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1 Electronic Transactions on Numerical Analysis. Volume 5, pp , Copyrigt 2003,. ISSN ETNA ON MULTIGRID FOR LINEAR COMPLEMENTARITY PROBLEMS WITH APPLICATION TO AMERICAN-STYLE OPTIONS C. W. OOSTERLEE Abstract. We discuss a nonlinear multigrid metod for a linear complementarity problem. Te convergence is improved by a recombination of iterants. Te problem under consideration deals wit option pricing from matematical finance. Linear complementarity problems arise from so-called American-style options. A 2D convectiondiffusion type operator is discretized wit te elp of second order upwind discretizations. Te properties of smooters are analyzed wit Fourier two-grid analysis. Numerical solutions obtained for te option pricing problem are compared wit reference results. Key words. linear complementarity problems, American-style options, nonlinear multigrid, projected Gauss- Seidel, iterant recombination, second-order upwind discretization, Fourier analysis AMS subject classifications. 65M55, 65F99, 90A09. Introduction. In tis paper, we discuss multigrid metods for solving a time-dependent 2D partial differential equation (PDE) arising in option pricing teory. Te problem considered is te computation of te value of an American-style option in a stocastic volatility setting. It leads to te solution of a convection-diffusion type PDE wit a free boundary. In [23], it as been sown tat for American-style options te teory of linear complementarity, as it was developed in te 970 s, applies. It is possible to rewrite te arising free boundary problem as a linear complementarity problem (LCP) of te form (.) (.2) (.3) Lu f x Ω u f 2 x Ω (u f 2 )(Lu f ) = 0 x Ω, plus boundary conditions, were L is a linear differential operator. Te option pricing context and te discretization of te LCP are discussed in Section 2. Te LCP formulation is beneficial for iterative solution, since te unknown boundary does not appear explicitly and can be obtained in a postprocessing step. In 983, Brandt and Cryer [3] proposed a multigrid metod for LCPs arising from free boundary problems. Te algoritm is a multigrid generalization of te projected SOR metod [7]. Due to te nonlinear caracter of te problem, te multigrid metod is based on te full approximation sceme [2], FAS, tat is often used for solving nonlinear PDEs. Te solution metod as terefore been called te projected full approximation sceme (PFAS) in [3]. In te original paper, te operator L in (.), (.3) was te nicely elliptic Laplace operator and fast convergence was presented. PFAS as already been successfully used in te financial community for American options wit stocastic volatility in [6]. Te smooter applied in [6] is, owever, somewat involved. It is based on te pointwise PSOR metod for te detection of te free boundary, followed by a partial linewise step in order to deal wit te stretced numerical grid occurring in te financial problem. As te free boundary is unknown and can be of general sape, te line relaxation may often need to be adapted. Te need to cange multigrid components like te smooter for optimal convergence of a new problem at and is sometimes considered as unsatisfactory. Tis is, for example, stated in [25] were an alternative formulation for American options wit a penalty function is proposed and an Received May 0, 200. Accepted for publication October 2, 200. Recommended by Irad Yavne. Fraunofer Institute for Algoritms and Scientific Computing (SCAI), D Sankt Augustin, Germany ( oosterlee@scai.fraunofer.de) 65

2 66 Accellerated multigrid for American-style options ILU preconditioned CG-type solver is considered. Tis solver can be considered as being of black-box type, but it is not ierarcical. It is our aim wit tis paper to give more insigt into te multigrid convergence for problems from option pricing. Fourier two-grid analysis [9] will be used to analyze favorable smooters and coarse grid correction components for te discrete operator under consideration. Te influence of different point- and linewise smooters, or of under- and overrelaxation parameters can be analyzed quantitatively in tis way, as presented in Section 3.2. At te same time, we introduce in Section 3 some recent developments in multigrid metods to te field of LCPs, making te algoritms more robust, like overrelaxation parameters and recombination tecniques. In te discretization of te operator arising in American-style option pricing by a second order upwind sceme, different complications for optimal multigrid convergence, suc as anisotropies and positive off-diagonal stencil elements in space-dependent operator coefficients, occur simultaneously. Wen pointwise Gauss-Seidel-type smooting is combined wit a standard coarse grid correction, certain error components may remain large. In suc a situation, it is possible to coose a more expensive smooter, suc as linewise relaxation or to cange te coarsening procedure, for example to semi-coarsening, or to improve te convergence by a Krylov subspace acceleration. We will coose te subspace acceleration approac ere. A well-known solution approac of te latter type for nonlinear equations is to apply global (Newton) linearization, solve te arising linear system wit a Krylov subspace metod, suc as te GMRES metod [8], and a multigrid preconditioner. Tis is not te approac followed in tis paper. We apply to te nonlinear problems a solution metod based on PFAS as te multigrid tecnique. Te Krylov subspace acceleration can be interpreted as a tecnique, in wic intermediate iterates are recombined in order to obtain an improved approximation, as explained in [20]. Wit tis metod many different nonlinear partial differential equations ave been solved, see, for example, [9, 6]. Te solver is related to te acceleration cycle presented in [4]. Here, we generalize tis metod to solving LCPs in Section 4. Numerical results are presented in Section Option basics and te Black-Scoles equation. Researc in option pricing teory concerns, among many oter issues, te computation of te value of an option during te life of an option contract. A famous equation for tis is te Black-Scoles partial differential equation. It represents a simple model for te values of two basic options, te so-called put option and call option. In te case of a European put option, te older of te option may sell at te expiration date, a prescribed time in te future, certain assets, like sares or stocks, at te exercise price. Te oter party of te option contract, te writer, must buy te asset, if te older decides to sell it. In te case of a European call option, te older as te rigt to purcase an asset on a certain date at a prescribed amount. Te writer is ten obliged to sell te asset. In tis paper, we concentrate on te put option. Options ave two main uses: speculation and edging. Wereas speculation migt be well-known, edging needs some more explanation. Let s consider a portfolio wit assets and put options. If te asset price s falls, (based on te definition above) te value of a put option increases: It is possible to sell assets at te expiration date for te exercise price, altoug te actual price falls. Te value of te portfolio terefore depends on te ratio between te number of assets and te number of put options in te portfolio. A ratio exists, wic results in no movement in te value of te portfolio. Tis ratio is instantaneously risk-free. Hedging means ere reducing risk (for example against falling asset prices) by combining options wit assets in a portfolio.

