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Modeling Implied VolatilityRongjiao JiInstituto Superior Técnico, Lisboa, PortugalNovember 2017AbstractWith respect to the valuation issue of a derivative’s contracts in finance, the volatility of the priceof the underlying asset is often unknown. Volatility is a measure of randomness, allowing us to assesshow uncertain the price movement is in the future.In this work we first derive the implied volatility for each contract, using the Black-Scholes formula.Since it is not possible to determine the implied volatility analytically, one needs to resort to numericalmethods. Here we propose to use the bisection method to compute the estimate value of the impliedvolatility. The determination of implied volatility is discussed afterwards, using the future price andthe asset price as input value in the Black-Scholes formula. In addition, two calculation methods arepresented, in order to increase the accuracy of the estimates obtained.Several models are presented for adjustment of the obtained values, namely models based on linearquantile regression and random forests. Using these models, we may forecast the implied volatility, andthen use these values to predict the price of an option contract in the future.Keywords: implied volatility, Black-Scholes formula, quantile regression, random foreststerion to a dataset composed by options, future contracts and discount values, and secondly builds thequantile linear and random forest for the computedimplied volatility, in order to generate a clear cognition of the option trading process and to predictthe Black-Scholes implied volatility.1. IntroductionBlack and Scholes [2] and Merton [11] proposedin 1973 the Black-Scholes model to fit the market prices of options. The Black-Scholes formulais applied to transform the market prices into anexpression in terms of implied volatility [9]. Implied volatility is the square root of the quadraticvariation of the asset’s log price process, definedas a parameter in the option pricing model BlackScholes formula that yields the price of a particularoption. Given observed option prices in the market, the value of implied volatility can be deducedby matching the corresponding Black-Scholes pricewith the market price [4]. The implied volatilityfrom the real market shows a pattern named volatility smile, influenced by time to maturity and strikeprice of the option.Different from the theoretical parametric models,such as stochastic volatility model [5] and jumpdiffusion model [12], non-parametric models basedon machine learning techniques, which has less restrictions in usual cases, have been developed forpricing options [14]. Additionally, quantile method[6], beyond the conditional mean, gives more information about the quantile distribution of response variable as a function of the explanatoryvariables. The combination of non-parametric models and quantile methods, in this way, has lessrestrictions on the required assumptions and canbring a new perspective for estimating the impliedvolatility [16].Our work firstly implements several financial cri-1.1. Dataset DescriptionThe dataset used in this work is real option tradesprovided by BNP Paribas bank, including information over three types of derivatives (option contracts, discount values and future contracts). Thesethree derivatives are all based on the same underlying asset, the EURO STOXX50 index, which is anEurope’s leading Blue-Chip index for the Eurozone.Specifically, as the main object, option contractsinclude trades in 424 different dates, ranged fromJanuary 02, 2014 to October 29, 2015. Almost allthe maturities of options concentrate on Friday, andthe months with more maturities are March, June,September and December. There are only ten distinct maturities of future contracts and these maturities spread over the third Friday of every quarterfrom 2014 to the mid year of 2016. These kind ofmaturities are denoted as ’major maturities’, andTriple Witching Phenomenon occurs at the majormaturities. At this time, the contracts for stock index futures, stock index options and stock optionsexpire on the same day. Trading activity is more active and volatility is larger because contracts thatare allowed to expire may necessitate the purchaseor sale of the underlying security.1

