QUADRATIC CONVERGENCE FOR VALUING AMERICAN OPTIONS USING A PENALTY METHOD

Transcription

1 QUADRATIC CONVERGENCE FOR VALUING AMERICAN OPTIONS USING A PENALTY METHOD P.A. FORSYTH AND K.R. VETZAL Abstract. The convergence of a penalty method for solvng the dscrete regularzed Amercan opton valuaton problem s studed. Suffcent condtons are derved whch both guarantee convergence of the nonlnear penalty teraton and ensure that the terates converge monotoncally to the soluton. These condtons also ensure that the soluton of the penalty problem s an approxmate soluton to the dscrete lnear complementarty problem. The effcency and qualty of solutons obtaned usng the mplct penalty method are compared wth those produced wth the commonly used technque of handlng the Amercan constrant explctly. Convergence rates are studed as the tmestep and mesh sze tend to zero. It s observed that an mplct treatment of the Amercan constrant does not converge quadratcally (as the tmestep s reduced) f constant tmesteps are used. A tmestep selector s suggested whch restores quadratc convergence. Key words. Amercan opton, penalty teraton, lnear complementarty AMS subject classfcatons. 65M12, 65M60, 91B28 Revsed: May 18, Introducton. The valuaton and hedgng of fnancal opton contracts s a subject of consderable practcal sgnfcance. The holders of such contracts have the rght to undertae certan actons so as to receve certan payoffs. The valuaton problem conssts of determnng a far prce to charge for grantng these rghts. A related ssue, perhaps of even more mportance to practtoners, s how to hedge the rs exposures whch arse from sellng these contracts. An mportant feature of such contracts s the tme when contract holders can exercse ther rghts. If ths occurs only at the maturty date of the contract, the opton s classfed as European. If holders can exercse any tme up to and ncludng the maturty date, the opton s sad to be Amercan. The value of a European opton s gven by the soluton of the Blac-Scholes PDE (see, e.g. 33). An analytcal soluton can be obtaned for cases wth constant coeffcents and smple payoffs. However, most optons traded on exchanges are Amercan. Such optons must be prced numercally, even for constant coeffcents and smple payoffs. Note also that the dervatves of the soluton are of nterest snce they are used n hedgng. More formally, the Amercan opton prcng problem can be posed as a tme dependent varatonal nequalty or a dfferental lnear complementarty problem (LCP). In current practce, the most common method of handlng the early exercse condton s smply to advance the dscrete soluton over a tmestep gnorng the constrant, and then to apply the constrant explctly. Ths has the dsadvantage that the soluton s n an nconsstent state at the begnnng of each tmestep (.e. a dscrete form of the LCP s not approxmately satsfed). As well, ths approach can obvously only be frst order correct n tme. On the other hand, ths explct applcaton of the constrant s computatonally very nexpensve. Ths wor was supported by the Natural Scences and Engneerng Research Councl of Canada, the Socal Scences and Humantes Research Councl of Canada, the Royal Ban of Canada, and the Fnancal Industry Solutons Center (FISC), an SGI/Cornell Unversty Jont Venture. In addton, Ratonal Software provded access to the Purfy software development tool under the SEED program. Department of Computer Scence, Unversty of Waterloo, Waterloo ON, Canada N2L 3G1, paforsyt@elora.math.uwaterloo.ca School of Accountancy, Unversty of Waterloo, vetzal@watarts.uwaterloo.ca 1

2 2 P.A. FORSYTH AND K.R. VETZAL Another common technque s to solve the dscrete LCP usng a relaxaton method 33. In terms of complexty, ths method s partcularly poor for prcng problems wth one space-le dmenson. A lower bound for the number of teratons requred to solve the LCP to a gven tolerance wth a relaxaton method would be the number of teratons requred to solve the unconstraned problem usng a precondtoned conjugate gradent method. Assumng that the mesh spacng n the asset prce S drecton s O( S) and that the tmestep sze s O( t), then the condton number of a dscrete form of the parabolc opton prcng PDE s O t/( S) 2. Let N be the number of tmesteps. If we assume that S = O( t) = 1/N, then the number of teratons requred per tmestep would be O(N 1/2 ). A multgrd method has been suggested n 5 to accelerate convergence of the basc relaxaton method. Although ths s a promsng technque, multgrd methods are usually strongly coupled to the type of dscretzaton used, and hence are complex to mplement n general purpose software. There are a large number of general purpose methods for solvng lnear complementarty problems 22, 7, 25. We can dvde these methods up to nto essentally two categores: drect methods, such as pvotng technques 7, and teratve methods, such as Newton teraton 25 and nteror pont algorthms 22. Some of these methods whch have been appled specfcally to Amercan opton prcng nclude lnear programmng 9, pvotng methods, 14, and nteror pont methods 15. As ponted out n 15, pvotng methods (such as Leme s algorthm 7) and LP approaches are not well equpped to handle sparse systems, especally n more than one dmenson (multfactor optons). Complementarty problems (both lnear and nonlnear) can be posed n the form of a set of nonlnear equatons. Varous nonsmooth Newton methods have been suggested for these types of problems 26, 27, 11, 19, 17. More recently, combnatons of nonsmooth Newton and smoothng methods have been proposed 20. It s well nown that an LCP (or equvalently, a varatonal nequalty) can be solved by a penalty method 10, 30, 24, 12, 8. In ths artcle, we wll explore some aspects of usng penalty methods for prcng Amercan optons. We wll restrct attenton to one dmensonal problems, whch are more amenable to analyss. However, we have successfully used penalty methods for two factor (two dmensonal) problems 37, 38. In ths wor, the nonlnear dscrete penalzed equatons are solved usng Newton teraton. Another approach whch also uses a Newton method has been suggested n 6. Note that relaxaton methods are frequently used to solve the dscrete penalzed nonlnear equatons 8. The advantage of the penalty method s that a sngle technque can be used for one dmensonal or mult-dmensonal problems, and standard sparse matrx software can be used to solve the Jacoban matrx. Ths technque can be used for any type of dscretzaton, n any dmenson, and on unstructured meshes. In partcular, there s no dffculty n handlng cases where the early exercse regon s multply-connected, as n 37. As well, a sngle method can be used to handle Amercan optons and other nonlneartes, such as uncertan volatlty and transacton cost models 33, 1. In addton, nonlneartes due to the use of flux lmters for drft-domnated problems 39 can also be handled easly. The objectve of ths artcle s to analyze the propertes of penalty methods for soluton of a dscrete form of a comparatvely smple problem: a sngle factor Amercan opton. In ths way, we hope to gan some nsght nto the use of penalty methods for more complex problems. A sngle factor Amercan opton can be posed as a var-

