arXiv subjects

Patrick Cheridito

Publications and source records attributed to Patrick Cheridito.

At least 37 records · Page 2Linked to original sources

arXiv2021

Solving high-dimensional optimal stopping problems using deep learning

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

cs.CE↗

arXiv2021

Deep splitting method for parabolic PDEs

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

math.NA↗

arXiv2021

Assessing asset-liability risk with neural networks

We introduce a neural network approach for assessing the risk of a portfolio of assets and liabilities over a given time period. This requires a conditional valuation of the portfolio given the state of the world at a later time, a problem that is particularly challenging if the portfolio contains structured products or complex insurance contracts which do not admit closed form valuation formulas. We illustrate the method on different examples from banking and insurance. We focus on value-at-risk and expected shortfall, but the approach also works for other risk measures.

q-fin.RM↗

arXiv2021

Representation of increasing convex functionals with countably additive measures

We derive two types of representation results for increasing convex functionals in terms of countably additive measures. The first is a max-representation of functionals defined on spaces of real-valued continuous functions and the second a sup-representation of functionals defined on spaces of real-valued Borel measurable functions. Our assumptions consist of sequential semicontinuity conditions which are easy to verify in different applications.

math.FA↗

arXiv2021

On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case.

math.PR↗

arXiv2021

A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions

Gradient descent optimization algorithms are the standard ingredients that are used to train artificial neural networks (ANNs). Even though a huge number of numerical simulations indicate that gradient descent optimization methods do indeed convergence in the training of ANNs, until today there is no rigorous theoretical analysis which proves (or disproves) this conjecture. In particular, even in the case of the most basic variant of gradient descent optimization algorithms, the plain vanilla gradient descent method, it remains an open problem to prove or disprove the conjecture that gradient descent converges in the training of ANNs. In this article we solve this problem in the special situation where the target function under consideration is a constant function. More specifically, in the case of constant target functions we prove in the training of rectified fully-connected feedforward ANNs with one-hidden layer that the risk function of the gradient descent method does indeed converge to zero. Our mathematical analysis strongly exploits the property that the rectifier function is the activation function used in the considered ANNs. A key contribution of this work is to explicitly specify a Lyapunov function for the gradient flow system of the ANN parameters. This Lyapunov function is the central tool in our convergence proof of the gradient descent method.

math.NA↗

arXiv2021

Non-convergence of stochastic gradient descent in the training of deep neural networks

Deep neural networks have successfully been trained in various application areas with stochastic gradient descent. However, there exists no rigorous mathematical explanation why this works so well. The training of neural networks with stochastic gradient descent has four different discretization parameters: (i) the network architecture; (ii) the amount of training data; (iii) the number of gradient steps; and (iv) the number of randomly initialized gradient trajectories. While it can be shown that the approximation error converges to zero if all four parameters are sent to infinity in the right order, we demonstrate in this paper that stochastic gradient descent fails to converge for ReLU networks if their depth is much larger than their width and the number of random initializations does not increase to infinity fast enough.

cs.LG↗

arXiv2021

Efficient approximation of high-dimensional functions with neural networks

In this paper, we develop a framework for showing that neural networks can overcome the curse of dimensionality in different high-dimensional approximation problems. Our approach is based on the notion of a catalog network, which is a generalization of a standard neural network in which the nonlinear activation functions can vary from layer to layer as long as they are chosen from a predefined catalog of functions. As such, catalog networks constitute a rich family of continuous functions. We show that under appropriate conditions on the catalog, catalog networks can efficiently be approximated with rectified linear unit-type networks and provide precise estimates on the number of parameters needed for a given approximation accuracy. As special cases of the general results, we obtain different classes of functions that can be approximated with ReLU networks without the curse of dimensionality.

math.NA↗

arXiv2020

Martingale optimal transport duality

We obtain a dual representation of the Kantorovich functional defined for functions on the Skorokhod space using quotient sets. Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. These sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given. Another important ingredient of our analysis is a regularized version of Jakubowski's $S$-topology on the Skorokhod space.

math.FA↗

arXiv2020

Pricing and hedging American-style options with deep learning

In this paper we introduce a deep learning method for pricing and hedging American-style options. It first computes a candidate optimal stopping policy. From there it derives a lower bound for the price. Then it calculates an upper bound, a point estimate and confidence intervals. Finally, it constructs an approximate dynamic hedging strategy. We test the approach on different specifications of a Bermudan max-call option. In all cases it produces highly accurate prices and dynamic hedging strategies with small replication errors.

