Hierarchical Learning For Option Implied Volatility Pricing

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Proceedings of the 54th Hawaii International Conference on System Sciences 2021Hierarchical Learning for Option Implied Volatility PricingHenry HanFordham Universityxhan9@fordham.eduAbstractMachine learning has been a popular optionimplied volatility pricing approach. It brings a goodgeneralization in pricing by avoiding building differentmodels for different options. However, it suffers from arelatively low prediction accuracy besides a modelselection issue. In this study, we propose a novelhierarchical learning approach to enhance machinelearning implied volatility pricing. It is designed for the‘learning-hard’ problem and boosts different machinelearning models’ performance for different option dataon behalf of moneyness besides identifying the optimallearning models. In particular, the proposedhierarchical learning can be an excellent way toenhance implied volatility pricing for the optiondatasets with more noise. In addition, we find out-ofthe-money options fit machine learning predictionbetter than the other options. This pioneering studyprovides a robust way to enhance implied volatilitypricing via machine learning and will inspire similarstudies in the future.1. IntroductionThe Black-Scholes-Merton (BSM) model has beenubiquitous in financial research since the 1980snotwithstanding critiques for its not all empirically validassumptions such as log-normal distributions of stockprices [1-2]. It not only inspires rigorous explorations inoption pricing but also continues to be a practical guidein trading for its useful approximation to reality [3]. Thevolatility in the BSM model is a type of forwardvolatility called implied volatility that measuresinvestor's confidence about the future risk of the stock.It is a future volatility of the stock but cannot beobserved directly from the historical data. It indicatesthe current market expectation of future stock volatilityand impacts market option prices seriously.Theoretically, implied volatility is the value that makesthe theoretical price of an option under an option pricingmodel equal to its current market price [4]. Tradersusually quote options by implied volatility rather thanURI: 0(CC BY-NC-ND 4.0)the price [5]. Implied volatility is also an importantindicator of the financial market: it decreases in bullishmarkets and increases in bearish markets. The wellknown VIX (Chicago Board Options Exchange (CBOE)volatility index) index is obtained by conductingimplied volatility pricing for S&P 500 index options bytallying up and averaging relevant implied volatilities.The VIX index itself is not only a tradable asset, but alsoa daily market indicator, i.e. 'investor fear index',measuring option market risks followed by variousinvestors and market participants. Moreover, highaccuracy implied volatility pricing plays an essentialrole in successful hedging and trading in the financialmarket, where trading is more and more dominated byhigh-frequency trading. The high-speed trading requiresoptions to be priced accurately as much as possible.Therefore, implied volatility needs to be pricedaccurately for the sake of trading and marketunderstanding.Implied volatility pricing remains an importantproblem in finance in the era of big data where tradingvolumes and speed grow exponentially. It demandsmore accurate implied volatility pricing for the sake ofoption pricing that affects corresponding equity hedgingprocedures. In this study, we propose a novelhierarchical learning algorithm to enhance machinelearning pricing to provide more accurate impliedvolatility prediction via a two-stage learning procedure.The proposed hierarchical learning works wellespecially for large amount of data or even big data. Itboosts state-of-the-art machine learning models’prediction at least 33.68% averagely on behalf ofmoneyness. Our study suggests that machine learningpricing performs better performance for OTM options,identifies Gradient boosting (GB) models outperformother peers in prediction, and provides an efficientprocedure for high-accuracy option pricing, which hasbeen challenging the implied volatility pricingcommunity [4,5,6].There are quite a lot of classic implied volatilitypricing methods rooted from the BSM model. They canbe classified as iteration and closed-form methods. Theformer solves a corresponding nonlinear equationnumerically and the latter seeks a closed-formula toPage 1573