3 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS 3 atonal nequalty, whch n turn can be expressed as a dscrete LCP problem at each tmestep. However, t s mportant to examne the penalty method n the context of the overall problem: we need to fnd tme accurate solutons to the varatonal nequalty. Consequently, we can expect that we have a good ntal guess for the soluton of the varatonal nequalty from the prevous tmestep. In fact, t s mportant to recall that smply advancng the soluton for a tmestep (gnorng the constrant) and applyng the constrant n an explct fashon, wll solve the tme dependent varatonal nequalty to O( t). Consequently, any method used to solve the LCP at each step should tae full advantage of a good ntal guess. We wll also study the convergence of these methods as the tmestep and mesh sze are reduced to zero. We wll determne suffcent condtons for monotone convergence of the penalty method n one dmenson. It may be possble to requre weaer condtons perhaps usng the methods n 24. However, opton prcng problems are typcally degenerate parabolc, and n non-conservatve form. Ths can be expected to complcate the methods n 24. In practce, we observe that the penalty method wors well for mult-factor optons 37, 38, and for nonlnear problems. In other words, although the condtons we derve are suffcent, they do not appear to be necessary. Consequently, t appears that the penalty method can be used for more general stuatons. In addton, we wll compare the penalty method (where the LCP s approxmately solved at each tmestep) wth an explct technque for handlng the Amercan constrant. Essentally, the method proposed n ths wor uses a nonsmooth Newton teraton 4 to solve the penalzed problem. A dsadvantage of the penalty method as formulated n ths wor s that the Amercan constrant s only satsfed approxmately, but snce ths error can be easly made to be much smaller than the dscretzaton error, ths does not appear to be a practcal dsadvantage. On the other hand, the advantages of ths approach are (under certan condtons): Ths method has fnte termnaton (n exact arthmetc),.e. for an terate suffcently close to the soluton, the algorthm termnates n one teraton. Ths s especally advantageous when dealng wth Amercan opton prcng, snce we have an excellent ntal guess from the prevous tmestep. In fact, as we shall see, for typcal grds and tmesteps, the algorthm taes, on average, less than two teratons per tmestep to converge. Fnte termnaton also mples that the number of teratons requred for convergence s nsenstve to the sze of the penalty factor (untl the lmts of machne precson s reached). The teraton s globally convergent usng full Newton steps. Of course, f we dd not have the advantage of havng a good ntal guess, a pvotng method 7 can be very effcently mplemented f the the coeffcent matrx s a trdagonal M-matrx. We emphasze here that the penalty method should not be regarded as a general purpose method for LCP problems. The real advantage of the penalty method s that ths technque taes full advantage of the fact that a good ntal terate s avalable, and we tae full advantage of sparsty, whch s mportant n multfactor problems. Another approach whch attempts to tae advantage of a good ntal terate combned wth standard sparse solvers s descrbed n 21. If we solve the LCP at each step (usng the penalty approach), and f constant tmesteps are used, we observe that second order convergence s not obtaned as the tmesteps and mesh sze tend to zero. Ths phenomenon can be explaned by examnng the asymptotc behavor of the soluton near the exercse boundary. A

4 4 P.A. FORSYTH AND K.R. VETZAL tmestep selector s developed whch restores second order convergence. Asymptotcally, the second order method s superor to the commonly used bnomal lattce technque 16. However, t s of practcal nterest to determne at what levels of accuracy a second order PDE method wll be computatonally more effcent than the lattce method. We present numercal comparsons to assst n ths determnaton. In the followng we wll restrct attenton to the Amercan put opton. However, these methods can be appled to dvdend payng calls, as well as complex optons whch nvolve solvng a set of one dmensonal Amercan-type problems embedded n a hgher dmensonal space. Examples of these types of optons nclude dscretely observed Asan optons 40, Parsan optons 31, shout optons 36, and segregated fund guarantees Formulaton. Consder an asset wth prce S whch follows the stochastc process ds = µsdt + σsdz (2.1) where µ s the drft rate, σ s volatlty, and dz s the ncrement of a Wener process. We wsh to determne the value V (S, t) of an Amercan opton where the holder can exercse at any tme and receve the payoff V (S, t). Denote the expry tme of the opton by T, and let τ = T t. Then the Amercan prcng problem can be formally stated as an LCP 33 LV 0 (V V ) 0 (LV = 0) (V V = 0) (2.2) where the notaton (LV = 0) (V V = 0) denotes that ether (LV = 0) or (V V = 0) at each pont n the soluton doman, and ( ) σ 2 LV V τ 2 S2 V SS + rsv S rv (2.3) and r s the rs free rate of nterest. A put opton s a contract whch gves the holder the rght to sell the asset for K (nown as the stre ). A call opton s smlar except that the holder has the rght to buy the asset for K. The payoff for a put s The boundary condtons are V (S) = V (S, τ = 0) = max(k S, 0). (2.4) V (S, τ) = 0 ; S, (2.5) LV = V τ rv ; S 0. (2.6) Condton (2.5) follows from the payoff (2.4), whle (2.6) s obvous gven (2.3). 3. The Penalty Method. The basc dea of the penalty method s smple. We replace problem (2.2) by the nonlnear PDE 10 V τ = σ2 2 S2 V SS + rsv S rv + ρ max(v V, 0), (3.1) where the postve penalty parameter ρ, ρ effectvely ensures that the soluton satsfes V V ɛ for ɛ > 0, ɛ 1.

