SHANNONS INFORMATION THEORY ITS APPLICATIONS IN DERIVATIVE PRICING - - PDF document

SHANNON’S INFORMATION THEORY ITS APPLICATIONS IN DERIVATIVE PRICING

ALEXANDER KUSHPEL AND JEREMY LEVESLEY

• Abstract. During the past two decades Lévy processes became very popular

in Financial Mathematics. Truncated Lévy distributions were used for mod- eling by Mantegna and Stanley [26], [27]. Later Novikov [29] and Koponen [16] introduced a family of in…nitely divisible analogs of these distributions. These models have been generalised by Boyarchenko and Levendorskii [6], and are known now as KoBoL models. Such models provide a good …t in many

• situations. Our aim is to shed a fresh light onto the pricing theory using regu-

lar Lévy processes of exponential type. We analyse pricing formula, construct and discuss several methods of approximation which are almost optimal in the sense of respective n-widths. Our approach has its roots in Shannon’s Information Theory.

• 1. introduction

To motivate the reader we start with the one dimensional situation. We shall study a common frictionless market consisting of a riskless bond and stock which is modeled by an exponential Lévy process St = eXt under a …xed equivalent martingale measure Q with a given constant riskless rate r > 0. Since in our model the stock does not pay dividends then the discounted stock price ertSt must be a martingale under Q. Consider a contract (European call option) which gives to its owner the right but not the obligation to buy the underlying asset for the …xed strike price K at the speci…ed expiry date T. We need to evaluate its price Fcall(St; t). In this case the payo¤ has the form (1.1) G(x) = (ex K)+; where for any a 2 R, (a)+ = maxfa; 0g, K is the strike price and x = ln St. In the classical Merton-Black-Scholes model the price of a stock follows the Geo- metric Brownian motion de…ned as St = exp(Xt), where Xt, t 0 is the Brownian motion with the probability density function pt(x) =

• 22t

1=2 exp

• (x t)2

22t

• for the increments Xt+t Xt and parameters and are known as drift and

volatility respectively. It is well-known that Merton-Black-Scholes theory becomes much more e¢cient if additional stochastic factors are introduced. Consequently, it is important to consider a wider family of Lévy processes. Stable Lévy processes have been used …rst in this context by Mandelbrot [25] and Fama [13].

Date: June 26, 2011. 1991 Mathematics Subject Classi…cation. 91G20, 60G51, 91G60, 91G80. Key words and phrases. Lévy process, n-widths, Wiener spaces, approximation.

1

2 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

From the 90th Lévy processes became very popular (see, e.g., [26], [27], [5], [6] and references therein).

• 2. basic definitions and results

Let x; y 2 Rn, x = (x1; :::; xn) ; y = (y1; :::; yn), x y be the usual scalar product in Rn, i.e., x y =

n

X

k=1

xkyk 2 R and jxj := (x x)1=2 : For a …nite measure on Rn we de…ne its Fourier transform as b (y) = F (y) = Z

Rn eixy (dx)

and its formal inverse (dx) = F1b (dx) = 1 (2)n Z

Rn eixyb

(y) dy: It is known that if is in…nitely divisible then there exists a unique continuous function : Rn! C such that (0) = 0 and e(y) = b (y) : Hence, the characteristic function of the distribution of Xt of any Lévy process can be represented in the form E

• eixXt

= et (x); where x 2 Rn, t 2 R+ and the function (x) is uniquely determined. This function is called the characteristic exponent. Vice versa, a Lévy process X = fXtgt2R+ is determined uniquely by its characteristic exponent (x). We say that a matrix A is nonnegative-de…nite (or positive-semide…nite) if xAx 0 for all x 2 Cn (or for all x 2 Rn for the real matrix), where x is the conjugate

• transpose. A matrix A is nonnegative-de…nite if and only if it arises as the Gram

matrix of some set of vectors v1; ; vn, i.e. A = (aij) = vj vi. The key role in our analysis plays the following classical result known as the Lévy-Khintchine formula which gives a representation of characteristic functions of all in…nitely divisible distributions. Theorem 1. Let X = fXtgt2R+ be a Lévy process on Rn. Then its characteristic exponent admits the representation (2.1) (y) = 1 2Ay y ib y Z

Rn

• 1 eiyx + iy xD(x)
• (dx);

where D(x) is the characteristic function of D := fx 2 Rn; jxj 1g, A is a symmetric nonnegative-de…nite n n matrix, b 2 Rn and (dx) is a measure on Rn such that (2.2) Z

Rn minf1; x xg(dx) < 1; (f0g) = 0:

Hence b (y) = e (y).

