American Options Valuation: A Parsimoniously Numerical Approach

Transcription

1 Amercan Optons Valuaton: A Parsmonously Numercal Approach Tzyy-Leng Horng * Department of Appled Mathematcs Feng Cha Unversty, Tachung, Tawan Chh-Yuan Ten Department of Mathematcs Unversty of York, York, Unted Kngdom The early exercse property of Amercan optons changes the orgnal Black-Scholes equaton to an nequalty that cannot be solved va tradtonal fnte dfference method. Therefore, fndng the early exercse boundary pror to spatal dscretzaton (dscretzaton on underlyng asset) s a must n each tme step. Ths overhead slows down the computaton and the accuracy of soluton reles on f the early exercse boundary can be accurately located. A parsmonously numercal method based on fnte dfference method and method of lnes (MOL) s proposed here to overcome ths dffculty n Amercan optons valuaton. Partcularly, our method averts the otherwse necessary procedure of locatng the optmal exercse boundary before applyng fnte dfference dscretzaton. Ths method s more concse, effcent, flexble to all knds of pay-off, and easer to mplement for fnancal practtoners when compared wth many other methods. Computaton of Amercan put, Amercan call wth dvdend, Amercan strangle, and two-factor Amercan basket put optons are demonstrated n the current artcle. Keywords: Amercan optons, fnte dfference method, method of lnes, free boundary, optmal exercse boundary, Amercan strangle opton, two-factor Amercan basket put opton. 1. Introducton In last two decades, the problem of prcng Amercan optons has been nvestgated extensvely both n numercal methods and analytcal approxmatons. These two approaches both encounter the arduous challenge whch s known as a free boundary Address correspondence to Prof. Tzyy-Leng Horng, Department of Appled Mathematcs, Feng Cha Unversty, Tachung, Tawan; e-mal: tlhorng@math.fcu.edu.tw.

2 value problem arsng form the early exercse feature of Amercan optons. The dffculty assocated wth the valuaton of Amercan optons stems from the fact that the optmal exercse boundary must be determned as a part of the soluton. Unfortunately, the early exercse boundary cannot be solved n a close form, and, consequently, nor can the opton s value. Up to now, revewng all relevant lteratures, most efforts have been exerted manly on locatng the free boundary that sets the doman for the Black-Scholes equaton. For analytc approxmatons, Johnson (1983) adopted an nterpolaton scheme to prce the Amercan put opton for a non-dvdend payng stock. MacMllan (1986) used a quadratc approxmaton, whch nvolved solvng an approxmate partal dfferental equaton (PDE) for the early exercse premum, the amount by whch the value of an Amercan opton exceeds a European one, to evaluate an Amercan put opton. Geske and Johnson (1984) obtaned a valuaton formula for Amercan put opton expressed n terms of a seres of compound-opton functons. Followng MacMllan (1986), MacMllan et al. (1999) presented an effcent and accurate approxmate formula for prcng Amercan optons on a dvdend payng stock. For numercal approxmatons, the most popular numercal methods for prcng Amercan optons can be classfed to lattce method, Monte Carlo smulaton and fnte dfference method. Sure, besdes fnte dfference methods, there are other popular numercal method based on dscretzaton for solvng PDEs lke fnte element method, boundary element method, spectral and pseudo-spectral methods and etc. Here we just use fnte dfference to stand for methods of ths knd. In fact, fnte dfference method ranks as the most popular one among ts knd n fnancal engneerng. The lattce method s smple and stll wdely used for evaluatng Amercan optons. It was frst ntroduced by Cox et al. (1979), and the convergence of the lattce method for Amercan optons s proved by Amn and Khanna (1994). The Monte Carlo method s also popular among fnancal practtoners. It s appealng, smple to mplement for prcng European optons, and especally has advantage of prcng mult-asset optons. For prcng Amercan optons, Monte Carlo method requests some further modfcaton. Fu (1994 a,b) prced Amercan-style optons by usng Monte Carlo method n conjuncton wth gradent-based optmzaton technques. Duck et al. (005) proposed a technque whch generates monotoncally varyng data to enhance the accuracy and relablty of Monte Carlo-based method n handlng early exercse features. The applcaton of fnte dfference method to prce Amercan optons can be frst found n the papers by Brennan and Schwartz (1977, 1978) and Schwartz (1977). Jallet et al. (1990) showed the convergence of the fnte dfference method. A comparson of