3 C. W. Oosterlee 67 To derive te Black-Scoles equation, one assumes a geometric Brownian motion stocastic differential equation as a model generating asset prices s to be valid, (2.) ds = s(σdw + µdt). Here, µdt is a deterministic return. Te volatility σ measures te standard deviation of te returns ds/s. Te random variable dw is assumed to be a Wiener process wit mean 0 and variance dt, so tat te mean of ds is µsdt and its variance is E[ds 2 ] E[ds] 2 = σ 2 s 2 dt. Furtermore, te equation as te Markov property: It does not depend on te past istory of te asset prices. Altoug asset prices are valid for discrete time, te PDE models are set up in continuous time wit limit dt 0. Te value of te option is denoted by u = u(s, t) ere. Te value u is influenced by te exercise price E, te expiration time T (0 t T ), te interest rate r (ere assumed to be constant), and te volatility σ. Itô s lemma makes it possible to andle te random term dw as dt 0 (analogously to Taylor s expansion for deterministic variables). Wit te insigt tat te random walks in s and u are driven by te same random variable dw, one can add in a portfolio an option wit value u to a number, Delta, of assets s in suc a way tat te portfolio is instantaneously risk free. By coosing := u/ s, te deterministic portfolio increment is obtained. At tis stage, te assumption tat tere are no arbitrage possibilities comes into play, wic means tat risk free profits tat are greater tan placing money at a bank are not allowed in te model. So, te instantaneously risk free portfolio must earn a rate of return tat equals te interest rate [0, 23]. Togeter wit te assumptions in tis model s framework of constant volatility, of no transaction costs for edging, no dividend payment and no taxes involved, tis finally results in te Black-Scoles partial differential equation for te value of an option, (2.2) Lu := u t + 2 σ2 s 2 2 u s 2 + rs u s ru = 0 In its basic form, (2.2) is a convection-diffusion-reaction type equation in one spacelike dimension, te s direction. Wit te terms s j j u/ s j, one recognizes an Eulerian differential equation tat can be transformed into a eat equation. Te Black-Scoles equation is a parabolic PDE wit boundary and final conditions. At expiry a European put option as te value E s for s < E. It is wortless if s > E, see Figure 2.. So, (2.3) u(s, T ) = max(e s, 0) =: f 2 (s). V E E S FIG. 2.. Final conditions for a put option. Te boundary condition for a put (see, for example [0, 23]) at s = 0 is u(0, t) = Ee r(t t). It represents te exponential growt for receiving an amount E at t = T wit

4 68 Accellerated multigrid for American-style options constant interest rates. Furtermore, u(s, t) 0 as s, because one obviously cannot gain by exercising te put option. American options. Wereas for European options exercise is only possible at te expiration date, te exercise of American options is permitted at any time during te life of an option, 0 < t T. Tis leads to te following considerations. It is well-known tat a European put option is, in a certain s range, less tan te pay-off function, u(s, t) < max(e s, 0) [23]. In te American context, tis would mean tat buying te option for u, selling te asset immediately for E, and buying te asset on te market for s, would result in a risk-free profit of E u s > 0 in tis s range, wic contradicts te arbitrage concept [0, 23]. Terefore, wen early exercise is permitted, a constraint (2.4) u(s, t) max(e s, 0) =: f 2 (s) must be imposed in order to avoid an arbitrage possibility. In tis s region, te value of te American put option is raised compared to te value of te similar option of European type, due to te constraint (2.4). Te addition of te constraint to te PDE (2.2) or to (2.7) gives rise to a free boundary problem: A special s value exists, te optimal exercise price s f, wit te following properties: on one side of s f it is beneficial to old te option, on te oter side, it is advantageous to exercise te option. s f (t) is time dependent and not known in advance. (Of course, optimal exercise also depends on oter important parameters and, most of all, on te option older s market strategy.) For an American put, if u > max(e s, 0) equation (2.2) or (2.7) olds (s > s f (t)); If u = E s, an inequality, (2.5) Lu < 0 olds (s s f (t)) wit L as in (2.2) or in (2.7). Based on te classical teory of linear complementarity, it is possible to formulate tis free boundary problem into an LCP of type (.),(.2),(.3), so tat te free boundary conditions need not be andled explicitly. Stocastic volatility. A generalization of te Black-Scoles equation is obtained if te restriction of constant volatility is replaced by te assumption of a stocastic volatility. Instead of (2.), te following stocastic differential equations are assumed to govern te asset price process s and its variance process y [], (2.6) ds = µsdt + ysdw dy = α(β y)dt + γ ydw 2 were w, w 2 represent standard Brownian motion wit correlation coefficient ρ [, ], γ is te volatility of te variance process, α, β > 0 determine te mean reversion (so tat te variance will drift back to some mean value β at a rate governed by α). Te volatility y can be sown to be positive valued provided tat γ 2 2αβ. Te assumption of te stocastic processes (2.6) leads to a 2D PDE problem for te value of an option in wic te variance y is, besides s and t, a tird variable (degree of freedom), see [, 25] for te derivation. Te resulting PDE reads, (2.7) Lu : = u t + 2 [s2 y 2 u s 2 + 2ργys 2 u s y + γ2 y 2 u y 2 ] + rs u s + [α(β y) λγ y] u ru = 0, Ω = {(s, y) s 0, y 0}, y were λ is te so-called market price of te risk function (for foreign currencies a nonzero constant is parameter of coice). It as been set to 0 ere, as in [6, 25]. For more details on