DERIVATIVE PRICING 3

The density of is known as the Lévy density and A is the covariance matrix. In particular, if A = 0 (or A = (aj;k)1j;kn, aj;k = 0) then the Lévy process is a pure non-Gaussian process and if = 0 the process is Gaussian. We say that the Lévy process has bounded variation if its sample paths have bounded variation on every compact time interval. A Lévy process has bounded variation i¤ A = 0 and Z

Rn min fjxj ; 1g (dx) < 1; (f0g) = 0

(see, e.g., [4], p.15). The systematic exposition of the theory of Lévy processes can be found in [10], [11], [12], [31], [2], [28].

• 3. characteristic exponents and density functions

De…nition 1. We say that the process is purely discontinuous if A = (aj;k), aj;k = 0, 1 j; k n and b = 0 in (2.1). Consider the one-dimensional case. It follows from (2.1) that for a purely dis- continuous process () can be written as (3.1) () = Z

R

• 1 eix + ix[1;1](x)
• (dx):

We shall be concerned just with purely discontinuous processes. Applying (3.1) we get the following representation for the density function pt(y); pt(y) = 1 2 Z

R

eiyE[eiXt]d = F1 et () (y) = 1 2 Z

R

eiyt ()d (3.2) = 1 2 Z

R

exp

• iy t

Z

R

• 1 eix + ix[1;1](x)
• (dx)
• d:

To …nd an explicit expression of the integral (3.2) we need to assume a very speci…c form of (dx). Consequently, an explicit representations in this case is a real rarity. For instance, Barndor¤ and Nielsen [3] obtained the Lebesgue density function for generalised hyperbolic distributions which depends on …ve parameters d (x; ; ; ; ; ) =

• 2 2=2

2 + (x )2(1=2)=2 (2)1=2 1=2+1K

• 2 2

K1=2

• 2 + 2
• x 21=2

e(x); where K (x) is the modi…ed Bessel function of the third kind, K (x) = 1 2 Z 1 1ex(1)=2d: Motivated by the idea to consider a wider class of models we need to decide at which stage we should fall into numerical methods. If we still want to be strapped with an explicit form of the characteristic exponent but still agree to apply numerical methods to compute the density function pt(y) then we apply the following standard

• trick. The sense of this manipulation is based on Cauchy’s theorem (see, e.g., [30]).

Let T be the expiry date, t < T and G (XT ) 0 be the terminal payo¤ at the

4 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

expiry date T, where G (x) is de…ned by (1.1). The price of the European call

• ption formally can be obtained from

ertFcall(St; t) = E

• erT G(Xt)jXt = x
• ;

where ex = St (see, e.g. [6], p. 104). It means that Fcall(St; t) can be formally written as (3.3) Fcall(St; t) = er Z

R

G(x + y)p(y)dy where := T t is the time to expiry.

• 4. kobol family

In this section we study characteristic exponents of KoBoL family. The idea is based on a simple observation. From the Lévy-Khintchine formula (3.1) it follows that it is possible to …nd () explicitly if we can compute explicitly the inverse Fourier transform of (dx): Therefore, it was suggested by the authors of [6] to consider the following form of (dx); (dx) = jxj ejxj; where and are …xed parameters. The following de…nitions are based on this ob-

• servation. We say that a Lévy process is a regular Lévy process of exponential type

if its density has at the origin a power type singularity and decays exponentially at the in…nity. The characteristic exponent () of a regular Lévy process of expo- nential type admits an analytic extension onto the strip = 2 (; +) continuous

• n the boundary and admits the representation

() = i + () ; where () c jj ; ! 1; = 2 (; +) : It is possible to show that for any regular Lévy process of exponential type the density function pt(y) is in…nitely di¤erentiable on R and exponentially decays with all of its derivatives as y ! 1. Let +(; ; dx) = (max fx; 0g)1 exdx and (; ; dx) = ( min f0; xg)1 exdx; where < 2 and > 0: De…nition 2. A Lévy process is called a KoBoL process of order < 2 if it is a purely discontinuous with the Lévy measure of the form (dx) = c++(; ; dx) + c(; +; dx); where c+ > 0; c > 0; < 0 < +: We call the order of the process, + and the steepness parameters and c+; c the intensity parameters of the process. The parameter (+ respectively) determines the rate of the exponential decay of the right (left respectively) tail of the density function. It is easy to see that the condition (2.2) is satis…ed, i.e., Z

