Option Pricing Under Stochastic Interest Rates

Transcription

1 I.J. Engineering and Manufacuring, 0,3, 8-89 ublished Online June 0 in MECS (hp:// DOI: 0.585/ijem Available online a hp:// Opion ricing Under Sochasic Ineres Raes Haowen Fang School of Business, Sun Ya-sen Universiy, Guangzhou, Guangdong rovince, China,5075 Absrac This paper reviews he research hisory of opion pricing, hen our model assumes ha he ineres rae subjec o a given Vasicek sochasic differenial equaions, using opion pricing by maringale mehod o sudy he sochasic ineres rae model of European opion pricing and obain he pricing formula. Finally, we compare he differences beween he sandard European opion pricing formulas and European opion pricing formula under sochasic ineres rae. Index Terms: Opion ricing; Sochasic Ineres Raes; Vasicek model, Brownian moions 0 ublished by MECS ublisher. Selecion and/or peer review under responsibiliy of he Research Associaion of Modern Educaion and Compuer Science.. Inroducion Black, Scholes (973) and Meron[] (973) showed in heir seminal papers ha a derivaive securiy can be priced by creaing a replicaing porfolio, i.e. a porfolio of primiive securiies which maches he payoff of he derivaive a mauriy. Since boh he replicaion porfolio and he derivaive offer he same payoff a mauriy, hey have o have he same price a any preceding ime. Deviaions from his equaliy lead o arbirage possibiliies. Hence, he pricing by duplicaion procedure inhibis arbirage by consrucion. Since hen he field of financial engineering has grown phenomenally. The Black Scholes Meron risk neuraliy formulaion of he opion pricing heory is aracive because he pricing formula of a derivaive deduced from heir model is a funcion of several direcly observable parameers (excep one, which is he volailiy parameer). The derivaive can be priced as if he marke price of he underlying asse s risk is zero. Deemple[8](005) reviewed he valuaion of American opions. Several semi-analyical approximaions for American opion prices have been proposed in he lieraure (Barone Adesi and Whaley[], 987; Broadie and Deemple[5], 996; Bunch and Johnson[6], 000). Alhough hese approaches are fas and accurae, hey can no easily be exended beyond he Black-Scholes model. I has been firmly esablished ha he Black-Scholes model is no consisen wih quoed opion prices. The lieraure advocaes he inroducion of sochasic volailiy or jump sore produce he implied volailiy smile observed in he marke. The inroducion of an addiional sochasic volailiy facor enormously complicaes he pricing of American opions. resenly, his can only be done by means of numerical schemes, which involve solving inegral equaions(kim[6],990; Huang, Subrahmanyam, and Yu[3], 996; Sullivan[0],000; Deemple and Tian[9],00), performing Mone Carlo simulaions(broadie and Glasserman[4], 997; Longsaff and Schwarz[7],00; Rogers[9], 00; * Corresponding auhor. address: fanghaowen@om.com

