Implied volatility formula of European Power Option Pricing

Transcription

1 Impli volatility fomula of Euopan Pow Option Picing Jingwi Liu * ing hn chool of Mathmatics an ystm cincs, Bihang Univsity, LMIB of th Ministy of Eucation,, Bijing, 009, P.R hina Abstact:W iv th impli volatility stimation fomula in Euopan pow call options picing, wh th payoff functions a in th fom of V T an V T 0 spctivly. Using quaatic Taylo appoimations, W vlop th computing fomula of impli volatility in Euopan pow call option an tn th taitional impli volatility fomula of hals J.oao, t al 996 to gnal pow option picing. An th Mont-alo simulations a also givn. ywos: Euopan Pow option; Impli volatility;taylo sis; Mont-alo simulation;. Intouction In cnt cas, financial ivativs picing attacts much mo attntion in both conomic an statistical fils. o th pactical pupos, impli volatility, which stimats th lvl of financial ivativ s isk, is a most impotant paamt in th Black-chols Euopan option picing mol an Mton s Euopan option picing mol [,]. As volatility is a masu of unctainty of th pic tn fo th futu, many woks ass th poblm, an vlop iffnt statgis. In 976, Latan an Rnlman suggst to us impli volatilitis in financial makts sach[3]. Butl an chacht 986 psnt an stimato of th Black-chols option picing foluma by Taylo sis pansion of th Black-chols fomula [4]. hauhuy 996 popos anoth Taylo pansion mtho to plac th Taylo pansion of Butl an chacht [5]. Using quaatic Taylo appoimations, oao an Mill 996 obtain a clos fomula of impli volatility stimation[6]. Utilizing th thi o Taylo sis pansion, Li 005 vlop a nw clos fomula of impli volatility [6]. An, th simulation sult of [7] show that Li s fomula is significantly btt than th oao Mill fomula. Howv, Li s fomula is also mo compl than oao Mill fomula. Euopan pow option picing is a hot sach fil of financial ivativ option picing [8]. In this pap, w iv a nw fomula to comput Euopan pow option impli volatility in th sach famwok of oao an Mill996[6], an giv clos fomula of impli volatility in th pow option picing famwok of Liu 007 [8]. Th st of th pap is oganiz as follows. In sction, Euopan pow call option picing fomula is intouc. In sction 3, th impli volatility stimation fomula a iv. In sction 4, th Mont-alo simulations a givn. Th conclusion is givn in sction 5.. Euopan pow option. lassical Euopan option picing fomula In th classical isk-nutal makt,, t, P, th pic of an asst t0 t at tim t is suppos to b a gomtic Bownian motion, * osponing autho: jwliu@buaa.u.cn

2 t t Bt t wh is isk-f intst at, is volatility, B t is stana Bownian motion an t Bs, 0 s t. At option piation tim T, payoff valu of th Euopan call option is V T, wh is th stik pic, T is th assts pic at tim T. By th no-abitag thoy, th valu of a taitional Euopan call option pic is stat as T t wh T t T t T t T t, T t o stuying mo convninc, w not T t as th tim th option pis, thn th fomula can b wittn as wh T t,. Euopan pow option picing fomula In o to ominat th comptition an attact mo customs, financial ngins us option thoy an analysis mthos to sign a vaity of options with iffnt chaactistics of nw vaitis. Accoing to th ns of th financial makt, th a many typs of innovativ options, pow options is a nw option typ. Pow option is a simpl non-lina paymnt options. W tak th pow call option fo th stuy, th a two paymnt foms fo -pow 0 option with option piationt an stik pic V 3 T V 4 T w nam fomula 3 as th fist Euopan pow call options, an fomula 4 as th scon Euopan pow call options. Th fist Euopan call pow option picing fomula of fomula 3 is as follows [9], t t wh, An, th scon Euopan pow call option picing fomula bas on fomula 4 is as follows [8], t t 6 5

3 3 wh, Obviously, fomula taks th spcial cas of fomula 5 an fomula 6 with. 3. Impli volatility fomula in Euopan pow option 3. Impli volatility fomula in fist Euopan pow option o th fist Euopan call pow options, w us th pansion of th nomal istibution function into th fomula5, an not,. W obtain / Etning to, w obtain Thn w can gt a quaatic quation about

4 Dnot 0 7, w gt W, omula 7 changs to + W =0 8 inc cofficint W coul not kp intical sign, th cas of lagst oot bcoms vy compl. Whil W 0 0, all al oots of fomula 8 a, as non-ngativ, th lagst oot is only if W 9 W Whil W 0 0, if th two al oots ist, th, as lagst oot is also non-ngativ. It coul b only if W 0 8W Impli volatility fomula in scon Euopan pow option imilaly, fo scon Euopan call pow options, w us th pansion of th nomal istibution function

5 5 into th fomula 6. Dnot,, w can gt / Epaning th fomula with, w can gt Thn, w can also gt a quaatic quation about, 0 Again, w not, thfo W Thn fomula changs to W + =0 Though th vaiabls of an a iffnt fom thos in sction 3., th fomula an fomula 8 kp intical fom. Th sam iscussion is as follows.