R

min

• 1; x2

c++(; ; dx) + c(; +; dx)

• < 1:

DERIVATIVE PRICING 5

Moreover, if < 1 then Z

R

min f1; jxjg

• c++(; ; dx) + c(; +; dx)
• < 1;

i.e., a KoBoL process is of …nite variation i¤ < 1: In what follows we shall adopt the standard notations, z = e ln z, where ; z 2 C such that z 62 (1; 0] and ln z denotes the branch of ln z de…ned on C n (1; 0] and such that that ln(1) = 0. Integrating by parts it is possible to show that if < 2; 6= 0; 1; then () = i + c+ () [() ( i)] +c ()

• + (+ + i)

: If = 0; then () = i + c+ [ln ( i) ln ()] +c [ln (+ + i) ln +] : If = 1; then () = i + c+ [() ln () ( i) ln ( i)] +c [+ ln + (+ + i) ln (+ + i)] ; where 2 R; c > 0; and < 0 < + (see [6], p.70). For larger and c we get a larger peak of the probability distribution. The parameters c+ and c control asymmetry of the probability distribution while and + determine the rate of exponential decay as ! 1:

• 5. theoretical background

Our main aim is to approximate the function Fcall de…ned on R. Remark that in [15] and [18] to approximate the integral (3.3) the authors use a piecewise linear approximation on a uniform mesh with step h > 0. Observe that such kind of ap- proximation has saturation of order O(h2) and can not re‡ect analytic properties

• f the characteristic exponent (). Possible use of polynomial splines of higher

degree, r 2 signi…cantly increase complexity of computations and gives the order

• f saturation O(hr+1) which is also too far from the optimal rate of convergence.

The same argument is applicable to the possible wavelet approximations. Since any sequence of algebraic polynomials is not uniformly convergent on R (except of sta- tionary sequences) then any continuous function which is not a polynomial can not be uniformly approximated. If we try to use rational approximations then we will be able to approximate just such continuous functions f(x) that limx!1 f(x) = c, where c 2 R is some constant, which is not su¢cient for our applications. To compare and determine which apparatus of approximation is better over a wide range of methods of approximation we will need to set up respective extremal

• problem. The key problem here is that the information regarding smoothness of

characteristic exponents () which are important in practical applications is given

• implicitly. This fact creates a range of signi…cant di¢culties of a fundamental na-
• ture. Of course, in such settings an analytic structure of the best possible algorithm

would be too complex to be useful in applications. Remark that in the periodic case the most natural (and in many important cases

• ptimal in the sense of the respective n-widths) method of approximation (3.3) is

the trigonometric approximation. In the case of approximation on the whole real line R the role of subspaces of trigonometric polynomials play functions from the Wiener spaces W(R), i.e., entire

6 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

functions from L2(R) whose Fourier transform has support [; ]. Such functions have an exponential type > 0. Remind that an entire function f(z) de…ned on the complex plane C can be represented as f(z) =

1

X

k=0

ckzk for any z 2 C: Assume that f(z) has such coe¢cients ck that limk!1 (k! jckj)1=k = < 1: Then for some constant M > 0 we have jf (z)j

1

X

k=0

jckj jzjk

1

X

k=0

1 k!

• jzj (k! jckj)1=kk

M

1

X

k=0

1 k! (jzj)k = Mejzj: We say that a function f(z) de…ned on the complex plane C is of exponential type > 0 if there exists a constant M such that for any 2 [0; 2), (5.1) jf (z)j Mer; z = rei in the limit of r ! 1. Remark that entire functions of exponential type as an apparatus of approximation was …rst considered by Bernstein [1]. In the n-dimensional settings (spread options) we need to approximate the inte- gral (5.2) V (x; t) := ert Z

Rn G(x + y)

Z

Rn eiyzt (z)dz:

To approximate such integrals we use entire functions f (z) : Cn ! C of n variables z = (z1;;zn) which satisfy the condition jf (z)j M exp n X

k=1

k jzkj ! ; 8z 2Cn; where M is a …xed constant. Here : = (1; ; n) is the exponential type of f (z) : To compare di¤erent methods of approximation we need to introduce Kolmogorov’s n-width of a symmetric with respect to the origin set A in a Banach space X, dn(A; X) := inf

LnX sup x2A

inf

y2Ln kx ykX;

where Ln runs over all collection of n-dimensional spaces in X. To be able to consider n-width of function classes de…ned on locally compact Abelian groups we use the notion of average dimension introduced by Tikhomirov [35]. Let M be a homogeneous locally compact space on which a group G acts. Assume that M is equipped with the invariant measure d and metric d (; ) with respect to G. For any …xed x 2 M de…ne M = M (x) := fz 2 M jd (x; z) g :

DERIVATIVE PRICING 7

Consider the collection of all subspaces L Lp (M) such that the operator P of restricting, P : L

• !

M (x) \ Lp (M) f 7 ! Pf is compact and the quantity K(; L; Lp(M)) := min fn 2 Z+ j dn(L \ Up(M); Lp(M)) < g ; where Up(M) := n g

• kgkp 1
• : We put K(; L; Lp(M)) = +1 if

fn 2 Z+ j dn(L \ Up(M); Lp(M)) < g = ?: It is easy to check that the function ! K(; L; Lp(M)) is non-decreasing for any > 0 and ! K(; L; Lp(M)) is non-increasing for any > 0: De…nition 3. Let ' : R+ ! R+ be a non-decreasing function. The quantity dim (L; Lp(M); ') := lim

!0 lim inf !1

K(; L; Lp(M)) ' () is called the 'average dimension of L in Lp(M): In particular, if M = R and ' () = 2, then the 'average dimension is called the average dimension and is denoted by dim (L; Lp(R)) : The idea of this general de…nition is based on Theorem 2. (Whitteker-Kotel’nicov-Shannon) Let f 2 W then f(x) = X

k2Z

f k

• sin ( (x k=))

(x k=) : It means that any function f 2 W can be recovered from its values on the set

• f points fk=gk2Z : It is shown in [35], p.367 that

dim (W; L2(R)) =

• which is the inverse of the distance between the mesh points fk=gk2Z : Similarly,

to recover any trigonometric polynomial tn (x) 2 Tn := lin f1; cos kx; sin kx; 1 k ng we need to know its values at 2n + 1 points f2k= (2n + 1) ; 1 k 2n + 1g : Re- mark that dim Tn = 2n + 1: Observe that Shannon [32]-[34] was the …rst who introduced the notion of average dimension in order to compare massivity of sets. His idea is based on the notion of entropy of a random object. Later, Kolmogorov de…ned an average dimension in terms of -entropy of a set of non-random functions. This line of research has been developed by Tikhomirov [35], [36], Din’ Zung [8], Le Chyong Tung [17], Magaril- Il’yaev [9], [19]-[24]. De…nition 4. Let A be centrally symmetric set, A Lp(M): The average Kol- mogorov -width is d (A; Lp(M)) := inf sup

f2A

inf

g2L kf gkLp(M) ;

where inf is taken over all subspaces of dimension > 0:

8 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

Let f be a measurable function on R. By f + we denote the non-increasing rearrangement of f, i.e., f +(t) := inff > 0 j () t; 0 < t < 1g; where () := measfx 2 Rjf(x) g: Let [ak; bk), 1 k N be a …nite system of disjoint intervals. Consider the family S of simple sets S representable in the form S = [N

k=1 [ak; bk), where N 2 N:

De…ne the Jordan measure m of S as m (S) = PN

k=1 (bk ak) : For a bounded

set B R de…ne its inner Jordan measure as m (B) = sup fm (S) jS B g and its outer Jordan measure as m (B) = inf fm (S) jS B g ; where the supremum and in…mum are taken over all simple sets S: If m (B) = m (B) then we say that B is Jordan measurable. We say that f 2 J (R) if for any > 0 the set ft 2 R j jf(t)j g is Jordan measurable. It is known [35] that Theorem 3. Let K 2 L2(R), FK 2 J (R), > 0. Then d(K U2(R); L2(R)) = (FK())+ : Consider a regular Lévy process of exponential type. To apply the last Theorem we make a standard change of contour of integration which is based on Cauchy’s

