The Computation of American Option Price Sensitivities using a Monotone Multigrid Method for Higher Order B Spline Discretizations

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1 The Computation of American Option Price Sensitivities using a Monotone Mutigrid Method for Higher Order B Spine Discretizations Markus Hotz October 26, 2004 Abstract In this paper a fast sover for discrete free boundary vaue probems which is based on hierarchica higher order discretizations is presented. The numerica method consists of a finite eement discretization with B spine ansatz functions of arbitrary degree combined with a monotone mutigrid method for the efficient soution of the resuting discrete system. In particuar, the potentia of the scheme in the fast and accurate computation of American stye option prices in the Back Schoes framework and of their derivatives with respect to the underying is investigated. Due to the higher order discretization, the derivatives, aso caed Greek etters, can be staby and accuratey determined via direct differentiation of the basis functions. Considering the vauation of pain vania American stock options, we show that our soution method is competitive to the best schemes proposed in the iterature when accurate approximations to the derivatives are required. It provides the first mutigrid approach based on higher order basis functions which is directy appicabe to American option pricing. Keywords: American options, Greek etters, free boundary vaue probems, inear compementary probems, finite eements, cardina higher order B spines, monotone mutigrid methods AMS-Cassification: 65M55, 65N30, 35J85 65D07, 65D99, 90C33 1 Introduction The overwheming majority of a traded options are of American stye. However, no genera cosed form soution for their vauation is known. Therefore, one has to resort to anaytica approximations or to numerica pricing methods. Here, one has to keep in mind that often not ony fair option prices are required, but aso accurate approximations to the derivatives of the option price, e.g., with respect to time or to the underying. The derivatives, aso caed Greek etters or Greeks, pay a crucia roe as hedge parameters in the anaysis of market risks. They can in contrast to option prices not directy be observed in the market. This fact is further increasing the demand for numerica schemes for their approximation. Usuay, the Greek etters are computed by numerica Institut für Numerische Simuation, Universität Bonn, Wegeerstr. 6, Bonn, Germany, hotz@ins.uni-bonn.de 1

2 differentiation of option vaues. Here, the option vaues must be approximated up to high accuracy in order to obtain stabe and reiabe resuts. In the Back Schoes framework [BS], the vauation of American options requires the soution of a particuar free boundary vaue probem. A finite difference or finite eement discretization eads to a discrete inear compementary probem which can aso be regarded as an obstace probem. Since the work of [BC], it is known that the most efficient sovers for this kind of probems are mutigrid techniques. However, due to inconsistent approximations of the free boundary on coarser grids, the scheme from [BC], caed PFAS is sometimes acking stabiity. Moreover, no convergence proof is known. This disadvantage coud be resoved by the work of Kornhuber [Ko1] with the introduction of monotone mutigrid methods (MMG). In the ast decade, they have been appied in [Ko1, Ko2, Kr] with great success to probems from continuum mechanics. Their key ingredients are sophisticated restriction operators for obstace functions which arise from a consistent handing of the free boundary, and a specia truncation of the basis functions. Unfortunatey, the method is restricted to continuous piecewise inear functions, aso caed hat functions. To our knowedge, this aso appies to a other mutigrid approaches (e.g., [BC, HM, Ho, Ko1, Ko2, Kr, Ma, Oo, Tai]) for inear compementary probems proposed in the iterature. Taking derivatives of approximations obtained by hat functions, however, wi ead to unstabe and miseading resuts. The main difficuty in generaizing monotone mutigrid methods to higher order basis functions is a suitabe handing of the obstace condition of the probem. Via the use of a B spine basis, this difficuty coud be recenty resoved in a companion paper [HK], which ed to the first generaization of the monotone mutigrid method to arbitrary smooth basis functions. Using arguments of [Ko1], goba convergence and optima compexity of the B spine based monotone mutigrid method coud be proved. By construction, one can expect specia robustness of the scheme and fu mutigrid efficiency in the asymptotic range. Moreover, the use of smooth basis functions admits a direct determination of the derivatives of the soution. In this paper, we appy the B spine based monotone mutigrid method in order to compute American option prices as we as their derivatives with respect to the underying up to high accuracy and investigate the performance of the scheme by numerica exampes. The paper is structured as foows. In Section 2, we appy a finite difference discretization in time and a higher order B spine-based finite eement discretization in space to the free boundary probem which describes the fair price of an American option in the Back Schoes framework. In Section 3, we generaize the projective Gauss Seide scheme to higher order basis functions and iustrate the monotone mutigrid method from [Ko1] as a fast sover for the discrete form. Finay, in Section 4, the potentia of the new scheme for the approximation of pain vania American option price sensitivities is demonstrated by numerica experiments. We show that our scheme is competitive to the best schemes proposed the iterature when accurate approximations to the derivatives of the option vaue are required. In Section 5, we indicate some possibe extensions of our scheme to higher dimensiona option pricing probems, to adaptive grid refinements and to higher order time discretizations. 2

