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.

Authors & year Features Outputs Benchmarks Performance measures Partition method Underlyings
Malliaris and Salchenberger [1993b, a] , , , , lagged and BS-IM MAE, MAPE, MSE Chronological S&P100. 6M
Hutchinson et al. [1994] BS-H, Linear MATE, PE, Chronological Simulation (BS);
S&P500. 5Y
Kelly [1994] CRR MAE, MTE, MSE, ? Individual stocks. 6M
Boek et al. [1995] , , , BS-H MAPE, ? AOSPI. 2Y
Miranda and Burgess [1995] ? Linear ? ? IBEX35. ?
Krause [1996] , , , , BS-H Chronological DAX. 3Y
Lachtermacher and Rodrigues Gaspar [1996] , , , , BS-H MAE, MAPE, MPE, MSE Random Individual stocks. 2M
Lajbcygier and Flitman [1996] , , BS-IH, KR, Linear MAE, Chronological AOSPI. 3Y
Lajbcygier et al. [1996a]222We were not able to obtain a copy of this paper. ? ? BS-?, BW ? ? AOSPI. ?
Lajbcygier et al. [1996b], Lajbcygier [2002] , , , BS-H, BW, Linear MAPE/MAE, MSE, Random AOSPI. 2Y
Liu [1996] 333The network learns the dynamics of the underlying iteratively and then relies on Monte-Carlo to determine option prices. BS-H MAE, MAX, MSE Chronological S&P500. 5Y444The network is trained on a five-year long stock price path, but uses only one day’s option price data.
Malliaris and Salchenberger [1996] , lagged , and others None MAE, MSE Chronological S&P100. 1Y
Niranjan [1996] , BS-H MSE ? FTSE100. 11M
Qi and Maddala [1996]555This paper relies on the PhD thesis Qi [1996]. , , , , open interest BS-H MAE, MSE, Random S&P500. 2M
Hanke [1997] , , ,666Additional GARCH parameters are also added as features. , None MSE Chronological Simulation (SV)
Herrmann and Narr [1997] , , , , BS-V MAE, ME, MSE, ? Simulation (BS); DAX. 1Y
Karaali et al. [1997] None None Chronological DEM volatility. 5Y
Lajbcygier and Connor [1997a, b] , BS-IH MAE, SR Chronological AOSPI. 1Y
Lajbcygier et al. [1997]\@footnotemark , ? ? ? ? AOSPI. ?
Ahmed and Swidler [1998] , , , volume None MAE, MSE Random Individual stocks. 3Y
Anders et al. [1998] , , , , , , BS-H, BS-V MAE, MAPE, ME, MSE, ? DAX. 3Y
Avellaneda et al. [1998] None %E ? USD-DEM. Several days
Garcia and Gençay [1998, 2000] , BS-H, Linear DM, MATE, MSE Chronological Simulation (BS);
S&P500. 8Y
White [1998] ? None MAE, MSE Random Simulation (BS)
Chen and Lee [1999] , , , , , volume BS-H, CRR MAE, MAPE, MSE Chronological Individual stocks. 1Y
Geigle and Aronson [1999]777This paper relies on the PhD thesis Geigle [1999]. , , , BS-H MAE, MAPE Chronological S&P500. 6Y
Hanke [1999a] BS-H MSE Chronological DAX. 1Y
Hanke [1999b] , , , BS-Cal MSE Chronological DAX. 10M
Ormoneit [1999] BS-H, BS-IH MATE, MSE, ? DAX. 9M
Tsaih [1999] , , , , BS-IH Sensitivity analysis Chronological Simulation (BS)
Briegel and Tresp [2000] , BS-?, lagged MSE ? FTSE100. 10M
Carelli et al. [2000] , None %E ? USD-DEM. Several days
de Freitas et al. [2000b, a] , BS-H ? FTSE100. 11M
Galindo-Flores [2000] , , Decision tree, Linear, Nearest neighbour MSE ? Simulation (BS)
Ghaziri et al. [2000] , , , , , open interest BS-H MSE ? S&P500. 2M
Raberto et al. [2000] , None None ? BUND. ?
Saito and Jun [2000]\@footnotemark ? ? BS-? ? ? S&P500. ?
White [2000] , , , BS-H MAE, MSE Random Simulation (BS);
Eurodollar. 7M
Yao et al. [2000] , , BS-H Chronological NIKKEI225. 1Y
Dugas et al. [2001, 2009] None MSE Chronological S&P500. 5Y
Gençay and Qi [2001] BS-H DM, MATE, MSE Chronological S&P500. 6Y
le Roux and du Toit [2001] , , , , None MSE Chronological Simulation (BS)
Meissner and Kawano [2001] , , BS-G MAE, MAPE, ME, MSE, ? Individual stocks. 8M
Schittenkopf and Dorffner [2001] Gaussian parameters888ANNs output parameters for a Gaussian mixture density as a model for the risk-neutral density. BS-H, CS MAE, MATE, ME, MSE Chronological FTSE100. 5Y
Andreou et al. [2002] , , , , and others , , BS-H, BS-V MdAE Chronological S&P500. 3Y
Billio et al. [2002] , , , BS-? MSE Chronological FTSE100. 1Y
Ghosn and Bengio [2002] , None MSE Chronological S&P500. 6Y
Healy et al. [2002] , , , , , spread, open interest, volume None MAE, ME, Random FTSE100. 5Y
Zapart [2002, 2003b] Lagged wavelet coefficients Wavelet coefficients999An ANN is used to predict the future volatility of the underlying. The volatility is represented in terms of wavelets and the underlying modelled as a binomial tree. BS-? MAE Chronological Individual stocks. 6M/1Y
Amilon [2003] , , , , lagged , BS-H, BS-IM ME, MTE, MSE Chronological OMX. 2Y
Carverhill and Cheuk [2003] , , , , HR CRR ?TE Chronological S&P500. 11Y
Gençay and Salih [2003] BS-H DM, MSE Chronological S&P500. 6Y
Healy et al. [2003, 2004]101010 These papers also derive prediction intervals for ANN estimates of option prices. , None MSE, Random FTSE100. 6Y
Lajbcygier [2003, 2004] None MAE, MSE, Chronological AOSPI. 3Y
Montagna et al. [2003] , None None ? Simulation (BS)
Zapart [2003a]111111This paper also treats the setup of Zapart [2002]. , , , BS-? MAE Chronological Individual stocks. ?
Bennell and Sutcliffe [2004] , , , open interest, volume , BS-IM MAE, ME, MPE, MSE Chronological FTSE100. 1Y
Choi et al. [2004] , , , BS-? ? Random KOSPI200. 1Y
Dindar and Marwala [2004] , None ? Random South Africa Foreign Exchange. 3Y
Morelli et al. [2004] , None ? ? Simulation (BS)
Pires and Marwala [2004a, b] , SVM MAX, ME ? Johannesburg Stock Exchange. 3Y
Xu et al. [2004] , , , , None Random FTSE100. 5Y
Charalambous and Martzoukos [2005] , , , correlations121212Correlations between underlyings. LA-10 MAE, MAX, MSE Chronological Simulation (BS)
Hamid and Habib [2005] None MAE, MSE ? S&P500. 12Y
Kakati [2005]\@footnotemark ? ? ? ? ? Individual stocks. ?
