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Higher Order Smoothing Schemes for Inhomogeneous Parabolic Problems with Applications to Nonsmooth Payoff in Option Pricing B.A. Wade

1

Center for Industrial Mathematics, University of Wisconsin–Milwaukee Milwaukee, Wisconsin 53201-0413, USA. E-mail: [email protected]

A.Q.M. Khaliq Department of Mathematical Sciences, Middle Tennessee State University Murfreesboro, TN 37132-0001, USA.

M. Yousuf Department of Mathematics and Physics, Alfred State College Alfred, NY 14802, USA

J. Vigo–Aguiar

2

Departamento de Matem´ atica Aplicada, Universidad de Salamanca 37003 Salamanca, Spain E-mail: [email protected] Abstract — A new family of numerical schemes for inhomogeneous parabolic partial differential equations is developed. The new family of algorithms utilizes diagonal Pad´e schemes combined with positivity–preserving Pad´e schemes as damping devices. We also utilize partial fraction decomposition to address problems with accuracy and computational efficiency in solving the higher order methods and to implement the algorithms in parallel. Numerical experiments are presented for various examples from financial mathematics, especially pricing options with nonsmooth payoffs. 2000 Mathematics Subject Classification: 65M12, 65M15, 65Y05, 65Y20. Keywords: Pad´e Scheme, parabolic problem, nonsmooth data, positivity, nonsmooth pay-off, Black-Scholes PDE. 1

Supported in part by grant number No. SAB2003-0266 from the Ministerio de Educaci´ on, Cultura y Deporte, Spain, and by the U.S. National Security Agency under Grant Agreement Number H98230-05-10062. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. 2 Supported by grants No. MTM2004-00295 from the Ministerio de Ciencia y Tecnolog´ıa and No. SAB2003-0266 from the Ministerio de Educaci´ on, Cultura y Deporte, Spain, as well as no. SA024/04 from Junta Castilla y Le´ on, Spain.

2

Wade, Khaliq, Yousuf, Vigo-Aguiar

1. Introduction For homogeneous parabolic partial differential equations (PDE) with nonsmooth initial data, a family of higher order accurate smoothing schemes has recently been developed by Wade et al. [22]. Convergence results and numerical experiments show that these schemes can be more robust than the well known Rannacher smoothing schemes [16, 17] with respect to spurious oscillations generated through high frequency components in nonsmooth initial or boundary data. Other papers related to the subject are [2, 3, 4, 5, 9, 16, 17, 20, 24], which, however, treat cases of nonsmooth data only for the homogeneous case. Robust schemes for inhomogeneous parabolic partial differential equations with nonsmooth initial, boundary, or forcing terms have only been lightly treated, for example, in [3, 4, 9]. In this article we develop a useful family of numerical schemes for advection– diffusion–reaction equations under conditions of low regularity by extending the aforementioned techniques of [22]. We then focus on some practical illustrations of the new family of schemes using important problems in the area of financial mathematics. Khaliq and Wade [9] have worked on a smoothing strategy for the Crank-Nicolson scheme using variable time steps for inhomogeneous equations. Using a rational approximation of the exponential function e−z , designed to have three distinct real poles, Serbin [19] has developed a fourth order A-stable scheme for inhomogeneous case, although only for smooth data. Voss and Khaliq [21] have also developed a fourth order scheme for inhomogeneous problems with smooth data using a rational approximation with four distinct real poles. In these papers the authors use partial fraction decomposition to deal with the difficulties in computing higher order matrix polynomials in the denominators, and to allow a parallel implementation. These schemes are designed to be implemented on serial or parallel machines. For the scheme given by Serbin [19], three systems must be solved, whereas the scheme developed Khaliq and Voss [21] requires four systems at each time level. In contrast to the aforementioned rational approximations, only one algebraic system arises if the diagonal (2, 2)-Pad´e is employed instead. However, one disadvantage of using diagonal Pad´e schemes is that they are not L-stable, in particular, their symbols converge to magnitude one at infinity, so that when the initial data is not smooth high frequency components in the error are not damped out. Here, we develop damping schemes, cf. [22], to the inhomogeneous case, being careful to preserve the essential properties of parallelizability and robustness for nonsmooth initial data enjoyed by the original family of schemes. We shall begin by mentioning a class of single step fully discrete numerical schemes described by Brenner et al. [3] and summarized in the book of Thom´ee [20, Ch. 9] since this will lead us most easily to the introduction of the new family of numerical schemes. This class of numerical methods achieves optimal order of convergence for smoothing initial data while avoiding imposing unsatisfactory boundary conditions on the forcing term f and some of its derivatives, cf. [20]. The inhomogeneous problem considered in [3, 20] satisfies certain regularity assumptions, and the problem of nonsmoothness of the initial data has not been sufficiently addressed. In the presence of nonsmooth data, the diagonal Pad´e schemes need to have some damping device to get optimal order convergence, cf. [11, 12, 16, 17, 22]. Our aim is to develop an implementation strategy to approximate more efficiently and accurately the solution of the linear inhomogeneous parabolic equation with nonsmooth initial data by synthesizing various existing ideas. Similar to the approach adopted by Wade et al. [22], our first

Smoothing Schemes for Inhomogeneous Parabolic PDEs

3

basic strategy is to use a positivity preserving Pad´e scheme at the start as a damping device followed by a diagonal Pad´e scheme. The second component of our strategy is a partial fraction decomposition of the Pad´e approximants using the algorithm developed by Gallopoulos and Saad [6] and Khaliq et al. [8]. With these approaches, we shall use partial fraction decomposition form of the schemes at each time step, which needs only the work of solving certain robust linear algebraic systems on parallel or serial machines. Even in serial, it is necessary to implement this split version in order to avoid conditioning problems from the higher matrix polynomials that would have to be inverted. That complex arithmetic is necessary adds only a small inconvenience and a fractional amount of extra computational time, which is compensated for by the robust performance of the higher order schemes.