5 C. W. Oosterlee 69 te stocastic volatility concept, we refer to te financial literature [, 0, 25]. Te following boundary conditions are proposed for a put option in [6], (2.8) (2.9) u(0, y, t) = f 2 (0, t), y 0, t [0, T ], u(s, 0, t) = f 2 (s, t), s 0, t [0, T ], wit f 2 defined by (2.3). Tese boundary conditions imply an immediate exercise of te put, in te case of a zero asset price and in te case of zero volatility. As for te oter two boundaries (s, y ), te computational domain is commonly truncated at finite values s max, y max respectively, at wic Neumann boundary conditions are imposed [6], (2.0) (2.) u(s max, y, t) = 0, y [0, y max ], t [0, T ] s u(s, y max, t) = 0, s [0, s max ], t [0, T ]. y Typically, y max is set to one, s max to two times E. Te computed values of te options are generally not very sensitive wit respect to te size of te truncation, as discussed in [6]. Summarizing, for American style options wit te stocastic volatility model (2.6),(2.7), te following LCP needs to be solved, (2.2) (2.3) (2.4) Lu(s, y, t) 0 u(s, y, t) f 2 (s) (u(s, y, t) f 2 (s))lu(s, y, t) = 0 wit f 2 defined by (2.4), L given by (2.7) and boundary conditions (2.8) (2.). 2.. Te discretization. Te stocastic volatility problem leads to te numerical solution of a 2D time-dependent problem wit an operator L of convection-diffusion type. Here we outline te discretization of te operator in (2.7), wic is in nonconservative form. Since we expect solutions witout steep gradients, it is possible to coose standard well-known discretization scemes. After a transformation t = T t, te equation wic is backward in time wit a final condition is transformed to an equation forward in time wit an initial condition. Tis puts a minus sign in front of te / t term in (2.7). For te time discretization, we consider te so-called backward difference formula BDF2 [8] sceme, (2.5) 3u (s, y, t + τ) 4u (s, y, t ) + u (s, y, t τ) 2τ = L u (s, y, t + τ). Te time discretization accuracy of tis implicit sceme is O(τ 2 ). We prefer tis discretization over te well-known Crank-Nicolson sceme (also called trapezoid rule), because of its better stability caracteristics. Te Crank-Nicolson sceme is not L-stable (see, for example, [9]), wereas BDF2 is. BDF2 as more favorable damping properties tan te Crank- Nicolson sceme. In a fortcoming paper discussing anoter type of option, we will sow tat te latter can result in undamped oscillations in important financial quantities, called te edge parameters. Te mixed derivative term is discretized by te O( 2 ) four-point discretization. Te second derivatives in bot directions are andled by te usual tree-point stencils. Linear second order upwind discretizations, like te upwind κ-discretizations [4], are sufficiently accurate for te discretization of te convective terms in te s and y directions. A D

6 70 Accellerated multigrid for American-style options upwind κ-discretization can be written as a combination of a central difference discretization plus a second-order dissipation term, wic is proportional to te tird derivative of u: ( u (s + s, y, t ) u (s s, y, t ) ( ) ) a 2 (au s ) = a + s ( κ) (2.6) 2 s 4 ( u (s s, y, t ) 3u (s, y, t ) + 3u (s + s, y, t ) u (s + 2 s, y, t ) ) for a < 0 (for example, a = rs after te transformation). In tis paper, we use te Fromm sceme for te discretization (i.e., κ = 0). Te well-known central discretization sceme κ = again leads to oscillations in financial quantities, see for example [26]. Te linear κ-sceme is a first coice for obtaining second-order accurate scemes wit a convection term. Te κ-scemes are, owever, not monotone, wic means tat tey ave to be modified (wit limiters) for problems containing strong gradients, like socks or boundary layers. In tat case, te BDF2 time discretization also needs to be modified in order to guarantee total variation diminising (TVD) solutions in space and time wit relatively large time steps. So-called partially implicit BDF blended scemes [2] from te family of implicit-explicit (IMEX) time discretization scemes are an alternative in tis situation. We will use tem in anoter financial setting in a fortcoming paper. Since we will not encounter extremely steep gradients in tis paper, te second order κ upwind scemes and BDF2 time discretization are fully satisfactory. Unrealistic oscillations in time or in (s, y)-space are not encountered. By setting s i = i s, y j = j y, i, j : 0,..., n, (i.e. coosing an equidistant grid Ω ), te stencil for te second order accurate discretization of (2.7) transformed into an equation forward in time reads in semi-discrete form (2.7) u (s, y, t ) t + wit elements µ,ν given by (2.8) wit 0,2, 0,, 3 s,0 0,0,0 2,0, 0,, 0, 2 u (s, y, t ) = 0,, = ργi j/4, a(2), = ργi j/4,, = ργi j/4, a(2), = ργi j/4, (s, y, t ) Ω (0, T ], 2,0 = ( κ)ri/4, a(2),0 = i 2 j y /2 + ( + κ)ri/4, 0, 2 = max(c( κ)/4, 0), a(2) 0,2 = min(c( κ)/4, 0),,0 = i2 j y /2 /4(5 3κ)ri, 0, = γ2 j/ y /2 min(0, c( κ)/4) max(0, c(5 3κ)/4), 0, = γ2 j/ y /2 + min(0, c(5 3κ)/4) + max(0, c( κ)/4), 0,0 = r + i2 j y + γ 2 j/ y + 3/4( κ)ri + 3 c ( κ)/4, c = (α(β j y ) λγ j y )/ y

