pricing american call options under the assumption of stochastic dividends

7/28/2019 Pricing American call options under the assumption of stochastic dividends

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S. Kruse, M. Mller

Pricing American call options un-der the assumption o stochasticdividends An application o the

Korn-Rogers model

Berichte des Fraunhoer ITWM, Nr. 158 (2009)


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Fraunhoer-Institut r Techno- und Wirtschatsmathematik ITWM 2009

ISSN 1434-9973

Bericht 158 (2009)

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Vorwort

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Pro. Dr. Dieter Prtzel-WoltersInstitutsleiter

Kaiserslautern, im Juni 2001


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Pricing American Call Options under the

Assumption of Stochastic Dividends -

An Application of the Korn-Rogers-Model

Susanne Kruse , Marlene Muller

Hochschule der Sparkassen-Finanzgruppe University of Applied Sciences Bonn, Ger-

many.

Fraunhofer Institute for Industrial and Financial Mathematics, Department of Finan-

cial Mathematics, Kaiserslautern, Germany.

This version April 15, 2009.

Abstract. In financial mathematics stock prices are usually modelled directly as

a result of supply and demand and under the assumption that dividends are paid

continuously. In contrast economic theory gives us the dividend discount model

assuming that the stock price equals the present value of its future dividends.

These two models need not to contradict each other - in their paper Korn and

Rogers (2005) introduce a general dividend model preserving the stock price to

follow a stochastic process and to be equal to the sum of all its discounted divi-

dends. In this paper we specify the model of Korn and Rogers in a Black-Scholes

framework in order to derive a closed-form solution for the pricing of American

Call options under the assumption of a known next dividend followed by several

stochastic dividend payments during the options time to maturity.

Key words: option pricing, American options, dividends, dividend discount

model, Black-Scholes model

JEL Classification: G13

Mathematics Subject Classification (2000): 60G44, 60H30

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Introduction

In finance stock prices are typically modeled directly and are assumed to follow a

geometric Brownian motion or more generally a semi-martingale without refer-

ring to the economic value of the payments obtained by possessing the stock. Even

more, sometimes the existence of dividend payments is simply ignored and, in par-

ticular, in option pricing the validity of certain key results depend in a crucial way

on the absence of dividends a prominent example is the price equality between

European and American calls on a non-dividend paying stock in the presence of a

non-negative interest rate. On the one hand, ignoring the dividend payments can

lead to serious pricing errors for derivatives. On the other hand, we could interpret

this as modeling the stock price evolution only as a result of supply and demand.

From an economic point of view, however, the price of a share of a company should

be equal to the present value of the future dividend payments. In general the twomodeling approaches need not to contradict each other. But there are situations,

where an explicit consideration of dividend payments is necessary, since otherwise

the price evolution of the share is not modeled in an adequate way. The typical sit-

uation is the time span between the announcement of the next dividend payment

and its actual payment date. Then the share price dynamics contains a certain

component which is deducted from the share price at the dividend payment time.

Thus, the dynamics of the share price has to differ from that during times of no

explicit dividend announcement.

The case of known dividends and the valuation of European options as well

as American call options has been widely discussed in the literature. Roll, Geske

and Whaley see [8], [9], [10], [15], [17] and [18] have solved the pricing problem

of an American call on a stock with one known dividend payment during its time

to maturity. Sterk [16] has verified the fit of the Roll-Geske-Whaley formula to

American call prices. In mathematical finance Geske [7] was the first to consider

uncertain dividends leading to a closed-form solution for European option prices in

an adjusted Black-Scholes framework. Following this introduction of an unknown

dividend Broadie et al. [3] as well as Chance et al. [4] examined the influence of

stochastic dividend payments on the price of an European option. Besides there are

a number of publications that are concerned with dividends and the derivation of

a market opinion such as for example [5]. Professionals in finance, see for example

[1], [2], [6] or [12], still take great interest in the question which model reflects

reality the best and offers consistent option pricing especially with American

options. Zhu [19] has tackled the problem of giving a closed-form solution for the

price of an American put option.

In order to include both approaches the following facts should be taken into

account when constructing a dividend model for stocks:

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The stock price is the present value of all future dividend payments.

The actual value of the next dividend payment and the stock price are closelyrelated.

There is no clear relation between the height of interest rates and the div-idend payments unless the fact that an investor expects the dividend yield

to be higher than the yield of a risk-less bond.

The closer one gets to the time of the dividend payment, randomness of thedividend reduces.

Based on these facts Korn and Rogers [14] have chosen to model all dividends by

a stochastic process and thereby derive the price of a stock S(t) paying dividends

D(ti) at future times ti > t as

S(t) = E

i=1

er(tit)D(ti)

. (1)

We note that this case also includes that the earlier dividend payments can be

known and therefore constant.

In this paper we use the basic idea of Korn and Rogers [14] in a Black-Scholes

framework which allows us to price American options on a dividend paying stock

in closed-form. The main contributions of our paper are the following:

(i) We transfer the model of Korn and Rogers into a Black-Scholes framework.

(ii) To illustrate the key idea of the proof we give closed-form solutions for just

one known respectively stochastic dividend during the options time to ma-

turity.

(iii) We derive a recursive algorithm for our valuation based on transferring back

the n-dimensional case to the (n 1)-dimensional case.

(iv) We achieve a closed-form solution to the pricing problem of American Call

options with several dividend payments during the time to maturity.

The remainder of the paper is structured as follows:

We use the dividend model in a lognormal framework, which is described in

detail in Section 1. To illustrate the idea, we discuss closed-form solutions to

American option pricing in Section 2 under the assumption that there is only

one dividend payment during the time to maturity of the corresponding option.

Furthermore, we distinguish between a stochastic and a known dividend payment.

In Section 3 we give closed-form solutions to the pricing of American Call options

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under the assumption of several dividends, where we distinguish two situations:

(1) strictly stochastic dividends, and (2) one known dividend followed by stochastic

dividend payments. Of course it is possible to combine these results to one, but

for the sake of a better understanding of the situation we present these cases

seperately.

1 Stochastic Dividends in the Korn-Rogers

Model

From now on we assume that the stock pays its dividends at equidistant times

of which the first l dividends D1, , Dl with payment dates 0 < t1, t1 +h , t1 + h(l 1) are known and therefore deterministic and the later dividendsDl+1, Dl+2, with payment dates t1+ hl,t1+ h(l +1), are stochastic. In theirpaper [14] Korn and Rogers assume that the stochastic dividends are of the form

Dl+1 = X(t1 + hl), , Dn = X(t1 + h(n 1)) (2)where X is an exponential Levy process scaled by some positive constant. Fur-

thermore they assume that holds

E

X(s)

Ft = e(ts) X(t) (3)for some < r and any 0 s t where r is the risk-free interest rate. Note thatthe representation of the stock price as the sum of the present values of its future

dividend paymentsS(t) = E

i=1

er(tit)Di

(4)

leads to the fact that we can write the stock price as

S(t) =

lm=1 e

r(t1+(m1)ht)Dm +X(t) e(r)(t1+lht)

1e(r)h

for t [0, t1)lm=k+1 e

r(t1+(m1)ht)Dm +X(t) e(r)(t1+lht)

1e(r)h

for t [t1 + h(k 1), t1 + hk),

1 k < lX(t) e

(r)(t1+kht)

1e(r)h for t [t1 + h(k 1), t1 + hk),

l k

(5)

since by Equation (3) holds

E

i>l

er(t1+(i1)ht)X(t1 + (i 1)h)Ft

= X(t)

e(r)(t1+lht)

1 e(r)h (6)

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for some t < t1 + hl as well as

E

i>k er(t1+(i1)ht)X(t1 + (i 1)h)Ft = X(t)e(r)(t1+kht)

1 e(r)h(7)

for t = t1 + hk t1 + hl.

So far we did not specify the dynamics of the dividend process. From now on

we assume that the dividend process X(t) follows a geometric Brownian motion

dX(t) = X(t)dt + X(t)dW(t), < r (8)

where r is the risk-free interest rate and the volatility of the dividend process X

and thereby of the stock price process S.

