Option Pricing via Multi-path Autoregressive Monte Carlo Approach
aa r X i v : . [ q -f i n . P R ] J un OPTION PRICING VIA MULTI-PATH AUTOREGRESSIVE MONTE CARLO APPROACH
Wei-Cheng Chen and Wei-Ho Chung
Research Center for Information Technology InnovationAcademia Sinica, Taipei, TaiwanEmail: [email protected], [email protected]
ABSTRACT
The pricing of financial derivatives, which requires massive calcu-lations and close-to-real-time operations under many trading andarbitrage scenarios, were largely infeasible in the past. However,with the advancement of modern computing, the efficiency has sub-stantially improved. In this work, we propose and design a multi-path option pricing approach via autoregression (AR) process andMonte Carlo Simulations (MCS). Our approach learns and incor-porates the price characteristics into AR process, and re-generatesthe price paths for options. We apply our approach to price weeklyoptions underlying Taiwan Stock Exchange Capitalization WeightedStock Index (TAIEX) and compare the results with prior practicedmodels, e.g., Black-Scholes-Merton and Binomial Tree. The resultsshow that our approach is comparable with prior practiced models. Index Terms — Financial derivative pricing, autoregressive pro-cess, monte carlo simulation, short-term option pricing
1. INTRODUCTION
Derivative is a financial product whose value is determined by anunderlying asset, such as stock, currency or commodity. As a pri-mary type of derivative, option is widely traded in financial marketfor a number of purposes, including speculation, hedging, spreadingand creating synthetic positions. According to investors’ personalperspective for future economy, two types of option are traded onthe market, i.e., Call Options and Put Options. A call option is anagreement that gives an investor the right to buy the underlying assetat a specified price within a specific time period. On the other hand,a put option is the right to sell a specified amount of an underlyingasset at a specified price within a period of time. Furthermore, op-tion investors can enter into either long position or short position onboth types of options under different trading strategies. However, themajor problem behind these trading purposes for option investors isto find option premium by using mathematical models.The field of option pricing was first introduced during nineteenthcenturies. Related techniques used in this area can be classified intotwo aspects, namely closed-form solutions and numerical solu-tions. In 1973, Black, Scholes [1] and Merton [2] provided thestate-of-art closed-form solution to European-style options pricingproblem in their seminal studies, known as the Black-Scholes-Merton (BSM) model. Their contributions have subsequently ledto a thriving growth in options trading via a mathematical tractabil-ity and legitimacy for the traders and institutional regulators. Forexample, Johnson and Shanno [3], and Hull and White [4] applied This work was supported in part by the Ministry of Science and Technol-ogy, Taiwan under grant numbers 104-2221-E-001 -008 -MY3, 105-2221-E-001 -009 -MY3, and 106-2218-E-002 -014 -MY4.