7 C. W. Oosterlee 7 from te (s, y) discretization explained above (wit discrete boundary conditions and te time discretization). An interesting aspect is tat all powers of s vanis after te discretization of (2.7), wic means tat different mes sizes in te s-direction do not play a role for te multigrid convergence factors. Tis is not typical for general convection-diffusion-reaction equations. 3. Te solution metod. We will discuss te multigrid solver in detail in different subsections, starting wit a discussion on suitable smooting metods for te operator (2.7) wit discretization (2.7) in te LCP setting. 3.. Smooter for convection-diffusion type operators. We use a projected version of te well-known lexicograpic pointwise Gauss-Seidel metod as a smooter. Suc an iteration consists of two partial steps. In a first step, a lexicograpic pointwise Gauss-Seidel iteration is applied to (2.2) wit te equality sign. In te second partial step, te solution is projected, so tat te constraint (2.3) is satisfied, (3.) û(s i, y j, t ) = 00 f (s i, y j, t ) µ Js,µ 0 (µ,µ 2) (0,0) µ Js,µ 0 µ 2 Jy, (µ,µ 2) (0,0) µ 2 0 µ 2 Jy, µ 2 0 µ µ 2 u(s i + µ, y j + µ 2, t ) µ µ 2 u(s i + µ, y j + µ 2, t ), (3.2) u(s i, y j, t ) = max{f 2, (s i, y j, t ), û(s i, y j, t )} (s i, y j ) Ω, were u denotes an unknown after a relaxation and û an unknown after a partial relaxation step, J s, J y are te integer index sets related to te nonzero stencil elements in (2.7). A Gauss-Seidel iteration will not give a good smooter for convection-dominated convectiondiffusion problems discretized wit κ-scemes. Multistage smooters, defect-correction approaces or KAPPA smooters [5] are commonly used for second order accurate upwind discretizations in convection-dominated problems. Here, owever, for te problem at and te well-known Gauss-Seidel relaxation metods can be used as smooters in multigrid, as we will sow by Fourier analysis in te following section. As te problem is not really convection-dominated, te above mentioned convergence difficulties do not occur Two-grid Fourier analysis. An important analysis tool for multigrid metods is Fourier analysis, see, for example [2], [9], [2]. It is, in fact, te main multigrid analysis possibility for nonsymmetric problems. We will perform a Fourier two-grid analysis to study te properties of a smooting metod and of te oter multigrid components in a two-grid metod. Te error v m = um u is transformed by te (m + )-st two-grid cycle as follows: (3.3) v m+ = M H v m, M H = S ν2 CH S ν ; CH = I I H(L H ) I H L. Te spectral radius ρ (M H) of te linear two-grid operator M H gives an indication of te asymptotic rate of te multigrid convergence. On a grid G := {x = (k s, k y ) : k s, k y Z}, we consider functions tat are linear combinations of te Fourier components ϕ (θ, x) = e ikθ = e i(ksθs+kyθy)

8 72 Accellerated multigrid for American-style options wit x G, k = (k s, k y ) Z 2 and frequencies θ = (θ s, θ y ) IR 2. Te Fourier space ε = span{e ikθ : θ Θ = ( π, π] 2 } contains any infinite grid function on G [9]. Te discrete solution u, te current approximation u m and te error v m (3.3) can be represented as linear combinations of te basis functions eikθ ε. ε can be divided into four-dimensional sub-spaces, te armonics (see Figure 3.): (3.4) ε θ = span{ϕ(θαsαy, x) = e ikθαsαy ; α s, α y {0, }}, x G, θ 00 Θ 00 := ( π/2, π/2] 2, θ αsαy := (θ s α s sign(θ s )π, θ y α y sign(θ y )π) π PSfrag replacements π x θ 0 π 2 π 2 x θ 00 π 2 π Θ 0,0 x θ π 2 x θ 0 π FIG. 3.. Hig (saded region) and low frequency regions of ε wit four armonics Te 2D coarse grid correction operator C H leaves te 4-dimensional space of armonics ε θ (3.4) wit an arbitrary θ Θ 00 = Θ 00 \ {θ : LH (2θ 00 ) = 0} invariant [9]. Te same invariance property is true for te smooters considered, C H : ε θ ε θ, S : ε θ ε θ (θ Θ 00 ). Hence M H is ortogonally equivalent to a block matrix consisting of 4 4 blocks, wic will be denoted by M H(θ) := M H ε (θ Θ 00 ). We can determine te spectral radius θ ρ (M H ) by calculating te spectral radii of 4 4 matrices: (3.5) ρ (M H ) = max θ Θ 00 ρ ( M H (θ)). To obtain te representation of te 4 4 blocks M H (θ) = S ν2 (I P ( L H ) R L ) S ν, te Fourier symbols of te multigrid operators for te armonic in ε θ are calculated. Te two-grid convergence properties of stencil (2.7) wit te coefficients depending on s and y are analyzed. Fourier analysis is, owever, only exact for linear operators wit constant (or frozen) coefficients. Terefore, we locally freeze te s and y terms in front of te derivatives in (2.7) and ceck te two-grid convergence factors for several relevant values of tese quantities. For tis purpose, we divide te unit square in 256 intervals s = y = /256 and vary i and j in (2.7) by 8 units, i, j : 0,..., 256. For eac (i, j)-set, we compute ρ(mh ) (3.5), wic brings many two-grid factors.

9 C. W. Oosterlee 73 In te analysis, we consider te steady equation ( / t = 0), since tis represents a worst case for multigrid convergence. Implicit time discretization brings a positive addition to te main diagonal operator element, wic is beneficial for te smooting properties. Te parameter set considered in (2.7) is te following: (3.6) α = 5, β = 0.6, γ = 0.9, ρ = 0., λ = 0, r = 0., We start te analysis of te lexicograpical point Gauss-Seidel smooting metod and te coarse grid correction consisting of injection as te restriction operator (te reason for tis is explained in Section 3.3. in te LCP context), bilinear interpolation as te prolongation and a direct discretization of te PDE on a standard coarsened grid, wit H = 2. Te number of smooting iterations is set to 2 ere. Remark: Lexicograpic smooting means tat for eac index te iteration proceeds from 0 to te maximal value. Te direction of te information propagation is, owever, opposite. Tis does not affect our results ere. On muc finer grids, owever, one may expect multigrid convergence difficulties for small y- and large s-values, since ten te convection term becomes more dominant. In tis situation, an anti lexicograpic order for smooting is appropriate. Te computed (s, y)-dependent two-grid convergence factors are grapically presented in te left-and part of Figure 3.2. Near te domain boundary s = 0, ρ(mh ) is observed wic is clearly unsatisfactory. For larger s-values, muc better two-grid factors are obtained. Te rigt-and side picture in Figure 3.2 sows te two-grid convergence results wit s-line Gauss-Seidel relaxation (i.e., lines wit j = const.). In tis case, only an isolated large factor close to (s, y) = (0, 0) is observed; most factors are about Tese two-grid s y s 0 y FIG Fourier two-grid convergence factors ρ(mh ) for different (s, y)-values; left: lexicograpic point Gauss-Seidel, rigt: s-line Gauss-Seidel. factors are confirmed by numerical multigrid experiments wit te components used in te two-grid analysis for equation (2.7) wit Diriclet boundary conditions on different domains Ω. On Ω = [0, ] 2, s = y = /256, we find poor multigrid convergence wit te lexicograpic point Gauss-Seidel smooter and a very satisfactory multigrid convergence factor of 0.35 wit te s-line Gauss-Seidel smooter (see Table 3.). Te isolated large value in te latter case can be considered as a boundary effect, due to te lack of ellipticity at te

