Time Domain Decomposition for European Options in Financial Modelling

Transcription

1 Contemporary Mathematcs Volume 218, 1998 B Tme Doman Decomposton for European Optons n Fnancal Modellng Dane Crann, Alan J. Daves, Cho-Hong La, and Swee H. Leong 1. Introducton Fnance s one of the fastest growng areas n modern appled mathematcs wth real world applcatons. The nterest of ths branch of appled mathematcs s best descrbed by an example nvolvng shares. Shareholders of a company receve dvdends whch come from the proft made by the company. The proceeds of the company, once t s taken over or wound up, wll also be dstrbuted to shareholders. Therefore shares have a value that reflects the vews of nvestors about the lkely dvdend payments and captal growth of the company. Obvously such value wll be quantfed by the share prce on stock exchanges. Therefore fnancal modellng serves to understand the correlatons between asset and movements of buy/sell n order to reduce rsk. Such actvtes depend on fnancal analyss tools beng avalable to the trader wth whch he can make rapd and systematc evaluaton of buy/sell contracts. There are other fnancal actvtes and t s not an ntenton of ths paper to dscuss all of these actvtes. The man concern of ths paper s to propose a parallel algorthm for the numercal soluton of an European opton. Ths paper s organsed as follows. Frst, a bref ntroducton s gven of a smple mathematcal model for European optons and possble numercal schemes of solvng such mathematcal model. Second, Laplace transform s appled to the mathematcal model whch leads to a set of parametrc equatons where solutons of dfferent parameterc equatons may be found concurrently. Numercal nverse Laplace transform s done by means of an nverson algorthm developed by Stehfast [4]. The scalablty of the algorthm n a dstrbuted envronment s demonstrated. Thrd, a performance analyss of the present algorthm s compared wth a spatal doman decomposton developed partcularly for tme-dependent heat equaton. Fnally, a number of ssues are dscussed and future work suggested. 2. European Optons One smple and nterestng fnancal model known as the European Opton has two types of contracts avalable namely, call optons and put optons. The holder 1991 Mathematcs Subject Classfcaton. Prmary 65D99; Secondary 65M06. Key words and phrases. Tme Doman Decomposton, Fnancal Applcatons. Scalablty tests n ths paper were performed by the frst two authors. 486 c 1998 Amercan Mathematcal Socety

2 TIME DOMAIN DECOMPOSITION IN FINANCIAL MODELLING 487 of a call opton has the rght, at a prescrbed tme known as the expry date, to purchase a prescrbed asset for a prescrbed amount usually known as the strke prce. Whle the other party of the contract must sell the asset f the holder chooses to buy t. On the other hand, the holder of a put opton has the rght, at the expry date, to sell the prescrbed asset at the strke prce. Whle the other party of the contract must buy the asset f the holder chooses to sell t [6]. Ths secton only examnes a European call opton. The stochastc background of the equaton s not dscussed n ths paper and readers should consult [6]. Let v(x, t) denotes the value of an opton where x s the current value of the underlyng asset and t s the tme. The value of the opton depends on σ(t), E, T and r(t) whch are, respectvely, known as the volatlty of the underlyng asset, the strke prce, the expry tme and the nterest rate. The Black-Scholes analyss for one ndependent varable leads to the famous Black-Scholes equaton [6], v t σ2 x 2 2 v (1) x 2 + rx v rv =0 Ω+ x where Ω + = {x : x 0}. In order to descrbe a European call opton, boundary condtons and fnal condtons are requred. Snce the call opton s worthless at x = 0 even f there s a long tme to reach expry, therefore t s sensble to have v(0,t) = 0. Snce the asset prce ncreases wthout bound, therefore t becomes lkely that the opton wll be exercsed and the magntude of the strke prce becomes less mportant. Therefore t s sensble to have v(s, t) s as s. At expry, f x>ethen one should exercse the call opton,.e. to hand over an amount E to obtan an seest wth x. However, f x<eat expry, one should not exercse the call opton. Snce the expry date s n the future, the fnal condton v(x, T )=max(x E,0) must be mposed. The soluton v for t<t s requred. The fnancal nterpretaton of the above model s as follows. Frst, the dfference between the return on an opton portfolo, whch nvolves the frst two terms, and the return on a bank depost, whch nvolves the last two terms, should be zero for a European opton. Second, the only parameter that affects the opton n a stochastc way s the volatlty σ(t) whch measures the standard devaton of the returns. Snce (1) s a backward equaton, one can transform t to a forward equaton by usng τ = T t and t leads to, v τ = 1 2 σ2 x 2 2 v (2) x 2 + rx v rv Ω+ x subject to boundary condtons v(0,τ) = 0andv(s, τ) s as s and ntal condtons v(x, 0) = max(x E,0). An analytc soluton may be derved f a change of varable s made where the Black-Scholes equaton s converted to a tme-dependent heat conducton equaton wth constant coeffcents [6]. However a feld method, such as fnte volume methods, s of more nterest for two reasons. Frst, there are many examples n mult-factor models such that a reducton of the tme dependent coeffcent to a constant coeffcent heat equaton s mpossble. Hence analytc form of solutons cannot be found. Second, the computatonal envronment at Greenwch s based on the fnte volume code PHYSICA [1] whch s the man research and development code for mult-physcs work. The code has capablty of solvng unsteady dffuson, convecton and radaton type of equatons. Fnancal modellng typcally requres large number of smulatons and hence computng resources and effcency of