Ko et al. [2005], Ko [2009] , , , Coefficients131313Coefficients for a linear regression that returns option prices. BS-H MATE ? TAIEX. 1Y/2Y
Lin and Yeh [2005] BS-H MAE, MSE ? TAIEX. 2Y
Pires and Marwala [2005] SVM MAX, ME, MSE ? ALSI. 3Y
Tung and Quek [2005] , , None MSE, Correlation141414Pearson correlation coefficient, a statistical measure to verify the goodness-of-fit between the predicted and desired function. Random GBP-USD. 1Y
Andreou et al. [2006]151515This paper relies on the PhD thesis Andreou [2008]. , , , , , , , , BS-Cal, BS-H, BS-V MAE, MSE Chronological S&P500. 3Y
Blynski and Faseruk [2006] , , , , BS-H, BS-IH MAE, MAPE, ME, MSE, ? S&P100. 7Y
Huang and Wu [2006], Huang [2008] , , SVM MAE, MAPE, MSE Chronological TAIEX. 9M
Jung et al. [2006] , , , C BS-IH MSE ? KOSPI200. 1Y
Kim et al. [2006] , SI MSE ? S&P500. 1M
Liang et al. [2006] 161616Various price estimations from parametric option pricing models. BS-?, CRR MAE Chronological Individual stocks. 5M
Mitra [2006] , None MAE, MSE Chronological NIFTY50. 1Y
Pande and Sahu [2006] , , , or(?) None ME, MSE, Correlation171717Correlation between the actual and computed prices. ? Individual stocks. 1Y
Teddy et al. [2006] , , None MSE, Correlation\@footnotemark Random GBP-USD. 1Y
Tzastoudis et al. [2006] BS-H MAE, Chronological S&P500. Several days
Wang [2006] , , , , BS-IH MAE, MSE, ? Individual stocks. 2M
Amornwattana et al. [2007] , , , , BS-H, BS-N MAE, MSE Chronological Individual stocks. 3M
Gençay and Gibson [2007] , , , , BS-G, BS-H, SV, SVJ MAE, MSE ? S&P500. 3Y
Gregoriou et al. [2007] , , , , , , None None Random FTSE100. 5Y
Healy et al. [2007] , , , , None Chronological FTSE100. ?
Thomaidis et al. [2007] BS-G, BS-H MAE, MSE Chronological S&P500. Several days
Zhou et al. [2007] , , , , BS-?, CRR MAE, MAPE, ME, MSE, Chronological Convertible bonds. 2Y
Andreou et al. [2008]\@footnotemark , , , , , kurtosis, skewness , , , BS-Cal, BS-H, BS-V, CS MAE, MATE, MdAE, MSE, MTE Chronological S&P500. 4Y
Chiu and Lin [2008] , , volume, and others None MSE Chronological Individual stocks. 1Y
Kakati [2008] , , , , , BS-G, BS-H, BS-IH MSE ? Individual stocks. Several days
Mostafa and Dillon [2008]181818This paper relies on the PhD thesis Mostafa [2011]. , BS-H, SV MAPE, MATE, MPE ? FTSE100. 2Y
Quek et al. [2008] lagged None None ? GBP-USD, Gold, Oil. 2Y
Saxena [2008] , , , BS-H MAE, ME, MPE, MSE, ? NIFTY50. 1Y
Teddy et al. [2008] , , BS-H MSE, Random GBP-USD. 9M
Tseng et al. [2008] , , None MAE, MAPE, MSE ? TAIEX. 2Y
Chen [2009] , , , , BS-H, SVM MAE, MSE Chronological S&P500. Several days
Gradojevic et al. [2009] BS-H DM, MSE, MSPE Chronological S&P500. 8Y
Leung et al. [2009] , , volume, open interest BS-IH, Linear, Polynomial ME Chronological Several currencies. 17Y
Liang et al. [2009] \@footnotemark CRR, SVM MAE, MAPE Chronological Individual stocks. 2Y
Martel et al. [2009] , , BS-H ME, MSE, MTE Chronological IBEX35. 2Y
Samur and Temur [2009] , None MAE, MSE, ? S&P100. Several days
Wang [2009a] , , , None MAE, MAPE, MSE ? TAIEX. 2Y
Wang [2009b] , , , , , None MAE, MAPE, MSE ? TAIEX. 2Y
Andreou et al. [2010]\@footnotemark BS-Cal, CS, SV, SVJ MAE, MATE, MdAE, MSE Chronological S&P500. 3Y
Barunikova and Barunik [2011] BS-H MAE, MAPE, MSE Random S&P500. 3Y
Gradojevic and Kukolj [2011] , , , BS-H DM, MAPE, MSE Chronological S&P500. 7Y
Liu and Zhang [2011] , ,191919More precisely, a Markov regime switching model is used to estimate the volatility. BS-H MAE, MSE Chronological Individual stocks. 2Y
Phani et al. [2011] BS-?, SVM MAE ? NIFTY50. 2Y
Tung and Quek [2011] None MAPE, MSE, Chronological HSI. 5Y
Wang [2011] , , , , SV, SVJ, SVM MAE, MAPE Chronological Several currencies. 7M
Ahn et al. [2012] Lagged , Greeks None Accuracy Chronological KOSPI200. 2Y
Chen and Sutcliffe [2012] , , HR BS-H MAE, ME, MSE Random Sterling futures. 2Y
Mitra [2012] , , , , BS-H ME, MSE Chronological NIFTY50. 3Y
Shin and Ryu [2012] , HR None MPE Chronological KOSPI200. 10Y
Wang et al. [2012] , , , , , , None MAE, MAPE, MSE Chronological TAIEX. 2Y
Chang et al. [2013] , or(?) None MAE, MAPE ? TAIEX. 2Y
Hahn [2013] , , , SV MAE, MAPE, MSE Chronological Individual stocks. 10Y
Can and Fadda [2014] , BS-H MAE Chronological S&P100. Several days
Lai [2014] , , KR, SI KS ? Simulation (BS, SV, SVJ)
Park et al. [2014] BS-H, SV MSE Chronological KOSPI200. 10Y
von Spreckelsen et al. [2014] , , None MSE, Chronological EUR-USD. 1M
Ludwig [2015] Quadratic MSE, ? S&P500. 12Y
Liu and Huang [2016] BS-H MAE, MAPE, ME, MSE ? HSI. 6Y
Montesdeoca and Niranjan [2016] , , volume None MSE Chronological FTSE100. ?;
Individual stocks. ?
Culkin and Das [2017] , None MSE, Chronological Simulation (BS)
Das and Padhy [2017] , , \@footnotemark BS-H, SVM MAE, MSE Chronological NIFTY50. 2Y
Fang and George [2017] None MSE, Chronological Simulation (BS); WTI. 1M
Palmer and Gorse [2017] , , , None MAE, MdAE, MAPE Chronological Simulation (BS)
Yang et al. [2017]202020This paper relies on the PhD thesis Zheng [2017]. BS-?, Kou, VG MAPE, MSE ? S&P500. 10Y
Ferguson and Green [2018] , , , correlations\@footnotemark None MSE Chronological Simulation (BS)
Ackerer et al. [2019] , , None MAPE, MSE Random S&P500. 1M
Buehler et al. [2019a, b] HR BS-I CVaR Chronological Simulation (BS, SV); S&P500. 5Y
Cao et al. [2019] , , underlying return HW MSE Random S&P500. 8Y
Jang and Lee [2019] ? BS-Cal, BW, KR, LSM, LV, SVJ, SVM MAE, MAPE, MPE, MSE ? S&P100. 9Y
Liu et al. [2019b] None MAE, MAPE, MSE Chronological Simulation (BS)
Liu et al. [2019c] , , BS-Cal, SVJ MAE, MATE, MPE, MSE Chronological DAX. 4Y
Karatas et al. [2019] , , , ? None MSE, Chronological Simulation (BS, SV, VG)
Palmer [2019] , , BS-I, LSM MAE, MAPE Chronological Simulation (BS)
Ruf and Wang [2019] , , , , Aggressor side212121A flag to indicate whether a transaction is induced by a buy or sell order. HR BS-I, HW, Linear MSE Chronological Simulation (BS, SV); S&P500. 8Y;
STOXX50. 3Y
Zheng et al. [2019] SSVI MAPE ? S&P500. 10Y
Table 1: This table summarises more than 150 papers that use ANNs as a nonparametric option pricing or hedging tool. These papers are compared in terms of 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. The performance measures marked bold are related to evaluations along multiple periods. We refer to Tables 25 for a dictionary of all abbreviations used here.