3 2 The Computation of American Price Sensitivities via a Higher Order Finite Eements Method In this section, the free boundary vaue probem which describes the fair price of an American option is introduced. A weak form is derived and discretized with finite differences in time and finite eements in space. This eads to a discrete inear compementary probem which is soved by a projective variant of the Gauss Seide reaxation. By using sufficienty smooth basis functions, the derivatives of the option vaue can be determined by direct differentiation of the finite eement ansatz functions. 2.1 The Pricing of American Options Options are financia contracts that give its owner the right, but not the obigation, to buy (ca option) respectivey to se (put option) an underying security (e.g. an asset) at specified times in the future for an agreed strike price K. European options can ony be exercised at the maturity date T, whereas American options can be exercised at any time at or before the maturity date. For a pain vania put option we have at time t = T a payoff of (1) H(S, T ) = max{0, K S(T )}. The enormous success and systematic trade of financia derivatives in the recent years ed to the need for fast and accurate pricing techniques. As options are getting more and more compex and their vauation more and more compicated, sophisticated computationa methods have been deveoped and continue to be an active fied of research. For traders not ony the vaue V = V (S, t) of an option at time t < T is important, but aso the derivatives, e.g., with respect to time, to the underying or to voatiity, as they pay a crucia roe in the anaysis of market risks. In this paper we restrict oursef to the space derivatives Deta := V S and Gamma := 2 V S 2. In the famous and we estabished Back Schoes mode [BS] the stochastic process generating the price S(t) of the underying asset is modeed as a geometric Brownian motion ds(t) = µ S(t) dt + σ S(t) db(t). As usua, B(t) denotes a one dimensiona standard Brownian motion. The parameters µ R and σ R + are the drift and the voatiity of the stochastic process S(t). Under additiona assumptions on the financia market and the centra assumption of absence of arbitrage, it is we known that the fair price of a European option satisfies the inear, instationary, backward partia differentia equation (2) L V := t V + σ2 2 S2 2 S 2 V + rs V rv = 0, S where r R + denotes the interest rate for a riskess investment. Taking suitabe boundary conditions into account, an anaytic soution of the Back Schoes equation (2) can be derived and is given by the famous Back Schoes formua [BS]. 3

4 As American options can in contrast to European ones be exercised at a times t T, their vauation is more invoved. For a t T not ony the option vaue must be determined but aso whether or not the option shoud be exercised. In the Back Schoes framework, the fair vaue of an American option can be determined by the soution of a free boundary vaue probem of the Back Schoes equation (2) in the domain R + [0, T ) (cf. [WHD]). We formuate it for the case of an American put option, i.e., for the payoff H(S, t) as in (1). Probem 2.1 (Free Boundary Probem) Find V = V (S, t) and S f = S f (t), such that with the boundary data the fina data L V (S, t) = 0 for S > S f and 0 t < T, V (S, t) = H(S, t) for S S f and 0 t < T, V (S, t) = 0 for S and 0 t < T, V (S, T ) = H(S, T ) for S 0 and the conditions, that V and V / S are continuous on the free boundary S f. Note that the boundary condition V (S, t) = K for S 0 is aready impied by the condition V (S, t) = H(S, t) for S S f. The free boundary S f, which is part of the soution, separates the part of the domain where it is optima to exercise the option immediatey from the part where it is favorabe to keep the option. It therefore describes the optima exercise price of the option. Note that the right of eary exercise eads to the condition V (S, t) H(S, t) for a S > 0 and 0 t T and can therefore aso be regarded as an ower obstace condition to the American option price V. Despite much effort, a cosed form soution of Probem 2.1 is not yet known except for the infinite horizon case [Mc]. Thus, the derivation of anaytica approximations or numerica methods for the vauation of American options is sti an active fied of research. Due to their fexibiity and simpicity the most common approaches used by financia institutions are binomia methods introduced by [CRR]. There are many extensions and improvements on this approach (e.g., [Bn, By, FG, BG2, LR]). Anaytica approximations incude the works [Mi, BW, BjS, Ro, Ge, Wh, GJ]. Other approaches incude Monte Caro simuation (see [BG3] for a review), the method of ines [Me, CF], penaty methods [FV] and techniques from inear optimization theory [DH]. A discretization of the free boundary probem by finite difference methods was initiay proposed by [BrS] and is aso considered in [GS, HW, TR]. Finite eement methods were used by [WHD, FVZ, ZS, ZFV]. Note that finite eement or finite difference approaches provide an approximation to the whoe surface, which is defined by a option vaues V (S, t) for S > 0 and 0 t T, whereas most of the other methods approximate just a certain singe vaue V (S 0, 0). The computation of derivatives of American option prices is expicity considered in [C, BG1, PV, R, WW]. In [WW], an anaytic approximation [BW], a finite difference method with hat functions [BrS] and a binomia tree in the variant of [LR], caed Leisen Reimer trees, are compared with regard to their performance in approximating American option price sensitivities. The author concudes that Leisen Reimer trees are the superior method. 4