In Table 1 we present the descriptions of relevantterms. In the original data set, some non-relevantinformation is discarded (such as the lot size, bidand ask size). Moreover we also include four variables which are marked by star and are relevant forthe rest of the study, namely: time to maturity,moneyness, constructed price and interest rate.Table 1: The descriptions of relevant terms which areprovided from three types of derivatives (option, future and discount). The number of observations foreach type is shown in brackets. The notation star(*)indicates that the marked variables are generated afterwards.TermDate tMaturity TOption (312339)TypeStrike KBid priceAsk price*Moneyness M*Time to maturity τ*Constructed price ŜDescriptionFigure 1: The tendencies of option prices of the call andcorresponding put options with strike price in ExampleSet. The enlargement of the crosspoint of lines is shownin the top right.Example SetThe beginning date of a derivative contract.The expiration date of a derivative contract.2014-01-032014-03-21Call options C or put options P .The price paid for the asset if the option is exercised.The highest price a buyer is willing to pay.The lowest price a seller is willing to accept.KDeduced by M KDS F and used to label options.The number of days from date to maturity.Constructed for the true market price S by Ŝ F D.Boththe options contracts were gathered at 17:15 CET.Following the ’Example Set’, the index price in Jan03, 2014 is 3074.43, while the constructed price is3062.73. The constructed price, by the way, is anon-standard terminology used in this work. Because it is deduced from the discounted future price,in theory it is supposed to be the best representation of the current asset price.The constructed price should worth the same ascurrent true price of underlying asset. Thus, wehave two different situations depending on whethera) Those options whose maturities belong to themajor maturities of future contracts. Thus the index price obtained can variate a bit from the constructed price within 45 minutes at very active trading days, in particular at the major maturities dueto Triple Witching Phenomenon.b) Those options whose maturities do not belongto the major maturities of future contracts, or theirmaturities do belong to the major maturities butthe influence of Triple Witching Phenomenon is limited. In this situation, the price of the underlyingasset is very similar to the index price.Thus the index price can be used as a supplementfor those options which have no information aboutthe relevant future price. It can give a rough direction for the investors although its accuracy is notassured.ShowninFigure 1773062.73Future Contracts (1153)Bid PriceThe highest price a buyer is willing to payAsk PriceThe lowest price a seller is willing to accept30613062Discount Value (7255)Discount Value DDiscount from future back to current time.Interest Rate rDeduced by D e r(T t) .0.99965.5 10 6Here we display specifically the trades (denotedas ’Example Set’) with trading date in January 03,2014 and maturity in March 21, 2014, as an exampleto explain the micro structure of the dataset andthe following operations. The information in detailis shown in the right side of Table 1.Combination of optionsDue to the Put-Call Parity, volatility is the samefor a call option and a put option with the samecombination of date, maturity and strike. Aftercombining the call with corresponding put options,we have now 107571 pairs of call and put options.Following our ’Example Set’, in Figure 1, we cansee that, under the condition of fixing other variables, for call options, when strike value goes up,the price of call option declines. Oppositely, theprice of put option increases. However, the rangeof call option prices is larger than the range of putoption prices. One of the possible reasons might bethat traders believe the market price would go up.Information of index priceEURO STOXX50 index is the asset under study,i.e. the objective financial asset that our threederivatives are associated with. The informationof historical index price can be easily obtained 1. Note that the newly obtained index prices werecaptured at 18:00 CET in each trading day while1.2. OutlineThe paper is structured as follows. Section 2 introduces most important criteria, the Put-Call Parity and Black-Scholes Formula. Section 3 interpretsquantile linear regression and the main idea of decision trees and random forest. Section 4 demonstrates how to calculate the implied volatility. Wegenerate the estimate prices as a function of implied volatility deduced by two forms of BlackScholes formula and two methods of calculation,and then derive the implied volatility by the Bisection Method, given the market price. Section 5mainly focuses on using quantile methods together1 From the website ’Yahoo X50E/history?period1 1388620800\&period2 1446076800\&interval 1d\&filter history\&frequency 1d2

with linear regression and random forests methodto model the subset which contains some significant features. The work concludes with Section 6,where we also point possible future work. We skipthe introduction of financial products (more detailsin [2] and [17]) and some statistical tools (details in[1], [13], [10] and [6]) due to the limitation of pages.Quantile regression focuses on the conditionalquantiles of Y given X x. Assume the distribution function for a real-valued response variableY is FY (y) P (Y y), then the τ -th quantile ofY is defined as the minimum value of y which satisfies F (y) τ , i.e., 1QY (τ ) FY(τ ) inf {y : F (y) τ }, 0 τ 1.Definition of quantile loss function2. Financial Theoretical OverviewGiven a quantile τ (0, 1), the L1 -norm quantileWe define that