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!!! EidgenossischeTechnische Hochschule

Zurich

Ecole polytechnique federale de ZurichPolitecnico federale di Zurigo

Swiss Federal Institute of Technology Zurich

Variational sensitivity analysis of parametricMarkovian market models

N. Hilber, C. Schwab and C. Winter

Research Report No. 2008-29October 2008

Seminar fur Angewandte MathematikEidgenossische Technische Hochschule

CH-8092 ZurichSwitzerland

Variational sensitivity analysis of parametric Markovian

market models

N. Hilber!, C. Schwab!, C. Winter!

October 8, 2008

Abstract

Parameter sensitivities of prices for derivative contracts play an important role inmodel calibration as well as in quantification of model risk. In this paper a unifiedapproach to the e!cient numerical computation of all sensitivities for Markovianmarket models is presented. Variational approximations of the integro-di"erentialequations corresponding to the infinitesimal generators of the market model di"eren-tiated with respect to the model parameters are employed. Superconvergent approx-imations to second and higher derivatives of prices w.r. to the price process’ statevariables are extracted from approximate, computed prices with low, C0 regularityby postprocessing. The extracted numerical sensitivities are proved to converge withoptimal rates as the mesh width tends to zero. Numerical experiments for uni- andmultivariate models with sparse tensor product discretization confirm the theoreticalresults.

Keywords: Markov process, Greeks, sensitivity, sparse tensor finite elements

1 Introduction

A key task in financial engineering is the fast and accurate calculation of sensitivities ofmarket models with respect to model parameters. This becomes necessary for examplein model calibration, risk analysis and in the pricing and hedging of certain derivativecontracts. Classical examples are variations of option prices with respect to the spotprice or with respect to time-to-maturity, the so-called “Greeks” of the model. For clas-sical, di"usion type models and plain vanilla type contracts, the Greeks can be obtainedanalytically (see [21]). With the trends to more general market models of jump-di"usiontype and to more complicated contracts, closed form solutions are generally not availablefor pricing and calibration. Thus, prices and model sensitivities have to be approximated

!Seminar for Applied Mathematics, ETH Zurich, 8092 Zurich, Switzerland

1

numerically. As model sensitivities are generally derivatives of the computed prices withrespect to model input parameters, a naive approach consists in numerically di"erenti-ating computed prices by, e.g., Finite Di"erence formulas. This results in extra work(e.g. due to multiple “forward” pricing runs) and, due to the generally low regularity ofFinite Di"erence or Finite Element approximations to prices of derivative contracts, insubstantial loss of accuracy in the computed sensitivities.

To obtain stable numerical procedures yielding approximate, numerically computablesensitivities for general Markovian market models and for general contracts which con-verge at the same rate as the computed option prices, additional analytical considerationsare necessary.

Most work in this direction has been devoted to Monte-Carlo methods (see [8, 13] andreferences therein) for di"usion and jump-di"usion models. This paper is focused ona more general class of Markov processes X, including stochastic volatility and multi-dimensional Levy models. A mesh-based approach is used to solve the correspondingpartial integro-di"erential equation (PIDE). A mesh-based approach is also described in[1] where automatic di"erentiation of a Finite Element code is used to approximate theGreeks.

In our approach, we distinguish between two classes of sensitivities. The sensitivity of thesolution u to variation of a model parameter, like the Greek Vega (!!u) and the sensitivityof the solution u to a variation of state spaces such as the Greek Delta (!xu). We showthat an approximation for the first class can be obtained as a solution of the pricingPIDE with a right hand side depending on u. For the second class, a finite di"erencelike di"erentiation procedure is presented which allows to obtain the sensitivities fromthe Finite Element forward price without additional forward solver.

The outline of the paper is as follows. We start by describing the problem setup. Firstwe explain the abstract framework and the variational discretization of the forward Kol-mogorov equation by Finite Element methods. Then, we derive for both classes ofsensitivities an algorithm to compute these by postprocessing the Finite Element solu-tion. It is shown that approximation of the sensitivities converge with the same rate asthe approximation of the option price. Finally, we give numerical examples for di"erentdimensions and models.

2 Variational option pricing

2.1 Parametric Markovian market models

Let (#,F , (Ft)t"0, P) be a filtered probability space satisfying the usual hypotheses. Weconsider the process X to model the dynamics of a single underlying, a basket or aunderlying and its ”background” volatility drivers in case of stochastic volatility models.For notational simplicity only we assume that the interest rate is zero. Let g be the

2

payo", T > 0 the maturity and Q an equivalent martingale measure (EMM) to P, i.e.Q ! P such that the process (Xt)t"0 is a Q-martingale. Since X is Markovian, the fairprice of European style contingent claim with underlying X is given by

u(t, x) = EQ!g(XT ) | Xt = x

".

If the function value u is su!ciently smooth, it is known to solve the backward Kol-mogorov equation

"!tu + Au = 0 in (0, T ) # Rd (2.1)

u(T, x) = g(x) in Rd, (2.2)

where A denotes the infinitesimal generator of X. We consider processes X where Asplits into the di"usive part AW , the drift part A" and the jump part AJ are given by

A = AW + A" + AJ , where

AW ["](x) = "1

2Q(x)D2"(x)

A"["](x) = $b(x),D"(x)% (2.3)

AJ ["](x) = "#

E

$"(x + #(x, z)) " u(x) " $#(x, z),D"(x)%1|z|#1

%$(dz).

Here, Q : Rd & Rd$d, b : Rd & Rd and # : Rd#E & Rd with the set of admissible jumpsE ' Rd \{0}. Furthermore, D and D2 are the di"erential operators D = (!xi)1#i#d, andD2 = (!xixj)1#i,j#d and $ denotes the compensator of a Poisson random measure on Esatisfying

&E min{1, |z|2}$(dz) < (.

Definition 2.1. We call a process X a parametric Markovian market model with ad-missible parameter set S#, if

(i) for all % ) S# X is a strong Markov process w.r. to a stochastic basis(#,F , (Ft)t"0, P)

(ii) the infinitesimal generator A of the semigroup generated by X has the form (2.3),and the mapping S# * % & {Q, b, $, #} is infinitely di"erentiable.

In (2.3), we assume that the coe!cients Q, b, # and $ do not depend on time t. Re-cently, Carr et al. in [10] considered Sato processes (self-similar additive processes) asdrivers for the underlying X. The authors introduce R-valued pure jump processes withLevy measure $ which has time-inhomogeneous Levy density k(z, t). Our approach tocompute prices and sensitivities is not restricted to time independent coe!cients butnaturally extends to the case when the coe!cients are time-inhomogeneous. We givesome examples of Markov processes X and their infinitesimal generators covered by ourapproach.