5 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS 5 4. Dscretzaton. We wll now dscretze equaton (3.1) and select a sutable form for the dscrete penalty term. Let V (S, τ n ) = V n be the dscrete soluton to equaton (3.1) at asset value S, and tme (gong bacwards) τ n. Applyng a standard fnte volume method wth varable tmeweghtng 37 then gves where ( V FV n+1 n+1 V n ) A τ FV n+1 = q n+1, (4.1) + (1 θ) γ j (V n+1 j V n+1 ) + Lj U V n+1 j+1/2 A rv n+1 j η j η + θ j η γ j (V n j V n ) + j η Lj U V n j+1/2 A rv n. (4.2) Fully mplct and Cran-Ncolson dscretzatons correspond to cases of θ = 0 and θ = 1/2 respectvely, and A = (S +1 S 1 )/2 η = { 1, + 1} τ = τ n+1 τ n γ j = σ2 S 2 2 S j S V n+1 j+1/2 = value of V at the face between nodes and j U = ( rs )î L j = { î f j = + 1 +î f j = 1 î = unt vector n the postve S drecton. (4.3) The dscrete penalty term q n+1 q n+1 = n equaton (4.1) s gven by { (A / τ)(v V n+1 )Large f V n+1 < V 0 otherwse, (4.4) where Large s the penalty factor (ths wll be related to the desred convergence tolerance below n 4.1). The face value V n+1 j+1/2 can be evaluated usng ether central weghtng or, to ensure non-oscllatory solutons, a flux lmter 39 { V n+1 j+1/2 = (V n+1 + V n+1 j )/2 f γ n+1 j + L j U /2 > 0 (4.5) second order flux lmter 39 otherwse. In general, for standard optons wth typcal values for σ, r, central weghtng can be used at most nodes (except perhaps as S 0). In order to determne suffcent condtons for the convergence of the nonlnear teraton for the penalzed Amercan

6 6 P.A. FORSYTH AND K.R. VETZAL equaton, we requre that the coeffcents of the dscrete equatons have certan propertes. We wll ensure that these condtons are satsfed by usng central or upstream weghtng. (In practce, we have observed that even f these condtons are not met, convergence of the penalty method s stll rapd 37). If we use central or upstream weghtng n the followng, then equaton (4.1) becomes V n+1 V n = (1 θ) + θ τ j η ( γj + β j ) (V n+1 j V n+1 τ j η ( γj + β j ) (V n j V n ) r τv n ) r τv n+1 + P n+1 (V V n+1 ), (4.6) where and where P n+1 = { Large f V n+1 < V 0 otherwse, (4.7) σ 2 S 2 γ j = 2A S j S { Lj U /2A β j = max( L j U, 0)/A f γ j + L j U /2 0 otherwse. For future reference, we can wrte the dscrete equatons (4.6) n matrx form. Let V n+1 = V0 n+1, V1 n+1,,..., Vm n+1 V n = V0 n, V1 n,..., Vm n, V = V0, V1,..., Vm, and ˆMV n = τ j η ( γj + β j ) (V n j V n ) r τv n. (4.8) Note that the frst and last rows of ˆM wll have to be modfed to tae nto account the boundary condtons. (An obvous method for applyng condtons ( ) results n the frst and last rows of ˆM beng dentcally zero except for postve entres on the dagonal.) In the followng, we wll assume that upstream and central weghtng are selected so that γ j + β j 0. Ths mples that the matrx ˆM s an M-matrx,.e. a dagonally domnant matrx wth postve dagonals and non-postve off-dagonals. Note that all of the elements of the nverse of an M-matrx are non-negatve. Let the dagonal matrx P be gven by { P (V n+1 Large f V n+1 < V and = j ) j = (4.9) 0 otherwse. We can then wrte the dscrete equatons (4.6) as I + (1 θ) ˆM + P (V n+1 ) V n+1 = I θ ˆM V n + P (V n+1 ) V. (4.10)

7 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS Soluton of the Dscrete LCP. The dscrete form of the LCP (2.2) can be wrtten as V n+1 FV n+1 0 V 0 (FV n+1 = 0) (V n+1 V = 0), (4.11) where F s gven by equaton (4.2). On the other hand, the dscrete soluton of the penalzed problem (4.10) has the property that ether or V n+1 V 0 ( q n+1 = 0 and FV n+1 = 0), (4.12) V n+1 V 0 ( q n+1 > 0 and FV n+1 > 0). (4.13) However, from equaton (4.11) the exact soluton of the dscrete LCP has V n+1 V = 0 at those nodes where FV n+1 > 0. In order to obtan an approxmate soluton of (4.11) wth an arbtrary level of precson, we need to show that the soluton of (4.10) satsfes V n+1 V 0 as Large for nodes where FV n+1 > 0. Ths follows f we can show that the term P n+1 (V V n+1 ) (4.14) n equaton (4.6) s bounded ndependent of Large. It s also desrable that the bound be ndependent of the tmestep and mesh spacng, so that Large can be chosen wthout regard to grd and tmestep sze. In Appendx A we determne suffcent condtons whch allow us to bound (4.14). The results can be summarzed as: Theorem 4.1 (Error n the penalty formulaton of the dscrete LCP). Under the assumptons that the matrx ˆM n equaton (4.10) s an M-matrx and that tmesteps are selected so that 1 θ τ ( γj + β ) j + r τ 0 (4.15) j η τ S < const. τ, S 0 (4.16) S = mn S +1 S then the penalty method for the Amercan put ( equaton (4.10) wth termnal condton (2.4) ) solves V n+1 (FV n+1 FV n+1 0 (4.17) V C = 0) Large ( V n+1 V ; C > 0 (4.18) C ) (4.19) Large where C s ndependent of Large, τ, and S. Note that condton (4.16) s not practcally restrctve, snce volaton of ths condton would result n an mbalance between tme and space dscretzaton errors.

