Proceedigs of the World Cogress o Egieerig 29 Vol II A Hybrid Fiite Differece Method for Valuig America Puts Ji Zhag SogPig Zhu Abstract This paper presets a umerical scheme that avoids iteratios to solve the oliear partial differetial equatio system for pricig America puts with costat divided yields. Upo applyig a frotfixig techique to the Black-Scholes partial differetial equatio, a predictor-corrector fiite differece scheme is proposed to umerically solve the discrete oliear scheme. I the compariso with the solutios from articles that cover zero divided ad costat divided yields cases, our results are foud accurate. The curret method is coditioally stable sice the Euler scheme is used, the covergecy property of the scheme is show by umerical experimets. Keywords:America Optios, Predictor-Corrector, Fiite Differece Method, Black-Scholes Equatio 1 Itroductio Optios are the most commo securities that are frequetly bought ad sold i today s fiacial markets. Uder the Black-Scholes partial differetial equatio (PDE) framework, Merto [1] casts the valuatio problem of America optios as a free-boudary problem i 1973. Ever sice the, there have bee two kids of approximatio methods i the literature, to solve the freeboudary problem associated with the valuatio of America optios. Oe approach is the aalytical approximatio method, e.g. the Quasi-aalytical formula [2]. The other oe is the umerical method, such as the Biomial Method [3], which are quite preferred by market practitioers, as they are usually much faster with acceptable accuracy. I the last decade, various umerical methods have bee preseted by usig the fiite differece method (FDM), to solve the pricig problems of America optios. For istace, Wu ad Kwok [5] use a multilevel FDM to solve the oliear Black-Scholes PDE after applyig a frotfixig techique [6], they adopt a so-called frot-fixig techique or Ladau trasform [6] to fix the optimal exercise boudary o a vertical axis. To tackle the oli- Cetre for Computatioal Fiace ad Ecoomic Agets, Uiversity of Essex, Uited Kigdom. Email:jzhagf@essex.ac.uk. The author gratefully ackowledges the fiacial support from the EU Commissio through MRTNCT263427 COMISEF for attedig this coferece. School of Mathematics ad Applied Statistics, Uiversity of Wollogog, Australia. Email: sogpig zhu@uow.edu.au ear ature of America optio pricig problems, which is explicitly exposed after applyig the frot-fixig techique [6] to the origial Black-Scholes PDE, they employ a two-level discretizatio scheme i time. However, sice the scheme is a multilevel discretizatio scheme, the iformatio at more tha oe time step is eeded at the begiig to start the computatio, which is referred to as the iitializatio for multilevel schemes i literature. The multilevel scheme of Wu ad Kwok [5] motivates us a simpler versio, while maitais the same level of computatioal accuracy. To avoid the iitializatio ad iteratio, we propose a oe-step scheme based o a predictio-correctio framework. The approach adopts a predictor-corrector fiite differece scheme at each time step to covert the oliear PDE to two liearized differece equatios associated with the predictio ad correctio phase respectively. The predictor is costructed by a explicit Euler scheme, whereas the corrector is desiged with the Crak- Nicolso scheme. The predictor is used oly to calculate the optimal exercise price, as the literature shows that it is far more difficult to calculate the optimal exercise price with a high accuracy. The predicted optimal exercise price is the corrected i the correctio phase together with the calculatio of the optio prices. The scheme maximizes the use of computatioal resources, as a high accuracy of the computed optio price is easy to achieve as log as a high accuracy ca be achieved i the computatio of the optimal exercise price. The efficiecy i the scheme results from the fact that oly oe set of liear algebraic equatios eeds to be solved at each time step. The paper is orgaized as follows. Sectio 2 itroduces the PDE system cocerig the valuatio of America put optios. Sectio 3 presets a predictor-corrector scheme for computig the optimal exercise prices ad the optio values. I Sectio 4, some umerical examples are give to demostrate the covergece ad accuracy of the ew scheme. Sectio 5 draws coclusios. 2 Partial Differetial Equatio System This paper cosiders a geeral case i which a costat divided yield is associated with the uderlyig asset ad adopt the PDE give i Merto [1]. Let V deote the value of a America put optio, which is a fuctio of ISBN:978-988-1821-1- WCE 29