5 One can generaize the standard Back Schoes mode by the assumption of stochastic voatiity (cf., e.g., [BR]). This way, a free boundary vaue probem of the above form is obtained, but in two space dimensions. Soutions via finite difference discretization combined with the PFAS mutigrid scheme from [BC] are discussed in [CP, Oo]. Since the work shows the need for higher order Back Schoes sovers, we aso mention the asymptotic mode [FPS] to price American options with stochastic voatiity, which is based on accurate approximations to the third derivative 3 V/ S 3 of the soution V of an one dimensiona Back Schoes mode. 2.2 Transformation and Weak Formuation The starting point of finite eement methods is a weak formuation of Probem 2.1. The first step in the derivation, which can be found in more detai in [Hz, WHD], is a reformuation into a inear compementary probem such that the free boundary does not show up expicity anymore. Then, the noninear transformations (3) S = Ke x, t = T 2 τ σ 2, V (S, t) = K e x 2 (q 1) ( 1 4 (q 1)2 +q)τ y(x, τ), with q := 2r/σ 2, are used to transform the Back Schoes equation (2) into the paraboic heat equation (4) y τ 2 y x 2 = 0 in the new variabe y(x, τ) in order to avoid numerica compications with the treatment of the convective term and to simpify the impementation. The transformation of the payoff function H(S, t) gives the transformed payoff function (5) g(x, τ) := e 1 4 (q+1)2τ max{e x 2 (q 1) e x 2 (q+1), 0}. The transformation of the boundary data eads to (6) im y(x, τ) = im g(x, τ), y(x, 0) = g(x, 0). x ± x ± To simpify the treatment of the boundary conditions, we reduce the probem (in contrast to [WHD]) to homogeneous boundary data by substituting (7) u(x, τ) := y(x, τ) g(x, τ). In the next step, the unbounded transformed domain R (0, σ 2 T/2] is substituted by the computationa domain Ω := I x I τ := [x min, x max ] (0, σ 2 T/2] R 2 with fixed vaues x min < 0 < x max. In [JLL], it is proved that the resuting ocaization error decreases uniformy for increasing Ω. Finay, a weak formuation can be derived by mutipying the transformed probem by a test function and integrating by parts. The obstace condition as we as the homogeneous 5

6 boundary and initia conditions are integrated in the definition of the set of admissibe test functions K := {v H(Ω) : v(x, τ) 0, v(x min, τ) = v(x max, τ) = 0, v(x, 0) = 0 for a (x, τ) Ω}, where H(Ω) denotes the Soboev space of a functions v = v(x, τ) L 2 (Ω), which are weaky differentiabe with respect to x and strongy differentiabe with respect to τ. The transformed soution u(x, τ) can now be determined by the soution of the foowing paraboic variationa inequaity, where, denotes the L 2 inner product in I x. Probem 2.2 (Weak Formuation) Find u K, such that u, v u + a(u, v u) f(v u) for a v K, τ where a : H(Ω) H(Ω) R and f : H(Ω) R are defined by a(u, v) := xmax x min x u x v dx, f(v) := xmax x min g τ v + g v x x dx. Since the biinear form a(, ) is symmetric and positive definite, Probem 2.2 admits a unique soution u K (cf. [EO]). From Probem 2.2 the American option vaue V can finay be derived by a back transformation of u via (7) and (3). 2.3 Discretization We discretize Probem 2.2 by finite differences in time and finite eements in space. Due to the demand of accurate approximations to the derivatives we use a B spine basis of arbitrariy order k for the space discretization. As we show in subsection 2.3.2, this basis aso has advantages in the handing of the obstace condition Time discretization Let τ m := m τ, m = 0,..., M, M N be an equidistant discretization of time interva I τ with step size τ := 1 2 σ2 T/M. Let further H 1 0 (I x) denote the Soboev space of functions with zero trace on the boundary and define u m := u(, τ m ) H 1 0 (I x). A finite difference discretization of the time derivative in Probem 2.2 by a θ scheme, which interpoates between an expicit (θ = 0) and an impicit (θ = 1) representation, eads to an eiptic variationa inequaity in each time step. Probem 2.3 (Semi discrete Form) Find u m+1 K, such that where a τ (u m+1, v u m+1 ) f m (v u m+1 ) for a v K K := { v H 1 0 (I x ) : v(x) 0 for a x I x } and a τ : H0 1(I x) H0 1(I x) R and f m : H0 1(I x) R are given by xmax a τ (u, v):= u v +θ τ u v xmax ] x min x x dx, f m (v):= τf(v)+ u m v+ [(θ 1) τ um v x min x x dx. Note that f m depends on the soution u m H 1 0 (I x) of the previous time step. Specific for θ = 1 2 the Crank Nichoson scheme is obtained. In [BHR] the reguarity um+1 H 5/2 ɛ is shown for arbitrary ɛ > 0. 6

7 2.3.2 A B spine based Finite Eement Discretization in Space Now, a higher order finite eement discretization is appied to Probem 2.3, which is based on cardina B spine functions. Let x i := x min + i h, i = 0,..., N 1, N N, be an equidistant discretization of the space interva I x := [x min, x max ] with step size h := (x max x min )/(N 1) and et S h H 1 0 (I x) be a finite dimensiona space of piecewise poynomias. If piecewise inear functions v h S h are used for the space discretization, the side condition (8) v h (x) 0 for a x I x from Probem 2.3 is obviousy satisfied if the set of pointwise inequaities (9) v h (x i ) 0 for a i = 0,..., N 1 hod. Trying to generaize this idea to piecewise functions v h of higher degree, one is confronted with the probem that for given x [x i, x i+1 ] the estimate (10) min {v h (x i ), v h (x i+1 )} v h (x) max {v h (x i ), v h (x i+1 )} is not vaid anymore. This shows that controing function vaues on grid points does not suffice to ensure a side condition of the form (8) in the higher order case and expains why higher order noda Lagrange basis functions are not suited for the probem at issue. Instead, we propose here a construction using B spines as higher order basis functions, which compares B spine expansion coefficients instead of function vaues, and heaviy profits from the fact that B spines are nonnegative. More information on finite eement methods for boundary vaue probems with B spines can, e.g., be found in [Hg]. There, a modification of the B spine basis, which eads to so caed web spines, is used to hande genera domains and genera boundary conditions. Due to the homogeneous boundary conditions (the soution is even zero in a neighborhood of the boundary as shown in [Hz]) and due to the rectanguar domain, such a modification, however, is not necessary for the probem under consideration. For readers convenience, we reca the reevant facts about B spines from [Bo]. Definition 2.4 (B spines) For k N and n = N + k 1 et h := {θ i } i=1,...,n+k be an equidistant expanded knot sequence in the interva I x with grid spacing h of the form (11) θ 1 =... = θ k = x min < θ k+1 <... < θ n < x max = θ n+1 =... = θ n+k with θ k+i = x i. Then the B spines N i,k, h of order k are recursivey defined by (12) { 1, if x [θi, θ N i,1, h (x) = i+1 ) 0, ese N i,k, h (x) = x θ i (k 1) h N i,k 1, h (x) + θ i+k x (k 1) h N i+1,k 1, h (x). for i = 1,..., n. 7