2. The Abstract PDE and Standard Numerical Schemes We shall assume an unbounded operator as infinitesimal generator of an analytic semigroup, whose generality affords an easier analysis for convergence estimates that work without dependence on the spatial mesh size. We adopt the treatment of V. Thom´ee, concisely summarized in [20, Ch. 3, 7, 8, 9]. Consider the following PDE:  ut (x, t) + A(x)u = f (x, t) x ∈ Ω, t ∈ 0, t = J, (1) u(x, t) = 0 x ∈ ∂Ω, t ∈ J, u(·, 0) = v on Ω, where Ω is a bounded domain in Rd with lipschitz boundary and A denotes the uniformly elliptic operator   X d d X ∂ ∂ ∂ A := − aj,k (x) + bj (x) + b0 (x). ∂x ∂x ∂x j k j j=1 j,k=1 The coefficients aj,k and bj are to be C ∞ (or sufficiently smooth) functions on Ω, aj,k = ak,j , b0 > 0, and for some c0 > 0 d X

j,k=1

aj,k (·)ξj ξk > c0 |ξ|2 ,

on Ω,

for all ξ ∈ Rd .

The initial value problem (2) is reset to be posed in a Hilbert space X, as follows, for the reason that it makes the analysis simpler. Consider now A to be a linear, selfadjoint, positive definite closed operator with a compact inverse T , defined on a dense domain D(A) ⊂ X. The operator A could represent any of {Ah }0 1 such that k(zI − A)−1 k 6 M |z|−1 ,

z ∈ Σα .

4

Wade, Khaliq, Yousuf, Vigo-Aguiar

It follows that −A is the infinitesimal generator of an analytic semigroup {e−tA }t>0 which is the solution operator for (2) below, cf. [14, 20]. There is a standard representation: Z 1 −tA E(t) := e = e−tz (zI − A)−1 dz, 2πi Λ

where Λ := {z ∈ C : | arg(z)| = θ}, oriented so that Im(z) decreases, for any θ ∈ (α, π2 ). The abstract initial value problem is as follows: ut + Au = 0,

t > 0,

u(0) = v ∈ X.

(2)

By the Duhamel principle the exact solution of (2) can be written as u(t) = E(t)v +

Zt 0

E(t − s)f (s)ds.

(3)

The solution then satisfies: u(t + k) = E(k)E(t)v + E(k)

Zt 0

Z1

= E(k)u(t) + k

0

Zt+k E(t − s)f (s)ds + E(t + k − s)f (s)ds, t

E(k − kτ )f (t + kτ )dτ,

Thus, to develop a numerical scheme we may utilize the representation u(tn+1 ) = e−kA u(tn ) + k

Z1

e−kA(1−τ ) f (tn + τ k) dτ,

(4)

0

where 0 < k 6 k, for some k, is the time step and tn = nk with 0 6 n 6 n = bt/kc. One must be careful to respect any lack of regularity in the data and design a computational procedure that is robust in its convergence properties. To approximate the solution of (4) in time using vn to denote the computed approximation of u(tn ), the scheme we consider has the following form: vn+1 = r(kA)vn + k

s X

Pi (kA)f (tn + τi k),

0 6 n 6 n, v0 = v,

(5)

i=1

which, by writing Rk f (tn ) =

Ps

i=1

Pi (kA)f (tn + τi k), can be rewritten as

vn+1 = r(kA)vn + kRk (tn ),

0 6 n 6 n, v0 = v.

The functions r (z) and {Pi (z)}si=1 are to be rational functions bounded on the spectrum of kA, uniformly in k. The distinct real numbers {τi }si=1 would typically be taken as Gaussian Quadrature points in the interval [0, 1]. For the time stepping scheme (5) to be accurate of order q, it is required that the scheme satisfy conditions developed by Brenner et al. [3]. The reader may consult [20, Ch. 9] to fill in various details omitted here for the sake brevity. The following result describes the accuracy and establishes some equivalence relations, which we then use later.

5

Smoothing Schemes for Inhomogeneous Parabolic PDEs

Proposition 1. [20, L. 9.1] The time discretization scheme (5) is accurate of order q if and only if r(z) = e−z + O(k q+1 ), z → 0, (6) and for 0 6 l 6 q,

s X i=1

l! τil Pi (z) = (−z)l+1

e−z −

l X (−z)j

j!

j=0

!

+ O(z q−l ),

z → 0.

(7)

Equation (7) is equivalent to s X

τil Pi (z)

i=1

=

Z1

sl e−z(1−s) ds + O(z q−l ),

z → 0.

(8)

0

It is computationally efficient to have r(z) and {Pi (z)}si=1 such that they share the same poles, which puts an additional constraint on the scheme. By considering, from [20], r(z) =

N (z) D(z)

and Pi (z) =

Ni (z) , D(z)

i = 1, 2, ..., s,

with N (z), D(z), and Ni (z) as polynomials, the scheme (5) formally can be rewritten as D(kA)vn+1 = N (kA)vn +

s X i=1

Ni (kA)f (tn + τi k).

It is shown in [20] that for the case s = q (s is the number of quadrature points and q is the accuracy of the scheme) this form as well as the conditions of the proposition can be achieved by choosing the rational functions r(z) satisfying (6), selecting distinct real numbers, say, by Gaussian Quadrature, {τi }m i=1 , and finally solving the system ! q l j X X l! (−z) τil Pi (z) = r(z) − , l = 0, 1, ..., q − 1, (9) (−z)l+1 j! i=1 j=0 for the coefficients of the Pi (z). This system (9) is of Vandermonde type (whose determinant is not zero, the τi are distinct [1]), which gives the rational functions {Pi (z)}qi=1 as linear combinations of the terms on the right hand side of (9). The only singularities are then those of r(z). Thus the denominators of the {Pi }qi=1 have the same factors as that of r(z). If r(z) is bounded for large z, then the right hand sides of (9) are small for large z, and the numerator of Pi (z) would be of lower degree than its denominator for each i. For the case when the number of quadrature points s is less than the order of the scheme q, an alternative formula similar to (9) is given in [20]. The accuracy conditions are reformulated by defining   X l m X (−z)j l! τil qi (z), l = 0, 1, ..., q − 1 r(z) − δl (z) = − (−z)l+1 j! j=0 i=1 and

  q X (−z)j q! r(z) − . δq (z) = (−z)q+1 j! j=0

6

Wade, Khaliq, Yousuf, Vigo-Aguiar

and requiring that δl (z) = O(z q−l ),

as z → 0, for l = 0, 1, ..., q.