Proceedigs of the World Cogress o Egieerig 29 Vol II the value of uderlyig asset S ad the time t. The value of a America put optio also depeds o the followig parameters: σ, the volatility of the uderlyig asset; T, the life time of the cotract; X, the strike price; r, the risk-free iterest rate; D, the divided yield. Without loss of geerality, we assume that both the riskfree iterest rate ad the divided yield be costats. The fuctios ca be easily modified for the cases whe they are some kow fuctios of time ad asset values. Sice America optios ca be decomposed ito its Europea couterparts plus a early exercise premium, this early exercise premium is associated with the extra right embedded i America optios i compariso with its Europea couterparts. Wilmott et al. [9] show that there are two boudary coditios of the optimal exercise price S = S f (t) for America optios: { V (Sf (t), t) = X S f (t), V S (S (1) f (t), t) = 1. To close the system, aother boudary coditio at the ed of large asset value, i.e. the payoff of the cotract at the expiry is ecessary, lim V (S, t) =, (2) S ad the termial coditio for a put optio is V (S, T ) = max{x S, }. (3) I summary, the differetial system for pricig America put optios ca be writte as: V t + 1 2 σ2 S 2 2 V S + (r D 2 )S V S rv =, V (S f (t), t) = X S f (t), V S (S f (t), t) = 1, (4) lim S V (S, t) =, V (S, T ) = max{x S, }. To solve the differetial system Eq. (4) effectively, we ormalize all variables i the system by itroducig the followig scale of variables, V = V X, S = S σ2 X, τ = (T t) 2, γ = 2r σ, 2 D = 2D σ, S 2 f (τ) = S f (T 2τ/σ 2 ) X. After ormalizig Eq. (4), droppig the primes, ad imposig the Ladau trasform [6], x = l S S f (τ), (5) the origial system becomes: P τ 2 P x + (γ D 1) P 2 x + γp = P 1 ds f (τ) x S f (τ) dτ, P (, τ) = 1 S f (τ), P x (, τ) = S f (τ), lim x P (x, τ) =, P (x, ) =. (6) After this rather simple maipulatio, the oliear ature of the problem is explicitly exposed i the ihomogeeous term o the right had side of Eq. (6), which cosists the product of the Delta of the ukow optio price uder the Ladau trasform, the time derivative of the ukow optimal exercise boudary S f (τ) ad its reciprocal. Oe should ote that we have replaced the ukow fuctio V (S, t) i Eq. (4), with a ew ukow fuctio P, which is defied as P (x, τ) = V (S(x, (τ)), τ) through the trasform defied i Eq. (5). This is to facilitate the itroductio of a relatio betwee P (, τ) ad the S f (τ) o the boudary x =, which is used to desig the predictor of the umerical scheme. Moreover, oe should also ote that the trasform i Eq. (5) oly holds if S f (τ) >. This coditio poses o problem sice it is easy to show that the S f (τ) for a America put optio is a mootoically decreasig fuctio of τ; the miimum value S f (τ) is the optimal exercise price of the correspodig perpetual cotract. For a perpetual America put o a costat divided yield payig asset, this value was show as follows: lim S f (τ) = η + η2 + 4γ τ 2 + η + η 2 + 4γ, (7) with η = γ D 1. It is the very trivial to show that S f (τ) > for ay η values. Therefore, the differetial system Eq. (5) defies a well-posed problem, other tha a well-kow sigular poit at τ = (see Barles et al. [1]). We ow propose a efficiet ad accurate umerical scheme to solve this system. 