Available online a www.sciencedirec.com ysems Engineering Procedia (11) 3 Valuaion of European Currency Opions in inancial Engineering huai Wang 1, Linyi Qian Eas China Normal Universiy, hanghai, 41, P China Absrac In his paper, we ry o solve he valuaion of currency opion in financial engineering. We use a generalized jump-diffusion sysem o describe he spo oreign Exchange (X) rae and apply regime swiching model o describe he domesic and foreign risk-free ineres rae and he appreciaion rae and he volailiy. And he regime swiching model is based on a coninuous ime finie sae Markov process. Under he minimal maringale measure, we obain a sysem of parial-differenial-inegral-equaions saisfied by he European currency opion prices. Our model provides he flexibiliy o model differen kinds of dynamics in X rae. A las, we presen a simulaion of opion pricing wih he special case of compound Poisson jump and we can find he effecs of he parameers on he prices. 11 Published by Elsevier B.V. 11 Published by Elsevier Ld. elecion and peer-review under responsibiliy of esheng ash Wu Keywords: Jump-diffusion sysem; Minimal maringale measure; Currency Opion pricing;egime swiching; inancial Engeering 1. Inroducion Currency opion is an imporan derivaive ool in foreign exchange marke and is pricing and hedging have also been a fundamenal subjec in financial engineering. Afer he seminal work of Black and choles [1] and Meron [] on European sock opion pricing, here were [3] and [4] as he pioneering works in his area. In heir models, hey applied he Geomeric Brownian Moion wih consan appreciaion rae and he volailiy and hen valuaed he opion under risk-neural measure. However, many following empirical sudies have defied he assumpion of Geomeric Brownian Moion of dynamics of he spo exchange rae and hey ask for more realisic models. As a resul, a lo of papers are devoed o he modificaion of he basic Geomeric Brownian Moion. irsly, in realiy he parameers of he spo X dynamics may depend on he sae of he economy. In his case, many sudies have inroduced he Markov egime wiching Model denoed by a finie sae Markov chain o capure he realiies. The swiching of economy saes can be caused by he srucural breaks of he economy siuaions and business cycles. The early work on he inroducion of regime swiching model includes Quand [5] and Goldfeld and Quand [6]. Ever since Hamilon [7] inroduced he regime swiching ime series o economics, he regime-swiching models have obained grea developmen in financial research, even in acuarial science. econdly, some rare evens (major poliical changes, a naural disaser in a major economy, or he release of unexpeced economic daa) may resul in he brusque variaions in spo X rae. ince he variaions canno be described by he usual diffusion models, we need o consider he jump diffusion model o capure he jump in raes. [8] applied he jump diffusion model for spo X rae. The improvemen of he abiliy of he model o explain he realiy is accompanied wih he difficuly o evaluae Corresponding auhor. Tel.: 861347633. E-mail address: wangshuaiecnu@gmail.com. 11-3819 11 Published by Elsevier B.V. doi:1.116/j.sepro.11.1.8
