Olivier Guéant:
From Theoretical Results to Real-World Applications in Bonds, FX, Commodities and Cryptocurrencies: An Overview on Market Making Models
Since the foundational work of Ho and Stoll, later refined by Avellaneda and Stoikov, algorithmic market-making models have advanced to include more realistic features, such as trade sizes, complex price dynamics, tiering, externalization, and market impact. These models have been applied to many OTC markets ranging from illiquid corporate bonds to highly liquid foreign exchange markets and cryptocurrencies (price-aware AMMs). This talk will review key developments in the field, highlighting both theoretical advancements and practical applications. It will also introduce a recent extension of these models to address challenges faced by precious metals dealers, who act as market makers in spot markets while hedging with futures. The mathematical framework leverages tools from stochastic optimal control, optimization, variational calculus, and stochastic filtering.
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References
[1] M. Avellaneda and S. Stoikov. High-frequency trading in a limit order book. Quantitative
Finance, 8(3):217–224, 2008.
[2] A. Barzykin, P. Bergault, and O. Guéant. Market-making by a foreign exchange
dealer. Risk, September 2022.
[3] A. Barzykin, P. Bergault, and O. Guéant. Dealing with multi-currency inventory
risk in foreign exchange cash markets. Risk, March 2023.
[4] A. Barzykin, P. Bergault, and O. Guéant. Market Making in Spot Precious Metals. Risk, Dec 2024.
[5] P. Bergault, D. Evangelista, O. Guéant, and D. Vieira. Closed-form approximations
in multi-asset market making. Applied Mathematical Finance, 28(2):101–142, 2021.
[6] O. Guéant. The Financial Mathematics of Market Liquidity: From optimal execution to market
making, volume 33. CRC Press, 2016.
[7] O. Guéant. Optimal market making. Applied Mathematical Finance, 24(2):112–154, 2017.
[8] O. Guéant, C. A. Lehalle, and J. Fernandez-Tapia. Dealing with the inventory risk:
a solution to the market making problem. Mathematics and financial economics, 7(4):477–507, 2013.
[9] T. Ho and H. R. Stoll. Optimal dealer pricing under transactions and return uncertainty.
Journal of Financial economics, 9(1):47–73, 1981.
Sara Svaluto-Ferro:
Tractable infinite dimensional models: theory and applications
During this course we introduce, study, and apply several stochastic models in infinite dimensions. We focus in particular to the class of infinite dimensional polynomial (jump-) diffusions, for which a set of ready-to-use mathematical tools have been developed.
The first part of the course is dedicated to the theoretical aspects. After a presentation of the martingale problem we move to the so-called moment formula, providing a representation of some expected quantities in terms of an infinite dimensional linear ODE. In some cases of interest such ODE reduces to a PDE, has a representation à la Feymann-Kac, or can even be solved explicitly, for some or all initial conditions.
In the second part we study two main applications: measure valued diffusions for energy markets and signature models in finance.
We introduce a framework that allows to employ (non-negative) measure-valued processes for energy market modeling, in particular for electricity and gas futures. Interpreting the process’ spatial structure as time to maturity, we show how the Heath-Jarrow-Morton (HJM) approach can be translated to this framework, thus guaranteeing arbitrage free modeling in infinite dimensions. We derive an analog to the HJM-drift condition and then treat in a Markovian setting existence of non-negative measure-valued diffusions that satisfy this condition. To analyze mathematically convenient classes we consider measure-valued polynomial and affine diffusions, where we can precisely specify the diffusion part in terms of continuous functions satisfying certain admissibility conditions. For calibration purposes, we illustrate how these functions can then be parameterized by neural networks yielding measure-valued analogs of neural SPDEs. By combining Fourier approaches or the moment formula with stochastic gradient descent methods, this then allows for tractable calibration procedures which we also test by way of example on market data.
Finally, we introduce the concept of signature of a semimartingale and present its main properties. In particular, we show that the signature process of a (classical) polynomial process is a polynomial process and illustrate how the moment formula applies in this case. We then move to signature SDEs whose characteristics are linear functions of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional tractable stochastic process.
Slides:
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References
Classics:
[1] C. Cuchiero, M. Keller-Ressel, and J. Teichmann (2012). Polynomial processes and their applications to mathematical finance. Finance Stoch. 16 711–740.
[2] D. Duffie D. Filipovic and W. Schachermayer. Affine processes and applications in finance. Annals of applied probability pages 984–1053 2003.
[3] D. Filipovic and M. Larsson (2016). Polynomial diffusions and applications in finance. Finance Stoch. 20 931–972. MR3551857
Measures:
[1] C. Cuchiero, L. Di Persio, F. Guida, S. Svaluto-Ferro, Measure-valued processes for energy markets, Mathematical Finance, 2024.
Link
[2] C. Cuchiero, L. Di Persio, F. Guida, S. Svaluto-Ferro, Measure-valued affine and polynomial diffusions, Stochastic Processes and their Applications, 175, 104392, 2024.
Link
Signatures:
[1] C. Cuchiero, G. Gazzani, S. Svaluto-Ferro, Signature-based models: theory and calibration, SIAM Journal on Financial Mathematics, 14(3), 910-957, 2023.