• theorem. Hence, we can assume that

G (y) e!y 2 CU2 (R) for some C > 0 and 0 < ! < : Lemma 1. Let = 2 [; +] and = x + iy then lim

=2[;+]; jj!1

() (jxj) = 1; where (x) = 8 < : ix () x c+ei=2 + cei=2 ; 0 < < 2; 6= 0; 1; i (x + c+x ln x cx ln x) + (c+ + c) x

2 ;

= 1; ix + (c+ + c) ln x; = 0: In particular, if c+ = c = c then (x) = 8 < : ix 2c () x cos (=2) ; 0 < < 2; 6= 0; 1; ix + cx; = 1; ix + 2c ln x; = 0: Corollary 1. d(pt(x)U2(R); L2(R)) 8 < : exp (4ct () cos (=2)) ; 0 < < 2; 6= 0; 1; exp (2ct) ; = 1; 4ct; = 0: In the subsequent sections we develop methods of approximation which have almost optimal rate of convergence.

DERIVATIVE PRICING 9

• 6. -deformation of entire basis functions of exponential type

Let m = (m1; ; mn) 2Zn and a =(a1; ; an) 2 Rn; ak > 0; 1 k n and A = diag

• a1

1 ; ; a1 n

• : Let

Qa := fx j x = (x1; ; xn) 2 Rn; jxkj ak; 1 k ng : Denote by Wa(Rn) the space of functions f 2 L2(Rn) such that supp Ff Qa: The next result which is a multidimensional generalisation of a well-known Whitekker-Kotel’nikov-Shannon theorem explains why Wiener spaces Wa(Rn) are so important. Observe that Whitekker-Kotel’nikov-Shannon theorem has its roots in the Information Theory …rst developed by Shannon [32], [33], [34]. We construct a family of linear operators Pa, Pa : C (Cn)

• !

W2a(Rn) f (z) 7 ! (Paf) (z) such that kPajC (Cn) ! C (Cn)k < 1 and (Paf) (z) = f (z) for any f (z) 2 Wa(Rn): Theorem 4. Let f (z) 2 Wa(Rn) and a : Rn ! R be any continuous function such that a (y) = 1 if y 2 Qa and a (y) = 0 if y 2 Rn n Q2a; then f (x) = X

m2Zn

f

• AmT

Jm;a (x) = X

m12Z

X

mn2Z

f

• m1

a1 ; ; mn an

• Jm1;;mn;a1;;an (x1; ; xn)

where Jm;a (x) = n Y

k=1

1 2ak ! (2)n F1a x i m a

• = 2n det A (Fa)
• x + m

a

• :

Consider a particular form of -deformation. Let (6.1) a (x) = a1;;an (x1; ; xn) :=

n

Y

k=1

• 1

ak xk

• ;

where (xk) := 8 > > > > < > > > > : 0; xk 1; 2xk + 2; 1 xk 1=2; 1; 1==2 xk < 1=2; 2xk + 2 1=2 xk < 1; 0; xk 1; ; and 1 k n: Corollary 2. Let a is de…ned by (6.1) then (Paf) (z) = f (z) for any f (z) 2 Wa(Rn) and kPajC (Cn) ! C (Cn)k < 2n + 3n:

10 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

• 7. methods of approximation

We start with the simplest method of approximation. The next result gives a linear projector, PQa : C (Cn)

• !

Wa (Rn) f (z) 7 !

• PQa f
• (z) :

Theorem 5. Let a = Qa. Assume that the integral V (x; t) is well-de…ned. Then the approximant V (x; t) for V (x; t) is given by V (x; t) = ert 2ndetA X

m2Zn

exp

• t
• AmT

(x; m; a); where (x; m; a) := Z

Qa

ei(m=a)y G(x + y)dy: Observe that in our notations p(y) = F1 et () (y): Applying Theorem 4 we get e (z) 2n det AQa (y) X

m2Zn

exp

• AmT

FQa z + AmT ; which implies the following approximation for the density function p(y) 2n det AQa (y) X

m2Zn

exp

• AmT

exp

• iAmT y
• Consider in more details the case n = 1; = [;] and G(x) = (ex K)+ :

In this case we have e (x+i!) 1 2 X

m2Z

exp

• m
• + i!