3

Example 2.2 (Multidimensional Levy model [16, 22]). The Markov process isgiven by the d dimensional Levy process X = (X1, . . . ,Xd) with characteristic triplet(Q, $Q, &) under the EMM Q. We assume that the Levy measure satisfies

&|z|>1 ezi$Q(dz) <

(, i = 1, . . . , d. Then, the coe!cients are given by

Q(x) = (Qij)1#i,j#d , b(x) = (&i)1#i#d , #(x, z) = z , (2.4)

with &i = 12Qii +

&Rd

$ezi " 1 " zi1|z|#1

%$(dz). The dependence structure of the Brow-

nian motion part of X is characterized entirely by its covariance matrix Q. The de-pendence structure of the purely discontinuous part of X can be described using Levycopulas. These were introduced in Tankov [22] and developed in Kallsen and Tankov [16].Analytic properties and wavelet discretization of the copula process’ generator were dis-cussed by Farkas et al. [11] . For the Clayton Levy copula with tempered stable marginswith CGMY [9] parameters the multidimensional Levy density is given by

k(x1, . . . , xd) = !1 . . . !dF |$1=U1(x1),...,$d=Ud(xd)k1(x1) . . . kd(xd) ,

with marginal Levy densities

ki(z) = Ci

'eGiz

|z|1+Yi1{z<0} +

e%Miz

|z|1+Yi1{z>0}

(

, Ci, Gi > 0,Mi > 1, Yi < 2 , (2.5)

marginal tail integrals

Ui(x) = Ci MYii $("Yi,Mix) 1{x>0} " Ci G

Yii $("Yi,"Gix) 1{x<0} ,

and Levy copula

F (x1, . . . , xd) = 22%d

'd)

i=1

|xi|%%

(% 1! $

%1{x1···xd"0} " (1 " %)1{x1···xd#0}

%,

where i = 1, . . . , d, ' > 0 and % ) [0, 1]. The Clayton copula density blends for xi + 0,i = 1, . . . , d the complete independence density (' = 0) and the complete dependencedensity (' & ().

Example 2.3 (Stochastic volatility model of Heston [14]). The Markov processX is of the form X = (S, Y ), where the R-valued process S describes the dynamics ofthe underlying and the R-valued process Y its volatility. Under a EMM Q, X satisfiesthe stochastic di"erential equation dXt = b(Xt)dt+%(Xt)dWt where (Wt) denotes a twodimensional Brownian motion and the coe!cients b,% are (see [14])

b =

*0

((m " Yt) " )(t, St, Yt)

+, % =

* ,YtSt 0

*+,

Yt *,

1 " +2,

Yt

+,

with ( > 0 the rate of mean reversion, m > 0 the long-run mean level of volatility, * ) R

and + ) ["1, 1] the instantaneous correlation. The function ) : [0, T ] # R+ # R+ & R

4

appearing in the second component of the drift b represents the price of volatility andreflects the incompleteness of this market model. The infinitesimal generator A of X isas in (2.3) where for x := (S, y) := (x1, x2) ) R+ # R+

b(x) =

*0

((m " x2) " )(t, x1, x2)

+, Q(x) = %%& =

*x2

1x2 *+x1x2

*+x1x2 *2x2

+, $ = 0.

Example 2.4 (Stochastic volatility model of BNS [3]). This stochastic volatilitymodel specifies the volatility (of the underlying) as a Ornstein-Uhlenbeck process drivenby a Levy subordinator Lt. The Markov process X = (S, Y ) = (eZ ,,2) under an EMMQ satisfies the SDE (see [3] and [19] )

dZt =$" )- "

1

2,2

t

%dt + ,tdWt + + dL&t

d,2t = "),2

t dt + dL&t,

where (Wt) is a Q-Brownian motion and (L&t) is a Q-Levy process. The parameterssatisfy *, µ, +,) ) R, ) > 0, + - 0 and cumulant transform - is -(+) =

&R+

(e'z "1)w(z)k(z)dz. Here, w : R+ & R+ satisfies

&R+

(,

w(z) " 1)2k(z)dz < (, and k isdensity of the Levy measure of Lt under the historical measure P. The infinitesimalgenerator A of X has the form as in (2.3), where for x := (x1, x2) ) R#R+ and z ) R+

the coe!cients are given by (with c :=&z#1 z$(dz))

b(x) =

*")- " +c " 1

2x2

")x2 " c

+, Q(x) =

*x2 00 0

+,

#(x, z) =

*+1

+z, $(dz) = )w(z)k(z)dz, E = R+ .

Note that the term $#(x, z),D"(x)%1|z|#1 appearing in AJ can be omitted here, since Lt

is a subordinator and hence has sample paths of finite variation.

We calculate the sensitivities of the solution u of (2.1)–(2.2) w.r. to parameters in theinfinitesimal generator A and w.r. to solution arguments x and t. We write A(%0) for afixed parameter %0 ) S# to emphasize the dependence of A on %0 and change the timeto time-to-maturity t & T " t in (2.1)–(2.2). For sensitivity computation (as well as fordomain truncation, cf. [18]), it will be crucial below to admit a non-trivial right handside. Accordingly, we consider from now on the forward parabolic problem

!tu + A(%0)u = f in (0, T ] # Rd (2.6)

u(0, x) = u0 in Rd, (2.7)

with u0 = g. For the numerical implementation we truncate the parabolic PIDE (2.6)–(2.7) to a bounded domain D ' Rd and impose boundary conditions on !D. Typically,D is d-dimensional hypercube, i.e. D =

-dk=1(ak, bk) for some ak, bk ) R, bk > ak,

k = 1, . . . , d. We approximate the solution to (2.6)–(2.7) by the Finite Element method,which is based on the variational formulation of (2.6)–(2.7).

5

2.2 Variational setting

With a parametric Markovian market model X in the sense of Definition 2.1 with pa-rameter set S# and infinitesimal generator A(%0) as in (2.3), %0 ) S#, we associate toA(%0) the Dirichlet form a(%0; ·, ·) : V # V & R via

a(%0;u, v) := $A(%0)u, v%V !$V , u, v ) V,

with domain Vd

.& H (dense embedding). We identify H with its dual H! and denote by

V ! the dual of V so that Vd

.& H != H! d.& V !. We denote by . · ., . · .V the norms in

H,V , by (·, ·) the inner product in H and by $·, ·%V !$V the duality pairing between V andits dual V !. L(V,W ) is the vector space of linear and continuous operators A : V & W .

