c 2008 International Press

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1 COMMUN MATH SCI Vol 6, No 3, pp c 28 Inernaional Press THE DERIVATIVES OF ASIAN CALL OPTION PRICES JUNGMIN CHOI AND KYOUNGHEE KIM Absrac The disribuion of ime inegrals of geomeric Brownian moion is no well undersood To price an Asian opion and o obain measures of is dependence on he parameers of ime, srike price, and underlying marke price, i is essenial o have he disribuion of ime inegral of geomeric Brownian moion and i is also required o have a way o manipulae is disribuion We presen inegral forms for key quaniies in he price of Asian opion and is derivaives (dela, gamma, hea, and vega) For example for any a>, Eˆ(A a) + = a+a 2 E where A = R exp(bs s/2)ds and M =exp(b /2) h (a+a ) 1 exp 2M a+a 2 a Key words Asian opion, derivaives of opion prices, geomeric Brownian moion, ime inegral AMS subjec classificaions Primary 91B28, 6J65; Secondary 6G99 i, 1 Inroducion The payoff of an Asian opion depends on he (geomeric or arihmeic) average of prices of a given risky asse over he pre-specified ime inerval Under he Black- Scholes framework, one assumes ha he price process {S, } of he risky asse follows ds =µs d+σs db, S > where µ and σ are given consans and {B, } is a sandard one dimensional Brownian moion In his seing, i is easy o undersand he geomeric average If 1 < 2 hen (σ σ S1 S 2 = S exp ) N 2 ] +(µ 2 )( 1+ 2 ) in disribuion, where N is a sandard normal random variable On he oher hand, he disribuion of an arihmeic average process is no well undersood A coninuous version of he arihmeic average is a ime inegral of a price process Using he Inverse Lapalce Transformaion, Yor 15] proved many ineresing ideniies relaed o he disribuion of geomeric Brownian moion, which gives us deeper undersanding of funcions of geomeric Brownian moion and useful informaion abou heir ime inegrals More deailed research for he relaion beween he ime inegral and an Asian opion was considered in 6] Using he join densiy of exp(b s)dw s,exp(b ), where B, W are independen Brownian moions given in 2], he momen generaing funcion of he ime inegral process was compued in 11] The mehod of changing measures was considered o analyze he properies of he ime inegral process, (see 7, 12, 13, 14]) In 5] he very useful ime reversing propery is used o analyze he ime inegral process Dufresne also provided a cerain form for he densiy funcion of he ime inegral of geomeric Brownian moion However he auhor poined ou Received: December 5, 27; acceped (in revised version): May 1, 28 Communicaed by David Cai Mahemaics Deparmen Florida Sae Universiy Tallahassee, FL 3236, USA (choi@mah fsuedu) Mahemaics Deparmen Florida Sae Universiy Tallahassee, FL 3236, USA (kim@mah fsuedu) 557

2 558 CALL DERIVATIVES he difficulies in using his formula in pracice, especially when he ime inegral is over a shor ime period, due o he slow convergence rae The payoff of a European syle fixed srike Asian opion is given by a funcion of he ime inegral of he price of he risky asse S ( + 1 S d κ), I I where κ is a fixed srike price and I is he pre-specified ime inerval wih he lengh I Under he risk neural measure Q, we may se he price process S given by a SDE, ds =σs db where σ is a consan depending on he risky asse and B is a Q-brownian moion Wihou loss of generaliy, we may also assume ha he ime inerval I =,τ] for some τ > I follows ha he price of a European syle Asian opion is given by (1 τ ) ] + e rτ E Q e r S exp{σb σ 2 /2}d κ τ Since σb =B σ 2 in law, we can rewrie he above quaniy as follows ( ) S σ 2 + τ τσ 2 e rτ E Q e B+(r/σ2 1/2) d σ2 κτ (11) S Noe ha he iniial price of he risky asse is no appeared in he ime inegral To obain he price of Asian opion and he derivaives wih respec o he asse price we need o undersand he following quaniies where A (µ) E (A (µ) a) +], d da E (A (µ) a) +], and d2 da 2 E (A (µ) a) +], (12) = exp(b s+(µ 1/2)s)ds, µ> and a> In his paper we show ha he quaniies in (12) can be expressed in erms of he expeced values of funcions of exponenial Brownian moion We believe ha hese expressions would provide he alernaive approach o simulae he Asian opion price and is greeks The simulaion resuls for Asian opion price and is greeks were considered by several auhors (for example see 1, 3, 4, 1]) Oher ypes of Asian opions are also considered When he srike price depends on he average price, i is called he floaing-srike Asian opion (see, for example, 8]) In 8], Henderson and Wojakowski show he very useful symmeries beween fixed-srike and floaing srike Asian opions They showed ha a he saring poin of he averaging period here exiss an equivalen relaion beween he floaing-srike Asian opion and he fixed srike Asian opion However, once he averaging period has begun, he floaing srike Asian opion can no be re-expressed as a fixed srike opion In Secion 2 we discuss he relaion beween he ime inegral and he exponenial Brownian moion We summarize he resul in 7] and presen he key proposiion In Secion 3 we discuss he price of an Asian opion and is derivaives (dela, gamma, hea, and vega) In Secion 4 we consider he (non-consan) sochasic shor rae Combining our resuls, we derive an inegral form of he call price This formula is given by he expeced value of a differeniable funcion

