arXiv subjects

Antonis Papapantoleon

Publications and source records attributed to Antonis Papapantoleon.

At least 19 recordsLinked to original sources

arXiv2025

Convergence of the generalization error for deep gradient flow methods for PDEs

The aim of this article is to provide a firm mathematical foundation for the application of deep gradient flow methods (DGFMs) for the solution of (high-dimensional) partial differential equations (PDEs). We decompose the generalization error of DGFMs into an approximation and a training error. We first show that the solution of PDEs that satisfy reasonable and verifiable assumptions can be approximated by neural networks, thus the approximation error tends to zero as the number of neurons tends to infinity. Then, we derive the gradient flow that the training process follows in the ``wide network limit'' and analyze the limit of this flow as the training time tends to infinity. These results combined show that the generalization error of DGFMs tends to zero as the number of neurons and the training time tend to infinity.

math.NA↗

arXiv2025

Stability of backward propagation of chaos

The purpose of the present paper is to introduce and establish a notion of stability for the backward propagation of chaos with respect to (initial) data sets. Consider, for example, a sequence of discrete-time martingales converging to a continuous-time limit, and a system of mean-field BSDEs that satisfies the backward propagation of chaos, i.e. converges to a sequence of i.i.d. McKean-Vlasov BSDEs. Then, we say that the backward propagation of chaos is stable if the system of mean-field BSDEs driven by the discrete-time martingales converges to the sequence of McKean-Vlasov BSDEs driven by the continuous-time limit. We consider the convergence scheme of the backward propagation of chaos as the image of the corresponding data set under which this scheme is established. Then, using an appropriate notion of convergence for data sets, we are able to show a variety of continuity properties for this functional point of view. Along the way, we also provide stability results for mean-field and McKean-Vlasov BSDEs, which are of interest in their own right, for numerical approximations of these equations.

math.PR↗

arXiv2025

A time-stepping deep gradient flow method for option pricing in (rough) diffusion models

We develop a novel deep learning approach for pricing European options in diffusion models, that can efficiently handle high-dimensional problems resulting from Markovian approximations of rough volatility models. The option pricing partial differential equation is reformulated as an energy minimization problem, which is approximated in a time-stepping fashion by deep artificial neural networks. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness, and adheres to a priori 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.

q-fin.CP↗

arXiv2024

Existence, uniqueness and propagation of chaos for general McKean-Vlasov and mean-field BSDEs

We consider backward stochastic differential equations (BSDEs) with mean-field and McKean-Vlasov interactions in their generators in a general setting, where the drivers are square-integrable martingales, with a focus on the independent increments case, and the filtrations are (possibly) stochastically discontinuous. In other words, we consider discrete- and continuous-time systems of mean-field BSDEs and McKean-Vlasov BSDEs in a unified setting. We provide existence and uniqueness results for these BSDEs using new a priori estimates that utilize the stochastic exponential. Then, we provide propagation of chaos results for systems of particles that satisfy BSDEs, i.e. we show that the asymptotic behaviour of the solutions of mean-field systems of BSDEs, as the multitude of the systems grows to infinity, converges to I.I.D solutions of McKean-Vlasov BSDEs. We introduce a new technique for showing the backward propagation of chaos, that makes repeated use of the a priori estimates, inequalities for the Wasserstein distance and the ``conservation of solutions'' under different filtrations, and does not demand the solutions of the mean field systems to be exchangeable or symmetric. Finally, we deduce convergence rates for the propagation of chaos, under advanced integrability conditions on the solutions of the BSDEs.

math.PR↗

arXiv2024

Improved model-free bounds for multi-asset options using option-implied information and deep learning

We consider the computation of model-free bounds for multi-asset options in a setting that combines dependence uncertainty with additional information on the dependence structure. More specifically, we consider the setting where the marginal distributions are known and partial information, in the form of known prices for multi-asset options, is also available in the market. We provide a fundamental theorem of asset pricing in this setting, as well as a superhedging duality that allows to transform the maximization problem over probability measures in a more tractable minimization problem over trading strategies. The latter is solved using a penalization approach combined with a deep learning approximation using artificial neural networks. The numerical method is fast and the computational time scales linearly with respect to the number of traded assets. We finally examine the significance of various pieces of additional information. Empirical evidence suggests that "relevant" information, i.e. prices of derivatives with the same payoff structure as the target payoff, are more useful that other information, and should be prioritized in view of the trade-off between accuracy and computational efficiency.