If we denote by S0 todays market price of the stock, this is leading to the

following representation of the stock price for t [0, t1)

S(t) =

S0

lm=1

Dmer(t1+h(m1))

e(r

12

2)t+W(t) +l

m=1

Dmer(t1+h(m1)t)

(9)

while for t [t1 + h(k 1), t1 + hk) and 1 k < l the ex-dividend stock priceequals

S(t) =

S0

l

m=1Dme

r(t1+h(m1))

e(r12

2)t+W(t) +

l

m=k+1Dme

r(t1+h(m1)t)

(10)

while for t [t1 + h(k 1), t1 + hk), l k holds

S(t) =

S0

lm=1

Dmer(t1+h(m1))

e(r)(kl)he(r

12

2)t+W(t). (11)

We note that this representation relies on the fact that announcement and payment

dates for the dividends coincide. It is possible to include a time difference between

the announcement and the payment of dividends, which would lead to higher

dimensional distribution functions. Nevertheless due to the fact that the dividend

payment date is the important date at which we distinguish between exercisingand holding the option we neglect this difference. Our representation contains the

case of strictly stochastic dividends (l = 0) as well as the case of n known dividends

(l = n) in some time interval [0, T] with 0 < t1 < < tn = t1 + (n 1)h t1

is given by

C0,1D (S0, 0, T , K ) =S0

0,11 (S0, 0) KerT0,12 (S0, 0) (16)

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where

0,11 (S0, 0) = N(d11) + e

(r)hNd12,d11,t1T (17)0,12 (S0, 0) = e

r(Tt1)N(d21) + N

d22,d21,

t1

T

(18)

with

d11 =ln

S0e

(r)h

S

+

r + 12

2

t1

t1and d21 = d

11

t1, (19)

d12 =ln

S0e

(r)h

K

+

r + 12

2

T

Tand d22 = d

12

T . (20)

N() is the standard normal cdf, N(, , ) is the bivariate normal cdf with correla-tion and S is the unique stock price such that holds

CBlack S choles(S, t1, T , K ) = S + D K (21)

with

D = S1 e(r)h

e(r)h. (22)

Proof: We note that the only time sensible for an early exercise of the American

call is the time t1 of the dividend payment. Analogously to the Roll-Geske-Whaley

formula (as in [13], see also [9], [15] and [17]) we find a critical stock price S

at time t1 such that the call should be exercised for S(t1) > S or should be hold

for S(t1) S. S is the stock price such that (21) holds. Hence we can writetodays option price as

E[ert1 C0,0D (S(t1), t1, T , K ) 1{S(t1)S}|F0]+ E[ert1 (S(t1) + D(t1) K) 1{S(t1)>S}|F0] (23)

where

C0,0D (S(t1), t1, T , K ) = S(t1)N(d1(t1)) Ker(Tt1)N(d2(t1)) (24)

with

d1(t1) =ln

S(t1)

K

+ (r + 12

2)(T t1)

T t1(25)

d2(t1) = d1(t1)

T t1 (26)

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is the Black-Scholes price of an European option with no dividend payment during

the time interval (t1, T) and

D(t1) = S(t1) 1 e(r

)h

e(r)h(27)

is the dividend payment at time t1. If we split up the above expectation, we first

get by using Equation (27)

E[ert1 (S(t1) + D(t1) K) 1{S(t1)>S}|F0] = S0N(d11) Kert1N(d21) (28)

with

d11 =ln

S0e

(r)h

S

+

r + 12

2

t1

t1and d21 = d

11

t1. (29)

Secondly we derive that

E[ert1 C(S(t1), t1, T , K ) 1{S(t1)S}|F0] (30)equals by plugging in the exact option price (24) at time t1

e(r)hS0N

d12,d11,

t1

T

KerTN

d22,d21,

t1

T

(31)

with

d11 =ln

S0e

(r)h

S

+

r + 12

2

t1

t1and d21 = d

11

t1 (32)

d12 =ln

S0e

(r)h

K

+

r + 12

2

T

Tand d22 = d

12

T (33)

using Lemma A.4, Corollary A.5 and Corollary A.8 for n = 1 and taking into

account, that we can rewrite d1(t1) and d2(t1) as

d1(t1) =ln

S(t1)

K

+ (r + 12

2)(T t1)

T t1(34)

=ln

S0e

(r)h

K

+ (r + 12

2)T 2t1

T t1+

1T t1

Wt1 (35)

= 11 + 11

Wt1t1

(36)

d2(t1) = lnS(t1)K + (r 122)(T t1)

T t1(37)

=ln

S0e

(r)h

K

+ (r 122)T

T t1+

1T t1

Wt1 (38)

= 21 + 21

Wt1t1

. (39)

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2.2 The Case of a Known Dividend

In this subsection we assume that there is one known constant dividend during

the options time to maturity. These assumptions are the same as for the Roll-Geske-Whaley formula. In the following we show, that our approach is consistent

with this well-known result:

Theorem 2.2 Roll-Geske-Whaley Formula. Assume that the stock pays a

known dividend D1 at time t1. The price C1,0D of an American Call with strike K

and maturity T > t1 is given by

C1,0D (S0, 0, T , K )

=

S0 D1ert1

1,01 (S0, 0) KerT1,02 (S0, 0) + D1ert1N(d21) (40)

where 1,01 (S0, 0) and

1,02 (S0, 0) are given by

1,01 (S0, 0) = N(d11) + N

d12,d11,

t1

T

(41)

1,02 (S0, 0) = er(Tt1)N(d21) + N

d22,d21,

t1

T

(42)

with

d11 =ln

S0D1ert1

S

+

r + 12

2

t1

t1and d21 = d

11

t1, (43)

d12 =

lnS0D1ert1

K + r + 122T

Tand d22 = d12 T (44)

where S is the unique stock price such that holds

CBlack S choles(S, t1, T , K ) = S + D1 K. (45)

Proof: The proof goes along the lines of the proof to the previous theorem taking

into account that under the assumption of a first known dividend the stock price

at time t1 is an ex-dividend stock price such that holds by Equation (11)

S(t1) =

S0 D1ert1

e(r

12

2)t+W(t). (46)

3 American Options in the Multi-Dividend Case

We finally focus on American call options on a stock with n dividend payments,

such that we have dividend payments at t1, t1+h,...,t1+h(n1) until the optionsmaturity at time T.

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3.1 The Case of n Stochastic Dividends

In the previous section we illustrated the structure of deriving a closed-form so-

lution under the assumption of a single dividend during the time to maturity. Inthe case ofn dividends this leads to the recursive calculation of n 1 multivariatenormal distributions. We note that the number of steps in the calculation process

of the option price is of order O(n4). However, in practice, this remains com-

putationally feasible since the number of dividend payments up to maturity T is

usually quite small.

Theorem 3.1 The price of an American call option with strike K and maturity

T on a dividend paying stock with market price S0 and n stochastic dividends at

times t1 < t1 + h S1 or

should be hold for S(t1) S1 . S1 is the stock price such that holds

C0,n1D (S

1 , t1, T , K ) = S

1 + D

1 K (65)

where D1 is the unknown dividend to be paid at time t1. Hence we can write theoption price at time t = 0 as

C0,nD (S0, 0, T , K ) = E

ert1

(S(t1) + D(t1) K) 1{S(t1)>S1}

F0+ E

ert1

C0,n1D (S(t1), t1, T , K ) 1{S(t1)S1}

F0

(66)

where C0,n1D (S(t1), t1, T , K ) is given by Equation (58) and

D(t1) = S(t1)1 e(r)h

e(r)h(67)

is the dividend payment at time t1. It is straightforward that the first term in

Equation (66) equals

ert1 E

(S(t1) + D(t1)K)1{S(t1)>S1} F0

= S0N(d

11)Kert1N(d21) (68)

with

d11 =ln

S0e

(r)h

S1

+ (r + 12

2)t1

t1and d21 = d

11

t1. (69)

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For the calculation of the second term we use the representation of the stock price

as in Equation (11) with l = 0 and rewrite

d1i (t1) =lnS(t1)e(r)ihS

i+1 + r + 122 ih

ih(70)

=ln

S0e

(r)(i+1)h

Si+1

+ (r + 12

2)(t1 + ih) 2t1

ih+

1ih

W(t1) (71)

= 1ni + 1ni

W(t1)t1

(72)

for i = 1,...,n 1 as well as

d1n(t1) =ln

S(t1)e

(r)(n1)h

K

+ (r + 12

2)(T t1)

T

t1

(73)

=ln

S0e(r)nhK

+ (r + 12

2)T 2t1

T t1+

1T t1

W(t1) (74)

= 11 + 11

W(t1)t1

. (75)

We note that the terms d2i (t1) and d2n(t1) can be rewritten in a similar way. Finally

we apply Lemma A.4 and Corollary A.5 in combination with Corollary A.7 and

Corollary A.8 to derive the required probabilities and recursively calculate the

correlation matrices C(i) from C(i1) for i = 2, . . . , n + 1. Hence we get that theprice at time t = 0 is given by (47).

3.2 The Case of a Known Dividend followed by Stochastic

Dividends

In the following theorem we modify the results of Section 3.1 to the realistic case

that the first coming dividend payment at t1 is known and the remaining n 1dividends are stochastic.