3 dfferent numercal methods for Amercan optons prcng was dscussed n Broade and Detemple (1996), Geske and Shastr (1985). Generally, there stll exst some dffcultes n usng these numercal methods. For fnte dfference method, the dffculty arses from the early exercse property, whch changes the orgnal Black-Scholes equaton to an nequalty that cannot be solved va tradtonal fnte dfference process. Therefore, fndng the early exercse boundary pror to spatal dscretzaton (dscretzaton on underlyng asset) s a must n each tme step. Ths overhead slows down the computaton and the accuracy of soluton reles on f the early exercse boundary can be accurately located. For lattce method and Monte Carlo smulaton, although they do not have the free boundary value problem, they would need some extra efforts to compute the Greeks and the optmal exercse boundary, whch are mostly desred n practce besdes opton valuaton. In ths paper, we propose a parsmonously numercal method based on fnte dfference method and method of lnes (MOL) to overcome the dffculty mentoned above n Amercan optons valuaton. Partcularly, our method averts the otherwse necessary procedure of locatng the optmal exercse boundary before applyng fnte dfference dscretzaton. Ths method s concse, effcent, flexble to all knds of pay-off, and easy to mplement when compared wth many other methods. The results also show that our method possesses the optmal accuracy ntrnsc to fnte dfference dscretzaton, and thereby make t a powerful tool for practtoners when evaluatng Amercan optons. Ths paper s organzed as follows. Secton s devoted to the descrpton of method of lnes, and a nd order optmal accuracy s shown n the error analyss of the computatonal result for vanlla European call opton. Secton 3 descrbes the core dea of our scheme of evaluatng Amercan opton through the sprt of Black-Scholes nequalty wth demonstratons through successful computaton of Amercan put, Amercan call wth dvdend, and Amercan strangle optons. Optmal accuracy s also shown n the example of Amercan put opton compared wth an exhaustve result from bnomal tree. Secton 4 depcts the scheme of locatng the optmal exercse boundary, and shows the optmal exercse boundares together wth the opton values for cases computed n Secton 3. The dual optmal exercse boundares n Amercan strangle opton are partcularly compared wth Charella and Zogas (005), and satsfactory agreement s observed. Secton 5 extends the current method to compute mult-asset Amercan optons wth a demonstraton of evaluatng a two-factor Amercan basket put opton. The opton value wth the assocated optmal exercse boundary s shown n the context. 3

4 . METHOD OF LINES Under the usual assumptons, Black and Scholes (1973) and Merton (1973) have shown that the prce V of any contngent clam wrtten on a stock, whether t s Amercan or European, satsfes the famous Black-Scholes equaton: 1 (.1) V V + σ S + ( r D) S V rv = 0, t S S where volatlty σ, the rsk-free rate r, and dvdend yeld D are all assumed to be constants. The value of any partcular contngent clam s determned by the termnal and boundary condtons. For an Amercan opton, notce that the PDE only holds n the not-yet-exercsed regon. At the place where the opton should be exercsed mmedately, the equalty sgn n (.1) would turn nto an nequalty one. That means the opton value V(S,t) at each tme follows ether V( S, t) = f( S, t) for the early exercsed regon or V 1 σ V ( ) V + S + r D S rv = 0 for the not-yet-exercsed regon, where t S S f ( St, ) s the payoff of an Amercan opton at tmet. There s a movng boundary that separates these two regons and makes the whole problem a free boundary problem for Black-Scholes equaton. Generally, ths early exercsed boundary s dffcult to locate and there s no smple closed-form expresson for t. Most fnte dfference methods nowadays exert ther efforts on locatng the early exercse boundary that s a must before fnte dfference dscretzaton can be appled to the not-yet-exercsed regon. The method of lnes (MOL), a popular method for solvng PDEs n engneerng, was frst promoted by Lskovets (1965). The dea s frst reducng a tme-dependent PDE to a system of ordnary dfferental equatons (ODEs) n tme va sem-dscretzaton n space. Then ths system of ODEs n tme can further be solved effcently by many well-developed ODE solvers. Ths methodology has been successfully appled to the valuaton of Amercan optons on common stock by Goldenberg and Schmdt (1995), Meyer and Van der Hoek (1994) and Carr and Faguet (1996), and s found to be accurate and effcent. To explan how MOL s employed, here we smply take valuaton of a European call as an example. Incorporatng wth nd order fnte dfference scheme, we can dscretze (.1) on a unform mesh of underlyng asset prce{ } nto a sem-dscrete system of ODEs n tme: N S = 0 (.) dv S V V V ( r D) S V V, 1,..., 1. rv dt σ + = = N ΔS ( ΔS ) The boundary condtons of (.1) are 4