We assume A(%0) ) L(V, V !) to be an elliptic, spatial operator given in weak form wherethe Dirichlet form a(·; ·, ·) : S# # V # V & R is continuous and satisfies a Gardinginequality: there exist non-negative constants ((%0),*(%0), &(%0) such that

|a(%0;u, v)| - ((%0).u.V .v.V , /u, v ) V, % ) S#, (2.8)

a(%0; v, v) + *(%0).v.2V " &(%0).v.2

H , /v ) V, %0 ) S#. (2.9)

Remark 2.5. (i) In general, the space V may depend on the parameter %0 and weshould write V#0

. For notational simplicity, we drop the subscript %0.

(ii) We can assume without loss of generality that &(%0) = 0 in (2.9) since by theexponential shift w := e%((#0))u we obtain !tw + A(%0)w + &(%0)u = e%((#0)tf andthe operator A(%0) + &(%0)I is coercive on V .

For f ) L2(0, T ;V !) and u0 ) H the weak formulation to the problem (2.6)–(2.7) isgiven by:

Find u ) L2(0, T ;V ) 0 H1(0, T ;V !) such that

(!tu(t, ·), v) + a$%0;u(t, ·), v

%= $f(t), v%V !$V , /v ) V , (2.10)

u(0, ·) = u0 .

Under the assumption (2.8)–(2.9) the operator A(%0) + &(%0)I ) L(V, V !) defines anisomorphism and (2.10) admits an unique solution.

We assume that V is Sobolev-type space with smoothness index r, i.e.

V = .Hr, .H0 = H = L2. (2.11)

Note that r depends on the order of the operator A(%0). We also assume that the solutionu(%0) to (2.10) has higher regularity in space, u(%0)(t) ) Hs ' .Hr for t ) (0, T ], whereHs is again a Sobolev-type space with smoothness index s.

Example 2.6. Consider the multivariate Levy copula model from Example 2.2.

6

(i) It can be shown similar to [18] that V = .Hr(D) with r = 1, if Q > 0. Here, forr + 0, the space .Hr(D) is given by .Hr(D) = {u|D | u ) Hr(R), u|R\D = 0}.

(ii) Now let Qij = bi = 0, 1 - i, j - d, and marginal Levy densities ki, 1 - i - d,as in (2.5). In [11] it was proved that for multivariate barrier contracts V isthe anisotropic Sobolev space V = .Hr(D), with r = (Y1/2, . . . , Yd/2). Here, forr = (r1, . . . , rd), ri + 0, i = 1, . . . , d, we denote the space .Hr(D) by .Hr(D) ={u | u ) C'

0 (D)} where u is the zero extension of u to Rd and the closure is taken

w.r. to the norm given by .u.2Hr(Rd)

=/d

j=1 .u.2H

rjj (Rd)

.

2.3 Variational discretization

Let Vh be a finite dimensional subspace Vh ' V consisting of continuous piecewise poly-nomials of degree p + 1 with dim Vh = N < (. The FE semi-discretization in (log) pricespace of (2.10) reads:

Find uh ) L2(J ;Vh) 0 H1(J ; (Vh)!) such that

(!tuh(t, ·), vh) + a(%0;uh(t, ·), vh) = $f(t), vh%V !$V / vh ) Vh , (2.12)

uh(0, ·) = u0,h , (2.13)

where u0,h is an approximation of u0 in Vh. The FE formulation (2.12)–(2.13) is equiva-lent to a large, but finite system of ODEs to be solved numerically on the time interval(0, T ). To this end, we fix a basis B := {&j}N

j=1 of Vh and let u denote the coe!cientvector of uh with respect to the basis B. Then, (2.12)–(2.13) is equivalent to:

Find u(t) ) RN such thatMu + Au = f(t) ,

where M and A are the so-called mass and sti"ness matrix given by

M =

*(&i,&j)

+

1#i,j#N

, A =

*a(%0;&j ,&i)

+

1#i,j#N

, (2.14)

as well as f(t) = $f(t),&j%1#j#N .

For the convergence analysis of the Finite-Element based pricing algorithms, we assumethe following approximation property of the space Vh: For all u ) Hs with r - s - p + 1there exists a uh ) Vh such that for 0 - / - r (with r as in (2.11))

.u " uh. eH" - Chs%).u.Hs (2.15)

We further assume the existence of a projector or an interpolant Ph : V & Vh whichsatisfies (2.15) with uh = Phu.

We give examples for the space Vh. In dimension d = 1, we consider Vh to be the waveletFinite Element space on a uniform mesh with mesh width h on D as proposed e.g in

7

[18]. In this setting, the projector Ph is defined by truncating the wavelet expansion ofu ) V . For problems in dimension d + 2, consider the sparse tensor space 0Vh as definede.g. in [26].

To discretize in time, we use the '-scheme. For M ) N define the time step 't = TM

and tm = m't, m = 0, . . . ,M . The fully discrete scheme reads: Find um+1h ) Vh,

m = 0, 1, . . . ,M " 1 such that$'t%1(um+1

h " umh ), v

%+ a(%0;u

m+%h , vh) = (fm+%, v), /v ) Vh , (2.16)

with u0h = u0,h. Here um+%

h := 'um+1h + (1" ')um

h and fm+% := 'f(tm+1) + (1" ')f(tm).In matrix form, (2.16) reads

$'t%1M + 'A

%um+1 =

$'t%1M" (1 " ')A

%um + fm+%, m = 0, 1, . . . ,M " 1,

where um is the coe!cient vector of umh with respect to the basis B of Vh.

3 Sensitivity analysis

For a parametric Markovian market model X in the sense of Definition 2.1 we distinguishtwo classes of sensitivities.

1. The sensitivity of the solution u to a variation S# * %s := %0 + s0%, s > 0, ofan input parameter %0 ) S#. Typical examples are the Greeks Vega (!!u), Rho(!ru) and Vomma (!!!u). Other sensitivities which are not so commonly used inthe financial community are the sensitivity of the price w.r. to the jump intensityor the order of the process that models the underlying. We show that the FiniteElement approximation to such sensitivities satisfies again the scheme (2.16) witha right hand side fm+% which depends on the approximation um+%

h of the pricingfunction u. We also show that the approximation of these sensitivities convergewith the same rate as uh.

2. The sensitivity of the solution u to a variation of arguments t, x. Typical examplesare the Greeks Theta (!)u), Delta (!xu) and Gamma (!xxu). Higher derivativeslike !xxxu are used in [12, Chapter 5] to approximate prices of European optionsunder stochastic volatility models. We show that these sensitivities can directly beobtained by postprocessing the Finite Element solution uh (2.12)–(2.13) withoutadditional runs. Again our numerical approximations of these sensitivities convergewith the same rate as uh.

3.1 Sensitivity w.r. to model parameters

Let C be a Banach space over a domain D ' Rd. C is the space of parameters orcoe!cients in the operator A and S# 1 C is the set of admissible coe!cients. We denote

8

by u(%0) the unique solution to (2.10) and introduce the derivative of u(%0) w.r. to%0 ) S# as the mapping D#0

u(%0) : C & V

.u(0%) := D#0u(%0)(0%) := lim

s(0+

1

s

$u(%0 + s0%) " u(%0)

%, 0% ) C.