q-fin.PR↗

arXiv2024

Quasi-Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi-Asset Options

Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. Fourier methods leverage the regularity properties of the integrand in the Fourier domain to accurately and rapidly value options that typically lack regularity in the physical domain. However, most of the existing Fourier approaches face hurdles in high-dimensional settings due to the tensor product (TP) structure of the commonly employed numerical quadrature techniques. To overcome this difficulty, this work advocates using the randomized quasi-MC (RQMC) quadrature to improve the scalability of Fourier methods with high dimensions. The RQMC technique benefits from the smoothness of the integrand and alleviates the curse of dimensionality while providing practical error estimates. Nonetheless, the applicability of RQMC on the unbounded domain, $\mathbb{R}^d$, requires a domain transformation to $[0,1]^d$, which may result in singularities of the transformed integrand at the corners of the hypercube, and hence deteriorate the performance of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on boundary growth conditions on the transformed integrand. The proposed transformation preserves sufficient regularity of the original integrand for fast convergence of the RQMC method. To validate our analysis, we demonstrate the efficiency of employing RQMC with an appropriate transformation to evaluate options in the Fourier space for various pricing models, payoffs, and dimensions. Finally, we highlight the computational advantage of applying RQMC over MC or TP in the Fourier domain, and over MC in the physical domain for options with up to 15 assets.

q-fin.CP↗

arXiv2024

Convergence rates for Backward SDEs driven by Lévy processes

We consider Lévy processes that are approximated by compound Poisson processes and, correspondingly, BSDEs driven by Lévy processes that are approximated by BSDEs driven by their compound Poisson approximations. We are interested in the rate of convergence of the approximate BSDEs to the ones driven by the Lévy processes. The rate of convergence of the Lévy processes depends on the Blumenthal--Getoor index of the process. We derive the rate of convergence for the BSDEs in the $\mathbb L^2$-norm and in the Wasserstein distance, and show that, in both cases, this equals the rate of convergence of the corresponding Lévy process, and thus is optimal.

math.PR↗

arXiv2024

A deep implicit-explicit minimizing movement method for option pricing in jump-diffusion models

We develop a novel deep learning approach for pricing European basket options written on assets that follow jump-diffusion dynamics. The option pricing problem is formulated as a partial integro-differential equation, which is approximated via a new implicit-explicit minimizing movement time-stepping approach, involving approximation by deep, residual-type Artificial Neural Networks (ANNs) for each time step. The integral operator is discretized via two different approaches: (a) a sparse-grid Gauss-Hermite approximation following localised coordinate axes arising from singular value decompositions, and (b) an ANN-based high-dimensional special-purpose quadrature rule. Crucially, the proposed ANN is constructed to ensure the appropriate asymptotic behavior of the solution for large values of the underlyings and also leads to consistent outputs with respect to a priori known qualitative properties of the solution. The performance and robustness with respect to the dimension of these methods are assessed in a series of numerical experiments involving the Merton jump-diffusion model, while a comparison with the deep Galerkin method and the deep BSDE solver with jumps further supports the merits of the proposed approach.

q-fin.CP↗

arXiv2023

Optimal Damping with Hierarchical Adaptive Quadrature for Efficient Fourier Pricing of Multi-Asset Options in Lévy Models

Efficiently pricing multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods are competitive compared to alternative techniques because the integrand in the frequency space often has a higher regularity than that in the physical space. However, when designing a numerical quadrature method for most Fourier pricing approaches, two key aspects affecting the numerical complexity should be carefully considered: (i) the choice of damping parameters that ensure integrability and control the regularity class of the integrand and (ii) the effective treatment of high dimensionality. We propose an efficient numerical method for pricing European multi-asset options based on two complementary ideas to address these challenges. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed optimization rule. Second, we employ sparsification and dimension-adaptivity techniques to accelerate the convergence of the quadrature in high dimensions. The extensive numerical study on basket and rainbow options under the multivariate geometric Brownian motion and some Lévy models demonstrates the advantages of adaptivity and the damping rule on the numerical complexity of quadrature methods. Moreover, for the tested two-asset examples, the proposed approach outperforms the COS method in terms of computational time. Finally, we show significant speed-up compared to the Monte Carlo method for up to six dimensions.

q-fin.CP↗

arXiv2023

Machine learning for option pricing: an empirical investigation of network architectures

We consider the supervised learning problem of learning the price of an option or the implied volatility given appropriate input data (model parameters) and corresponding output data (option prices or implied volatilities). The majority of articles in this literature considers a (plain) feed forward neural network architecture in order to connect the neurons used for learning the function mapping inputs to outputs. In this article, motivated by methods in image classification and recent advances in machine learning methods for PDEs, we investigate empirically whether and how the choice of network architecture affects the accuracy and training time of a machine learning algorithm. We find that the generalized highway network architecture achieves the best performance, when considering the mean squared error and the training time as criteria, within the considered parameter budgets for the Black-Scholes and Heston option pricing problems. Considering the transformed implied volatility problem, a simplified DGM variant achieves the lowest error among the tested architectures. We also carry out a capacity-normalised comparison for completeness, where all architectures are evaluated with an equal number of parameters. Finally, for the implied volatility problem, we additionally include experiments using real market data.

q-fin.CP↗

arXiv2023

Stability of backward stochastic differential equations: the general case

In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The current result extends earlier contributions in the literature of stability of BSDEs and unifies several frameworks for numerical approximations of BSDEs and their implementations.

math.PR↗

arXiv2022

Model-free bounds for multi-asset options using option-implied information and their exact computation