8 We aso use the abbreviation N i,k = N i,k, h, when ony one grid is considered. It is known that B spines have the properties supp N i,k [θ i, θ i+k ] (oca support), N i,k (x) 0 for a x I x (nonnegativity) and N i,k C k 2 (I x ) (differentiabiity). Moreover, the set Σ h := {N 1,k,..., N n,k } constitutes a ocay independent and unconditionay stabe basis with respect to to Lp, 1 p, for the finite dimensiona space N k, h := span Σ h of the spines of order k. Lemma 2.5 If the B spine coefficients v i, g i of two B spine functions v h, g h N k, h satisfy v i g i for a i = 1,..., n, then v h (x) g h (x) hods for a x I x. Proof: Using the representation v h = n v i N i,k, g h = i=1 n g i N i,k and the nonnegativity N i,k (x) 0 for a x I x, we deduce v h (x) g h (x) = i=1 n (v i g i ) N i,k (x) 0 for a x I x. i=1 Here and beow, we use the subscript i in v i = (v h ) i to denote B spine expansion coefficients and bodface etters v to denote B spine coefficient vectors v := (v 1,..., v n ) T R n. By appying Lemma 2.5 with g h = 0, the side condition (8) can now be ensured for B spine functions of genera order k simiar to (9) by the conditions (13) v i 0 for a i = 1,..., n. Choosing S h := N k, h the space of B spines of order k and approximating the exact soution u m+1 H0 1(I x) of Probem 2.3 by a discrete function u m+1 h S h with the B spine coefficient vector u m+1 R n, the obstace condition u m+1 0 in I x of Probem 2.3 can now be repaced by the pointwise side condition u m+1 0. Foowing the computations of [Hz, WHD], one therefore obtains the foowing discrete form of Probem 2.2, which has to be soved in each time step. Probem 2.6 (Discrete Variationa Inequaity) Find 0 u m+1 R n, such that (14) (v u m+1 ) T ( C u m+1 b m) 0 hod for a 0 v R n. Here the matrix C = (C i,j ) R n n is defined by (15) C i,j := xmax x min N i,k N j,k dx + θ τ xmax x min where N i,k := N i,k/ x. The right hand side b m is given by where (16) r m i := b m := τ r m + (B + (θ 1) τa) u m xmax x min N i,k N j,k dx =: B i,j + θ τa i,j g τ (x, τ m) N i,k (x) + g x (x, τ m) N i,k (x)dx. 8

9 For the soution of Probem 2.6 the error estimates u m+1 u m+1 h 1 = O(h) and u m+1 u m+1 h 1 = O(h 3/2 ɛ ) in the H 1 Soboev norm are proved in [BHR], provided S h is the space of continuous piecewise inear resp. piecewise quadratic functions. Note that Probem 2.6 can be equivaenty written in the form of the inear compementary probem (17) C u m+1 b u m+1 0 (u m+1 ) T ( C u m+1 b ) = 0 and be regarded as an obstace probem with the zero function as obstace (cf. [EO]). Remark 2.7 Note that this soution approach can immediatey be adapted to the pricing of European and Bermudan options (cf. [WHD]). In each time step, where prior exercise of the option is not possibe, the variationa inequaity (14) must just be repaced by the corresponding variationa equaity. In the case of equidistant grids, simpe expicit formuas for the entries A i,j and B i,j of the stiffness matrix A and the mass matrix B from (15) can be found in [Hg]. To compute the vector r m R n at time step m from (16), the transformed payoff function g(x, τ m ) H 1 (I x ) at time step τ m is approximated by a function gh m S h and expanded as n (18) gh m (x) = gi m N i,k (x). i=1 By substituting this representation into (16), one derives the discrete right hand side (19) r m = ( C g m+1 (B + (θ 1) τa) g m) / τ. By the approximation properties of B spines (see [Bo]) the resuting approximation error is of same order as the discretization error. Aternativey, an expicit formua for the computation of r m in the specia case of equidistant grids and the function g from (5) is given in [Hz]. 2.4 Approximations to the Greek Letters The use of smooth basis functions in the finite eement approach is motivated by the possibiity to determine the space derivatives of the soution by direct differentiation of the ansatz functions. This way, numerica differentiation can be avoided and a much higher accuracy can be expected. This wi indeed be confirmed in Section 4 by numerica exampes. Lemma 2.8 For fixed τ, et y(x, τ) be the soution of the heat equation (4) with boundary data (6). Then the Greeks etters Deta and Gamma are given by the identities = e x 2 (q+1) ( 1 4 (q 1)2 +q)τ (y (x, τ) 12 ) (q 1) y(x, τ), V (S, t) S 2 V (S, t) S 2 = e x 2 (q+3) ( 1 4 (q 1)2 +q)τ ( y (x, τ) q y (x, τ) + 1 ) 4 (q2 1) y(x, τ) /K. 9