huai Wang and Linyi Qian / ysems Engineering Procedia (11) 3 3 he derivaives. An imporan propery of regime swiching model or jump diffusion model or heir combinaion is he incompleeness of he financial marke. I means ha he coningen claim canno be hedged perfecly. Harrison and Kreps [9], Harrison and Pliska [1, 11] found he relaionship beween he absence of arbirage opporuniy and he exisence of he equivalen maringale measure. They showed ha here would be infiniely many equivalen maringale measures and he problem was how o selec a consisen measure from ha se of measures. öllmer and ondermann [1], öllmer and chweizer [13], chweizer [14] and avis [15] provided differen mehods o choose a pricing measure on basis of differen opimizaion crieria. In his paper, we follow risk minimizing mehod proposed by öllmer and chweizer [13]. In an incomplee marke, for non-aainable claims, i's desirable o find he sraegies wih he minimizaion of fuure risk exposure a any ime in he fuure. öllmer and ondermann [1] firs inroduced he risk minimizing hedging o address he pricing of he opions in an incomplee marke and his was furher sudied by öllmer and chweizer [13] in which hey provided he öllmer and chweizer decomposiion of he discouned asse price. In his paper, we consider he pricing of he European currency opions when he dynamics of spo X rae are driven by a generalized Markov-modulaed jump-diffusion model using he risk minimizing mehods o choose he equivalen maringale measure. Our work is differen from he previous exising sudies. [16] sudied he currency opion pricing under wo-facor Markov-modulaed sochasic volailiy model. I used he Esscher Transform o choose he equivalen maringale measure which was differen from our minimal maringale measure. And here was no jump par in heir model. [17] presened he formula of European currency opion price when he spo foreign exchange rae followed he fracional Brownian moion wih jumps. [18] sudied he European currency opion valuaion wih spo foreign exchange rae described by a diffusion model wih compound Poisson jumps and regime swiching. They also used he Esscher Transform o selec he equivalen maringale measure. urhermore, since we consider he opion pricing under he generalized form of he dynamics of spo X rae, our work will provide he flexibiliy o he marke paricipans o choose he special models hey need o describe spo X rae in markes. And under he minimal maringale measure, we derive he sysem of coupled parial differenial inegral equaions saisfied by he opion prices. A las, we do he numerical experimen of opion prices when we choose he compound Poisson process as he concree jump par in our original model and we consider he qualiaive behavior of he price wih differen parameers. This paper is srucured as follows. ecion presens he general formula of spo X rae and he derivaion of minimal maringale measure. ecion 3 gives he sysem of coupled parial differenial inegral equaions. In ecion 4, we discuss he simulaion resuls condiioned on he concree jump process. A las, ecion 5 concludes he whole paper.. The Model In his secion we inroduce wo basic elemens of he model: he spo X rae and he minimal maringale measure. We consider only wo currencies in X marke, called he domesic currency and he foreign currency. Le denoe he spo X rae process, represening he domesic price of one uni of foreign currency a ime. Then we le he dynamic of follow he generalized jump-diffusion regime swiching model. We also assume ha he insananeous marke ineres raes in domesic and foreign markes are conrolled by a coninuous-ime finie-sae Markov chain. And hen we can inroduce he dynamics of he spo X rae. ix a complee probabiliy space (,, P ), where P is a real-world probabiliy. T < denoes a fixed, finie ime horizon. Le X : = { X } T be a coninuous-ime Markov chain on (,, P ) aking values in a finie sae space :={1,,, n}. We inerpre he sae of X as he sae of he economy. We suppose ha he risk-free ineres rae, appreciaion rae and volailiy of he risky asse depend on he sae of he economy. The Markov chain X is governed by P ( X = j X = i ) = o ( ) i j, ij (1) n where ij, i j ; and ii = ij. Le = { ij} denoe he generaing Q -marix of he chain. I would be convenien o wrie j i, (1) j = 1 in an equivalen way where X } is represened as a sochasic inegral wih {