8 8 P.A. FORSYTH AND K.R. VETZAL Condton (4.15) s trvally satsfed for any tmestep sze f a fully mplct method (θ = 0) s used. However, f Cran-Ncolson tmesteppng s used, condton (4.15) essentally requres the boundedness of τ/( S) 2. Ths tmestep condton arses snce we requre that (I θ ˆM)V n be bounded. Ths s essentally a requrement on the smoothness of the dscrete soluton usng Cran-Ncolson tmesteppng. Note that for pure parabolc problems, careful analyss s requred 28 to obtan quadratc convergence estmates for Cran-Ncolson methods (wthout restrctons on τ/( S) 2 ). In fact, n 28, t s necessary to tae a fnte number of fully mplct steps ntally, n order to smooth rough data. Ths approach wll be used n our numercal examples, and wll be dscussed n detal n later sectons. In our numercal experments, we routnely volate condton (4.15). As a chec on the soluton, we montor the quantty max Amercan error = max n, max0, (V V n ) max(1, V ). (4.20) Ths s a measure of the maxmum relatve error n enforcng the Amercan constrant usng the penalty method. Ths quantty wll be small f the quantty (4.14) s bounded, and Large s suffcently large. As long as we use the modfcaton to Cran-Ncolson tmesteppng suggested n 28, we observe that the a posteror error chec (4.20) s ndeed small. It remans an open queston f we can remove condton (4.15) for the tmesteppng method suggested n 28 (Cran-Ncolson wth a fnte number of fully mplct steps). In practce, we can use the followng heurstc argument to estmate the sze of Large n terms of the relatve accuracy requred. In equaton (4.10), suppose that (1 θ) ˆMV n+1 and (I θ ˆM)V n are bounded ndependent of Large. Then, as Large, equaton (4.10) reduces to for nodes where V n+1 V n+1 ( ) Large V (4.21) 1 + Large < V. If V 0, then we have V n+1 V V 1 Large. (4.22) Therefore, f we requre that the LCP be computed wth a relatve precson of tol for those nodes where V n+1 < V then we should have Large 1/tol. Note that n theory, f we are tang the lmt as S, τ 0, then we should have ( Large = O 1 mn ( S) 2, ( τ) 2 ). (4.23) Ths would mean that any error n the penalzed formulaton would tend to zero at the same rate as the dscretzaton error. However, n practce t seems easer (to us at any rate) to specfy the value of Large n terms of the requred accuracy. In other words, we specfy the maxmum allowed error n the dscrete penalzed problem. We then reduce S, τ untl the dscretzaton error s reduced to ths level of accuracy.

9 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS 9 5. Penalty Iteraton. We wll use Newton teraton to solve the dscrete nonlnear equatons (4.10). Of course, due regard must be pad to the dscontnuous dervatve whch appears n the penalty term. More formally, we are solvng the nonsmooth equaton (4.10) usng a generalzed Newton teraton 4, 27, 25. We wll defne the dervatve of the penalty term, whch s requred n the Newton teraton as { P n+1 (V V n+1 ) Large f V n+1 < V V n+1 = (5.1) 0 otherwse, whch s a partcular choce of a member of the generalzed Jacoban of equaton (4.10). Consequently, a generalzed Newton teraton appled to equaton (4.10) yelds the followng algorthm. Let (V n+1 ) be the th estmate for V n+1. For notatonal convenence, we wll defne P P ( (V n+1 ) ) and V (V n+1 ). If V 0 = V n, then For = 0,... untl convergence f Penalty Amercan Constrant Iteraton I + (1 θ) ˆM + P V +1 = (V n+1 ) +1 (V n+1 ) max max(1, (V n+1 < tol ) +1 ) I θ ˆM V n + P V (5.2) or P +1 = EndFor. It s worthwhle at ths pont to determne the complexty of the above teraton, compared to an explct evaluaton of the Amercan constrant. Assumng that all of the coeffcents are stored, that Cran-Ncolson tmesteppng s used wth nonconstant tmesteps, and that there are I nodes n the S drecton, the frst teraton of the penalty algorthm requres () 6I multples to evaluate the rght hand sde of equaton (5.2), where we have made the pessmstc assumpton that P V requres I multples. Ths step also determnes the entres n ˆM, assumng all possble quanttes are precomputed and stored. () 2I multply/dvdes to factor the matrx n equaton (5.2). () 3I multply/dvdes for the forward and bac solve. (v) I dvdes for the convergence test. (Ths s also pessmstc, snce we can sp the test on the frst teraton, or f no constrant swtches have occurred.) Ths gves a total of 12I multply/dvdes for the frst penalty teraton, and 7I multply/dvdes for subsequent teratons. If constant tmesteps are used, 4I multples are needed to evaluate the rght hand sde of (5.2), leadng to a total of 10I multply/dvdes for the frst penalty teraton, and 7I multply/dvdes for subsequent teratons. If an explct method s used to evaluate the constrant, then there s only one matrx solve per tmestep. To be precse here, an explct method for handlng the constrant s P qut Explct Amercan Constrant Tmestep ˆM I + (1 θ) ˆV n+1 = I θ ˆM V n (5.3) V n+1 = max( ˆV n+1, V ).