Option price
Option price given by the Black-Scholes formula; see Table 3 for the different meanings of X
Option price given by -step multi-dimensional lattice scheme
HR Hedging ratio
Strike price
Stock price
Interest rate
Gamma: second-order sensitivity of option price with respect to underlying price
Delta: sensitivity of option price with respect to underlying price
Vega: sensitivity of option price with respect to volatility
Rho: sensitivity of option price with respect to interest rate
Volatility from calibration (e.g., constant across strikes and maturities)
GARCH–generated volatility
Historical volatility
Implied volatility
Implied historical volatility
At-the-money implied volatility
Volatility obtained from Kalman filter
Macroeconomic variables that contribute the most to volatility, determined by principle component analysis
Volatility index such as VIX and DVAX
Time to maturity
Table 2: This table presents notations and abbreviations for features and outputs, used in Table 1.
BS-Cal Black-Scholes formula with calibrated volatility
BS-G Black-Scholes formula with GARCH-generated volatility
BS-H Black-Scholes formula with historical volatility
BS-I Black-Scholes formula with contract-specific implied volatility
BS-IH Black-Scholes formula with historical implied volatility
BS-IM Black-Scholes formula with at-the-money implied volatility
BS-K Black-Scholes formula with volatility obtained from Kalman filter
BS-N Black-Scholes formula with ANN-generated volatility
BS-V Black-Scholes formula with volatility index, such as VIX or VDAX
BW Barone-Adesi and Whaley [1987] pricing method
CRR Cox et al. [1979] model
CS Corrado and Su [1996] model
HW Hull and White [2017] model
Kou Kou [2002]’s jump diffusion model
KR Kernel regression
LA-n n-step multi-dimensional lattice scheme
Linear Linear regression on features
LSM Longstaff and Schwartz [2001] method
LV Local volatility model
Quadratic Quadratic regression on features
SI Spline interpolation
SSVI Surface stochastic volatility inspired model, see Gatheral and Jacquier [2014]
SV Stochastic volatility models, such as Heston [1993] or GARCH
SVJ Stochastic volatility with jumps model, see Bates [1996] or Carr et al. [2003]
SVM Support vector machine
VG Variance Gamma model, see Madan et al. [1998]
Table 3: This table presents abbreviations for various benchmarks, used in Table 1.
DM Diebold and Mariano test
KS Kolmogorov and Smirnov two-sample test
MAE Mean absolute error
MAPE Mean absolute percentage error
MAX Maximum error
MdAE Median absolute error
ME Mean error
MPE Mean percentage error
MSE Mean squared error
Coefficient of determination
SR Sharpe ratio of a trading ratio
%E Sample-wise percentage error
CVaR Conditional value-at-risk
MATE Mean absolute tracking error
MTE Mean tracking error
PE Prediction error
Table 4: This table presents abbreviations and definitions for performance measures, used in Table 1. Here, is the estimated option price / implied volatility / portfolio value, is the target value, is the average of target values, and denotes the number of samples. Moreover, , also called tracking error, denotes the terminal value at of a hedged option portfolio starting with zero wealth. All performance measures marked bold are related to evaluations along multiple periods.
ALSI South African All Share Index
AOSPI Australian All Ordinaries Share Price Index
BUND German treasury bond
DAX German stock index
DEM Deutsche Mark
FTSE100 UK Financial Times Stock Exchange 100 index
HSI Hong Kong Heng Seng Index
IBEX35 Spanish stock index
KOSPI200 Korea Composite Stock Price Index
NIFTY50 Indian National Stock Exchange Fifty
NIKKEI225 Japanese stock index
OMX Swedish stock index
S&P100 US Standard & Poor’s 100
S&P500 US Standard & Poor’s 500
STOXX50 Eurozone stock index
TAIEX Taiwanese stock index
WTI US Light Sweet Crude Oil Futures
Table 5: This table presents abbreviations for various stock market indices and other underlyings, used in Table 1. For the shortcuts used to describe simulation data, we refer to Table 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.

References

  • Ackerer et al. [2019] D. Ackerer, N. Tagasovska, and T. Vatter. Deep smoothing of the implied volatility surface. SSRN 3402942, 2019.
  • Ahmed and Swidler [1998] P. Ahmed and S. Swidler. Forecasting properties of neural network generated volatility estimates. In Decision Technologies for Computational Finance, pages 247–258, 1998.
  • Ahn et al. [2012] J. J. Ahn, D. H. Kim, K. J. Oh, and T. Y. Kim. Applying option Greeks to directional forecasting of implied volatility in the options market: an intelligent approach. Expert Systems with Applications, 39(10):9315–9322, 2012.
  • Albanese et al. [2019] C. Albanese, S. Crépey, R. Hoskinson, and B. Saadeddine. XVA analysis from the balance sheet. Retrieved on October 25, 2019 from https://math.maths.univ-evry.fr/crepey/, 2019.
  • Amilon [2003] H. Amilon. A neural network versus Black–Scholes: a comparison of pricing and hedging performances. Journal of Forecasting, 22:317–335, 2003.
  • Amornwattana et al. [2007] S. Amornwattana, D. Enke, and C. H. Dagli. A hybrid option pricing model using a neural network for estimating volatility. International Journal of General Systems, 36(5):558–573, 2007.
  • Anders et al. [1998] U. Anders, O. Korn, and C. Schmitt. Improving the pricing of options: a neural network approach. Journal of Forecasting, 17(5-6):369–388, 1998.
  • Andreou [2008] P. C. Andreou. Parametric and Nonparametric Functional Estimation for Options Pricing with Applications in Hedging and Trading. PhD thesis, University of Cyprus, 2008.
  • Andreou et al. [2002] P. C. Andreou, C. Charalambous, and S. H. Martzoukos. Critical assessment of option pricing methods using artificial neural networks. In International Conference on Artificial Neural Networks, pages 1131–1136, 2002.
  • Andreou et al. [2006] P. C. Andreou, C. Charalambous, and S. H. Martzoukos. Robust artificial neural networks for pricing of European options. Computational Economics, 27(2-3):329–351, 2006.
  • Andreou et al. [2008] P. C. Andreou, C. Charalambous, and S. H. Martzoukos. Pricing and trading European options by combining artificial neural networks and parametric models with implied parameters. European Journal of Operational Research, 185(3):1415–1433, 2008.
  • Andreou et al. [2010] P. C. Andreou, C. Charalambous, and S. H. Martzoukos. Generalized parameter functions for option pricing. Journal of Banking & Finance, 34(3):633–646, 2010.
  • Avellaneda et al. [1998] M. Avellaneda, A. Carelli, and F. Stella. Following the Bayes path to option pricing. Journal of Computational Intelligence in Finance, 1998.
  • Barone-Adesi and Whaley [1987] G. Barone-Adesi and R. E. Whaley. Efficient analytic approximation of American option values. The Journal of Finance, 42(2):301–320, 1987.
  • Barucci et al. [1996] E. Barucci, U. Cherubini, and L. Landi. No-arbitrage asset pricing with neural networks under stochastic volatility. In Neural Networks in Financial Engineering: Proceedings of the Third International Conference on Neural Networks in the Capital Markets, pages 3–16, 1996.
  • Barucci et al. [1997] E. Barucci, U. Cherubini, and L. Landi. Neural networks for contingent claim pricing via the Galerkin method. In Computational Approaches to Economic Problems, pages 127–141. 1997.
  • Barunikova and Barunik [2011] M. Barunikova and J. Barunik. Neural networks as semiparametric option pricing tool. Bulletin of the Czech Econometric Society, 18, 2011.