Theorem 3.2 The price of an American call option with strike K and maturity

T on a dividend paying stock with market price S0 and a deterministic dividendD1at timet1 and n 1 stochastic dividends at times t1 + h < < t1 + (n 1)h < Tduring maturity is given by

C1,n1D (S0, 0, T , K )

=

S0 D1ert1

1,n11 (S0, 0) KerT1,n12 (S0, 0) + D1ert1N(d21) (76)

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where

1,n11 (S0, 0) = N(d11) +

n

i=1 e(r)(i1)hNi+1 d1i+1; C(i+1) (77)1,n12 (S0, 0) = e

r(Tt1)N(d21) +n1i=1

er(T(t1+ih))Ni+1

d2i+1; C

(i+1)

+Nn+1

d2n+1; C

(n+1)

(78)

and where for i = 1, , n and a = 1, 2 we define

dai+1 = (d

ai+1,dai , ,da2 ,da1) (79)

with

d1i =ln (S0D1ert1)e(r)(i1)hSi + (r + 122)(t1 + (i 1)h)

t1 + (i 1)h(80)

d2i = d1i

t1 + (i 1)h (81)

as well as

d1n+1 =

ln

(S0D1ert1)e(r)(n1)h

K

+ (r + 12

2)T

T(82)

d2n+1 = d1n+1

T (83)

and withS1 ,

S2

,

, Sn such that for

i= 1

,

, n

C0,niD (S

i , t1 + (i 1)h , T , K ) = Si + Di K (84)

where D1 = D1 is the fixed dividend at time t1 and Di , i 2, are the unknown

dividends paid at times t1 + (i 1)h which equal

Di = Si

1 e(r)he(r)h

. (85)

Proof: By Theorem 3.1 we know that the price of the American option at time t1

equals

C0,n1D (S(t1), t1, T , K ) = S(t1)

0,n11 (S(t1), t1) KerT0,n12 (S(t1), t1) (86)

with 0,n11 (S(t1), t1) and 0,n12 (S(t1), t1) as defined by Equation (48) up to

Equation (79). Analogously to the argumentation in Theorem 3.1 we can now find

a critical stock price S1 at time t1 such that the call should be exercised forS(t1) > S

1 or should be hold for S(t1) S1 . S1 is the stock price such that holds

C0,n1D (S

1 , t1, T , K ) = S

1 + D1 K (87)

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where D1 is the now known dividend to be paid at time t1. As before we can write

the option price at time t = 0 as

C1,n1D (S0, 0, T , K ) = E

ert1 (S(t1) + D1) K) 1{S(t1)>S1} F0+ E

ert1

C0,n1D (S(t1), t1, T , K ) 1{S(t1)S1}

F0

(88)

where C0,n1D (S(t1), t1, T , K ) is given by Equation (86). We note that by usingthe representation of the ex-dividend stock price at time t1 as in Equation (11)

S(t1) =

S0 D1ert1

e(r12

2)t+W(t). (89)

the proof is an analogue to the proof of Theorem 3.1.

Remark. We note that if we allow several known dividend payments during

time to maturity, the option price can not be computed in this iterative way.

Alternatively one could use the argument as in [13], that only the last known

dividend payment is relevant and use its payment date as time t1 in the above

recursion. Of course one would still have to take all the previous payments to this

date into account as in representation (9) to (11).

Conclusion

Since the introduction of the Black-Scholes formula in 1973 the pricing of Europeanoptions under the assumption that the stock price follows a geometric Brownian

motion is quite well understood. Not only has this formula been established as a

market standard for these types of options but is also a model most professional

market participants are quite familiar with. In this paper we used the general

dividend model of Korn and Rogers in a Black-Scholes framework in order to price

American Call options in closed-form. We derived closed-form solutions in the case

of multiple stochastic dividends during time to maturity of the option which might

be following a constant first dividend payment. We also showed that the model

is consistent not only with the Black-Scholes price but also with the well-known

Roll-Geske-Whaley formula for the price of an American Call in the presence of a

known dividend payment.

References

[1] Bos, R.; Gairat A.; Shepelleva, A. (2003): Dealing with discrete dividends,

Risk, 16 (1), 109-112.

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[2] Bos, M.; Vandermark, S. (2002): Fitnessing fixed dividends, Risk, 15 (9),

157-158.

[3] Broadie, M., Detemple, J.; Ghysels, E.; Torres, O. (2000): American optionswith stochastic dividends and volatility: a non-parametric investigation, Jour-

nal of Econometrics, 94, 53-92.

[4] Chance, D. M.; Kumar, R.; Rich, D. (2002): European option pricing with

discrete stochastic dividend, Journal of Derivatives, 9 (3), 39-45.

[5] Decamps, J.; Villeneuve, S. (2007): Optimal dividend policy and growth

option, Finance and Stochastic, 11, 3-27.

[6] Frishling, V. (2002): A discrete question, Risk, 15 (1), 115-116.

[7] Geske, R. (1978): The pricing of options with stochastic dividend yield, Jour-nal of Finance, 33 (2), 617-625.

[8] Geske, R. (1979): The valuation of compound options, Journal of Financial

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American call options on stocks with known dividends, Journal of Financial

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nomics, 9, 213-215.

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ume Two, Cambridge University Press.

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the discrete dividend problem, Wilmott Magazine, 9, 37-47.

[13] Hull, J.C. (2008): Options, Futures and Other Derivatives, Prentice Hall.

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[15] Roll, R. (1977): An analytic valuation formula for unprotected American call

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16


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[19] Zhu, S.P. (2005): A Closed-form Exact Solution for the Value of American Putand its Optimal Exercise Boundary, Noise and Fluctuations in Econophysics

and Finance, 5848, 186 - 199.

A Appendix

We denote by the transpose symbol, by 0n the n-dimensional null vector, byIn the n n identity matrix and by Nn(z1, . . . , zn; C) the n-variate (standard)normal cumulative distribution function (cdf) with correlation matrix C at values

z1, . . . , zn.

A.1 Results on the Multivariate Normal Distribution

We first give some useful results on block matrices and multivariate normal distri-

butions:

Lemma A.1 Let X N(, 2) be a random variable and let Y =(Y1, . . . , Y n)

N(0, C) be a random vector where C is a correlation matrix.Assume that X and Y are independent. Further, denote by fX the probability

density function (pdf) ofX and consider a vector of constants = (1, . . . , n).

Then holds

Nn(1x + 1, . . . , nx + n, C) fX (x) dx

= P(Y1 1X + 1, . . . , Y n nX + n, X )= P(Z1 1, . . . , Z n n, X )

where Zj = Yj j X. Here, the vector Z = (Z1, . . . , Z n) and X are jointlynormal with

Z

X

N

,

C + 2 2

2 2

.

Proof: Since X and Y are independent they follow a joint normal distribution

Y

X

N

0n

,

C 0n0n

2

=: N(, ) .

17


22/32

Furthermore we have

ZX = HYX with H = In 0n 1

.Thus, the vector consisting of Z and X has expection and variancecovariance

matrix

H =

and HH =

C + 2 2 2 2

.With the help of some well-known properties for block matrices (see for example

[11], pp. 417419) we can now derive

det(HH) = 2det(C) ,

(HH)1 =

C1 C1C1 2 + C1

.Let z denote the vector (z1, . . . , zn)

. The joint density of Z and X is given by

fZ,X (z, x) =1

det(2 HH) exp

1

2

z +

x

(HH)1

z +

x

Since we have that

z + x

(HH)1

z + x

= (z + x)C1(z + x) +

(x )22

.

it follows

P(Z1 1, . . . , Z n n, X )

=

1

. . .

n

fZ,X (z, x) dz dx

=

fX (x)

1

. . .

n

1det(2C)

exp

1

2(z + x)C1(z + x)

dz dx

=

fX (x) N(1x + 1, . . . , nx + n, C) dx .

Corollary A.2 Under the same assumptions as in Lemma A.1 it follows that

Nn(1x + 1, . . . , nx + n, C) e+x fX (x) dx

= e++12

22P(Z1 1, . . . , Z n n, X )18


23/32

where the vector Z = (Z1, . . . , Z n) and X are jointly normal with

ZX N( + 2) + 2 , C + 2

2

2 2 .

Lemma A.3 Let again X N(, 2) be a random variable and let Y =(Y1, . . . , Y n)

N(0, C) be a random vector where C is a correlation matrix withelements cjk . Assume and X and Y are independent and let further U be the

vector consisting of

Y1 1X , . . . , Y n nX, and X .

Then the correlation matrix ofU is given by the (n + 1) (n + 1) matrix C whichhas off-diagonal elements

cjk =

cjk+jk2

1+2j21+2

k2

for j = k and j, k = 1, . . . , n ,j1+2j

2for k = n + 1 and j = 1, . . . , n ,

k1+2

k2

for j = n + 1 and k = 1, . . . , n ,

(90)

Proof: Consider again = (1, . . . , n). From the proof of Lemma A.1 we know

thatZ = (Y1 1X , . . . , Y n nX) N

, C + 2 and

U =

Z

X

N

,

C + 2 2 2 2

=: N(u, u) .

Denoting by Du the diagonal matrix having the elements of u on the diagonal,

we calculate the correlation matrix C by multiplying out D1/2u uD

1/2u .

Lemma A.4 LetfX be the standard normal pdf and let C be a correlation matrix.

Then holds

Nn(1x + 1, . . . , nx + n, C) fX (x) dx

= Nn+1

1

1 + 21, . . . ,

n1 + 2n

, , C

where C is the correlation matrix given in Equation (90).

19


24/32

Proof: The proof follows immediately by standardizing the Zj and X from

Lemma A.1, their correlation matrix C is given in Lemma A.3.