10 10 P.A. FORSYTH AND K.R. VETZAL For constant tmesteps (assumng that all coeffcents are precomputed and stored), () 3I multply/dvdes are requred to evaluate the rght hand sde of equaton (5.3), assumng that P = 0; () assumng that the matrx s factored once and the factors stored, 3I multply/dvdes are requred for the forward and bac solve; gvng a total of 6I multply/dvdes per tmestep. For non-constant tmesteps, () 5I multply/dvdes are requred to evaluate the rght hand sde of equaton (5.3), assumng that P = 0; () 2I multply/dvdes are requred to factor the matrx; () 3I multply/dvdes are requred for the forward and bac solve; gvng a total of 10I multply/dvdes per tmestep. 6. Convergence of the Penalty Iteraton. Recall that the basc penalty algorthm (5.2) can be wrtten as I + (1 θ) ˆM + P V +1 = I θ ˆM V n + P V. (6.1) In Appendx B, we prove the followng result: Theorem 6.1 (Convergence of the nonlnear teraton of the penalzed equatons). Under the assumptons that the matrx ˆM n equaton (4.10) s an M-matrx, then The nonlnear teraton (5.2) converges to the unque soluton to equaton (4.10), for any ntal terate V 0. The terates converge monotoncally,.e. V +1 V for 1. The teraton has fnte termnaton,.e. for an terate suffcently close to the soluton of the penalzed problem (4.10), convergence s obtaned n one step. In 37, t was demonstrated expermentally that usng a smooth form of the penalty functon (4.4) dd not ad convergence of the soluton of the nonlnear equatons. Intutvely, ths s somewhat surprsng. It mght be expected that the nonsmooth penalty functon (4.4), whch has a dscontnuous dervatve, mght cause oscllatons durng the teratons. However, the above result concernng monotonc convergence explans why the penalty teraton wors so well, even wth a non-smooth dervatve. Snce V +1 V for 1, n the worst case we have V 0 V, V 1 < V V, p > V for some p 2. No further constrant swtches wll occur. In other words, for any gven node, the teraton wll not oscllate between V > V +1 and V < V ( 1). Note that V 0 can be arbtrary, but that V 1 s gven by the soluton to equaton (6.1). After V 1 s determned, the terates ncrease monotoncally. It s nterestng to observe the connecton between the penalty teraton and a pvotng method for solvng the LCP. If we let the set of nodes where V n+1 satsfy = 0 be denoted by κ, then as ponted out n 7, we can regard a pvotng FV n+1 method as a technque for determnng κ n a systematc way. Once κ s nown, then we can order the nodes κ frst, and those nodes where V n+1 = V last, and solve the resultng system. In the case where ˆM s an M matrx, then the pvotng method 7 becomes very smple. At the th pvotng step, a node s placed n κ. Any further pvotng operatons wll not remove that node from κ p, p >. In the termnology of LCP algorthms, once a node has become basc, t wll never become nonbasc as the algorthm proceeds. In ths case, t s clear the pvotng algorthm termnates n at most number of pvotng steps equal to the sze of the matrx.

11 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS 11 Each teraton of the penalty method carres out a sortng step. Those nodes n κ are labelled by havng P = 0. If the coeffcent matrx s an M matrx, once a node s placed n κ, t never leaves κ on subsequent teratons. (Note that ths s true only for 1.) The next teraton smply moves nodes that are not n κ to κ, or termnates. Ths property would hold for a problem n any dmenson, provded the coeffcent matrx s an M matrx. If we have a good estmate for κ 0, and the number of nodes whch move nto or out of κ s small, then the convergence of the penalty method wll be rapd. There also appears to be a connecton between the approach used here and n 21. In 21, each teraton conssts of obtanng an ntal estmate of constraned nodes usng a few projected relaxaton teratons, then the remanng nodes are solved usng a sparse teratve method. It s argued n 21 that ths s a good method f a good estmate of the constraned nodes s avalable. In some sense the penalty method combnes these two steps, snce an outcome of the sparse solve n the penalty teraton s an ndcaton of whether or not a node s stll constraned. More nterestngly, we observe that the penalty teraton s rapdly convergent for mult-factor optons wth non-zero correlaton 38. The dscretzed equatons n ths case are not M-matrces, and n the convecton domnated case, the dscrete equatons are nonlnear. 7. Numercal Examples. In order to carry out a careful convergence study, we need to tae nto consderaton the fact that the payoff functon (2.4) has only pecewse smooth dervatves. Ths can cause problems f Cran-Ncolson tmesteppng s used. Specfcally, oscllatory solutons can be generated 41. For example, f we consder a smple European put opton, then we now that the asymptotc soluton near the expry tme τ = 0 and close to the stre K s 33 ( put value = O τ 1/2). (7.1) Ths would suggest that V ttt = O(τ 5/2 ). The local fnte dfference truncaton error for a Cran-Ncolson step (near τ = 0) would then be O V ttt ( τ) 3. If we set τ = τ (the frst step) then the local error would be O( τ) 1/2, resultng n poor convergence. Fortunately, ths analyss s a bt too smplstc. The behavor of the soluton n equaton (7.1) s due to the non-smooth payoff near K, whch causes V SS to behave (near τ = 0, S = K) as O ( τ 1/2) 33. Ths large value of V SS causes a very rapd smoothng effect due to the parabolc nature of the PDE. Consequently, f an approprate tmesteppng method s used, we can expect the ntal errors to be damped very qucly. However, there s a problem wth Cran-Ncolson tmesteppng. Cran-Ncolson s only A-stable, not strongly A-stable. Ths means that some errors are damped very slowly, resultng n oscllatons n the numercal soluton. Snce a fnte volume dscretzaton n one dmenson can be vewed as a specal type of fnte element dscretzaton, we can appeal to the fnte element analyss n 28. Ths analyss was specfcally drected towards the case of parabolc PDEs wth non-smooth ntal condtons. Essentally, n 28 t s shown that f we tae constant tmesteps wth a Cran-Ncolson method, then second order convergence (n tme) can be guaranteed f () after each non-smooth ntal state, we tae two fully mplct tmesteps, and then use Cran-Ncolson thereafter (payoffs wth dscontnuous dervatves qualfy as non-smooth); and () the ntal condtons are l 2 projected onto the space of bass functons. In our case, ths means that the ntal condton should be approxmated by contnuous, pecewse lnear bass functons.