3 488 DIANE CRANN ET AL. algorthms are very mportant n order to make evaluaton and decson before the agreement of a contact. Wth the present day hgh performance computng and/or dstrbuted computng, parallel algorthms offer effcent numercal solutons to the equaton gven by (2). 3. Tme Doman Decomposton For tme varyng σ(t) and r(t), t s possble to make sutable coordnate transformaton to the Black-Scholes equaton n order to obtan a tme ndependent lke heat equaton [6]. Hence the method descrbes n ths secton may then be appled. Here, a method s descrbed whch focuses on tme ndependent coeffcents σ and r. Takng Laplace transform of (2) and takng ntegraton by parts to the left-hand-sde of the transformed equaton, one obtans the parametrc equaton 1 2 σ2 x 2 d2 u (3) dx 2 + rxdu dx (r + λ j)u = v(x, 0) Ω + subject to boundary condtons u(0; λ j )=0andu(s; λ j )= s λ j.hereu(x, λ j )sthe Laplace transform of v(x, t) andλ j s a dscrete set of transformaton parameters defned by (4) λ j = j ln 2,j=1, 2,,m τ where m s requred to be chosen as an even number [5]. An approxmate nverse Laplace transform [4] may be used to retreve v(x, t) accordng to v(x, τ) ln 2 m (5) w j u(x, λ j ) τ where mn(j,m/2) w j =( 1) m/2+j k=(1+j)/2 j=1 k m/2 (2k)! (m/2 k)!k!(k 1)!(j k)!(2k j)! s known as the weght factor. Each of the above m parametrc equatons may be rewrtten as d (6) dx (1 2 σ2 x 2 du dx + d dx ((r σ2 )xu) (2r σ 2 + λ j )u = v(x, 0) The computatonal doman has a unform mesh and fnte volume method s appled to (6) whch leads to (7) ( 1 S 2 σ2 x 2 du dx +(r σ2 )xu)ds (2r σ 2 + λ j )udω = v(x, 0)dΩ Ω Ω The resultng system of lnear equatons s solved n a local area network whch conssts of P workstatons. There are two possble mplementatons as follow. Frst, the soluton at a partcular tme τ s beng sought. For the case when m = P, one would expect deal load balancng. The total computng tme, t A1, usng the present scheme can be estmated as t A = t 1 + t a /P where t 1 s the computng tme for solvng one parametrc equaton and t a s the correspondng computng tme for numercal nverse Laplace transforms gven by (5) and s assumed to be equally spreaded across the P workstatons. For the case when m>p, one would expect just a slght out of load balance for the reason that m s