5 V( S, t) 1, as S, 0 t T, S V(0, t) = 0, 0 t T. Both boundary condtons above exhbt lnear behavor close to boundares, and hence they can be ncorporated nto ths system of ODEs by compensatng (.3) dv ( ) V + r D S V = V + rv dt ΔS 0 0, dvn 3V N 4V N 1+ VN (.4) = ( r D) SN + rv dt ΔS N, for (.) wth S beng truncated to S whch s specfed by Smax = 3K or max S max = 4K consderng both computatonal effcency and acceptable boundary error. (.-4) poses as an ordnary dfferental ntal value problem n tme and can be solved by many effcent ODE solvers whch have been developed n hundreds of years. Actually, tradtonal fnte dfference method ncorporatng wth explct Euler scheme n tme ntegraton s equvalent to choose forward Euler scheme as the ODE solver n MOL. Lkewse, popular Crank-Ncolsen fnte dfference scheme s equvalent to select the trapezodal rule as the ODE solver n MOL. As noted n Meyer and Van der Hoek (1997), the free boundary ntally moves wth nfnte speed but slows down very quckly. Therefore, ths stff problem would be hard to mplement effcently by plan fnte dfference methods. To evaluate Amercan opton effcently, t would request a self-adjustng-n-tme-step solver for tme ntegraton, and ths can be done easly by novel ODE solvers nowadays featurng self-adjustng varable step sze and order (VSVO). Many VSVO type ODE solvers have been collected n popular MATLAB ODE sute developed by Shampne and Rechelt (1997). In the current study, we selected ode3 from the sute to be our demonstratng ODE solver. Ode3 mplements an explct Runge-Kutta (,3) par of Bogack and Shampne (1989). It features by an adaptve step sze controlled by specfed error tolerance and s an effcent on-step solver for moderately stff ODE s. More detals of ths solver can be found n Bogack and Shampne (1989). To examne the accuracy of MOL wth nd order fnte dfference dscretzaton n space, here we apply (.-4) to evaluate European call opton wth termnal condton gven by the pay-off V( S, T) = max( S K,0). The specfc parameter values are r = 10%, σ = 40%, D = 0, T = 1, K = 1/5and stock prce ranges from S = 0 to S = 1. Here we truncate the nfnte doman of S by fve tmes of the exercse prce K to assure that the boundary error due to truncaton wll not 5

6 domnate the overall accuracy. In order to examne the order of accuracy n space, we lmt the maxmum tme step to be as small as error wll not domnate the total error ether. 5 here for ode3 so that the tme Table.1 reports the error analyss of our computaton. The frst part n ths table lsts the numercally calculated call prce, Delta and Gamma at the exercse prce. The absolute errors are obtaned by comparng the numercal solutons wth analytc solutons. We can observe that the absolute error cuts down n quarter as the spatal resoluton doubled. Ths observaton exsts not only n opton value tself but also n Delta and Gamma, whch demonstrates a perfect nd order of accuracy n space. The second part lsts the maxmum absolute error (MAE) on whole stock prce range and reports the locaton where the MAE occurred. The MAE of opton prce together wth the assocated Delta and Gamma values agan demonstrates perfect nd order of accuracy and MAE happens around the exercse prce as expected except at N=1600, where the MAE devates from nd order of accuracy wth MAE occurrng at the rght hand sde boundary (a sgn of boundary error due to truncaton domnatng the space error). Ths result shows MOL wth fnte dfference dscretzaton s both accurate and 4 robust. Here, the maxmum tme step s set to be as small as on purpose so that the tme error would not overrde the space error n order to check for order of accuracy n space. In practcal calculaton, ths tme step s self-adaptve to meet a specfed tolerance of tme error, whch makes the computaton very effcent. TABLE.1 Error Analyss for European Call Opton (The parameter values are r = 10%, σ = 40%, T = 1, K = 1/5and stock prce ranges from S = 0 to S = 1. The maxmum tme step s set to 4 n ode3.) At the Money Theoretcal Prce Theoretcal Delta N MOL Prce Absolute Error N MOL Delta Absolute Error

