BOUNDS FOR THE PRICE OF A EUROPEAN-STYLE ASIAN OPTION IN A BINARY TREE MODEL

Transcription

1 BOUNDS FOR THE PRICE OF A EUROPEAN-STYLE ASIAN OPTION IN A BINARY TREE MODEL HUGUETTE REYNAERTS, MICHELE VANMAELE, JAN DHAENE ad GRISELDA DEELSTRA,1 Departmet of Applied Mathematics ad Computer Sciece, Faculty of Scieces, Ghet Uiversity, Krigslaa 281 S9, 9000 Ghet, Belgium huguette.reyaerts@uget.be Departmet of Applied Ecoomics, Faculty of Ecoomics ad Applied Ecoomics, Catholic Uiversity Leuve, Naamsestraat 69, 3000 Leuve, Belgium Departmet of Mathematics, ISRO ad ECARES, Uiversité Libre de Bruxelles, CP 120, 1050 Brussels, Belgium Abstract Ispired by the ideas of Rogers ad Shi 1995, Chalasai, Jha & Varikooty 1998 derived accurate lower ad upper bouds for the price of a Europea-style Asia optio with cotiuous averagig over the full lifetime of the optio, usig a discrete-time biary tree model. I this paper, we cosider arithmetic Asia optios with discrete samplig ad we geeralize their method to the case of forward startig Asia optios. I this case with daily time steps, the method of Chalasai et al. is still very accurate but the computatio ca take a very log time o a PC whe the umber of steps i the biomial tree is high. We derive aalytical lower ad upper bouds based o the approach of Kaas, Dhaee & Goovaerts 2000 for bouds for stop-loss premiums of sums of depedet radom variables, ad by coditioig o the value of the uderlyig at the exercise date. The comootoic upper boud correspods to a optimal superhedgig strategy. By puttig i less iformatio tha Chalasai et al. the bouds lose some accuracy but are still very good ad they are easily computable ad moreover the computatio o a PC is fast. We illustrate our results by differet umerical experimets ad compare with bouds for the Black & Scholes model 1973 foud i aother paper Vamaele et al We otice that the itervals of Chalasai et al. do ot always lie withi the Black & Scholes itervals. We have proved that our bouds coverge to the correspodig bouds i the Black & Scholes model. Our umerical illustratios also show that the hedgig error is small if the Asia optio is i the moey. If the optio is out of the moey, the price of the superhedgig strategy is ot as adequate, but still lower tha the straightforward hedge of buyig oe Europea optio with the same exercise price. Keywords Comootoicity, Asia optios, superhedgig strategy. 1. INTRODUCTION The biomial tree model of the Cox-Ross-Rubestei CRR 1979 model ca be cosidered as a discrete-time versio of the Black & Scholes B&S 1973 model. The Europea optio prices have well-kow formulae i this model. For example whe the life time T of the optio is divided ito N time steps of legth T/N, we have for a Europea call optio: ECK, T = e rt N N p 1 p N S0u d N K +, 1 where r is the risk-free rate of iterest, St is the price of the uderlyig asset at time t, p the risk-eutral probability that the price goes up with a factor u = 1/d = expσ T/N with σ the volatility of the uderlyig. It is kow that p = ert /N d u d. Although i the 1 This research was carried out while the author was employed at the Ghet Uiversity 1