3 The Predictor-Corrector FDM Scheme This sectio presets the predictor-corrector scheme. We propose to solve the oliear PDE i differetial system Eq. (6) i two phases withi a time step, a predictio phase i which a iitial guess of the S f (τ) is worked out before its fial value is calculated together with the optio value P (x, τ) i the correctio phase of the scheme. Begiig with trucatig the bouded x domai, as well as the time domai τ, the computatioal domai is discretized with uiformly spread M + 1 grids placed i the x directio ad N + 1 grids i the τ directio (i.e., M ad N are the umber of steps i these two directios, respectively). For the easiess of presetatio, we deote the step legth i the x directio by x = x max M ad that i the τ directio by = τexp N, i which τ exp is the ormalized teor of the cotract with respect to half of the variace of the uderlyig asset, i.e., τ exp = T σ 2 /2. Cosequetly, the value of ukow fuctio P at a grid poit is deoted by Pm with the superscript deotig the th time step ad the subscript m deotig the mth log-trasformed asset grid poit. To facilitate the umerical computatio, we derive a ad- ISBN:978-988-1821-1- WCE 29
Proceedigs of the World Cogress o Egieerig 29 Vol II ditioal boudary coditio to costruct our predictorcorrector scheme. This coditio is ot idepedet from all those boudary coditios prescribed i Eq. (6). Rather, it is derived by makig use of the PDE i Eq. (6) as well as the boudary coditios that have already made the system closed. Firstly, we take a partial derivative with respect to τ o both sides of the first boudary coditio i Eq. (6), which yields P τ (, τ) = ds f (τ). (8) dτ I fact, oe easily shows that Eq. (8) is cosistet with the coditio V τ (S f (τ), τ) = i Buch ad Johso s paper [7]. The, if we evaluate the PDE i Eq. (6) at x =, utilizig Eq. (8) ad the secod boudary coditio i Eq. (6), we obtai 2 P x 2 (D + 1)S f (τ) + γ =, if τ >. (9) x= Eq. (9) reveals a relatioship of the put optio price ad the optimal exercise price at ay time, except o the expiry day. This relatio is importat to our scheme i elimiatig the value of the ukow fuctio defied o the fictitious grid poit ear the boudary x =, whe the secod-order cetral differece scheme is applied. The reaso that it is oly valid for τ > is the iheret sigular behavior of the Black-Scholes PDE at τ = (see Barles et al. [1]). Applyig a secod-order cetral differece scheme to the equatio, oe has the asset price discretizatio i the x directio. Eq. (9) ad the boudary coditios i Eq. (6) is writte as P 1 2P + P 1 x 2 (D + 1)S f + γ =, (1) ad P P 1 P 1 P = 1 S f, 2 x = S f, M =, Pm =, (11) respectively. Upo elimiatig the fictitious odal value P 1 from Eq. (1) ad the secod equatio i Eq. (11), we obtai a relatio betwee S f ad P 1 at the ( + 1)th time step as P1 = α βs f, (12) i which α = 1 + γ 2 x2 ad β = 1 + x + D+1 2 x2. Eq. (12) is used i the predictor ad corrector costructio. Predictor: The predictor is costructed by usig the explicit Euler scheme to calculate a guessed value of S f, which is deoted as ˆ S f. Applyig the explicit Euler scheme to the PDE i Eq. (6) results i ˆP 1 P 1 γp 1 = P 2 P 2 x P 2 2P1 + P x 2 (γ D 1) P 2 P 2 x + 1 Sˆ f Sf, (13) S f which is coupled with Eq. (12) to geerate the S ˆ f value. The boudary coditio of ˆP used i the corrector is also predicted here; with the calculated Sˆ f value, ˆP is calculated from the first equatio i Eq. (11), which is othig but the payoff fuctio. Like the predicted S ˆ f value, this predicted boudary value of ˆP will also be corrected oce the S ˆ f is corrected i the followig corrector scheme. Corrector: The corrector is based o the Crak- Nicolso scheme, applied to the liearized PDE i Eq. (6). The liearlizatio is desiged with a alteratig term beig valued at the curret time step i compariso with that i the predictor. I the predictor, we let the time derivative of the S f i the oliear ihomogeeous term be valued at the curret time step, whereas ow we let the asset price derivative of P be valued at the curret time step through the Crak-Nicolso scheme. This alteratig approach, ispired by the idea of the ADI approach i solvig two dimesioal time-depedet PDEs [11], has a advatage of reducig the umerical errors iduced i the predictio-correctio process. The fiite differece scheme used for the corrector is P m Pm P m+1 + Pm 2 2Pm + Pm 1 + P m+1 2Pm + Pm 1 2 x 2 + γ P m (γ D 1) P m+1 P m 1 + P m+1 Pm 1 + 4 x = P m+1 P m 1 + P m+1 Pm 1 2 4 x Sˆ f + Sf S ˆ f Sf. (14) I Eq. (14), m value starts from 1 to M 1, which idicates that M 1 equatios are solved simultaeously to obtai the corrected optio values at the ( + 1)th time step. P1 is obtaied upo solvig Eq. (14). The, by meas of Eq. (12), the ewly-obtaied P1 is used to correct the S f, which is the used to correct the P value before it is used i the calculatio of the ext time step. Ad Eq. (14) ca be writte i matrix form which is a more codesed way for Matlab computatio. This predictor-corrector process is repeated util the expiry time is reached. We solve these matrix equatios i Matlab, Versio 7 o a Itel P4 machie. ISBN:978-988-1821-1- WCE 29