3 JUNGMIN CHOI AND KYOUNGHEE KIM The ime inegral of exponenial Brownian moion For µ R le us denoe by A (µ) a ime inegral of an exponenial Brownian moion wih drif µ A (µ) := expb s +(µ 1/2)s]ds When µ= we simply use A wihou a superscrip For and y > we se M :=exp(b /2) and M R := y 1 =2 d 1/2A d log(1 y 2 A ) I is no hard o see ha he process R saisfies he following SDE dr =R db 1 2 R2 d, R = y (21) up o an explosion ime τ :=inf{ :A =2/y} By considering a sopping ime τ n =inf{:r n} and a Girsanov densiy process R 1 {<τn}, we define a new measure Q under which B =B R s 1 {s<τn}ds=b 2log(1 y 2 A τ n ) is a sandard Brownian moion Le us define M =exp( B /2) and Ã = M s ds I follows ha and M = M (1 y 2 A τ n ) 2, 1+ y 1 2Ã τ n = 1 y 2 A (22) τ n M τn R τn = (23) y Ã τ n The following proposiion is a simple modificaion of Proposiion 33 in 7] Proposiion 21 If f(x,z) is a Borel measurable funcion and y > hen E f(m,a );A < 2 ] y ( )] =e y M E f( (1+ y 2 A ) 2, A M 1+ y 2 A )exp y A (24) Proof The proof of he proposiion is essenially same as he proof of he Proposiion 33 in 7], so we give a brief skech (for he deailed proof we refer o 7]) For fixed n and y > i is no hard o see ha ( τn exp(r +y)1 {τn>} =exp R s db s 1 τn ) R 2 2 sds saisfies a Novikov condiion Thus we have Ef(M,A );τ n >]=E Q f(m,a )exp( R y)1 τn>]

4 56 CALL DERIVATIVES Since he even {τ n >} is he same as {max s Ms /(y 1 +1/2Ãs)<n}, by leing n he righ hand side becomes E Q f(m,a )exp( R y)] Use (22) and (23) we can rewrie he limi of he righ hand side in erms of M,Ã and R Since we are only ineresed in he quaniy, we remove he ilde and ge (24) Le us discuss cerain ineresing choices of f in Proposiion 21 Simple choices like a consan funcion or power funcions allows us o have various relaions beween a simple maringale M and is ime inegral We consider he ones which direcly relae o he problem of pricing he European syle Asian opion and esimaing he sensiiviies Firs we consider a consan funcion f(x,z) 1 Using a consan funcion, i is easy o see ha he lef hand side of (24) is a probabiliy disribuion of he ime inegral of a geomeric brownian moion By seing a=2/y i follows ha Lemma 22 For a posiive a>, we have ( )] PrA <a]=e 2/a M E exp a/2+1/2a For ν R we have M ν exp(ν/2 ν 2 /2)=exp(νB ν 2 /2) and hus By seing f(x,z)=x ν we ge EM ν exp(ν/2 ν 2 /2);A 2/y]=PrA (ν) 2/y] (25) Lemma 23 For a real number ν and a posiive real a>, we have PrA (ν) a] =a 2ν e 2/a E (a+a (ν) ) 2ν exp ( )] 2exp(B +(ν 1/2)) a+a (ν) (26) g(a) a Fig 21 The densiy funcion of A Le g denoe he probabiliy densiy funcion for A I is known ha he densiy funcion for A is coninuous and posiive Figure 21 is he resul of he simulaion (1 simulaions for each a = 1,2,,2) using he ideniy given in he following Lemma 24 The n-h momens of he ime inegral process A can be compued wih