2 CRR-model, America optio pricig turs out easily umerically see e.g. Hull 1989, the pricig of Asia optios remais a ope questio. Chalasai et al have worked out a umerical recipe to obtai a very accurate price-iterval for the Asia optio price i the CRR-model. Their method is based upo groupig paths with the same geometric stock-price average that ed at the same stock price. This is realised by coditioig o some well chose vector Z = XT, χt, where XT deotes the umber of icreases util time T ad χt is defied by χt = N l=1 Xl T/N. Deotig by AV T the average stock price at time T i the CRR-model with N time steps, i.e. AV T = 1 N N S0u X T/N d N X T/N 2 it follows by Jese s iequality ad from Rogers ad Shi 1995 that: E[E[AV T Z] K + ] E[AV T K + ] E[E[AV T Z] K + ] + ε 3 where ε = 1 2 E [ Var [AV T Z] ]. 4 Multiplyig by the discout factor e rt relatio 3 prevails a lower ad a upper boud for the Asia optio price. Chalasai et al. stipulate that for N large eough say 30 to 40 those bouds give a precise price-iterval for the Asia optio i the Black ad Scholes framework with a cotiuous averagig over the whole lifetime of the optio. I what follows we focus o Europea-style Asia optios with discrete samplig which are forward startig, i.e. the averagig has ot yet started at the begiig of the lifetime of the optio ad takes oly the daily prices ito accout durig the time iterval [T + 1, T ]. Recallig that XT i stads for the umber of icreases util time T i, i = 0, 1,..., 1, we deote this arithmetic average i the CRR-model with time step oe day as 1 S = 1 S0u XT i d T XT i 5 ad the price of the Asia optio is give by AC, K, T = e rt E[ 1 S K +], 6 where we work uder the risk-eutral probability p = er d u d. The method of Chalasai et al ca easily be geeralized to this case of forward startig Asia optios ad is very accurate for the CRR-model. However, the computatios of the bouds are very time cosumig o a PC i particular whe the umber of steps i the biomial tree is high. Note that the umber of time steps caot be chose freely sice it is at least equal to the umber of samplig days of the lifetime of the optio, for example 60 or 120 days. Thus whe we average say for example over the last 10 days, we have to keep track of the whole tree of the uderlyig asset for the whole lifetime of the optio. The paper is composed as follows. I sectio 2 lower ad upper bouds for the price of arithmetic Asia optios with discrete samplig are derived. I sectio 3 the covergece of those bouds to the correspodig bouds i the Black & Scholes settig is cosidered. I sectio 4 we commet a umerical example, with special attetio to the speed of the calculatios o PC. Sectio 5 cocludes. 2

3 2. BOUNDS FOR THE PRICE OF ARITHMETIC ASIAN OPTIONS WITH DISCRETE SAMPLING 2.1 COMONOTONIC BOUNDS AND BOUNDS WITH CONDITIONING ON THE FINAL VALUE OF THE UNDERLYING LOWER BOUND I order to obtai a lower boud i the biomial case, we coditio o the value of the asset price at the fial time T, that is equivalet to coditioig o the umber XT of icreases of the price process util maturity date T : AC, K, T e rt E[E[S XT ] K +] T = e rt E[ST i XT = ] K P [XT = ]. 7 Sice l ST i/s0 = XT i l u d + T i l d with XT i a biomial radom variable with parameters T i ad p, E[ST i XT = ] = ad the lower boud is foud to be: e rt T T p 1 p T mi,t i l=max0, i mi,t i l=max0, i + T i i S0u l d T i l l l T 8 T i i S0u l d T i l l l T K ROGERS AND SHI UPPER BOUND A upper boud ca be deduced by applyig the approximatio error 4 to the case of the average 1 S with XT as coditioig variable: where E[Var 1/2 [ 1 S XT ]] = T Var 1/2 [ 1 S XT = ]P [XT = ] 10 Var[ 1 S XT = ] = E[ 1 S 2 XT = ] E[ 1 S XT = ]2. 11 The secod term o the right-had side ca be obtaied from 8 while the first term ca be expressed as 1 2 E[ S XT = ] = 1 2 E [ S 2 T i XT = ] k=i+1 3 E [ST ist k XT = ], 12

4 where the first term i the last expressio is obtaied aalogously as i 8: E [ S 2 T i XT = ] = ad where the secod term equals E[ST ist k XT = ] = mi,t k s=max k,0 mit i s, s,k i t=max i s,0 mi,t i l=max0, i T i i S0 2 u 2l d 2T i 2l l l T 13 S0 2 u 2s+t d 2T i+k 2s+t k s k i s+t t i T. 14 Due to the fact that for some values of the coditioig variable XT, some terms vaish i the differece E[ 1 S K + XT ] E[ 1 S XT ] K +, ad thus ca be omitted, oe obtais a improved approximatio error depedet o K: where εk = T S mi <K S max, >K S mi = S0 Var 1/2 [ 1 S XT = ]P [XT = ], 15 d T i ad S max, = S mi u 2, 16 leadig to a improved upper boud for AC, K, T cosistig of the lower boud 9 plus the approximatig error εk i 15. This idea is i fact a atural oe ad Nielse ad Sadma 2003 used a similar idea i the B&S-settig INVERSE DISTRIBUTION FUNCTIONS Let X be a radom vector with cumulative distributio fuctio cdf hereafter F X. The the usual iverse fuctio of F X is the o-decreasig ad left-cotiuous fuctio X q, defied by X q = if {x R F Xx q}, q [0, 1], 17 with if = + by covetio; for ay α [0, 1], the o-decreasig α-mixed iverse fuctio of F X is defied as follows: α X q = α 1+ q + 1 α F q, q 0, 1, 18 X with the o-decreasig ad right-cotiuous fuctio + X p defied as X + X q = sup {x R F Xx q}, q [0, 1]. 19 4