10 74 Accellerated multigrid for American-style options corner point, tat is not observed in te actual numerical experiment wit Diriclet conditions at te particular boundary. For oter computational domains Ω, away from te s = 0 axis, we obtain a muc improved multigrid convergence wit te point Gauss-Seidel smooter, see Table 3.. We can expect tis from te ρ(mh )-values in te left-and side picture in Figure 3.2. Table 3. also presents te effect of an overrelaxation parameter ω =.2 on TABLE 3. Multigrid convergence of te PDE (2.7) on a grid wit varying domains and overrelaxation for pointand linewise Gauss-Seidel iteration (ν = ν 2 = ). Smooter: lex. point Gauss-Seidel s-line GS Overrelaxation / Domain Ω: [0, ] 2 [0.2, ] 2 [0.4, ] 2 [0.6, ] 2 [0, ] 2 ω = ω = te multigrid convergence, since tis can bring improved convergence for several elliptic PDE problems of anisotropic-type [22, 24]. Te numerical convergence wit te relaxation parameter as also been validated by Fourier analysis. Te reason for concentrating interest so muc on te results wit te point smooter is te following. Due to te occurring free boundary in te LCP problem, large parts at te left-and side of a domain will not be processed explicitly by te multigrid smooting and coarse grid correction. At tese parts, te second constraint (2.3) olds wit equality sign. For te grid points, were te smooting will typically be applied for te option pricing problem under consideration, te very satisfactory two-grid convergence factors are found in te left-and side picture of Figure 3.2 and Table 3.. Te convergence factors in tis region are, in fact, very similar for pointwise and linewise smooting. A problem wit linewise smooting for LCPs is, as explained in [6], tat te lines sould end at te free boundary for good smooting and convergence. Terefore a detection mecanism must be incorporated into te smooting metod, wic is in [6] te pointwise Gauss-Seidel iteration. For equidistant grids and te LCP problem setting, owever, te pointwise smooter is fully satisfactory. From (2.8), we already observed tat all powers of s vanis in te discretization of (2.7). For te multigrid convergence tis means tat also on severely stretced but equidistant grids in te s-direction, as we find tem ere wit s max = 20 or s max = 200 in (2.0),(2.), very similar convergence factors are observed by Fourier analysis and by te numerical experiments as te ones in Figure 3.2 and Table 3.. Te convergence factors increase drastically, owever, if we deal wit larger values of y and substantially fewer grid points in y- tan in s-direction. Figure 3.3 sows te two-grid factors for 32 points in y-direction, y = /32 ( s = /256). Tis case is a limit case for te convergence of te point smooter; more points in y-direction give satisfactory convergence, less points worse convergence. Figure 3.3 also sows tat te s-line smooter will give worse convergence in tis situation, similar to te point smooter. Domains oter tan [0, ] 2 will not improve te convergence ere, since te worst factors are now found at te rigt-and side domain boundary. An alternating line Gauss-Seidel smooter, for wic ρ(mh ) is presented in Figure 3.3c, brings good convergence. Tese Fourier analysis results are all confirmed by numerical experiments Te Multigrid metod for linear complementarity problems. Te fundamental idea of multigrid for nonlinear PDEs of te form (3.7) Nu = f

11 C. W. Oosterlee (a) s y (b) s y (c) 0 s y FIG Fourier two-grid factors for a grid wit s = /256, y = /32, (a) wit point Gauss-Seidel smooter, (b) wit s-line Gauss-Seidel, (c) alt. line Gauss-Seidel. is te same as tat for linear equations. First, te errors of te solution ave to be smooted so tat tey can be approximated on a coarser grid. Ten, a nonlinear analog of te linear defect equation is transferred to te coarse grid. In te nonlinear case, te (exact) defect equation on Ω is given by ( ) (3.8) N u m + vm N u m = d m, wit u m te approximation of te solution after relaxation in te mt multigrid cycle, vm error and d m te corresponding defect. Tis equation is approximated on Ω H by ( ) (3.9) N H u m H + vm H N H u m H = d m H, te were v H m is te coarse grid approximation of te error vm. Not only is te defect dm transferred to te coarse grid by some restriction operator I H, but also te relaxed approximation u m itself by a restriction operator ÎH. On te coarse grid Ω H, we deal wit te problem N H w H = f H, were w H = u m H + vm H and were te rigt-and side f H is defined by f H := I H(f N u m ) + NHÎH um. Te coarse grid corrections v H are interpolated back to te fine grid, were te fine grid errors are smooted again. Te generalization from two grids to multigrid is done recursively. If N and N H are linear operators, te FAS metod is equivalent to te (linear) multigrid correction sceme. It was sown in [3], tat a variant of FAS, te projected full approximation sceme, PFAS can be used to solve linear complementarity problems of te form (2.2),(2.3),(2.4). PFAS is based on te projected Gauss-Seidel smooter (3.),(3.2). We now explain coarse grid correction parts for te problem (2.2),(2.3), (2.4) in some more detail.