Stochastic Volatility model to do option pricing. This model as-sumes that volatility has random processes and fluctuations overtime. Consequently, similar studies include those proposed by E.Stein and J. Stein [5], and Heston [6]. Moreover, Cox [7] derivedthe well-known Constant Elasticity of Variance model which isextended by Schroder [8]. However, closed-form solutions serveas a suitable model if financial instruments have simple structuresand assumptions. On the other hand, numerical methods such asLattice approach, Monte Carlo Simulations (MCS) method, andFinite Difference method were commonly used to price derivativeswith complex structure and products with path-dependent charac-teristic, e.g., Boyle [9], Hull and White [10], and Brennan andSchwartz [11]. Since Boyle demonstrated how to price European-style options using MCS method [12], MCS method has becomeone of the most important techniques in numerical solutions. Forinstance, Boyle, Broadie and Glasserman [13] proved the ability toapply MCS method to evaluate American-style options. Further-more, Longstaff and Schwartz provided Least Squares Monte Carlomethod to deal with option pricing when underlying asset followsa jump-diffusion process [14]. In the past, the major problem ofnumerical solutions was that the pricing results were not exact dueto computational inefficiency; however, with the advancement ofmodern computing and related researches, e.g., Kim and Byun [15],and Wang and Kao [16], this problem has substantially improved.Recently, the growing trend of High Frequency Trading has ledinvestors into seeking profit within a shorter period. In consequence,short-duration options have substantially draw great attention be-cause of their high volatility that provides investors a better opportu-nity to earn extra profit. However, there is a paucity of literature onthe discussion of short-term options pricing, e.g., Andersen, Fusariand Todorov [17]. Hence, we would like to investigate the prob-lem of options pricing when option period is less than or equal to aweek. In order to analyze the problem, we choose weekly options(TXOW) which issued by Taiwan Futures Exchange (TFE) and con-nected with Taiwan Stock Exchange Capitalization Weighted StockIndex (TAIEX). Moreover, TXOW is an European-style option thatcan only be exercised at the option’s expiration date. In our ap-proach, we utilize the correlation of day-by-day price change by de-signing a multi-path simulation algorithm to extract the informationvia autoregression (AR) process and MCS method [18]. Then, wecompare our results with two kinds of commonly used models: (1)Black-Scholes-Merton (BSM) and, (2) Binomial Tree (BT). In con-clusion, the model we built (Multi-path Autoregression Monte CarloApproach, MAMC) shows comparable performance as well as othercommonly practiced models.The rest of this paper is structured as follows: Section 2 de-scribes the datasets used in this paper; Section 3 provides anoverview of MAMC model; Section 4 contains description of bothBSM model and BT model, and the indicators for performance mea-urement; Finally, the simulation results obtained in this work areexamined in Section 5 and the conclusions of this paper are writtenin Section 6.
2. DATA STRUCTURE
In this paper, we use two sets of data that contain information ofoptions and underlying assets. First, we obtained a list of TXOWthat was issued between January 7, 2015 to December 21, 2016.Furthermore, ten call options and ten put options with strike pricerelatively close to prior trading day’s TAIEX closing price, were se-lected on each issue date. These information provide issue date, duedate, strike price, and its market price on each trading day for eachoption. Second, we collect a series of TAIEX’s daily closing pricefrom 2014 to 2016 through Google Finance API. Details of chosenoptions are described in Panel A (2015) and Panel B (2016) of TA-BLE 1.
Table 1 . Details of option dataType MoneynessPanel A: 2015 All Call Put ITM NTM OTMTotal 794 397 397 317 160 317Percent (%) 100 50.0 50.0 39.9 20.2 39.9Type MoneynessPanel B: 2016 All Call Put ITM NTM OTMTotal 798 399 399 319 160 319Percent (%) 100 50.0 50.0 40.0 20.0 40.0* In-the-money (ITM) call/put options are options with strike pricelower/larger than underlying asset’s current price.** Near-the-money (NTM) call/put options are options with neareststrike price to underlying asset’s current price.*** Out-of-the-money (OTM) call/put options are options with strikeprice larger/lower than underlying asset’s current price.
