Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

Abstract

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black-Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. Moreover, these two methods solve the backward dynamic programming problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite this model is only bidimensional, the whole history of the process impacts on the price, and handling all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.Pricing American options is clearly a crucial question of finance but also a challenging one since computing the optimal exercise strategy is not an evident task. This issue is even more exacting when the underling of the option is a multi-dimensional process, such as a baskets of d assets, since in this case the direct application of standard numerical schemes, such as finite difference or tree methods, is not possible because of the exponential growth of the calculation time and the required working memory.Common approaches in this field can be divided in four groups: techniques which rely on recombinant trees to discretize the underlyings (see [4], [11] and [24]), techniques which employ regression on a truncated basis of L 2 in order to compute the conditional expectations (see [28] and [32]), techniques which exploit Malliavin calculus to obtain representation formulas for the conditional expectation (see [1], [3], [9], and [27]) and techniques which make use of duality-based approaches for Bermudan option pricing (see [21], [26] and [31]). Recently, Machine Learning algorithms (Rasmussen and Williams [33]) and Deep Learning techniques (Nielsen [30]) have found great application in this sector of option pricing. Neural networks are used by Kohler et al. [25] to price American options based on several underlyings. Deep Learning techniques are nowadays widely used in solving large differential equations, which is intimately related to option pricing. In particular, Han et al. [20] introduce a Deep Learning-based approach that can handle general high-dimensional parabo...

DNN Expression Rate Analysis of High-Dimensional PDEs: Application to Option Pricing

“…While we admit that the European maximum option pricing problem for uncorrelated assets constitutes a rather special problem, the proofs in this paper develop several novel deep neural network approximation results of independent interest that can be applied to more general settings where a low-rank structure is implicit in highdimensional problems. For mostly numerical results on machine learning for pricing American options, we refer to [16]. Lastly we note that after a first preprint of the present paper was submitted, a number of research articles related to this work have appeared [13,14,17,19,[24][25][26]28,36].…”

Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities

“…Recently, several new stochastic approximation methods for certain classes of high-dimensional nonlinear PDEs have been proposed and studied in the scientific literature. In particular, we refer, e.g., to [11,12,26,29,30,53] for BSDE-based approximation methods for PDEs in which nested conditional expectations are discretized through suitable regression methods, we refer, e.g., to [10,39,41,42] for branching diffusion approximation methods for PDEs, we refer, e.g., to [1][2][3][6][7][8]13,14,16,17,21,24,25,31,[34][35][36]40,43,48,50,52,[54][55][56][57][58]60,62,63] for deep learning based approximation methods for PDEs, and we refer to [4,5,20,28,46,47] for numerical simulations, approximation results, and extensions of the in…”

“…For MLP approximation methods it has been recently shown in [4,45,46] that such algorithms do indeed overcome the curse of dimensionality for certain classes of gradient-independent PDEs. Numerical simulations for deep learning based approximation methods for nonlinear PDEs in high dimensions are very encouraging (see, e.g., the above named references [1][2][3][6][7][8]13,14,16,17,21,24,25,31,[34][35][36]40,43,48,50,52,[54][55][56][57][58]60,62,63]) but so far there is only partial error analysis available for such algorithms (which, in turn, is strongly based on the above-mentioned error analysis for the MLP approximation method; cf. [44] and, e.g., [9,23,32,33,36,49,51,61,62]).…”

JDOI Variance Reduction Method and the Pricing of American-Style Options

“…Additionally, the recent advances in machine learning has incentivized research in this particular area and allowed for extensions of these techniques to high-dimensional problems (cf. [KKT10], [BCJ19], [BCJ20], [GMZ20], [RW20]). The present article follows this stream of the literature and provides a variance-reduction technique that can be applied on top of numerous Monte Carlo based algorithms, therefore providing a powerful way to speed up these methods.…”