for a fixed date t, an European call regression is used to minimize the loss function [7]:option (whose call option price is C(T, K) and thenPcorresponding put option price is P (T, K)) with asL(yi , ŷi )τ (5)ρτ (yi ŷi ),set price St , maturity T and strike price K is definedi 1as a contingent claim with payoff max{ST K, 0}where ŷi is the estimate values and ρτ (yi ŷi )at maturity. The related future price for the under- (named as check function in [6]) is defined as:lying asset is F (t, T ), discount value is D(t, T ) and(τ (yi ŷi ),yi ŷi 0interest rate is r. Put-Call Parity describes theρτ (yi ŷi ) . (6)relationship for different portfolios, shown as: (1 τ )(yi ŷi ), otherwiseC(T, K) P (T, K) St Ke r(T t) ,The quantile regression is going to be discussedin more detail in Section 3.1.1, and in Section 3.1.2estimation methods based on decision trees and random forest are briefly reviewed.or: C(T, K) P (T, K) [F (t, T ) K]D(t, T ).The Black-Scholes Formula [2] proposes theestimate value of the call option price C(T, K) andput option price P (T, K) asC(T, K) St Φ(d1 ) Ke r(T t) Φ(d2 ),(1)3.1.1Linear regressionP (T, K) St Φ( d1 ) Ke r(T t) Φ( d2 ),S ln( Kt ) (r 0.5σ 2 )(T t) d1 , d2 d1 σ T t,σ T t(2)Linear quantile regression fits a conditional quantileof the response variable by a linear function x β.In Table 2 we compare the general quantile linearregression with the most common alternatives: ordinary least squares regression and least absolutedeviation regression [15].where Φ(·) is the cumulative distribution functionof the standard normal distribution, and σ is thevolatility of returns of the underlying asset.Another common form for Black-Scholes formulabased on future price and discount value for call Table 2: Comparison between ordinary least squares,least absolute deviation and quantile regression methoption price C(T, K) and put option price P (T, K)ods.are proposed as:Ordinary Least Squares (OLS)C(T, K) D(t, T )(F (t, T )Φ(d1 ) KΦ(d2 )),(3)P (T, K) D(t, T )[Φ( d2 )K Φ( d1 )F (t, T )],F (t,T ) ln( K ) 0.5σ 2 (T t) d1 , d2 d1 σ T t.σ T t(4)Conditional mean function E(Y X x) x β.nPLoss function:(yi xi β)2 ,i 1minimize the sum of square of residuals.nPEstimates β̂ arg min(yi xi β)2 .p 1β R3. Statistical Theoretical OverviewIn this section we do a brief overview of quantileregression and random forests.i 1Least Absolute Deviation (LAD)Conditional median function Qτ 0.5 (Y X x) x β(τ 0.5).nPLoss function: yi xi β ,i 1minimize the sum of absolute errors.Pn Estimates β̂ arg mini 1 yi xi β ,.p 13.1. Quantile Regression Methodŷ RQuantile regression is a generalization of linear reτ-QuantileRegressiongression where it models a quantile of interest as aConditional quantile function Qτ (Y X x) x β(τ ).linear function of the explanatory variables [6]. It isnPLoss function:ρτ (yi xi β),more robust then ordinary least squares, and it hasi 1minimize the sum of weighted absolute errors.shown that it can lead to good results when therenPare complex relations between variables.Estimates β̂ β̂(τ ) arg minρτ (yi xi β).β Ri 1Definition of quantileSuppose that there is a dataset with n observations, {yi , xi } for i 1, . . . , n, where yi is re3.1.2 Tree-based regressorssponse variable and the explanatory variables xi (xi1 , . . . , xip ), and β (β0 , β1 , . . . , βp ) is the coef- If the relationship between response and explanaficients vector.tory variables is highly non-linear or complicated,ip 13

the tree-based models may give better results andexplanations than linear regression.Decision TreeDecision tree is a non-parametric supervisedlearning method in the form of a tree structure. Itsplits a dataset from the entire space (denoted asR) into several non-overlapping regions (denoted asRj , j 1, . . . , J) from the top of the tree to eachleaf in the bottom by a series of binary if-then rules.These rules identify distinct regions where the observations inside share the most homogeneous responses to explanatory variables. Due to the factthat it is difficult to take every possible partition ofthe dataset into J regions into account, we need toapply a top-down, greedy recursive binary splittingapproach to split successively further down to theleaf and to choose the best split at each step of treebuilding process. In each internal node, the datasetis split and explanatory variables are judged to minimize the prediction error. The leaves, named terminal nodes, represent the final division of regions.At a leaf node, the mean of the response values assigned to that node is the predicted value returnedby the decision tree.This work uses CART, namely Classificationand Regression Tree algorithm (please check [3]for more details). In order to perform this approach, for a explanatory variable Xj , there is acutpoint s such that the variable can be split totwo regions, either the region where Xj is smallerthan s, i.e. {Xj s} or the region where Xjis greater or equal to s, i.e. {Xj s}. Thebest cutpoint for each variable Xj , j 1, . . . , pis chosen such that the