Pricing 50ETF in the Way of American Options Based on Least Squares Monte Carlo Simulation

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com 1 School of Sciece, Najig Uiversity of Sciece ad Techology, Najig, People s Republic of Chia Correspodece: Shuai Gao, School of Sciece, Najig Uiversity of Sciece ad Techology, Najig, 10094,

1 Pricig 50ETF i the Way of America Optios Based o Least Squares Mote Carlo Simulatio Shuai Gao 1, Ju Zhao 1 Applied Fiace ad Accoutig Vol., No., August 016 ISSN E-ISSN Published by Redfame Publishig URL: 1 School of Sciece, Najig Uiversity of Sciece ad Techology, Najig, People s Republic of Chia Correspodece: Shuai Gao, School of Sciece, Najig Uiversity of Sciece ad Techology, Najig, 10094, People s Republic of Chia. Received: May 4, 016 Accepted: May 31, 016 Available olie: Jue 3, 016 doi: /afa.vi.1657 URL: Abstract 50ETF appears o the Chiese stock market o 9th February,015, the cotracts are Europea Optios ad the optios are priced by B-S model.50etf is the oly oe optio that ca be traded, there are o America Optios i Chiese stock market. This paper studies 50ETF pricig aalysis i accordace with the way of America Optio. We use Least Squares Mote Carlo Simulatio to price 50ETF ad aalyze them, give the umerical results by matlab program. This issue is worth studyig, because the paper studies 50ETF, ad price it i the way of America Optios, we try to employ Mote Carlo Simulatio to solve this problem i chia ad the results of the paper ca erich the optio products i the stock market of Chia. Keywords: 50etf, America optio, least squares mote carlo simulatio 1 Itroductio The egotiable securities of SSE 50 exchage-traded Idex securities ivestmet fuds are called 50ETF, Stock code:510050,huaxia Fud Maagemet compay is the fud maager, it is a fiacial product that based o a certai type of uderlyig assets.50etf Optio, its uderlyig assets are Huaxia 50ETF(510050).50ETF appears o the market o 9th February,015 ad it is the oly oe optio that ca be traded i Chiese stock market.50etf is the oly oe optio that ca be traded, there are o America Optios i Chiese stock market, i this paper, we study 50ETF,we use Least Squares Mote Carlo Simulatio to price 50ETF, the price is valuable ad the results ca give some referece to the Chiese stock market, so the researchers ca desig more fiacial derivatives ad it is also meaigful for the developmet of optios i Chia. The earliest optio pricig theory was proposed by Bachelier (1990),the 65 years later, Samuelso (1965) recosidered the problem of optio pricig, he proposed the uderlyig assets follow Gometric Browia Motio.The Black-Scholes (1973) ad Merto (1973) studied risk eutral pricig, the famous model B-S model was created by them. The Sharp (1978) studied perivative securities pricig, accordig to the self-fiacig strategy, Biomial tree method ca price optios very well. Wiggis (1983) geeralized the Black-Scholes-Merto model. The models were proved price Europea Optios very well, America optios pricig was uder cosideratio all the time. Util 1990s,Boyle, Broadie, ad Glasserma (1997) give the Mote Carlo method. Logstaff ad Schwartz (001) proposed the Least Squares Mote Carlo simulatio, so the optio pricig theory has developed very well. I practice, Mote Carlo simulatio method is very suitable for pricig the optios that are path-depedet or multivariate related, because the traditioal Mote Carlo simulatio uses forward solutio method, at this poit, we ca ot calculate the expected retur of the optios at every momet, the optios are cotiued to hold. So the proceeds of the optios that exercise immediately ad the expected retur of the optios that cotiue to hold ca ot be compared, ad we caot decide whe to exercise the optios. Therefore, at the begiig, Mote Carlo simulatio method is cosidered that ca ot be used for pricig America optios, with the developmet of mathematical fiace, Mote Carlo simulatio has developed, there appears some Mote Carlo simulatio algorithm simulatio of America Optio pricig. Sice the 1990s, dyamic programmig priciple ad digitmap aalysis are leaded ito Mote Carlo simulatios, the America optios are successfully priced. The rest of the paper is structured as follows. We itroduce some steps of Least Squares Mote Carlo simulate optio 71

