Neural networks for option pricing and hedging:
a literature review^{0}^{0}footnotetext: We thank Marc Chataigner, Stéphane Crépey, Antoine Jacquier, and Martin Larsson for comments on an early version of this note.
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]^{1}^{1}1Hahn [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 socalled 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 ANNbased 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 socalled 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.

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  BSIM  MAE, MAPE, MSE  Chronological  S&P100. 6M  
Hutchinson et al. [1994]  BSH, Linear  MATE, PE,  Chronological  Simulation (BS);  
S&P500. 5Y  
Kelly [1994]  CRR  MAE, MTE, MSE,  ?  Individual stocks. 6M  
Boek et al. [1995]  , , ,  BSH  MAPE,  ?  AOSPI. 2Y  
Miranda and Burgess [1995]  ?  Linear  ?  ?  IBEX35. ?  
Krause [1996]  , , , ,  BSH  Chronological  DAX. 3Y  
Lachtermacher and Rodrigues Gaspar [1996]  , , , ,  BSH  MAE, MAPE, MPE, MSE  Random  Individual stocks. 2M  
Lajbcygier and Flitman [1996]  , ,  BSIH, KR, Linear  MAE,  Chronological  AOSPI. 3Y  
Lajbcygier et al. [1996a]^{2}^{2}2We were not able to obtain a copy of this paper.  ?  ?  BS?, BW  ?  ?  AOSPI. ? 
Lajbcygier et al. [1996b], Lajbcygier [2002]  , , ,  BSH, BW, Linear  MAPE/MAE, MSE,  Random  AOSPI. 2Y  
Liu [1996]  ^{3}^{3}3The network learns the dynamics of the underlying iteratively and then relies on MonteCarlo to determine option prices.  BSH  MAE, MAX, MSE  Chronological  S&P500. 5Y^{4}^{4}4The network is trained on a fiveyear 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]  ,  BSH  MSE  ?  FTSE100. 11M  
Qi and Maddala [1996]^{5}^{5}5This paper relies on the PhD thesis Qi [1996].  , , , , open interest  BSH  MAE, MSE,  Random  S&P500. 2M  
Hanke [1997]  , , ,^{6}^{6}6Additional GARCH parameters are also added as features.  ,  None  MSE  Chronological  Simulation (SV) 
Herrmann and Narr [1997]  , , , ,  BSV  MAE, ME, MSE,  ?  Simulation (BS); DAX. 1Y  
Karaali et al. [1997]  None  None  Chronological  DEM volatility. 5Y  
Lajbcygier and Connor [1997a, b]  ,  BSIH  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]  , , , , ,  ,  BSH, BSV  MAE, MAPE, ME, MSE,  ?  DAX. 3Y 
Avellaneda et al. [1998]  None  %E  ?  USDDEM. Several days  
Garcia and Gençay [1998, 2000]  ,  BSH, Linear  DM, MATE, MSE  Chronological  Simulation (BS);  
S&P500. 8Y  
White [1998]  ?  None  MAE, MSE  Random  Simulation (BS)  
Chen and Lee [1999]  , , , , , volume  BSH, CRR  MAE, MAPE, MSE  Chronological  Individual stocks. 1Y  
Geigle and Aronson [1999]^{7}^{7}7This paper relies on the PhD thesis Geigle [1999].  , , ,  BSH  MAE, MAPE  Chronological  S&P500. 6Y  
Hanke [1999a]  BSH  MSE  Chronological  DAX. 1Y  
Hanke [1999b]  , ,  ,  BSCal  MSE  Chronological  DAX. 10M 
Ormoneit [1999]  BSH, BSIH  MATE, MSE,  ?  DAX. 9M  
Tsaih [1999]  , , , ,  BSIH  Sensitivity analysis  Chronological  Simulation (BS)  
Briegel and Tresp [2000]  ,  BS?, lagged  MSE  ?  FTSE100. 10M  
Carelli et al. [2000]  ,  None  %E  ?  USDDEM. Several days  
de Freitas et al. [2000b, a]  ,  BSH  ?  FTSE100. 11M  
GalindoFlores [2000]  , ,  Decision tree, Linear, Nearest neighbour  MSE  ?  Simulation (BS)  
Ghaziri et al. [2000]  , , , , , open interest  BSH  MSE  ?  S&P500. 2M  
Raberto et al. [2000]  ,  None  None  ?  BUND. ?  
Saito and Jun [2000]\@footnotemark  ?  ?  BS?  ?  ?  S&P500. ? 
White [2000]  , , ,  BSH  MAE, MSE  Random  Simulation (BS);  
Eurodollar. 7M  
Yao et al. [2000]  , ,  BSH  Chronological  NIKKEI225. 1Y  
Dugas et al. [2001, 2009]  None  MSE  Chronological  S&P500. 5Y  
Gençay and Qi [2001]  BSH  DM, MATE, MSE  Chronological  S&P500. 6Y  
le Roux and du Toit [2001]  , , , ,  None  MSE  Chronological  Simulation (BS)  
Meissner and Kawano [2001]  , ,  BSG  MAE, MAPE, ME, MSE,  ?  Individual stocks. 8M  
Schittenkopf and Dorffner [2001]  Gaussian parameters^{8}^{8}8ANNs output parameters for a Gaussian mixture density as a model for the riskneutral density.  BSH, CS  MAE, MATE, ME, MSE  Chronological  FTSE100. 5Y  
Andreou et al. [2002]  , , , , and others  , ,  BSH, BSV  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 coefficients^{9}^{9}9An 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  ,  BSH, BSIM  ME, MTE, MSE  Chronological  OMX. 2Y 
Carverhill and Cheuk [2003]  , , ,  , HR  CRR  ?TE  Chronological  S&P500. 11Y 
Gençay and Salih [2003]  BSH  DM, MSE  Chronological  S&P500. 6Y  
Healy et al. [2003, 2004]^{10}^{10}10 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]^{11}^{11}11This paper also treats the setup of Zapart [2002].  , , ,  BS?  MAE  Chronological  Individual stocks. ?  
Bennell and Sutcliffe [2004]  , , , open interest, volume  ,  BSIM  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]  , , , correlations^{12}^{12}12Correlations between underlyings.  LA10  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]  , , ,  Coefficients^{13}^{13}13Coefficients for a linear regression that returns option prices.  BSH  MATE  ?  TAIEX. 1Y/2Y 
Lin and Yeh [2005]  BSH  MAE, MSE  ?  TAIEX. 2Y  
Pires and Marwala [2005]  SVM  MAX, ME, MSE  ?  ALSI. 3Y  