For the purpose of construction of the schemes when s < q, it is shown in [20, Lemma 9.2] that this is equivalent to the condition (6) plus δl (z) = 0,

as z → 0, for l = 0, 1, ..., s − 1

and a moment condition Z1

w(τ )τ j dτ = 0,

0

on the quadrature points, with ω(τ ) ≡ functions {Pi (z)}si=1 in [20] is s X

τil Pi (z)

i=1

l! = (−z)l+1

for j = 0, 1, ..., q − s − 1

Qs

i=1 (τ

r(z) −

− τi ). The formula to obtain the rational

l X (−z)j j=0

j!

!

,

l = 0, 1, ..., s − 1.

(10)

3. Pad´ e Schemes for Inhomogeneous Problems Using Pad´e schemes as rational function approximations of the matrix exponential functions in (4, 5) is an important and common tool. Let Pn,m (z) and Qn,m (z) be two polynomials of degree n and m, respectively, defined as follows [10]: Pn,m (z) = and

Qn,m (z) =

n X

j=0 m X j=0

(m + n − j)!n! (−z)j , (m + n)!j!(n − j)! (m + n − j)!m! (z)j . (m + n)!j!(m − j)!

The following useful properties of (n, m)−Pad´e are given, for example, in Lambert [10]: The (n, m)−Pad´e approximant Rn,m to e−z is: (i) A-acceptable if n = m, (ii) A0 -acceptable if n 6 m, and (iii) L-acceptable if n = m − 1 or n = m − 2. For example, the Backward Euler scheme is R0,1 (z) = (1 + z)−1 , the Crank-Nicolson scheme is R1,1 (z) = (1 − 21 z)(1 + 12 z)−1 , and other higher order schemes of present interest include the following: 1 1 1 2 1 R0,3 (z) = (1 + z + z 2 + z 3 )−1 , R1,2 (z) = (1 − z)(1 + z + z 2 )−1 , 2 6 3 3 6 1 1 2 1 1 2 −1 R2,2 (z) = (1 − z + z )(1 + z + z ) , 2 12 2 12 1 4 1 5 −1 1 2 1 3 z ) , R0,5 (z) = (1 + z + z + z + z + 2 6 24 120 2 1 3 3 1 R2,3 (z) = (1 − z + z 2 )(1 + z + z 2 + z 3 )−1 , 5 20 5 20 60 1 2 1 3 1 1 1 3 −1 1 z )(1 + z + z 2 + z ) . R3,3 (z) = (1 − z + z − 2 10 120 2 10 120

Smoothing Schemes for Inhomogeneous Parabolic PDEs

7

We shall design our schemes for inhomogeneous problems based on the positivity preserving (0, 2m − 1)–Pad´e schemes as damping devices. These practical Pad´e schemes do well in approximating the exponential function in the sense that their symbols capture the properties of positivity and monotone convergence to zero at infinity of e−x , x > 0— thus avoiding the amplification or introduction of oscillations in high frequency components of the error. Although the first subdiagonal (m-1, m)–Pad´e schemes are also L-stable and sometimes have a smaller constant in the local truncation error, their symbols change sign or their derivatives do, which can cause unphysical oscillations due to amplification in the error, depending on the relative sizes of the diffusion rate, the spatial mesh size and time steps. As an example, let us begin by mentioning the backward Euler scheme for inhomogeneous problems. Using the rational approximation
r(z) = R0,1 (z) and τ1 = 21 , this first order scheme 1 is vn+1 = R0,1 (kA) vn + kP1 (kA)f tn + 21 k , for which P1 (z) = 1+z . Next, we shall give the more complicated versions, involving higher order schemes. These standard schemes form the basis for the new family of damping schemes of this. 3.1. A fourth Order Scheme for Inhomogeneous Problems √

Let r(z) = R2,2 (z), and define τ1 = 3−6 3 and τ2 = order 2. Then the system (10) reduces to:

√ 3+ 3 , 6

the Gaussian quadrature points of

1 P1 (z) + P2 (z) = − (r (z) − 1) , z 1 τ1 P1 (z) + τ2 P2 (z) = 2 (r (z) − 1 + z) , z

(11)

which results in a fourth order scheme as follows: vn+1 = R2,2 (kA) vn + kP1 (kA) f (tn + τ1 k) + kP2 (kA)f (tn + τ2 k) where

√

√

1 1 − 63 z 1 1 + 63 z P1 (z) = 1 2 , P2 (z) = 1 2. 2 1 + 12 z + 12 2 1 + 12 z + 12 z z

The above mentioned scheme performs with fourth order accuracy only if the initial data has sufficient regularity. To achieve good convergence and robust performance with diagonal Pad´e schemes some damping treatment is necessary. The idea is to use positivity preserving (0, 2m − 1)-Pad´e schemes for initial damping and then use diagonal (m, m)-Pad´e schemes after that. That the global accuracy of the (0, 2m − 1)-Pad´e schemes is one less does not matter because the damping scheme is used only two times, and the local truncation error, one higher than the global, then truly affects the accuracy. Next, we develop the details for an important damping scheme in our proposed fourth order numerical method. 3.2. A Third Order Scheme for Inhomogeneous Problems Taking the rational approximation r(z) = R0,3 (z), a good choice for its positivity properties, and using the Gaussian quadrature points τ1 and τ2 as above, we construct the following

8

Wade, Khaliq, Yousuf, Vigo-Aguiar

third order scheme: vn+1 = R0,3 (kA) vn + kP1 (kA) f (tn + τ1 k) + kP2 (kA)f (tn + τ2 k) 11+ P1 (z) = 2 P2 (z) =

11+ 2

1 6 1 6

√
√
3 − 3 z + 61 1 − 3 z 2 , 1 + z + 21 z 2 + 16 z 3 √
√
3 + 3 z + 61 1 + 3 z 2 . 1 + z + 21 z 2 + 16 z 3