Corollary A.5 Under the same assumptions as in Lemma A.4 it follows that

Nn(1x + 1, . . . , nx + n, C) e+x fX (x) dx

= e+12

2

Nn+1

1 + 1

1 + 21, . . . ,

n + n1 + 2n

, , C

where C is the correlation matrix given in Equation (90).

Proof: The proof follows immediately by standardizing the Zj and

X from Corol-

lary A.2, the correlation matrix C is still that given in Lemma A.3.

A.2 Recursive Calculation of the Distribution

From Lemma A.3 we see that the (n + 1) (n + 1) correlation matrix C of U is afunction of the nn correlation matrix C and the parameters , 1, . . . , n. To im-plement the n-dividend case, we consider a sequence of time points t1, t2, . . . , tn+1

and let the parameters = n, j = nj depend on these time points. The

following lemma shows how to calculate

C(n+1) =

1 c(n+1)12 . . . c

(n+1)1,d+1

c(n+1)12 1 . . . c

(n+1)2,n+1

......

. . ....

c(n+1)1,n+1 c

(n+1)2,n+1 . . . 1

in dependence of C(n) which can then be used to compute the sequence

{C(n+1)}n=1,2,... recursively.

Lemma A.6 For a sequence of time points t1 < t2 < . . . < tm+1 define

j = tmj+1 for j = 1, . . . , m ,and

jk =1

tmk+2 tmj+1 for all j = 1, . . . , m ; k = 1, . . . , j and j k ,

and consider the recursion to obtain C(m+1) from C(m) according to the C Cmapping from Lemma A.3 starting at C(1) = 1. Then the off-diagonal elements of

the (m + 1) (m + 1) correlation matrix C(m+1) are obtained by:

20


25/32

(a) case m = 1

c(2)12 =

1111 + 21121 = t1

t2

(b) case m > 1

c(m+1)j,m+1 =

mjm1 + 2mj

2m

=

t1

tmj+2for j = 1, . . . , m

c(m+1)jk =

c(m)jk + mjmk

2m

1 + 2mj2m

1 + 2mk

2m

for j = 1, . . . , m; k = j + 1, . . . , m

=c(m)jk

tmj+2 t1tmk+2 t1 + t1

tm

j+2

tm

k+2.

The following two corollaries define the recursive algorithm to calculate the cor-

relation matrices for the n-dividend cases in Theorems 3.1 and 3.2. We here use

dividends paid at equidistant time points starting at t1. Corollary A.7 explains

how to compute the matrices C(i+1), i < n, whereas Corollary A.8 considers the

calculation of C(n+1).

Corollary A.7 Consider the sequence{t1, t2 = t1 + h , . . . , ti+1 = t1 + ih}, i < n.We then obtain

j = t1 + (i j)h for j = 1, . . . , i ,jk =

1(j k + 1)h for all j = 1, . . . , i ; k = 1, . . . , j and j k ,

and the off-diagonal elements of C(i+1) are given by

(a) case i = 1

c(2)12 =

1111 + 211

21

=

t1

t1 + h,

(b) case i > 1

c(i+1)j,i+1 =

ij i1 + 2ij

2i

= t1t1 + (i j + 1)h for j = 1, . . . , i ,

c(i+1)jk =

c(i)jk + ijik

2i

1 + 2ij 2i

1 + 2ik

2i

for j = 1, . . . , i; k = j + 1, . . . , i .

=c(i)jk

(i j + 1)h

(i k + 1)h + t1

t1 + (i j + 1)h

t1 + (i k + 1)h.

21


26/32

Corollary A.8 Consider the sequence {t1, t2 = t1 + h , . . . , tn = t1 + (n 1)h,tn+1 = T}. We now have

j = t1 + (n j)h for j = 1, . . . , n ,j1 =

1T t1 (n j)h

,

jk =1

(j k + 1)h for all j = 2, . . . , n ; k = 2, . . . , j and j k ,

and the off-diagonal elements of C(n+1) are given by

(a) case n = 1

c(2)12 =

111

1 + 21121=

t1

T,

(b) case n > 1

c(n+1)1,n+1 =

n1n1 + 2n1

2n

=

t1

T,

c(n+1)1k =

c(n)1k + n1nk

2n

1 + 2n12n

1 + 2nk

2n

for k = 2, . . . , n .

=c(n)jk

T t1

(n k + 1)h + t1

T

t1 + (n k + 1)h

c(n+1)

j,n+1

=nj n1 + 2nj2n =

t1

t1 + (n j + 1)hfor j = 2, . . . , n ,

c(n+1)jk =

c(n)jk + nj nk

2n

1 + 2nj2n

1 + 2nk

2n

for j = 2, . . . , n; k = j + 1, . . . , n .

=c(n)jk

(n j + 1)h

(n k + 1)h + t1

t1 + (n j + 1)h

t1 + (n k + 1)h.

22


27/32

Published reports o theFraunhoer ITWM

The PDF-les o the ollowing reportsare available under:www.itwm.raunhoer.de/de/zentral__berichte/berichte

1. D. Hietel, K. Steiner, J. StruckmeierA Finite - Volume Particle Method orCompressible Flows(19 pages, 1998)

2. M. Feldmann, S. SeiboldDamage Diagnosis o Rotors: Applicationo Hilbert Transorm and Multi-Hypothe-sis TestingKeywords: Hilbert transorm, damage diagnosis,

Kalman ltering, non-linear dynamics

(23 pages, 1998)

3. Y. Ben-Haim, S. SeiboldRobust Reliability o Diagnostic Multi-

Hypothesis Algorithms: Application toRotating MachineryKeywords: Robust reliability, convex models, Kalman l-

tering, multi-hypothesis diagnosis, rotating machinery,

crack diagnosis

(24 pages, 1998)

4. F.-Th. Lentes, N. SiedowThree-dimensional Radiative Heat Transerin Glass Cooling Processes(23 pages, 1998)

5. A. Klar, R. WegenerA hierarchy o models or multilane vehicu-lar tracPart I: Modeling(23 pages, 1998)

Part II: Numerical and stochastic investigations(17 pages, 1998)

6. A. Klar, N. SiedowBoundary Layers and Domain Decomposi-tion or Radiative Heat Transer and Diu-sion Equations: Applications to Glass Manu-acturing Processes(24 pages, 1998)

7. I. Choquet

Heterogeneous catalysis modelling andnumerical simulation in raried gas fowsPart I: Coverage locally at equilibrium(24 pages, 1998)

8. J. Ohser, B. Steinbach, C. LangEcient Texture Analysis o Binary Images(17 pages, 1998)

9. J. OrlikHomogenization or viscoelasticity o theintegral type with aging and shrinkage(20 pages, 1998)

10. J. MohringHelmholtz Resonators with Large Aperture(21 pages, 1998)

11. H. W. Hamacher, A. SchbelOn Center Cycles in Grid Graphs(15 pages, 1998)

12. H. W. Hamacher, K.-H. KerInverse radiation therapy planning -a multiple objective optimisation approach(14 pages, 1999)

13. C. Lang, J. Ohser, R. HilerOn the Analysis o Spatial Binary Images(20 pages, 1999)

14. M. JunkOn the Construction o Discrete EquilibriumDistributions or Kinetic Schemes(24 pages, 1999)

15. M. Junk, S. V. Raghurame RaoA new discrete velocity method or Navier-Stokes equations(20 pages, 1999)

16. H. NeunzertMathematics as a Key to Key Technologies(39 pages (4 PDF-Files), 1999)

17. J. Ohser, K. SandauConsiderations about the Estimation o theSize Distribution in Wicksells CorpuscleProblem(18 pages, 1999)

18. E. Carrizosa, H. W. Hamacher, R. Klein,S. Nickel

Solving nonconvex planar location prob-lems by nite dominating setsKeywords: Continuous Location, Polyhedral Gauges,

Finite Dominating Sets, Approximation, Sandwich Algo-

rithm, Greedy Algorithm

(19 pages, 2000)

19. A. BeckerA Review on Image Distortion MeasuresKeywords: Distortion measure, human visual system

(26 pages, 2000)

20. H. W. Hamacher, M. Labb, S. Nickel,T. Sonneborn

Polyhedral Properties o the UncapacitatedMultiple Allocation Hub Location ProblemKeywords: integer programming, hub location, acility

location, valid inequalities, acets, branch and cut

(21 pages, 2000)

21. H. W. Hamacher, A. SchbelDesign o Zone Tari Systems in PublicTransportation(30 pages, 2001)

22. D. Hietel, M. Junk, R. Keck, D. TeleagaThe Finite-Volume-Particle Method orConservation Laws(16 pages, 2001)

23. T. Bender, H. Hennes, J. Kalcsics, M. T. Melo,S. Nickel

Location Sotware and Interace with GISand Supply Chain ManagementKeywords: acility location, sotware development,

geographical inormation systems, supply chain man-

agement

(48 pages, 2001)