5 JUNGMIN CHOI AND KYOUNGHEE KIM 561 simple compuaion However, i is known ha A has a heavy ail probabiliy and hus knowing all ineger momens does no give a probabiliy densiy funcion In Figure 21 we can see he heavy ail probabiliy Lemma 24 For a real number a>, he probabiliy densiy funcion g for A saisfies g (a)= 2 a 2 PrA a] 2 a 2 PrA(1) a] Furhermore for a real posiive number ν >, he probabiliy densiy funcion g (ν) saisfies A (ν) g (ν) (a)= 2 a 2 PrA(ν) a] 2 a 2 eν PrA (ν+1) a] ( )] +2νa 2ν 1 e 2/a E (a+a (ν) ) 2ν 1 2exp(B +(ν 1/2)) exp Proof From Lemma 22 we have g (a)= d da PrA <a] = 2 a 2 PrA a] 1 2 e 2/a E M (a/2+1/2a ) 2 exp ( a+a (ν) M a/2+1/2a Since M =exp(b /2) we use M as a Girsanov densiy funcion for he second erm Under he new measure B /2 is a sandard Brownian moion and hus he second erm is equal o he following quaniy: 1 2 e 2/a E ( )] 1 (a/2+1/2a (1) ) exp 2exp(B +/2) 2 a+a (1) Comparing wih Lemma 23, we can see ha he above quaniy is he same as 2/a 2 PrA (1) a] For non-zero ν, we do he same compuaion using Lemma 23 Furhermore we have Proposiion 25 For any a>, b> and > we have ( ) PrM <b,a <a]=e 2/a 2M E exp ;M b(1+ 1 a+a a A ) ] 2 Proof Consider f(x,z)=1{x<b} in equaion (24) By seing a=2/y we ge an inegraion form for he join probabiliy densiy funcion for M and A The simulaion resul of Proposiion 25 for =a is shown on he lef of he following figure and he join disribuion for =1 is shown in he righ The firs quaniy of ineres on he derivaives of Asian opions is he expeced value of he maximum funcion of he ime inegral subraced by a consan Using Proposiion 21 we have he following resul: Theorem 26 For any a> and ν R, we have E (A (ν) a) +] = eν 1 a+e ν/2 ν2/2 a 2ν+2 E ν M ν exp (a+a ) 2ν+1 )] for ( 2M 2 )] a+a a

6 562 CALL DERIVATIVES b a b a 15 2 Fig 22 Join disribuion of M and A In paricular, if ν = we have E (A a) +] = a+a 2 E ( (a+a ) 1 2M exp 2 )] (27) a+a a Proof Le us se f(x,z)=x ν (2/y z) The lef hand side of (24) becomes E2/y A ;A <2/y]=E2/y A ] E(A 2/y) + ] Since 2/y A /(1+y/2A )= 1/(2/y+A ), he second resul comes direcly from (24) Using ideniy (25) we can change he drif erm of he exponenial Brownian moion o ge he firs resul 35 LHS RHS 3 Average of he Asian opion price Number of simulaions Fig 23 Expeced value of max funcion : a=4,=5 The above figure is he simulaion resul of he second ideniy (27) in Theorem 26 We se a=4 and =5 The doed line is obained using he lef hand side of he equaion (27), ha is, we simulae A using Mone Carlo mehods and ake he maximum funcion The solid line is from he righ hand side of equaion (27) Wih only 5 simulaions, we can observe ha he convergence is faser for he solid line,