model implied volatility by using at-the-money (ATM)option price as an initial point [5].An ATM option means its strike price is identicalto the market price of the stock. Similarly, ITM andOTM both describes the moneyness of options. ITMmeans the stock’s market price of a call (put) option isabove (below) its strike price. An ITM option has apositive intrinsic value, which is the difference betweenthe market price and the strike price of the option. OTMrefers to a call (put) with a strike price higher (lower)than the market price of the stock. Both OTM and ATMoptions only have time value rather than intrinsic value.OTM options need some time to be profitable for optionbuyers. OTM, ITM, and ATM can be well separatedunder knowledge-based visualization.Figure 1. The PCA visualization of OTM and ITM/ATMamong the 6041 call options of the 2017 option dataset.Figure 1 visualizes OTM and ITM/ATM of 6041calls of the 2017 option dataset used in this study intotwo well-separated groups under PCA. Since ATMoptions count only 21 samples, we group it with ITMthat has 4328 samples into the ITM/ATM group.Generally, an ATM will appear as the boundary pointbetween the ITM and OTM in PCA visualization.The iteration methods include different types ofroot-finding methods that range from Newton-Raphsonto Brent-Dekker method as well as their variants [6,7].Some may suffer from a slow convergence to findimplied volatilities, some cannot guarantee aconvergence, and some improved ones have quitecomplicated implementations [6-9]. Furthermore, theycannot handle deep in-the-money (ITM) and deep outof-the-money (OTM) option pricing well [4,6].The deep ITM call/put options have strike pricesmuch lower/higher than the market stock price. Forexample, a call whose strike price is at least 10 lowerthan the market stock price will be a typical deep ITMcall. Their option prices are highly sensitive to thechange of their stock prices instead of impliedvolatilities. The deep ITM options are usually preferredby long-term investors for its high intrinsic value. Onthe other hand, the deep OTM options have the strikeprices significantly higher /lower for call/put than thestock price. Deep OTM options are considered as high-risk ones because they are more likely to bring highpayoffs or losses.The closed-form approximation methods seek anapproximated formula to calculate implied volatilitydirectly. They are extremely fast in implied volatilitypricing compared to the iteration methods. They employTaylor expansions on an at-the-money (ATM) optionprice point to obtain an approximated implied volatilityformula [10]. Brenner and Subrahmanyam, Corrado andMiller, Chambers and Nawala made several successfulattempts to optimize the approximation form of impliedvolatility [10-13]. However, the methods may performpoorly on out-of-the-money (OTM) options either [12].The closed-form and iteration methods have wellestablished theoretical background. However, they aremodel-driven methods and cannot take advantage oflarge amount of available data in the market. Actually,a large option dataset can even be a nightmare for theslow-convergent iteration methods. Since differentmodels have to be developed for different types ofoptions, they face the challenges from the high modelingcomplexity and trade-off between the theoretical marketassumptions and implied volatility pricing accuracy.More technically, both are not able to handle OTMoption implied volatility pricing well.During the recent two decades, the applications ofmachine learning have been developing rapidly andactively in predicting implied volatilities [14-17].Machine learning implied volatility pricing is differentfrom the model-driven methods for its data-driven.Almost all machine learning models do not rely ontheoretical assumptions about markets and options.Instead, they dig knowledge and learn impliedvolatilities from input option data and construct animplied volatility prediction function.Malliaris and Salchenberger apply neural networksto predict S&P 100 implied volatility by using pastvolatilities and other