3. MULTI-PATH AUTOREGRESSION ALGORITHM3.1. Model Overview
In this section, we give the details of the construction and processingsteps in our model. First, we consider TAIEX as a discrete-timeasset with closing price S t at time t , and S t − at time t − . In orderto estimate the regularities and patterns, we have to generate serialprice return data with the rate of price changes. Therefore, we definethe change rate y t as the logarithm of the ratio of closing price: y t = ln S t S t − , t = 1 , , , .... (1)Hence, closing price at time t can be represented as S t . That is: S t = S t − ∗ exp( y t ) , t = 1 , , , .... (2)Second, we assume that price return on each day consists oftwo parts: an expected return as a dependent variable affected byprevious price change which generated through AR process and anunexpected return whose value are computed by performing MCS method on a stochastic process. On top of that, we estimate both pa-rameters by using underlying asset’s closing price based on N -dayrolling horizon statistics. After describing the concept in our model,we apply this model to price the premium of TXOW which issues attime t . Before pricing the option, we set a training dataset of price return N days before time t for regression process. Then, we calculatelogarithmic price changes with (1) and define price return series Y N,t as: Y N,t = { y t − i | i = 1 , , , ..., N } . (3)After that, we compute first-order autocorrelation parameter α for entire Y N,t series under Least Square Method (LSM). By usingthe computation results, we can write the expected return y t as: y t = α ∗ y t − , (4)where y t represents expected return on time t . Next, we would like to compute unexpected return which caused byunpredicted factors, such as temporary market shocks and investorirrationality. In previous equations, we defined price return consistsof expected and unexpected return. That is: y t = y t + θ t , t = 1 , , , .... (5)in which y t is the real return for TAIEX at time t . θ t is the un-expected return, which is the difference between real return y t andexpected return y t . Meanwhile, we consider the series of unexpectedreturn over the past N days Θ N,t as: Θ N,t = { θ t − i | i = 1 , , , ..., N } . (6)By using historical volatility of unexpected return, we can estimatefuture price change at time t + 1 as follows: y t +1 = α ∗ y t + ε, (7)where ε is a white noise ∼ N (0 , σ N,t ) .Then, we generate the expected price of underlying asset at theexpiration date of an option by performing (7) T times which equalsto the day remaining before options expiration. That is: S T = S t ∗ T Y t =1 exp( y t ) , (8)where S T is the closing price of underlying asset at the expirationdate of an option. Finally, we perform previous processes through MCS for U timesand subtract option’s strike price from each outcome. Then, we dis-count the value with risk-free rate r and calculate the expectationof all traces with (9) and (10), which generate the premium of calloptions and put options. C = 1 U U X i =1 e − rT max ( S T,i − K, , (9) = 1 U U X i =1 e − rT max ( K − S T,i , , (10)where U represents the number of simulation paths in MCS method. K is the strike price of the option. T is the number of days remain-ing before options expiration. S T,i stands for i -th underlying asset’sexpected closing price after T days. Risk-free rate r is 12-monthscertificate deposit rate announced by Central Bank of Republic ofChina (Taiwan) Moreover, we exclude the problem of dividend pay-ment from this paper.
4. MODEL COMPARISON
In this section, we present the equations of two practiced models,namely Black-Scholes-Merton (BSM) and Binomial Tree (BT),when models are used to price European-style options on a non-dividend-paying index. Equations are given as follows: C BSM = S t N ( d ) − Ke − rT N ( d ) , (11a) P BSM = Ke − rT N ( − d ) − S t N ( − d ) , (11b) d = ln( S t K ) + ( r + σ ) Tσ √ T , (11c) d = d − σ √ T , (11d)where N ( x ) is the cumulative probability distribution functionfor a standardized normal distribution. S t refers to closing price attime t . K stands for exercise price of the option. r is the annualrisk-free rate using 12-months certificate deposit rate announced byCentral Bank of Republic of China (Taiwan). T is option’s annual-ized time to expiry. σ represents annualized volatility. C BSM and P BSM denote the price for call options and put options generatedfrom BSM model.