12 12 P.A. FORSYTH AND K.R. VETZAL No Smoothng Rannacher Smoothng Nodes Tmesteps Value Change Rato Value Change Rato Table 7.1 Value of a European put, σ =.8, T =.25, r =.10, K = 100, S = 100. Exact soluton (to seven fgures): Change s the dfference n the soluton from the coarser grd. Rato s the rato of the changes on successve grds. However, consder the case of a smple payoff such as that for a put opton. Although ths has a dscontnuous dervatve at K, no smoothng s requred provded we have a node at K. Ths s because we have a pecewse lnear representaton of the ntal condton, consstent wth the mpled bass functons used n the fnte volume method. In the case of a dscontnuous ntal condton, smoothng s necessary snce ths s not n the space of contnuous pecewse lnear bass functons. Fnally, we remar that although second order convergence does not guarantee that the soluton s non-oscllatory, n practce the above methods wor well. We can demonstrate the effectveness of the smple dea of tang two fully mplct methods at the start and Cran-Ncolson thereafter (whch we wll henceforth refer to as Rannacher smoothng 28) for a European put opton wth nown soluton. We wll use the rather extreme value of σ =.8 for llustratve purposes. Results are provded n Table 7.1, whch demonstrates that the soluton wth no smoothng converges erratcally as the grd spacng and tmestep sze are reduced. In contrast, the smoothed soluton shows quadratc convergence. The reason for the poor convergence of the non-smoothed runs can be be explaned by examnng plots of the value V, delta (V S ), and gamma (V SS ), as shown n the left sde of Fgure 7.1. (Recall that t s of practcal mportance to determne delta and gamma for hedgng purposes 16). Note that although the value appears smooth, oscllatons appear n delta (near the stre) and are magnfed n gamma. The same problem was run usng Rannacher smoothng, and the results are shown n the rght sde of Fgure 7.1. The oscllatons n delta and gamma have dsappeared. All subsequent runs wll use Rannacher smoothng. It mght appear approprate to use a tmesteppng method wth better error dampng propertes, such as a second order BDF method 2. However, our experence wth ths method for complex Amercan style problems (see 35) was poor. We conjecture that ths s due to a lac of smoothness n the tme drecton, causng problematc behavor for multstep methods. Ths effect wll be addressed n some detal below. 8. Implct and Explct Handlng of the Amercan Constrant. We wll now compare an mplct treatment of the Amercan constrant (usng the penalty technque) wth an explct treatment (see pseudo-code (5.3)). In these examples we use constant tmesteps, a convergence tolerance of tol = 10 6 (see pseudo-code (5.2)), and consequently a value of Large = Two volatlty values were used n these examples: σ =.2,.8. We truncate the computatonal doman at S = S max, where condton (2.5) s appled. The grd for σ =.2 used S max = 200, whle the grd for σ =.8 used S max = Both grds were dentcal for 0 < S < 200. The grd for σ =.8 added addtonal nodes for 200 < S < 1000.

13 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS European Put Opton Value 50 European Put Opton Value Opton Value Opton Value Asset Prce Asset Prce 0.1 European Put Opton Delta 0.1 European Put Opton Delta Opton Delta Opton Delta Asset Prce Asset Prce 0.5 European Put Opton Gamma 14 x 10 3 European Put Opton Gamma Opton Gamma 0.1 Opton Gamma Asset Prce Asset Prce Fg European put, σ =.8, T =.25, r =.10, K = 100. Cran-Ncolson tmesteppng, grd wth 135 nodes. Left: no smoothng, rght: Rannacher smoothng. Top: opton value (V ), mddle: delta (V S ), bottom: gamma (V SS ). Table 8.1 compares results for mplct (penalty method) and explct handlng of the Amercan constrant wth constant tmesteps. Frst, we note that for the penalty method the total number of nonlnear teratons s roughly the same across the two values of σ at each refnement level. Ths ndcates that the volatlty parameter has lttle effect on the number of teratons requred. Now consder the results for the case where σ =.2. Tang nto account the wor per unt accuracy, the mplct method s slghtly superor to the explct technque. However, note that the mplct method does not appear to be convergng quadratcally (the rato of changes s about 3 nstead of 4, whch we would expect for quadratc convergence). The explct method appears to be convergng at a frst order rate (rato of 2). Now consder the hgh volatlty

14 14 P.A. FORSYTH AND K.R. VETZAL Nodes Tmesteps Iters Value Change Rato Wor (flops) explct constrant, σ = mplct constrant, σ = explct constrant, σ = mplct constrant, σ = Table 8.1 Value of an Amercan put opton, T =.25, r =.10, K = 100, S = 100. Iters s the total number of nonlnear teratons. Change s the dfference n the soluton from the coarser grd. Rato s the rato of the changes on successve grds. Constant tmesteps. Rannacher smoothng used. Wor s measured n terms of number of multply/dvdes. (σ =.8) results. Tang nto account the total wor, t would appear that n ths case the explct method s a lttle better than the mplct method. The latter seems to have an error rato of about 2.9, whle the explct method has a somewhat lower convergence rate. As an addtonal accuracy chec, for all runs we also montored the quantty (4.20), whch s a measure of the maxmum relatve error n enforcng the Amercan constrant usng the penalty method. As well, we also montored the sze of the normalzed resdual of equaton (4.10) for all unconstraned nodes (.e. those nodes were P = 0). More precsely K n max lnear solver error = max n, κ n B n ( ˆM K n+1 = I + (1 θ) V n+1 ( B n+1 = max I θ ˆM V n) n κ n I θ ˆM V n) n κ n+1 = { P n+1 = 0} (8.1) In Table 8.2, we gve the statstcs for a sngle run, varyng the Large parameter (equaton (4.9) ). Note that the value of Large = 10 6 results n a maxmum relatve error n enforcng the Amercan constrant of Consequently, snce ths s well below the tme and spatal dscretzaton errors, we wll use ths value for all subsequent tests. It s nterestng to see that the number of teratons s ndependent