12 76 Accellerated multigrid for American-style options Te LCP coarse grid correction. Suppose tat te error v m := u u m is smoot after relaxation. Te following LCP olds for v m : L v m d m, x Ω, v m + um f 2,, x Ω, (v m + u m f 2, )(L v m d m )= 0, x Ω, wit defect: d m = f, L u m. A smoot error vm can be approximated on a coarse grid witout any essential loss of information. Te LCP coarse grid equation for te coarse grid approximation of te error v H m is terefore defined in PFAS by: L H v m H I H d m, v m H + ÎH um f 2,H, ( v H m + ÎH um f 2,H)(L H v H m IH dm )= 0. Since te problem is nonlinear and we are solving inequalities, we solve for a full approximation u m H := vm H + IH um but interpolate only vm H back to te fine grid. A relevant difference between multigrid metods for equations and inequalities follows from te requirement tat, in te case of a converged solution on a fine grid u m u, corrections from te coarse grid equation sould be zero. Ten, we need I H v m H = I H(u m H ÎH u m ) = 0, leading to u m H = ÎH um (assuming operator I H keeps nonzero quantities nonzero). If for a fine grid LCP wit a converged solution we consider te coarse grid correction, it leads to te following requirements [3] on te restriction operators, (3.0) I H (f, L u m ) 0, Î H u m f 2,H, (ÎH u m f 2,H ) T I H (f, L u m )= 0. For equations, since f, L u m 0 in tis situation, tese requirements are satisfied for any reasonable coice of I H and ÎH. For LCPs, we coose for bot restriction operators straigt injection in order to satisfy te requirements (3.0). Te injection operator is in a certain sense constraint preserving. (Tese requirements do not, owever, prevent us from using any residual transfer in te interior, away from te free boundary). Te prolongation operator IH, bilinear interpolation, is applied only for unknowns on te inactive points, as tis resulted in te best convergence in [3] (for detail, see M6 in section 5 of [3]): u m um + I H vm H if um > f 2,, u m um elsewere (u m = f 2,). Tis combination of restriction and prolongation does not satisfy te well-known rule [2, 9] on te orders of te transfer operators, for example, for PDEs wit second derivatives. Moreover, it is known tat problems wit Neumann boundary conditions, as tey appear for te option pricing problem in te stocastic volatility setting, converge rapidly wit so-called modified full weigting operators at te boundary [9]. Tese boundary transfer operators cannot be used ere, because of te requirements (3.0). From tese points of view, te

13 C. W. Oosterlee 77 coarse grid correction part migt be not as powerful as it is commonly for multigrid for elliptic PDEs. Extra investment in smooting (more iterations, line or ILU smooters, etc.), or oter approaces for convergence improvement migt be necessary for fast and robust convergence. Te robustness of PFAS is also discussed in [3], were a convergence proof for a PFAS competitor, te so-called monotone multigrid metod is presented. Te example on wic tis discussion is based is not, owever, a decisive example for possible robustness problems of PFAS for te LCPs as we will sow in Section 5.. Notwitstanding tis discussion, we present an improvement for te robustness of PFAS by using it as a preconditioner for an acceleration metod in Section Acceleration of Multigrid by Iterate Recombination (Multigrid as a Preconditioner). In tis section, we discuss multigrid used as a preconditioner for Krylov subspace metods. From te multigrid point of view, multigrid as a preconditioner can also be interpreted as an acceleration of multigrid by iterate recombination. Tis interpretation easily allows generalizations, for example, to nonlinear problems and also to linear complementarity problems. Let u 0 be an initial approximation for solving L u = f and d 0 = f L u 0 its defect. Te Krylov subspace K m is defined by Km := span[d0, L d 0,..., Lm d 0 ]. Tis subspace can also be represented by (4.) K m = span[u u 0, u 2 u,..., u m u m ] = span[u 0 um, u um,..., um u m ], were te u m = (I L )u m + f are te iterates from te well-known Ricardson s iteration. Tese representations are easily obtained by induction using u u0 = d0, ui+ Te Krylov subspace approximation u m,acc u 0 + K m u i = (I L )(u i ui ). = u 0 + span[d 0, L d 0,..., L m d 0 ] is ten caracterized by finding approximations u m,acc for m =, 2,... wit minimal defect in a suitable norm. Te various Krylov subspace metods differ in te way te minimization is carried out. If we use te 2 norm for minimization, we obtain GMRES [8]. In te same way as te classical single grid iterative metods can be used as preconditioners, it is also possible to use a linear multigrid metod (algebraic multigrid, for example) as a preconditioner. Multigrid acceleration by iterate recombination and multigrid preconditioning lead to very similar algoritms: We searc for an improved solution based on te second representation of te subspace in (4.). In order to find an improved approximation u m,acc, we consider a linear combination of te m + latest approximations u m i, i = 0,, m (assuming m m), (4.2) u m,acc = u m + m i= α i (u m i u m ). For linear equations, te corresponding defect, d m,acc = f L u m,acc, is given by (4.3) d m,acc = dm + m i= α i (d m i d m ),

14 78 Accellerated multigrid for American-style options were d m i = f L u m i. In order to obtain an improved approximation u m,acc, te parameters α i are determined in suc a way tat te defect d m,acc is minimized, for example wit respect to te l 2 -norm 2. Tis is a classical defect minimization problem. Here, we solve te system of normal equations. Te work for solving te minimization problem is small. In general, for suc minimization problems, it may appen tat one as to deal wit an extremely ill-conditioned matrix. However, in tis setting m can often be cosen small, for example, 5 or less. Suc small matrices are usually still satisfactorily conditioned, so tat te matrix can be solved directly. In te linear case, it is not important weter te accelerated unknowns are cosen for u (4.2) or te multigrid iterates. It merely implies different weigts α i in te recombination process. We always use te accelerated iterates. Tis saves some storage and it is beneficial in te nonlinear case discussed below. Te resulting iterative metod is outlined in Figure 4., were te current approximation u m is replaced by : recombination : smooting : coarsest grid treatment FIG. 4.. Recombination of multigrid iterates u m,acc. Wit tis approximation replacement, te next multigrid cycle is performed resulting in a new iterate u m+. Te recombination (4.2) is again carried out wit te latest iterates u m+ i, i = 0,, m and so on. It is possible to generalize te idea of iterate recombination to nonlinear situations, were a nonlinear multigrid metod, suc as FAS is used. In tis case, te defect relation (4.3) does not old exactly, due to te nonlinearity. u m is ten only substituted by um,acc, if te defect d m,acc of um,acc is not muc larger tan te defect dm (as described below). Here, we generalize te defect minimization approac to linear complementarity problems. Tis is done very similarly to te generalization to nonlinear PDEs. PFAS will be used as te underlying multigrid metod, wose iterates are recombined. Te defect minimization for obtaining an improved solution is based on equation (.3) in te system of linear complementarity, d m,acc = (u m,acc f 2, )(Lu m,acc f, ). We obtain solutions wit an improved defect tat satisfy constraint (.2), as all recombined iterates do. Te points were u = f 2 in te linear combination are te intersection of te sets of tese contact points amongst all te vectors participating in te linear combination wit nonzero coefficients. Te linear combination cannot increase te number of contact points. Te convergence of te metod to a solution tat also satisfies (.) is still based on te convergence of PSOR in [7]. If all te vectors participating in te linear combination satisfy (.) at any particular point, ten so does any linear combination (by te linearity of L). Te resulting metod can be interpreted as a projected Gauss-Seidel metod, accelerated in terms of te coarse grid correction by a projected FAS multigrid metod and accelerated furter in terms of an outer iteration by a recombination tecnique. Due to te nonlinearity, criteria for replacing te PFAS solution u m by te accelerated solution u m,acc are needed. We use te following selection criteria:

15 C. W. Oosterlee 79. Te norm of d m,acc is not larger tan dm and te m intermediate defects dm i : (4.4) d m,acc 2 < min i m ( dm i 2, d m 2) 2. u m,acc is not too close to any of te intermediate solutions unless a considerable decrease of te defect norm is acieved. Criterion 2 is necessary to prevent stagnation in te convergence. Te same criteria are used for restarting te Krylov subspace, as is presented in [20]. Restarting takes place if te criteria are not satisfied in two consecutive iterations. 5. Numerical Results. In tis section, we present some numerical experiments. We measure te size of te defect of equation (2.4) in te LCP system. In addition to tis defect, we also cecked te difference between te previous and a current solution, as in [3]. Bot measures sow a very similar convergence istory. Indicated in te Tables 5.3 and 5.4 below is te average number of cycles per time step to reduce te defect in (2.4) in te infinity norm by five orders of magnitude, (5.) d m (s, y, t) d 0 (s, y, t) 0 5. Tis quantity is also presented if te recombination is applied (in tat case, te defect after te recombination is given by d m ). As an appetizer, before we consider te option pricing problem, we briefly discuss an LCP from [3] based on te Laplace operator. In [3], a very poor PFAS convergence was presented. We will consider tis problem wit PFAS in more detail also including te recombination tecnique and sow a very satisfactory convergence. 5.. Linear complementarity model problem based on Laplace operator. In [3], te following optimization problem modelling elasto-plastic torsion of a cylindrical bar in a model region Ω = (0, ) (0, ) is considered, (5.2) u ss + u yy 2C, x Ω, u d(x, Ω), x Ω, (u d(x, Ω))(u ss + u yy 2C) = 0, x Ω, wit boundary cond. u = 0, x Ω, were te d(x, Ω)-operator measures te distance from a grid point x = (s, y) to te domain boundary Ω. Wit parameter C = 0, te problem (5.2) was considered to be very difficult in [3]. Figure 5. presents level curves of te solution (left-and side figure) and te small region of inactive points, were u ss + u yy = 2C is valid (rigt-and side picture). Tis region, wose size depends on parameter C, represents te plastic region, wereas te active points, were te second constraint wit equality sign is valid, represents te elastic region (see [7] for details). In [3] te V-cycle PFAS convergence wit only one lexicograpic projected Gauss-Seidel smooting iteration as been presented. Tis convergence is not satisfactory as is confirmed by te upper curve at te top of Figure 5.2, were te same multigrid components as in [3] are used. Tis result is, owever, not a real surprise. Even for linear elliptic PDEs, a multigrid V-cycle wit one smooting iteration, injection as te restriction operator and a direct discretization of te PDE on coarse grids must be considered wit caution [9]. An injection-based coarse grid correction must be supplied wit more smooting iterations and wit more coarse grid processing as confirmed in te upper picture in Figure 5.2: An F(2,0)- cycle wit only sligtly more work on coarser grids leads to very fast multigrid convergence.

16 80 Accellerated multigrid for American-style options FIG. 5.. Level curves and inactive set for C = 0. Tese results are computed on a grid. By only looking at te V(,0)-cycle convergence, an incorrect impression of te quality of PFAS migt easily lead to unnecessarily considering oter solution metods. In te lower picture in Figure 5.2, we present te convergence of accelerated multigrid, were 3 iterates are recombined after eac multigrid cycle. A faster and more robust convergence is obtained in tis case. Table 5. finally presents average reduction factors for different grid sizes and multigrid cycles. Especially te F-cycles sow a very satisfactory convergence, even witout te recombination tecnique. TABLE 5. Multigrid and accelerated multigrid convergence ( m = 3) for te 2D model LCP and various multigrid cycles. Grid Cycles V(2,0) V(2,0) + acc. F(2,0) F(2,0) + acc. F(,) An American-style option. Wit te components introduced in Section 3, we now solve an LCP problem from American-style options wit stocastic volatility. Te PFAS multigrid metod is based on te F(2,2)-cycle wit pointwise lexicograpic Gauss-Seidel smooting. Te parameter set considered is again (5.3) α = 5, β = 0.6, γ = 0.9, ρ = 0., λ = 0, r = 0. and exercise price E = 0. Te expiration date is set to T = For tis set of parameters reference solutions for te American put option prices are presented in [5] and [25]. First, we discuss te accuracy of te discretization by comparing our numerical results on different grid sizes wit te reference results. A truncated domain Ω = [0, s max ] [0, y max ] is used wit s max = 20 and y max =. Te grids consist of 256 cells in te s-direction; in te y-direction four sizes are considered wit 32, 64, 28 and 256 cells. Table 5.2 compares te solution u obtained at y = and y = 0.25 respectively wit te solutions in [5] and [25]. Te results agree very well, especially wit tose in [25], in wic te difference to te oter reference result was discussed. Tis difference migt be due to te stretced grid considered in [5] or due to te interpolation tecnique and te first order accurate discretization employed tere in some parts of te domain.