24. H. W. Hamacher, S. A. TjandraMathematical Modelling o EvacuationProblems: A State o Art(44 pages, 2001)

25. J. Kuhnert, S. TiwariGrid ree method or solving the PoissonequationKeywords: Poisson equation, Least squares method,

Grid ree method

(19 pages, 2001)

26. T. Gtz, H. Rave, D. Reinel-Bitzer,K. Steiner, H. Tiemeier

Simulation o the ber spinning processKeywords: Melt spinning, ber model, Lattice Boltz-

mann, CFD

(19 pages, 2001)

27. A. ZemitisOn interaction o a liquid lm with an obstacleKeywords: impinging jets, liquid lm, models, numeri-

cal solution, shape

(22 pages, 2001)

28. I. Ginzburg, K. SteinerFree surace lattice-Boltzmann method tomodel the lling o expanding cavities byBingham FluidsKeywords: Generalized LBE, ree-surace phenomena,

interace boundary conditions, lling processes, Bing-

ham viscoplastic model, regularized models

(22 pages, 2001)

29. H. NeunzertDenn nichts ist r den Menschen als Men-schen etwas wert, was er nicht mit Leiden-schat tun kannVortrag anlsslich der Verleihung desAkademiepreises des Landes Rheinland-Palz am 21.11.2001Keywords: Lehre, Forschung, angewandte Mathematik,

Mehrskalenanalyse, Strmungsmechanik

(18 pages, 2001)

30. J. Kuhnert, S. TiwariFinite pointset method based on the projec-tion method or simulations o the incom-pressible Navier-Stokes equationsKeywords: Incompressible Navier-Stokes equations,

Meshree method, Projection method, Particle scheme,

Least squares approximation

AMS subject classication: 76D05, 76M28

(25 pages, 2001)

31. R. Korn, M. KrekelOptimal Portolios with Fixed Consumptionor Income StreamsKeywords: Portolio optimisation, stochastic control,

HJB equation, discretisation o control problems

(23 pages, 2002)

32. M. KrekelOptimal portolios with a loan dependentcredit spreadKeywords: Portolio optimisation, stochastic control,

HJB equation, credit spread, log utility, power utility,

non-linear wealth dynamics

(25 pages, 2002)

33. J. Ohser, W. Nagel, K. SchladitzThe Euler number o discretized sets on thechoice o adjacency in homogeneous latticesKeywords: image analysis, Euler number, neighborhod

relationships, cuboidal lattice

(32 pages, 2002)

34. I. Ginzburg, K. SteinerLattice Boltzmann Model or Free-Suracefow and Its Application to Filling Process inCasting


28/32

Keywords: Lattice Boltzmann models; ree-surace phe-

nomena; interace boundary conditions; lling pro-

cesses; injection molding; volume o fuid method; in-

terace boundary conditions; advection-schemes; up-

wind-schemes

(54 pages, 2002)

35. M. Gnther, A. Klar, T. Materne, R. Wegen-er

Multivalued undamental diagrams and stopand go waves or continuum trac equationsKeywords: trac fow, macroscopic equations, kinetic

derivation, multivalued undamental diagram, stop and

go waves, phase transitions

(25 pages, 2002)

36. S. Feldmann, P. Lang, D. Prtzel-WoltersParameter infuence on the zeros o net-work determinantsKeywords: Networks, Equicoactor matrix polynomials,

Realization theory, Matrix perturbation theory

(30 pages, 2002)

37. K. Koch, J. Ohser, K. SchladitzSpectral theory or random closed sets andestimating the covariance via requencyspaceKeywords: Random set, Bartlett spectrum, ast Fourier

transorm, power spectrum(28 pages, 2002)

38. D. dHumires, I. GinzburgMulti-refection boundary conditions orlattice Boltzmann modelsKeywords: lattice Boltzmann equation, boudary condis-

tions, bounce-back rule, Navier-Stokes equation

(72 pages, 2002)

39. R. KornElementare FinanzmathematikKeywords: Finanzmathematik, Aktien, Optionen, Port-

olio-Optimierung, Brse, Lehrerweiterbildung, Mathe-

matikunterricht

(98 pages, 2002)

40. J. Kallrath, M. C. Mller, S. NickelBatch Presorting Problems:Models and Complexity ResultsKeywords: Complexity theory, Integer programming,

Assigment, Logistic s

(19 pages, 2002)

41. J. LinnOn the rame-invariant description o thephase space o the Folgar-Tucker equationKey words: ber orientation, Folgar-Tucker equation, in-

jection molding

(5 pages, 2003)

42. T. Hanne, S. NickelA Multi-Objective Evolutionary Algorithmor Scheduling and Inspection Planning inSotware Development ProjectsKey words: multiple objective programming, project

management and scheduling, sotware development,

evolutionary algorithms, ecient set

(29 pages, 2003)

43. T. Borteld , K.-H. Ker, M. Monz,A. Scherrer, C. Thieke, H. Trinkaus

Intensity-Modulated Radiotherapy - A LargeScale Multi-Criteria Programming ProblemKeywords: multiple criteria optimization, representa-

tive systems o Pareto solutions, adaptive triangulation,

clustering and disaggregation techniques, visualization

o Pareto solutions, medical physics, external beam ra-

diotherapy planning, intensity modulated radiotherapy

(31 pages, 2003)

44. T. Halmann, T. WichmannOverview o Symbolic Methods in Industrial

Analog Circuit DesignKeywords: CAD, automated analog circuit design, sym-

bolic analysis, computer algebra, behavioral modeling,

system simulation, circuit sizing, macro modeling, di-

erential-algebraic equations, index

(17 pages, 2003)

45. S. E. Mikhailov, J. OrlikAsymptotic Homogenisation in Strengthand Fatigue Durability Analysis o Compos-ites

Keywords: multiscale structures, asymptotic homoge-nization, strength, atigue, singularity, non-local con-

ditions

(14 pages, 2003)

46. P. Domnguez-Marn, P. Hansen,N. Mladenovi c , S. Nickel

Heuristic Procedures or Solving theDiscrete Ordered Median ProblemKeywords: genetic algorithms, variable neighborhood

search, discrete acility location

(31 pages, 2003)

47. N. Boland, P. Domnguez-Marn, S. Nickel,J. Puerto

Exact Procedures or Solving the DiscreteOrdered Median ProblemKeywords: discrete location, Integer programming

(41 pages, 2003)

48. S. Feldmann, P. LangPad-like reduction o stable discrete linearsystems preserving their stabilityKeywords: Discrete linear systems, model reduction,

stability, Hankel matrix, Stein equation

(16 pages, 2003)

49. J. Kallrath, S. NickelA Polynomial Case o the Batch PresortingProblemKeywords: batch presorting problem, online optimization,

competetive analysis, polynomial algorithms, logistics

(17 pages, 2003)

50. T. Hanne, H. L. TrinkausknowCube or MCDM Visual and Interactive Support orMulticriteria Decision MakingKey words: Multicriteria decision making, knowledge

management, decision support systems, visual interac-

es, interactive navigation, real-lie applications.

(26 pages, 2003)

51. O. Iliev, V. LaptevOn Numerical Simulation o Flow ThroughOil FiltersKeywords: oil lters, coupled fow in plain and porous

media, Navier-Stokes, Brinkman, numerical simulation

(8 pages, 2003)

52. W. Drfer, O. Iliev, D. Stoyanov, D. VassilevaOn a Multigrid Adaptive Renement Solveror Saturated Non-Newtonian Flow inPorous MediaKeywords: Nonlinear multigrid, adaptive renement,

non-Newtonian fow in porous media

(17 pages, 2003)

53. S. Kruse

On the Pricing o Forward Starting Optionsunder Stochastic VolatilityKeywords: Option pricing, orward starting options,

Heston model, stochastic volatility, cliquet options

(11 pages, 2003)

54. O. Iliev, D. StoyanovMultigrid adaptive local renement solveror incompressible fowsKeywords: Navier-Stokes equations, incompressible fow,

projection-type splitting, SIMPLE, multigrid methods,

adaptive local renement, lid-driven fow in a cavity

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55. V. StarikoviciusThe multiphase fow and heat transer inporous mediaKeywords: Two-phase fow in porous media, various

ormulations, global pressure, multiphase mixture mod-

el, numerical simulation(30 pages, 2003)

56. P. Lang, A. Sarishvili, A. WirsenBlocked neural networks or knowledge ex-traction in the sotware development processKeywords: Blocked Neural Networks, Nonlinear Regres-

sion, Knowledge Extraction, Code Inspection

(21 pages, 2003)

57. H. Kna, P. Lang, S. ZeiserDiagnosis aiding in RegulationThermography using Fuzzy LogicKeywords: uzzy logic,knowledge representation,

expert system

(22 pages, 2003)

58. M. T. Melo, S. Nickel, F. Saldanha da GamaLargescale models or dynamic multi-commodity capacitated acility locationKeywords: supply chain management, strategic

planning, dynamic location, modeling

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59. J. OrlikHomogenization or contact problems withperiodically rough suracesKeywords: asymptotic homogenization, contact problems

(28 pages, 2004)