We also introduce the derivative of A(%0) w.r. to %0 ) S#

.A(0%)" := D#0A(%0)(0%)" := lim

s(0+

1

s

$A(%0 + s0%)" "A(%0)"

%, " ) V, 0% ) C.

We assume that .A(0%) ) L(.V , .V !) with .V a real and separable Hilbert space satisfying

.V 1 Vd

.& H != H! d.& V ! 1 .V !.

We further assume that there exists a real and separable Hilbert space V 1 .V such that.Av ) V !, /v ) V .

We have the following relation between D#0u(%0)(0%) and u.

Lemma 3.1. Let .A(0%) ) L(.V , .V !), /0% ) C and u(%0) : (0, T ] & V , %0 ) S# be theunique solution to

!tu(%0) + A(%0)u(%0) = 0 in (0, T ) # Rd (3.1)

u(%0)(0, ·) = g(x) in Rd. (3.2)

Then .u(0%) solves

!t.u(0%) + A(%0).u(0%) = " .A(0%)u(%0) in (0, T ) # Rd (3.3)

.u(0%)(0, ·) = 0 in Rd (3.4)

Proof. Since u(%0)(0) = g does not depend on %0 its derivative w.r. to % is 0. Now let%s := %0 + s0%, s > 0, 0% ) C. Subtract from the equation !tu(%s)(t) +A(%s)u(%s)(t) = 0equation (3.1) and divide by s to obtain

!t1

s

$u(%s)(t) " u(%0)(t)

%+

1

s

$A(%s)"A(%0)

%u(%s)(t) +

1

sA(%0)

$u(%s)(t) " u(%0)(t)

%= 0.

Taking lims(0+ gives equation (3.3).

We associate to the operator " .A(0%) the Dirichlet form .a(0%; ·, ·) : .V # .V & R which isgiven by

".a(0%;u, v) = "$ .A(0%)u, v

%.

The variational formulation to (3.3)-(3.4) reads: Find .u(0%) ) L2(0, T ;V )0H1(0, T ;V !)such that .u(0%)(0, ·) = 0 in H and such that

(!t.u(0%)(t, ·), v) + a$%0; .u(0%)(t, ·), v

%= ".a

$0%;u(%0)(t, ·), v

%, /v ) V. (3.5)

Note that (3.5) has an unique solution .u(0%) ) V due to the assumptions on a(%0; ·, ·),.A and u(%0) ) V .

9

Example 3.2 (CGMY model). We consider an one-dimensional tempered stable Levyprocess X with CGMY density k as in (2.5). According to (2.3)–(2.4), its infinitesimalgenerator A has the form

A["] = "1

2,2!xx" +

*1

2,2 + c

+!x" "

#

R

1"(x + z) " "(x) " z1{|z|#1}!x"(x)

2k(z)dz

where the constant c depends only on k via c :=&

R

$ez " 1 " z1{|z|#1}

%k(z)dz to ensure

that eX is a martingale. The weak formulation for the price of European style contingentclaim is as in (2.10) with f = 0.

For the sensitivity of the price w.r. to the volatility , the set of admissible parametersS# is S# = R+ with % = ,. We have

.A(0,)" = "0,,0!xx" + 0,,0!x" ) L(V, V !),

with 0, ) R = C. The Dirichlet form .a(0,; ·, ·) appearing in the weak formulation (3.5)of .u(0,) is given by

.a(0,;",1) = 0,,0(!x", !x1) + 0,,0(!x",1).

For the sensitivity of the price w.r. to the jump intensity parameter Y of the Levyprocess X we let 0 < Y < 2. Then, we have S# = (0, 2) with % = Y and

.A(0Y )" = "0Y

#

R

1"(x + z) " "(x) " z!x"(x)

2.k(z)dz ) L(.V , .V !)

where the kernel .k is given by

.k(z) := " ln |z|k(z).

It is easy to check that due to Y < 2 in (2.5)&|z|#1 z2.k(z)dz < (,

&|z|>1

.k(z)dz < (. In

this setting, .V = V = .H1(D), if , > 0, and .V = .HY/2+*(D) ' .HY/2(D) = V , /2 > 0, if, = 0 and if the drift has been removed by a change of variables as in [18].

The fully discrete scheme to find an approximation to .u(0%) in (3.5) is:

Given .u0h = 0, for m = 0, 1, . . . ,M " 1 find .um+1

h ) Vh such that$'t%1(.um+1

h " .umh ), v

%+ a(%0; .um+%

h , vh) = ".a$0%;um+%

h , v%, /v ) Vh (3.6)

or in matrix form$'t%1M + 'A

%.um+1 =

$'t%1M" (1 " ')A

%.um " .A

$'um+1 + (1 " ')um

%,

where .A is matrix of the Dirichlet form .a(0%; ·, ·) in the basis B,

.A =

*.a(0%;&j ,&i)

+

1#i,j#N

. (3.7)

The resulting algorithm is illustrated as pseudo code in Table 1. Here, we denote byy 2 solve(B, x) the output of a generic (exact or approximate) solver for a linear systemBx = y.

10

Choose %0 ) S#, 0% ) C.

Calculate the matrices M, A and .A according to (2.14) and (3.7).Let u0 be the coe!cient vector of u0

h in the basis B of Vh.Set .u0 = 0.For j = 0, 1, . . . ,M " 1

u1 2 solve$'t%1M + 'A, ('t%1M " (1 " ')A)u0

%

Set f := .A('u1 + (1 " ')u0).u1 2 solve

$'t%1M + 'A, ('t%1M " (1 " ')A).u0 " f)

%

Set u0 := u1, .u0 := .u1

Next j

Table 1: Algorithm to compute sensitivities w.r. to model parameters

3.2 Convergence rates for sensitivities w.r. to model parameters

In this section we establish convergence rates for the sequence {.um}M%1m=0 of sensitivities

w.r. to model parameters as the discretization parameter h in (2.12)–(2.13) tends tozero. We show that the computed sensitivities converge essentially at the same rate asthe computed prices. For notational simplicity the subscript %0 is omitted. We definethe energy norm

.u.a :=,

a(u, u) ! .u.V

which is, by (2.8) and (2.9), equivalent to the norm . · .V . For f ) V !h , we let

.f.! := sup0)=vh*Vh

(f, vh)

.vh.a.

