Iranian In. J. Sci. 6(), 005, p.15-5 An Exended Model of Asse Price Dynamics M. H. Nojumi Deparmen of Maemaical Sciences, Sarif Universiy of Tecnology P.O. Box 11365-9415, Teran, Iran. e-mail: nojumi@sina.sarif.edu (received: 17/5/005 ; acceped: 5/10/005) Absrac An exended model of asse price dynamics for modeling socasic upward and downward jumps in asse prices is developed, and e modified Black-Scoles soluion for value of vanilla opions is derived. Te cange in volailiy is idenified in deail using e Iô inegrals and Iô formulas. Keywords: Socasic jump, price dynamics, opion pricing, volailiy, Black-Scoles Analysis, vanilla opions 1. Inroducion A grea variey of financial producs are offered and raded in financial markes, of wic opions are one of e mos frequenly raded. A call opion is a conrac beween wo paries by wic e buyer of e opion (also called e older of e opion) as e opion, no an obligaion, o buy a specified (by quaniy and qualiy) asse for a specified price a specified daes. If e cooses o do so, en e seller of e opion (also called e wrier of e opion) as e obligaion o sell e asse. A pu opion is a similar conrac wi e difference a e buyer of e opion as e opion o sell e asse, and if e cooses o do so, en e seller of e opion as e obligaion o buy e asse. In e European opion, e rig o buy or sell can be exercised only a e expiraion ime of e opion. In American opion, is rig can be exercised a any ime ill expiry. In pracice, mos opions are of Bermudan ype in wic e rig o buy or sell can be exercises only a cerain pre-specified daes.
14 Nojumi M.H. IIJS, 6 (Ma.), 005 Hence, an opion pus e older in an advanaged posiion wi respec o e wrier. Tis advanaged posiion as money value. Hence, o become e older of an opion, one mus pay some premium o purcase e opion from e wrier. Te problem of opion pricing is o deermine ow muc premium one sould pay o purcase an opion; in oer words, wa e curren price of e opion is, given is parameers (asse price movemen, exercise daes, exercise prices, ec.). Te European opion is e simples opion o analyze and closedform formulas are available for is value. Aloug in pracice opions are rarely European, e resuls for e European opion are valuable since ey can be generalized o obain resuls for values of oer ypes of opions (Hull, 1997; Nefci, 1996; Wilmo e al., 1995). Te underlying asse can be anying. One can consider an asse, including cas, bonds, socks, ineres raes, foreign currency excange raes, anoer opion, various agriculural commodiies like wea or coffee, even ings like weaer condiion, resul of spors maces, ec. (Hull, 1997; Kolmann and Tang, 001). An imporan feaure of asse prices is eir socasic movemens, wic in urn make prices of financial derivaives socasic. Some asses display even fracal-like igly socasic beavior (Peer, 1994).. Incorporaion of Jump Variaions in an asse price are of wo general ypes: expeced and socasic (Nefci, 1996). Te Black-Scoles analysis assumes a lognormal coninuous socasic grow; described by e socasic differenial equaion ds = µ Sd + σ SdW; wi { W : 0} a Wiener process. A fundamenal limiaion of is model is a i does no capure e socasic jumps in asse price; a penomenon wic appens in e real world. A socasic process suiable for modeling rare evens is e Poisson process (Ross, 003). Wi { U : 0} a Poisson process wi rae λ wic can be esimaed by observing e marke, e equaion
An Exended Model of Asse Price Dynamics 15 ds = ad + σ dw + γ du, wi a, σ and γ funcions of S and, models socasic jumps as well bu as wo major problems: We are ineresed in capuring all average beavior of S in e deerminisic erm ad, and ave e oer wo erms capure purely socasic beavior of e asse price. Te Wiener erm adw as zero mean as desired. However, e Poisson erm γ du does no ave is feaure since U as mean λ 0. Te Poisson couning process is non-decreasing. In e ime inerval [, + ] e process U does no cange wi e probabiliy 1 λ + o( ), increases by one uni wi e probabiliy λ + o( ), and increases by more an one uni wi e probabiliy o( ). Hence wi γ a nonnegaive funcion, only e upward jumps in asse price are modeled. Aloug coosing a funcion γ wic akes on bo posiive and negaive values, downward jumps are also modeled, e firs problem remains unresolved. Te suggesion made ere o simulaneously solve bo e above problems is o inroduce anoer Poisson process { D : 0} wi e same rae λ ino e model in e following way: ds = ad + σ dw + γ d( U D ); wi a, σ, and γ funcions of S and. In every inerval [, + ], bo U and D are Poisson random variables wi mean λ, ence U D as zero mean, as desired. Te second issue is resolved as well even wi γ a nonnegaive funcion since in e ime inerval [, + ] e process U D says e same wi probabiliy λ + (1 λ) ( ) + o ( ), increases by one uni wi probabiliy λ(1 λ)( ) o ( ) +,