4. The New Smoothing Schemes IPSP(m) In this section we will present the new family of smoothing/damping schemes in general. Similar to the homogeneous case, we use the idea of combining diagonal Pad´e schemes Rm,m with the positivity preserving Pad´e schemes R0,2m−1 . Because of the positivity preserving property of these starting schemes, we shall use the notation IPSP(m) for our inhomogeneous positively smoothed Pad´e schemes. Each IPSP(m) scheme is of order 2m. The proposed family of smoothing schemes is as follows:  m P  Pi (kA)f (tn + τi k), 0 6 n < p;  R0,2m−1 (kA)vn + k i=1 (IPSP(m)) vn+1 = (12) m P   Rm, m (kA)vn + k Pi (kA)f (tn + τi k), n > p, i=1

where the Pi are solutions of equation (10) when r = R0,2m−1 or Rm,m , respectively. Here, p > 2 is the number of initial damping steps, where p can always be taken as 2 independent of m. Taking p > 2, however, can sometimes result in a bit more damping, for example, p = 4 occasionally works better. Much larger values of p do not give significant improvement. For √example, the √ fourth order scheme, using second order Gaussian quadrature points 3− 3 3+ 3 τ1 = 6 , τ2 = 6 , has the following form: 1 1 + kA + (kA)2 + 2  1 = 1 + kA + (kA)2 + 2

v1 = v2



with P1 (kA) and P2 (kA) as

−1 1 v0 + kP1 (kA)f (t0 + τ1 k) + kP2 (kA)f (t0 + τ2 k), (kA)3 6 −1 1 (kA)3 v1 + kP1 (kA)f (t1 + τ1 k) + kP2 (kA)f (t1 + τ2 k), 6

√
√
  −1 3− 3 1− 3 1 1 1 2 1+ kA + (kA) 1 + kA + (kA)2 + (kA)3 P1 (kA) = 2 6 6 2 6 √
√
  −1 3+ 3 1+ 3 1 1 1 P2 (kA) = , 1+ kA + (kA)2 1 + kA + (kA)2 + (kA)3 2 6 6 2 6 and for n > 2 vn+1 =



−1 1 1 1 1 1 − kA + (kA)2 1 + kA + (kA)2 vn + kP1 (kA)f (tn + τ1 k) 2 12 2 12 +kP2 (kA)f (tn + τ2 k),

Smoothing Schemes for Inhomogeneous Parabolic PDEs

9

where P1 (kA) and P2 (kA) are  1 1− P1 (kA) = 2  1 1+ P2 (kA) = 2

√

 3 kA 1 + 6 √  3 kA 1 + 6

−1 1 1 (kA) + (kA)2 2 12 −1 1 1 . (kA) + (kA)2 2 12

4.1. Partial Fraction Form of the IPSP(m) Schemes A problem now arises, how to accurately and efficiently compute these solutions despite the presence of higher order matrix polynomials that must be inverted. These schemes are designed to share the same denominator, and we may take advantage of recent advances in parallelization of Pad´e schemes, albeit for a somewhat different application, developed by a number of authors, particularly, by Gallopoulos and Saad [6], Serbin [19], Khaliq et al. [8], and Ruesch et al. [18]. Allowing the use of complex arithmetic, it has become possible to implement more efficiently the higher order Pad´e schemes in parallel or in serial by using the partial fraction technique. Even in serial, it is best to employ these splittings for the calculation of the inverse in order to avoid disproportionate roundoff error. The partial fraction form of the rational functions r(z) = Rn,m (z) and {Pi (z)}m i=1 requires us to consider two cases, n < m for subdiagonal Pad´e schemes and n = m for diagonal Pad´e schemes. If n < m, then we utilize   q1 q1 +q2 X X wj wj Rn,m (z) = +2 < z − cj z − cj j=1 j=q +1 1

and the corresponding {Pi (z)}m i=1 take the form   q1 q1 +q2 X X wij wij Pi (z) = < +2 , z − c z − c j j j=1 j=q +1

i = 1, 2, ..., m.

1

Here, Rn,m as well as the Pi have q1 real and 2q2 nonreal poles {cj } , with q1 + 2q2 = n, and 0 0 where wj = Pn,m (cj ) /Qn,m (cj ) and wij = Ni (cj ) /D (cj ) . For the case when m = n, we utilize   q1 +q2 q1 X X wj wj m +2 < Rm,m (z) = (−1) + z − cj z − cj j=q +1 j=1 1

and

  q1 +q2 q1 X X wij wij +2 < Pi (z) = , z − c z − c j j j=q +1 j=1

i = 1, 2, ..., m.

1

The following algorithm results: Parallel Algorithm

1. For i = 1, ..., q1 + q2 , solve (kA − ci I) yi = wi vs + 2a. (If n < m) Compute vs+1 =

q1 P

i=1

yi + 2

q1P +q2

i=q1 +1

m P

j=1

< (yi ) ;

kwij f (ts + τj k) ;

10

Wade, Khaliq, Yousuf, Vigo-Aguiar

2b. (If n = m) Compute vs+1 = (−1)m vs +

q1 P

yi + 2

i=1

q1P +q2

i=q1 +1

< (yi ) .

In order to construct some complete examples of smoothing schemes in their partial fraction form, we have computed the poles and weights of the schemes (0, 3)-Pad´e, and (2, 2) -Pad´e approximations of e−z . I. For the (0, 3)−Pad´e scheme, we have q1 = q2 = 1 and c1 w1 w11 w21

= = = =

−1.5960716379833, c2 = −0.7019641810083 − i1.807339494452, 1.475686517795720, w2 = −0.7378432588979 + i0.365017840801, 0.25964745169791, w12 = −0.3128364277412 + i0.472314917248, 0.66492666056455, w22 = 0.3505493716099 − i0.04941905457189.

The algorithm works out to be vs+1 = y1 + 2 0. Then there exists a constant C = C(t) such that kvn − u(tn )k 6 Ck

q



t−q n kvk

+ tn

q−1 X l=0

Sl +

Ztn

kf

(q)

0



kds ,

0 6 n 6 n, 0 < k 6 k.

