Neural networks for option pricing and hedging: a literature reviewWe thank Marc Chataigner, Stéphane Crépey, Antoine Jacquier, and Martin Larsson for comments on an early version of this note.

# Neural networks for option pricing and hedging: a literature review00footnotetext: We thank Marc Chataigner, Stéphane Crépey, Antoine Jacquier, and Martin Larsson for comments on an early version of this note.

Johannes Ruf Department of Mathematics, London School of Economics and Political Science. Email: j.ruf@lse.ac.uk    Weiguan Wang Department of Mathematics, London School of Economics and Political Science. Email: w.wang34@lse.ac.uk
###### Abstract

Neural networks have been used as a nonparametric method for option pricing and hedging since the early 1990s. Far over a hundred papers have been published on this topic. This note intends to provide a comprehensive review. Papers are compared in terms of input features, output variables, benchmark models, performance measures, data partition methods, and underlying assets. Furthermore, related work and regularisation techniques are discussed.

## 1 Introduction

Beginning with Malliaris and Salchenberger [1993b] and Hutchinson et al. [1994], more than one hundred papers in the academic literature concern the use of artificial neural networks (ANNs) for option pricing and hedging. This work provides a review of this literature. The motivation for this summary arose from our companion paper Ruf and Wang [2019]. There we continue the discussions of this note; in particular, of potentially problematic data leakage when training ANNs to historic financial data.

This paper is organised in the following way. Section 2 features Table 1, a summary of the literature that concerns the use of ANNs for nonparametric pricing (and hedging) of options. Section 3 provides a list of recommended papers from Table 1. Section 4 provides an overview of related work where ANNs are applied in the context of option pricing and hedging, but not necessarily as nonparametric estimation tools. Section 5 briefly discusses various regularisation techniques used in the reviewed literature.

## 2 ANN based option pricing and hedging in the literature

Bennell and Sutcliffe [2004], Chen and Sutcliffe [2012], and Hahn [2013]111Hahn [2013] also surveys the use of ANNs to predict realised volatility. Here we do not aim to do so. provide extensive literature surveys on the application of ANNs to option pricing and hedging problems. Here we complement these surveys with additional and more recent papers.

Table 1 summarises a large part of the literature and compares six relevant characteristics. They are features (or so-called explanatory variables), outputs of the ANN, benchmark models, data partition between training and test sets, and the underlyings along with the time span of the data. In Table 1, we only list papers that study an ANN’s performance for the option pricing and hedging problem with a somehow statistical perspective. Other papers have different approaches, e.g., a computational perspective, and hence do not fit naturally in the table. These papers are discussed separately in Section 4.

Let us explain how to read Table 1. It summarises six relevant characteristics that describe how each paper treats the pricing/hedging problem. The columns ‘Features’ and ‘Outputs’ show explanatory features given to the ANN as inputs and outputs, respectively. Table 2 explains notations and abbreviations used for these columns. The ‘Benchmarks’ column lists non ANN-based techniques with which an ANN is compared. Table 3 explains the corresponding abbreviations. Table 4 presents abbreviations and definitions for the ‘Performance measures’ column, which summarises how an ANN (and its benchmarks) are evaluated in each paper. The performance measures marked bold are related to evaluations along multiple periods. Table 5 explains abbreviations for the underlying assets used in each study and listed in the ’Underlyings’ column.

Here an ‘executive summary’ of Table 1:

• There exist two ways of using the stock price and option strike as inputs to an ANN. Sometimes they are used as two separate features. Other times, only their ratio (the so-called moneyness) is used as an input. Recent research uses the second approach more often than early research. See also Subsection 2.1 for a discussion of this point.

• There are many different choices of volatility estimates concerning input features and benchmarks. The conclusions drawn often depend on this choice. Subsections 2.1 and 2.3 provide more details on this point.

• Most papers focus on estimating option prices, around fifteen ten papers (10% of all papers listed) on estimating implied volatilities, and very few deal with the hedging problem directly; see also Subsection 2.2.

• In some studies, data is partitioned into a training and a test set in a way that violates the underlying time series structure. This introduces information leakage and underestimates the generalization error of the ANN. This is further discussed in Subsection 2.4.

For the reader interested in a small selection of all these papers, we refer to Section 3.