Tung and Quek [2005]  , ,  None  MSE, Correlation^{14}^{14}14Pearson correlation coefficient, a statistical measure to verify the goodnessoffit between the predicted and desired function.  Random  GBPUSD. 1Y  
Andreou et al. [2006]^{15}^{15}15This paper relies on the PhD thesis Andreou [2008].  , , , , ,  , , ,  BSCal, BSH, BSV  MAE, MSE  Chronological  S&P500. 3Y 
Blynski and Faseruk [2006]  , ,  , ,  BSH, BSIH  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  BSIH  MSE  ?  KOSPI200. 1Y 
Kim et al. [2006]  ,  SI  MSE  ?  S&P500. 1M  
Liang et al. [2006]  ^{16}^{16}16Various 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, Correlation^{17}^{17}17Correlation between the actual and computed prices.  ?  Individual stocks. 1Y 
Teddy et al. [2006]  , ,  None  MSE, Correlation\@footnotemark  Random  GBPUSD. 1Y  
Tzastoudis et al. [2006]  BSH  MAE,  Chronological  S&P500. Several days  
Wang [2006]  , , , ,  BSIH  MAE, MSE,  ?  Individual stocks. 2M  
Amornwattana et al. [2007]  , , ,  ,  BSH, BSN  MAE, MSE  Chronological  Individual stocks. 3M 
Gençay and Gibson [2007]  , , , ,  BSG, BSH, 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]  BSG, BSH  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  , , ,  BSCal, BSH, BSV, 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]  , , , , ,  BSG, BSH, BSIH  MSE  ?  Individual stocks. Several days  
Mostafa and Dillon [2008]^{18}^{18}18This paper relies on the PhD thesis Mostafa [2011].  ,  BSH, SV  MAPE, MATE, MPE  ?  FTSE100. 2Y  
Quek et al. [2008]  lagged  None  None  ?  GBPUSD, Gold, Oil. 2Y  
Saxena [2008]  , , ,  BSH  MAE, ME, MPE, MSE,  ?  NIFTY50. 1Y  
Teddy et al. [2008]  , ,  BSH  MSE,  Random  GBPUSD. 9M  
Tseng et al. [2008]  , ,  None  MAE, MAPE, MSE  ?  TAIEX. 2Y  
Chen [2009]  , , , ,  BSH, SVM  MAE, MSE  Chronological  S&P500. Several days  
Gradojevic et al. [2009]  BSH  DM, MSE, MSPE  Chronological  S&P500. 8Y  
Leung et al. [2009]  , , volume, open interest  BSIH, Linear, Polynomial  ME  Chronological  Several currencies. 17Y  
Liang et al. [2009]  \@footnotemark  CRR, SVM  MAE, MAPE  Chronological  Individual stocks. 2Y  
Martel et al. [2009]  ,  ,  BSH  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  BSCal, CS, SV, SVJ  MAE, MATE, MdAE, MSE  Chronological  S&P500. 3Y  
Barunikova and Barunik [2011]  BSH  MAE, MAPE, MSE  Random  S&P500. 3Y  
Gradojevic and Kukolj [2011]  , , ,  BSH  DM, MAPE, MSE  Chronological  S&P500. 7Y  
Liu and Zhang [2011]  , ,^{19}^{19}19More precisely, a Markov regime switching model is used to estimate the volatility.  BSH  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  BSH  MAE, ME, MSE  Random  Sterling futures. 2Y  
Mitra [2012]  , , , ,  BSH  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]  ,  BSH  MAE  Chronological  S&P100. Several days  
Lai [2014]  , ,  KR, SI  KS  ?  Simulation (BS, SV, SVJ)  
Park et al. [2014]  BSH, SV  MSE  Chronological  KOSPI200. 10Y  
von Spreckelsen et al. [2014]  , ,  None  MSE,  Chronological  EURUSD. 1M  
Ludwig [2015]  Quadratic  MSE,  ?  S&P500. 12Y  
Liu and Huang [2016]  BSH  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  BSH, 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]^{20}^{20}20This 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  BSI  CVaR  Chronological  Simulation (BS, SV); S&P500. 5Y  
Cao et al. [2019]  , , underlying return  HW  MSE  Random  S&P500. 8Y  
Jang and Lee [2019]  ?  BSCal, 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]  , ,  BSCal, SVJ  MAE, MATE, MPE, MSE  Chronological  DAX. 4Y  
Karatas et al. [2019]  , , , ?  None  MSE,  Chronological  Simulation (BS, SV, VG)  
Palmer [2019]  , ,  BSI, LSM  MAE, MAPE  Chronological  Simulation (BS)  
Ruf and Wang [2019]  , , , , Aggressor side^{21}^{21}21A flag to indicate whether a transaction is induced by a buy or sell order.  HR  BSI, HW, Linear  MSE  Chronological  Simulation (BS, SV); S&P500. 8Y; 
STOXX50. 3Y  
Zheng et al. [2019]  SSVI  MAPE  ?  S&P500. 10Y  
Option price  
Option price given by the BlackScholes formula; see Table 3 for the different meanings of X  
Option price given by step multidimensional lattice scheme  
HR  Hedging ratio 
Strike price  
Stock price  
Interest rate  
Gamma: secondorder 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  
Atthemoney 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  
BSCal  BlackScholes formula with calibrated volatility 
BSG  BlackScholes formula with GARCHgenerated volatility 
BSH  BlackScholes formula with historical volatility 
BSI  BlackScholes formula with contractspecific implied volatility 
BSIH  BlackScholes formula with historical implied volatility 
BSIM  BlackScholes formula with atthemoney implied volatility 
BSK  BlackScholes formula with volatility obtained from Kalman filter 
BSN  BlackScholes formula with ANNgenerated volatility 
BSV  BlackScholes formula with volatility index, such as VIX or VDAX 
BW  BaroneAdesi 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 
LAn  nstep multidimensional 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] 
DM  Diebold and Mariano test  
KS  Kolmogorov and Smirnov twosample 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  Samplewise percentage error 