Link
[2] C. Cuchiero, G. Gazzani, J. Möller, S. Svaluto-Ferro, Joint calibration to SPX and VIX options with signature-based models, Mathematical Finance, 2024.
Link
[3] C. Cuchiero, J. Teichmann, S. Svaluto-Ferro, Signature SDEs from an affine and polynomial perspective, 2023. Link
Short lectures
Kristoffer Andersson:
Deep xVA under a structural model
In this talk, we present a structural default model for portfolio-wide valuation adjustments (xVAs), modeled as a system of coupled BSDEs. These equations are organized into four interdependent layers: i) clean values, ii) initial margin and ColVA, iii) CVA/DVA/MVA, and iv) FVA. The nonlinear dependencies between these layers mean that deeper layers rely on adjustments from earlier layers. A naive Monte Carlo approach would require nested averaging across all four layers, which is computationally infeasible.
To overcome this, we apply the deep BSDE method iteratively, layer by layer, breaking the nested structure. Each equation in the current layer is approximated using a neural network, with outputs from earlier layers serving as inputs. Initial margin is modeled as a quantile and computed via deep quantile regression, while a change of measure technique captures the impact of rare but critical default events by the bank or the counterparty.
This framework significantly reduces computational complexity and scales effectively to high-dimensional, non-symmetric portfolios. We will outline the methodology and present numerical results demonstrating its effectiveness in realistic scenarios.
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Zhipeng Huang:
Convergence of the Markovian iteration for coupled FBSDEs via a differentiation approach
In this talk, we present the Markovian iteration for coupled forward-backward stochastic differential equations (FBSDEs) with fully coupled forward drift. The convergence of this scheme for FBSDEs coupled with the Y process was previously studied by Bender and Zhang (2008), and it is noticed that the main challenge of having Z coupling lies in controlling the Lipschitz constant of the decoupling field uniformly in time steps and iterations when applying the fixed-point argument. In this paper, we address this difficulty through a differentiation approach for the Z process. As a result, we establish the well-posedness of the discretization of the FBSDE with fully coupled drift and obtain the convergence of the Markovian iteration for this type of equation. Finally, we provide several numerical examples to illustrate our theoretical results.
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Balint Negyesi :
A Deep BSDE approach for the simultaneous pricing and delta-
gamma hedging of large portfolios consisting of high-dimensional multi-
asset Bermudan options
Direct financial applications of the methods introduced in Negyesi et al. (2024), Negyesi (2024) are presented. Therein, a new discretization of (discretely reflected) Markovian backward stochastic differential equations is given which involves a Gamma process, corresponding to second-order sensitivities of the associated option’s price. The main contributions of this work is to apply these techniques in the context of portfolio risk management. Large portfolios of a mixture of European and Bermudan derivatives are cast into the framework of discretely reflected BSDEs. The resulting system is solved by a neural network regression Monte Carlo method proposed in the aforementioned papers. Numerical experiments are presented on high-dimensional portfolios, consisting of several European, Bermudan and American options, which demonstrate the robustness and accuracy of the method. The corresponding Profit-and-Loss distributions indicate an order of magnitude gain in the number of rebalancing dates needed in order to achieve a certain level of risk tolerance, when the second-order conditions are also satisfied. The hedging strategies significantly outperform benchmark methods both in the case of standard delta- and delta-gamma hedging.
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Jasper Rou:
A time-stepping deep gradient flow method for option pricing
In this research, we consider a novel deep learning approach for pricing options by solving partial differential equations (PDE). More specifically, we consider a Time-stepping Deep Gradient Flow method, where the PDE is solved by discretizing it in time and writing it as the solution of minimizing a variational problem. A neural network approximation is then trained to solve this minimization using stochastic gradient descent. This method reduces the training time compared to for instance the Deep Galerkin Method. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness and adheres to known bounds for option prices. The accuracy and efficiency of the proposed method is assessed in a series of numerical examples, with particular focus in the lifted Heston model. Furthermore, we prove two things. First, that there exists a neural network converging to the solution of the PDE. This proof consists of three parts: 1) convergence of the time stepping; 2) equivalence of the solution of the discretized PDE and the minimizer of the variational formulation and 3) the approximation of the minimizer by a neural network by using a version of the universal approximation theorem. Second, we prove that when training the network we converge to the correct solution. This proof consists of two parts: 1) as the number of neurons goes to infinity we converge to some gradient flow and 2) as the training time goes to infinity this gradient flow converges to the solution.
- Ivo Richert (Kiel University): Estimation of dynamically recalibrated affine and polynomial models in finance
- Felix Sachse (Technische Universität Dresden): Term structure shapes in the Svensson family
- Wojciech Świercz (Wrocław University of Economics and Business): Stock Market Prediction on High-Frequency Data Using LDE-Net Model
- Radosław Szostak (Wrocław University of Economics and Business): Neural-SDE Model for High-Frequency Stock Market Index Forecasting
- Henrik Valett (Kiel University): Parameter estimation for polynomial processes