4 huai Wang and Linyi Qian / ysems Engineering Procedia (11) 3 respec o a Poisson random measure (see Ghosh e al. [19]). or i, j, i j, le ij be consecuive (w.r.. o lexicographic ordering on ) lef closed righ open inervals of he real line, each having lengh ij. efine a funcion h : by j i h ( i, z ) = if z ij oherwise. () Then d X = h ( X, z ) p ( d, d z ), (3) where p ( d, d z ) is a Poisson random measure wih inensiy d m ( d z), where m ( d z) is he Lebegue measure on. We use p ( d, d z ) o denoe he corresponding compensaed maringale measure. Our X marke has hree underlying elemens: he domesic bond B = ( B ) T, he foreign bond B = ( B ) T and he spo X rae = ( ) T, which are radable coninuously. Le r : [, ), r : [,) denoe he domesic and foreign risk-free ineres raes; ha is, if he regime X = i, hen he insananeous ineres raes in domesic marke and foreign marke are r ( i) and r ( i) separaely. Thus he ineres rae process r( X ) is also an irreducible Markov chain aking values in {r(1),r(),, r ( n ) }. In order o simplify he noaion, we wrie r, and for r ( X ), ( X ), ( X ). o he price processes of he domesic and foreign bonds are given by he P dynamic d B d B = r = r B B d d (4) (5) We assume ha he spo X rae is given by d = d dw f ( y ) N ( d y, d ), (6) where : describes he drif coefficien and : (, ) is he volailiy, N(.,.) is a Poisson random measure wih inensiy measure v ( d y ) d and f ( ) is any suiable funcion f :. By a direc applicaion of I ô 's formula, we can obain = 1 exp{ { s s} d s sd W ( ) l n ( f ( y ) 1) N (d y,ds)}. (7) s Le denoe he spo X rae a ime discouned by he curren value of domesic bond. Namely, B = B = exp ( r u r u ) du (8) The dynamic of he discouned spo X rae saisfies d = ( ( r r ) ) d d W f ( y ) N ( d y, d ). has he following decomposiion
huai Wang and Linyi Qian / ysems Engineering Procedia (11) 3 5 = M A, (9) where M = u u d W u u f ( y ) N ( d y, d u ), u u u u A = ( ( r r ) f( yv ) (d y)) du, and N ( d y, d u ) = N ( d y, d u ) v ( d y ) d u. Then M = ( M ) T is a square-inegrable maringale for which M =, and A = ( A ) T is a predicable process of finie variaion for which A = ue o Lemma 3.1 of chweizer [], we have he following Lemma. Lemma.1 The minimal maringale measure Q is given by d Q T 1 T = exp{ u u u dw u u T u u du d P ln(1 u u f ( y )) N (dy,du ) T u u f ( y ) v ( d y ) d u } (1) where u ( ru ru) f( yv )( d) y u = [ f ( yv ) (dy)] u u Then W ˆ ( ) = W ( ) u u u du is a Q Wiener process, and he L e vy measure under he measure Q is v ˆ ( d y, d u ) = ( 1 u u f ( y ) ) v ( d y ). 3. European Opion Pricing In his secion, we will presen he valuaion of European currency opion under he minimal maringale measure and ge he sysem of parial differenial inegral equaions. The payoff is g ( T ) a mauriy T. Then he condiional price and he discouned condiional price given = and X = X are given by V (, T,, X ) = E Q [ e T ( r r u u ) d u g ( T ) ], and V (, T,, X ) = e ( r r u u ) d u V (, T,, X ).