options market factors [14].Gavrishchaka and Banerjee forecast stock marketvolatility using support vector machines (SVM),suggesting the efficiencies of working with highdimensional inputs [15]. Yang and Lee predict theimplied volatility distribution by using Bayesian kernelmachines [16]. Zeng and Klabjan design an onlineadaptive primal support vector regression model toexplore the implied volatility surface and realizedynamically updating support vectors to improve itsefficiency [17].The increasing data volumes in the financial marketcall for a data-driven way to exploit a large amount ofdata in implied volatility pricing. The machine learningapproaches meet such an urgent demand. It makes itpossible to derive an appropriate prediction model bylistening to ‘data talks’ and achieve ‘more data, betterprediction’. It can fully exploit the impacts of allPage 1574

possible variables on implied volatilities in a learningprocedure rather than a few ones specified by the model.It can do implied volatility pricing for almost any typeof options given enough data. It brings a built-in pricinggeneralization for different types of options and avoidsthe modeling complexities to build different pricingmodels. On the other hand, it enables implied volatilitiesmore interpretable for the real market. Impliedvolatilities can be easily interpreted as a function of a setof variables such as option price, stock price, strikeprice, and other related ones under a machine learningpricing.However, different challenges remain though moreand more machine learning methods are employed inimplied volatility pricing [14-16]. Since the strongnonlinearity between implied volatility and its affectingfactors from the market with a stochastic nature, impliedvolatility prediction is a ‘learning-hard’ problem thathas a relatively low prediction accuracy for almost allmachine learning models. How to enhance machinelearning prediction accuracy remains an urgentchallenge.Besides, it remains unknown which machinelearning models can ‘fit’ option data well though manyhave been employed [14,16,17]. It is also unknown howoptions in different moneyness status behave under themachine learning approaches. How OTM, ITM, andATM options react differently in machine learningpricing? The answer to the query will unveil latentknowledge in implied volatility pricing on behalf ofmoneyness.In this study, we propose a novel hierarchicallearning algorithm to improve the accuracy of machinelearning pricing. Unlike traditional learning, it presentsa hierarchical learning scheme for data via a two-stagelearning procedure. The potentially well-performed datawill have a priority in learning. It boosts each machinelearning model’s prediction by providing a selectivemechanism to pick high-quality data involved inlearning. It seeks a partial but better learning result aswell as provides an adaptive learning scheme for thepotentially poorly performed data. It is particularlysuitable for solving the learning-hard problem such asimplied volatility prediction. We apply the proposedalgorithm to benchmark option datasets under state-ofthe-art machine learning models. The proposed methodboosts each model’s prediction at least 33.68%averagely on behalf of moneyness.Furthermore, our study answers the unsolvedqueries in fintech: which machine learning models fitoption implied volatility pricing better and How optionswith different moneyness statuses react differently inmachine learning pricing? We identify gradientboosting (GB) as the best model for implied volatilitypricing for its leading performance over its peers. Ourstudy suggests that machine learning pricing performsbest for OTM options though traditional models usuallyfail on the high-risk options [4,5,6].Our study also presents a new option moneynessclassification by grouping options as OTM, ITM, andNTM (near-the-money) for the sake of the robustness ofmachine learning pricing. The NTM options includeATM options and the shallow OTM and ITM options.An OTM/ITM option is an NTM if the absolutedifference between its strike prices and the stock pricesis less than 0.5. Such classification groups theboundary moneyness options into the same group so thatITM and OTM options are more representative inlearning for their similar behavior in the