16 Nojumi M.H. IIJS, 6 (Ma.), 005 decreases by one uni wi probabiliy λ(1 λ)( ) + o ( ), increases by more an one uni or decreases by more an one uni wi probabiliy o ( ). 3. Iô s Formula for Derivaive Price Ignoring all e od ( ) erms, e differenial of V( S, ), e price of a financial derivaive based on e underlying asse price S a ime, is dv = d + ds + ds. V V 1 V S S In deerminisic calculus, one would reain only e firs wo erms since all oer erms are od ( ). However, in socasic calculus e ird erm canno be ignored since i conains an Od ( ) erm. To idenify is erm, we consider ds = a d + σ dw + γ dj + aσddw + aγddj + σγdw dj ; wi J := U D. A fundamenal propery of e Wiener process is dw = d (Nefci, 1996). To invesigae e naure of dj using e noion of e Iô inegral, wi an arbirarily seleced T > 0, we pariion e ime inerval [0, T ] ino n subinervals of equal leng = T/ n and, wi := j; j = 01,,, n and J := J J, j = 01,,, n 1, we consider e quaniy j j+ 1 j Ω n := E J j Z ; 0 j n 1 wi E e expecaion operaor, and Z e consan suc a j
An Exended Model of Asse Price Dynamics 17 Now lim Ω = 0. n 0 n n 4 j 0 j n 1 Ω = E J + Z J j ( Jk ) Z J j 0 j< k n 1 0 j n 1 + 0 j n 1 4 j = E J + Z ( J ) E J j k Z E J j 0 j< k n 1 0 j n 1 +. By independen incremens propery of e Poisson process, for j k e random variables J j and Jk are independen, so ( J ) ( J ) j k j k E J = E J E. Moreover, by saionary incremens propery of e Poisson process, for every j = 0, 1, n, 1, e random variable J j := J J j+ 1 j as e same probabiliy disribuion as e random variable J = J Poisson( λ). Hence j+ 1 j
18 Nojumi M.H. IIJS, 6 (Ma.), 005 Ω = E J + Z n 0 j n 1 4 EJ Z EJ 0 j< k n 1 0 j n 1 + 4 ( 1) = ne J + n n E J nze J + Z. Lemma. Te Momen generaing funcion of a Poisson disribuion wi rae λ is z MGF X( z) = exp[ λ ( e 1 )]. Proof. k k z e zx zk λ λ λ λ MGF X( z) = E e = e e e =. k! k! k= 0 k= 0 Having e momen generaing funcion of J, we can deermine e required quaniies E J E J and 4 : z z λ λ( ) dmgf X ( z ) = e exp[ e 1 ] d d d ( ) MGF X ( ) z 1 z exp[ z z = λe + λe λ e 1 ] 3 MGF X 3 ( ) z 1 3 z z exp[ z z = λe + λe + λ e λ( e 1 ) ] 4 MGF X 3 3 4 ( ) z 1 7 z 6 z z exp[ z z = λe + λe + λ e + λ e λ( e 1 ) ]
An Exended Model of Asse Price Dynamics 19 Since U and D are bo Poisson processes wi rae λ, we ave dmgf U EU ( ) = ED ( ) = (0) = λ d MGF U EU = ED = (0) = λ (1 + λ ) d EU ED 4 4 4 d MGF U EU = E D = 4 (0) = λ 1+ 7λ+ 6λ + λ 3 3 3 MGF U 3 (0) λ 1 3 = = = + λ + λ 3 3 So ( ) ( ) ( ) ( ) 0 E J = E U D = E U E D = ; ( ) = E J E U D = EU EU E D + E D = λ(1 + λ) ( λ) = λ ( ) 4 4 = E J E U D = EU 4EU ED + 6EU ED 4 3 4EU ED + ED 3 4 = λ 1+ 7λ+ 6λ + λ 3 3 + + + + λ) 8λ 1 3λ λ 6 λ (1 = λ(1+ 6 λ) [ ]
0 Nojumi M.H. IIJS, 6 (Ma.), 005 Wi ese resuls, and by λ = T, we obain Ω n = nλ(1+ 6 λ) + 4 n( n 1) λ 4nλZ + Z = Z 4λTZ+ 4λT+ λt+ 8λT Terefore, noing a leing n is equivalen o leing 0, lim Ω = n 4 + 4 + n Z λtz λ T λt Te value of Z making is limi zero is Z = λt ± i λt We ave us sown a T 0 ( U D ) d = λt ± i λt T λ = λ i ± d. 0 For large e imaginary erm is negligible. We ave us proved Proposiion. For large compared o λ suc a λ 1 λ or equivalenly 8λ e variable U D saisfies e following equaion wi good approximaion ( ) d U D = λd. Now we can idenify e significan Od ( ) erms in ds :