60. A. Scherrer, K.-H. Ker, M. Monz,F. Alonso, T. Borteld

IMRT planning on adaptive volume struc-tures a signicant advance o computa-tional complexityKeywords: Intensity-modulated radiation therapy

(IMRT), inverse treatment planning, adaptive volume

structures, hierarchical clustering, local renement,

adaptive clustering, convex programming, mesh gener-

ation, multi-grid methods

(24 pages, 2004)

61. D. KehrwaldParallel lattice Boltzmann simulationo complex fowsKeywords: Lattice Boltzmann methods, parallel com-

puting, microstructure simulation, vir tual material de-

sign, pseudo-plasti c fuids, liquid composite moulding

(12 pages, 2004)

62. O. Iliev, J. Linn, M. Moog, D. Niedziela,V. Starikovicius

On the Perormance o Certain IterativeSolvers or Coupled Systems Arising in Dis-cretization o Non-Newtonian Flow Equa-tionsKeywords: Perormance o iterative solvers, Precondi-

tioners, Non-Newtonian fow

(17 pages, 2004)

63. R. Ciegis, O. Iliev, S. Rie, K. SteinerOn Modelling and Simulation o DierentRegimes or Liquid Polymer Moulding


29/32

Keywords: Liquid Polymer Moulding, Modelling, Simu-

lation, Inltration, Front Propagation, non-Newtonian

fow in porous media

(43 pages, 2004)

64. T. Hanne, H. NeuSimulating Human Resources inSotware Development ProcessesKeywords: Human resource modeling, sotware process,

productivity, human actors , learning curve

(14 pages, 2004)

65. O. Iliev, A. Mikelic, P. Popov

Fluid structure interaction problems in de-ormable porous media: Toward permeabil-ity o deormable porous mediaKeywords: fuid-structure interaction, deormable po-

rous media, upscaling, linear elasticity, stokes, nite el-

ements

(28 pages, 2004)

66. F. Gaspar, O. Iliev, F. Lisbona, A. Naumovich,P. Vabishchevich

On numerical solution o 1-D poroelasticityequations in a multilayered domainKeywords: poroelasticity, multilayered material, nite

volume discretization, MAC type grid

(41 pages, 2004)

67. J. Ohser, K. Schladitz, K. Koch, M. NtheDiraction by image processing and its ap-plication in materials scienceKeywords: porous microstructure, image analysis, ran-

dom set, ast Fourier transorm, power spectrum, Bar-

tlett spectrum

(13 pages, 2004)

68. H. NeunzertMathematics as a Technology: Challengesor the next 10 YearsKeywords: applied mathematics, technology, modelling,

simulation, visualization, optimization, glass processing,

spinning processes, ber-fuid interaction, trubulence

eects, topological optimization, multicriteria optimiza-tion, Uncertainty and Risk, nancial mathematics, Mal-

liavin calculus, Monte-Carlo methods, virtual material

design, ltration, bio-inormatics, system biology

(29 pages, 2004)

69. R. Ewing, O. Iliev, R. Lazarov, A. NaumovichOn convergence o certain nite dierencediscretizations or 1D poroelasticity inter-ace problemsKeywords: poroelasticity, multilayered material, nite

volume discretizations, MAC type grid, error estimates

(26 pages,2004)

70. W. Drfer, O. Iliev, D. Stoyanov, D. Vassileva

On Ecient Simulation o Non-Newto-nian Flow in Saturated Porous Media with aMultigrid Adaptive Renement SolverKeywords: Nonlinear multigrid, adaptive renement,

non-Newtonian in porous media

(25 pages, 2004)

71. J. Kalcsics, S. Nickel, M. SchrderTowards a Unied Territory Design Approach

Applications, Algorithms and GIS IntegrationKeywords: territory desgin, political districting, sales

territory alignment, optimization algorithms, Geo-

graphical Inormation Systems

(40 pages, 2005)

72. K. Schladitz, S. Peters, D. Reinel-Bitzer,A. Wiegmann, J. Ohser

Design o acoustic trim based on geometricmodeling and fow simulation or non-woven

Keywords: random system o bers, Poisson

line process, fow resistivity, acoustic absorption,

Lattice-Boltzmann method, non-woven

(21 pages, 2005)

73. V. Rutka, A. WiegmannExplicit Jump Immersed Interace Methodor virtual material design o the eectiveelastic moduli o composite materialsKeywords: virtual material design, explicit jump im-

mersed interace method, eective elastic moduli,

composite materials

(22 pages, 2005)

74. T. HanneEine bersicht zum Scheduling von BaustellenKeywords: Projektplanung, Scheduling, Bauplanung,

Bauindustrie

(32 pages, 2005)

75. J. LinnThe Folgar-Tucker Model as a Di eretialAlgebraic System or Fiber OrientationCalculationKeywords: ber orientation, FolgarTucker model, in-

variants, algebraic constraints, phase space, trace sta-

bility

(15 pages, 2005)

76. M. Speckert, K. Dreler, H. Mauch,A. Lion, G. J. Wierda

Simulation eines neuartigen Prsystemsr Achserprobungen durch MKS-Model-lierung einschlielich RegelungKeywords: virtual test rig, suspension testing,

multibody simulation, modeling hexapod test rig, opti-

mization o test rig conguration

(20 pages, 2005)

77. K.-H. Ker, M. Monz, A. Scherrer, P. Sss,F. Alonso, A. S. A. Sultan, Th. Borteld,D. Crat, Chr. Thieke

Multicriteria optimization in intensity

modulated radiotherapy planningKeywords: multicriteria optimization, extreme solu-

tions, real-time decision making, adaptive approxima-

tion schemes, clustering methods, IMRT planning, re-

verse engineering

(51 pages, 2005)

78. S. Amstutz, H. AndrA new algorithm or topology optimizationusing a level-set methodKeywords: shape optimization, topology optimization,

topological sensitivity, level-set

(22 pages, 2005)

79. N. Ettrich

Generation o surace elevation models orurban drainage simulationKeywords: Flooding, simulation, urban elevation

models, laser scanning

(22 pages, 2005)

80. H. Andr, J. Linn, I. Matei, I. Shklyar,K. Steiner, E. Teichmann

OPTCAST Entwicklung adquater Struk-turoptimierungsverahren r GieereienTechnischer Bericht (KURZFASSUNG)Keywords: Topologieoptimierung, Level-Set-Methode,

Gieprozesssimulation, Gietechnische Restriktionen,

CAE-Kette zur Strukturoptimierung

(77 pages, 2005)

81. N. Marheineke, R. WegenerFiber Dynamics in Turbulent FlowsPart I: General Modeling Framework

Keywords: ber-fuid interaction; Cosserat rod; turbu-

lence modeling; Kolmogorovs energy spectrum; dou-

ble-velocity correlations; dierentiable Gaussian elds

(20 pages, 2005)

Part II: Spec ic Taylor DragKeywords: fexible bers; k-e turbulence model; -

ber-turbulence interaction scales; air drag; random

Gaussian aerodynamic orce; white noise; stochastic

dierential equations; ARMA process

(18 pages, 2005)

82. C. H. Lampert, O. WirjadiAn Optimal Non-Orthogonal Separation othe Anisotropic Gaussian Convolution FilterKeywords: Anisotropic Gaussian lter, linear ltering, ori-

entation space, nD image processing, separable lters

(25 pages, 2005)

83. H. Andr, D. StoyanovError indicators in the parallel nite ele-ment solver or linear elasticity DDFEMKeywords: linear elasticity, nite element method, hier-

archical shape unctions, domain decom-position, par-

allel implementation, a posteriori error estimates

(21 pages, 2006)

84. M. Schrder, I. Solchenbach

Optimization o Transer Quality inRegional Public TransitKeywords: public transit, transer quality, quadratic

assignment problem

(16 pages, 2006)

85. A. Naumovich, F. J. GasparOn a multigrid solver or the three-dimen-sional Biot poroelasticity system in multi-layered domainsKeywords: poroelasticity, interace problem, multigrid,

operator-dependent prolongation

(11 pages, 2006)

86. S. Panda, R. Wegener, N. Marheineke

Slender Body Theory or the Dynamics oCurved Viscous FibersKeywords: curved viscous bers; fuid dynamics; Navier-

Stokes equations; ree boundary value problem; asymp-

totic expansions; slender body theory

(14 pages, 2006)

87. E. Ivanov, H. Andr, A. KudryavtsevDomain Decomposition Approach or Auto-matic Parallel Generation o Tetrahedral GridsKey words: Grid Generation, Unstructured Grid, Delau-

nay Triangulation, Parallel Programming, Domain De-

composition, Load Balancing

(18 pages, 2006)

88. S. Tiwari, S. Antonov, D. Hietel, J. Kuhnert,R. Wegener

A Meshree Method or Simulations o In-teractions between Fluids and FlexibleStructuresKey words: Meshree Method, FPM, Fluid Structure

Interaction, Sheet o Paper, Dynamical Coupling

(16 pages, 2006)