4 TIME DOMAIN DECOMPOSITION IN FINANCIAL MODELLING 489 possbly not an ntegral multple of P. It s possble to estmate the total computng tme as (8) t A1 = m P t 1 + t a /P The case m<p s not of nterest because the actve workstatons become a subset of the local area network. Second, the solutons at P partcular tmes τ k, k = 1, 2,,P are beng sought. In ths stuaton, each workstaton looks after the solutons of m parametrc equatons and the correspondng nverse Laplace transform at a partcular tme τ k. Hence the total computng tme, t A2, may be estmated as (9) t A2 = mt 1 + t a In order to check the scalablty of the algorthm, a cell-centred fnte volume scheme s appled to the constant coeffcent heat equaton, 2 u = 1 u (10) 1 <x<1, 1 <y<1 k t subject to unt boundary condtons along the whole boundary and zero ntal condton. A unform 16 x 16 grd where the set of dscrete equaton s solved by Gauss-Sedel teraton. Solutons at eght tme values, τ = 0.1, 0.2, 0.5, 1, 2, 4, 10 and 20, are sought. For the purpose of demonstraton, a network of 4 T800 transputers were used as the hardware platform. The computng tmes for P = 1, 2 and 4 are respectvely 2537, 1309 and 634 seconds. The speed-up rato for usng two and four processors are thus 1.94 and 4 respectvely. More results about ths test problem can be found n [2] and experence shown n the paper suggests that m = 8 provdes suffcent accuracy for the model test. Note that the value of m determnes the accuracy of the nverse Laplace transform and hence t depends on the mesh sze or the number of grd pont. In general, m = N/2whereN s the total number of grd ponts n a two-dmensonal problem [2]. Note also that the scalablty property as shown n ths secton also apples to eqn (6). 4. A Comparson wth Spatal Doman Decomposton In order to fnd out the sutablty of the proposed algorthm for a dstrbuted computng envronment, a comparson wth a spatal doman decomposton method s examned n ths secton. The spatal doman decomposton method s smlar to the one developed by Dawson et al [3] for unsteady heat conducton equaton. The problem descrbed n (2) s parttoned nto P subdomans so that a coarse mesh of mesh sze H = s/p s mposed wth nteror boundary of the subdomans beng the same as the nodal ponts of the coarse mesh. In order to determne the nteror boundary values of each of the subdomans, an explct scheme derved from usng a central dfference method along the spatal axs and a forward dfference method along the temporal axs s as follows, v n =( 1 2 σ2 x 2 t H 2 rx t 2H )vn 1 1 +(1 2 σ2 x 2 t H 2 + rx t 2H )vn 1 +1 (11) +(1 σ 2 x 2 t H 2 r t)vn 1