10 Proof: Using the transformations (3) and the identity S/ x = Ke x, the assertion can easiy be verified by the product rue of differentiation. For τ = τ m, y(x, τ m ) and the partia derivatives y (j) (x, τ m ) can be obtained from the finite eement soution u m h (x) of Probem 2.6 using (7) and (18). The function vaues um h (x) and the derivatives (u m h )(j) (x) can efficienty and staby be determined by the we known recursion formuas for the vauation of B spines and their derivatives (cf. [Bo]), provided B spines of order k j + 2 are used. 3 Fast Soution of the discrete form In this section we present a fast sover for Probem 2.6. In the first subsection, the projective Gauss Seide scheme from [Cr] is generaized to a B spine basis. This scheme is then used as smoothing component within a mutigrid scheme to obtain convergence rates which are independent of the grid spacing h. 3.1 B spine based Projective Gauss Seide Schemes In order to emphasize the h dependency, we adapt our notation and write the discrete inear compementary probem (17) in the operator form (20) L h u h f h, u h g h, (u h g h )(L h u h f h ) = 0 with f h, g h, u h S h and the inear operator L h. Note that the zero obstace function in (17) is repaced by a genera discrete obstace g h S h, and that the time index m is omitted for the sake of carity. Since the operator L h, which corresponds to the matrix C from (15), is symmetric and positive definite, the inear compementary probem (20) can be soved by the projective Gauss Seide scheme, provided S h denotes the space of continuous piecewise inear functions (cf. [Cr]). Given an iterate u ν h in the ν th iteration, a standard Gauss Seide sweep ū ν h := S (uν h ) is suppemented by a projection u ν+1 h := P ū ν h in order to satisfy the side condition u ν+1 h g h. For each grid point x i I x, the projection of a piecewise inear function v h is usuay performed by (21) P v h (x i ) := max{v h (x i ), g h (x i )}. In order to generaize the projection to higher order functions v h, one is, however, again confronted by the probem that the estimate (10) is not vaid anymore. Once more, the difficuty can be resoved by the use of a B spine basis. In that case, the projection can be reaized by Lemma 2.5 simiar to (21) but invoving B spine coefficients v i by setting (22) P v i := max{v i, g i } 10

11 for i = 1,..., n. Because of the tensor product structure of the discrete soution set {v R n : v i g i for i = 1,..., n} R n, the convergence of the resuting projective Gauss Seide scheme foows using the same arguments as in [Cr]. Furthermore, under the assumption of no degeneracy (cf. [Cr, EO]), which is satisfied for Probem 2.6, it foows that the contact set of the soution u h, defined by a coefficients u i for which equaity hods, is identified after a finite number of iterations. This impies that the asymptotic convergence rate of the projective Gauss Seide scheme is of the order 1 O(h 2 ), as it is we known for the Gauss Seide scheme in the unconstrained case. Therefore, one suffers under unsatisfactoriy sow convergence rates for sma grid spacings h. 3.2 The Monotone Mutigrid Agorithm (MMG) For boundary vaue probems, it is we-known that the disadvantage of the h dependency of the Gauss Seide reaxation can be overcome by mutigrid techniques. This aso appies to variationa inequaities and inear compementary probems as it is demonstrated by a wide range of iterature (e.g. [BC, HM, Ho, Ko1, Ko2, Kr, Ma, Oo, CP, Tai]). However, a cited works are restricted to discretizations with piecewise inear functions. This gap was cosed in [HK], where the monotone mutigrid method (MMG) from [Ko1] coud be generaized to smooth basis functions by using a B spine basis. If we introduce a nested sequence of finite dimensiona spaces S 1 S 2,..., S L H 1 0 (I x ) with equidistant grids := h, = 1,..., L, L N, and grid spacings h 1 := 2 h, the monotone mutigrid method can be impemented as a variant of a standard mutigrid scheme by adding a projection step as in (22) and empoying specia restriction operators r, r for the inter grid transfer. Agorithm 3.1 MMG Let u ν := uν h S be a given approximation in the ν th cyce on eve 1. Then, the MMG agorithm consists of the foowing steps: 1. A priori smoothing: u ν,1 := (P S) η 1 (u ν ). 2. Coarse grid correction: d := f L u ν,1, f 1 := rd, g 1 := r(g u ν,1 ), L 1 := rl p. If = 1, exacty sove the inear compementary probem L h 1 v 1 f 1, v 1 g 1, (v 1 g 1 )(L 1 v 1 f 1 ) = 0. If > 1, do γ steps of MMG 1 with start vaue u 0 1 := 0 and soution v 1. Set u ν,2 := u ν,1 + p v A posteriori smoothing: u ν+1 := (P S) η 2 (u ν,2 ). The number of a priori and a posteriori smoothing steps is denoted by η 1 and η 2, respectivey. For γ = 1 one obtains a V cyce, for γ = 2 a W cyce iteration. To appy Agorithm 11