An Exended Model of Asse Price Dynamics 1 ds = a d + σ dw + γ dj + aσddw + aγddj + σγdw dj ( σ λγ λρσγ) = + + d; wi 1 ρ 1 e insananeous coefficien of correlaion beween W and U D, wic ere we define via e equaion dw d( U D ) = λ ρd. I is reasonable o assume a under normal operaion of e marke, economic facors a conribue o upward socasic jumps in price of a paricular asse are independen from ose responsible for downward socasic jumps in e price of e same asse. Also, an economic facor is no likely o cause bo an upward jump and a downward jump in e asse price. Hence i is reasonable o assume saisical independence of U and D However, one may expec correlaion beween W and U D. A coninuous socasic movemen of e asse price may rigger facors a cause socasic jumps in eier direcion. So bo posiive and negaive values of e correlaion coefficien ρ are likely. Te Iô s formula for e financial derivaive on e underlying asse price S en akes e form 1 V dv = d + ds + ds S S = d+ ( ad + σdw + γd( U D) ) S 1 V + ( σ + λγ + λρσγ) d ; S or
Nojumi M.H. IIJS, 6 (Ma.), 005 ( σ λγ λρσγ) 1 V dv = + + + + a d S S + σ dw + γ d( U D). S S 4. Te Updaed Black-Scoles Equaion Te Black-Scoles dela-edged porfolio Π:= V S S is sill risk-free since, keeping inerval [, + d], dπ= dv ds S or S ( σ λγ λρσγ) fixed during e infiniesimal ime 1 = + + + + + σ dw + γ d( U D) S S ( ad + σ dw + γ d ( U D ) ) S V V V a S S ( σ λγ λρσγ) 1 V dπ= + + + d. S d In e absence of arbirage opporuniies, wic is e usual case in e financial markes, e reurn from is porfolio sould be equal o reurn from e risk-free invesmen of amoun Π ; a is, r Π d, wi r e insananeous risk-free ineres rae a ime. We us arrive a
An Exended Model of Asse Price Dynamics 3 e parial differenial equaion saisfied for e financial derivaive price V( S, ) : ( σ λγ λρσ γ ) + ( S, ) + ( S, ) + ( S, ) ( S, ) + rs rv = 0 V 1 V V S S In e special case of e asse price aving consan log-normal expeced grow rae, log-normal coninuous socasic cange, and log-poisson jump socasic cange; a is, as (, ) = µ S, σ( S, ) = σs, γ( S, ) = γs wi µ, σ, and γ consans, e asse price dynamics follows e socasic differenial equaion ds = µ S d + σs dw + γs d( U D ) Wi risk-free ineres rae also assumed consan r, e parial differenial equaion saisfied by e financial derivaive price V( S, ) is 1 V + ( σ + λγ + λρσγ) S + rs 0 rv =. S S Tis is e Black-Scoles equaion wi σ replaced by e effecive volailiy ˆ σ given by ˆ σ := σ + λγ + λρσγ ; Tree special cases are ˆ σ = σ + λγ in case of full posiive correlaion ( ρ = 1) ˆ σ = σ λγ in case of full negaive correlaion ( ρ = 1) ˆ σ = σ + λγ in case of no correlaion ( ρ = 0 ). A sock a as price S a ime and pays a coninuous dividend yield q beaves like a non-dividend paying sock wic as price
4 Nojumi M.H. IIJS, 6 (Ma.), 005 qt ( ) e S a ime. Hence e price of European call and pu opions on is sock are given by e Black-Scoles formulas (Hull, 1997; Nefci, 1996) wi σ replaced by ˆ σ ; a is, and wi CS (, ) = e SNd ( ) e KNd ( ) qt ( ) rt ( ) 1 PS, = e KN d e SN d rt ( ) qt ( ) ( ) ( ) ( 1) 1 S 1 1 log d := r q ˆ σ ( T ) ˆ + +, σ T K 1 S 1 log d := r q ˆ σ ( T ) ˆ + σ T K = d ˆ σ T 1 Here T and K are e expiraion ime and exercise price of e opion, respecively, and N is e cumulaive disribuion funcion of e sandard normal disribuion: 1 d s / Nd ( ) = e ds. π Te price of a European call or pu opion on a sock index, like S&P 500, can be obained by seing as q e coninuous reurn rae from a sock index. Similarly, e price of a European call or pu opion on a foreign currency can be obained by replacing every q in e above formulas by e risk-free ineres rae in e corresponding foreign counry.
An Exended Model of Asse Price Dynamics 5 Acknowledgmen Te auor wises o ank e Deparmen of Maemaical Sciences and e Researc Council of Sarif Universiy of Tecnology for providing scienific and financial suppor. References Hull, J.C. (1997) Opions, Fuures, and Oer Derivaives, Tird Ediion, Prenice Hall. Kolmann, M., and Tang, S. (001) Maemaical Finance, Proceedings of e Worksop on Maemaical Finance Researc Projec, Konsanz, Germany, Ocober 5-7, 000, Birkäuser Verlag. Nefci, S. (1996) An inroducion o e maemaics of financial derivaives, Academic Press, London. Peer, E. (1994) Fracal marke analysis: applying caos o e eory o invesmen and economics, Jon Wiley & Sons, New York. Ross, S. (003) Inroducion o probabiliy models, Eig Ediion, Academic Press, London. Wilmo, P., Howison, S., and Dewynne, J. (1995) Te maemaics of financial derivaives, Cambridge Universiy Press, Cambridge.