q-fin.CP↗

arXiv2020

Deep optimal stopping

In this paper we develop a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such, it is broadly applicable in situations where the underlying randomness can efficiently be simulated. We test the approach on three problems: the pricing of a Bermudan max-call option, the pricing of a callable multi barrier reverse convertible and the problem of optimally stopping a fractional Brownian motion. In all three cases it produces very accurate results in high-dimensional situations with short computing times.

math.NA↗

arXiv2018

Robust expected utility maximization with medial limits

In this paper we study a robust expected utility maximization problem with random endowment in discrete time. We give conditions under which an optimal strategy exists and derive a dual representation for the optimal utility. Our approach is based on a general representation result for monotone convex functionals, a functional version of Choquet's capacitability theorem and medial limits. The novelty is that it works under nondominated model uncertainty without any assumptions of time-consistency. As applications, we discuss robust utility maximization problems with moment constraints, Wasserstein constraints and Wasserstein penalties.

q-fin.PM↗

arXiv2018

BSDE formulation of combined regular and singular stochastic control problems

In this paper we study a class of combined regular and singular stochastic control problems that can be expressed as constrained BSDEs. In the Markovian case, this reduces to a characterization through a PDE with gradient constraint. But the BSDE formulation makes it possible to move beyond Markovian models and consider path-dependent problems. We also provide an approximation of the original control problem with standard BSDEs that yield a characterization of approximately optimal values and controls.

math.OC↗

arXiv2017

Duality formulas for robust pricing and hedging in discrete time

In this paper we derive robust super- and subhedging dualities for contingent claims that can depend on several underlying assets. In addition to strict super- and subhedging, we also consider relaxed versions which, instead of eliminating the shortfall risk completely, aim to reduce it to an acceptable level. This yields robust price bounds with tighter spreads. As examples we study strict super- and subhedging with general convex transaction costs and trading constraints as well as risk-based hedging with respect to robust versions of the average value at risk and entropic risk measure. Our approach is based on representation results for increasing convex functionals and allows for general financial market structures. As a side result it yields a robust version of the fundamental theorem of asset pricing.

q-fin.MF↗

arXiv2017

BSE's, BSDE's and fixed point problems

In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed point problem in a space of random vectors. This makes it possible to employ general fixed point arguments to find solutions. For instance, Banach's contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean-Vlasov type equations and equations with coefficients of superlinear growth.

math.PR↗

arXiv2016

Duality for increasing convex functionals with countably many marginal constraints

In this work we derive a convex dual representation for increasing convex functionals on a space of real-valued Borel measurable functions defined on a countable product of metric spaces. Our main assumption is that the functionals fulfill marginal constraints satisfying a certain tightness condition. In the special case where the marginal constraints are given by expectations or maxima of expectations, we obtain linear and sublinear versions of Kantorovich's transport duality and the recently discovered martingale transport duality on products of countably many metric spaces.

math.FA↗

arXiv2015

Multidimensional quadratic and subquadratic BSDEs with special structure

We study multidimensional BSDEs of the form $$ Y_t = ξ+ \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s $$ with bounded terminal conditions $ξ$ and drivers $f$ that grow at most quadratically in $Z_s$. We consider three different cases. In the first one the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver $f$ in $Z_s$ is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending the solution recursively.

math.PR↗

arXiv2014

Conditional Analysis on R^d

This paper provides versions of classical results from linear algebra, real analysis and convex analysis in a free module of finite rank over the ring $L^0$ of measurable functions on a $σ$-finite measure space. We study the question whether a submodule is finitely generated and introduce the more general concepts of $L^0$-affine sets, $L^0$-convex sets, $L^0$-convex cones, $L^0$-hyperplanes, $L^0$-half-spaces and $L^0$-convex polyhedral sets. We investigate orthogonal complements, orthogonal decompositions and the existence of orthonormal bases. We also study $L^0$-linear, $L^0$-affine, $L^0$-convex and $L^0$-sublinear functions and introduce notions of continuity, differentiability, directional derivatives and subgradients. We use a conditional version of the Bolzano-Weierstrass theorem to show that conditional Cauchy sequences converge and give conditions under which conditional optimization problems have optimal solutions. We prove results on the separation of $L^0$-convex sets by $L^0$-hyperplanes and study $L^0$-convex conjugate functions. We provide a result on the existence of $L^0$-subgradients of $L^0$-convex functions, prove a conditional version of the Fenchel-Moreau theorem and study conditional inf-convolutions.

math.FA↗