In the following, we compare and classify papers listed in Table 1 in terms of features, outputs, performance measures and benchmarks, data partition methods, underlying assets and time span.

### 2.1 Features

To estimate the option price, the underlying price and the strike price are two indispensable variables. Two ways of feeding these two variables into an ANN as input have been suggested. One way is to use the underlying price and strike price separately. An alternative is to use a ratio (i.e., moneyness) instead. For several reasons, the alternative way is favourable:

• Using moneyness instead of the stock price and the strike price separately reduces the number of inputs and thus makes the training of the ANN easier.

• Many parametric models assume that the statistical distribution of the underlying asset’s return is independent of the level of the underlying. Hence, the option pricing function is homogeneous of degree one with respect to the underlying stock price and the strike price, so that only moneyness is needed to learn the function. Incorporating this assumption into the ANN can potentially reduce overfitting; see Hutchinson et al. [1994], Lajbcygier and Connor [1997a, b], Anders et al. [1998], and Garcia and Gençay [1998, 2000].

• Moneyness is a stationary input feature in contrast to the stock price and the strike price. Using it helps generalisation and reduces overfitting; see Ghysels et al. [1998] and Garcia and Gençay [1998, 2000].

Bennell and Sutcliffe [2004] undertake a systematic experiment on various choices of input features, including underlying price, strike price, moneyness, and on choices of outputs, including option price and option price divided by strike.

Apart from the underlying price and the strike price, volatilities are also widely used as input features. This can be done in several different ways. The most relevant ones are the following:

• Using historical volatility estimates as features.

• Using volatility indices such as VIX as features.

• Using implied volatilities as features.

• Using GARCH forecasts of (realised or implied) volatility as features.

Table 2 lists further volatility features. The choices of features by the different papers are worked out in the ‘Features’ column of Table 1. There exist also several papers that do not use any volatility-type feature as input for their ANNs. A few papers, e.g., Blynski and Faseruk [2006], Andreou et al. [2008], or Wang [2009b], compare different volatility features. It is now believed that implied volatility is more effective as a feature than historical volatility in order to predict option prices.

Some papers investigate whether additional features can help the ANN with prediction. To name a few, Ghaziri et al. [2000] and Healy et al. [2002] incorporate option open interests. Samur and Temur [2009] study whether the inclusion of variance improves the performance of the ANN. Montesdeoca and Niranjan [2016] explore the potential prediction power of trading volume, option interest, and other variables. Cao et al. [2019] investigate the benefit from using the underlying return.

### 2.2 Outputs

The papers of Table 1 can also be categorised in terms of their outputs:

• The most common output is the option price. Depending on whether moneyness is used, or underlying price and strike price are used separately, the output can be the option price or the option price divided by the strike price. Some papers also investigate the ANN’s ability when it is trained to learn the so-called bias; i.e., the difference between market price and a price estimated by a parametric model. Such an ANN is called hybrid ANN; see, for example, Boek et al. [1995] or Lajbcygier and Connor [1997a, a]. While most of the early papers train their ANNs to fit prices, Garcia and Gençay [2000] train to prices, but validate to hedging errors in order to determine the network size that gives the lowest hedging error. Andreou et al. [2010] emphasize the relevance of choosing the right loss function when interested in the hedging task.

• Another type of output is the implied volatility. The obtained implied volatilities can be converted to option prices by the Black-Scholes formula. Mostafa and Dillon [2008] compare ANNs that output option prices to ANNs that output implied volatilities. More recently, Liu et al. [2019b] evaluate an ANN’s ability to approximate the inverse of the Black-Scholes formula.

• The third kind of output (always denoted by HR in Table 1) is a sensitivity or a hedging ratio. Only a few papers discuss such an architecture for an ANN. The first papers are Carverhill and Cheuk [2003], Chen and Sutcliffe [2012], and Shin and Ryu [2012]. More recently, Buehler et al. [2019a, b] and Ruf and Wang [2019] follow up on this line of research. Buehler et al. [2019b] consider also the hedging of exotic options such as barrier options.

We could have also added the so-called calibration papers to Table 1, which construct ANNs to map prices to specific model parameters or vice versa. Instead we decided to dedicate Section 4.1 below to these papers.