  • Bates [1996] D. S. Bates. Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options. The Review of Financial Studies, 9(1):69–107, 1996.
  • Bayer and Stemper [2018] C. Bayer and B. Stemper. Deep calibration of rough stochastic volatility models. arXiv:1810.03399, 2018.
  • Bayer et al. [2019] C. Bayer, B. Horvath, A. Muguruza, B. Stemper, and M. Tomas. On deep calibration of (rough) stochastic volatility models. arXiv:1908.08806, 2019.
  • Beck et al. [2019] C. Beck, S. Becker, P. Cheridito, A. Jentzen, and A. Neufeld. Deep splitting method for parabolic PDEs. arXiv:1907.03452, 2019.
  • Becker et al. [2019] S. Becker, P. Cheridito, and A. Jentzen. Deep optimal stopping. Journal of Machine Learning Research, 20(74):1–25, 2019.
  • Bennell and Sutcliffe [2004] J. Bennell and C. Sutcliffe. Black–Scholes versus artificial neural networks in pricing FTSE 100 options. Intelligent Systems in Accounting, Finance & Management: International Journal, 12(4):243–260, 2004.
  • Billio et al. [2002] M. Billio, M. Corazza, and M. Gobbo. Option pricing via regime switching models and multilayer perceptrons: a comparative approach. Rendiconti per gli Studi Economici Quantitativi, 2002:39–59, 2002.
  • Blynski and Faseruk [2006] L. Blynski and A. Faseruk. Comparison of the effectiveness of option price forecasting: Black–Scholes vs. simple and hybrid neural networks. Journal of Financial Management & Analysis, 19(2):46–58, 2006.
  • Boek et al. [1995] C. Boek, P. Lajbcygier, M. Palaniswami, and A. Flitman. A hybrid neural network approach to the pricing of options. In Proceedings of ICNN’95–International Conference on Neural Networks, volume 2, pages 813–817. IEEE, 1995.
  • Briegel and Tresp [2000] T. Briegel and V. Tresp. Dynamic neural regression models. Retrieved on August 29, 2019 from https://epub.ub.uni-muenchen.de/1571/, 2000.
  • Buehler et al. [2019a] H. Buehler, L. Gonon, J. Teichmann, and B. Wood. Deep hedging. Quantitative Finance, 19(8):1271–1291, 2019a.
  • Buehler et al. [2019b] H. Buehler, L. Gonon, J. Teichmann, B. Wood, B. Mohan, and J. Kochems. Deep hedging: hedging derivatives under generic market frictions using reinforcement learning. SSRN 3355706, 2019b.
  • Can and Fadda [2014] M. Can and Š. Fadda. A nonparametric approach to pricing options learning networks. Southeast Europe Journal of Soft Computing, 3(1), 2014.
  • Cao et al. [2019] J. Cao, J. Chen, and J. C. Hull. A neural network approach to understanding implied volatility movements. SSRN 3288067, 2019.
  • Carelli et al. [2000] A. Carelli, S. Silani, and F. Stella. Profiling neural networks for option pricing. International Journal of Theoretical and Applied Finance, 3(02):183–204, 2000.
  • Carr et al. [2003] P. Carr, H. Geman, D. B. Madan, and M. Yor. Stochastic volatility for Lévy processes. Mathematical Finance, 13(3):345–382, 2003.
  • Carverhill and Cheuk [2003] A. P. Carverhill and T. H. Cheuk. Alternative neural network approach for option pricing and hedging. SSRN 480562, 2003.
  • Chan-Wai-Nam et al. [2019] Q. Chan-Wai-Nam, J. Mikael, and X. Warin. Machine learning for semi linear PDEs. Journal of Scientific Computing, 79(3):1667–1712, 2019.
  • Chang et al. [2013] T.-Y. Chang, Y.-H. Wang, and H.-Y. Yeh. Forecasting of option prices using a neural network model. Journal of Accounting, Finance & Management Strategy, 8(1):123–136, 2013.
  • Charalambous and Martzoukos [2005] C. Charalambous and S. H. Martzoukos. Hybrid artificial neural networks for efficient valuation of real options and financial derivatives. Computational Management Science, 2(2):155–161, 2005.
  • Chen and Sutcliffe [2012] F. Chen and C. Sutcliffe. Pricing and hedging short sterling options using neural networks. Intelligent Systems in Accounting, Finance and Management, 19(2):128–149, 2012.
  • Chen [2009] J. Chen. Learning the Black–Scholes formula via support vector machines. In Recent Advances in Statistics Application and Related Areas, 2nd Conference of the International Institute of Applied Statistics Studies, volume 1&2, pages 756–760, 2009.
  • Chen and Lee [1999] S.-H. Chen and W.-C. Lee. Pricing call warrants with artificial neural networks: the case of the Taiwan derivative market. In IJCNN’99. International Joint Conference on Neural Networks. Proceedings (Cat. No. 99CH36339), volume 6, pages 3877–3882. IEEE, 1999.
  • Chiu and Lin [2008] D.-Y. Chiu and C.-C. Lin. Exploring internal mechanism of warrant in financial market with a hybrid approach. Expert Systems with Applications, 35(3):1237–1245, 2008.
  • Choi et al. [2004] H.-J. Choi, H.-S. Lee, G.-S. Han, and J. Lee. Efficient option pricing via a globally regularized neural network. In International Symposium on Neural Networks, pages 988–993, 2004.
  • Corrado and Su [1996] C. J. Corrado and T. Su. Skewness and kurtosis in S&P 500 index returns implied by option prices. Journal of Financial Research, 19(2):175–192, 1996.
  • Cox et al. [1979] J. C. Cox, S. A. Ross, and M. Rubinstein. Option pricing: a simplified approach. Journal of Financial Economics, 7(3):229–263, 1979.
  • Culkin and Das [2017] R. Culkin and S. R. Das. Machine learning in finance: the case of deep learning for option pricing. Journal of Investment Management, 15(4):92–100, 2017.
  • Das and Padhy [2017] S. P. Das and S. Padhy. A new hybrid parametric and machine learning model with homogeneity hint for European-style index option pricing. Neural Computing and Applications, 28(12):4061–4077, 2017.
  • de Freitas et al. [2000a] J. F. G. de Freitas, M. Niranjan, and A. H. Gee. Hierarchical Bayesian models for regularization in sequential learning. Neural Computation, 12(4):933–953, 2000a.
  • de Freitas et al. [2000b] J. F. G. de Freitas, M. Niranjan, A. H. Gee, and A. Doucet. Sequential Monte Carlo methods to train neural network models. Neural Computation, 12(4):955–993, 2000b.
  • Dimitroff et al. [2018] G. Dimitroff, D. Röder, and C. Fries. Volatility model calibration with convolutional neural networks. SSRN 3252432, 2018.
  • Dindar and Marwala [2004] Z. A. Dindar and T. Marwala. Option pricing using a committee of neural networks and optimized networks. In 2004 IEEE International Conference on Systems, Man and Cybernetics (IEEE Cat. No. 04CH37583), volume 1, pages 434–438. IEEE, 2004.
  • Dugas et al. [2001] C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia. Incorporating second-order functional knowledge for better option pricing. In Advances in Neural Information Processing Systems, pages 472–478, 2001.
  • Dugas et al. [2009] C. Dugas, Y. Bengio, F. Bélisle, C. Nadeau, and R. Garcia. Incorporating functional knowledge in neural networks. Journal of Machine Learning Research, 10(Jun):1239–1262, 2009.
  • E et al. [2017] W. E, J. Han, and A. Jentzen. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Communications in Mathematics and Statistics, 5(4):349–380, 2017.