market. Ourresult indicates that the implied volatilities of the ITMoptions can be more unpredictable than the OTM andNTM options though they have relatively small pricemove percentages in the market.2. Hierarchical learning (HL) pricingHierarchical learning (HL) is a novel genericacceleration algorithm for any supervised machinelearning. It consists of two stages: selective learning andadaptive learning. The selective learning stage is toeliminate ‘bad guys’ from training and test data. The‘bad guys’ are the options with or potentially with poorperformance under a given machine learning model. Wename them as bad points, problematic points orpotentially problematic points in our algorithm. Thenthe following learning is built upon the good-qualitydata. The adaptive learning stage builds customizedtraining data for each ‘bad guy’ dropped in the selectivelearning stage to conduct learning adaptively to enhancelearning performance. In our context, the bad guys arethose options that negatively affect the prediction: theyare options with or potentially with poor impliedvolatility pricing results. The proposed hierarchicallearning has a specific algorithm to identify them evenbefore learning.2.1. Machine learning pricing evaluationsMachine learning implied volatility pricing isessentially a regression procedure where impliedvolatility is a response variable. Although R2-basedmodel evaluation theoretically works for generalregression models, it cannot disclose real differencesbetween two models involved in pricing, i.e. their R2values can be close or the same but their realperformance can be quite different. Thus, we use meansquare error (MSE) and prediction error to evaluate alearning model’s performance for their more concreteperformance. The prediction error (Err) evaluates thePage 1575

performance of the model on an individual data point(option).' 𝐼𝑉 𝐸𝑟𝑟 𝐼𝑉(1)' is thewhere 𝐼𝑉 is the true implied volatility, and 𝐼𝑉predicted implied volatility. The mean square error(MSE) represents the average performance of the modelfor n options. The less MSE value signifies the betterperformance of the model.𝑀𝑆𝐸 !" '# . "#%!-𝐼𝑉# 𝐼𝑉(2)2.2 Selective learning: the 1st stage ofhierarchical learningHierarchical learning assumes we have enoughtraining and test data in learning. It has two stages:selective learning and adaptive learning. It has threebasic inputs: a machine learning model 𝜃, training data"𝑋 {𝑢# , 𝑣# }&# , and test data 𝑋′ {𝑥# , 𝑦# }# , where 𝑢# and𝑥# are a training and test option, and 𝑣# and 𝑦# are theircorresponding implied volatilities. But the impliedvolatilities in the test data are supposed to be unknown.We introduce selective learning as follows first.Unlike traditional learning using the whole trainingand test data unselectively, selective learning eliminatesthe potentially poorly performed options by theproposed probing learning and nearest neighbor searchto obtain good-quality training and test data that fit themodel 𝜃 better. Selective learning consists of probelearning, nearest neighbor search, and data clean threemain components.Probing learning is a warm-up learning procedureon the ‘whole training data’ to find initial ‘bad guys’under the machine learning model 𝜃. It views trainingdata as an entirety to perform machine learning. Toconduct probing learning, we first randomly splittraining data into two parts: train-train 𝑋''()#" and traintest 𝑋''* ' according to the threshold ratio specified for,- % ,the training and test data size: 𝜏 "% " :. For & " - % 0 - example, if 𝜏 80%: 20%, then 𝑋''()#" and 𝑋''* ' willcount 80% of training and 20% test samplesrespectively. The 80% data splitting will make surethere are enough training data available in the selectivelearning stage.Problematic dataset collection. The machinelearning model 𝜃 is fitted by 𝑋''()#" to predict theimplied volatility for each entry of the train-test data𝑋''* ' . Since all the true implied volatilities are knownfor 𝑋''* ' , we can identify the first group of ‘bad guys’/‘bad points’, i.e. problematic points, whose absoluteerrors are top-ranked (e.g., 90th percentile) among theabsolute errors of all entries in 𝑋''* ' , i.e.,𝑆1 {𝑥: 𝐸𝑟𝑟(𝑥) 𝐸𝑟𝑟2 , 𝑥 𝑋''* ' }(2)where 𝐸𝑟𝑟! is the δth percentile of all train-test absoluteerrors (generally δ 90). The problematic