(13)

where Sl = sups6tn |f (l) (s)|2q−2l Proof. Note, by [20] the solution operator Ek = r(kA) is stable in H, where r is either of R0,2m−1 or Rm,m . The starting scheme is vn = rsn (kA)v + k

n−1 X

rsn−1−j (kA)Rks f (tj ),

for n = 1, 2,

(14)

j=0

and with rs = R0,2m−1 . The main scheme, with rm = Rm,m , is as follows: n−2 vn = r m (kA)rs2 (kA)v + k

1 X

n−2 rm (kA)rs1−j (kA)Rks f (tj )

j=0

+k

n−1 X

n−1−j rm (kA)Rkm f (tj ),

n > 3.

(15)

j=2

Also, using E(t) = e−tA , the exact solution of equation (2) can be written as: u(tn ) = E(tn )v + k

1 X

E(tn−1−j )Ik f (tj ) + k

j=0

where Ik f (tj ) =

n−1 X

E(tn−1−j )Ik f (tj ),

j=2

Z1 0

E(k − sk)f (tj + sk)ds.

(16)

12

Wade, Khaliq, Yousuf, Vigo-Aguiar

The error E n = vn − u(tn ), for n > 2, can be written as: n−2 E n = rm (kA)rs2 (kA)v − E(tn )v 1 X
n−2 rm (kA)rs1−j (kA)Rks f (tj ) − E(tn−1−j )Ik f (tj ) +k j=0

+k

n−1 X

n−1−j rm (kA)Rks f (tj ) − E(tn−1−j )Ik f (tj )

j=2

=

E0n

n + Esn + Em ,

(respectively)




where E0n is the error for the corresponding homogeneous equation, Esn is the error for the n inhomogeneous part of the starting scheme, and the error Em is due to the inhomogeneous part of the main scheme. The error term E0n can be approximated by the established result [22, Theorem 3.1] as follows:
n−2 kE0n k = k rm (kA)rs2 (kA) − E(tn ) k 6 Ck q t−q (17) n kvk

n−1−j n After adding and subtracting rm (kA)Ik f (tj ) in the error term Em and rearranging its terms, we arrive at

n Em

= k =

n−1 X

j=2 n Em1 +

n−1−j rm (kA) n Em2 .




− E(tn−1−j ) Ik f (tj ) + k

(respectively)

n−1 X j=2

n−1−j rm (kA)(Rkm − Ik )f (tj )

(18)

Following the approach given by Thom´ee [20, Theorem 9.1] we have the following estimate n n for Em1 and Em2 , Rt n kEm1 k 6 Ck q t2n |f |2q ds. (19) which is bounded by the right hand side of (13). Also, n kEm2 k 6

n−1 X

Ck q+1

j=2

q−1 X l=0

t n−1 Zj+1 X kf (q) kds. |f (l) (tj )|2q−2l + Ck q

(20)

j=2 t j

Incorporating (19) into the right hand side of (20), we therefore arrive at the following estimate for the main scheme: n kEm k 6

n−1 X

Ck q+1

j=2

q−1 X l=0

|f (l) (tj )|2q−2l + Ck q

t n−1 Zj+1 X j=2 t j

kf (q) kds.

(21)

Using a similar approach, we can also develop the estimate for the starting scheme as: kEsn k 6

1 X j=0

Ck q+1

q−1 X l=0

|f (l) (tj )|2q−2l + Ck q

t 1 Zj+1 X j=0 t j

kf (q) kds,

0 6 n 6 2.

(22)

Smoothing Schemes for Inhomogeneous Parabolic PDEs

13

Combining (21) and (22), kEsn k

+

n kEm k

6

n−1 X

Ck

q+1

j=0

6 Ck q tn

q−1 X l=0

q−1 X l=0

(l)

|f (tj )|2q−2l + Ck

Sl + Ck q

Ztn

q

t n−1 Zj+1 X j=0 t j

kf (q) kds,

kf (q) kds.

(23)

0

where Sl = sups6tn |f (l) (s)|2q−2l , and (17) together with (23) complete the proof.

6. Numerical Experiments In this section we demonstrate the performance of the new algorithm on some important examples form financial mathematics, using a Black-Scholes PDE model. Consider the following nonself-adjoint advection-diffusion type equation [23] with spatially variable coefficients: 1 ∂2V ∂V ∂V + σ 2 S 2 2 + rS − rV = 0. (24) ∂t 2 ∂S ∂S Here S represents the price of the underlying asset (serving as a space variable), and V (S, t) the value of the option at time t before the expiry time T ; parameters σ, r and E are the volatility of the underlying asset, the interest rate, and the exercise/strike price of the option, respectively. One can find the derivation, background, and technical details of this model in [7, 13, 23]. To determine V (S, t) uniquely we must specify other conditions that involve information about the particular option, e.g., initial and boundary conditions. Our first example is of a so-called butterfly option which has three strike prices. The payoff for this option has three corners at strike prices. The second example is of a binary call option whose payoff has a jump discontinuity at the strike price. Convergence results for Examples 6.1 and 6.2 are obtained using the fourth order PSP(2) smoothing scheme. In our first experiment, we employ a spatial discretization using a fourth order central difference scheme and compute combined fourth order convergence in space and time. In our second experiment, we compute with a very small spatial mesh so that there is relatively no error in the x variable to show fourth order convergence in time. All the numerical schemes are implemented in serial, except we use the split versions described earlier. Without these splittings the higher polynomial functions of the matrices that must be inverted would cause numerical difficulties. The Delta of an option is an important parameter in pricing and hedging of that option. It is the rate of change of the option value with respect to the asset price, [7, 23]. In our examples, we compute the Delta of an option using the second order central difference formula: k V k − Vi−1 ∂V (Si , tk ) ≈ i+1 ∂S Si+1 − Si−1 Another important parameter used in hedging strategies is the Gamma of an option. It is the rate of change of Delta with respect to asset price. If Gamma is small, Delta changes slowly,

14

Wade, Khaliq, Yousuf, Vigo-Aguiar

and adjustments to keep a portfolio Delta neutral need to be made relatively infrequently. We compute the Gamma of an option using a second order central difference formula: k k Vi+1 − 2Vik + Vi−1 ∂2V (S , t ) ≈ , i k ∂S 2 h2

(h = Si+1 − Si ).