12 3.1 to the vauation of American options (i.e., to Probem 2.6), we choose the different components as foows: The spaces S := N k, are defined as the spaces of B spines of order k as motivated before. The a priori and the a posteriori smoothing steps P S are reaized as in (22) by the projective Gauss Seide scheme. The proongation operator p = p +1 : S S +1 is defined by the B spine refinement reation (cf. [Bo]) N i,k, = k ( ) k 2 1 k N 2i 1+j,k, +1. j j=0 Foowing [Ha2], we choose the restriction r = r +1 : S +1 S, which is used to transfer the defect d to coarser grids, as the adjoint of the proongation p. In our case, one obtains the weighted restriction r = 1 2 p. In order to obtain the coarse grid obstace function g 1 := r(g u ν,1 ), a specia restriction operator r = r +1 : S +1 S shoud be used, which differs from r in genera and eads to monotone approximations of the obstace. As we show in the next section, the monotonicity of restriction operators eads to admissibe new iterates u ν,2 in the sense that the side condition (23) u ν,2 is satisfied. This is one of the underying ideas of monotone mutigrid methods and eads to specia robustness of the scheme. g Remark 3.2 The convergence of the scheme can be significanty further acceerated if the coarse grid basis functions are adapted in each iteration step to the actua position of the free boundary by a suitabe truncation operator. This eads to the truncated version (TrMMG) of the monotone mutigrid method (cf. [Ko1, HK]) The Construction of Monotone Obstace Approximations In this section we summarize the main resuts from [HK] with regard to the construction of monotone restriction operators r. The construction is not obvious for B spines of genera order k. It is based on the nonnegativity and on the refinement properties of B spines. In the foowing, we fix two eves and + 1 and expand the ower obstace function S := g u ν,1 S and its approximation S := g 1 S 1 with B spine coefficient vectors c R n and c R n 1 as n S = c i N i,k, =: c T N k,, S = i=1 n 1 i=1 c i N i,k, 1 =: c T N k, 1, where n depends on the number of grid points in and n 1 = (n 1 + k)/2. Definition 3.3 (Monotone Coarse Grid Approximation) A function S S 1 is caed an upper monotone coarse grid approximation to S S if S(x) S(x) hods for a x I x. Remark 3.4 The condition (23) is satisfied if g 1 is an upper monotone coarse grid approximation to g u ν,1, since u ν,2 := u ν,1 + p v 1 u ν,1 + p g 1 u ν,1 + g u ν,1 g. 12

13 For hat functions such approximations are constructed in [Ma] and [Ko1]. For B spines of genera order k monotone coarse grid approximations can be obtained by the foowing proposition, which we cite from [HK]. Proposition 3.5 The B spine L k := q T k N k, 1 with expansion coefficients (24) q k,i := max { c 2i k,..., c 2i } for i = 1,..., n 1 is a monotone upper coarse grid approximation to the B spine S = c T N k,. Within a monotone mutigrid scheme, it can be expected that better approximations of the obstace function on coarse grids ead to more efficient coarse grid corrections and thus to a faster convergence. As shown in [HK], the approximations L k can further be improved via a inear optimization formuation. By Fourier Motzkin Eimination in the case k = 2 and a new optimization agorithm, caed OCGC, in the case k > 2 (approximate) soutions can be obtained in optima O(n ) operations. These ead to the foowing coarse grid approximations S which we cite ony for the cases k = 2, 3, 4 for the sake of simpicity. The genera formua and a proofs can be found in [HK]. Proposition 3.6 The B spine S = c T N k, 1 with recursivey defined coefficients c 1 := q 2,1 and c i := max{2 c 2i q 2,i+1, c 2i 1, 2 c 2i 2 c i 1 } for i = 2,..., n in the case k = 2, c 1 = q 3,1 and c i := max { 4 c 2i 3 3 c i 1, 4 3 c 2i c i 1, 4 3 c 2i q 3,i+1, 4 c 2i 3 q 3,i+1 } in the case k = 3, and c 1 = q 4,1 and { c i := max 8 c 2i 4 6 c i 1 c i 2, 2 c 2i 3 c i 1, 4 3 c 2i c i q i+1, } 2 c 2i q i+1, 8 c 2i 6 q i+1 q i+2 in the case k = 4, where q k,i is defined as in (24), is an upper monotone coarse grid approximation to the ower obstace S = c T N k,. It is an improvement of the approximation L k from Proposition 3.5 in the sense that c q k hods. Remark 3.7 In the specia case k = 2 our B spine based monotone mutigrid scheme corresponds to the mutigrid method from [Ma] if the obstace transfer is performed as described in Proposition 3.5. Moreover, if the coarse grid approximations are constructed according to Proposition 3.6, one recovers the monotone mutigrid scheme from [Ko1]. For iustration, in Figure 1 ower obstace functions g 1 := r(g u ν,1 ) are dispayed for eves = 2, 3, 4 as they typicay arise if the MMG agorithm is appied to American option vauation with continuous piecewise inear (eft) and C 1 smooth piecewise quadratic (right) basis functions. The obstace functions restrict the size of the coarse grid correction v 1 such that the new iterate u ν,2 = u ν,1 + p v 1 is admissibe, i.e., equa or above the transformed payoff function g. The restriction operator r is chosen as derived in Proposition 3.6. Note that the vaue x = 0, where the coarse grid correction is east restricted, corresponds to the strike price S = K by back transformation. 13