6 huai Wang and Linyi Qian / ysems Engineering Procedia (11) 3 By I ô 's differeniaion rule, B B V V = ( r r ) VT (,,, X)d d B B d (, T,, X ) B V c 1 B V c d d B B B B [ VT (,, f( y), X ) V (, T,, X ) ] N ( d, d y ) [ (,,, (, )) V (, T,, X )] p (d,dz ). B VT X hx z B (11) ince V,, N ( d, d y ) v ˆ( d y ) d, p ( d, d z ) are maringales under Q, all bounded variaion erm in (11) mus be idenical o zero. Then he price V saisfies he following P..I.E.: V V ( r r) V ( ) 1 V [ (,, ( ), ) V T f y X V (, T,, X ) ] v ˆ( d, d y ) [ V (, T,, X h ( X, z ) ) V (, T,, X )] m (d,dz ) =. (1) wih erminal condiion V ( T, T,, i ) = V ( T ), i = 1,,, n. Le V i = V (, T,, i),for each i = 1,,, n and V = ( V1, V,, V N ). Then he n -coupled P..I.E.s saisfied by V are V V ( ri ri ) V ( i i ) 1 V [ V (, T, f ( y ), i ) i V (, T,, i ) ] v ˆ( d, d y ) [ V (, T,, i h ( i, z )) (, T,, i )] m (d,dz ) =. V wih erminal condiion V ( T, T,, i ) = V ( T ), i = 1,,, n. 4. imulaion In his secion, we perform he numerical experimen o sudy he European call opion prices wih differen parameers in he general model wih he special jump ype. Le f ( y ) = e x p ( y ) 1 and v( dy) = GY ( dy) where is he inensiy parameer of N( ) and G Y ( y) is any fixed disribuion funcion. Then from (7) we have = = 1 exp{ { s s} d s sd W ( ) y N ( d y, d s ) } s 1 exp{ { s s} d s sd W ( ) Y i} (14) s N ( ) i= (13)
huai Wang and Linyi Qian / ysems Engineering Procedia (11) 3 7 where Y i I. I.. G ( ) Then (7) becomes he usual compound Poisson jump diffusion process wih he jump size Y i following any given disribuion. In his secion, we selec normal disribuion o simulae he jump size. irsly,we use he Euler forward scheme o discreize he log reurn process in (4). We divide he inerval [, T ] ino JT subinervals of equal lengh = 1 / J,where [( j 1), j ]( j = 1,,, JT) is he j h inerval. Then we can ge he Euler forward discreizaion of log reurn of asse price. In his case, we suppose ha he Markov chain X has wo saes,"good" sae and "Bad" sae, denoed by number 1 and separaely, and he ransiion probabiliies of he wo sae are 11 =.7, 1 =.3, 1 =., =.8 Then we consider some special values for oher parameers. Le he lengh of subinerval is = 1/ 5. 1 = 1 uppose r.6, r =. when domesic economy is "Good" and r., r =.6 when domesic economy is "Bad"; =.5 and =. ; =.1 and =.3. The mean and variance of he normal 1 1 = disribuion of jump size are Y =.5, Y =.1. The iniial price is 1 and he iniial economy sae is X = 1. or each of he fixed mauriies T = 1,3,5 we consider differen srike prices K =.8,1.,1..Wih each pair of ( T, K), we consider differen jump inensiy parameer =,1 and hen, we can sudy he effecs of he jump on he opion prices. igure 1: Opion prices vs rike Price wih T=1 year igure : Opion prices vs rike Price wih T=3 years
8 huai Wang and Linyi Qian / ysems Engineering Procedia (11) 3 igure 3: Opion prices vs rike Price wih T=5 year igure 4: Opion prices vs mauriy wih rike Price K=.8 igure 5: Opion prices vs mauriy wih rike Price K=1 igure 6: Opion prices vs mauriy wih rike Price K=1. igure 1,igure and igure 3 depic he plos of he European call currency opion prices agains srike prices for differen fixed mauriies T = 1,3,5. rom he hree differen figures, we can find ha he opion prices decrease wih srike prices regardless of he jump pars. This is consisen wih he common sense since he higher srike price makes i harder o execue he call opions for he policy holder wih no respec o he dynamics of spo X rae. We can also find ha he call opion prices wih jump are always higher han he counerpars. The reason is ha he mean of he disribuion of jump size in he simulaion assumpion is posiive and he spo X rae is more likely o have he posiive jump. Consequenly, i will have higher erminal rae han corresponding par of he dynamics wihou jump,which resuls in he higher opion price a he iniial ime. And we can also find ha he difference beween opion price wih jump and one wihou jump becomes larger as mauriies become longer. This can also be explained by he more posiive jumps in he case wih jump. igure 4, igure 5 and igure 6 depic he plos of he European call currency opion prices agains mauriies for differen fixed srike prices K =.8,1,1..rom he hree differen figures, we can find ha he opion prices increase wih he mauriies regardless of he jump pars. This is consisen wih he common sense. We can sill find ha he call opion prices wih jump are always higher han he counerpars. The reason is he same as he previous case. And we can sill find ha for differen fixed srike prices, he difference beween opion price wih jump and one wihou jump becomes larger as mauriies increase. This is consisen wih he findings in igure 1, igure and igure 3.