7 JUNGMIN CHOI AND KYOUNGHEE KIM 563 which was expeced because of he discarded simulaions of he lef hand side (when A is smaller han a) Now le us consider he derivaives of E(A (ν) a) + ] wih respec o a Since we have ha and I follows ha EA (ν) a;a (ν) a]= a PrA (ν) u]du E(A a) + ] = EA a] EA a;;a a] = a+ a E(A (ν) a) + ] = 1 ν (eν 1) a+ d da E(A(ν) a) + ]= 1+PrA (ν) a], and PrA u]du a d 2 PrA (ν) u]du da 2 E(A(ν) a) + ]=g (ν) (a) (28) The following heorems are he direc applicaions of Lemma 22, Lemma 23 and Lemma 24 Theorem 27 The firs and second derivaives of E(A a) + ] wih respec o a are given by following equaions: ( )] d da E(A a) + ] = 1+e 2/a 2exp(B /2) E exp, a+a d 2 da 2 E(A a) + ] = 2 ( )] 2exp(B /2) a 2 e 2/a E exp a+a 1 ( )] M 2 e 2/a E (a/2+1/2a ) 2 exp M a/2+1/2a Theorem 28 The firs and second derivaives of E(A (ν) a) + ] wih respec o a are given by following equaions: ( )] d da E(A(ν) a) + ] = 1+a 2ν e 2/a E d 2 (a+a (ν) ) 2ν exp 2exp(B +(ν 1/2)) a+a (ν) da 2 E(A(ν) a) + ] = 2 a 2 PrA(ν) a] 2 a 2 eν PrA (ν+1) a] ( )] +2νa 2ν 1 e 2/a E (a+a (ν) ) 2ν 1 2exp(B +(ν 1/2)) exp a+a (ν), 3 Sensiiviies Since A saisfies A +s =A +M Ã s for,s> where Ã is an independen copy of A, pricing he fixed srike Asian opion a ime > is essenially idenical as he pricing he opion a he beginning of he averaging period

8 564 CALL DERIVATIVES One imporan aciviy in Financial Markes is managing risk One way o measure he risk in he opion is esimaing he Greek leers such as dela, gamma, hea, ec In his secion we use ideniies obained in Secion 2 o ge he Greek leers of a European Syle Asian opion under he Black-Scholes framework The price process of he risky asse follows under he risk-neural measure ds =σs db +rs d, where B is a sandard Brownian moion under he risk-neural measure, r is a consan ineres rae and σ is a consan volailiy of given asse The expression (11) is equal o he price of an European syle Asian opion wih he expiraion dae τ and he srike price κ : ( ) Call= S σ 2 + τ τσ 2 e rτ E e B+(r/σ2 1/2) d σ2 κτ (31) S Theorem 31 The price of an Asian call a ime = is given by Call = S e rτ 1 τ e rτ a/σ 2 + r /σ 2 )τ/2 ( a + e(r r2 2r/σ2 +2 M r/σ2 σ σ 2 E 2 τ (a+a σ 2 τ) exp 2Mσ 2 τ 2 ) ]], 2r/σ2 +1 a+a σ2 τ a where a= σ2 κτ S, τ is he expiraion dae, κ is he srike price Proof The resul comes direcly from he combinaion of Theorem 26 and Equ (31) The dela is he rae of change of he price of he opion wih respec o he price of he underline asse Thus we have = Call κ e rτ d S S da E(A(r) σ 2 τ a)+ ] (32) a= σ 2 κτ S Using Theorem 26 and Theorem 27 we have Theorem 32 The dela of an Asian call opion a ime = is given by = Call + κ ( ]) e rτ 1 Pr A (r) S S σ 2 τ σ2 κτ S The gamma Γ is he rae of change of he dela wih respec o he price of underlying asse Thus he direc compuaion using Equ (32) and he resuls in Theorem 26, Theorem 27 and Theorem 32 we have Theorem 33 The gamma Γ of an Asian call opion a ime = is given by Γ = σ2 κ 2 ( τ S 3 e rτ g (r) σ 2 ) κτ τσ, 2 S where g (r) is a coninuous probabiliy densiy funcion of A (r) given in Lemma 24