The main result of this section is the following Theorem. The proof is given in theAppendix A.1

Theorem 3.3. Let the assumptions of Lemma A.5 and Lemma A.1 be fulfilled. Thenthere holds

..uM " .uMh .2 + 't

M%1)

m=0

..um+% " .um+%h .2

V

- C)

v*{u,eu}

3('t)2

& T0 .v(/).2

!d/ ' ) [0, 1]

('t)4& T0 ....v (/).2

!d/ ' = 12

+ Ch2(s%r))

v*{u,eu}

# T

0.v(/).2

Hs"r d/

+ Ch2(s%r) max0#t#T

.u(t).2Hs .

Theorem 3.3 shows that if the error between the exact and the approximate price satisfies.um"um

h . = O(hs%r)+O(('t)+), the error between the exact and approximate sensitivitypreserves the same convergence rates both in space and time, i.e. ..um".um

h . = O(hs%r)+O(('t)+).

11

3.3 Sensitivity w.r. to solution arguments

Let u be the solution of the variational problem (2.10). We discuss the computation of

D,u = -|#|

-#1x1

···-#dxd

u for arbitrary multi index ( ) Nd0. The approximation of derivatives

of solutions to elliptic and parabolic partial di"erential equations in the context of Fi-nite Elements is well studied [2, 6, 5, 7, 20, 23, 24]. In these papers, derivatives areapproximated by applying di"erence operators or local averaging operators to the FiniteElement solution of the problem. We follow this approach. For µ ) Zd a multi-integerand h ) R+ we define the translation operator T µ

h "(x) = "(x + µh) and the forwarddi"erence quotient !h,j"(x) = h%1(T

ej

h "(x) " "(x)), where ej , j = 1, . . . , d, denotes thej-th standard basis vector in Rd. For ( ) Nd

0 we denote by !,h " = !,1

h,1 · · · !,dh,d" and by

D,h the di"erence operator of order n + 0

D,h" :=

)

(,|,|=n

C(,,T (h !,

h ".

Definition 3.4. The di"erence operator D,h of order |(| = n and mesh width h is called

an approximation to the derivative D, of order s ) N0 if for any D0 ' D there holds

.D," " D,h". eHr(D0) - Chs.".Hs+r+n(D),/" ) Hs+r+n(D). (3.8)

Given a basis B := {&j}Nj=1 of Vh, the action of D,

h to vh ) Vh can be realized asmatrix-vector multiplication vh 3& D,

hvh, where

D,h =

$D,

h&1, · · · ,D,h&N

%) RN$N (3.9)

and vh is the coe!cient vector of vh w.r. to basis B, respectively.

Example 3.5. Let Vh = span{"j(x) | 1 - j - N}, "j(x) = max{0, 1 " h%1|x " jh)|},h = 1

N+1 , be the space of piecewise linear continuous functions on [0, 1] vanishing at theend points 0, 1. For (,*, & ) R and µ ) N0 we denote by diag±µ((,*, &) the matrices

diag%µ((,*, &) =

4

56· · · 0 ( * & 0 · · ·

( * &. . .

. . .. . .

7

89

4µ + 1 " th column

and diagµ((,*, &) = (diag%µ((,*, &))&. Then the matrices Qh of the forward di"erencequotient !h and Tµ of the translation operator T µ

h respectively are given by

Qh = h%1diag0(0,"1, 1), Tµ = diagµ(0, 1, 0).

Hence, for example, we have for the centered finite di"erence quotient

D2h"(x) = h%2("(x + h) " 2"(x) + "(x " h))

12

Choose %0 ) S#.Calculate the matrices M, A and D,

h according to (2.14) and (3.9).Let u0 be the coe!cient vector of u0

h in the basis B of Vh.For j = 0, 1, . . . ,M " 1

u1 2 solve$'t%1M + 'A, ('t%1M" (1 " ')A)u0

%

Set u0 := u1

Next jSet v := D,

hu1

Table 2: Algorithm to compute w.r. to arguments of solution

of order 2 in one dimension D2h = T%1Q2

h = h%2diag0(1,"2, 1).

Now let Vh = Vh 5 · · · 5 Vh be the d-fold tensor product of Vh. Then the matrix D,h is

given by

D,h =

)

(,|,|=n

C(,,T(15 · · · 5T(d

Q,1

h 5 · · ·5 Q,dh .

In Table 2 the algorithm how to obtain an approximation to the derivative D,u(T, x)at maturity T is illustrated. The vector v ) RN is the coe!cient vector of D,

huMh in the

basis B of Vh.

3.4 Convergence rates for sensitivities w.r. to solution arguments

For simplicity, we shall assume in this section that the function # : Rd#E & Rd appearingin (2.3) depends only on z, i.e. # : E & Rd.

We have the following convergence result for the approximation of sensitivities w.r. tosolution arguments. Its proof can be found in the Appendix A.2.

Theorem 3.6. Let the assumptions of Lemma A.1 be fulfilled and assume that u(x, t) is

su!ciently smooth in [0, T ] # D. Assume that the approximation !.hu0

h is quasi-optimalin L2(D) for all * - (. Assume further that D,

h approximates D, in the sense ofDefinition 3.4. Then there holds

.D,uM " D,huM

h .2 + 'tM%1)

m=0

.D,um+% " D,hum+%

h .2V

- C

3('t)2

& T0 .u(/).2

!d/ ' ) [0, 1]

('t)4& T0 ....u (/).2

!d/ ' = 12

+ Ch2(s%r)# T

0.u(/).2

Hs"rd/

+ Ch2(s%r) max0#t#T

.u(t).2Hs .

Remark 3.7. (i) Note that we can not get higher convergence rates than s " r, evenif u has higher regularity (u(t) ) Hs+r+n).

13

(ii) Theorem 3.6 shows that arbitrary derivatives of u can be approximated with thesame rate as u itself, provided u is su!ciently smooth.

4 Numerical examples

In this section we compute various sensitivities for di"erent models. We mainly choosemodels where the price is known in closed form such that we are able to compute theerrors between the exact price/sensitivities and their Finite Element approximations. InTheorems 3.3 and 3.6 these discretization errors are estimated in the energy norm. In thenumerical examples, however, we measure the errors in the L' norm. This choice stemsfrom the fact that the above mentioned closed form solutions are not given explicitly asfunctions of x and t, which turns the computation of .u(T, x) " uM

h .V very expensiveand only approximative.

We measure the L' norm of the error on a subset D0 of the computational domain D attime to maturity t = T . In all computations, we choose wavelet Finite Element spacesspanned by continuous wavelets of polynomial degree p = 1. For problems in dimensiond + 2, we choose the sparse grid spaces 0Vh to reduce the computational complexity of theapproximations. In the '-scheme, we let ' = 1

2 and choose the time step 't su!cientlysmall.

The experimental convergence rates are obtained by the least square method applied tothe data (log h, log eh), where eh := .u(T, x) " uM

h .L#(D0).