Proceedigs of the World Cogress o Egieerig 29 Vol II 4 Numerical Examples Although the Crak-Nicolso scheme for the corrector is ucoditioally stable [11], our predictor-corrector fiite differece scheme is oly coditioally stable sice the explicit Euler scheme for the predictor is coditioally stable. I this sectio, the coditioal stability of our approach, as well as the accuracy shall be verified empirically. 4.1 Discussio o Covergece For the liearized system, the proof of the cosistecy is trivial through the applicatio of Taylor s expasio ad thus is omitted here. A theoretical proof the stability for the liearized system, o the other had, is ot so trivial because of the presece of the sigularity at τ = (see Barles et al. [1]). Therefore, we establish a stability criterio empirically. Based o prelimiary umerical experimets, we were coviced that the stability criterio 1 should be imposed i the selectio of time step x 2 legth for a give grid size i the x directio. Betwee the optio price ad the optimal exercise price, the latter is far more difficult to calculate accurately; oce the S f (τ) is determied accurately, the calculatio of the optio price itself is straight forward. Therefore, i this subsectio we focus o the calculatio of the S f (τ) first. The example we chose for our umerical tests has bee used by researchers for the discussio of America puts o a asset without ay divided paymet [4, 5]. The relevat parameters are: the strike price X = $, the iterest rate r = 1%, the volatility of the uderlyig asset σ = 3% ad the teor of the optio beig oe year. I this subsectio, we focus oly o the zero-divided case, i.e., we set the costat divided yield to zero. For the coveiece of those readers who prefer to see the results i dimesioal form, all results preseted i this sectio are those associated with the origial dimesioal quatities before the ormalizatio process was itroduced. Firstly, we examied a poit-wise covergece by focusig o a specific poit of the S f value first. As a idicator, the differeces of S f values at a specific time to expiry, say 1 year, are calculated with time step size beig cosecutively halved. Table 1 shows the differeces of the computed S f values with the total umber of grid poits i the x directio beig fixed to 51, while the umber of time step itervals is cosecutively doubled from 2 to 32 (the time step size is cosecutively halved). Oe should ote that i Table 1, the differece refers to the absolute chage i S f values whe the time step size is halved, while the ratio refers to the ratio of successive differeces. Theoretically, the order of covergece is related to calculated ratio by ratio = 2 k, i which k is the order of covergece. Clearly, whe the grid size i the x directio is fixed, the ratios of the differeces of two S f values at τ = 1 year with two cosecutive calculatios of Table 1: Ratios for the order of covergece i time Time steps S f ($) differece ratio 2 76.126 4 76.121.46 76.12.125 3.261 16 76.12.6 2.72 32 76.119.3 2.36 Table 2: Ratios for the order of covergece i asset price Grid itervals S f ($) differece ratio 76.12 76.1.317 2 76.16.1254 2.927 4 76.164.3342 3.68 76.164.1114 3. time step legth beig halved ideed approach 2, which idicates that our scheme is ideed of the first order i time. The we fixed the time step size to = τ max 16 istead ad examie the ratios of the of the differeces of two S f values at τ = 1 year with the two cosecutive calculatios of x grid legth beig halved, we fid that these ratios are close to 3, as show i Table 2. This idicates that the order of covergece i the x directio is certaily higher tha oe but lower tha theoretically predicted 2d order covergece of the Crak-Nicolso scheme. Oe plausible reaso for this is that the