  • Fang and George [2017] Z. Fang and K. George. Application of machine learning: an analysis of Asian options pricing using neural network. In 2017 IEEE 14th International Conference on e-Business Engineering (ICEBE), pages 142–149. IEEE, 2017.
  • Fecamp et al. [2019] S. Fecamp, J. Mikael, and X. Warin. Risk management with machine-learning-based algorithms. arXiv:1902.05287, 2019.
  • Ferguson and Green [2018] R. Ferguson and A. Green. Deeply learning derivatives. SSRN 3244821, 2018.
  • Galindo-Flores [2000] J. Galindo-Flores. A framework for comparative analysis of statistical and machine learning methods: an application to the Black–Scholes option pricing model. Computational Finance 1999, pages 635–660, 2000.
  • Garcia and Gençay [1998] R. Garcia and R. Gençay. Option pricing with neural networks and a homogeneity hint. In Decision Technologies for Computational Finance, pages 195–205, 1998.
  • Garcia and Gençay [2000] R. Garcia and R. Gençay. Pricing and hedging derivative securities with neural networks and a homogeneity hint. Journal of Econometrics, 94(1-2):93–115, 2000.
  • Gatheral and Jacquier [2014] J. Gatheral and A. Jacquier. Arbitrage-free SVI volatility surfaces. Quantitative Finance, 14(1):59–71, 2014.
  • Geigle [1999] D. S. Geigle. An Artificial Neural Network Approach to the Valuation of Options and Forecasting of Volatility. PhD thesis, Nova Southeastern University, 1999.
  • Geigle and Aronson [1999] D. S. Geigle and J. E. Aronson. An artificial neural network approach to the valuation of options and forecasting of volatility. Journal of Computational Intelligence in Finance, 7(6):19–25, 1999.
  • Gençay and Gibson [2007] R. Gençay and R. Gibson. Model risk for European-style stock index options. IEEE Transactions on Neural Networks, 18(1):193–202, 2007.
  • Gençay and Qi [2001] R. Gençay and M. Qi. Pricing and hedging derivative securities with neural networks: Bayesian regularization, early stopping, and bagging. IEEE Transactions on Neural Networks, 12(4):726–734, 2001.
  • Gençay and Salih [2003] R. Gençay and A. Salih. Degree of mispricing with the Black–Scholes model and nonparametric cures. Annals of Economics and Finance, 4:73–101, 2003.
  • Ghaziri et al. [2000] H. Ghaziri, S. Elfakhani, and J. Assi. Neural networks approach to pricing options. Neural Network World, 10(1):271–277, 2000.
  • Ghosn and Bengio [2002] J. Ghosn and Y. Bengio. Multi-task learning for option pricing. Retrieved on October 29, 2019 from https://cirano.qc.ca/files/publications/2002s-53.pdf, 2002.
  • Ghysels et al. [1998] E. Ghysels, V. Patilea, É. Renault, and O. Torrès. Nonparametric methods and option pricing. In D. Hand and S. Jacka, editors, Statistics in Finance, chapter 13, pages 261–282. John Wiley & Sons, 1998.
  • Gradojevic and Kukolj [2011] N. Gradojevic and D. Kukolj. Parametric option pricing: a divide-and-conquer approach. Physica D: Nonlinear Phenomena, 240(19):1528–1535, 2011.
  • Gradojevic et al. [2009] N. Gradojevic, R. Gençay, and D. Kukolj. Option pricing with modular neural networks. IEEE Transactions on Neural Networks, 20(4):626–637, 2009.
  • Gregoriou et al. [2007] A. Gregoriou, J. Healy, and C. Ioannidis. Hedging under the influence of transaction costs: an empirical investigation on FTSE 100 index options. Journal of Futures Markets, 27(5):471–494, 2007.
  • Hahn [2013] J. T. Hahn. Option Pricing Using Artificial Neural Networks: An Australian Perspective. PhD thesis, Bond University, 2013.
  • Halperin [2017] I. Halperin. QLBS: Q-learner in the Black-Scholes (-Merton) worlds. arXiv:1712.04609, 2017.
  • Hamid and Habib [2005] S. A. Hamid and A. Habib. Can neural networks learn the Black-Scholes model?: A simplified approach. Retrieved on September 9, 2019 from https://academicarchive.snhu.edu/bitstream/handle/10474/1662/cfs2005-01.pdf, 2005.
  • Han et al. [2018] J. Han, A. Jentzen, and W. E. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34):8505–8510, 2018.
  • Hanke [1997] M. Hanke. Neural network approximation of option pricing formulas for analytically intractable option pricing models. Journal of Computational Intelligence in Finance, 5(5):20–27, 1997.
  • Hanke [1999a] M. Hanke. Adaptive hybrid neural network option pricing. Journal of Computational Intelligence in Finance, 7(5):33–39, 1999a.
  • Hanke [1999b] M. Hanke. Neural networks versus Black-Scholes: an empirical comparison of the pricing accuracy of two fundamentally different option pricing methods. Journal of Computational Intelligence in Finance, 5:26–34, 1999b.
  • Healy et al. [2002] J. Healy, M. Dixon, B. Read, and F. Cai. A data-centric approach to understanding the pricing of financial options. The European Physical Journal B, 27(2):219–227, 2002.
  • Healy et al. [2003] J. V. Healy, M. Dixon, B. J. Read, and F. F. Cai. Confidence in data mining model predictions: a financial engineering application. In IECON’03. 29th Annual Conference of the IEEE Industrial Electronics Society (IEEE Cat. No. 03CH37468), volume 2, pages 1926–1931. IEEE, 2003.
  • Healy et al. [2004] J. V. Healy, M. Dixon, B. J. Read, and F. F. Cai. Confidence limits for data mining models of options prices. Physica A: Statistical Mechanics and its Applications, 344(1-2):162–167, 2004.
  • Healy et al. [2007] J. V. Healy, M. Dixon, B. J. Read, and F. F. Cai. Non-parametric extraction of implied asset price distributions. Physica A: Statistical Mechanics and its Applications, 382(1):121–128, 2007.
  • Henry-Labordère [2017] P. Henry-Labordère. Deep primal-dual algorithm for BSDEs: applications of machine learning to CVA and IM. SSRN 3071506, 2017.
  • Henry-Labordère [2019] P. Henry-Labordère. Generative models for financial data. SSRN 3408007, 2019.
  • Hernandez [2017] A. Hernandez. Model calibration with neural networks. Risk Magazine, pages 1–5, June 2017.
  • Herrmann and Narr [1997] R. Herrmann and A. Narr. Neural networks and the evaluation of derivatives: some insights into the implied pricing mechanism of german stock index options. Retrieved on August 29, 2019 from http://finance.fbv.kit.edu/download/dp202.pdf, 1997.
  • Heston [1993] S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2):327–343, 1993.
  • Horvath et al. [2019] B. Horvath, A. Muguruza, and M. Tomas. Deep learning volatility. arXiv:1901.09647, 2019.
  • Huang [2008] S.-C. Huang. Online option price forecasting by using unscented Kalman filters and support vector machines. Expert Systems with Applications, 34(4):2819–2825, 2008.
  • Huang and Wu [2006] S.-C. Huang and T.-K. Wu. A hybrid unscented Kalman filter and support vector machine model in option price forecasting. In International Conference on Natural Computation, pages 303–312, 2006.
  • Huh [2019] J. Huh. Pricing options with exponential Lévy neural network. Expert Systems with Applications, 127:128–140, 2019.
  • Hull and White [2017] J. Hull and A. White. Optimal delta hedging for options. Journal of Banking & Finance, 82:180–190, 2017.
  • Huré et al. [2019] C. Huré, H. Pham, and X. Warin. Some machine learning schemes for high-dimensional nonlinear PDEs. arXiv:1902.01599, 2019.