17 C. W. Oosterlee MULTIGRID CONVERGENCE V(,0)-CYCLE V(2,0)-CYCLE F(2,0)-CYCLE e-06 e-08 e MG WITH RECOMBINATION (3 ITERANTS) V(,0)-CYCLE V(2,0)-CYCLE F(2,0)-CYCLE e-05 e-06 e-07 e-08 e-09 e FIG Multigrid convergence wit different cycles (upper picture) and convergence of multigrid wit acceleration (lower picture), 3 recombined iterates, grid. Figure 5.3 presents te moving free boundary for tis problem in pictures for te t values 0, 0.05, 0., 0.5, 0.2. On te left-and side of tis free boundary, te active points are found, i.e., te constraint (2.3) wit equality sign is valid. Table 5.3 presents te average number of multigrid iterations necessary to reduce te initial defect eac time step by 5 orders of magnitude on te different grid sizes. In te left-and side part of te table, te multigrid convergence witout te recombination is presented. It can be seen tat for problem (2.7) wit parameter set (5.3) fewer grid points in te y-direction lead to worse multigrid convergence as discussed in te Fourier analysis section. Te overrelaxation parameter elps in tis case (see te results) to improve te convergence: instead of 46.5 iterations per time step, 27.6 are needed wit ω =.4. Te rigt-and side part of Table 5.3 sows te results wit te same multigrid algoritm accelerated by a recombination of te 3 previous iterates. A muc better convergence is obtained now, especially in te case wit only 32 grid points in te y-direction, wereas te convergence improvement on te grid is not impressive. Tis confirms te general impression tat (almost)

18 82 Accellerated multigrid for American-style options (t = 0.00) (t = 0.05) (t = 0.0) (t = 0.5) (t = 0.20) (all t ) FIG Te moving free boundary for te American put option. optimal multigrid metods are difficult to improve by Krylov subspace acceleration, wereas in cases in wic one of te multigrid components (ere it is te smooter) is not optimal, te acceleration by te recombination tecnique can give a more robust and faster convergence. It is also sown tat te combination of te best overrelaxation in te left-and side of Table 5.3 wit te recombination tecnique does not result in te best algoritm in te rigt-and side. Tis as also been observed in [22], were a multigrid metod wit pointwise relaxation used as a preconditioner is analyzed by Fourier analysis for model anisotropic equations. Te best coice ere seems to be an overrelaxation parameter of. combined wit te recombination tecnique.

19 C. W. Oosterlee 83 TABLE 5.2 American put option values wit stocastic volatility for y = and y = 0.25 compared wit reference values on different grid sizes at t = 0, E = 0. y = Asset price Grid (smaller time step) ref [25] ref [5] y = 0.25 Asset price Grid (smaller time step) ref [25] ref [5] TABLE 5.3 Average number of cycles needed for 5 orders of defect reduction over 20 time steps on different stretced meses. Grid F(2,2)-cycle F(2,2)- wit recombination ω = ω =. ω =.2 ω =.4 ω = ω =. ω =.2 ω =

20 84 Accellerated multigrid for American-style options Finally, Table 5.4 presents te accelerated multigrid convergence wit m = 3 for te parameter set (5.3) wit E = 00 and, terefore, wit a 0 times larger domain, s max = 200. As expected from te Fourier analysis, te convergence results are very similar to te results in Table 5.3. TABLE 5.4 Average number of cycles needed for 5 orders of defect reduction over 20 time steps on different stretced meses. Metod Grid F(2,2)- wit recomb., m = 3, ω = Remark: Te metod described ere also works well for te call option wit dividend payment. Altoug te free boundary is in a completely different part of te computational domain ten, for te call option te solution is simply zero in te problematic part. It is, of course, still beneficial for te convergence to cut off te domain near s = 0 as it was sown in Table Conclusions. In tis paper we ave presented a multigrid solution metod for linear complementarity problems. Te metod is based on te projected full approximation sceme. It is combined wit a recombination of iterates convergence acceleration tecnique. By providing muc detail about te different parts of te solution metod, we ope to give insigt into its expected convergence. Te smooter and te oter components in te multigrid metod ave been analyzed by means of Fourier analysis for te main discrete operator appearing in te LCP. Te Gauss-Seidel lexicograpic point smooter was cosen ere. Wit te acceleration tecnique, fast convergence is obtained for an option pricing problem on grids wit different grid sizes. Te error of te discretization is determined by comparison wit reference solutions. Te smooter proposed in [6] consisting of a point SOR metod for te free boundary detection followed by a modified line Gauss-Seidel smooter for stretced grids for first order upwind and central discretizations of convective terms used. Te Fourier analysis results in Figure 3.3 sowed, owever, tat only te multigrid metod based on an alternating line smooter was robust. Wit suc a more involved smooter, a recombination tecnique is not necessary for fast convergence. Altoug we only considered one parameter set (3.6) in tis paper, we used te Fourier analysis (not sown ere) to investigate te parameter range for wic te conclusions remain valid. Te sensitivity wit respect to variations of te (important) parameter r is, for example in its relevant range, not at all significant. Te same is true for te oter parameters. By retaining a point smooter for tis 2D problem, we constructed a fast and ceap solver, tat can serve as a basis for treating oter, iger dimensional, problems in te option pricing context. Acknowledgement Te autor gratefully acknowledges R. Wienands for providing is Fourier analysis software package LF A00 2D and is assistance in using it. REFERENCES [] C.A. Ball and A. Roma, Stocastic volatility option pricing. J. Financial Quantitative Anal., 29: , 994. [2] A. Brandt, Multi level adaptive solutions to boundary value problems. Mat. Comp., 3: , 977. [3] A. Brandt and C.W. Cryer, Multigrid algoritms for te solution of linear complementarity problems arising from free boundary problems, SIAM J. Sci Comput., 4: , 983.