point setincludes the options with poor prediction errors underthe model 𝜃. The 90th percentile choice can guarantee toobtain poorly-performed ‘bad points’ under a single MLmodel rather than using several ones under a relaxedcutoff.Nearest neighbor search (NNS) finds the secondgroup ‘bad guys’, i.e. potentially problematic points,which are close to the first group of ‘bad guys’ 𝑆1 . Theyare defined as the union of the points close to eachoption 𝑥 𝑆1 in the whole training data 𝑋 and test data𝑋 3 . For any point 𝑥 𝑆1 , the nearest neighbor search isemployed to find its k nearest neighbors in the trainingdata 𝑋 according to a distance measure,'()#"𝑆4,6 @𝑎: 𝐷(𝑥, 𝑎) 𝜏6 , 𝑥 𝑆1 D(3)It answers the query: ‘which ones are the closestneighbors to a problematic point 𝑥,’ where 𝜏6 is thedistance to find the k nearest neighbors. Suppose anoption 𝑥 is marked as a ‘bad guy’ in the probing learningunder a learning machine, say gradient boosting (GB),then the NNS calculates its distances to the other pointsin the training dataset, sorts the distances from thesmallest to the largest, and identifies the top k (e.g. k 5)neighbors with the smallest distances to it. Similarly, itspotentially problematic point set in the test dataset canbe obtained by finding 𝑘 3 neighbors of 𝑥,'* '𝑆4,6% @𝑎: 𝐷(𝑥, 𝑎) 𝜏6 % , 𝑥 𝑆1 D(4)where 𝑘 3 is usually set as 𝑘. Thus, the sets of potentiallyproblematic points in the training and test are 𝑋6 '* ''()#"and 𝑋 3 6 % 4 8' 𝑆4,6 4 8' 𝑆4,6% , respectively.We employ the Manhattan distance in the nearestneighbor search for its advantage over other measures(e.g. the Euclidean distance) in identifying potentiallyproblematic points for option data. The important reasonwhy the Manhattan distance outperforms other peerdistances is that it will not blur or offset some variableswith small values (e.g. volatility, implied volatility, timeto maturity, or even some option prices) and distinguishoption similarity better. For example, the Euclideandistance will let the variables with the large values (e.g.strike price) dominate the distance calculation for theirlarge values and shadow the contributions of thevariables with small values in its squaring calculation.As a result, the ‘true differences’ between options maynot be measured well and the quality of nearest neighborsearch will be affected. The superior implied volatilitypricing results under the Manhattan distance to thoseunder the other distances in our study also support it.Data clean cleans and updates training and test databy removing all the ‘bad points’: 𝑋9:*)" 𝑋 𝑋6 𝑆1Page 1576

and 𝑋 3 9:*)" 𝑋 3 𝑋6 % , where 𝑋6 and 𝑋6 % are thesecond group of potentially problematic points from thetraining and test dataset respectively and 𝑆1 is the firstgroup of problematic points from the training dataset.The cleaned training data 𝑋9:*)" will be used topredict the implied volatilities for the cleaned test data𝑋 3 9:*)" . Since the problematic points perform poorlyunder the machine learning model 𝜃 and its associatedpoints potentially to have a similar behavior under themodel, it is reasonable to believe that removing themand their nearest neighbors in the training and testdataset will enhance the following machine learningpricing. The algorithm 1 describes the details of theproposed selective learning.Algorithm 1 Selective learningInput:Training data 𝑋 {𝑢" , 𝑣" }#"Test data 𝑋′ {𝑥" , 𝑦" } "Machine learning model 𝜃Training data partition threshold 𝜏The percentile cutoff 𝛿The number of nearest neighbors 𝑘, 𝑘′Output:Cleaned training data 𝑋%&'( Cleaned test data 𝑋 ) %&'( // Probing learning1. //Partition training data as train-train and train-test under a2.3.4.5.6.7.8.9.10.//threshold 𝜏𝑋** (" , 𝑋**',* ��𝑛𝑖𝑛𝑔𝐷𝑎𝑡𝑎(𝑋, 𝜏)//training machine learning model 𝜃 by train-train data𝜃 𝑓𝑖𝑡(𝜃, 𝑋** (" )//Predict implied volatilities for train-test data𝑣C 𝜃. 𝑝𝑟𝑒𝑑𝑖𝑐𝑡(𝑋**',* )// Calculate absolute error for all entries in train-test data𝐸𝑟𝑟 𝑣C 𝑣 // Identify problematic points (bad points)𝑆- ��𝑖𝑐𝑃𝑜𝑖𝑛𝑡𝑠(𝐸𝑟𝑟, 𝛿)// Nearest neighbor search11. // Find potentially bad points in training and test data12.𝑋. 𝑋.) {}13. For 𝑥 𝑆14.15.𝑋. . , 𝑘, 𝑋))𝑋 ( ) ! . , 𝑘 ( , 𝑋 ()716. // Clean training and test data17. 𝑋%&'( 𝑋 𝑋. 𝑆18. 