  • Hutchinson et al. [1994] J. M. Hutchinson, A. W. Lo, and T. Poggio. A nonparametric approach to pricing and hedging derivative securities via learning networks. The Journal of Finance, 49(3):851–889, 1994.
  • Itkin [2019] A. Itkin. Deep learning calibration of option pricing models: some pitfalls and solutions. arXiv:1906.03507, 2019.
  • Jacquier and Oumgari [2019] A. Jacquier and M. Oumgari. Deep PPDEs for rough local stochastic volatility. arXiv:1906.02551, 2019.
  • Jang and Lee [2019] H. Jang and J. Lee. Generative Bayesian neural network model for risk-neutral pricing of American index options. Quantitative Finance, 19(4):587–603, 2019.
  • Jung et al. [2006] K.-H. Jung, H.-C. Kim, and J. Lee. A novel learning network for option pricing with confidence interval information. In International Symposium on Neural Networks, pages 491–497, 2006.
  • Kakati [2005] M. Kakati. Pricing and hedging performances of artificial neural net in Indian stock option market. The ICFAI Journal of Applied Finance, 11(1):62–73, 2005.
  • Kakati [2008] M. Kakati. Option pricing using Adaptive Neuro-Fuzzy System (ANFIS). ICFAI Journal of Derivatives Markets, 5(2), 2008.
  • Karaali et al. [1997] O. Karaali, W. Edelberg, and J. Higgins. Modelling volatility derivatives using neural networks. In Proceedings of the IEEE/IAFE 1997 Computational Intelligence for Financial Engineering, pages 280–286. IEEE, 1997.
  • Karatas et al. [2019] T. Karatas, A. Oskoui, and A. Hirsa. Supervised deep neural networks (DNNs) for pricing/calibration of vanilla/exotic options under various different processes. arXiv:1902.05810, 2019.
  • Kelly [1994] D. L. Kelly. Valuing and hedging American put options using neural networks. Retrieved on August 29, 2019 from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.721.8497&rep=rep1&type=pdf, 1994.
  • Kim et al. [2006] B.-H. Kim, D. Lee, and J. Lee. Local volatility function approximation using reconstructed radial basis function networks. In International Symposium on Neural Networks, pages 524–530, 2006.
  • Ko et al. [2005] P. Ko, P. Lin, W. Chien, and Y. Cheng. Hedging derivative securities based on the neural network coefficient model. In Proceedings of the Eighth Joint Conference on Information Sciences, pages 1163–1166, 2005.
  • Ko [2009] P.-C. Ko. Option valuation based on the neural regression model. Expert Systems with Applications, 36(1):464–471, 2009.
  • Kohler et al. [2010] M. Kohler, A. Krzyżak, and N. Todorovic. Pricing of high-dimensional American options by neural networks. Mathematical Finance, 20(3):383–410, 2010.
  • Kolm and Ritter [2019] P. N. Kolm and G. Ritter. Dynamic replication and hedging: A reinforcement learning approach. The Journal of Financial Data Science, 1(1):159–171, 2019.
  • Kondratyev and Schwarz [2019] A. Kondratyev and C. Schwarz. The market generator. SSRN 3384948, 2019.
  • Kou [2002] S. G. Kou. A jump-diffusion model for option pricing. Management Science, 48(8):1086–1101, 2002.
  • Krause [1996] J. Krause. Option pricing with neural networks. In Proceedings of the Fourth European Congress on Intelligent Techniques and Soft Computing, volume 3, pages 2206–2210, 1996.
  • Lachtermacher and Rodrigues Gaspar [1996] G. Lachtermacher and L. Rodrigues Gaspar. Neural networks in derivative securities pricing forecasting in Brazilian capital markets. In Neural Networks in Financial Engineering: Proceedings of the Third International Conference on Neural Networks in the Capital Markets, pages 92–97, 1996.
  • Lai [2014] W.-N. Lai. Comparison of methods to estimate option implied risk-neutral densities. Quantitative Finance, 14(10):1839–1855, 2014.
  • Lajbcygier [2002] P. R. Lajbcygier. Comparing conventional and artificial neural network models for the pricing of options. In Neural Networks in Business: Techniques and Applications, pages 220–235. IGI Global, 2002.
  • Lajbcygier [2003] P. R. Lajbcygier. Improving option pricing with the product constrained hybrid neural network. In Artificial Neural Networks and Neural Information Processing, pages 615–621, 2003.
  • Lajbcygier [2004] P. R. Lajbcygier. Improving option pricing with the product constrained hybrid neural network. IEEE Transactions on Neural Networks, 15(2):465–476, 2004.
  • Lajbcygier and Connor [1997a] P. R. Lajbcygier and J. T. Connor. Improved option pricing using artificial neural networks and bootstrap methods. International Journal of Neural Systems, 8(04):457–471, 1997a.
  • Lajbcygier and Connor [1997b] P. R. Lajbcygier and J. T. Connor. Improved option pricing using bootstrap methods. In Proceedings of International Conference on Neural Networks, volume 4, pages 2193–2197. IEEE, 1997b.
  • Lajbcygier and Flitman [1996] P. R. Lajbcygier and A. Flitman. A comparison of non-parametric regression techniques for the pricing of options using an optimal implied volatility. In Decision Technologies for Financial Engineering: Proceedings of the Fourth International Conference on Neural Networks in Capital Markets, pages 201–213, 1996.
  • Lajbcygier et al. [1996a] P. R. Lajbcygier, C. Boek, A. Flitman, and M. Palaniswami. Comparing conventional and artificial neural network models for the pricing of options on futures. NeuroVe$t Journal, 4(5):16–24, 1996a.
  • Lajbcygier et al. [1996b] P. R. Lajbcygier, C. Boek, M. Palaniswami, and A. Flitman. Neural network pricing of all ordinaries SPI options on futures. In Neural Networks in Financial Engineering: Proceedings of the Third International Conference on Neural Networks in the Capital Markets, 1996b.
  • Lajbcygier et al. [1997] P. R. Lajbcygier, A. Flitman, A. Swan, and R. J. Hyndman. The pricing and trading of options using a hybrid neural network model with historical volatility. NeuroVe$t Journal, pages 27–41, 1997.
  • le Roux and du Toit [2001] L. J. le Roux and G. S. du Toit. Emulating the Black & Scholes model with a neural network. Southern African Business Review, 5(1):54–57, 2001.
  • Leung et al. [2009] M. T. Leung, A.-S. Chen, and R. Mancha. Making trading decisions for financial-engineered derivatives: a novel ensemble of neural networks using information content. Intelligent Systems in Accounting, Finance & Management, 16(4):257–277, 2009.
  • Liang et al. [2006] X. Liang, H. Zhang, and J. Yang. Pricing options in Hong Kong market based on neural networks. In International Conference on Neural Information Processing, pages 410–419, 2006.
  • Liang et al. [2009] X. Liang, H. Zhang, J. Xiao, and Y. Chen. Improving option price forecasts with neural networks and support vector regressions. Neurocomputing, 72(13-15):3055–3065, 2009.
  • Lin and Yeh [2005] C.-T. Lin and H.-Y. Yeh. The valuation of Taiwan stock index option price—comparison of performances between Black-Scholes and neural network model. Journal of Statistics and Management Systems, 8(2):355–367, 2005.
  • Liu and Huang [2016] D. Liu and S. Huang. The performance of hybrid artificial neural network models for option pricing during financial crises. Journal of Data Science, 14(1):1–18, 2016.
  • Liu and Zhang [2011] D. Liu and L. Zhang. Pricing Chinese warrants using artificial neural networks coupled with Markov regime switching model. International Journal of Financial Markets and Derivatives, 2(4):314–330, 2011.
  • Liu [1996] M. Liu. Option pricing with neural networks. In Progress in Neural Information Processing, volume 2, pages 760–765, 1996.