CVaR  Conditional valueatrisk  
MATE  Mean absolute tracking error 


MTE  Mean tracking error 


PE  Prediction error 


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 
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].
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 volatilitytype 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 socalled 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 BlackScholes 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 BlackScholes 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 socalled 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).^{22}^{22}22Several 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 valueatrisk (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 BlackScholes 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 atthemoney) appear in the literature. Blynski and Faseruk [2006] compare historical realised and historical implied volatility for the BlackScholes benchmark.
Note that for the pricing task, the BlackScholes formula with contractspecific 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 BlackScholes formula with contractspecific implied volatility is a valid benchmark.
In addition to the BlackScholes 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 twofactor regression model.
2.4 Data partition methods
An ANN needs to be trained on a training set (insample) and then tested on a test set (outofsample). 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 socalled ‘oddeven 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 timeinhomogeneity 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 closetooptimal solution is available as a benchmark, such as for the BlackScholes 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.^{23}^{23}23As 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.

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

Carverhill and Cheuk [2003] (# citations: 15) first propose an ANN that outputs hedging strategies directly, instead 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 deltavega 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.^{27}^{27}27At 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 BlackScholes 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 offline, thus significantly accelerating option pricing. For instance, Hernandez [2017] uses an ANN to calibrate a singlefactor HullWhite model. Dimitroff et al. [2018], McGhee [2018] and Liu et al. [2019a] calibrate stochastic volatility models, and Stone [2019] and Bayer et al. [2019]^{28}^{28}28For 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 tailormade regularisation effect as it replaces a nonparametric estimation task by the task of estimating a model with usually less than 510 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 BlackScholes PDE. E et al. [2017], Han et al. [2018], and Beck et al. [2019] utilize ANNs to solve highdimensional 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 highdimensional) problems. One case study involves the pricing of European options on 100 defaultable underlying assets. There are several recent papers, such as HenryLabordère [2017], Sirignano and Spiliopoulos [2018], ChanWaiNam 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 highdimensional 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 valueatrisk 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 nonparametric simulation tool for stock prices; see, for example, HenryLabordère [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:
At the training state the following techniques are being used:
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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