5 2.1.4 COMONOTONIC UPPER BOUND I fiacial ad actuarial situatios oe ecouters quite ofte radom variables of the type S = Y i where the terms Y i are ot mutually idepedet, but the multivariate distributio fuctio of the radom vector Y = Y 0, Y 1,..., Y is ot completely specified because oe oly kows the margial distributio fuctios of the radom variables Y i. Kaas et al use comootoic couterparts to derive bouds for such sums. The comootoic couterpart S c of S with S = ST i, is the radom variable S c = ST i U U = d 0, 1-uiform. 20 The reasoig of Simo et al leads to the fact that the price of a Asia optio ca be bouded above by a optimal liear combiatio of Europea vailla call optios: AC, K, T 1 e ri EC [ α ST i F S ck, T i ] 21 where EC, is defied i 1 ad where α is determied by α S c F S ck = K. 22 From 22 ad the geometric iterpretatio see Dhaee et al of stop-loss premiums, oe obtais that for ay stochastic process St; t 0 for the risky asset, the followig iequality holds AC, K, T e rt e rt [ ] E ST i ST i F S + ck [ K F 1 S c F S ck ] 1 F S ck. 23 We apply this iequality i the case of the biary tree model where F, T i = P [ST i S0u d T i ] = for i = 0,..., 1. Hece, for 0 < q 1, we fid that T i p l 1 p T i l, 24 l=0 T i ST i q = S0u d T i I F 1,T i<q F,T i, 25 where I deotes the idicator fuctio, ad therefore { } F S ck = sup q 0, 1] ST i q K = sup q 0, 1] T i u d T i I F 1,T i<q F,T i K S

6 Further, we remark that due to comootoicity see Kaas et al S F c S ck = ST i F S ck T i = S0u d T i I F 1,T i<fs c K F,T i. 27 Clearly, F S ck will be oe of the cumulative distributio fuctios F, T i. Therefore, we first cocetrate o the orderig amog them for differet ad i. We cosider the cdf-values F, T i as elemets F i, = 0,..., T i, i = 0,...,, i a T + 1 -matrix F. The oly the elemets F i with T i are take ito accout. It is obvious that F i = 1 for = T i ad that F i > F i for >. Oe ca show that also F i > F i for i > i ad that F i = 1 pf,i+1 + p,i+1 =,i+1 < F i < F,i I other words, walkig through rows i oe colum of the matrix F, the value of the cdf icreases. The same happes whe walkig trough colums i oe row. This implies that give a cdf-value F, T i there are precisely two elemets i each colum such that F, T i lies betwee these elemets. Therefore, the double sum i 26 reduces to a sum with at most terms. I fact the problem i 26 turs dow to determie by a algorithm the largest q = F i, T i such that u i d T i i K/S0. 29 Note that at the same time, the sum i 25 reduces to oe term u i d T i i. Substitutig the computed relatios ito 23, we obtai a value for the upper boud. As metioed i Albrecher et al. 2003, this comootoic upper boud 21 correspods to the price of a static superhedgig strategy cosistig of buyig ad holdig Europea call optios o the same uderlyig asset, ad that mature at the respective exercise dates. Our umerical illustratio will show that the hedgig error is small if the Asia optio is i the moey. If the optio is out of the moey, the price of the superhedgig strategy is ot adequate, but still lower tha the straightforward hedge of buyig oe Europea optio with the same exercise date. The comootoic hedge is usefull because Europea call optios are typically available o the market. Moreover, amog all superhedgig strategies which are costructed as liear combiatios of Europea calls o the same uderlyig asset, the comootoic superhedge is optimal i the sece that it is the oe with the smallest cost. 2.2 IMPROVED COMONOTONIC UPPER BOUND We assume from ow o that b l=a x l = 0 for a > b. Let S u = i=1 ST i XT U U d = 0, 1-uiform. 30 6