  • Liu et al. [2019a] S. Liu, A. Borovykh, L. A. Grzelak, and C. W. Oosterlee. A neural network-based framework for financial model calibration. Journal of Mathematics in Industry, Forthcoming, 2019a.
  • Liu et al. [2019b] S. Liu, C. W. Oosterlee, and S. M. Bohte. Pricing options and computing implied volatilities using neural networks. Risks, 7(1):1–22, 2019b.
  • Liu et al. [2019c] X. Liu, Y. Cao, C. Ma, and L. Shen. Wavelet-based option pricing: an empirical study. European Journal of Operational Research, 272(3):1132–1142, 2019c.
  • Longstaff and Schwartz [2001] F. A. Longstaff and E. S. Schwartz. Valuing American options by simulation: a simple least-squares approach. The Review of Financial Studies, 14(1):113–147, 2001.
  • Lu and Ohta [2003a] J. Lu and H. Ohta. A data and digital-contracts driven method for pricing complex derivatives. Quantitative Finance, 3(3):212–219, 2003a.
  • Lu and Ohta [2003b] J. Lu and H. Ohta. Digital contracts-driven method for pricing complex derivatives. Journal of the Operational Research Society, 54(9):1002–1010, 2003b.
  • Ludwig [2015] M. Ludwig. Robust estimation of shape-constrained state price density surfaces. The Journal of Derivatives, 22(3):56–72, 2015.
  • Madan et al. [1998] D. B. Madan, P. P. Carr, and E. C. Chang. The variance Gamma process and option pricing. Review of Finance, 2(1):79–105, 1998.
  • Malliaris and Salchenberger [1993a] M. Malliaris and L. Salchenberger. Beating the best: a neural network challenges the Black-Scholes formula. In Proceedings of 9th IEEE Conference on Artificial Intelligence for Applications, pages 445–449. IEEE, 1993a.
  • Malliaris and Salchenberger [1993b] M. Malliaris and L. Salchenberger. A neural network model for estimating option prices. Journal of Applied Intelligence, 3(3):193–206, 1993b.
  • Malliaris and Salchenberger [1996] M. Malliaris and L. Salchenberger. Using neural networks to forecast the S&P100 implied volatility. Neurocomputing, 10(2):183–195, 1996.
  • Martel et al. [2009] C. G. Martel, M. D. G. Artiles, and F. F. Rodriguez. A financial option pricing model based on learning algorithms. In Proceedings of the World Multiconference on Applied Economics, Business and Development, pages 153–157, 2009.
  • McGhee [2018] W. A. McGhee. An artificial neural network representation of the SABR stochastic volatility model. SSRN 3288882, 2018.
  • Meissner and Kawano [2001] G. Meissner and N. Kawano. Capturing the volatility smile of options on high-tech stocks—a combined GARCH-neural network approach. Journal of Economics and Finance, 25(3):276–292, 2001.
  • Miranda and Burgess [1995] F. G. Miranda and N. Burgess. Intraday volatility forecasting for option pricing using a neural network approach. In Proceedings of 1995 Conference on Computational Intelligence for Financial Engineering, page 31. IEEE, 1995.
  • Mitra [2006] S. K. Mitra. Improving accuracy of option price estimation using artificial neural networks. SSRN 876881, 2006.
  • Mitra [2012] S. K. Mitra. An option pricing model that combines neural network approach and Black Scholes formula. Global Journal of Computer Science and Technology, 12(4), 2012.
  • Montagna et al. [2003] G. Montagna, M. Morelli, O. Nicrosini, P. Amato, and M. Farina. Pricing derivatives by path integral and neural networks. Physica A: Statistical Mechanics and its Applications, 324(1-2):189–195, 2003.
  • Montesdeoca and Niranjan [2016] L. Montesdeoca and M. Niranjan. Extending the feature set of a data-driven artificial neural network model of pricing financial options. In 2016 IEEE Symposium Series on Computational Intelligence (SSCI), pages 1–6. IEEE, 2016.
  • Morelli et al. [2004] M. J. Morelli, G. Montagna, O. Nicrosini, M. Treccani, M. Farina, and P. Amato. Pricing financial derivatives with neural networks. Physica A: Statistical Mechanics and its Applications, 338(1-2):160–165, 2004.
  • Mostafa [2011] F. Mostafa. Applications of Neural Networks in Market Risk. PhD thesis, Curtin University, 2011.
  • Mostafa and Dillon [2008] F. Mostafa and T. Dillon. A neural network approach to option pricing. WIT Transactions on Information and Communication Technologies, 41:71–85, 2008.
  • Niranjan [1996] M. Niranjan. Sequential tracking in pricing financial options using model based and neural network approaches. In Advances in Neural Information Processing Systems, pages 960–966, 1996.
  • Ormoneit [1999] D. Ormoneit. A regularization approach to continuous learning with an application to financial derivatives pricing. Neural Networks, 12(10):1405–1412, 1999.
  • Palmer [2019] S. Palmer. Evolutionary Algorithms and Computational Methods for Derivatives Pricing. PhD thesis, University College London, 2019.
  • Palmer and Gorse [2017] S. Palmer and D. Gorse. Pseudo-analytical solutions for stochastic options pricing using Monte Carlo simulation and breeding PSO-trained neural networks. In European Symposium on Artificial Neural Networks, Computational Intelligence and Machine Learning, pages 365–370, 2017.
  • Pande and Sahu [2006] A. Pande and R. Sahu. A new approach to volatility estimation and option price prediction for dividend paying stocks. In WEHIA 2006–1st International Conference on Economic Sciences with Heterogeneous Interacting Agents; 15–17 June 2006, University of Bologna, Italy, 2006.
  • Park et al. [2014] H. Park, N. Kim, and J. Lee. Parametric models and non-parametric machine learning models for predicting option prices: empirical comparison study over KOSPI 200 index options. Expert Systems with Applications, 41(11):5227–5237, 2014.
  • Phani et al. [2011] B. Phani, B. Chandra, and V. Raghav. Quest for efficient option pricing prediction model using machine learning techniques. In The 2011 International Joint Conference on Neural Networks, pages 654–657. IEEE, 2011.
  • Pires and Marwala [2004a] M. M. Pires and T. Marwala. American option pricing using multi-layer perceptron and support vector machine. In 2004 IEEE International Conference on Systems, Man and Cybernetics (IEEE Cat. No. 04CH37583), volume 2, pages 1279–1285. IEEE, 2004a.
  • Pires and Marwala [2004b] M. M. Pires and T. Marwala. Option pricing using Bayesian neural networks. In Fifteenth Annual Symposium of the Pattern Recognition Association of South Africa, pages 161–166, 2004b.
  • Pires and Marwala [2005] M. M. Pires and T. Marwala. American option pricing using Bayesian multi-layer perceptrons and Bayesian support vector machines. In IEEE 3rd International Conference on Computational Cybernetics, pages 219–224. IEEE, 2005.
  • Qi [1996] M. Qi. Financial Applications of Generalized Nonlinear Nonparametric Econometric Methods (Artificial Neural Networks). PhD thesis, Ohio State University, 1996.
  • Qi and Maddala [1996] M. Qi and G. Maddala. Option pricing using artificial neural networks: the case of S&P 500 index call options. In Neural Networks in Financial Engineering: Proceedings of the Third International Conference on Neural Networks in the Capital Markets, pages 78–91, 1996.
  • Quek et al. [2008] C. Quek, M. Pasquier, and N. Kumar. A novel recurrent neural network-based prediction system for option trading and hedging. Applied Intelligence, 29(2):138–151, 2008.
  • Raberto et al. [2000] M. Raberto, G. Cuniberti, M. Riani, E. Scales, F. Mainardi, and G. Servizi. Learning short-option valuation in the presence of rare events. International Journal of Theoretical and Applied Finance, 3(03):563–564, 2000.