6.1. A Butterfly Spread A Butterfly Spread is a combination of three options with three strike prices, in which one contract is purchased with two outside strike prices and two contracts are sold at the middle strike price. We will use a call option in this example, solving the following PDE model: ∂V 1 ∂2V ∂V + σ 2 S 2 2 + rS − rV = 0, ∂t 2 ∂S ∂S with payoff at expiry being

0 6 S 6 100, t ∈ [0, 0.5],

(25)

V (S, T ) = max(S − E1 , 0) − 2 max(S − E2 , 0) + max(S − E3 , 0) where E1 , E2 , and E3 are the three strike prices with E1 < E2 < E3 and E2 = (E1 + E3 )/2. This is a homogeneous problem with corners in the initial data, i.e. the payoff function, at E1 , E2 , and E3 ; its Delta has three jump discontinuities. The behavior of Delta at expiry can be summarized with the following conditions:  0,    ∂V 1, = lim− −1,  t→T ∂S   0,

for 0 6 S < E1 ; for E1 6 S < E2 ; for E2 6 S < E3 ; for S > E3 .

(26)

Tables 1 and 2 show the order of convergence for the fourth order undamped scheme (2,2)-Pad´e (# 1) and the fourth order smoothing scheme PSP(2) (# 2). The convergence order is evaluated at the strike price E2 = 0.5. It is clear from Table 1 that the scheme is not showing fourth order of convergence, while Table 2 shows the expected order of convergence. The column ”Order” represents the exponent on k in the ratio of the error sequence, computed in the standard fashion as the ratio of the logarithms of the ratios between the errors and step sizes, respectively. As there is no analytic solution available, the error has been developed relative to a reference solution computed separately on an extremely fine grid with the backward Euler scheme. The reference solution at the strike price E2 = 0.5 is ≈ 0.02102705683555. Figure 1 is the graph of the payoff function for the Butterfly Spread. Time evolution graphs shown in Figures 2 and 3 are obtained using (2,2)-Pad´e and PSP(2), respectively. Figure 2 shows oscillations at the three strike prices E1 , E2 , and E3 since there is irregularity at these points. These oscillations are recovered using our smoothing scheme as shown in Figures 3. Figures 4 and 5 show the time evolution plots of Delta for the Butterfly Spread using (2,2)-Pad´e and PSP(2) respectively. Figure 4 shows large oscillations at the three strike prices E1 , E2 , and E3 due to a jump discontinuity at these points. Again these solutions are accurately recovered using our smoothing scheme as shown in Figure 5. The parameters used for the Figures 2–5 are as follows: σ = 0.5, r = 0.1, ∆t = 0.2, ∆S = 1.0 and three strike prices E1 = 40, E2 = 50 E3 = 60.

Smoothing Schemes for Inhomogeneous Parabolic PDEs

15

Table 1. Convergence results for the Butterfly Spread using the undamped (2,2)-Pad´e scheme. Parameter values are: x0 = 0, xn = 1, T = 0.5, r = 0.1, and σ = 0.5.

∆S 0.0010000 0.0005000 0.0002500 0.0001250 0.0000625

∆t 0.2500 0.1250 0.0625 0.0313 0.0156

Strike Price (E2 ) 0.03111779 0.02612218 0.02356926 0.02229881 0.02166293

Error (E2 ) 1.009e-002 5.095e-003 2.542e-003 1.272e-003 6.359e-004

Order

— 0.98584 1.00304 0.99926 1.00001

Table 2. Convergence results for the Butterfly Spread using the smoothing scheme PSP(2). We have used the same parameters as in Table 1.

∆S 0.0010000 0.0005000 0.0002500 0.0001250 0.0000625

∆t 0.2500 0.1250 0.0625 0.0313 0.0156

Strike Price (E2 ) 0.02122575 0.02104830 0.02102885 0.02102719 0.02102707

Error (E2 ) 1.987e-004 2.124e-005 1.789e-006 1.325e-007 8.707e-009

Order

— 3.22565 3.56937 3.75493 3.92811

10

9

8

7

Payoff

6

5

4

3

2

1

0

0

10

20

30

40

50 Asset Price

60

70

80

90

100

Figure 1. The Payoff function of the Butterfly Spread cf. [23, p. 259].

Wade, Khaliq, Yousuf, Vigo-Aguiar

5 4.5 4

Option Value

3.5 3 2.5 2 1.5 1 0.5 0 1

0.5

0

Time

0

10

20

30

40

50

60

70

80

90

100

Asset Price

Figure 2. The Butterfly Spread using the undamped (2,2)-Pad´e scheme.

5 4.5 4 3.5 Option Value

16

3 2.5 2 1.5 1 0.5 0 1

0.5

0 Time

0

10

20

30

40

50

60

70

80

90

100

Asset Price

Figure 3. The Butterfly Spread using the smoothing scheme PSP(2).

Smoothing Schemes for Inhomogeneous Parabolic PDEs

1 0.8 0.6 0.4

Delta

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 1

0.5

0

Time

0

10

20

30

40

50

60

70

80

90

100

Asset Price

Figure 4. The Delta of the Butterfly Spread using the undamped (2,2)-Pad´e scheme.

1 0.8 0.6 0.4

Delta

0.2 0 −0.2 −0.4 −0.6 −0.8 −1 1

0.5

Time

0 0

10

20

30

40

50

60

70

80

90

100

Asset Price

Figure 5. The Delta of the Butterfly Spread using the smoothing scheme PSP(2).