7 Theoretcal Gamma N MOL Gamma Absolute Error Whole Stock Prces Range N Maxmum Abs. Error Prce Delta Gamma Locaton Maxmum Abs. Error Locaton Maxmum Abs. Error Locaton VALUATION OF AMERICAN OPTIONS Amercan optons dffer from European ones by that the holder can select to exercse at any tme before the expry date. Ths early exercse feature of Amercan optons causes the free boundary problem of Black-Scholes equaton. So far all fnte dfference methods for prcng Amercan optons, ncludng those employng MOL mentoned above, request to fnd out the optmal exercse boundary frst n each tme step. Ths procedure s a must, and accurate locaton of ths optmal exercse boundary s crucal to the overall accuracy. That s why many analytcal mathematcans study ths subject and compete on dervng a better analytc approxmaton of optmal exercse boundary that s crucal to fnte dfference methods. However, our current method does not request ths optmal exercse boundary at all n each tme step, and ths optmal exercse boundary can be extracted from the numercal results afterwards f wanted. Due to the savng of overhead on computng ths optmal exercse boundary, our method s much more effcent compared wth other numercal methods. The key dea of our method s to modfy (.) to 7

8 (3.1) dv mn S V V V ( r D) S V V,0, 1,..., 1. rv dt σ + = = N ( ΔS ) ΔS Ths dea s based on the fact that the value of an Amercan opton, when evolvng backward n tme, can not allow smaller than ts fnal pay-off at any tme before expraton. If t s smaller than ts fnal pay-off, the opton should then be early exercsed at that spot. Ths sprt can also be observed n evaluatng Amercan optons by bnomal tree. To elaborate our pont, for those nodes that do not need early exercse, ther opton values would be governed by the backward tme evoluton of Black-Scholes equaton and dv 1 V+ 1 V + V 1 V+ 1 V 1 = mn σ S ( r D) S,0 + rv dt ( S ) S Δ Δ 1 V+ 1 V + V 1 V+ 1 V 1 = σ S ( r D) S. + rv ΔS ( ΔS ) For opton values at other nodes that would fall below ther fnal pay-off f followng the evoluton of Black-Scholes equaton, t would request an early exercse and then dv 1 V 1 V V 1 V 1 V 1 mn S ( r D) S,0 0. rv dt σ + = = ( ΔS ) ΔS (3.1) actually can deduce the Black-Scholes nequalty for Amercan optons: V 1 V + σ S + ( r D) S V rv 0. t S S Solvng (3.1) s straght forward and easy to mplement n the frame work of MOL. Extractng the optmal exercse boundary, f wanted, can be done afterwards from the dv result by re-computng,, at each tme step, and locatng the sngle zero of dt through nterpolaton. Ths trajectory of sngle zero would be the optmal exercse boundary. The detals of computng ths optmal exercse boundary wll be dscussed n a later secton below. One may argue that, followng the sprt of bnomal tree, one can easly ntegrate the Black-Scholes equaton at each tme step just by conventonal fnte dfference technque such as forward Euler or Crank-Ncolsen methods, and then replace the opton value by ts fnal pay-off at those nodes that need an early exercse. Ths may look lke a smpler way to evaluate Amercan optons. However, as mentoned before, the free boundary moves extremely fast near the expry date, whch would request much fner mesh n tme near the expry date. Ths makes conventonal fnte dfference technque wth fxed tme step very neffcent n computaton. If the mesh of tme s not approprately resolved near the expry date, t may cause large errors n both 8 dv dt