2 pricig i Sectio.I Sectio 3,we price 50ETF i the way of America Optios ad give aalysis of the results simulated by matlab program.. Least Squares Mote Carlo Simulatio As a result of the itroductio of ew securities ad ew sources of fiacig, various iovative techiques have bee used by researchers ad traders i order to determie the price of the securities (Asari & Riasi, 016; Riasi, 015). However, oe of the most effective methods for pricig the securities is least squares Mote Carlo simulatio. Whe pricig America Optios, it is ecessary that make a choice whether to cotiue to hold optios or to exercise the optios o the lead time. The resultig value of exercisig optio is usually easy to get, some researchers, icludig Logstaff ad Schwartz, i 001,they proposed to choose the best match betwee the value of optio that cotiue to hold ad certai related variables by the Least Squares method. The Mote Carlo method has become the stadard method for pricig America optios, its basic priciple: I a limited discrete time poit, simulate the data o sample path at each time accordig to the uderlyig asset price. Obtai the expected retur of the optios that cotiue to hold by Least Squares method, comparig it ad the proceeds of the optios that exercise immediately, if the former is less tha the latter, exercise the optios immediately, else if the former is bigger tha the latter, cotiue to hold the optios. Some steps of Least Squares Mote Carlo simulate optio pricig Firstly, geerate the sample price path of the uderlyig asset S( t T) S( t) e ( ) T ti T (1) Secodly, solve the optio iversely from the expiratio date,obtai the optimal executio time ad optio earigs o each path, at time i,calculate the optimal executio time ad optio earigs o each path.for example,put optio,its itrisic value o sample path is j j j I ( S ) max( X S,0) i i i () amog them, S is the asset prices of sample path j, i is the executive time, X is executive price. Due to America optio ca be exercised i advace, whe choose the optimal executio time, it is ecessary to compare the expected retur of the optios that cotiue to hold ad the istat earigs of the optios that exercise i advace, choose the bigger oe f ( S ) max( I ( S ), E( e f ( S ) S )) j j j j r t j j j i i i i i (3) Here, the expected retur is the earigs of the optios that cotiue to hold o the cotidio S, regressio method ca give a sample polyomial about the curret price of the uderlyig asset S,ormally, quadratic polyomial, ad use it to calculate the expected retur approximately, that is E( e r t f j 1( j 1) j ) ( j ) j i Si Si a Si bsi c (4) S is regarded as the value of the horizotal axis, let the future earigs o the path be the value of the vertical axis, the ca get the coefficiet aalogously about the above equatio by Least Squares. Startig from the due date, obtai the optimal executio time ad optio earigs o each path. O the due date, for the America put optios, the optios are exercised whe the optios become a premium, the earigs are max( X S,0),time N is the due data, ad cotiue to aalyze the time N 1,if the optio becomes a premium o the sample path, compare the expected retur of the optios that cotiue to hold up to the due date ad the proceeds of the optios that exercise immediately, if the former is less tha the latter, exercise the optios immediately, else if the former is bigger tha the latter, cotiue to hold the optios. The optio ca be exercised whe it becomes a premium, it is a prerequisite, at time N 1,just aalyze the data o the sample path, the optio is a premium, the expected retur of the optios that cotiue to hold ca be got by Least Squares aalogously E[ y ] e max( X S,0) a( S ) bs c j r t j j j N 1 N N 1 N 1 (5) 7