9 JUNGMIN CHOI AND KYOUNGHEE KIM (dela) 75 Γ (gamma) Time o expiraion Time o expiraion Θ (hea) ν (vega) Time o expiraion Time o Expiraion Fig 31 Derivaives vs he ime o expiraion: S = κ = σ = 1 In Figure 31 we use Mone Carlo mehods o ge Greek leers,γ, Θ and ν We ake he risk free rae r= and boh he iniial sock price and he srike price are equal o 1 We also se for simpliciy σ =1 Noice ha we plo greek leers vs ime o expiraion Since we compued greeks a he beginning of he averaging period, ime o expiraion is he same as he lengh of he averaging period In he upper lef figure we can observe ha as ime o expiraion increases, he value of increases The plo of Γ wih respec o ime o expiraion is given in he upper righ figure Since Γ is he firs derivaive of wih respec o he sock prices, i reflecs our observaion in he plo of The lower lef figure is he plo of Θ and he lower righ figure is he plo of ν Since all parameers are consans, he call price saisfies he differenial equaion I follows ha Call +rs Call S σ2 S 2 Call S 2 =rcall Θ +rs σ2 S Γ =rcall where Θ is he rae of change of he price of he opion a ime = wih respec o ime and Call is he price of he call opion a ime Theorem 34 The hea Θ of an Asian call opion a ime = is given by Θ =rcall rs 1 2 σ2 S Γ where Call, and Γ are given in he previous Theorems 31 3

10 566 CALL DERIVATIVES Now le us discuss he vega ν The vega ν is he rae of change of he price of he opion wih respec o he volailiy of he underlying asse The call price of European syle Asian opion is given in (31) Thus we have I follows ha Call σ = 2 σ Call+ 2S 2 σ e rτ+rσ τ/2 E M σ2 τ κ ; A (r) S σ 2 τ > σ2 κτ = 2 σ Call+ 2S 2 σ e rτ+rσ τ/2 E 2S 2 σ e rτ+rσ τ/2 E Call σ = 2 σ Call+ 2S 2 σ e rτ+rσ τ/2 2κ σ e rτ+rσ2 τ/2 M σ 2τ κs ] S M σ2 τ κ ] ;A (r) S σ 2 τ < σ2 κτ S ( 1 E M σ 2 τ;a (r) ( 1 P A (r) σ 2 τ < σ2 κτ S By seing ν =1 in Equ (25) in Lemma 22, we see ha Therefore we have ] 1 E M σ2 τ ;A (r) σ 2 τ < σ2 κτ =P S A (r+1) σ 2 τ > σ2 κτ S ]) σ 2 τ < σ2 κτ S ]) (33) Theorem 35 The vega ν of an asian call opion a ime = is given by ν = 2 σ Call + 2S ] 2 σ e rτ+rσ τ/2 P A (r+1) σ 2 τ > σ2 κτ 2κ σ e rτ+rσ2 τ/2 P S A (r) σ 2 τ > σ2 κτ S Las wo quaniies in he above heorem can be obained using Lemma 23 4 Non-consan ineres raes In his secion we exend our mehod o he sochasic insananeous risk-free ineres rae r() In fuure work we will consider sochasic ineres raes and (or) sochasic volailiy A presen we show ha he call price of he European syle Asian opion can be wrien as he inegral of a differeniable funcion of parameers The money marke accoun g T a ime T is given by g T =e R T r()d Thus he price of he European syle Asian Call opion a ime = is following: ( Call = E Q e R τ 1 τ ) ] + r()d S d κ τ = E Q e R τ r()d ( 1 τ τ ] ] ) ] S e σb σ2 /2+ R + r(s)ds d κ ]