errors itroduced i the predictor somehow reduced the order of covergece i the x directio a bit, so that ow it is of a order of oe ad half rather tha two. Havig discussed the poit-wise covergece, we tested the covergece of the ew scheme o the etire solutio of S f. We first ra our code with a extremely fie grid, e.g., the N ad M are set up as 12,4 ad 1,, respectively. Naturally, this takes a log time to compute. But, oce the S f values are computed o this fie grid, we used these values as the referece values to verify the covergece of computed S f values based of some coarse grid. To measure the overall differece betwee the results of the coarse grid ad those of the fiest grid, we use two error measures. The root mea square absolute errors (RMSAE), which is usually referred as root mea square errors. I order to tell relative errors, a modificatio of root mea square errors is used here, we refer it as the root mea square relative errors (RMSRE). The two measures are defied respectively as RMSAE = 1 I I (ã i a i ) 2, (15) i=1 ISBN:978-988-1821-1- WCE 29
Proceedigs of the World Cogress o Egieerig 29 Vol II.8.7 Optimal Exercise Price ($) RMSAE.5.4.3.2 Solutio of curret method Solutio of Zhu s Method O 95.6 9 85.1 Time Steps 75 1 2 9 7 6 4 3 2.1.2.3 1.4.5.6.7.8.9 1 Time to Expiry (Year) Grid Numbers Figure 1: The RMSAE with Icreased Grid umbers Figure 3: Compariso: Two Optimal Exercise Boudaries.1.9 95 the curret solutio the solutio of perpetual case with D=.8 Optimal Exercise Price ($) RMSRE.7.6.5.4.3.2.1 9 85 75 7 Time Steps 1 2 9 7 6 4 3 2 1 65 Grid Numbers 2 4 6 8 1 12 14 16 18 2 Time to Expiry (year) Figure 2: The RMSRE with Icreased Grid Numbers Figure 4: Optimal Exercise Prices with A Log Teor v u I u 1 X a i ai RM SRE = t ( )2, I i=1 ai lead to a uiform order of covergece for the calculatio of the optimal exercise prices. (16) where a i s are the odal Sf values associated with coarse grid; ai s are the Sf values associated with fiest grid ad I is the umber of sample poits used i the RMSAE ad RMSRE. I the followig experimets, I was set to be for all the results show i the followig diagrams. By demostratig the RMSAE ad RMSRE, we obtai a overall measure of the covergece to make sure what we observed from aalyzig the order of covergece previously based o oe poit oly is also true for other grid poits. Figure 1 ad 2 show the RMSAE ad RMSRE respectively, for the Sf values whe the umber steps i the x directio ad the τ directio are gradually icreased. As ca be clearly see from these figures, the RMSAE reduces by early 1 folds whe the grid size chages from N = 1 ad M = 1 (with RM SAE =.254) to N = 2 ad M = (with RM SAE =.28). I fact, the differece betwee the results obtaied with a coarse grid ad those obtaied with the fiest grid is better reflected by the RMSRE, which shows very similar tred as that of the RMSAE; whe the umber of grid has icreased to N = 2 ad M =, the RMSRE has reached.3%, which is quite a acceptable accuracy i compariso with the solutio based o a extremely refied grid. This cofirms that aalysis of covergece order preseted earlier ca be exteded to other grid poits as well. Therefore, we are cofidet that the scheme ca ISBN:978-988-1821-1- 4.2 Discussio o Accuracy This subsectio proves that the umerical solutio of the liearized PDE does coverge to that of the origial oliear PDE system. We firstly compare our results with Zhu s semi-closed solutio i the o-divided case [4]. If oe ca demostrate that the coverged solutio approaches Zhu s solutio, it is cofidet to say that the liearizatio process we took before the predictor-corrector scheme was applied. Figure 3 shows such a compariso with the optimal exercise boudaries beig computed by usig the curret scheme with N = 12, 4, M = 1, ad Zhu s solutio [4]. As it ca be see from this figure, the two results agree with each other almost perfectly, especially whe the time to expiry icreases. A close examiatio reveals that the curret approach slightly uderestimates the Sf values whe the time is close to expiry. Whe the time to expiry icreases from to 1 year, the uder-estimatio gradually improves from a roughly 2.16% at the time to expiry (T t) beig.67 year, to.5% at the time to expiry beig 1 year. Give the presece of the well-kow sigularity at the expiry [1], which is ot possible for ay umerical scheme to deal with, the performace of the proposed umerical scheme is certaily very satisfactory. Aother test that a good umerical scheme must pass is WCE 29