5 490 DIANE CRANN ET AL. Here subscrpts denote the mesh ponts on the coarse mesh and superscrpts n denotes the tme step at t = n t. The choce of t must satsfy the coarse grd restrcton for an explct scheme whch s t mn( σ2 x 2 (12) x H 2 + r) 1 (σ 2 (P 1) 2 + r) 1 Note here that t should be of the order of h 2 where h s the grd sze of the fne mesh. Therefore the total number of tme steps nvolved n the present calculaton s T (σ 2 (P 1) 2 + r). It s reasonable to assume that the computng tme for obtanng the soluton at a new tme step usng the fne mesh and a fnte dfference scheme s the same as the computng tme for obtanng the soluton of a parametrc equaton gven n (3). Therefore the computng tme for marchng one tme step forward usng the classcal spatal doman decomposton wth P subdomans on P processors s t 1 /P. The total parallel computng tme can be estmated as (13) t B = T (σ 2 (P 1) 2 + r)( t 1 P + t b P ) where t b s the overheads for obtanng nteror boundary condtons and s assumed to be equally spreaded across the P workstatons. It s natural to requre t A1 <t B for any advantage of the proposed tme doman decomposton scheme to be happened when comparng wth the classcal spatal doman decomposton, and hence one would requre m P t 1 < T (σ2 (P 1) 2 +r) P t 1,.e. m P < T (σ2 (P 1) 2 + r) (14) P When m s an ntegral multple of P, one obtans (15) m<t(σ 2 (P 1) 2 + r) In other words, one requres the number of parameters n nverse Laplace transforms to be smaller than the number of tme steps nvolved n spatal doman decomposton. Note that the domnant term n the nequalty s obvously T. Snce typcal values of m s usually much smaller than s/h for an acceptable accuracy of nverse Laplace transform [2], the nequalty (15) s easly satsfed wth typcal ranges of 0.05 <σ<0.45 and 0.6 <r<1.1. Therefore for a gven value of P,the nequalty offers only a very mld restrcton. Hence the tme doman decomposton method proposed n ths paper has advantage over the classcal spatal doman decomposton method for European call optons. From (15), we have mt 1 <T(σ 2 (P 1) 2 ) t1 P whch combnes wth (13) to gve t o1 < T (σ2 (P 1) 2 + r) (16) t o P The rato governs the number of tme levels τ k to be allowed n each workstaton n order that the nequalty t A2 <t B remans vald. Supposng each parametrc equaton has N grd ponts nvolved n the dscretsaton, the total number of floatng pont operatons nvolved n (5) can be easly counted as 2N + Nm.Also, supposng some of floatng pont operatons n (11) can be done once for all, the total number of floatng pont operatons for the update of nteror boundary condtons

6 TIME DOMAIN DECOMPOSITION IN FINANCIAL MODELLING 491 s counted as 18P. Hence (16) become (17) m< 18T N (σ2 (P 1) 2 + r) 2 It can be checked that the nequalty s easly satsfed wth the above typcal ranges of σ and r. 5. Conclusons A parallel algorthm based on Laplace tranform of the tme doman nto a set of parametrc equatons s developed for European call opton. Dstrbuted computng may be appled to solve the parametrc equatons concurrently. An nverse Laplace transform based on Stehfast method s appled to retreve the soluton. The method s compared wth classcal spatal doman decomposton. A prelmnary analyss shows that the proposed method has advantage over the spatal doman decomposton method. References 1. Centre for Numercal Modellng and Process Analyss - Unversty of Greenwch, Physca - user gude, 1995, Beta Release AJ Daves, J Mushtaq, Radford LE, and Crann D, The numercal Laplace transform soluton method on a dstrbuted memory archtecture, Applcatons of Hgh Performance Computng n Engneerng V (H Power and JJ Casares Long, eds.), Computatonal Mechancs Publcatons, 1997, pp CN Dawson, Q Du, and TF Du Pont, A fnte dfference doman decomposton algorthm for the numercal soluton of the heat equaton, Math Comp 57 (1991), H Stehfast, Numercal nverson of Laplace transforms, COMMACM13 (1970), D V Wdder, The Laplace transform, Prnceton Unversty Press, Prnceton, P Wlmott, S Howson, and J Dewynne, The mathematcs of fnancal dervatves, Press Syndcate of the Unversty of Cambrdge, New York, Department of Mathematcs, Unversty of Hertfordshre, Hatfeld, UK E-mal address: D.Crann@herts.ac.uk Department of Mathematcs, Unversty of Hertfordshre, Hatfeld, UK E-mal address: A.J.Daves@herts.ac.uk School of Computng & Mathematcal Scences, Unversty of Greenwch, Wellngton Street, Woolwch, London SE18 6PF, UK E-mal address: C.H.La@greenwch.ac.uk Quanttatve Research and Tradng Group, the Chase Manhattan Bank, 125 London Wall, London EC2Y 5AJ, UK E-mal address: Swee.Leong1@chase.com