F[;] x + m

• and the density function can be approximated as

p (y) = 1 2 e!y Z

R

eiyx (x+i!)dx 1 2 e!y[;] (y) X

m2Z

exp

• m
• + i!
• exp
• im

y

• :

In particular, if ! = 0 then p (y) 1 2 [;] (y) X

m2Z

exp

• m
• exp
• im

y

• :

Consider the KoBoL exponent () = i + c+ () [() ( i)] + c ()

• + (+ + i)

with the parameters = 0:5; c+ = c = 1; + = 5; = 5; = 0:019721: We present several approximants of p(y) for di¤erent and di¤erent values of parameters and the number of terms R in P

n2Z (i.e., instead of P n2Z we consider

P

jnjR).

DERIVATIVE PRICING 11

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 0.2 0.4 0.6 0.8 1.0

x y

p

1 (x) ; = 1; = 20; R = 100

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

x y

p

0:5 (x) ; = 0:5; = 10; R = 50

12 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

• 20
• 15
• 10
• 5

5 10 15 20 1 2 3 4

x y

p

0:1 (x) ; = 0:1; = 10; R = 50

Consider the case + 6= :In particular, let = 0:5; c+ = c = 0:6506; + = 1:9458; = 11:0187; = 0:39563:

• 20
• 15
• 10
• 5

5 10 15 20 0.2 0.4 0.6 0.8 1.0

x y

p

1 (x) ; = 1; = 20; R = 100

DERIVATIVE PRICING 13

• 20
• 15
• 10
• 5

5 10 15 20 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

x y

p

0:5 (x) ; = 0:5; = 20; R = 100

Consider approximation of two dimensional density functions. It is well known that spread options are an important class of derivative contracts written on mul- tiple assets which are used in a wide range of …nancial markets [7]. Let Sjt; j = 1; 2 are two asset price processes then the basic spread option with strike K and ma- turity T is the contract that pays (S1T S2T K)+ : In the Black-Scholes model St = (S1t; S2t) where Sjt = Sj0 exp r 2

j

2 ! t + jW j

t

! ; 1; 2 > 0 and W 1

t ; W 2 t are Brownian motions. The joint characteristic function of

XT = (ln S1T ; ln S2T ) is exp

• iuXT
• (u; T) ; where u = (u1; u2)

(u; T) = (u1; u2; T) = exp iu

• rTe 2T

2 T T 2 uuT ! ; :=

• 2

1

12 12 2

2

• ; 2 := diag :=
• 2

1; 2 2

• ;

e = (1; 1) ; jj < 1 and u

0 denotes the matrix transpose. Let us …x the parameters

as in [14], p. 13. In this case we have r = 0:1; T = 1:0; = 0:5; 1 = 0:2; 2 = 0:1 and (u; T) simpli…es as (u1; u2; T) = exp

• 0:02u2

1 0:01u1 + 0:08iu1 0:005 u2 2 + 0:095 iu2

• :

We use 1 = 2 = 10 and the truncation parameters R1 = R2 = 20:

14 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

• 4

4

• 2
• 0.5

y x

2

• 4
• 2

4 0.0 2

z

1.0 0.5 1.5 2.0 2.5

1 = 2 = 10; R1 = R2 = 20: The approximant F

call (St; t) for Fcall (St; t) is given by

F

call (St; t) = er (I1 (x) I2 (x)) ;

where I1 (x) = ex+(1!) 2 X

m2Z

(1)m e (m=+i!) 1 ! + im= e!xK1! 2 X

m2Z

Kim=e (m=+i!) 1 ! + im= eimx=: I2 = Ke! 2 X

m2Z

(1)m

im

• ! e ( m

+i!)

K1!e! 2 X

m2Z

Kim=

im

• ! e ( m

+i!)eimx=:

The next statement indicates an exponential rate of convergence F

call ! Fcall as

! 1: Theorem 6. Let M := max fexp ( (T t) (x + i!)) jx 2 Rg ; := min f+ 1 ; 1 + g ; > 0 and ! = 1 + : Then jFcall F

callj exp (r + !x) 80MK1!

22 (! 1) ()1= exp () :

DERIVATIVE PRICING 15

References

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16 ALEXANDER KUSHPEL AND JEREMY LEVESLEY

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• ry of cubature formulae and computational mathematics (Proc. Conf. on di¤erential equa-

tions and computational mathematics), Nauka, Novosibirsk 1980, 183–188. Department of Mathematics, University of Leicester E-mail address: ak412@le.ac.uk, jl1@le.ac.uk