  • Ruf and Wang [2019] J. Ruf and W. Wang. Neural network based discrete hedging. Working paper, 2019.
  • Saito and Jun [2000] S. Saito and L. Jun. Neural network option pricing in connection with the Black and Scholes model. In Proceedings of the Fifth Conference of the Asian Pacific Operations Research Society, 2000.
  • Samur and Temur [2009] Z. I. Samur and G. T. Temur. The use of artificial neural network in option pricing: the case of S&P 100 index options. International Journal of Social, Behavioral, Educational, Economic, Business and Industrial Engineering, 3(6):644–649, 2009.
  • Saxena [2008] A. Saxena. Valuation of S&P CNX Nifty options: comparison of Black-Scholes and hybrid ANN model. In Proceedings SAS Global Forum, 2008.
  • Schittenkopf and Dorffner [2001] C. Schittenkopf and G. Dorffner. Risk-neutral density extraction from option prices: improved pricing with mixture density networks. IEEE Transactions on Neural Networks, 12(4):716–725, 2001.
  • Shin and Ryu [2012] H. J. Shin and J. Ryu. A dynamic hedging strategy for option transaction using artificial neural networks. International Journal of Software Engineering and its Applications, 6(4):111–116, 2012.
  • Sirignano and Spiliopoulos [2018] J. Sirignano and K. Spiliopoulos. DGM: a deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 375:1339–1364, 2018.
  • Stone [2019] H. Stone. Calibrating rough volatility models: a convolutional neural network approach. Quantitative Finance, pages 1–14, 2019.
  • Taudes et al. [1998] A. Taudes, M. Natter, and M. Trcka. Real option valuation with neural networks. Intelligent Systems in Accounting, Finance & Management, 7(1):43–52, 1998.
  • Teddy et al. [2006] S. D. Teddy, E.-K. Lai, and C. Quek. A brain-inspired cerebellar associative memory approach to option pricing and arbitrage trading. In International Conference on Neural Information Processing, pages 370–379, 2006.
  • Teddy et al. [2008] S. D. Teddy, E.-K. Lai, and C. Quek. A cerebellar associative memory approach to option pricing and arbitrage trading. Neurocomputing, 71(16-18):3303–3315, 2008.
  • Thomaidis et al. [2007] N. S. Thomaidis, V. S. Tzastoudis, and G. Dounias. A comparison of neural network model selection strategies for the pricing of S&P500 stock index options. International Journal on Artificial Intelligence Tools, 16(06):1093–1113, 2007.
  • Tsaih [1999] R. Tsaih. Sensitivity analysis, neural networks, and the finance. In IJCNN’99. International Joint Conference on Neural Networks. Proceedings (Cat. No. 99CH36339), volume 6, pages 3830–3835. IEEE, 1999.
  • Tseng et al. [2008] C.-H. Tseng, S.-T. Cheng, Y.-H. Wang, and J.-T. Peng. Artificial neural network model of the hybrid EGARCH volatility of the Taiwan stock index option prices. Physica A: Statistical Mechanics and its Applications, 387(13):3192–3200, 2008.
  • Tung and Quek [2005] W. L. Tung and C. Quek. GenSo-OPATS: a brain-inspired dynamically evolving option pricing model and arbitrage trading system. In 2005 IEEE Congress on Evolutionary Computation, volume 3, pages 2429–2436. IEEE, 2005.
  • Tung and Quek [2011] W. L. Tung and C. Quek. Financial volatility trading using a self-organising neural-fuzzy semantic network and option straddle-based approach. Expert Systems with Applications, 38(5):4668–4688, 2011.
  • Tzastoudis et al. [2006] V. S. Tzastoudis, N. S. Thomaidis, and G. D. Dounias. Improving neural network based option price forecasting. In Hellenic Conference on Artificial Intelligence, pages 378–388, 2006.
  • Vidales et al. [2019] M. S. Vidales, D. Siska, and L. Szpruch. Unbiased deep solvers for parametric PDEs. arXiv:1810.05094, 2019.
  • von Spreckelsen et al. [2014] C. von Spreckelsen, H.-J. von Mettenheim, and M. H. Breitner. Steps towards a high-frequency financial decision support system to pricing options on currency futures with neural networks. International Journal of Applied Decision Sciences, 7(3):223–238, 2014.
  • Wang et al. [2012] C.-P. Wang, S.-H. Lin, H.-H. Huang, and P.-C. Wu. Using neural network for forecasting TXO price under different volatility models. Expert Systems with Applications, 39(5):5025–5032, 2012.
  • Wang [2006] H.-W. Wang. Dual derivatives spreading and hedging with evolutionary data mining. Journal of American Academy of Business, 9:45–52, 2006.
  • Wang [2011] P. Wang. Pricing currency options with support vector regression and stochastic volatility model with jumps. Expert Systems with Applications, 38(1):1–7, 2011.
  • Wang [2009a] Y.-H. Wang. Nonlinear neural network forecasting model for stock index option price: hybrid GJR–GARCH approach. Expert Systems with Applications, 36(1):564–570, 2009a.
  • Wang [2009b] Y.-H. Wang. Using neural network to forecast stock index option price: a new hybrid GARCH approach. Quality & Quantity, 43(5):833–843, 2009b.
  • White [2000] A. White. Pricing Options with Futures-Style Margining: a Genetic Adaptive Neural Network Approach. Garland Publishing, 2000.
  • White [1998] A. J. White. A genetic adaptive neural network approach to pricing options: a simulation analysis. Journal of Computational Intelligence in Finance, 6(2):13–23, 1998.
  • Wiese et al. [2019a] M. Wiese, L. Bai, B. Wood, and H. Buehler. Deep hedging: learning to simulate equity option markets. arXiv:1911.01700, 2019a.
  • Wiese et al. [2019b] M. Wiese, R. Knobloch, R. Korn, and P. Kretschmer. Quant GANs: deep generation of financial time series. arXiv:1907.06673, 2019b.
  • Xu et al. [2004] L. Xu, M. Dixon, B. A. Eales, F. F. Cai, B. J. Read, and J. V. Healy. Barrier option pricing: modelling with neural nets. Physica A: Statistical Mechanics and its Applications, 344(1-2):289–293, 2004.
  • Yang et al. [2017] Y. Yang, Y. Zheng, and T. M. Hospedales. Gated neural networks for option pricing: rationality by design. In Association for the Advancement of Artificial Intelligence, pages 52–58, 2017.
  • Yao et al. [2000] J. Yao, Y. Li, and C. L. Tan. Option price forecasting using neural networks. Omega, 28(4):455–466, 2000.
  • Ye and Zhang [2019] T. Ye and L. Zhang. Derivatives pricing via machine learning. SSRN 3352688, 2019.
  • Zapart [2002] C. Zapart. Stochastic volatility options pricing with wavelets and artificial neural networks. Quantitative Finance, 2(6):487–495, 2002.
  • Zapart [2003a] C. Zapart. Beyond Black–Scholes: a neural networks-based approach to options pricing. International Journal of Theoretical and Applied Finance, 6(05):469–489, 2003a.
  • Zapart [2003b] C. Zapart. Statistical arbitrage trading with wavelets and artificial neural networks. In 2003 IEEE International Conference on Computational Intelligence for Financial Engineering, pages 429–435. IEEE, 2003b.
  • Zheng [2017] Y. Zheng. Machine Learning and Option Implied Information. PhD thesis, Imperial College London, 2017.
  • Zheng et al. [2019] Y. Zheng, Y. Yang, and B. Chen. Gated deep neural networks for implied volatility surfaces. arXiv:1904.12834, 2019.
  • Zhou et al. [2007] W. Zhou, M. Yang, and L. Han. A nonparametric approach to pricing convertible bond via neural network. In Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing (SNPD 2007), volume 2, pages 564–569. IEEE, 2007.
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