15 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS 15 Large Total Iteratons Value max lnear solver max Amercan error (8.1) error (4.20) *** *** *** *** Table 8.2 Test of varyng the penalty parameter Large (equaton (4.9) ). Amercan put opton, T =.25, r =.10, K = 100, sgma =.8, value at S = 100. Nodes = 269, tmesteps = 100. Iters s the total number of nonlnear teratons. Constant tmesteps. Rannacher smoothng used. tol = 1/Large (equaton( 5.2) ). ndcates teraton faled to converge, due to machne precson lmtatons. of the value of Large. Ths s a result of the fnte termnaton property of the teraton. We can see from Table 8.2 that the upper lmt to Large s determned by machne precson. Consequently, we can solve the dscrete LCP problem to arbtrary accuracy (lmted by machne precson) wth a fxed number of nonlnear teratons. 9. Analyss of Constant Tmestep Examples. In terms of approxmately solvng the dscrete LCP (4.11), the penalty method performs as the analyss predcts. The number of nonlnear teratons per tmestep s typcally of the order , ndependent of the volatlty, for reasonable tmesteps. The a posteror chec (4.20) (a maxmum relatve error of 10 9, wth tol = 10 6 n terms of satsfacton of the dscrete LCP constrant) ndcates that the error ntroduced by the penalty method s qute small. Ths error s a functon of tol (pseudo-code (5.2)), and hence can be adjusted to the desred level. In these examples we have volated condton (4.15), whch ndcates that ths condton s suffcent but not necessary for boundng the sze of the penalty term ndependent of t, S. However, the results are dsappontng n terms of the convergence of the dscretzaton of the LCP. We do not observe quadratc convergence for the mplct handlng of the Amercan constrant. An error rato of about 2.8 would be consstent wth global tmesteppng convergence at a rate of O( τ) 3/2. Now, from 29, 32, we now that the value of an Amercan call opton (where the underlyng asset pays a proportonal dvdend) behaves le V = const. + O ( τ 3/2) near the exercse boundary and close to the expraton of the contract (τ 0). Ths would gve a value for V τττ n ths regon of V τττ = Oτ 3/2. (9.1) It appears that the behavor of the Amercan put near the exercse boundary and close to expry s 23 V = const. + O(τ log τ) 3/2 const. + O(τ 1 ɛ ) 3/2 ) ; ɛ > 0, ɛ 1, τ 0. (9.2) In the followng we wll gnore the ɛ n equaton (9.2) and assume that the behavor of V τττ s gven by equaton (9.1). From equaton (9.1), the local tme truncaton error for Cran-Ncolson tmesteppng s (near the exercse boundary) ( τ) 3 local error = O. (9.3) τ 3/2

16 16 P.A. FORSYTH AND K.R. VETZAL Assumng that the global error s of the order of the sum of the local errors, from equaton (9.3) we obtan =1/ τ global error = O ( τ) 3 O( τ) 3/2, (9.4) ( τ) 3/2 =1 whch s consstent wth the observed rate of convergence. Now, nstead of tang constant tmesteps, suppose we tae tmesteps whch satsfy max( V n+1 V n ) d, (9.5) where d s a specfed constant. In order to tae the lmt to convergence, at each grd refnement we wll halve both the grd spacng and d. It s reasonable to assume that the maxmum change over a tmestep (at least near τ = 0) wll occur near K. So, from equaton (7.1), τ V n+1 = max( V n+1 V n n+1 ) O. (9.6) τ n Therefore, from equatons (9.5) and (9.6), we have τ n+1 = Od τ n. (9.7) Assumng a local error of the form (9.3), and usng equatons (9.3) and (9.7), ths gves a local error wth the varable tmesteps satsfyng equaton (9.5) as ( ) ( τ) n+1 3 d 3 (τ n ) 3/2 local error = O = O = O(d 3 ). (9.8) (τ n ) 3/2 (τ n ) 3/2 Ths mples a global error (wth O(1/d) tmesteps) of global error = O(d 2 ). (9.9) Therefore, suppose that we tae varable tmesteps consstent wth (9.5). Then at each refnement stage, where we double the number of grd nodes, and double the number of tmesteps (by halvng d), we should see quadratc convergence. Note that we should reduce the ntal tmestep τ 0 by four at each refnement. We mae no clam that the above analyss of the tme truncaton error s n any way precse, but only suggestve of an approprate tmesteppng strategy. 10. A Tmestep Selector. The tmestep selector used s based on a modfed form of that suggested n 18. Gven an ntal tmestep τ n+1, then a new tmestep s selected so that τ n+2 = mn dnorm V (S,τ n + τ n+1 ) V (S,τ n ) max(d, V (S,τ n + τ n+1 ), V (S,τ n ) ) τ n+1, (10.1) where dnorm s a target relatve change (durng the tmestep) specfed by the user. The scale D s selected so that the tmestep selector does not tae an excessve number of tmesteps n regons where the value s small (for optons valued n dollars, D = 1 s typcally approprate). In equaton (10.1), we have normalzed the factor used to

17 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS 17 estmate the new tmestep. Ths s smply to avod slow tmestep growth for large values of the contract. Ths could be a problem wth call optons, for example, where the computatonal doman s truncated at a large value of S. If we dd not examne the relatve changes over a tmestep, then t s possble that the tmestep would be lmted by large absolute changes n the soluton (whch would occur as S ), even though the relatve changes were small. Snce V (S = K, τ 0) 0, we expect that the denomnator of equaton (10.1) wll tae ts largest value near S = K, snce V ncreases rapdly there. Consequently, the tmestep selector (10.1) wll approxmately enforce the condton that Hence we wll have V n+1 D dnorm. (10.2) τ n+1 = O(dnorm τ n ), (10.3) so that we should see a global error of O(dnorm) 2, whch follows from equaton (9.9). Note that tmestep selector (10.1) estmates the change n the soluton at the new tmestep based on changes observed over the old tmestep. Some adjustments can be made to ths smple model f a more precse form for the tme evoluton of the soluton s assumed, but we prefer (10.1) snce t s smple and conservatve. In practce, we select a ( τ) 0 for the coarsest grd, and then ( τ) 0 s cut by four at each grd refnement. There s not much of a penalty for underestmatng a sutable ( τ) 0 snce the tmestep wll ncrease rapdly f the estmate s too conservatve. In the followng runs, we used values of ( τ) 0 = 10 3 and dnorm =.2 on the coarsest grd. The value of dnorm was reduced by two at each grd refnement. 11. Varable Tmestep Examples. Table 11.1 presents results for the cases consdered n Table 8.1, but ths tme usng the tmestep selector (10.1). In ths case, the mplct method appears to be a clear wnner n terms of flops per unt accuracy. Use of varable tmesteps actually seems to degrade the convergence of the explct method. Ths can be explaned by loong at the tmestep hstory. The tmestep selector uses small tmesteps at the start, and then taes large steps at the end. Note that the average tmestep sze (total tme dvded by number of tmesteps) s larger for the varable tmestep run compared to the constant tmestep run (Table 8.1). Ths clearly negatvely mpacts the explct method, whch seems to show a frst order rate of convergence. On the other hand, the mplct method appears to exhbt close to quadratc convergence. Fgure 11.1 shows value, delta, and gamma for the σ =.2 case, usng both explct and mplct treatments of the Amercan constrant. Although the value and delta appear smlar for both cases, there are clearly large oscllatons n the gamma near the early exercse boundary for the explct method. The mplct method does show some small oscllatons near the exercse boundary. However, ths s due to the use of Cran-Ncolson tmesteppng, as noted n 6. These oscllatons dsappear f fully mplct tmesteppng s used, as shown n Fgure Comparson Wth Bnomal Lattce Methods. It s nterestng to compare the results here wth those obtaned usng the bnomal lattce method, whch s commonly used n fnance 33. In Appendx C, we show that ths technque s smply an explct fnte dfference method on a log-transformed grd. Consequently, the truncaton error s O( τ), where the total number of steps s N = O1/( τ).

