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Pricing Options and Computing Implied Volatilities using Neural Networks

Shuaiqiang Liu
Cornelis W. Oosterlee
1,2 and
Sander M. Bohte
Delft Institute of Applied Mathematics (DIAM), Delft University of Technology, Building 28, Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands
Centrum Wiskunde & Informatica, Science Park 123, 1098 XG Amsterdam, The Netherlands
Author to whom correspondence should be addressed.
Risks 2019, 7(1), 16;
Submission received: 8 January 2019 / Revised: 3 February 2019 / Accepted: 6 February 2019 / Published: 9 February 2019


This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function approximators, this method trains an optimized ANN on a data set generated by a sophisticated financial model, and runs the trained ANN as an agent of the original solver in a fast and efficient way. We test this approach on three different types of solvers, including the analytic solution for the Black-Scholes equation, the COS method for the Heston stochastic volatility model and Brent’s iterative root-finding method for the calculation of implied volatilities. The numerical results show that the ANN solver can reduce the computing time significantly.

1. Introduction

In computational finance, numerical methods are commonly used for the valuation of financial derivatives and also in modern risk management. Generally speaking, advanced financial asset models are able to capture nonlinear features that are observed in the financial markets. However, these asset price models are often multi-dimensional, and, as a consequence, do not give rise to closed-form solutions for option values.
Different numerical methods have therefore been developed to solve the corresponding option pricing partial differential equation (PDE) problems, e.g. finite differences, Fourier methods and Monte Carlo simulation. In the context of financial derivative pricing, there is a stage in which the asset model needs to be calibrated to market data. In other words, the open parameters in the asset price model need to be fitted. This is typically not done by historical asset prices, but by means of option prices, i.e., by matching the market prices of heavily traded options to the option prices from the mathematical model, under the so-called risk-neutral probability measure. In the case of model calibration, thousands of option prices need to be determined in order to fit these asset parameters. However, due to the requirement of a highly efficient computation, certain high quality asset models are discarded. Efficient numerical computation is also increasingly important in financial risk management, especially when we deal with real-time risk management (e.g., high frequency trading) or counterparty credit risk issues, where a trade-off between efficiency and accuracy seems often inevitable.
Artificial neural networks (ANNs) with multiple hidden layers have become successful machine learning methods to extract features and detect patterns from a large data set. There are different neural network variants for particular tasks, for example, convolutional neural networks for image recognition LeCun et al. (2010) and recurrent neural networks for time series analysis Lipton (2015). It is well-known that ANNs can approximate nonlinear functions Cybenko (1989); Hornik (1991); Hornik et al. (1990), and can thus be used to approximate solutions to PDEs Lagaris et al. (1998); Sirignano and Spiliopoulos (2018). Recent advances in data science have shown that using deep learning techniques even highly nonlinear multi-dimensional functions can be accurately represented LeCun et al. (2015). Essentially, ANNs can be used as powerful universal function approximators without assuming any mathematical form for the functional relationship between the input variables and the output. Moreover, ANNs easily allow for parallel processing to speed up evaluations, especially on Graphics Processing Units (GPUs) Oh and Jung (2004) or even Tensor Processing Units(TPUs) Jouppi et al. (2017).
We aim to take advantage of a classical ANN to speed up option valuation by learning the results of an option pricing method. From a computational point of view, the ANN does not suffer much from the dimensionality of a PDE. An “ANN solver” is typically decomposed into two separate phases, a training phase and a test (or prediction) phase. During the training phase, the ANN “learns” the PDE solver, by means of the data set generated by the sophisticated models and corresponding numerical solvers. This stage is usually time consuming, however, it can be done off-line. During the test phase, the trained model can be employed to approximate the solution on-line. The ANN solution can typically be computed as a set of matrix multiplications, which can be implemented in parallel and highly efficiently, especially with GPUs or TPUs. As a result, the trained ANN delivers financial derivative prices, or other quantities, efficiently, and the on-line time for accurate option pricing may be reduced, especially for involved asset price models. We will show in this paper that this data-driven approach is highly promising.
The proposed approach in this paper attempts to accelerate the pricing of European options under a unified data-driven ANN framework. ANNs have been used in option pricing for some decades already. There are basically two directions. One is that based on the observed market option prices and the underlying asset value, ANN-based regression techniques have been applied to fit a model-free, non-parametric pricing function, see, for example, Hutchinson et al. (1994); Yao et al. (2000); Gencay and Qi (2001); Garcia and Gençay (2000). Furthermore, the authors of Dugas et al. (2001); Yang et al. (2017) designed special kernel functions to incorporate prior financial knowledge into the neural network while forecasting option prices.
Another direction is to improve the performance of model-based pricing by means of ANNs. The interest in accelerating classical PDE solvers via ANNs is rapidly growing. The papers Han et al. (2017); Weinan et al. (2017); Beck et al. (2017) take advantage of reinforcement learning to speed up solving high-dimensional stochastic differential equations. The author of Sirignano and Spiliopoulos (2017) proposes an optimization algorithm, the so-called stochastic gradient descent in continuous time, combined with a deep neural network to price high-dimensional American options. In Fan and Mancini (2009) the pricing performance of financial models is enhanced by non-parametric learning approaches that deal with a systematic bias of pricing errors. Of course, this trend takes place not only in computational finance, but also in other engineering fields where PDEs play a key role, like computational fluid dynamics, see Hesthaven and Ubbiali (2018); Raissi and Karniadakis (2018); Sirignano and Spiliopoulos (2018); Tompson et al. (2016). The work in this paper belongs to this latter direction. Here, we use traditional solvers to generate artificial data, then we train the ANN to learn the solution for different problem parameters. Compared to Lagaris et al. (1998) or Sirignano and Spiliopoulos (2018), our data-driven approach finds, next to the solutions of the option pricing PDEs, the implicit relation between variables and a specific parameter (i.e., the implied volatility).
This paper is organized as follows. In Section 2, two fundamental option pricing models, the Black-Scholes and the Heston stochastic volatility PDEs, are briefly introduced. In addition to European option pricing, we also analyze robustness issues of root-finding methods to compute the so-called implied volatility. In Section 3, the employed ANN is presented with suitable hyper-parameters. After training the ANN to learn the results of the financial models for different problem parameters, numerical ANN results with the corresponding errors are presented in Section 4.

2. Option Pricing and Asset Models

In this section, two asset models are briefly presented, the geometric Brownian motion (GBM) asset model, which gives rise to the Black-Scholes option pricing PDE, and the Heston stochastic volatility asset model, leading to the Heston PDE. We also discuss the concept of implied volatility. We will use European option contracts as the examples, however, other types of options can be taken into consideration in a similar way.

2.1. The Black-Scholes PDE

A first model for asset prices is GBM,
d S t = μ S t d t +