2 Opion ricing Under Sochasic Ineres Raes 83 Haugh and Kogan[],004), or discree he parial differenial equaion(brennan and Schwarz[3], 977; Clarke and arro[7], 999; Ikonen and Toivanen[5], 007). The early exercise premium of he American pu opion depends on he cos of carry deermined by ineres raes. Consequenly, he volailiy of ineres raes does affec he decision o exercise his opion a any poin in ime. This fac is recognized in he lieraure dealing wih models wih sochasic ineres raes(ho,sapleon, and Subrahmanyam[], 997; Menkveld and Vors[8],00;Deemple and Tian[9], 00). This lieraure, however, considers only wo-facor exensions of he Black-Scholes model assuming ha he volailiy of he underlying asse is consan. In his paper, we assume ha he ineres rae subjec o a given Vasicek sochasic differenial equaions, by using maringale mehod o sudy he sochasic ineres rae model of European opion pricing and obain he pricing formula. The paper is organized as follows. In Secion we describe he assumpions of he opion model, using maringale mehod, by solving a second order parabolic parial differenial equaion, we obain he European opion pricing formula. In Secion 3 we compare he differences beween he sandard European opion pricing formulas and European opion pricing formula under sochasic ineres rae.. European opion pricing Formula The sandard BS model makes he following assumpions: he marke is fricionless (i. e. no ransacion coss or axes and no penalies for shor selling); he marke operaes coninuously, he risk-free ineres rae r is a known consan; he asse price follows Geomerical Brownian Moion(GBM) wih consan volailiy σ > 0 and pays no dividends; opions and derivaives are European (i. e. no early exercise) and expire a ime T wih a payoff ha depends only on T ; he marke is arbirage free. Under he assumpion of GBM, he asse price saisfies a sochasic differenial equaion (sde) of he form d ( d db ) x, 0 0 B is a sandard Brownian moion under a measure (called he real- where is he growh rae of he asse. The erm world measure). Theorem (Iô s Lemma) Le d ( x) d ( x, ) db saisfy he sde x where C, funcion. Then V(, ) saisfies he sde dv V V V d V db x xx x and le V( x, ) be any where subscrips on V denoe parial derivaives. I is someimes more insrucive o wrie his las sde in he equivalen form dv V V d V d xx x Iô s Lemma is he main ool used o solve sde s. For example, he sde () for GBM can be solved by aking dy Y ( ) log( / x0) Y ( ) d db. The sde for hen becomes, and his is readily inegraed o give he represenaion d x Z Z N 0 exp{( ) } (0,) Now, le s deduce he euro-opions pricing formula under sochasic ineres rae. Assume he asse price saisfies

3 84 Opion ricing Under Sochasic Ineres Raes GBM d (, ) r d r db () The ineres rae is given by Vasicek Model[] dr a( r ) d ( r, ) db () { B : 0} where { B : 0}, are sandard Brown moions, cov( db, db ) d ( ) (3) V (,, ) Le V r denoe he price of he call European opion, V ( K), K is he srike price. We will find he opion price. Using hedging echnical we derive funcion V(, r, ) saisfy he appropriae sde, and obain he porfolio V Choose share of sock and share of zero-coupon, he porfolio is risk-free in he period [, d].i s also mean if choose appropriae and, hen we can ge d dv d d, (4) which is risk-free, and hen d r d r V d (5) (, ; ) Here r T is he price of zero-coupon, and saisfies a sochasic differenial equaion (sde) of he form[] d r d V () db (Vasicek Model) d rd C () r db or (C-I-R Model). By Iô s Lemma, (4) can be wrien as V V V V d d r r V V + d dr r r d r (6) The hird erm of he righ equaion can be subsiue as r a( r ) d r (7) In order o eliminae he risk, le V V /, r, considering (7) and (), we obain

4 Opion ricing Under Sochasic Ineres Raes 85 V V V V V r r r V a( r) rv 0 ( r,, [0, T ]) r (8) as T V(,, T) ( K) ( r, ), (9) We know by he maringale mehod opion pricing heory[0], here exis a maringale measure Q, such ha T Q rd V E e ( T K) r( ) r, ( ) (0) To ransform valuaion of uni of accoun, ha zero-coupon ( r, ; T) as a new uni of accoun, and a corresponding price sysem inroduced V, V () Based on uni conversion valuaion heory, he equivalen maringale measure exiss, such ha (0) can be rewrien as U Q V E ( V r( ) r, ( ) ) U Q T = E K ( ), ( ) T U Q = E T K ( ), ( ) () In he las equaion, we use he fac T. () showed ha he proceeds a he ime T, funcion V can be ransformed ino only depends on. In he Vasicek d rd M () db model, zero-coupon price processes saisfies sochasic differenial equaion, d ln r V ( ) d V ( ) db so, According o (), d ln r d db As for,such ha d ln d ln d ln ( ) d db ( ) db V This shows ha for he Vasicek model, he sochasic differenial equaions of and r, so we can pu () rewrien as is no longer significan conain wih