4.1 One-dimensional models

We consider Example 2.2 with d = 1 for two models: (i) the Black-Scholes model [4] and(ii) Variance Gamma model [17] with parameters (,, $,3), i.e one has in (2.5) Y = 0,C = $%1, M = ($µ+)%1 and G = ($µ+)%1 where µ+ = 1

2

,32 + 2,2$%1+ /

2 , µ% = µ+"3.For both models, we consider a European put with strike K = 1 and maturity T = 0.1,and we calculate the Greeks Delta, ' = Du = -u

-S , and Gamma, $ = D2u = -2u-S2 . For

the Black-Scholes model we additionally compute the Vega, V = .u(0,) = -u-! . We choose

for both models the parameter , = 0.4 as well as $ = 0.04, 3 = "0.2 for (ii). Using theanalytic solution for the Black-Scholes model [4] and the Variance Gamma model [17],we can compute the error of the FEM solution. The convergence rates in .·.L#([0.5,2])are shown in Figure 1.

As predicted in Theorem 3.3 and 3.6, all Greeks convergence with the same rate as theprice u itself. Here, since the degree of polynomials in Vh is p = 1, the rate is s = p+1 = 2,even when the error is measured in the L'-norm.

As a further illustration of the method, we compute the sensitivity w.r. to the order Y.u(0Y ) = -u

-Y as explained in Example 3.2. We consider a European call with strike K = 1

14

10!3

10!2

10!1

10!6

10!5

10!4

10!3

10!2

10!1

100

Mesh Width

Err

or

p = 2.0

Value

Delta

Gamma

Vega

10!3

10!2

10!1

10!6

10!5

10!4

10!3

10!2

10!1

100

Mesh Width

Err

or

p = 2.0

Value

Delta

Gamma

Figure 1: Convergence rates of Greeks for a European put in the Black-Scholes (left) andVariance Gamma (right) model.

and maturity T = 0.5 and choose the model parameters C = 1, G = 12, M = 10 and%0 = Y = 1. The functions u(T, S) and .u(0Y )(T, S) for 0Y = 1 are shown in Figure 2.

0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Stock

Value

Payoff

Sensitivity wrt Y

Figure 2: Computed sensitivity of a European call w.r. to the jump intensity parameterY in the CGMY model.

4.2 Multi-variate models

We consider the Heston stochastic volatility model and a three dimensional basket option.

4.2.1 Heston model

We calculate the sensitivities .u(0+) and .u(0() w.r. to correlation + of the Brownianmotions that drive the underlying and the volatility and the rate of mean reversion ((see Example 2.3). To this end, we consider a European call with strike K = 1 and

15

maturity T = 12 . The model parameters for both sensitivity runs are ) = 0, , = 0.5

and m = 0.06. Additionally, for the sensitivity w.r. to + we let +0 = "0.5, 0+ = 1 and( = 2.5. For the sensitivity w.r. to ( we set (0 = 2.5, 0( = 1 and + = "0.5. Wecompare the FEM solution with the closed form solution given in [14]. The convergencerates in .·.L#([e"0.25,e0.75]$[0.24,1.2]) are shown in Figure 3.

10!4

10!3

10!2

10!1

10!5

10!4

10!3

10!2

10!1

s = 1.36

Mesh Width

Err

or

Value

Sensitivity wrt rho

Sensitvity wrt alpha

Figure 3: Convergence rates of sensitivities .u(0+), .u(0() for a European call in the Hestonstochastic volatility model.

The experimental convergence rate s is s 6 1.36 for u and .u(0+) and s 6 1.43 for.u(0(). This confirms the theoretical finding of Theorem 3.3 that computed prices andsensitivities convergence with the same rate. In Figure 4 the sensitivities w.r. to + and( are shown.

0.51

1.52 0.2

0.4

0.6

0.8

1

1.2

!0.015

!0.01

!0.005

0

0.005

0.01

0.015

y

S

u

0.51

1.52

0.20.4

0.60.8

11.2

!0.025

!0.02

!0.015

!0.01

!0.005

0

0.005

Sy

u

Figure 4: Computed sensitivities for a European call w.r. to model parameters + and (:.u(0+) (left) and .u(0() (right) in the Heston stochastic volatility model.

16

4.2.2 Basket option

We again need an analytic solution to compare our FEM solution with. Therefore, we

choose g(S) =:-d

i=1 Si " K;+

where for the multidimensional Black-Scholes model an-

alytic solutions can be found by reducing the problem to an one-dimensional problem [15,Section 7.5]. We consider the dimension d = 3, strike K = 1 and maturity T = 0.1, and

we calculate the Greeks Delta, '1 = -u-S1

, and Gamma, $11 = -2u-S2

1

. The parameters are

, = 0.4 and + = 0. The convergence rates in .·.L#([0.5,2]3) are shown in Figure 5.

10!2

10!1

10!2

10!1

100

101

102

Mesh Width

Err

or

p = 1.8

Value

Delta1

Gamma11

Figure 5: Convergence rates of Greeks of a basket option in a three dimensional Black-Scholes model.

The experimental convergence rate s is s 6 1.82 for '1 and s 6 1.56 for $11.

A Appendix

A.1 Proof of Theorem 3.3

We first recall the following stability result for '-scheme from [25, Proposition 4.1]. Wedenote by ) the constant

) := sup0)=vh*Vh

.vh.2

.vh.2!.

Lemma A.1. If 12 - ' - 1, assume that 0 < C1 < 2, C2 + (2 " C1)%1. If 0 - ' < 1

2assume that

, := 't(1 " 2')) < 2, 0 < C1 < 2 " ,, C2 +1 + (4 " C1),

2 " , " C1.

Then the sequence {umh }M

m=0 generated by the '-scheme (2.16) satisfies the stability esti-

17

mate

.uMh .2 + C1't

M%1)

m=0

.um+%h .2

a - .u0h.

2 + C2'tM%1)

m=0

.fm+%.2!. (A.1)

We will also need the following convergence result proved in [25, Theorem 5.4].

Lemma A.2. Let the assumptions of Lemma A.1 be fulfilled. Assume that u(x, t) issu!ciently smooth in [0, T ]#D and assume that the approximation property (2.15) holds.Assume further that the approximation u0,h ) Vh of u0 is quasi-optimal in L2(D). Thenthe sequence {um

h }M%1m=0 in (2.16) satisfies

.uM " uMh .2 + 't

M%1)

m=0

.um+% " um+%h .2

V

- C

3('t)2

& T0 .u(/).2

!d/ ' ) [0, 1]

('t)4& T0 ....u (/).2

!d/ ' = 12

+ Ch2(s%r) max0#t#T

.u(t).2Hs

+ Ch2(s%r)# T

0.u(/).2

Hs"rd/ .