9 the opton value and optmal exercse boundary. Under ths stuaton, MOL employng VSVO-type error-control ODE solvers would be a much more superor method. Here, we use Amercan put opton, Amercan call opton wth dvdend and Amercan strangle opton to demonstrate how our parsmonous approach would work Amercan Put Opton In Fgure 3.1, the optmal exercse boundary S, () t separates the underlyng asset doman n two regons, contnuous and stoppng regons. On the stoppng regon [0, S ( t)) [0, T], V( S, t) = KP S, whle on the contnuous regon f, P [ S ( t), ) [0, T], V( S, t) needs to satsfy the followng free-boundary problem: f, P V 1 V V + σ S + ( r D) S rv = 0, S f, P( t) < S, 0 t <T, t S S V( S, T) = max( KP S,0), S 0, VS ( S ),, (),, (), 0, f P f P t t = KP Sf P t t T (3.) V lm = 1, 0 t T, S Sf, P() t S rkp Sf, P( T) = mn KP,. D Naturally, fndng out the optmal exercse boundary would be a must before ntegratng (3.). Instead, our parsmonous method solves (3.1) wth lnear boundary condtons (.3-4) and the followng termnal condton f P V( S, T) = max( K S,0). P V ( S, t) Stoppng Regon S S ( ) f, P t Contnuaton Regon S P f, ( t) < S max( K P S,0) 9 () K 0 S t P f, P S

10 FIGURE 3.1. Illustraton of contnuaton regon, stoppng regon and optmal exercse boundary for Amercan put opton. The specfc parameter values adopted here n our computaton are r = 10%, σ = 40%, D = 0, T = 1, K = 1/5, and stock prce ranges from S = 0 to S = 1 wth the P nfnte doman of S beng truncated by fve tmes of the exercse prce K P. The error analyss of our result s reported n Table 3.1. The frst part n ths table lsts the calculated opton prce at the exercse prce. For calculatng absolute error, we calculated a bnomal tree wth exhaustve N = 10,000 tme steps and used the result as the exact soluton V ( S,0). Here, we agan lmt the maxmum tme step to be so exa that the total error would be domnated only by the spatal error. The column of absolute error agan shows perfect 0 N nd 4 order convergence. In the second part, the MAE, max V( S,0) V ( S,0), shows that the order of accuracy deterorates slghtly from nd exa order, and the locaton of MAE s close to the free boundary. Ths deteroraton may be due to the fact that V fals to twce dfferentable at the optmal exercse boundary. As we know, error analyss of fnte dfference s based on Taylor expanson. nd order of accuracy on the whole range of S would request the justfcaton of twce dfferentablty on all S. TABLE 3.1 Error Analyss for Amercan Put Opton (The parameter values are r = 10%, σ = 40%, T = 1, K P = 1/5and prce of underlyng 4 asset ranges from S = 0 to S = 1. The maxmum tme step s set to n ode3.) At the Money N Bnomal Prce MOL Prce Absolute Error Whole Underlyng Prce Range N Maxmum Abs. Error Locaton Free Boundary

11 Amercan Call Opton wth dvdend In Fgure 3., the optmal exercse boundary S f, C() t separates the underlyng asset doman to contnuous and stoppng regons. On the stoppng regon [ S ( t), ) [0, T], V( S, t) = S KC, whle on the contnuous regon [0, Sf, C( t)] [0, T], V( S, t) needs to be solved from the followng free-boundary problem for Amercan call optons: V 1 V V + σ S + ( r D) S rv = 0, 0 < S < S f, C( t), 0 t <T, t S S V( S, T) = max( S KC,0), S 0, VS ( S ),, (),, (), 0, f C f C t t = Sf C t KC t T (3.3) V lm = 1, 0 t T, S Sf, C() t S rkc Sf, C( T) = max KC,. D The above optng prcng problem can be solved exactly n the same way as prcng Amercan put opton before only wth the termnal condton changed to f, C V( S, T) = max( S K C,0). 11

12 V ( S, t) Contnuaton Regon S < S ( ) f, C t Stoppng Regon S S ( ) f, C t max( S KC,0) 0 K C S f, C ( t) S FIGURE 3.. Illustraton of contnuaton regon, stoppng regon and optmal exercse boundary for Amercan call opton wth dvdend. Table 3. reports the error analyss of Amercan call opton prcng. The specfc parameter values are r = 9%, σ = 40%, D = 10%, T = 1, K = 1/5 and stock prce ranges agan from S = 0 to S = 1. Same as before, the nfnte doman of S s truncated by fve tmes of the exercse prce K C. The frst part n ths table lsts the calculated opton prce at the exercse prce. Agan, for calculatng absolute error, we calculated a bnomal tree wth exhaustve N = 10,000 tme steps and used the result as the exact soluton. Here, both the local error at the exercse prce and MAE show perfect nd order of convergence, wth MAE happenng near exercse prce nstead of optmal exercse boundary. C TABLE 3.. Error Analyss for Amercan Call Opton wth Dvdend (The parameter values were r = 9%, D = 10%, σ = 40%, T = 1, K C = 1/5and prce of underlyng asset ranges from S = 0to S = 1. The maxmum tme step s set to4 n ode3.) At the Money N Bnomal Prce MOL Prce Absolute Error