From different versions of BT model, we choose the version createdby Cox, Ross and Rubinstein, which implies the concept of Risk-Neutral Valuation [19]. By using discrete-time approximation to acontinuous-time geometric Brownian motion, we can calculate op-tion price C BT and P BT as follows: C BT = e − rT ∗ ( p ∗ S t u + (1 − p ) ∗ S t d ) , (12a) P BT = C BT − S + Ke − rT , (12b) p = e rT − du − d , (12c) u = e σ √ T , d = 1 u , (12d)where p represents the probability of the underlying stock to in-crease following geometric Brownian motion with parameters r and σ . C BT and P BT are the price for call options and put options gen-erated from BT model. The other parameters have the same assump-tion as BSM model. Furthermore, annualized volatility used in BSM model and BTmodel is calculated with N days of historical closing price as follows: σ = s P Ni =1 ( y t − i − y ) N − ∗ √ N , (13)where y t stands for the logarithm of the ratio of closing price at time t . y (= N P Ni =1 y t − i ) denotes the average price returns for the past N days. Finally, we assess the pricing ability among three models withfive indicators: Mean error, Standard deviation, NTD Root MeanSquared Error (RMSE) (14), Symmetric Mean Absolute PercentageError (SMAPE) (15), and Absolute Percentage Error (APE) (16).Equations are shown as follows:
RMSE ( NT D ) = vuut Q X i =1 ( O market − O model ) Q , (14)
SMAP E (%) = 2 Q Q X i =1 | O market − O model | O market + O model ∗ , (15) AP E (%) = 1 Q ∗ O market Q X i =1 | O market − O model | ∗ , (16)where Q stands for the total number of options. O market is option’smarket price. O model denoted the option’s closing price calculatedby each model. O market represents the average of all options’ mar-ket price.
5. NUMERICAL RESULTS AND DISCUSSION5.1. Training Period Determination
In financial market, the fluctuations of volatility can be categorizedinto short-, mid-, and long-term periods. These fluctuations werecaused by secular trend, irregular fluctuation or periodical adjust-ment, e.g., seasonal and cycling variation. Furthermore, previousliterature showed that these fluctuations are the reasons that lead tovolatility clustering among various financial time series [20]. In or-der to adjust volatility clustering, investors usually use a specific pe-riod of time, like a month (21 days), a quarter (63 days) or a year(252 days) when calculating annualized volatility. This process pro-vides investors a reasonable and proper assumption of generatingthe volatility that reflects market conditions. Therefore, the annu-alized volatility applied to BSM model and BT model is calculatedfrom daily historical closing prices based on 252-day rolling horizonstatistics and the value of N in MAMC model is set to 252 as well. After determining the training period, we use MAMC model to priceoptions which was issued in 2015 and 2016. These options werealso evaluated via practiced models. The results of option pricing in2015 and 2016 were shown separately in TABLE 3 and TABLE 4.Moreover, we classified options based on their (A) Type (Call/Put), able 2 . Performance measurement in three models (2015), U=50000Mean error STD RMSE (NTD) SMAPE (%) APE (%)Option type MAMC BSM BT MAMC BSM BT MAMC BSM BT MAMC BSM BT MAMC BSM BTAll -5.33 -5.60 -12.71 -0.33 15.00 -11.16 -11.69 -25.10 22.73 -4.84 -5.10 -16.36 -5.98 -5.49 -5.75 -9.19
Table 3 . Performance measurement in three models (2016), U=50000Mean error STD RMSE (NTD) SMAPE (%) APE (%)Option type MAMC BSM BT MAMC BSM BT MAMC BSM BT MAMC BSM BT MAMC BSM BTAll 5.50 3.85 -3.56 -3.53 -5.21 -20.46 -6.46 26.30 -1.76 20.76 -1.57 16.41
6. CONCLUSIONS
In this paper, we provide a compound model for option pricing viaAR process and MCS method. With fine results yielded by MAMCmodel, we prove the feasibility of applying this methodology to priceshort-term options. However, some disadvantages exist in MAMCmodel when pricing call options. These results are the direction ofour future researches. First, we will probe into order selection on ARprocess in order to improve MAMC model’s performance. Second,MAMC model shows worse results when evaluating call options in2016 than in 2015. The reasons of worse outcomes might need fur-ther investigations. Finally, MCS method serves as the basis of ad-vanced Machine Learning models (e.g., ANN and Deep Learning).Using these models in the field of option pricing could be anotherinteresting direction. . REFERENCES [1] F. Black and M. Scholes. The pricing of options and corporateliabilities.
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