We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested in the computation of model-free upper and lower bounds for their arbitrage-free prices. We work in a completely realistic setting, in that we only assume the knowledge of traded prices for other single- and multi-asset derivatives, and even allow for the presence of bid-ask spread in these prices. We provide a fundamental theorem of asset pricing for this market model, as well as a superhedging duality result, that allows to transform the abstract maximization problem over probability measures into a more tractable minimization problem over vectors, subject to certain constraints. Then, we recast this problem into a linear semi-infinite optimization problem, and provide two algorithms for its solution. These algorithms provide upper and lower bounds for the prices that are $\varepsilon$-optimal, as well as a characterization of the optimal pricing measures. These algorithms are efficient and allow the computation of bounds in high-dimensional scenarios (e.g. when $d=60$). Moreover, these algorithms can be used to detect arbitrage opportunities and identify the corresponding arbitrage strategies. Numerical experiments using both synthetic and real market data showcase the efficiency of these algorithms, while they also allow to understand the reduction of model risk by including additional information, in the form of known derivative prices.

math.OC↗

arXiv2021

Detection of arbitrage opportunities in multi-asset derivatives markets

We are interested in the existence of equivalent martingale measures and the detection of arbitrage opportunities in markets where several multi-asset derivatives are traded simultaneously. More specifically, we consider a financial market with multiple traded assets whose marginal risk-neutral distributions are known, and assume that several derivatives written on these assets are traded simultaneously. In this setting, there is a bijection between the existence of an equivalent martingale measure and the existence of a copula that couples these marginals. Using this bijection and recent results on improved Fréchet-Hoeffding bounds in the presence of additional information on functionals of a copula by Lux and Papapantoleon [18], we can extend the results of Tavin [33] on the detection of arbitrage opportunities to the general multi-dimensional case. More specifically, we derive sufficient conditions for the absence of arbitrage and formulate an optimization problem for the detection of a possible arbitrage opportunity. This problem can be solved efficiently using numerical optimization routines. The most interesting practical outcome is the following: we can construct a financial market where each multi-asset derivative is traded within its own no-arbitrage interval, and yet when considered together an arbitrage opportunity may arise.

q-fin.PR↗

arXiv2019

Stability results for martingale representations: the general case

In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi--left--continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems.

math.PR↗

arXiv2018

Model-free bounds on Value-at-Risk using extreme value information and statistical distances

We derive bounds on the distribution function, therefore also on the Value-at-Risk, of $φ(\mathbf X)$ where $φ$ is an aggregation function and $\mathbf X = (X_1,\dots,X_d)$ is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of $\mathbf X$, is available. In order to include this information in the computation of Value-at-Risk bounds, we utilize a reduction principle that relates this problem to an optimization problem over a standard Fréchet class, which can then be solved by means of the rearrangement algorithm or using analytical results. Second, we assume that the copula of $\mathbf X$ is known on a subset of its domain, and finally we consider the case where the copula of $\mathbf X$ lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Fréchet--Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Fréchet--Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.

q-fin.RM↗

arXiv2018

Existence and uniqueness results for BSDEs with jumps: the whole nine yards

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.

math.PR↗

arXiv2018

Marginal and dependence uncertainty: bounds, optimal transport, and sharpness

Motivated by applications in model-free finance and quantitative risk management, we consider Fréchet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for these Fréchet classes that extend previous results in the related literature. These proofs are based on representation results for increasing convex functionals and the explicit computation of the conjugates. We show that the dual transport problem admits an explicit solution for the function $f=1_B$, where $B$ is a rectangular subset of $\mathbb R^d$, and provide an intuitive geometric interpretation of this result. The improved Fréchet--Hoeffding bounds provide ad-hoc upper bounds for these Fréchet classes. We show that the improved Fréchet--Hoeffding bounds are pointwise sharp for these classes in the presence of uncertainty in the marginals, while a counterexample yields that they are not pointwise sharp in the absence of uncertainty in the marginals, even in dimension 2. The latter result sheds new light on the improved Fréchet--Hoeffding bounds, since Tankov [30] has showed that, under certain conditions, these bounds are sharp in dimension 2.

math.PR↗

arXiv2018

Expansion formulas for European quanto options in a local volatility FX-LIBOR model

We develop an expansion approach for the pricing of European quanto options written on LIBOR rates (of a foreign currency). We derive the dynamics of the system of foreign LIBOR rates under the domestic forward measure and then consider the price of the quanto option. In order to take the skew/smile effect observed in fixed income and FX markets into account, we consider local volatility models for both the LIBOR and the FX rate. Because of the structure of the local volatility function, a closed form solution for quanto option prices does not exist. Using expansions around a proxy related to log-normal dynamics, we derive approximation formulas of Black--Scholes type for the price, that have the benefit of giving very rapid numerical procedures. Our expansion formulas have the major advantage that they allow for an accurate estimation of the error, using Malliavin calculus, which is directly related to the maturity of the option, the payoff, and the level and curvature of the local volatility function. These expansions also illustrate the impact of the quanto drift adjustment, while the numerical experiments show an excellent accuracy.

q-fin.PR↗