14 obstace at eve 2 obstace at eve 3 obstace at eve 4 grid at eve 2 grid at eve 3 grid at eve x obstace at eve 2 obstace at eve 3 obstace at eve 4 grid at eve 2 grid at eve 3 grid at eve x Figure 1: Lower coarse grid obstace functions g for the coarse grid correction and grids, = 2, 3, 4 as they typicay arise when the MMG agorithm is appied to American option pricing with continuous, piecewise inear (eft) and C 1 smooth, piecewise quadratic finite eement ansatz functions (right). The restriction operators r from Proposition 3.5 as we as from Proposition 3.6 can immediatey be modified such that they can be used in the truncated version TrMMG from Remark 3.2. In [HK] it is shown by numerica experiments that the truncated version combined with the obstace approximations according to Proposition 3.6 is the fastest convergent variant. When appied to American option pricing, the scheme requires ony one or two smoothing steps on each refinement eve. It recovers asymptotic convergence rates which are independent of the grid size h. In the case k = 3 the convergence rates are bounded by about 0.27 for one smoothing step and 0.17 for two smoothing steps on each refinement eve. 4 Numerica Resuts In this section we investigate the performance of our scheme by numerica exampes. In the first two subsections, we compute fair prices of short and ong term pain vania American Put options and its derivatives with respect to the stock price. In particuar, we anayze the infuence of basis functions of different smoothness. In subsection 4.3, we then compare the performance of our scheme to the performance of Leisen Reimer trees [LR] in the approximation of the second derivative Gamma. 4.1 Pain Vania American Put Option Prices In our first experiment we consider an American put option with strike price K = 100, maturity T = 0.5 and an underying stock with voatiity σ = 0.4 which pays no dividends. The interest rate is assumed to be r = 6%. Foowing [AC], the vaue V (S, 0) obtained from the average of a 1000 step and a 1001 step binomia method is regarded as exact. Against this benchmark we compute the pointwise errors of our B spine based finite eement scheme (B-FEM) with 14

15 Stock price S = 80 S = 90 S = 100 S = 110 S = 120 Comp. time exact V (S, 0) Sec. B-FEM, k = Sec. B-FEM, k = Sec. B-FEM, k = Sec. Tabe 1: Short term American Put vaue (K = 100, σ = 0.4, d = 0.0, r = 0.06, T = 0.5), and pointwise errors of B-FEM scheme with N = 128 and M = 16 space and time steps. Stock price S = 80 S = 90 S = 100 S = 110 S = 120 Comp. time exact V (S, 0) Sec. B-FEM, k = Sec. B-FEM, k = Sec. B-FEM, k = Sec. Tabe 2: Long term American Put vaue (K = 100, σ = 0.4, d = 0.02, r = 0.06, T = 3), and pointwise errors of B-FEM scheme with N = 128 and M = 64 space and time steps. C 0 smooth, piecewise inear B spines (k = 2), C 1 smooth, piecewise quadratic B spines (k = 3), C 2 smooth, piecewise cubic B spines (k = 4). We used an uniform grid, θ = 1 and the space interva I x = [ 5, 5]. Since the scheme is much more sensibe to the the number of space steps N compared to the number of times steps M, we chose N = 128 and M = 16. This eads to a overa number of 2, 048 unknowns. The resuts are isted in Tabe 1. The speed of the scheme is measured in computationa time in seconds and dispayed in the ast coumn. The computations were performed on a dua Inte(R) Xeon(TM) CPU 3.06GHz workstation. No cacuation took onger then 0.01 seconds. The computation of the five option vaues by the average of a 1000 step and a 1001 step binomia method took 3.45 seconds. Note, that finite eement or finite difference approaches provide an approximation to the whoe surface V (S, t) with S 0 and 0 t T, whereas tree methods have to be restarted for every singe vaue V (S 0, 0). In Tabe 2 we ist the errors, which resut from the vauation of the same American put, but with much onger maturity T = 3 and a dividend payment of d = 2% of the underying. Due to the onger maturity, we chose a arger number of time steps M = 64. As the errors are of simiar size for different orders k we concude that for the computation of the option vaue the use of piecewise inear functions (k = 2) in the finite eement scheme suffice. This, however, is no onger true if aso derivatives of the option price with respect to the underying have to be computed as we show in our next numerica exampes. 4.2 American Option Price Sensitivities We now consider an pain vania American ca option with parameters (25) K = 10, σ = 0.6, r = 2.5%, d = 0.0, T = 1. 15