We now estimate the error emh := .um " .um

h , where we set .um := .u(tm). We write

emh = .um " Ph.um

< => ?#m

+ Ph.um " .umh< => ?

$mh

= %m + 4mh .

Since %m can be estimated by the approximation property (2.15), we now focus on 4mh .

Lemma A.3. Assume .u ) C1([0, T ];H). Then {4mh }M%1

m=0 satisfy the '-scheme: Given40h = 0, for m = 0, 1, . . . ,M " 1

('t)%1(4m+1h " 4m

h , vh) + a(4m+%h , vh) = (rm, vh), /vh ) Vh, (A.2)

with weak residual rm : Vh & R given by rm =/4

j=1 rmj where

(rm1 , vh) :=

$('t)%1(.um+1 " .um) " .u

m+%, vh%,

(rm2 , vh) :=

$('t)%1(Ph.um+1 " Ph.um) " ('t)%1(.um+1 " .um), vh

%,

(rm3 , vh) := a

$Ph.um+% " .um+%, vh

%,

(rm4 , vh) := .a(0%;um+% " um+%

h , vh).

Proof. We proceed as in the proof of Lemma 5.1 in [25]. We first recall that .u )C1([0, T ];H) implies

(.um+%

, v) + a(.um+%, v) = ".a(0%;um+% , v), /v ) V. (A.3)

18

Inserting the definition of 4mh into the left hand side of equation (A.2) yields

('t)%1(4m+1h " 4m

h , vh) + a(4m+%h , vh)

=$('t)%1[(Ph.um+1 " .um+1

h ) " (Ph.um " .umh )], vh

%+ a(Ph.um+%, vh) " a(.um+%

h , vh)

=$('t)%1(Ph.um+1 " Ph.um), vh

%+ a(Ph.um+%, vh) "

$('t)%1(.um+1

h " .umh ), vh

%

" a(.um+%h , vh)

(3.6)=$('t)%1(Ph.um+1 " Ph.um), vh

%+ a(Ph.um+%, vh)

+ .a(0%;um+% " um+% + um+%h , vh)

(A.3)=$('t)%1(Ph.um+1 " Ph.um) " .u

m+%, vh%

+ a(Ph.um+% " .um+%, vh)

" .a(0%;um+% " um+%h , vh)

By stability (A.1) we have

Corollary A.4. Let .u ) C1([0, T ];H). Then, under the assumptions of Lemma A.1,there holds

.4Mh .2 + C1't

M%1)

m=0

.4m+%h .2

a - .40h.

2 + C2'tM%1)

m=0

.rm+%.2!. (A.4)

We estimate the quantities .rmj .!, j = 1, . . . , 4. For j = 1, 2, 3 the estimates can be

found in [25, Section 5]. For clarity, we restate them.

.rm1 .! - C

@AAAB

AAAC

('t)12

*& tm+1

tm..u(/).2

!d/

+ 12

' ) [0, 1]

('t)32

*& tm+1

tm.....u (/).2

!d/

+ 12

' = 12

(A.5)

.rm2 .! - C('t)%

12 hs%r

*# tm+1

tm

..u(/).2Hs"r d/

+12

(A.6)

.rm3 .! - Chs%r..um+%.Hs (A.7)

To estimate rm4 , we assume that the bilinear form .a(0%, ·, ·) is continuous on V # V .

Hence|(rm

4 , vh)| - .(.um+% " um+%h .a.vh.a

We obtain

Lemma A.5. Assume .u(x, t) is su!ciently smooth in [0, T ]#D and assume that .a(0%, ·, ·)is continuous on V # V . Then there holds

.rm.! -3)

j=1

.rmj .! + C.um+% " um+%

h .a,

with .rmj .! given by (A.5)–(A.7).

19

We are able to prove Theorem 3.3.

Proof (of Theorem 3.3). By definition emh = %m + 4m

h , m = 0, 1, . . . ,M " 1. Hence

.eMh .2+'t

M%1)

m=0

.em+%h .2

a - 2

*.%M.2+'t

M%1)

m=0

.%m+%.2a

++2

*.4M

h .2+'tM%1)

m=0

.4m+%h .2

a

+

The first term can be estimated by the approximation property (2.15). For the secondterm we have by the stability (A.4)

.4Mh .2 + 't

M%1)

m=0

.4m+%h .2

a - .40h.

2 + C2'tM%1)

m=0

.rm+%.2!.

Using the estimates for .rm+%.! of Lemma A.5 and the convergence result Lemma A.2for the sequence {um

h } finishes the proof.

A.2 Proof of Theorem 3.6

We estimate the error emh := D,um " D,

humh . We consider the splitting

emh = (D,um " PhD,um)

< => ?#m

+ (PhD,um " PhD,hum)

< => ?0m

h

+ (PhD,hum " D,

humh )

< => ?$mh,#

.

In order for D,hum and D,

humh being well defined, we extend um and um

h by zero to all ofRd.

If um is su!ciently smooth such that D,um ) Hs, %m can be estimated using theapproximation property (2.15). If we further assume that the projector Ph : V & Vh isuniformly stable (i.e. there exists a constant C independent of h such that .Phv.V -C.v.V ), the term $m

h can be estimated using the approximation property (3.8) of D,h .

It therefore remains to estimate 4mh,,.

We shall need the following subspace of Vh. For D0 '' D we denote by Vh(D0) the space

Vh(D0) = {v ) Vh | supp v ! D0} ' Vh.

We may assume that D0 is such that " ) Vh(D0) implies D,h" ) Vh. It is obviously

su!cient to consider 4mh,, := Ph!,

h um " !,h um

h .

Lemma A.6. Assume u ) C1([0, T ];H). Assume that the operators Ph and !,h commute.

Then {4mh,,}

M%1m=0 satisfy for any ( ) Nd

0, |(| = n + 0 the '-scheme: Given 40h,, =

Ph!,h u0 " !,

h u0h, for m = 0, 1, . . . ,M " 1

('t)%1(4m+1h,, " 4m

h,,, vh) + a(4m+%h,, , vh) = (rm, vh), /vh ) Vh(D0), (A.8)

20

with weak residual rm : Vh & R given by

rm =4)

j=1

rmj

where

(rm1 , vh) := ("1)|,|

$('t)%1(um+1 " um) " um+%, !,

h vh

%,

(rm2 , vh) := ("1)|,|

$('t)%1(Phum+1 " Phum) " ('t)%1(um+1 " um), !,

h vh

%,

(rm3 , vh) := ("1)|,|a

$Phum+% " um+%, !,

h vh%,

(rm4 , vh) := "

#

D

)

.<,

C,,.

D d)

i,j=1

(!,%.h Qij)

$T,%.

h !.hDi(Phum+% " um+%

h )%Djvh

+d)

i=1

(!,%.h bi)

$T,%.

h !.hDi(Phum+% " um+%

h )%vh

Edx

with C,,. :=

*(

( " *

+.