huai Wang and Linyi Qian / ysems Engineering Procedia (11) 3 9 5. Conclusion In his paper, we sudy he pricing of European currency opions in financial engineering when he dynamics of he spo X rae are driven by he generalized Markov-modulaed jump-diffusion sysem, wih he jump par described by he random measure. We use he coninuous ime Markov chain o drive he regime swiching of he ineres rae and appreciaion rae and volailiy of he asse. Then we find he minimal maringale measure and derive he sysem of parial differenial inegral equaions saisfied by he opion price we are searching for. This aricle provides marke paricipans wih flexibiliy o consruc differen and concree regime-swiching jumpdiffusion models in pracice. And from he simulaion resuls, we can also ge he behavior of he price wih differen parameers consisen wih he heory. 6. Copyrigh All auhors mus sign he Transfer of Copyrigh agreemen before he aricle can be published. This ransfer agreemen enables Elsevier o proec he copyrighed maerial for he auhors, bu does no relinquish he auhors' proprieary righs. The copyrigh ransfer covers he exclusive righs o reproduce and disribue he aricle, including reprins, phoographic reproducions, microfilm or any oher reproducions of similar naure and ranslaions. Auhors are responsible for obaining from he copyrigh holder permission o reproduce any figures for which copyrigh exiss. Acknowledgemens We would like o hank Professor Weian Zheng for his suppor and commens o his paper. We are also graeful o he paricipans of inancial Engineering eminar a Eas China Normal Universiy and he Inernaional Conference on Acuarial cience and elaed ields. eferences 1.. Black.and M. choles, The pricing of opions and corporae liabiliies. Journal of Poliical Economy 81(1973) 637-659...C. Meron, The heory of raional opion pricing. Bell Journal of Economics and Managemen cience 4(1973) 141-183. 3. N. Biger and J. Hull, The valuaion of currency opions. inancial Managemen 1 (1983) 4-8. 4. M. Garman and. Kohlhangen, oreign currecny opion values. Journal of Inernaional Money and inance (1983)39-53. 5..E. Quand, The esimaion of parameers of linear regression sysem obeying wo separae regimes. Journal of he American aisical Associaion 55(1958)873-88. 6..M. Goldfeld and.e. Quand, A markov model for swiching regressions. Journal of Economerics 1(1973)3-16. 7. J.. Hamilon, A new approach o he economic analysis of nonsaionary ime series and he business cycle. Economerica 57() (1989 )357-384. 8. K. hasri and K. Weheyavivorn, The valuaion of currency opions for alernae sochasic processes. Journal of inancial esearch 1(1987) 83-93. 9. J.M. Harrison and.m. Kreps, Maringales and arbirage in muliperiod securiies markes. Journal of Economic Theory (1979) 381-48. 1. J.M. Harrison and.. Pliska, Maringales and sochasic inegrals in he heory of coninuous rading. ochasic Processes and Their Applicaions 11 (1981) 15-8. 11. J.M. Harrison,.. Pliska, A sochasic calculus model of coninuous rading: complee markes. ochasic Processes and Their Applicaions 15 (1983) 313-316. 1. H. öllmer and. ondermann, Hedging of coningen claims under incomplee informaion.. In: Conribuions o Mahemaical Economics. Norh Holland, Amserdam, (1986)5-3. 13. H. öllmer and M. chweizer., Hedging of coningen claims under incomplee informaion. In: Applied ochasic Analysis. Gordon and Breach, London, (1991) 389-414. 14. M. chweizer, Approximaion pricing and he variance-opimal maringale measure. Annals of Probabiliy 4 (1996) 6-36.
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