way around intrinsic instabilities via parsimonious structures and a collateral market criterion, Quantitative Finance 2002, Risk Waters Group, London, UK, November 26, 2002.

J.G.M. Schoenmakers, Endogenous interest rates in asset markets, 2nd World Congress of the Bachelier Finance Society, June 12  15, 2002, Crete, Greece, June 14, 2002.

J.G.M. Schoenmakers, Kalibrierung im LIBOR Modell, Reuters AG, Düsseldorf, March 11, 2002.

J.G.M. Schoenmakers, Correlation structure in LIBOR market models, calibration to caps and swaptions, Technical University of Delft, Netherlands, May 8, 2001.

J.G.M. Schoenmakers, Term structure dynamics endogenously induced by multiasset markets, Conference Risk 2001 Europe, April 10  11, 2001, Paris, France, April 10, 2001.

J.G.M. Schoenmakers, HJM term structure dynamics from a multi asset market; finite factor models, Hamburger StochastikTage 2000, March 21  24, 2000, Universität Hamburg, March 21, 2000.

J.G.M. Schoenmakers, HJM term structure dynamics from a multi asset market; finite factor models, WIASKolloquium, Berlin, May 15, 2000.

J.G.M. Schoenmakers, Stable calibration of multifactor LIBOR market models via a semiparametric correlation structure, "`ICBI 2000 Conference"', December 6  7, 2000, Genf, Switzerland, December 7, 2000.

J.G.M. Schoenmakers, Stable implied calibration of multifactor LIBOR models by semiparametric correlation structure, Risk Conference Math Week 2000, November 13  17, 2000, New York, USA, November 15, 2000.
External Preprints

CH. Bayer, M. Fukasawa, N. Shonosuke , On the weak convergence rate in the discretization of rough volatility models, Preprint no. arXiv:2203.02943, Cornell University, 2022, DOI 10.48550/arXiv.2203.02943 .

P. Friz, S. Gerhold, A. Gulisashvili, S. Sturm, On refined volatility smile expansion in the Heston model, Preprint no. arXiv:1001.3003, Cornell University Library, arXiv.org, 2010.