### 2.3 Performance measures and benchmarks

When evaluating the performance of ANNs, common statistical measures are mean absolute error (MAE), mean absolute percentage error (MAPE), and mean squared error (MSE).222222Several papers use equivalent versions of the measures in Table 4. For example, sometimes root mean squared error is used instead of mean squared error. For consistency, in Table 1, we have made the corresponding adjustments. These are related to evaluations over a single period, in terms of pricing or hedging. Some papers also propose to evaluate the ANN’s performance over multiple periods. For instance, Hutchinson et al. [1994] introduce the mean absolute tracking error (MATE) and prediction error (PE), which appear also in many later papers. Buehler et al. [2019a] introduce the conditional value-at-risk (CVaR) for evaluating hedging strategies.

An ANN’s performance should also be compared to a benchmark, for example, a parametric pricing model. The most widely used benchmark is the Black-Scholes formula, which requires a volatility as input. As Table 1 summarises a historic volatility estimate is used the most often. Also certain implied volatilities (e.g., historical or at-the-money) appear in the literature. Blynski and Faseruk [2006] compare historical realised and historical implied volatility for the Black-Scholes benchmark.

Note that for the pricing task, the Black-Scholes formula with contract-specific implied volatility is not a suitable benchmark as by definition it prices options without errors. However, if the task is to predict hedging ratios instead of option prices, the Black-Scholes formula with contract-specific implied volatility is a valid benchmark.

In addition to the Black-Scholes formula, other widely used parametric benchmarks are stochastic volatility pricing models; e.g., used in Gençay and Gibson [2007], Jang and Lee [2019], or Liu et al. [2019b]. Ruf and Wang [2019] observe that if a benchmark is chosen that incorporates both delta and vega hedging then an ANN does not outperform even a simple two-factor regression model.

For American type options, benchmarks used are the Barone-Adesi and Whaley [1987] pricing method (e.g., Lajbcygier [2002]), and the Cox-Ross-Rubinstein model (e.g., Chen and Lee [1999]).

### 2.4 Data partition methods

An ANN needs to be trained on a training set (in-sample) and then tested on a test set (out-of-sample). There exist several ways to partition a data set into such a training and test set. The first way is chronologically. That is, the early data constitutes the training set, and the late data constitutes the test set. Table 1 indicates that most of the papers follow this approach. However, some studies violate this time structure in the data by choosing a different way to partition the data. Violations can be introduced by randomly partitioning the data into a training and a test set or by using a so-called ‘odd-even split.’

Random partitioning breaks the time structure and introduces information leakage between the training set and the test set. When an ANN is trained on a training set constructed in such a way, the error on the test set underestimates the generalisation error of the ANN. Yao et al. [2000] and our companion paper Ruf and Wang [2019] provide more discussion on this point.

Some papers only work with independent draws from various distributions, and therefore do not involve any time series structure. Although these papers randomly partition the whole data set into a training and test set, no time structure is violated. Hence, in Table 1, we classify this approach as chronological partition.

A related issue is the existence of time-inhomogeneity in financial data; in particular, volatility changes over time. When working with real data, some papers use a rolling window method to tackle this issue, especially when the time range is long and volatilities are not included as input features. Such papers include Hutchinson et al. [1994], Dugas et al. [2009], and others. However, it remains an open question how big window sizes need to be.

### 2.5 Underlying assets and time span

Both simulation data and real data can be used to train an ANN for a specific problem. Simulation data is much easier to work with, since it is free of noise and sometimes a close-to-optimal solution is available as a benchmark, such as for the Black-Scholes and Heston models. For instance, le Roux and du Toit [2001], Morelli et al. [2004], and Karatas et al. [2019] investigate an ANN’s performance on simulation data. Most other papers use either both simulation and real data or only real data. Options on S&P500 have been studied by the largest number of papers, since they are the most liquidly traded options. Options on FTSE100 and S&P100 have also been studied in several papers. We refer to Table 5 for a more complete list of all the underlyings being used.

Some papers focus on American option pricing and hedging. Underlyings for American options are usually individual stocks. Papers involving American options include Kelly [1994], Chen and Lee [1999], Meissner and Kawano [2001], Pires and Marwala [2004a], Pires and Marwala [2005], and Amornwattana et al. [2007]. As elaborated in Subsection 4.3, American options can also be priced differently by ANNs, via learning the value function or optimal stopping rule in a dynamic programming setting; see Kohler et al. [2010] and Becker et al. [2019].