16 In this scenario, the vaue of an American and European Ca are identica, such that the Back Schoes formua [BS] can be used to obtain benchmark vaues for the option price and for the Greek etters. For the numerica computations, we used an uniform grid, θ = 1/2, the space interva I x = [ 5, 5] and B spine ansatz functions of orders k = 2, 3, 4 as in the ast subsection. If the basis functions are sufficienty smooth, the derivatives of the soution V are determined by direct differentiation via Lemma 2.8. In the other case numerica differentiation is used. On the eft hand side of Figure 2, 3 and 4, we dispayed the L 2 errors at time t = 0 which arise if we compute the option vaue, Deta and Gamma with our scheme, respectivey, for different numbers N = M of unknowns. Due to the Crank Nichoson time discretization quadratic convergence is the best that can be expected. One can see that the option vaue is computed in quadratic convergence for a orders k, but not the derivatives. The j th derivative (j) V/ S (j) is ony determined in quadratic convergence if basis functions of order k j + 2 are used. For the same choice of basis functions, we dispayed on the right hand side of Figures 2, 3 and 4 the corresponding distributions of the pointwise error at time t = 0 in the case N = M = 275. The much more accurate approximation of Deta and Gamma obtained by the use of higher order basis functions and direct differentiation is ceary visibe. 1 k=2 k=3 k= k=2 k=3 k= L2-error pointwise error e e number of unknowns stock price S Figure 2: Mean square errors for M = N (eft) and distribution of the pointwise errors for N = M = 275 (right) which arise in the computation of the option vaue at time t = 0 with parameters from (25). 4.3 Comparison to other Schemes In [WW], a comparison of various pricing methods eads to the concusion that Leisen Reimer trees [LR] are the superior method for the approximation of American option price sensitivities. Thus, we chose this scheme as a benchmark for our finite eement scheme. We compare the pointwise error in the computation of Gamma 2 V/ S 2 (K, 0) of an American option with the parameters from (25). The resuts are dispayed on the eft hand side of Figure 5 for an American ca option and on the right hand side for an American put option, respectivey. In the first case, an exact benchmark vaue can be obtained by partia 16

17 1 k=2 k=3 k= k=2 k=3 k= L2-error pointwise error e e number of unknowns stock price S Figure 3: Mean square errors for M = N (eft) and distribution of the pointwise errors for N = M = 275 (right) which arise in the computation of Deta at time t = 0 with parameters from (25). differentiation of the Back Schoes formua. In the other case, we approximated Gamma by numerica differentiation of the average of a 20,000 step and a 20,001 step binomia method [CRR]. Using the parameter set (25), we obtain 2 V/ S 2 (K, 0) Against this benchmark vaue we computed the pointwise errors of our scheme (B-FEM) with basis functions of order k = 2 and k = 4 and the pointwise errors of Leisen Reimer trees. Reca that the costs, which are potted on the x axis, of a binomia scheme with n steps are of order O(n 2 ), whereas the costs of our finite eement scheme with M time and N space steps add up to O(N M). One can see in Figure 5 that Leisen Reimer trees outperform the finite eement scheme with hat functions (k = 2). Both schemes exhibit a pointwise convergence rate of about ρ = 1/2. In contrast, the finite eement scheme with piecewise cubic functions (k = 4) attains a much better convergence rate of neary ρ = 1. Whie this coud be expected for American ca options without dividends, where the soution is known to be smooth for a t < T, it is remarkabe for American put options, where the soution is known to have a jump in the second derivative on the free boundary. 5 Concuding Remarks In this paper we presented a finite eement method which is based on higher order B spine discretizations for the approximation of American option prices and their space derivatives Deta and Gamma. The method was suppemented by an monotone mutigrid scheme for the efficient soution of the discrete form in order to achieve convergence rates which are independent of the grid spacing h. Appying the scheme on uniform grids to the pricing of pain vania American options, our numerica experiments ead to the concusion that just for the computation of the option vaue an increase of the poynomia degree of the ansatz functions is not advantageous. This, however, is no onger true if aso the derivatives of the option vaue with respect to the underying are required. Then, the correct choice of the order k of the ansatz functions 17

18 0.1 k=2 k=3 k= k=2 k=3 k= L2-error pointwise error e e e number of unknowns stock price S Figure 4: Mean square errors for M = N (eft) and distribution of the pointwise errors for N = M = 275 (right) which arise in the computation of Gamma at time t = 0 with parameters from (25). (i.e., k = j + 2 for the j th derivative) eads to a much higher accuracy and a much faster convergence. This way, our scheme, which provides the first mutigrid approach which is appicabe to higher order discretizations of free boundary vaue probems, is competitive to the best schemes proposed in the iterature. We wish to point out that the soution approach can, in principe, be generaized to higher spatia dimensions and thus aso to higher dimensiona option pricing probem, as ong as partia differentia equations can be derived for the option vaue, ike, e.g, in the case of American options with stochastic voatiity (cf. [BR, CP, Oo]) or in the case of convertibe bonds with stochastic interest rate (cf. [BN, WHD]). Moreover, though the refinement reations of B spines become more compicated (cf. [Bo]), a resuts can be generaized to non uniform grids. It can be expected that a finer grid in the most interesting region near the strike price and a further refined grid near the free boundary woud enhance the performance of the scheme significanty (cf. [PH, CP]). Using a suitabe error estimator, as e.g. in [Ko2], it woud then aso be possibe to perform the grid refinement adaptivey. Due to the Crank Nichoson time discretization at most quadratic convergence of our scheme can be expected. This imitation coud be overcome by a higher order discretization of the time, e.g. by a Runge Kutta scheme or by higher order finite eements. Then, aso Theta, the derivative of the option vaue with respect to time, coud be approximated with much higher accuracy. Acknowedgments. I woud ike to thank Michae Griebe for pointing out the probem of deriving MMG methods based on higher order basis functions and their possibe appication to the computation of American options. I aso thank Angea Kunoth for her suggestion to use a B spine basis and for her continuous encouragement and support throughout my dipoma theses. I am further gratefu to Thomas Gerstner, whom I owe most of my knowedge about option pricing, and to Rof Krause for hepfu discussions on monotone mutigrid methods. Thanks aso go to Matthias Reimers, who provided to me usefu tips and his 18

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