tree has the lowest sumof square of residuals. We define two regionsR1 (j, s) {Xj : Xj s} and R2 (j, s) {Xj : Xj s},, andwe wantpair ofP to find the bestP(j, s) to minimize(yi ŷR1 )2 (yi ŷR2 )2 ,i:xi R1 (j,s)is split using the best variable among a subset ofvariables which is randomly chosen. This strategyturns out to perform well. It is robust and preventsover-fitting [3].The algorithm for generating a random forestregression model is as follow [8]:1. Before building each tree, firstly draw several samples from training set by bootstrap resampling method(with replacement) from the original data. Here we denote the number of samples as ntree , indicating thatntree trees will be built.2. For each selected sample, grow a regression decision tree without pruning. At each node of a decisiontree, a random subset of mtry variables is chosen fromthe entire p variables, as the target to be split at thisnode, rather than choosing the best split among all variables. Only one of these mtry variables can be used togenerate the best split rule and corresponding subregions at this node. Thus for the selected training setsin step 1, ntree trees are fully grown and combined intoa random forest.3. Predict new observations by aggregating the predictions of this ntree trees. For regression, the prediction value of random forests for a new data point is theaveraged response of all the trees.Quantile Random ForestThe key difference between quantile regressionforest and random forest is that: in each tree, random forest keeps only the mean response values ofthe observations that fall into each leaf node, andneglects all other information. In contrast, quantileregression forests keeps recording every quantile response values in the leaf node, not just their means,and assesses the conditional distribution based onthis broader and more comprehensive information.In prediction, each observation in test set will gothrough every tree and get a set of prediction valuesof the same size as the number of trees. Inside ofthis set, different quantiles of the prediction valuescan be reached, and we can even set a prediction interval with lower (higher) quantile prediction valueas lower (upper) bound.i:xi R2 (j,s)where ŷR1 (ŷR2 ) represents the mean response overall observations in R1 (j, s) (R2 (j, s)).Finally, it is known that decision trees can causethe over-fitting problem if the tree is too complexand full of details, leading to bad performance onprediction of new observations. It is necessary toset constraints on tree size or to prune the growntree. (Note that in this work, we mainly use decision trees following the random forest method mentioned in next subsection. Therefore the pruning isnot necessary and its introduction is omitted here.)3.1.34. Computation and Analysis of ImpliedVolatilityThe main goal of this section is to demonstrate howto calculate the implied volatility from the datasetintroduced in Section 1.1 and analyze the results.4.1. Calculation ProcessesThanks to Put-Call Parity, we can relate call options prices with put options prices for the sameunderlying asset, strike, date and maturity. Basedon the combination of call and put options we didin Section 1.1, as a matter of choice, we have decided to focus on call options prices and thus the implied volatility mentioned afterwards refers to calloptions.If we define a call option price generated byRandom ForestRandom forest is an ensemble learning method generated by generating many decision trees on selectedtraining samples. Random forests employs randomness each time it selects a different bootstrap sampleof the train set for building each tree, and each node4

Black-Scholes formula with the only unknown parameter implied volatility σ as FBS (σ), it should beequal to the true market value Cmarket of the calloption contract with the same asset, date, maturityand strike price: FBS (σ) Cmarket .We need to invert the Black-Scholes formula, because it is a non-linear function which does not havea closed form solution for implied volatility. Therefore we need to resort to numerical approximationsand we consider the bisection method. Then weneed to estimate:σ̂ arg min[FBS (σ) Cmarket ].σFigure 3: The framework of computation of impliedvolatility.(7)4.2. Analysis of the computed implied volatilityWe start our analysis by showing the impliedvolatilities computed from the contracts in the subset mentioned in Section 1.1, which is named ’Example Set’. Afterwards, we focus on the most reliable result, IVF , and present some plots and descriptive analysis.What is more, in fact, the market price for an option contract Cmarket is not provided in the dataset.We have information concerning bid and ask prices,hereby denoted by Cbid and Cask respectively, asthe range of Cmarket (also known as bid-ask spread[Cbid , Cask ]). The true price of the option, Cmarket ,may not exactly fall in this range, although mostlikely it will match inside. Therefore we proposethe following two alternative methods and show theideas briefly