## 3 Recommended papers

Among the many papers of Table 1, we would like to highlight a few. Such a selection is clearly personal and subjective. Despite the subjective selection, we believe that this list might serve as a good starting point to get an overview of this field. We also provide a Google Scholar citation count.232323As of October 3, 2019. As mentioned before, Table 1 focuses only on those papers that use ANNs to estimate option prices and related variables. Recently there have been many interesting and promising developments in the use of ANNs for calibration purposes or as computational tools. These papers are not included here, but Section 4 provides some pointers to this literature.

Among the following highlighted papers, some are the first to propose innovative solutions. Others investigate the problem in a systematic way.

• Hutchinson et al. [1994] (# citations: 749) is one of the first papers and the most highly cited one to use ANNs to estimate option prices. They introduce a methodology to evaluate the hedging performance over multiple periods, applied by many papers later on.

• Lajbcygier and Connor [1997a] (# citations:242424This count includes the number of citations for Lajbcygier and Connor [1997b]. 51) is one of the first papers that propose to learn the difference between model prices and observed market option prices.

• Anders et al. [1998] (# citations: 106) compare the performance of ANNs and of the Black-Scholes benchmark when using different volatility estimates.

• Garcia and Gençay [2000] (# citations:252525This count includes the number of citations for Garcia and Gençay [1998]. 210) incorporate a homogeneity hint for the ANN. Hence, this is one of the first papers that embed financial domain knowledge into the construction of an ANN.

• Carverhill and Cheuk [2003] (# citations: 15) first propose an ANN that outputs hedging strategies directly, instead of option prices.

• Bennell and Sutcliffe [2004] (# citations: 83), Chen and Sutcliffe [2012] (# citations: 12), and Hahn [2013] (# citations: 9) provide three extensive literature surveys.

• Dugas et al. [2009] (# citations:262626This count includes the number of citations for Dugas et al. [2001]. 172) first design an ANN architecture that enforces no-arbitrage conditions such as convexity of option prices.

• Andreou et al. [2010] (# citations: 19) combines an ANN with parametric models to learn functions that return implied model parameters. Such an ANN essentially calibrates parametric models.

• Buehler et al. [2019a] (# citations: 23) develop a novel framework for hedging a portfolio of derivatives in the presence of market frictions, and allow convex risk measures as loss functions. Their framework allows pricing and hedging without observing option prices.

As this is a subjective selection, we also would like to highlight our companion paper Ruf and Wang [2019], which provides a new benchmark based on delta-vega hedging and discusses data leakage issues.

## 4 Related papers

In the last few years, many novel techniques have been developed to apply ANNs to tasks arising in option pricing beyond the nonparametric estimation of prices and hedging ratios. In this section we provide a few pointers to this rapidly developing literature.272727At times it was not always clear cut to us whether a paper should be included in Table 1 or in this section. For example, the calibration papers of Section 4.1 could have been put into Table 1 as mentioned in Section 2.2. Similarly, Barucci et al. [1996, 1997], discussed in Section 4.2, learn the Black-Scholes model and hence could have been put into Table 1.

### 4.1 Calibration

As already mentioned in Section 3, Andreou et al. [2010] propose an ANN that returns implied model parameters. Hence, the ANN essentially calibrates parametric models. We observe a recent surge of the application of ANN to calibration. In this approach option prices are first mapped to a parametric model, which is then used to determine option prices. This approach can move the computationally heavy calibration off-line, thus significantly accelerating option pricing. For instance, Hernandez [2017] uses an ANN to calibrate a single-factor Hull-White model. Dimitroff et al. [2018], McGhee [2018] and Liu et al. [2019a] calibrate stochastic volatility models, and Stone [2019] and Bayer et al. [2019]282828For more details, see also Bayer and Stemper [2018] and Horvath et al. [2019]. calibrate rough volatility models. Itkin [2019] highlights some pitfalls in the existing approaches and proposes resolutions that improve both performance and accuracy of calibration.

Going the ‘indirect’ way via first calibrating a model and then using it to determine the hedging ratio has at least two advantages. First, it provides additional interpretability as only the calibration step is replaced by an ANN. This can be important for a financial entity subject to regulatory requirements. Second, it provides an arguably strong tailor-made regularisation effect as it replaces a nonparametric estimation task by the task of estimating a model with usually less than 5-10 parameters.

