Equivalent Martingale Measure in Asian Geometric Average Option Pricing
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1 Journal of Mahemaical Finance, 4, 4, ublished Online Augus 4 in SciRes hp://wwwscirporg/journal/jmf hp://dxdoiorg/436/jmf4447 Equivalen Maringale Measure in Asian Geomeric Average Opion ricing Yonggang Zhu School of Science, China hree Gorges Universiy, Yichang, China zygpf@63com Received 9 May 4; revised 8 July 4; acceped 7 July 4 Copyrigh 4 by auhor and Scienific Research ublishing Inc his work is licensed under he Creaive Commons Aribuion Inernaional License (CC BY) hp://creaivecommonsorg/licenses/by/4/ Absrac he general siuaion of he Black-Scholes Opion ricing Model was discussed under he assumpion of he arbirage-free marke, and he pricing of Asian geomeric average opions wih fixed srike price was analyzed a any valid ime Consequenly, he price formula of he Asian geomeric average opions was drawn using he equivalen maringale measure and he significance of he sudy was also indicaed Keywords Asian Geomeric Average Opions, Equivalen Maringale Measure, Black-Scholes Opion ricing Model, Srike rice Inroducion Asian opion, also known as he average price of opions, was one of he derivaives of he sock opions, and was firsly inroduced by he American Bankers rus Company (Bankers rus) in okyo, Japan, on he basis of he lessons learned from he opion implemenaions, such as real opions, virual opions and sock opions I was a kind of exoic opions, which was he mos acive one in financial derivaive marke, wih he difference of he limiaion of he exercise price from he usual sock opion, ha is, is exercise price was he average secondary marke price of he sock price implemened during he curren six monhs In his paper, afer he Black-Scholes [] Opion ricing Model was fully undersood, he pricing of Asian opions was discussed: I was assumed ha he underlying asse price was driven by he geomeric Brownian moion, ha is, lognormal disribuion By using he random variables wih he same Second momen driven by he lognormal disribuion o approximae he arihmeic average of he underlying asse price, he approximae soluion of he arihmeic average price of Asian pu and call opion wih fixed exercise price was obained, and he applicaion of he equivalen maringale measure in he pricing of financial derivaives was furher expanded How o cie his paper: Zhu, YG (4) Equivalen Maringale Measure in Asian Geomeric Average Opion ricing Journal of Mahemaical Finance, 4, hp://dxdoiorg/436/jmf4447
2 Y G Zhu [] Model and Formulas Generally, he sock marke could be described as a probabiliy space wih a -sream, ha is, ( Ω,,) ; I was supposed ha he marke could mee he following condiions: () he marke was an efficien fricionless marke including wo asses: one was he risk-free asses, known as he bonds, whose price process was denoed by B,( > ) ; anoher was he risky asses, called socks, he S, > hey saisfied he following formula separaely: price process was denoed by ( ) ds = µ S d S d W, () B db = rd, () where, µ denoes he expecaion of he yield rae, denoes firm-value process volailiy, denoes ime o expiraion of opion, r denoes he risk-free ineres rae and hey all are consans dw denoes he insananeous incremen of he Brownian moion under he probabiliy measurer a ime ; () Securiy rading is coninuous and here are no ransacions coss or axes; (3) here are no dividends o be payoff during opions being held Ω,, be a probabiliy space and n be an increasing chain of - Definiion Le ( ) fields spanning, which = { ΦΩ, } and ( W, ) algebra of spanning W Le measures saisfy: and se be a sandard Brownian moion be a - W s W d rµ rµ rµ rµ = exp d d exp d s = d Z =, we know ha Z is a maringale since d defined a maringale measure equivalen o measure, where ( ) r µ E exp ds < hen measure is E denoes he expecaion of random variable under probabiliy measure [3] Lemma he dynamics of he share price under probabiliy measure : S = Sexp r W, where S denoes he share price now and W W W W N, rove Because he share price process saisfy he formula (), using Io s heorem, we have: =, ( ) d ln S = r d d W, hen we ge he dynamics of he share price under probabiliy measure : (3) exp S = S r W where W W W W N, r µ Le W = W ds, by definiion and Girsanov s heorem we ge ha he random process W, Ω,, and: = and ( ) ( ) is a Brownian moion on ( ) r µ dw = dw d (5) (4) 35
3 Y G Zhu ( ) ( ), E Z I = A A F (6) A where E ( ) denoes he expecaion of random variable in probabiliy measure and ( ) probabiliy of random variable in measure, ge: hus we have ha under probabiliy measure : he proof is compleed S S Lemma Le X ln =, Y ln =, S S denoes he I is an indicaor funcion of se A Subsiuing (5) ino (3), we A d ln S = r d d W, S = S r W exp µ = r, hen he disribuion funcion of (, ) µ x µ x µ y { X xy, y} = N e N ( d ) Definiion Using a bond as he denominaed uni, exp ( ) was he value process of discouned asses facor, and S = B( ) S( ) 3 Asian Geomeric Average Opions ricing X Y is: B = r s s was he process of he discoun In general, for he coningen claim, he risk-neural pricing principle [4] was obained as followed heorem I was supposed ha he marke was arbirage-free, so ha he value of he process of any asse V ( ) a ime was: ( ( ) ) ( ) = ( ) exp ( ) d V E V r s s F Considering one bearish Asian opion, is reurn a expiraion ime was: V ( ) ( K J ) J = exp ln S d p =, where, Under he condiions of arbirage-free marke and from heorem, he price a ime was: ( exp( d ) ) exp( ( ) d ) ( ) ( ) ( ) ( ) ( ) V = E V r s s F = r s s E K J F p p o ge he specific expression of ( ) p F Made hereby, herefore, V, he key was o obain he disribuion of J under he condiion Y = ln S d, hen J e Y =, and: ln ln d dw S( ) = S s ln S ln S r s s ds s dw s ( ) = ( ) ( ) ( ) ( ) 36
4 Y G Zhu ( ) Y = ln S d ln S d ln S d = = ln S ( ) ( ) ( ) ( ) ( ) ( ) d ln S r s s ds s dw s s dw s d = ln S d ln S dsd dw dw d Wrien as: µ = ln S d ln S dsd dw, hen, η = dw d, µ ( ) e η = J heorem η = dw d was driven by he normal disribuion N (, ), where, ( ) = s ds Noe: I could be deduced from he Lemma heorem 3 I was supposed ha he marke was arbirage-free, hen he price of Asian pu opions V = K J a any valid ime was: p ( ) ( ) where, N( ) rove ds ln K µ µ ln K µ Vp ( ) = e KN e N x = e d x π he proof was compleed ds ln K µ µ ln K µ Vp ( ) = e KN e N (( e ) ) (( e ) ) x= µ ds µ η = e E K F ds x η = e E K F y ds ln K µ µ y ( K ) = e e e dy π ds ln K µ = e KN e Similarly, he price of Asian call opions ( ) ( ) c ln N µ K µ V = J K a any ime could be obained: 37
5 Y G Zhu where, N( ) ds ln K ln K Vc ( ) e e µ N µ KN µ = x = e d x π Deducion For Asian geomeric average opions, he pariy relaionship beween call opion and pu opion was: ds µ Vc( ) Vp( ) = e e K Deducion When he ineres rae r and he volailiy of sock reurns Acknowledgemens µ = ln S r, = ln S r = 3 were consan, here was: Naural Research Foundaion of Educaion Bureau of Hubei rovince of China, NO, Yichang Municipal Science and echnology lanning rojec, NO A-3-8 References [] Black, F and Scholes, M (973) he ricing of Opions Corporae Liabiliies Journal of oliical Economy, 8, hp://dxdoiorg/86/66 [] Zhu YG (9) Applicaions of Equivalen Maringales Model in ricing Warrans roceedings of Conference on Inernaional Insiue of Applied Saisics Sudies, Recen Advance in Saisic Applicaion and Relaed Areas, Sydney, 9- [3] Kong, FL (998) Large Number Laws for Banach Space-Valued Amar Aca Mahemaica Sinica, 4, [4] Marin, B (997) Finacial Calculus Cambridge Universiy ress, Cambridge 38
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