7 the we obtai a improved comootoic upper boud, aalogously as i 21-23: with AC, K, T T e rt P [XT = ] = e rt T { E[ ST i ST i XT = FS u XT =K XT = ] + K S u XT = FS u XT =K 1 FS u XT =K } T p 1 p T mi,t i l=max0, i T i l i T l S0u l d T i l ST i XT = FS u XT =K + K S u XT = FS u XT =K 1 FS u XT =K } 31 By the comootoicity property, we have that S u XT = q = ST i XT =q 0 < q 1 32 ST i XT = q = mi,t i s=max0, i S0u s d T i s I F s 1,T i<q F s,t i, 33 where F s, T i = 0 s < max0, i s T i i l l T max0, i s < mi, T i l=max0, i 1 s mi, T i. 34 Hece, the cdf of S u give the evet XT = ca be foud from F S u XT =K = sup q 0, 1] mi,t i s=max0, 1 u s d T i s I F s 1,T i<q F s,t i K S0. 35 As for the comootoic upper boud, F S u XT = will be oe of the distributio fuctios F s, T i or will be zero i case of a empty set. The cdf s F s, T i ca also be ordered i a matrix which has similar properties as for the comootoic upper boud. Thus, a cdf-value F s, T i lies precisely betwee two elemets i each colum ad the sum 33 reduces to oe term ad hece the double sum 32 to at most terms. Oce we foud for each fixed, the largest q = F s, T i by a search algorithm, we ca easily compute 33, 32 ad hece obtai the value of the improved comootoic upper boud. 7

8 3. CONVERGENCE Whe we cosider steps with legth equal to a fractio of oe day, say 1/m day, ad let m go to ifiity, we ca prove that the lower boud ad the comootoic upper ad improved comootoic upper boud coverge to the correspodig bouds for a Europea-style arithmetic forward startig Asia optio with discrete samplig i the Black & Scholes settig, i.e. the uderlyig asset follows a geometric Browia motio, see Vamaele et al The proof follows the ideas of covergece of the CRR-model to the B&S-model. 4. A NUMERICAL EXAMPLE We illustrate our results by some umerical examples. We cosider the case of a Europeastyle arithmetic forward startig Asia optio with discrete samplig that i practice is used for protectio agaist price maipulatio. For this purpose, averagig over the last 10 days of the life time of the optio is sufficiet. Ituitively it is clear that this is a case where coditioig o the fial value of the uderlyig asset will perform well. Whe oe takes a loger averagig period ito accout, the weight of the fial value dimiishes ad hece also the quality of the bouds. This ca be see by comparig the results for a expiratio time T = 120 days but with differet averagig periods, amely = 10 i table 1 ad = 30 i table 3. I table 2 we report for the case of a expiratio time T = 60 days ad = 10 averagig days. I all cases, the risk-free iterest rate r equals 9% yearly. The iitial stock price S0 is fixed at 100. Further, we cosider several exercise prices K i the rage of 90 to 110 by steps of 5 or 10. We also study the ifluece of differet volatilities σ =0.2, 0.3 ad 0.4 o the quality of the bouds. From tables 1-3 we observe the followig: 1. The method of Chalasai is accurate for the CRR-model sice the lower LBC ad upper boud UBC coicide. 2. The lower boud LB with coditioig o the fial value of the uderlyig asset together with the comootoic upper boud CUB or with the improved comootoic upper boud ICUB gives a price-iterval which cotais the boud LBC=UBC. 3. The Rogers ad Shi approach for the upper boud LB +εk is of very bad quality i the sese that it leads to very large upper bouds that i all cases are larger tha CUB ad ICUB. 4. The relative error ICUB-LB/LB icreases with icreasig exercise price K which implies that the bouds are better for optios i-the-moey tha out-of-the-moey. 5. The absolute differece as well as the relative differece betwee the value LBC ad LB icreases with volatility or with, icreasig umber of averagig days for a fixed T or with decreasig life time of the optio for a fixed. 6. Chalasai et al remark that For N = 30 resp. N = 40 our algorithm computes the lower ad upper bouds i about 5 resp. 20 secods o a Su Ultra- SPARC workstatio. Speed is ideed very importat for traders who have to make their decisios i a few secods. But ormally they do t use a Su UltraSPARC but a PC. Note moreover that for the forward startig case the umber of steps N i the biomial tree equals T which is 60 or 120 i our example while, the umber of averagig dates, equals 10 or 30. This implies a higher CPU-time with their approach tha for the case T = N = 30 or 40 they reported. It turs out that i the forward startig case the calculatio time of our bouds o a Petium III with 128Mb Ram is much shorter. For example the lower boud is very fast, it eeded aroud 0.35 CPU-time secods whe T = 60 ad = 10, up to 4.6 CPU-time secods whe T = 120 ad = 30. The calculatio time of the superhedgig strategy equals 0.6 T = 60 ad = 10 to 2.9 T = 120 ad = 30 CPU-time secods. 8