13 Whole Underlyng Prce Range N Maxmum Abs. Error Locaton Free Boundary Amercan Strangle To demonstrate the flexblty of our method that can be appled to any knd of pay-off functon, here we apply our method to the opton prcng of an Amercan strangle poston, unusually seen on the market, studed frst by Charella and Zogas (005). Amercan strangle pay-off can be comprehended as a combnaton of Amercan put and call, and there would be two free boundares when opton evolves backward n tme. In Fgure 3.3, the optmal exercse boundares S f, P() t and S f, C() t separates the underlyng asset doman nto contnuous and stoppng regons. On the stoppng regon [0, S f, P( t)) [0, T], V( S, t) = KP S, and [ S f, C( t), ) [0, T], V( S, t) = S KC, whle on the contnuous regon [ S ( t), S ( t)] [0, T], V( S, t) s governed by the f, P f, C followng free-boundary problem for Amercan strangle: (3.4) V 1 V V + σ S + ( r D) S rv = 0, S f, P() t < S < Sf, C(), t t S S 0 t < T, V( S, T) = max( KP S,0) + max( S KC,0), S 0, V( Sf, P( t), t) = KP Sf, P( t), 0 t T, V( Sf, C( t), t) = Sf, C( t) KC, 0 t T, V V lm = 1, lm = 1, 0 t T, S Sf, P() t S S Sf, C() t S rkp rkc Sf, P( T) = mn KP,, Sf, C( T) = max KC,. D D Though the pay-off of Amercan strangle s a combnaton of put and call postons, ts current value would not just be the sum of the assocated Amercan put and call. Ths 13

14 s chefly because the dual optmal exercse boundares, S f, p and S, f c, are not ndependent. Charella and Zogas (005) frst derved a complcated coupled ntegral equatons for these dual early exercse boundares. Then they solved the Black-Scholes equaton for the opton value n the contnuous regon by Crank-Ncolsen fnte dfference method. Usng our parsmonous method, t needs only to change the termnal condton to be the followng pay-off functon V( S, T) = max( K S,0) + max( S K,0), P C ( S t) V, S f, P( t) < S < S f, C ( t) max( K P S,0) max( S K,0 C ) S S ( ) f, P t S S ( ) f, C t S () t KP KC f, P S f, C ( t) FIGURE 3.3. Illustraton of contnuaton regon, stoppng regon and dual optmal exercse boundares for Amercan strangle opton. and follow the procedure same as above. Table 3.3 reports our result of Amercan strangle prcng. The parameter values r = 5%, σ = 0%, D = 10%, T = 1, K = 1, K = 1.5 are adopted from Table 1 of P C Charella and Zogas (005). The nfnte doman n S s truncated by fve tmes of n the current computaton. In ths table, we quote the Crank-Ncolson fnte dfference result from the Table 1 of Charella and Zogas (005), where they employed 1460 tme steps and space-nodes n ther fnte dfference calculaton. We mght as well treat ths exhaustve soluton as exact. The Amercan strangle prces calculated by our parsmonous approach wth varous number of space nodes and the assocated absolute errors are lsted n the exhbt. Here we dd not lmt maxmum tme step to a very small number as before (just chose ordnary for the maxmum tme step) n ode3. The 14 K C