3 Therefore, whe decidig whether to exercise the optio i advace, oly compare the expected retur of the optios that cotiue to hold ad its itrisic value at that time. Similarly, the expected retur of the optios that cotiue to hold ca also be obtaied at the other time. The optios o the sample path ca be exercised at the oly time or ot be exercised. From the iitial state, cosider the time N, if the optimal executio time is time N,at time N 1,if exercise the optio, so the optimal executio time is time N 1, else the optimal executio time will ot be chaged, repeat the steps above. Due to every path has oly oe optimal executio time, fially, the expected retur of the optios will be got. Thirdly, the expected retur of the optios o every sample path ca be discouted by the risk-free rate, the take the mea, so the expected retur of the optios will be got, after simulatig the optio price may times, the optimal executio time ad optio earigs o each path ca be obtaied, fially, take the mea, so the America optio price ca be simulatly by Least Squares Mote Carlo Simulatio. 3. The Study of 50ETF by America Optios Pricig 3.1 Backgroud The egotiable securities of SSE 50 exchage-traded Idex securities ivestmet fuds are called 50ETF, Stock code: , Huaxia Fud Maagemet compay is the fud maager, the optio just like the commo optio we kow, it is a fiacial product based o a certai type of uderlyig assets.50etf Optio, its uderlyig assets are Huaxia 50ETF(510050),oe cotract of 50ETF optio represets Huaxia 50ETF.50ETF appears o the market o 9th February, 015, call optio; put optio, four expiry times, five prices that the optio be exercise, 40 cotracts i total. The expiry time is March, April, Jue ad September. The closig price of 50ETF is.91 yua o 6th February, 015. Accordig to the spacig of exercisig the optios, five strike price of the optios are.0 yua,.5 yua,.30 yua,.35 yua ad.40 yua. 3. Data preparatio(employ Mote Carlo Simulatio to price 50ETF) S:.91 yua; K:.0 yua,.5 yua,.30 yua,.35 yua ad.40 yua; T:if Jue or September is chose as the expiry time,the the time will be four moths or seve moths, so T will be years or years; r: risk-free rate,here choose the three-year treasury bods iterest rates that issues i 015 i Chia, the value is 4.9%; : the volatility of the uderlyig asset,use historical data to estimate volatility, the observatio data of the stock price is usually withi a fixed time, defiitio i 1: the times of the observatio data; S :the stock prices that eded at the i-th time zoe, i 0,1,..., ; :the legth of time ad its uit: year let Si ui l( ), i 1,... S u i is the estimated value of stadard deviatio, s is s 1 1 ( u i u ) or s 1 1 ui ( ui ),u is the mea of u i. is the stadard deviatio of i 1 ( 1) u,so the variable s is the estimated value of,ad itself ca be estimated as,here s. is the umber of stock tradig 73

4 days withi a year, geerally, set the umber of days of tradig days per year 5. The article chooses the data whose uderlyig asset is Huaxia 50ETF (the historical data is from 5th Jauary,015 to 15th Demcember,015),use the above model, the estimated value of the volatility is Pricig America-style Optios by Mote Carlo Simulatio Pricig with the expiry time is April ad the strike price is.30 yua For example, the America Put Optio are priced by Least Squares Mote Carlo model,april is chose as the expiry time, the strike price is.30 yua. Usig the Least Squares Mote Carlo model, the accuracy of the results deped o the times that simulated,the more times, the results will be more precise, simulate the optio 10 times ad 1500 paths, the results are as follows Table 1.The value of America put optio ad call optio Frequecy Put optio Call optio Average The results are simulated by the Least Squares Mote Carlo,they are radom,simulate the optio 10 times ad take the average of the results,the value of America put optio is yua ad the value of America call optio is yua Pricig America-style Optios with all expiry time ad strike prices 50ETF Optio,four expiry time,five strike prices,study the America Put ad Call Optios,forty prices of the optio will be obtaied by matlab program.just like.3.1,each price is simulated may times. The forty results are simulated as follows: Table. The forty prices of the put ad call optios obtaied by matlab program Time March April Jue September Strike Price (Yua) Put Call Put Call Put Call Put Call The forty prices of the optios are obtaied by matlab program. To get precise results, each price is simulated more tha 74

10 times by matlab program, so the price is valuable ad the results ca give some referece to the Chiese stock market, so the researchers ca desig more fiacial derivatives ad it is also meaigful for

The value of 50ETF optios chage with the stike price K,March is chose as the expiry time The Value Of 50ETF America Call ad Put Optio Put Optio Call Optio 0.35 0.3 0.5 0. 0.15 0.1 0.

30 yua For the America Optio, five factors affect the value of the optio: the price of the uderlyig asset, the strike price, the expiry time, the volatility of the uderlyig asset ad risk-free rate.