89. R. Ciegis , O. Iliev, V. Starikovicius, K. SteinerNumerical Algorithms or Solving Problemso Multiphase Flows in Porous MediaKeywords: nonlinear algorithms, nite-volume method,

sotware tools, porous media, fows

(16 pages, 2006)

90. D. Niedziela, O. Iliev, A. LatzOn 3D Numerical Simulations o ViscoelasticFluids


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Keywords: non-Newtonian fuids, anisotropic viscosity,

integral constitutive equation

(18 pages, 2006)

91. A. WintereldApplication o general semi-innite Pro-gramming to Lapidary Cutting ProblemsKeywords: large scale optimization, nonlinear program-

ming, general semi-innite optimization, design center-

ing, clustering

(26 pages, 2006)

92. J. Orlik, A. Ostrovska

Space-Time Finite Element Approximationand Numerical Solution o HereditaryLinear Viscoelasticity ProblemsKeywords: hereditary viscoelasticity; kern approxima-

tion by interpolation; space-time nite element approxi-

mation, stability and a priori estimate

(24 pages, 2006)

93. V. Rutka, A. Wiegmann, H. AndrEJIIM or Calculation o eective ElasticModuli in 3D Linear ElasticityKeywords: Elliptic PDE, linear elasticity, irregular do-

main, nite dierences, ast solvers, eective elas-

tic moduli

(24 pages, 2006)

94. A. Wiegmann, A. ZemitisEJ-HEAT: A Fast Explicit Jump HarmonicAveraging Solver or the Eective HeatConductivity o Composite MaterialsKeywords: Stationary heat equation, eective ther-

mal conductivity, explicit jump, discontinuous coe-

cients, virtual material design, microstructure simula-

tion, EJ-HEAT

(21 pages, 2006)

95. A. NaumovichOn a nite volume discretization o thethree-dimensional Biot poroelasticity sys-tem in multilayered domains

Keywords: Biot poroelasticity system, interace problems,nite volume discretization, nite dierence method

(21 pages, 2006)

96. M. Krekel, J. WenzelA unied approach to Credit Deault Swap-tion and Constant Maturity Credit DeaultSwap valuationKeywords: LIBOR market model, credit risk, Credit De-

ault Swaption, Constant Maturity Credit Deault Swap-

method

(43 pages, 2006)

97. A. DreyerInterval Methods or Analog Circiuts

Keywords: interval arithmetic, analog circuits, toleranceanalysis, parametric linear systems, requency response,

symbolic analysis, CAD, computer algebra

(36 pages, 2006)

98. N. Weigel, S. Weihe, G. Bitsch, K. DrelerUsage o Simulation or Design and Optimi-zation o TestingKeywords: Vehicle test rigs, MBS, control, hydraulics,

testing philosophy

(14 pages, 2006)

99. H. Lang, G. Bitsch, K. Dreler, M. SpeckertComparison o the solutions o the elasticand elastoplastic boundary value problemsKeywords: Elastic BVP, elastoplastic BVP, variational

inequalities, rate-independency, hysteresis, linear kine-

matic hardening, stop- and play-operator

(21 pages, 2006)

100. M. Speckert, K. Dreler, H. MauchMBS Simulation o a hexapod based sus-pension test rigKeywords: Test rig, MBS simulation, suspension,

hydraulics, controlling, design optimization

(12 pages, 2006)

101. S. Azizi Sultan, K.-H. KerA dynamic algorithm or beam orientationsin multicriteria IMRT planningKeywords: radiotherapy planning, beam orientation

optimization, dynamic approach, evolutionary algo-

rithm, global optimization

(14 pages, 2006)

102. T. Gtz, A. Klar, N. Marheineke, R. WegenerA Stochastic Model or the Fiber Lay-downProcess in the Nonwoven ProductionKeywords: ber dynamics, stochastic Hamiltonian sys-

tem, stochastic averaging

(17 pages, 2006)

103. Ph. Sss, K.-H. KerBalancing control and simplicity: a variableaggregation method in intensity modulatedradiation therapy planningKeywords: IMRT planning, variable aggregation, clus-

tering methods

(22 pages, 2006)

104. A. Beaudry, G. Laporte, T. Melo, S. NickelDynamic transportation o patients in hos-pitalsKeywords: in-house hospital transportation, dial-a-ride,

dynamic mode, tabu search

(37 pages, 2006)

105. Th. HanneApplying multiobjective evolutionary algo-rithms in industrial projectsKeywords: multiobjective evolutionary algorithms, dis-

crete optimization, continuous optimization, electronic

circuit design, semi-innite programming, scheduling

(18 pages, 2006)

106. J. Franke, S. HalimWild bootstrap tests or comparing signalsand imagesKeywords: wild bootstrap test, texture classication,

textile quality control, deect detection, kernel estimate,

nonparametric regression

(13 pages, 2007)

107. Z. Drezner, S. NickelSolving the ordered one-median problem inthe planeKeywords: planar location, global optimization, ordered

median, big triangle small triangle method, bounds,

numerical experiments(21 pages, 2007)

108. Th. Gtz, A. Klar, A. Unterreiter,R. Wegener

Numerical evidance or the non-existing osolutions o the equations desribing rota-tional ber spinningKeywords: rotational ber spinning, viscous bers,

boundary value problem, existence o solutions

(11 pages, 2007)

109. Ph. Sss, K.-H. KerSmooth intensity maps and the Borteld-

Boyer sequencerKeywords: probabilistic analysis, intensity modulatedradiotherapy treatment (IMRT), IMRT plan application,

step-and-shoot sequencing

(8 pages, 2007)

110. E. Ivanov, O. Gluchshenko, H. Andr,A. Kudryavtsev

Parallel sotware tool or decomposing andmeshing o 3d structuresKeywords: a-priori domain decomposition, unstruc-

tured grid, Delaunay mesh generation

(14 pages, 2007)

111. O. Iliev, R. Lazarov, J. WillemsNumerical study o two-grid precondition-ers or 1d elliptic problems with highlyoscillating discontinuous coecientsKeywords: two-grid algorithm, oscillating coecients,

preconditioner(20 pages, 2007)

112. L. Bonilla, T. Gtz, A. Klar, N. Marheineke,R. Wegener

Hydrodynamic limit o the Fokker-Planck-equation describing ber lay-down pro-cessesKeywords: stochastic dierential equations, Fokker-

Planck equation, asymptotic expansion, Ornstein-

Uhlenbeck process

(17 pages, 2007)

113. S. RieModeling and simulation o the pressing

section o a paper machineKeywords: paper machine, computational fuid dynam-

ics, porous media

(41 pages, 2007)

114. R. Ciegis, O. Iliev, Z. LakdawalaOn parallel numerical algorithms or simu-lating industrial ltration problemsKeywords: Navier-Stokes-Brinkmann equations, nite

volume discretization method, SIMPLE, parallel comput-

ing, data decomposition method

(24 pages, 2007)

115. N. Marheineke, R. WegenerDynamics o curved viscous bers with sur-ace tensionKeywords: Slender body theory, curved viscous bers

with surace tension, ree boundary value problem

(25 pages, 2007)

116. S. Feth, J. Franke, M. SpeckertResampling-Methoden zur mse-Korrekturund Anwendungen in der BetriebsestigkeitKeywords: Weibull, Bootstrap, Maximum-Likelihood,

Betriebsestigkeit

(16 pages, 2007)

117. H. KnaKernel Fisher discriminant unctions a con-

cise and rigorous introductionKeywords: wild bootstrap test, texture classication,textile quality control, deect detection, kernel estimate,

nonparametric regression

(30 pages, 2007)

118. O. Iliev, I. RybakOn numerical upscaling or fows in hetero-geneous porous mediaKeywords: numerical upscaling, heterogeneous porous

media, single phase fow, Darcys law, multiscale prob-

lem, eective permeability, multipoint fux approxima-

tion, anisotropy

(17 pages, 2007)

119. O. Iliev, I. RybakOn approximation property o multipointfux approximation method


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Keywords: Multipoint fux approximation, nite volume

method, elliptic equation, discontinuous tensor coe-

cients, anisotropy

(15 pages, 2007)

120. O. Iliev, I. Rybak, J. WillemsOn upscaling heat conductivity or a class oindustrial problemsKeywords: Multiscale problems, eective heat conduc-

tivity, numerical upscaling, domain decomposition

(21 pages, 2007)

121. R. Ewing, O. Iliev, R. Lazarov, I. Rybak

On two-level preconditioners or fow inporous mediaKeywords: Multiscale problem, Darcys law, single

phase fow, anisotropic heterogeneous porous media,

numerical upscaling, multigrid, domain decomposition,

ecient preconditioner

(18 pages, 2007)

122. M. Brickenstein, A. DreyerPOLYBORI: A Grbner basis rameworkor Boolean polynomialsKeywords: Grbner basis, ormal verication, Boolean

polynomials, algebraic cr yptoanalysis, satisabilit y

(23 pages, 2007)

123. O. WirjadiSurvey o 3d image segmentation methodsKeywords: image processing, 3d, image segmentation,

binarization

(20 pages, 2007)