15 errors show fast convergence to the exact soluton. The computatonal effcency of our method can be especally noted by that our N=800 results almost match wth every dgt of the exact solutons that used N=10000 space nodes n Charella and Zogas (005). We conclude that our parsmonous method can be easly and robustly appled to all knds of pay-off functons and have a satsfactory accuracy wth an economc spatal resoluton. TABLE 3.3. Error Analyss for Amercan Strangle Opton (The parameter values are r = 5%, σ = 0%, D = 10%, T = 1, K P = 1, K C = 1.5 and prce of underlyng asset ranges from S = 0 to S = 7.5. The maxmum tme step s set to n ode3. The numbers n the parentheses are the absolute error.) Method, N S CN, MOL, ( ) (0.0005) ( ) ( ) ( ) MOL, ( ) ( ) ( ) ( ) ( ) MOL, ( ) ( ) ( ) ( ) (0.0000) MOL, ( ) ( ) ( ) (0.0000) ( ) 4. ESTIMATING THE OPTIMAL EXERCISE BOUNDARY Dfferent from other methods, the optmal exercse boundary s not requested at each tme step for our method. However, t can be extracted from the numercal result afterwards f wanted. We frst re-compute ( ) V / t,, at each tme step by (.). Then the optmal exercse boundary V S opt would be the sngle zero of ( St, ) = 0 t as mentoned before. Instead of dong tme-consumng root fndng, S opt can be smply ( ) nterpolated by V / t,. The way s to see S as functon of V / t nstead, ( ) snce V t vs. S s monotonc across S. We can then nterpolate to fnd / opt Sopt 15

16 through S vs. ( V / t). There are other more accurate ways of locatng S lke opt utlzng Delta value for example. Takng Amercan put as an example, we can V extrapolate to fnd S opt that approxmates ( S opt, t ) = 1 through s ( V / s) n the contnuous regon near optmal exercse boundary. Fgure 4.1 shows S vs. respectvely the optmal exercse boundary S () t = S () t of the Amercan put opton opt f, P n Table 3.1, S () t = S () t of the Amercan call opton wth dvdend n Table 3., opt f, C and the dual optmal exercse boundares S f, P() t and S f, C() t of the Amercan strangle opton n Table 3.3. Here, we also computed Amercan strangle opton wth r = 10%, σ = 0%, D = 5%, T = 1, K = 1, K = 1.1 and compare our dual free P boundares wth ther counterparts n Fgure and 3 of Charella and Zogas (005) n Fgure 4.. Obvously, our result agrees very well wth Charella and Zogas (005). C 16

17 Fgure 4.1. From top to bottom, optmal exercse boundares together wth opton value are shown for Amercan put opton, Amercan call opton wth dvdend, and Amercan strangle opton. 17

18 Fgure 4.. Comparson of current dual optmal exercse boundares wth Charella and Zogas (005) 5. VALUATION OF MULTI-ASSET AMERICAN OPTIONS Our current method can be extended drectly to evaluate mult-asset Amercan optons. Takng mult-asset Amercan basket put opton as an example. The equaton s (5.1) SS V r D S rv S = 0, n n n V 1 V + ρσσ j j j + ( ) t = 1 j= 1 S Sj = 1 subject to the pay-off functon n (5.) V( S1, S,, SN, T) = max K αs,0, = 1 wth n = 1 α = 1. Lteratures dscussng numercal methods for mult-asset Amercan optons are rare. Nelsen et al. (000) used penalty method to compute mult-asset Amercan optons. Ther method s only frst order of accuracy n both tme and space (underlyng asset prce). Fasshauer et al. (004) employed novel meshless method to do the same thng. Though a computatonal result for two-factor Amercan basket put opton was shown, nether artcle above ponted out the optmal exercse boundary. Besdes, these two methods are far too complcated for most fnancal practtoners to 18

19 mplement. Here we also demonstrate our parsmonous method wth a result for two-factor Amercan basket put opton as shown n Fgure 5.1. In addton to the opton value, we also computed ts optmal exercse boundary afterwards n terms of the contour V / t =0 and show that n Fgure 5.1. Fgure 5.1. Two-factor Amercan basket put opton value at t=t (top) and t=0 (bottom) 19