5 86 Opion ricing Under Sochasic Ineres Raes = U Q V E T K ( ) This shows ha for he new price sysem {, V } V V(, ), funcion correlaion holds. To solve problem (8) and (9), we draw a new ransformaion of independen variables, y ( r, ; T ) and a new unknown funcion denoes as V (, r, ) V ( y, ) ( r, ; T ) According o primarily compuaions, V V V V y y V V V y r r y r V V y V V V V y y r r y r y r V V y r y r V V y Subsiue hem ino (8), and divided by ( r, ; T ), such ha V V y r r y V + a( r) r y r r y y + a( r) r V 0 r r ( r, ; T ) Considering ransformaion (3) and he funcion saisfies he following second order parabolic pde s Cauchy problem a( r) r 0 ( r, [0, T ]) r r ( r, T ). (5) Then, we immediaely find ha funcion V( y, ) saisfies he equaion (3) (4)

6 Opion ricing Under Sochasic Ineres Raes 87 V V ( ) y 0 r y, V (, r, T ) V ( y, T ) ( y K) and he definiely soluion condiion is ( r, T; T) where K is he opion s srike price, ( ) ( ) ( ) () r The soluion of problem (5) can be expressed by he general Black-Scholes formula V( y, ) yn( d) KN( d) (6) y T ln ( ) d K d T ( ) d (7) T d d ( ) d (8) Reverse o he original variables, r, and unknown funcion V by he ransformaion (3) and (4), such ha (6) and (7) become V (, r, ) ( r, ; T ) V, ( r, ; T ) d * * * = N( d) K( r, ; T ) N( d) (9) T ln ln ( r, ; T ) ( ) d K T ( ) d (0) T * * d d ( ) d () This formula was firs proposed by Meron in 973, when he was no received he random model of shorerm ineres rae r, bu direcly saring from he zero-coupon, assuming o mee he geomeric Brownian moion, under he maringale measure, i is described by he following Sochasic differenial equaions, d rd db () { B : 0} where is sandard Brown moion, is he zero-coupon s (bonds) volailiy. Thus, European call opion pricing formula is * * V(,, )= N( d) KN ( d) (3) which

7 88 Opion ricing Under Sochasic Ineres Raes d * T ln ln ( ) d K T ( ) d T * * d d ( ) d ( ) (4) 3. Conclusions Compared wih he sandard European opion pricing formula and European opion under sochasic ineres rae, here ( ) are only wo differences: one is zero-coupon replaced by r T e ; anoher is ha using insead of sock price volailiy. Excep ha he pricing formulas is exacly he same form. Analysis from he acual markes, means zero-coupon s (bonds) volailiy is far smaller han sock marke s () volailiy, bu in general (), is monoonic decreasing, and lim ( ) 0 T. and in fac has he minor 0 difference. Because in general, sock prices and bond prices are posiively correlaed, hen. Therefore, if, by equaion (4) we know holds. Therefore under sochasic ineres raes, he price of an opion bu have slighly decreased. If a shor-erm ineres rae model is given, only for he Vasicek model and Hull-Whie model[4], European opion pricing formula has a simple form of he Meron formula (3). For C-I-R model, he corresponding zero-coupon sochasic r model, he flucuaions in he rae of enry also including, so i can no wrie (), so by pricing uni conversion of lower dimension han Number of purposes, such as los ha possible syle of (4). References []Barone-Adesi, G., Whaley, R., Efficien analyical approximaion of American opion values. Journal of Finance 4, 987, []Black, F., Scholes, M., The pricing of opions and corporae liabiliies. Journal of oliical Economy 8, 973, [3]Brennan, M., Schwarz, E., The valuaion of American pu opions. Journal of Finance 3, 977, [4]Broadie, M., Glasserman,., ricing American-syle securiies using simulaion. Journal of Economic Dynamics and Conrol, 997, [5]Broadie, M., Deemple, J. American opion valuaion: new bounds, approximaions, and a comparison of exising mehods. Review of Financial Sudies 9,, 996, 50. [6]Bunch, D., Johnson, H., 000. The American pu opion and is criical sock price. Journal of Finance 55, [7]Clarke, N., arro, K., Muligrid for American opion pricing wih sochasic volailiy. Applied Mahemaical Finance 6, 999, [8]Deemple, J.,. American-Syle Derivaives: Valuaion and Compuaion. Chapman & Hall/CRC Financial Mahemaics Series, Boca Raon, FL. 005 [9]Deemple, J., Tian, W., The valuaion of American opions for a class of diffusion processes. Managemen Science 48, 00,

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