𝑋′%&'( 𝑋′ 𝑋.)The selective learning stage gives betterapproximate learning by removing ‘bad data’ that doesnot match the learning machine before learning is made.It can be used independently to handle the learning-hardproblems by finding approximate solutions.2.3 Adaptive learning: the 2nd stage ofhierarchical learningThe selective learning stage brings a new learningmechanism by fitting a learning machine with highquality data to find partial but better approximatesolutions instead of the whole solutions. But sometimeswe may not be able to simply disregard the ‘bad-guys’in learning. They can be urgent options in pricing.We propose the secondary stage of hierarchicallearning: adaptive learning to handle this situation. Theadaptive learning constructs a local individual trainingdataset for each ‘bad guy’ dropped from the test data andconducts learning adaptively.The basic idea is to generate a local training dataset𝑆4 for each bad point 𝑥 𝑋63 by searching its m (e.g.m 20) nearest neighbors in the training dataset 𝑋 basedon a distance (e.g. ‘manhattan’ distance). The localtraining dataset of 𝑥 consists of its nearest neighbors, i.e.proximity points, which will have more advantages topredict its implied volatility than a genetic trainingdataset. The local training dataset 𝑆4 is employed to fita machine learning model before predicting the impliedvolatility of 𝑥. Such adaptive learning can enhance theprediction of each ‘bad guy’ because of a morecorrelated local training dataset construction. It is notedthat the machine learning models in the adaptive stageshould not be those models that require a large trainingdataset (e.g. deep neural networks) [18].2.4 Model selection and extreme case handleIt is noted that hierarchical learning can reuse theknown training results, which include the known ‘badpoints’ and an updated better-quality training dataset fora given set of new options arriving. Hierarchicallearning only needs to identify the ‘buddies’ of theknown ‘bad points’ in the arriving test dataset and obtaina better-quality test dataset for the selective learningstage. The identified potentially bad points will enter theadaptive learning stage. However, training needs to beredone for the customized small training dataset for eachpotentially bad point in the test dataset that generallycontains m 20 samples in the adaptive learning stage.It is recommended to use gradient boosting (GB) orsimilar extra-tree models in hierarchical learning for thesake of accuracy and computing speed [19-20].Hierarchical learning does not require to use the samelearning model in the selective and adaptive learningstages. But the adaptive learning stage needs to choosethe models that work well for a small dataset (e.g.Page 1577

support vector machines (SVM) or GB) [19,22]. BothDNN and SVM are not recommended in hierarchicallearning because the former suffers from a long trainingtime and the latter may encounter the scalar issue forlarge datasets in the selective learning stage.The extreme case can be both training and testdatasets can be quite small. The number of ‘bad-guys’identified in probing learning can be few. It is suggestedto only use the adaptive learning step under the situationfor the sake of efficiency, where an individuallycustomized training dataset will be built dynamically foreach test entry.2.5 ML models for hierarchical learningAs mentioned before, hierarchical learning is ageneric method applied to any machine learning models.We employ six state-of-the-art models to evaluate itsperformance. They include k-nearest neighbor (k-NN),support vector machines (SVM), random forests (RF),gradient boosting (GB), extra tree (ET), and deep neuralnetworks (DNN) [18-23]. We focus on the GB modelfor its importance in this study and the page limit.Gradient Boosting (GB) seeks an implied volatilityprediction function by optimizing the predictionfunctions of weak learners along with the negativegradient directions of the loss function. Unlike baggingensemble learning that averages the prediction functionsfrom the independent weak learners, the weak learns ofGB are no longer independent [19]. On the other hand,each weak learner is added sequentially to the procedureto ‘boost’ learning results. GB learns its predictionfunction in an iteration model,* 𝑓H6 𝑓H6;! 