20 and the assocated optmal exercse boundary. The parameters used here are r = 4.5%, σ 1 = 5%, σ = 35%, D1 = D = 0, ρ 1 = 0.65, α 1 = 0.58, α = 0.4, T = 1, K = CONCLUSION We have ntroduced an effcent numercal method to evaluate Amercan style optons n ths artcle. Ths parsmonous method s much easer to mplement compared wth those numercal methods requestng the early exercse boundary n advance at each tme step. Not requestng the early exercse boundary makes t flexble to sut all knds of pay-off functon. The optmal exercse boundary can be easly extracted afterwards from the computed opton values at each tme step f wanted. Varous ways lke nterpolatng on the Theta or Delta values can do the purpose effcently. Most of all, ths effcent method can be drectly extended to evaluate Amercan optons wth multple underlyng assets, and trace the optmal exercse boundary afterwards easly. REFERENCES Amn, K. and A. Khanna (1994): Convergence of Amercan Opton Values from Dscrete- to Contnuous Tme Fnancal models, Mathematcal Fnance, 4, Black, F. and M. Scholes (1973): The Prcng of Optons and Corporate Labltes, Journal of Poltcal Economy, 81, Bogack, P. and L. F. Shampne (1989): A 3() par of Runge-Kutta formulas, Appled Mathematcs Letters,, 1-9. Brennan, M. and E. Schwartz (1977): The Valuaton of Amercan Put Optons, Journal of Fnance, 3, Brennan, M. and E. Schwartz (1978): Fnte Dfference Methods and Jump Processes Arsng n the Prcng of Contngent Clams: A synthess, Journal of Fnancal Quanttatve Analyss, 13, Broade, M. and J. Detemple (1996): Amercan Opton Valuaton: New Bounds, Approxmatons, and a Comparson of Exstng Methods, Revew of Fnancal Studes, 9,

21 Charella, C. and A. Zogas (005): Evaluaton of Amercan strangles, Journal of Economc Dynamcs and Control, 9, Carr, P. and D. Faguet (1996): Fast Accurate Valuaton of Amercan Optons, workng paper, Cornell Unversty. Cox, J. C., S. A. Ross, and M. Rubnsten (1979): Opton Prcng: A Smplfed Approach, Journal of Fnancal Economcs, 7, Duck, P. W., D. P. Newton, M. Wddcks, and Y. Leung (005): Enhancng the Accuracy of Prcng Amercan and Bermudan Optons, Journal of Dervatve, 1, Fasshauer, G. E., A. Q. M. Khalq, and D. A. Voss (004): Usng Meshfree Approxmaton for Mult-Asset Amercan Opton Problems, Journal of Chnese Insttute of Engneers, 7, Fu, M. C. (1994a): Optmzaton Usng Smulaton: a Revew, Annals of Operaton Research, 53, Fu, M. C. (1994b): A Tutoral Revew of Technques for Smulaton Optmzaton, n Proceedngs of the 1994 Wnter Smulaton Conference, Geske, R. and H. E. Johnson (1984): The Amercan Put Optons Valued Analytcally, Journal of Fnance, 39, Geske, R. and K. Shastr (1985): Valuaton by Approxmaton: A Comparson of Alternatve Opton Valuaton Technques, Journal of Fnancal Quanttatve Analyss, 0, Goldenberg, D. and R. Schmdt (1995): Estmatng the Early Exercse Boundary and Prcng Amercan Optons, workng paper, Rensselaer Polytechnc Insttute. Jallet, P., D. Lamberton, and B. Lapeyre (1990): Varatonal Inequaltes and the Prcng of Amercan Optons, Acta Appled Mathematcs, 1, Johnson, H. (1983): An Analytc Approxmaton for the Amercan Put Prce, Journal of Fnancal Quanttatve Analyss, 18,

22 Ju, N. and R. Zhong (1999): An Approxmate Formula for Prcng Amercan Optons, Journal of Dervatves, 7, Lskovets, O. A. (1965): Method of Lnes, Journal of Dfferental Equatons, 1, MacMllan, L. W. (1973): Analytc Approxmaton for the Amercan Put Opton, Advances n Futures and Optons Research, 1, Merton, R. (1973), Theory of Ratonal Opton Prcng, Bell Journal of Economcs and Management Scence, 4, Meyer, G. H. and J. Van der Hoek (1994): The Evaluaton of Amercan Optons wth the Method of Lnes, workng paper, Georga Insttute of Technology. Nelson, B. F., O. Skavhaug, and A. Tveto (000): Penalty Method for the Numercal Soluton of Amercan Mult-Asset Opton Problems, preprnt , Department. of Informatcs, Unversty of Oslo. Schwartz, E. S. (1977): The Valuaton of Warrants: Implementng a New Approach, Journal of Fnancal Economcs, 4,