### 4.2 Solving partial differential equations

The option pricing problem sometimes involves solving a partial differential equation (PDE). Barucci et al. [1996, 1997] use the Galerkin method and ANNs for solving the Black-Scholes PDE. E et al. [2017], Han et al. [2018], and Beck et al. [2019] utilize ANNs to solve high-dimensional semilinear parabolic PDEs. They propose to reformulate the PDEs using backward stochastic differential equations, and the gradient of the unknown solutions is approximated by ANNs. Their numerical results suggest that the method is effective for a wide variety of (possibly high-dimensional) problems. One case study involves the pricing of European options on 100 defaultable underlying assets. There are several recent papers, such as Henry-Labordère [2017], Sirignano and Spiliopoulos [2018], Chan-Wai-Nam et al. [2019], Huré et al. [2019] , Jacquier and Oumgari [2019], and Vidales et al. [2019], who have developed this application of ANNs further.

### 4.3 Approximating value functions in optimal control problems

ANNs can be used to approximate value functions that appear in dynamic programming, for example arising in the American option pricing problem; see for example Ye and Zhang [2019]. Kohler et al. [2010] use ANNs to estimate continuation values for high-dimensional American option pricing. Becker et al. [2019] use ANNs for optimal stopping problems by learning the optimal stopping rule from Monte Carlo samples. ANNs have also been proposed to approximate the value function of a dynamic program for real option pricing, see Taudes et al. [1998].

In this context, we also mention Fecamp et al. [2019], who use an ANN as a computational tool to solve the pricing and hedging problem under market frictions such as transaction costs.

### 4.4 Further work

Albanese et al. [2019] use an ANN to compute the conditional value-at-risk and expected shortfall necessary for certain XVA computations, by solving a quantile regression.

We would like to also mention Halperin [2017] and Kolm and Ritter [2019] who suggest a reinforcement learning methodology to take market frictions into account for the option pricing task.

Finally, generative ANNs have been suggested recently as a non-parametric simulation tool for stock prices; see, for example, Henry-Labordère [2019], Kondratyev and Schwarz [2019], and Wiese et al. [2019b]. Such simulation engines could then be used for option pricing and hedging, a direction still to be explored systematically. Just after finishing this survey, Wiese et al. [2019a] proposed a generative ANN for option prices (instead of stock prices).

## 5 Digression: regularisation techniques

As the advance of hardware allows for bigger ANNs to be built, regularization techniques have become more important as part of the ANN training. Such techniques include , dropout, early stopping, etc.; see Ormoneit [1999], Gençay and Qi [2001], Gençay and Salih [2003], and Liu et al. [2019b]. Complementing these universal regularisations, several papers embed financial domain knowledge into ANNs, either at the stage of architecture design or training. Let us here also mention the suggested feature design by Lu and Ohta [2003a, b], who consider the pricing of exotic options and suggest to use digital option prices as features.

For the architecture design the following has been suggested:

• Homogeneity hint. Garcia and Gençay [1998, 2000] incorporate a homogeneity hint by considering an ANN consisting of two parts, one controlled by moneyness and the other controlled by time-to-maturity.

• Shape-restricted outputs. Dugas et al. [2001, 2009], Lajbcygier [2004], Yang et al. [2017], Huh [2019], and Zheng et al. [2019] enforce certain no-arbitrage conditions such as monotonicity and convexity of the ANN pricing function by fixing an appropriate architecture.

At the training state the following techniques are being used:

• Data augmentation. Yang et al. [2017] and Zheng et al. [2019] create additional synthetic options to help with the training of ANNs.

• Loss penalty. Itkin [2019] and Ackerer et al. [2019] add various penalty terms to the loss function. Those terms present no-arbitrage conditions. For example, parameter configurations that allow for calendar arbitrage are being penalised.

In the context of ANN training, we would like also to mention Niranjan [1996], de Freitas et al. [2000a, b], and Palmer [2019]. These papers propose and examine novel training algorithms for ANNs and illustrate them in the context of option hedging; these algorithms include the extended Kalman filter, sequential Monte Carlo, and evolutionary algorithms.

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