2 4 6 12 14 16 1 2 Proceedigs of the World Cogress o Egieerig 29 Vol II Optio Value ($) 9 7 6 4 3 2 1 Curret Approach o Oosterlee et al. s Grid Stretchig Method (25) 5 Coclusio This paper presets a ew predictor-corrector scheme to umerically tackle America put optio pricig with costat divided yields. The key feature of the curret scheme is its high efficiecy sice there is either iteratio or iitializatio required. Through a couple of umerical examples, we have demostrated the covergecy ad accuracy of the proposed scheme. Refereces Uderlyig Asset Price ($) Figure 5: The Optio Value with D = 5%, T = 1 year that the optimal exercise price asymptotically approaches to that of its correspodig perpetual couterpart whe the lifetime of the put optio becomes ifiite. I this extreme case, it was reported i the literature that some approaches lead to a oscillatory ad o-mootoic optimal exercise price whe the lifetime of a optio is very log. We have prologed the lifetime to 2 years to artificially make the optio i this example a log-lifetime optio. Agai, usig the fiest grid N = 12, 4 ad M = 1,, We calculated the S f (τ) ad plotted its value agaist the theoretical perpetual optimal exercise price give i Eq. (7), as show i Figure 4. Clearly, the umerical solutio exhibits a ice asymptotical approach to the optimal exercise price of the correspodig perpetual put optio; o oscillatio was observed at all. This shows that our scheme is very stable ad ca be used for for log-lifetime optios as all. 4.3 Optio Prices i Costat Divided Yield Cases This subsectio presets optio prices from the curret method for costat divided yields case discussio. The relevat optio parameters used i the followig example are the same as those used i the o-divided case, except the costat divided yield D is ow set at 5%. The results preseted i this sectio were obtaied usig a grid resolutio of N = 2 ad M =. Figure 5 shows a compariso of the optio values calculated by usig the curret approach ad the oes from Oosterlee et al. [12], who employ the so-called Grid Stretchig Method. The optio values i Figure 5 are plotted agaist the uderlyig asset prices at time to expiry beig 1 year. The agreemet betwee the two appears to be excellet, reiforcig the fact that oce the optimal exercise price ca be accurately calculated, the accurate calculatio of the optio price itself aturally follows. [1] Merto, R.C., The Theory of Ratioal Optio Pricig Joural of Ecoomics ad Maagemet Sciece, V1, pp. 141 183, 1973 [2] Kim, I. J., The Aalytic Valuatio of America Puts The Review of Fiacial Studies, V3, pp. 547 572, 199 [3] Cox, J., S. Ross, M. Rubistei, Optio Pricig - A Simplified Approach Joural of Fiacial Ecoomics, V7, pp. 229 236, 1979 [4] Zhu, Sogpig, A exact ad explicit solutio for the valuatio of America put optios Quatitative Fiace, V6, pp. 229 242, 26 [5] Wu, Lixi., Kwok, Y. K. A frot-fixig Fiite Differece Method for the valuatio of AmericaOptios Joural of Fiacial Egieerig, V6, pp. 83 97, 1997 [6] H. G. Ladau, Heat Coductio i a Meltig Solid Quarterly Applied Mathematics, V8, pp. 81 95, 19 [7] Buch, D. S., Johso, H., The America Put Optio ad Its Critical Stock Price The Joural of Fiace, V5, pp. 2333 2356, 2 [8] Tavella, Domigo, Pricig Fiacial Istrumets, the fiite differece method, Joh Wiley ad Sos, Ic., 2 [9] Wilmott, Paul., Dewye, Jeff., Howiso, Sam. Optio Pricig, Oxford Fiacial Press, 1993 [1] Barles, Guy., Burdeau, Julie, Romao, Marc., Samsce, Nicolas. Critical Stock Price Near Expiratio Mathematical Fiace, V5, pp. 77 95, 1995 [11] Golub, Gee H., Ortega, James M. Scietific Computig ad Differetial Equatios, Academic Press, Ic., 1992 [12] Oosterlee, Corelis W., Leetvaar, Coeraad C.W., Huag, Xizheg Accurate America Optio Pricig by Grid Stretchig ad High Order Fiite Differeces, Workig papers, DIAM, Delft Uiversity of Techology, the Netherlads, 25 ISBN:978-988-1821-1- WCE 29