18 18 P.A. FORSYTH AND K.R. VETZAL Nodes Tmesteps Iters Value Change Rato Wor (flops) explct constrant, σ = mplct constrant, σ = explct constrant, σ = mplct constrant, σ = Table 11.1 Value of an Amercan put opton, T =.25, r =.10, K = 100, S = 100. Iters s the total number (over all tme steps) of nonlnear teratons. Change s the dfference n the soluton from the coarser grd. Rato s the rato of the changes on successve grds. Varable tmesteps. Rannacher smoothng used. Wor s measured n terms of number of multply/dvdes. The bnomal lattce method requres about 3/2N 2 flops (countng only multples, and assumng all necessary factors are precomputed). Note that we obtan the value of the opton at t = 0 only at the sngle pont S0, 0 n contrast to the PDE methods whch obtan values for all S 0, S max. As a result, the methods are not drectly comparable. Nevertheless, assumng that we are only nterested n obtanng the soluton at a sngle pont, t s nterestng and useful to compare these two technques. Gven N = O1/( τ), the complexty of the bnomal method s O(N 2 ). Snce the error n the lattce method s O( τ) = O(1/N), we have error bnomal lattce = O (complexty) 1/2. (12.1) Suppose nstead that we use an mplct fnte volume method wth Cran-Ncolson tmesteppng, and that the penalty method s employed for handlng the Amercan constrant. The complexty of ths approach s O(N 2 ), where we have assumed that N = O1/( S) (note that ths s the case f we use the tmestep selector (10.1) and dnorm = O( S)). It s also assumed that the the number of nonlnear teratons per tmestep s constant as S 0, whch s observed as long as dnorm = O( S). When tmesteps are selected usng (10.1), we have observed quadratc convergence. Ths mples error mplct fnte volume = O ( N 2) = O (complexty) 1. (12.2) Therefore the mplct fnte volume method s asymptotcally superor to the bnomal lattce method, even f the soluton s desred at only one pont.

19 QUADRATIC CONVERGENCE FOR AMERICAN OPTIONS Amercan Put Opton Value 50 Amercan Put Opton Value Opton Value Opton Value Asset Prce 0 Amercan Put Opton Delta Asset Prce 0 Amercan Put Opton Delta Opton Delta Opton Delta Asset Prce 0.3 Amercan Put Opton Gamma Asset Prce 0.07 Amercan Put Opton Gamma Opton Gamma Opton Gamma Asset Prce Asset Prce Fg Amercan put, σ =.2, T =.25, r =.10, K = 100. Cran-Ncolson tmesteppng, Rannacher smoothng, varable tmesteps, grd wth 433 nodes. Left: explct constrant, rght: mplct constrant. Top: opton value (V ), mddle: delta (V S ), bottom: gamma (V SS ). It s nterestng to determne at what levels of accuracy we can expect the mplct PDE method to become more effcent than the bnomal method. Table 12.1 gves the results for a bnomal lattce soluton (algorthm (C.2)) for the problems solved earler usng an mplct PDE approach. Ths table should be compared to Table For further ponts of comparson, we also computed solutons to the problem used n 13. We used two versons of the problem n 13, one wth an expry tme of T = 1 and the other wth T = 5. Fgure 12.1 summarzes the convergence of both the bnomal lattce and PDE methods for all four problems. The absolute error s computed by tang the exact soluton as obtaned by extrapolatng the PDE soluton down to zero grd and tmestep sze, assumng quadratc behavor. The

20 20 P.A. FORSYTH AND K.R. VETZAL 0.07 Amercan Put Opton Gamma Opton Gamma Asset Prce Fg Gamma (V SS ) of an Amercan put, σ =.2, T =.25, r =.10, K = 100. Fully mplct tmesteppng, Rannacher smoothng, varable tmesteps. Grd wth 433 nodes used. Constrant mposed mplctly. Tmesteps Value Change Rato Wor (flops) σ = σ = Table 12.1 Bnomal lattce method. Value of an Amercan put, T =.25, r =.10, K = 100, S = 100. Change s the dfference n the soluton from the coarser grd. Rato s the rato of the changes on successve grds. Wor s measured as the number of multples. PDE method becomes more effcent than the bnomal lattce method at tolerances between dependng on the problem parameters. These crossover ponts occur at tolerances whch would be used n practce. Note that n these comparsons, we are puttng the best possble lght on the bnomal lattce method, snce we gnore the fact that we obtan much more nformaton wth the mplct PDE technque. 13. Applcaton of Penalty Methods to More General Problems. As derved n the Appendces, a suffcent condton for monotone convergence of the penalty teraton s that the dscretzed dfferental operator s an M-matrx. In practce, we have found that ths condton s not necessary for rapd convergence of the penalty teraton. For example n 37, 38, we have appled the penalty method to Amercan optons wth stochastc volatlty, convertble bonds (whch have Amercan type maxmum and mnmum constrants), and Amercan optons on two assets, wth good results. In ths case, the dscretzed dfferental operator was not an M-matrx, and f a flux lmter was used, the dscretzed dfferental operator was nonlnear.