6 Whil W 0 0, all al oots of fomula a, as non-ngativ, th lagst oot is only if W W Whil W 0 0, if th two al oots ist, th, as lagst oot is also non-ngativ. It coul b W only if W uthmo, th fomula8 with will b th cosponing fomula of oao an Mill s sult 996 in [6]. 4. umical imulation Lt th oiginal pic of th unlying asst 0 at tim t 0, option piation att, tu tun stana volatility 5%, isk-f intst at 0.00, fo th stik pic, w st 0. 9 Discount,. 0 Paity,. Pmium, an {0.4,0.6,0.8,.0,.,.4,.6,.8,.0} spctivly. Th calculation stps a as follows:. Th unlying asst pic t is simulat accoing to th fomula, wh, th T Bownian motion Bt 0,, 00. o ach, th call option pic t of two kins of Euopan option pow mols a calculat accoing to th fomula 5 an 6 spctivly un th alization of B,..., B... Accoing to th fomula 9 0 an fomula 3 4, w calculat th impli volatility i i, i,, at th tim T T ; W fin th ins to flct th anom compl of ou pimnt. #{ i 0, i,,..., } n, which mans th istnc of oots in fomula 8. L ˆ i, wh L L #{ i 0, i,,..., }, which mans th avag impli i volatility fo on simulation. 6

7 L i =, which masus th ivgnc g of volatility stimation L i in on simulation. 3. Rpat th pimnt fom stp to stp fo M=00 tims, an th avag sults of n,, a pot in th Tabl an Tabl. Tabl. Impli volatility stimation of fist Euopan call pow option n n n Tabl. Impli volatility stimation of scon Euopan call pow option n n n om th abov simulations, w can conclu that th n in flct th succssful 7

8 stimation pobability in Euopan call pow option picing mol, as th compl of stochastic nvionmnt, th n with 0.9 is small than that of an.0 in both two kin of Euopan call pow option. Th accuacy stimation of volatility of pow option pic is slight high in ang of than th cas of. Howv, th is still th cas that th volatility stimation is mo accuat than that with, fo ampl, in Tabl, whn 0.4, 0.9,though its n in is vy low. Th pimntal sults patly suppot th conclusion of [8]. om futh invstigation in ou sach show that if moifying th valu of to guaant 0 as iscuss in [6], th volatility viation g will is. Thfo, w pot th n in to flct th ffctivnss of fomula8 with pow option pic. An, th accuacy of volatility stimation invsly flcts th fitting g with iffnt pow option. om Tabl an Tabl, w can conclu that th ists pow option mol btt than taitional option pic in impli volatility stimation. An, th appciat pow in slction will b ou futh sach intst. 5. onclusion In this pap, with th quaatic Taylo appoimations popos by oao an Mill 996, w iv th clos fomula of impli volatility in two kin of Euopan call pow option picing, th simulation with Mont-alo mtho also show th ffctivnss of ou mol in impli volatility stimation. Th futu wok will focus on th pow option picing slction an appli ou mol to al option ata application. Rfncs [] Black, chols M. Th picing of options an copoat liabilitis [J]. Jounal of Political Economy, 973, 8 3 : [] Mton R. Th Thoy of Rational Option Picinsa[J]. Bll Jounal of conomic managmnt scinc, 973, 4: [3] Latan, H.A., Rnlman R.J.. tana viation of stock pic atios impli by option pmia [J], Jounal of inanc. 976, 3: [4] Butl J.., chacht B. Unbias Estimation of th Black-chols omula [J]. Jounal of inancial Economics, 986, 5 3 : [5] hauhuy M. M. An Appoimatly Unbias Estimato fo Th Thotical Black-chols Euopan all Valuation[J]. Bulltin of Economic Rsach, 989, [6] oao,.j., Mill, J, T.W.. A not on a simpl, accuat fomula to comput impli stana viations [J]. Jounal of Banking &inanc. 996, 0: [7] Li.. A nw fomula fo computing impli volatility[j]. Appli Mathmatics an omputation.005, 70: [8] Liu J.W. Th tatistical Poptis of Paamts an Impli Volatility fom Euopan Pow unction all Option. Application of tatistics an Managmnt [J].007,66: [9] Wang Y.J, Zhou.W., Zhang Y.Th Picing of Euopan Pow Options[J].Jounal of Gansu cinc. 005, 7 :