Proof. Recall that u ) C1([0, T ],H) implies

(um+%, v) + a(um+%, v) = (fm+%, v), /v ) V. (A.9)

Let vh ) Vh(D0). Inserting 4mh,, in the '-scheme yields

('t)%1(4m+1h,, " 4m

h,,, vh) + a(4m+%h,, , vh)

=$('t)%1(Ph!,

h um+1 " Ph!,h um), vh

%+ a(Ph!,

h um+%, vh)

"1('t)%1((!,

h um+1h " !,

h umh ), vh) + a(!,

h um+%h , vh)

2.

By the discrete Leibniz rule, we have

('t)%1((!,h um+1

h " !,h um

h ), vh) + a(!,h um+%

h , vh)

= ("1)|,|$('t)%1(um+1

h " umh ), !,

h vh

%+ ("1)|,|a(um+%

h , !,h vh) + R,

h(um+%h )

(2.16)= ("1)|,|(fm+%, !,

h vh) + R,h(um+%

h )

(A.9)= ("1)|,|

$um+%, !,

h vh

%+ ("1)|,|a

$um+%, !,

h vh

%+ R,

h(um+%h ).

Here, we denote by R,h(um+%

h ) the residual term

R,h(um+%

h ) := "#

D

)

.<,

C,,.

D d)

i,j=1

(!,%.h Qij)

$T,%.

h !.hDiu

m+%h

%Djvh

+d)

i=1

(!,%.h bi)

$T,%.

h !.hDiu

m+%h )

%vh

Edx

21

Utilizing once more the discrete Leibniz rule and using that the operators Ph and !,h

commute yields

$('t)%1(Ph!,

h um+1 " Ph!,h um), vh

%+ a(Ph!,

h um+%, vh)

= ("1)|,|$Phum+1 " Phum, !,

h vh

%+ ("1)|,|a(Phum+%, !,

h vh) + R,h(Phum+%).

The representation of rm in (A.8) is now obvious.

Remark A.7. Note that the residual rm4 in the Lemma A.6 satisfies rm

4 = 0 if thecoe!cients Q, b in (2.3) are constant, as it is the case for the Multidimensional Levymodel in Example 2.2.

By the stability result Lemma A.1 we obtain the stability estimate for 4mh,, in (A.8)

.4Mh,,.

2 + C1'tM%1)

m=0

.4m+%h,, .2

a - .40h,,.

2 + C2'tM%1)

m=0

.rm+%.2!. (A.10)

We estimate the residuals .rmj .!. For j = 1, 2, 3 these are the same as Lemma A.5 (with

u replacing .u, see also [25, Section 5]. To estimate .rm4 ., we use again that the operators

Ph and !,h commute. We find

.rm4 .! - C

)

.<,

.!.h (Phum+% " um+%

h ).a = C)

.<,

.4m+%h,. .a.

Rewriting the estimate (A.10) shows that for any ( ) Nd0 there holds

.4Mh,,.2 + C1't

M%1)

m=0

.4m+%h,, .2

a

- .40h,,.

2 + 4C2'tM%1)

m=0

3)

j=1

.rm+%j .2

a + C'tM%1)

m=0

)

.<,

.4m+%h,. .2

a. (A.11)

Since (A.11) holds for an arbitrary (but fixed) ( ) Nd0, we may iterate the inequality

until * = 0 to obtain

.4Mh,,.

2 + C1'tM%1)

m=0

.4m+%h,, .2

a - C(()'tM%1)

m=0

.rm+%1 .2

a + .rm+%2 .2

a + .rm+%3 .2

a + .4m+%h,0 .2

a

+ C(())

.#,

.40h,..

2. (A.12)

22

Proof (of Theorem 3.6). We have for emh = D,um " D,

humh and M + 1

.eMh .2 + 't

M%1)

m=0

.em+%h .2

a

- 3

D.%M.2 + 't

M%1)

m=0

.%m+%.2a + .$M

h .2 + 'tM%1)

m=0

.$m+%h .2

a + .4Mh,,.

2

+ 'tM%1)

m=0

.4m+%h,, .2

a

E.

If D,u(t) ) Hs for t ) [0, T ], the first term can be estimated with the approximationproperty (2.15). The second term is estimated using the uniform stability of the projectorPh and the approximation property (3.8) of D,

h (provided u(t) ) Hs+r+n for t ) [0, T ]).For the last term we have by (A.12) and the fact that 't.4m+%

h,0 .a can be estimated also

by the quantities .rm+%j .!, j = 1, 2, 3

.4Mh,,.

2 + 'tM%1)

m=0

.4m+%h,, .2

a - C)

.#,

.40h,..

2 + C'tM%1)

m=0

3)

j=1

.rm+%j .2

a.

Now conclude by using (A.5)–(A.7) to bound .rm+%j .! (replacing .u by u), the quasi-

optimality of !.hu0

h and the approximation property (2.15) to estimate .40h,...

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25

Research Reports

No. Authors Title

08-29 N. Hilber, C. Schwab,C. Winter

Variational sensitivity analysis of parametricMarkovian market models

08-28 K.Schmidt, S. Tordeux Asymptotic modelling of conductive thinsheets

08-27 R. Hiptmair, P.R. Kotiuga,S. Tordeux

Self-adjoint curl operators

08-26 N. Reich Wavelet compression of anisotropic integro-di!erential operators on sparse tensor prod-uct spaces

08-25 N. Reich Anisotropic operator symbols arising frommultivariate jump processes

08-24 N. Reich Wavelet compression of integral operatorson sparse tensor spaces: Construction,Consistency and Asymptotically OptimalComplexity

08-23 F. Liu, N. Reich andA. Zhou

Two-scale finite element discretizations for in-finitesimal generators of jump processes infinance

08-22 M. Bieri and Ch. Schwab Sparse high order FEM for elliptic sPDEs

08-21 M. Torrilhon andR. Jeltsch

Essentially optimal explicit Runge-Kuttamethods with application to hyperbolic-parabolic equations

08-20 G. Dahlquist andR. Jeltsch

Generalized disks of contractivity for explicitand implicit Runge-Kutta methods

08-19 M. Karow and D. Kressner On the structured distance to uncontroll-ability

08-18 D. Kressner,Ch. Schroeder andD.S. Watkins

Implicit QR algorithms for palindromic andeven eigenvalue problems

08-17 B. Kagstroem, D. Kressner,E.S. Quintana-Ort andG. Quintana-Ort

Blocked algorithms for the reduction toHessenberg-Triangular form revisited

08-16 R. Granat, B. Kagstroemand D. Kressner

Parallel eigenvalue reordering in real Schurforms