in Figure 2:Method 1: Assume that Cmarket 12 (Cbid Cask ), and compute the implied volatility from thisnew value.Method 2: First, compute the implied volatilityσbid ( σask ) using Cbid (Cask ) respectively as theinput variable each time, and compute the resultingimplied volatility as the average of σbid and σask .4.2.1Analysis based on the exampleThe ’Example Set’ contains 42 contracts of optionsfrom Jan 3, 2014 to March 1, 2014, and the computed implied volatility is shown in Figure 4.Figure 2: The framework of computation Method 1(m1) and Method 2 (m2).Next, two forms of Black-Scholes formula (by using asset price in Equation 1 and by using future price in Equation 3) are utilized to generatethe implied volatility based on two combinations ofthe data sets. The specific formula is chosen depending on whether the price of asset or the futureis involved.In Figure 3 we show the framework of how weorganize the related data sets, and as a result, thereare in total four sets of implied volatility, noted asIVFm1 , IVFm2 , IVSm1 and IVSm2 , where the subscriptF (S) means that we use future price (asset price),and the superscript m1 (m2) means that we haveused method 1 (method 2).Figure 4: Different types of implied volatility based onthe Example set. Implied volatility computed by future price, asset price and constructed price are shownin different colors. The results computed by method 1(denoted as m1 in the legend) are denoted by the notation ’o’, while the ones by method 2 (denoted as m2)are denoted by the notation ’ ’.Note that although the constructed price is de5

duced from future prices, here the constructed price difference between IVF (m1) and IVF (m2) smalleris used as input in the same form of Black-Scholes than 0.001. Here the cut-off point 0.001 is chosenformula as index price.manually by visualization.We can see that the computation methods do notcause much difference. What is more, the impliedvolatilities based on future price and constructedprice are similar as expected. However, the computed implied volatility based on index price is notvery accurate especially when the implied volatilityis larger in the major maturities, due to the factthat index price can variate quite a bit from deduced constructed price at the last 45 minutes. Ina nutshell, the implied volatility based on futureprice is the most accurate and reliable result. Onlywhen the future price is not available, which meansthat the maturity of an option is not one of the major maturities, the implied volatility based on theindex price can be considered as a supplement.4.2.2Analysis on the entire IVFNow it is the time to have a general look of the entire implied volatility based on future price, IVF ‘.Pair plot shown in Figure 5 illustrates the correlation of seven variables as follows: time to maturity, strike, constructed price, IV (m1), IV (ask),IV (bid) and IV (m2). The strike seems to followa normal distribution, while the constructed pricehas two obvious normal distributions combined. Alltypes of implied volatility obtained in this datasetfollow similar distributions (almost normal distribution with right heavy tail) and they are highlycorrelated.Figure 6: Comparison of implied volatility derived byboth methods. Most of points marked black fall aroundthe diagonal line. Only a small set of option contracts(only 0.65% of the whole dataset) is marked red.After analysis, we find that the observations inthe unstable set all belong to deep ’in-the-money’options or deep ’out-of-the-money’ options, whichare hard to be exercised successfully in reality. Wedecide not to focus on the unstable set anymore,and continue our study base on the stable set. Thedataset mentioned afterwards refers to the stableset in the computed implied volatility with futureprice as input, denoted by IVFstable .5. Modeling the Implied VolatilityNow we move forward, trying to fit the dataset concerned by regression models.5.1. The process of choosing a subsetDue to the fact that one of the biggest challenges ofthis work is that the observations are highly mixedFigure 5: Pair plot of relevant variables. Here we and overlapped, clear patterns of implied volatilityplot the seven variables as follows: time to maturity, are hard to attain.strike, constructed price, IV (m1), IV (ask), IV (bid)and IV (m2).Stable and Unstable dataset in IVFBased on the phenomenon shown in Figure 6,we define two datasets as ’Stable’ set (markedblack) and ’Unstable’ set (marked red) accordingto whether or not the observations are insensitiveto the computation methods, i.e. whether the computed implied volatility remains relatively the samewhen the computation method changes. Here we Figure 7: The range of trading dates for every majordefine the meaning of ’relatively the same’ as the maturities.6