17

18

Wade, Khaliq, Yousuf, Vigo-Aguiar

6.2. A Digital Call Option The preceding example was of homogeneous type (f = 0); we now move to consider a nonhomogeneous problem in financial mathematics. Here, one boundary condition (31) is time-dependent, which can be equivalently reformulated as a nonhomogeneous problem. In this example we solve the Black-Scholes PDE model ∂C 1 2 2 ∂ 2 C ∂C + σ S + rS − rC = 0, ∂t 2 ∂S 2 ∂S

0 6 S 6 80, t ∈ [0, 0.5],

(27)

for the case when the payoff is not continuous. The payoff function for this call option, also called ‘cash-or-nothing,’ is given as:  A, for S > E; lim− C(S, t) = (28) 0, for S < E. t→T where the constant A > 0 represents the payoff amount at expiry. Discontinuities in the payoff reduce the order of convergence, which is restored by averaging the payoff as:  for S > E;  A, lim− C(S, t) = A/2, for S = E; (29)  t→T 0, for S < E.

When S = 0, the asset remains at zero for all later times and hence the payoff is zero. This gives the boundary condition C(0, t) = 0,

for all 0 6 t 6 T.

(30)

When S is large, the option is almost certain to pay off the amount A. Therefore after discounting for interest, we assume C(S, t) ≈ Ae−r(T −t) ,

S → ∞.

(31)

The analytic solution for this digital option, as given in [7, 23], is:

where

C(S, t) = Ae−r(T −t) N (d2 ),

(32)

log(S/E) + (r − 12 σ 2 )(T − t) √ d2 = σ T −t

(33)

and N (·) is the cumulative distribution function for a standardized normal random variable: 1 N (x) = 2π

Zx

1 2

e− 2 s ds.

(34)

−∞

The Delta of the digital call option, partial derivative of (32) with respect to S, has form: Ae−r(T −t) N 0 (d2 ) ∂C √ = ∂S σS T − t

(35)

Smoothing Schemes for Inhomogeneous Parabolic PDEs

19

Table 3. Convergence results for the Digital Call option using the (2,2)-Pad´e. Parameters are: x 0 = 0, xn = 1, T = 0.5, r = 0.05, and σ = 0.2.

∆S 0.0001 0.0001 0.0001 0.0001 0.0001

∆t 0.5000 0.2500 0.1250 0.0625 0.0313

Strike Price (E2 ) 0.5279693254 0.5301633433 0.5312449746 0.5317849008 0.5320547600

Error (E2 ) 4.355e-003 2.161e-003 1.080e-003 5.399e-004 2.701e-004

Order

— 1.01082 1.00120 1.00002 0.99948

Table 4. Convergence results for the Digital Call option using PSP(2).

∆S 0.0001 0.0001 0.0001 0.0001 0.0001

∆t 0.5000 0.2500 0.1250 0.0625 0.0313

Strike Price (E2 ) 0.5322132399 0.5323137134 0.5323239089 0.5323247500 0.5323248117

Error (E2 ) 1.116e-004 1.110e-005 9.066e-007 6.549e-008 3.796e-009

Order

— 3.32913 3.61423 3.79113 4.10875

and the behavior of Delta at expiry can be summarized as [7]:  0, ∂C  ∞, lim =  t→T − ∂S 0,

for S > E; for S = E; for S < E.

(36)

Tables 3 and 4 show the order of convergence for the fourth order undamped scheme (2,2)Pad´e the smoothing scheme IPSP(2), respectively, at T = 1. It is evident that the ordinary (2,2)-Pad´e scheme is not showing the expected fourth order, while IPSP(2) does. The convergence is computed at the strike price E where the payoff has a jump discontinuity. The analytic solution at the strike price is approximately 0.53232481545376 Figure 6 is the graph of the payoff function for the digital call option. The time evolution graphs of the digital call option, shown in Figures 7 and 8, are obtained using the (2,2)-Pad´e and PSP(2) schemes, respectively. A discontinuity in the payoff function results in spurious oscillation in the (2,2)-Pad´e scheme, which are eliminated by the smoothing scheme PSP(2). Figures 9 and 10 are the time evolution graphs of the Delta function using (2,2)-Pad´e and PSP(2) schemes respectively. The parameter used to obtained these graphs are as follows: σ = 0.3, r = 0.05, ∆t = 0.1, and ∆S = 0.2667. Figure 11 shows the graph of the Gamma function using the (2,2)-Pad´e scheme combined with the smoothing scheme PSP(2). 6.3. European Call Option (Dimensionless Form) Considering the Black-Scholes PDE model (24) with V (0, t) = 0, V (S, t) ∼ S, as S → ∞, and V(S, T) = max(S − E, 0)

Wade, Khaliq, Yousuf, Vigo-Aguiar

1.1 1 0.9 0.8 0.7

Payoff

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1

0

10

20

30

40 Asset Price

50

60

70

80

Figure 6. The Payoff function of the Digital Call option.

1

0.8

Option Value

20

0.6

0.4

0.2

0 0.5 0.4

80 0.3

60 0.2

40 0.1

Time

20 0

0

Asset Price

Figure 7. The Digital Call Option using the undamped scheme (2,2)-Pad´e.

Smoothing Schemes for Inhomogeneous Parabolic PDEs

1

Option Value

0.8

0.6

0.4

0.2

0 0.5 0.4

80 0.3

60 0.2

40 0.1

20 0

Time

0

Asset Price

Figure 8. The Digital Call Option using the smoothing scheme PSP(2).

0.3 0.25

Option Value

0.2 0.15 0.1 0.05 0 −0.05 0.5 0.4

80 0.3

60 0.2

40 0.1

Time

20 0

0

Asset Price

Figure 9. The Delta of the Digital Call Option using the undamped scheme (2,2)-Pad´e.

21

22

Wade, Khaliq, Yousuf, Vigo-Aguiar

0.2

Option Value

0.15

0.1

0.05

0 0.5 0.4

80 0.3

60 0.2

40 0.1

20 0

Time

0

Asset Price

Figure 10. The Delta of the Digital Call Option using the smoothing scheme IPSP(2).

−3

5

x 10

PSP(2) (2,2)−Pade 4

3

2

Gamma

1

0

−1

−2

−3

−4

−5 25

30

35

40 Asset Value

45

50

55

Figure 11. Comparison of the Gamma values for the Digital Call option using the undamped (2,2)-Pad´e scheme and the smoothing scheme IPSP(2). The parameters used for this graph are σ = 0.3, r = 0.05, ∆t = 0.02, and ∆S = 0.5.