11 JUNGMIN CHOI AND KYOUNGHEE KIM 567 Le us consider a discree model for he shor rae r n+1 r n =θ(r n ) +ρ(r n ) W n, r()=r n for n, n+1 ) (41) where W is a sandard Brownian moion (no necessarily independen o B ), =, r n =r( n ), n =n, = n+1 n, W n =W n+1 W n for n=,1,2, and θ and ρ are real valued funcions From he fac ha Brownian moion has independen incremens, for some <T <τ we have τ e σb σ2 /2+ R r(s)ds d= T +e σbt σ2 T/2+ R T e σb σ2 /2+ R r(s)ds d r(s)ds τ T e σ B σ 2 /2+ R r(s+t)ds d (42) where B is an independen Brownian moion o B Le us se m:=max{n: n <τ} I follows ha using he above Equ (42) we have ( ]] Call = E Q E Q e R τ 1 τ r()d S e σb σ2 /2+ R + d κ) r(s)ds τ F m ( τ m + ]] = E Q E Q C () C 1 () e σ B σ 2 /2+r m d C 2 ()) F m where C ()=e R m r()d e rm(τ m) S /τ C 1 ()=e σbm σ2 m/2+ R m r(s)ds C 2 ()=κτ/s m e σb σ2 /2+ R r(s)ds d and r m, C j (), j =,1,2 are measurable in F m Thus we have ( τ m Call = E Q C ()C 1 ()E Q e σ B σ 2 /2+r m d C ) + ]] 2() C 1 () F m (43) (44) Using Theorem 26 he inner inegral can be rewrien as: M rm/σ2 σ 2 (τ m) φ 1 (r m )+φ(r m )E exp (a+ãσ 2 (τ m)) 2rm/σ2 +1 where a=σ 2 C 2 ()/C 1 (), and ( ) 2 Mσ 2 (τ m) 2 a+ãσ 2 (τ m) a M =exp( B /2), Ã (ν) = φ 1 (r m,)=(e rm(τ m) 1)/r m a/σ 2 exp( B +(ν 1/2)) φ 2 (r m,)=e rm(τ m)/2 r2 m (τ m)/(2σ2) a 2rm/σ2 +2 /σ 2 (45)

12 568 CALL DERIVATIVES I follows ha he call price is given by he following expecaion of a differeniable funcion of parameers such as S,σ,: Theorem 41 The price of an Asian call a ime = is given by Call = EC ()C 1 ()(φ 1 (r m )+φ(r m )M)], where τ is he expiraion dae, κ is he srike price, C j (), j =,1,2 are given in (43), φ j, j =1,2 are given in (45) and M rm/σ2 σ 2 (τ m) M=E exp (a+ãσ 2 (τ m)) 2rm/σ2 +1 ( ) 2 Mσ2 (τ m) 2 a+ãσ 2 (τ m) a Acknowledgemen We wish o hank he referee for helpful commens on his paper, and Kiseop Lee for finding an error in an earlier version REFERENCES 1] J Andreasen, The pricing of discreely sampled Asian and lookback opions: a change of numeraire approach, J Comp Finance, 1, 15 36, ] P Bougerol, Examples de héprèmes locaux sur les groups résolubles, Ann Ins H Poincaré Sec B (NS), 19, , ] M Broadie and P Glasserman, Esimaing securiy price derivaives using simulaion, Managemen Science, 42, , ] SL Chung, M Shackleon and R Wojakowski, Efficien quadraic approximaion of floaing srike Asian opion values, Finance, 24, 49 62, 23 5] D Dufresne, The inegral of geomeric Brownian moion, Adv in Appl Probab, 33, , 21 6] H Geman and M Yor, Asian opions, Bessel processes and perpeuiies, Mah Finance, 2, , ] V Goodman and K Kim: Exponenial maringales and ime inegrals of Brownian moion, preprin, 26, mahpr/ ] V Henderson and R Wojakowski, On he equivalence of floaing and fixed-srike asian opions, J Appl Prob, 39, , 22 9] JC Hull, Opions, Fuures, and Oher Derivaives, 6h Ediion, Prenice Hall, 26 1] N Ju, Pricing Asian and baske opions via Taylor expansion, J Comp Finance, 5, 79 13, 22 11] K Kim, Momen generaing funcion of he inverse of inegral of geomeric Brownian moion, Proc Amer Mah Soc, 132, , 24 12] H Masumoo and M Yor, An analogue of Piman s 2M X heorem for exponenial Wiener funcionals, par I: a ime-inversion approach, Nagoya Mah J, 159, , 2 13] H Masumoo and M Yor, An analogue of Piman s 2M X heorem for exponenial Wiener funcionals, par II: he role of he generalized inverse Gaussian laws, Nagoya Mah J, 162, 65 86, 21 14] H Masumoo and M Yor, A relaionship beween Brownian moions wih opposie drifs via cerain enlargmens of he Brownian filraion, Osaka J Mah 38, ] M Yor, On some exponenial funcionals of Brownian moion, Adv in Appl Probab, 24, , 1992