Table 3: An overview in terms of the response variable together with explanatory variables. Both notations and descriptions are displayed representatively.Figure 7 shows, respectively, the range of thetrading dates for those contracts which expire at tendistinct major maturities appearing in the dataset.The trading dates seem to concentrate on roughlythree months before each maturity. Note that theinformation of options was gathered from January2, 2014 to October 29, 2015. It is worth to mentionthat the records in the original dataset lacks theinformation from March 21, 2014 to May 16, 2015,due to unclear reasons.According to our knowledge for financial activities, the movement of the market and actions oftraders tend to have periodic variations. It is considered to be a good choice for studying the contracts with maturity at September 18, 2015as our current target, as it firstly contains relatively complete information, holding 17.7% of thetotal contracts. Secondly it is the last maturitybefore the end of the records, supposedly containing the most comprehensive information throughout the period of records.VariablesDescriptionsResponseYTransformed implied volatilityCovariateX1X2X3X4X5StrikeTime to maturitytime to maturity strike1/ time to maturitystrike2sult. In this case, we set two quantiles 0.05 and 0.95to create a 90% prediction interval, then calculatethe response values in 5% quantile and 95% quantile and use them as the boundaries values. Theobservations in test set can be checked later on howcorrect it is for their true values to be contained inthe prediction interval.The general expressions, coefficients and evaluations of three linear models are shown in Table 4.Specifically, the models are generated as the following steps:5.2. Regression ModelingAs mentioned before, the implied volatility is influenced by time to maturity and strike price of theoption. Thus our goal here is to generate alternative parametric models for implied volatility basedon time to maturity and strike price.We first separate the dataset into train set (75%of the entire observations) and test set (25% of theentire observations). Here we come up with bothlinear regression models and random Forest modelto fit the chosen dataset by train set and comparethe results at the end by test set.Response variable (implied volatility) for trainand test are positive-skewed, so we adopt Box-Coxmethod to calibrate the distribution of the impliedvolatility into normalization. The transformationparameter λ 1.59 generated by train set is applied on response variable for both train and testsets to keep the transformed implied volatility staying in the same scale.As the initial model, M odel 1 is generated based ona simple addition of two original explanatory variables,time to maturity and strike. Taken into considerationof the relationship between strike and time to volatility,which seems to follow a non-linear curve. It indicatesthat the interaction of these two explanatory variablesmight be vital, so we combine the interaction term intothe initial model and create M odel 2. Two more termsare added in M odel 3, which are the reciprocal valueof time to maturity and the square of strike. Impliedvolatility seems to be inversely proportional to the timeto maturity. What is more, the phenomenon of ’volatility smile’ is visible, showing the square of strike andimplied volatility are quite related.Table 4: Summary of explanatory variables on threeregression functions. The evaluation of adjusted R2 fortrain and multiple R2 for train and test are shown atthe bottom of each case.EstimateStd. Errort valuePr( t )Model 1: Yi β0 β1 X1,i β2 X2,i i ,5.2.1β0β1β2Linear 3685.63-131.42-57.680.00000.00000.0000Multiple R2 : 0.8186, Adjusted R2 : 0.8185We start now to fit linear models. Table 3 providesan overview in terms of the response variable together with original explanatory variables and onesgenerated afterwards.The coefficient of determination R2 , as an effective performance statistic, is used to measure thegoodness of the fitted models.In fact, there is a more robust way to estimatethe performance of models by prediction intervalsfor linear methods, because there is no requirementfor the assumption of normally distributed residuals. In particular, using prediction interval canexplain better and visualize straightforward the re-Model 2: Yi β0 β1 X1,i β2 X2,i β3 X3,i i 00.00000.0000Multiple R2 : 0.8541, Adjusted R2 : 0.8540Model 3: Yi β0 β1 X1,i β2 X2,i β3 X3,i β4 X4,i β5 X5,i i 00.00000.

Di erent from the theoretical parametric models, such as stochastic volatility model [5] and jump-di usion model [12], non-parametric models based on machine learning techniques, which has less re-strictions in usual cases, have been developed for pricing options [14]. Additionally, quantile method [6], beyond the conditional mean, gives more in-

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stochastic volatility models for option pricing. A notable example of an attempt to find analytic formulas for option prices under stochastic volatility is (Fouque et al., 2000a). Even so, there are no simple formulas for the price of options on stochastic-volatility-driven

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