Smoothing Schemes for Inhomogeneous Parabolic PDEs

23

1

0.8

u

0.6

0.4

0.2

0 5 4 0.2

3

0 −0.2

2

−0.4

1

−0.6 −0.8

0

t

x

Figure 12. Time evolution graph of a European call option using (2,2)-Pad´e.

and applying the following transformation S = Eex ,

t=T−

τ

1 2, σ 2

C = Eu(x, τ ).

a dimensionless and forward in time form of (24) is obtained, cf. [23]. We will solve the transformed equation using parameter values from [13], which is o the form ∂u ∂2u ∂u = 2 + (c − 1) − cu, t0 6 t 6 tf , a6x6b (37) ∂t ∂x ∂x

where c= 1 rσ2 , r = 0.065, σ = 0.8, a = ln 25 , b = ln 75 , t0 = 0, tf = 5, and with initial 2 condition u(x, 0) = max(exp(x) − 1, 0) and boundary conditions u(a, t) = 0,

u(b, t) =

7 − 5 exp(−kt) . 5

Figures 12 and 13 are the surface plots of European Call Option in dimensionless form using (2,2)-Pad´e and PSP(2), respectively. The parameter values used are σ = 0.8, r = 0.01, T = 5.0, ∆t = 0.25, and ∆S = 0.02. At x = 0 the payoff has a corner and therefore its derivative is discontinuous there. The graph for the (2,2)-Pad´e shows some unwanted oscillations at x = 0 where as the graph of PSP(2) shows no oscillations. The oscillations at x = 0 become worse for the delta function shown in Figure 14. Again, these oscillations are recovered by PSP(2), as shown in Figure 15.

Wade, Khaliq, Yousuf, Vigo-Aguiar

1

0.8

u

0.6

0.4

0.2

0 5 4 0.2

3

0 −0.2

2

−0.4

1

−0.6 0

t

−0.8 x

Figure 13. Time evolution graph of a European call option using PSP(2).

1.4 1.2 1 0.8 du

24

0.6 0.4 0.2 0 5 4 0.2

3

0 −0.2

2

−0.4

1 t

−0.6 0

−0.8 x

Figure 14. Time evolution graph of the Delta of a European call option using (2,2)-Pad´e.

Smoothing Schemes for Inhomogeneous Parabolic PDEs

25

1.4 1.2 1

du

0.8 0.6 0.4 0.2 0 5 4 0.2

3

0 −0.2

2

−0.4

1 t

−0.6 0

−0.8 x

Figure 15. Time evolution graph of Delta of a European call option using PSP(2).

In this experiment we consider a well known problem given in a number of papers, for example Voss and Khaliq [21] and Serbin [19]: ∂u ∂u2 = + tν (νx(1 − x) + 2t), ∂t ∂t2 u(0, t) = 0, u(1, t) = 0, t > 0, u(x, 0) = 0, 0 < x < 1.

(38)

This problem has a nontrivial forcing term, although smooth initial data. We present it at this time to demonstrate that the advantages of the scheme presented in this article are not only for problems with nonsmooth initial data, but also include more efficient computational procedures and parallelizability. In addition, to show flexibility in the companion spatial discretization, we here employ a spatial discretization with the Chebyshev spectral method. Since the initial data is smooth, the (2,2)-Pad´e scheme shows fourth order convergence, yet the IPSP(2) scheme performs about the same and also is more robust to be used with confidence for more difficult PDE problems. Thome´e [20] has also developed a fourth order scheme for inhomogeneous problems with smooth data using a (2,2)-Pad´e approximation of the matrix exponential function. That scheme in its original form, however, requires inverting higher order matrix polynomial functions, which present computational difficulty. Our modified scheme using the partial fraction technique not only does not require inverting higher order matrix polynomials but also performs computationally more efficiently. For fourth order convergence, it is only required to solve simple Backward Euler like problems, albeit with complex arithmetic.

26

Wade, Khaliq, Yousuf, Vigo-Aguiar

Table 5. Convergence results for the Example [19] using the (2,2)-Pad´e scheme. Parameter values are: x0 = 0, xn = 1, T = 1, and ν = 3.

∆x 0.010000 0.005000 0.002500 0.001250 0.000625

∆t 0.05000 0.02500 0.01250 0.00625 0.00313

Err. (2, 2) 2.984e-006 1.840e-007 1.147e-008 7.164e-010 4.478e-011

Ord. (2,2)

Err. IPSP(2)

Ord. IPSP(2)

—

2.932e-006 1.833e-007 1.146e-008 7.162e-010 4.478e-011

— 3.99935 4.00037 3.99964 3.99930

4.01916 4.00379 4.00104 3.99975

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[18] M.F. Reusch, L. Ratzan, N. Pomphrey, and W. Park, Diagonal Pad´e Approximations for Initial Value Problems, SIAM J. Sci. & Stat. Comp. 9 (1988), 829–838. [19] S. Serbin, A Scheme for Parallelizing Certain Algorithms for the Linear Inhomogeneous Heat Equation, SIAM J. Sci. & Stat. Comp. 13 (1992), 449–458. [20] V. Thom´ee, Galerkin Finite Element Methods for Parabolic Problems, Ser. Comp. Math. 25, SpringerVerlag, Berlin, 1997. [21] D. Voss and A.Q.M. Khaliq, Time–Stepping Algorithms for Semidiscretized Linear Parabolic PDEss Based on Rational Approximations with Distinct Real Poles, Adv. Comput. Math. 6 (1996), 353–363. [22] B.A. Wade, A.Q.M. Khaliq, M. siddique, and M.Yousuf, Smoothing with Positivety–Preserving Pad´e Schemes for Parabolic Problems with Nonsmooth data, Numerical Methods for Partial Differential Equations, (2005) 21(3), 553-573. [23] P. Wilmott, S. Howison, and J. Dewynne,The Mathematics of Financial Derivatives Vol. A and B, Cambridge University Press, 1995. [24] Y. Yan, Smoothing Properties and Approximation of Time Derivatives for Parabolic Equations: Constant Time Steps, IMA J. Numer. Anal. 23 (2003), 465–487.