9 Table 1: Bouds whe T = 120 ad = 10. T σ K LB LBC LBBS UBBS ICUB CUB LB = UBC +εk Table 2: Bouds whe T = 60 ad = 10. T σ K LB LBC LBBS UBBS ICUB CUB LB = UBC +εk Table 3: Bouds whe T = 120 ad = 30. T σ K LB LBC LBBS UBBS ICUB CUB LB = UBC +εk

10 I the tables, we icluded also a price iterval [LBBS,UBBS] for the correspodig Asia optio i the Black & Scholes settig, see Vamaele et al We may coclude that: 1. The price iterval [LB,ICUB] i the CRR-model cotais the price iterval [LBBS, UBBS] ad hece cotais ot oly the price of the Asia optio i the CRR-model but also i the B&S-model. 2. The method of Chalasai leads to a accurate price i the CRR-model but ot i the B&S-model. It gives oly a lower boud i the latter case. 5. CONCLUSIONS I this paper, we have cocetrated o the CRR-model with daily time step ad we have derived price itervals for arithmetic Asia optios with discrete samplig which are forward startig. Our method is based o comootoicity ad o coditioig o the fial value of the uderlyig asset. Moreover, it ca be show that our upper ad lower bouds i the CRR-model coverge to those of the B&S-model. We also geeralized the method of Chalasai et al to this settig. Their method turs out to be still very accurate but the computatios are very time cosumig. We further oticed that the Chalasai et al. results form a lower boud for the price of a arithmetic Asia optio with discrete samplig i the B&S-model, but that this lower boud is of a lesser quality tha the B&S lower bouds foud i Vamaele et al Moreover, amog all superhedgig strategies which are costructed as liear combiatios of Europea calls o the same uderlyig asset ad with exercise dates correspodig to the averagig dates, the comootoic superhedge is optimal i the sece that it is the oe with the smallest cost. As a coclusio, we ca suggest that if the cotiuous B&S-model is a acceptable model, the oe could better work with the B&S-bouds, see Vamaele et al If oe prefers to work i the CRR-model itself for pricig arithmetic Asia optios with discrete samplig which are forward startig, the it turs out that o a PC the calculatio time of our bouds is much shorter tha the oe of the bouds based o the geeralized method of Chalasai et al. ad is therefore recommeded. A iterestig questio is to look at America-style Asia optios i the CRR-model. ACKNOWLEDGEMENTS G. Deelstra, H. Reyaerts ad M. Vamaele would like to ackowledge the fiacial support by the BOF-proect of Ghet Uiversity. J. Dhaee ackowledges the fiacial support of the Oderzoeksfods K.U. Leuve GOA/02: Actuariële, fiaciële e statistische aspecte va afhakelikhede i verzekerigs- e fiaciële portefeuilles. REFERENCES [1] H. Albrecher, J. Dhaee, M. Goovaerts ad W. Schouters, Static Hedgig of Asia Optios uder Levy Models: The Comootoicity Approach, Research Report, Departmet of Applied Ecoomics, Kuleuve, OR 0365, [2] F. Black ad M. Scholes, The Pricig of Optios ad Corporate Liabilities, Joural of Political Ecoomy, 7, [3] P. Chalasai, S. Jha ad A. Varikooty, Accurate approximatios for Europea-style Asia optios, Joural of Computatioal Fiace, 14, [4] J. Cox, S. Ross ad M. Rubestei, Optio pricig. A simplified approach, Joural of Fiacial Ecoomics, 81,

11 [5] J. Dhaee, M. Deuit, M.J. Goovaerts, R. Kaas ad D. Vycke, The cocept of mootoicity i actuarial scieces ad fiace: theory, Isurace: Mathematics & Ecoomics, 31, [6] J. Hull, Optios, Futures ad other Derivative Securities, Pretice-Hall, Eglewood Cliffs, New Jersey [7] R. Kaas, J. Dhaee ad M.J. Goovaerts, Upper ad lower bouds for sums of radom variables, Isurace: Mathematics & Ecoomics, 27, [8] J.A. Nielse ad K. Sadma, Pricig Bouds o Asia optios, Joural of Fiacial ad Quatitative Aalysis, 382, [9] L.C.G. Rogers ad Z. Shi, The value of a Asia optio, Joural of Applied Probability, 32, [10] S. Simo, M.J. Goovaerts ad J. Dhaee, A easy computable upper boud for the price of a arithmetic Asia optio, Isurace: Mathematics & Ecoomics, 26, [11] M. Vamaele, G. Deelstra, J. Liiev, J. Dhaee ad M.J. Goovaerts, Bouds for the price of discretely sampled arithmetic Asia optios, workig paper, Ghet Uiversity