124. S. Zeytun, A. GuptaA Comparative Study o the Vasicek and theCIR Model o the Short RateKeywords: interest rates, Vasicek model, CIR-model,

calibration, parameter estimation

(17 pages, 2007)

125. G. Hanselmann, A. SarishviliHeterogeneous redundancy in sotware

quality prediction using a hybrid BayesianapproachKeywords: reliability prediction, ault prediction, non-

homogeneous poisson process, Bayesian model aver-

aging

(17 pages, 2007)

126. V. Maag, M. Berger, A. Wintereld, K.-H.Ker

A novel non-l inear approach to minimalarea rectangular packingKeywords: rectangular packing, non-overlapping con-

straints, non-linear optimization, regularization, relax-

ation

(18 pages, 2007)

127. M. Monz, K.-H. Ker, T. Borteld, C. ThiekePareto navigation systematic multi-crite-ria-based IMRT treatment plan determina-tionKeywords: convex, interactive multi-objective optimiza-

tion, intensity modulated radiotherapy planning

(15 pages, 2007)

128. M. Krause, A. ScherrerOn the role o modeling parameters in IMRTplan optimizationKeywords: intensity-modulated radiotherapy (IMRT),

inverse IMRT planning, convex optimization, sensitiv-

ity analysis, elasticity, modeling parameters, equivalent

uniorm dose (EUD)(18 pages, 2007)

129. A. WiegmannComputation o the permeability o porousmaterials rom their microstructure by FFF-StokesKeywords: permeability, numerical homogenization,

ast Stokes solver

(24 pages, 2007)

130. T. Melo, S. Nickel, F. Saldanha da GamaFacility Location and Supply Chain Manage-ment A comprehensive reviewKeywords: acility location, supply chain management,

network design

(54 pages, 2007)

131. T. Hanne, T. Melo, S. NickelBringing robustness to patient fowmanagement through optimized patienttransports in hospitalsKeywords: Dial-a-Ride problem, online problem, case

study, tabu search, hospital logistics

(23 pages, 2007)

132. R. Ewing, O. Iliev, R. Lazarov, I. Rybak,J. Willems

An ecient approach or upscaling proper-ties o composite materials with high con-trast o coecientsKeywords: eective heat conductivity, permeability o

ractured porous media, numerical upscaling, brous

insulation materials, metal oams

(16 pages, 2008)

133. S. Gelareh, S. NickelNew approaches to hub location problemsin public transport planningKeywords: integer programming, hub location, trans-

portation, decomposition, heur istic

(25 pages, 2008)

134. G. Thmmes, J. Becker, M. Junk, A. K. Vai-kuntam, D. Kehrwald, A. Klar, K. Steiner,

A. WiegmannA Lattice Boltzmann Method or immisciblemultiphase fow simulations using the LevelSet MethodKeywords: Lattice Boltzmann method, Level Set

method, ree surace, multiphase fow

(28 pages, 2008)

135. J. OrlikHomogenization in elasto-plasticityKeywords: multiscale structures, asymptotic homogeni-

zation, nonlinear energy

(40 pages, 2008)

136. J. Almquist, H. Schmidt, P. Lang, J. Deitmer,

M. Jirstrand, D. Prtzel-Wolters, H. BeckerDetermination o interaction betweenMCT1 and CAI I via a mathematical andphysiological approachKeywords: mathematical modeling; model reduction;

electrophysiology; pH-sensitive microelectrodes; pro-

ton antenna

(20 pages, 2008)

137. E. Savenkov, H. Andr, O. IlievAn analysis o one regularization approachor solution o pure Neumann problemKeywords: pure Neumann problem, elasticity, regular-

ization, nite element method, condition number

(27 pages, 2008)

138. O. Berman, J. Kalcsics, D. Krass, S. NickelThe ordered gradual covering locationproblem on a network

Keywords: gradual covering, ordered median unction,

network location

(32 pages, 2008)

139. S. Gelareh, S. NickelMulti-period public transport design: Anovel model and solution approachesKeywords: Integer programming, hub location, public

transport, multi-period planning, heuristics

(31 pages, 2008)

140. T. Melo, S. Nickel, F. Saldanha-da-GamaNetwork design decisions in supply chain

planningKeywords: supply chain design, integer programming

models, location models, heuristics

(20 pages, 2008)

141. C. Lautensack, A. Srkk, J. Freitag,K. Schladitz

Anisotropy analysis o pressed point pro-cessesKeywords: estimation o compression, isotropy test,

nearest neighbour distance, orientation analysis, polar

ice, Ripleys K unction

(35 pages, 2008)

142. O. Iliev, R. Lazarov, J. Willems

A Graph-Laplacian approach or calculatingthe eective thermal conductivity o com-plicated ber geometriesKeywords: graph laplacian, eective heat conductivity,

numerical upscaling, brous materials

(14 pages, 2008)

143. J. Linn, T. Stephan, J. Carlsson, R. BohlinFast simulation o quasistatic rod deorma-tions or VR applicationsKeywords: quasistatic deormations, geometrically

exact rod models, variational ormulation, energy min-

imization, nite dierences, nonlinear conjugate gra-

dients

(7 pages, 2008)

144. J. L inn, T. StephanSimulation o quasistatic deormations us-ing discrete rod modelsKeywords: quasistatic deormations, geometrically

exact rod models, variational ormulation, energy min-

imization, nite dierences, nonlinear conjugate gra-

dients

(9 pages, 2008)

145. J. Marburger, N. Marheineke, R. PinnauAdjoint based optimal control using mesh-less discretizationsKeywords: Mesh-less methods, particle methods, Eul-

erian-Lagrangian ormulation, optimization strategies,

adjoint method, hyperbolic equations(14 pages, 2008

146. S. Desmettre, J. Gould, A. SzimayerOwn-company stockholding and work eortpreerences o an unconstrained executiveKeywords: optimal portolio choice, executive compen-

sation

(33 pages, 2008)

147. M. Berger, M. schrder, K.-H. KerA constraint programming approach or thetwo-dimensional rectangular packing prob-lem with orthogonal orientations

Keywords: rectangular packing, orthogonal orienta-tions non-overlapping constraints, constraint propa-

gation

(13 pages, 2008)


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148. K. Schladitz, C. Redenbach, T. Sych,M. Godehardt

Microstructural characterisation o openoams using 3d imagesKeywords: virtual material design, image analysis, open

oams

(30 pages, 2008)

149. E. Fernndez, J. Kalcsics, S. Nickel,R. Ros-Mercado

A novel territory design model arising inthe implementation o the WEEE-DirectiveKeywords: heuristics, optimization, logistics, recycling

(28 pages, 2008)

150. H. Lang, J. Linn

Lagrangian eld theory in space-time orgeometrically exact Cosserat rodsKeywords: Cosserat rods, geometrically exact rods,

small strain, large deormation, deormable bodies ,

Lagrangian eld theory, variational calculus

(19 pages, 2009)

151. K. Dreler, M. Speckert, R. Mller,Ch. Weber

Customer loads correlation in truck engi-neering

Keywords: Customer distribution, saety critical compo-nents, quantile estimation, Monte-Carlo methods

(11 pages, 2009)

152. H. Lang, K. Dreler

An improved multiaxial stress-strain correc-tion model or elastic FE postprocessingKeywords: Jiangs model o elastoplasticity, stress-strain

correction, parameter identication, automatic dier-

entiation, least-squares optimization, Coleman-Li algo-

rithm

(6 pages, 2009)

153. J. Kalcsics, S. Nickel, M. Schrder

A generic geometric approach to territorydesign and districtingKeywords: Territory design, districting, combinatorial

optimization, heuristics, computational geometry

(32 pages, 2009)

154. Th. Ftterer, A. Klar, R. Wegener

An energy conserving numerical scheme orthe dynamics o hyperelastic rodsKeywords: Cosserat rod, hyperealstic, energy conserva-

tion, nite dierences

(16 pages, 2009)

155. A. Wiegmann, L. Cheng, E. Glatt, O. Iliev,S. Rie

Design o pleated lters by computer sim-ulationsKeywords: Solid-gas separation, solid-liquid separation,

pleated lter, design, simulation

(21 pages, 2009)

156. A. Klar, N. Marheineke, R. Wegener

Hierarchy o mathematical models or pro-duction processes o technical textilesKeywords: Fiber-fuid interaction, slender-body theory,

turbulence modeling, model reduction, stochastic di-

erential equations, Fokker-Planck equation, asymptotic

expansions, parameter identication

(21 pages, 2009)

157. E. Glatt, S. Rie, A. Wiegmann, M. Kneel,E Wegenke

Keywords: metal wire mesh, structure simulation,

model calibration, CFD simulation, pressure loss

(7 pages, 2009)

158. S. Kruse, M. Mller

Pricing American call options under the as-sumption o stochastic dividends An ap-plication o the Korn-Rogers modelKeywords: option pricing, American options, dividends,

dividend discount model, Black-Scholes model

(22 pages, 2009)

Status quo: April 2009

pricing american call options under the assumption of stochastic dividends

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