𝑟6 *,-3. Results3.1 Data acquisition and preprocessingWe developed python option acquisitionsoftware: OptionGlean for this study. It is designedspecifically to retrieve option data from Yahoo //www.nasdaq.com). The OptionGlean requiresticker names as input besides accepting user-specifiedinput about options such as type, expiration, moneyness,and exchanges, etc. Three option datasets acquired byusing OptionGlean include options in 2015, 2017, and2018. We name the corresponding datasets as optiondata 2015, 2017, and 2018, which contain 25701, 14251,and 36646 options respectively. The options can be anytype of options in the market. They are traded in Nasdaqand NYSE and are not high-frequently traded options.Figure 2 compares the p.d.f.s of the four variables ofthe three datasets: option prices, and stock & strike priceratio (S/K), implied volatility, and time to maturitybesides their boxplots of the 2017 option dataset. Thep.d.f.s illustrate the 2015 and 2017 option datasets sharemore similarity between each other. We skip othervariables such as option type, ask, bit, and volumes inthe plots.(5)where 𝐿(. ) is the loss function of the learning model.The GB prediction function is initially formulated as theform of a weighted sum of decision functions ℎ# (𝑥) ofthe weak learners 𝑓H(𝑥) #%! 𝛾# ℎ# (𝑥), where theweights 𝛾# grow in each step when a new weak learneris introduced. It is further optimized in gradientlearning. The samples are no longer equally likely to beselected to train a weak learner in GB. Instead, thosewith larger predation errors are more likely to beselected for training, because GB learns from mistakescommitted by the weak learners in the previousiterations. As a result, GB does not demonstrateoverfitting robustness as RF [21]. There are differentloss functions, but we choose the least square for itsmathematical efficiency.Figure 2. The comparisons of p.d.f.s of option prices, stockand strike price ratio (S/K), implied volatility, and time tomaturity of the three datasets besides their boxplots for the2017 option dataset.We separate each option dataset into three groupsaccording to ‘updated’ moneyness: in-the-money(ITM), near-the-money (NTM), and out-of-the-money(OTM). Unlike traditional ways, we define OTM as calloptions with stock price 𝑆 𝐾 𝛿, where threshold 𝛿 isselected as 50 cents in this study, or puts satisfying 𝐾 𝑆 𝛿. Similarly, ITM options are those calls and putssatisfying 𝑆 𝛿 𝐾 𝑆 𝛿. The near-the-money(NTM) includes at-the-money (ATM) options and theshallow out-of-the-money (OTM) and in-the-money(ITM) options.Page 1578

The reason to introduce a threshold in moneynessclassification is to enhance machine learning pricinggeneralization. Traditional classification may not matchthe market reality very well. A very shallow OTM orITM option can be equivalent to an ATM option becauseof a trade commission. Adding a threshold in optionmoneyness classification can make machine learningresults in pricing close to the market reality.Furthermore, using NTM data rather than only ATMdata can prevent the failure of machine learning becausethe small size of ATM data can lead to a learning failure.The call and put distributions generally arebalanced for the 2015 and 2017 option data. But they areimbalanced for the 2018 OTM and ITM data. The OTMdata has 13849 and 6501 calls and puts, but the ITM datahas 4443 calls and 10553 puts. Compared to the OTMand ITM data, the NTM data has the least amount ineach dataset. For example, the 2015 NTM data consistsof 728 calls and 777 puts; the 2017 NTM data onlyconsists of only 444 calls and 471 puts; the 2018 NTMdata has 684 calls and 618 puts.3.2 Selective learning pricingpotential ‘bad points’ under the ‘manhattan’ distance.The neighborhood sizes in NNS are set as 𝑘 𝑘 ) 10.Figure 3 compares the MSE values of selectivelearning under the six models for the OTM, ITM, andNTM datasets. The MSE values of all the models underselective learning are lower or even much lower thanthose of the original ones. This suggests theeffectiveness of the proposed selective learning

option implied volatility pricing well. During the recent two decades, the applications of machine learning have been developing rapidly and actively in predicting implied volatilities [14-17]. Machine learning implied volatility pricing is dif

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