5 10 times by matlab program, so the price is valuable ad the results ca give some referece to the Chiese stock market, so the researchers ca desig more fiacial derivatives ad it is also meaigful for the developmet of optios i Chia Aalysis ad commets The Value of 50ETF America Call ad Put Optio Put Optio Call Optio Chart 1. The value of 50ETF optios chage with the stike price K,March is chose as the expiry time The Value Of 50ETF America Call ad Put Optio Put Optio Call Optio March April Jue September Chart. The value of 50ETF optios chage with the expiry time T, the stike price is.30 yua For the America Optio, five factors affect the value of the optio: the price of the uderlyig asset, the strike price, the expiry time, the volatility of the uderlyig asset ad risk-free rate. Seeig from the forty results above, uder the same coditios, with the icrease of strike price, the value of the America Put Optio will icrease while the value of the America Call Optio will decrease, the Chart 1 ca explai this pheomeo; similarly, uder the same coditios, with the icrease of time, both of the value of the America Put Optio ad Call Optio will icrease, the Chart ca explai this pheomeo. I fact, may methods ca price optios, Mote Carlo simulatio method is very suitable for pricig optios that are path-depedet or multivariate related. If pricig America-style Optios By Mote Carlo Simulatio, the results may be radom, it is ecessary to simulate may times to make sure that the results precise. So the Mote Carlo Simulatio ca be improved to cut dow radomess, it eeds further study. 75

6 4. Coclusio This paper studies 50ETF pricig aalysis i accordace with the way of America Optio. We use Least Squares Mote Carlo Simulatio to price 50ETF ad aalyze them, give the umerical results by matlab program. Alouthgh may methods ca price optios, geerally, Mote Carlo simulatio method is widely used ad is very suitable for pricig America optios. I Chiese stock market, 50ETF is the oly oe optio that ca be traded, there are o other optios, the optio is Europea Optio. Except Chia, it is also a problem i some other stock markets, the paper studies 50ETF, ad price it i the way of America Optios, the results of the paper ca give some guidace ad help to researchers, it ca guide them to desig ew fiacial derivatives ad it is also beeficial for developmet of optios i Chia. Refereces Asari, A., & Riasi, A. (016). A Ivestigatio of Factors Affectig Brad Advertisig Success ad Effectiveess, Iteratioal Busiess Research, 9(4), Berard, C., & BoyleMote, P. P. (011). Carlo Methods for Pricig Discrete Parisia Optios, 17(3), Boyle, P. P. (1977). Optios: A Mote Carlo approach, Joural of Fiacial Ecoomics, 4, Boyle, P., Broadie, M., & Glasserma, P. (1997). Mote Carlo methods for security pricig, Joural of Ecoomic Dyamics & Cotrol, 1(8-9), Cescato,C. D., & Lemgruber, E. F. (011). Valuatio of America iterest rate optios by the Least-Squares Mote Carlo method, Pesquisa Operacioal, 31(3), Clémet, E., Lamberto, D., & Protter, P. (00). A Aalysis of a Least Squares Regressio Method for America Optio Pricig, Fiace & Stochastics, 6, Corporatio, H. P. (014). Radomized Biomial Tree ad Pricig of America-Style Optios, 1, 1-6. Jia, Q. (010). Pricig America Optios usig Mote Carlo Methods, Departmet of Mathematics. Liu, Q. (008). Pricig America Optios by Caoical Least-Squares Mote Carlo, Joural of Futures Markets, 30(), Logstaff, F. A. (001). Valuig America Optios by Simulatio: A Simple Least-Squares Approach, Review of Fiacial Studies, 14, Riasi, A. (015). Competitive Advatages of Shadow Bakig Idustry: A Aalysis Usig Porter Diamod Model, Busiess Maagemet ad Strategy, 6(), Wu, Z. (01). Pricig America Optios usig Mote Carlo Method, Numerical Aalysis. Yeug, J. A. (010). Path-Depedet Optios Pricig: A Quasi Mote Carlo Simulatio Approach with MATLAB, Ssr Electroic Joural. Zhao, Q., Liu, G., & Gu, G. (013). Variace Reductio Techiques of Importace Samplig Mote Carlo Methods for Pricig Optios, Joural of Mathematical Fiace, 4, This work is licesed uder a Creative Commos Attributio 3.0 Licese. 76

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