double barrier with arbitrary payoffs

8/3/2019 Double Barrier With Arbitrary Payoffs

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A New Approach to Pricing Double-

Barrier Options with Arbitrary Payoffs

and Exponential BoundariesPeter Buchen

a& Otto Konstandatos

b

a Discipline of Finance, The University of Sydney, Sydney, Australiab

School of Finance and Economics, University of Technology,

Sydney, Australia

Available online: 06 Nov 2009

To cite this article: Peter Buchen & Otto Konstandatos (2009): A New Approach to Pricing Double-

Barrier Options with Arbitrary Payoffs and Exponential Boundaries, Applied Mathematical Finance,

16:6, 497-515

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A New Approach to PricingDouble-Barrier Options with ArbitraryPayoffs and Exponential Boundaries

PETER BUCHEN* & OTTO KONSTANDATOS**

*Discipline of Finance, The University of Sydney, Sydney, Australia, **School of Finance and Economics,

University of Technology, Sydney, Australia

(Received 15 May 2008; in revised form 12 December 2008)

ABSTRACT We consider in this article the arbitrage free pricing of double knock-out barrieroptions with payoffs that are arbitrary functions of the underlying asset, where we allow exponen-tially time-varying barrier levels in an otherwise standard BlackScholes model. Our approach,reminiscent of the method of images of electromagnetics, considerably simplifies the derivation ofanalytical formulae for this class of exotics by reducing the pricing of any double-barrier problem tothat of pricing a related European option. We illustrate the method by reproducing the well-knownformulae of Kunitomo and Ikeda (1992) for the standard knock-out double-barrier call and putoptions. We give an explanation for the rapid rate of convergence of the doubly infinite sums for

affine payoffs in the stock price, as encountered in the pricing of double-barrier call and put optionsfirst observed by Kunitomo and Ikeda (1992).

KEY WORDS: Exotic options, double-barrier options, method of images, parity relations ofdouble-barrier options

1. Introduction

Options with barrier features have become quite common instruments in various

derivative markets, especially in over-the-counter trades and FX markets. In fact,

the growth in their use has been so dramatic that many authors have long considered

the single-barrier options to be non-exotic (see Carr, 1995).

The major reason why barrier options have become so popular is because they can in

fact offer an equivalent level of protection when used as a hedge but are cheaper than the

plain vanilla calls and puts in the same circumstances. For example, one can use a down-

and-in put with the barrier set at a low level as an inexpensive way to protect against a large

drop in the underlying asset price, compared with a standard put option on the underlying.

It is relatively straightforward to price and hedge single-barrier options, and in fact

valuation formulae have been in the literature for a long time. Merton (1973) gave the

pricing formula for an option with a continuously monitored lower knock-out

Applied Mathematical Finance,

Vol. 16, No. 6, 497515, December 2009

Correspondence Address: Otto Konstandatos, School of Finance and Economics, University of Technology,

P.O. Box 123, Broadway, New South Wales, Sydney, Australia. Email: [email protected]

1350-486X Print/1466-4313 Online/09/06049719# 2009 Taylor & Francis

DOI: 10.1080/13504860903075480
mailto:[email protected]:[email protected]


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boundary, whereas a full treatment for various types of weakly path-dependent

options was presented in Goldman et al. (1979). More recently, Rich (1994) and

Rubinstein and Reiner (1991) also tackled the pricing of European single-barrier

options, including the knock-in barrier calls and puts, by the calculation of discounted

expectations under the equivalent martingale measure (EMM).

The innovation has continued, for example Heynen and Kat (1994) gave pricing

formulae for partial barrier options, where the barrier monitoring window is restricted

to a subset of the life of the option, while in Buchen (2001) the author introduced a

technique which he termed the method of images for the BlackScholes (BS) equation to

price options with single flat barriers. The method in Buchen (2001) should not be

confused with what is normally referred to as the method of images, which is tradi-

tionally used to determine the fundamental solution (namely the Greens function) of the

Heat equation for the so-called first boundary value problem, as outlined in Wilmott

et al. (1995) in applications to barrier options. In Buchen and Konstandatos (2005), the

authors extended the methods of Buchen (2001) to price lookback options as well.It is quite natural to extend the vanilla barrier options to incorporate an upper level

and a lower level, in which case we obtain a double-knock-out option, or more simply

a double-barrier option. As in the single-barrier case, one could speak of knock-in

double-barriers, which expire worthless unless one of the two barrier levels are reached

before expiry.

Several approaches have been taken to analytically price knock-out double-barrier

call and put options. The earliest approach is that of Kunitomo and Ikeda (1992),

where the authors derive the probability density function for the stock price staying

between two exponentially time-varying (i.e. curved) boundaries, and then use this to

price the double-knock-out call and put option prices. The result, as may be expected,is expressed as an infinite sum of normal distribution functions. Alternatively, the

authors Geman and Yor (1996) approached the problem by deriving an expression for

the Laplace transformations of the double knock-out call and put options and then

numerically inverting these expressions to recover the required prices, although they

only considered the case of time-independent boundaries.

In this article, we will consider the arbitrage free pricing of an arbitrary double-knock-

out barrier option, with exponential time-varying upper and lower barrier levels, in an

otherwise standard BS model. By standard BS model, we mean the option is written on

a non-dividend paying asset whose price follows geometrical Brownian motion of

constant volatility . The risk-free interest rate is also assumed to be constant andequal to r, and it is elementary to add in a continuous dividend yield.The central result of this article is to demonstrate how a double-knock-out barrier

option with an arbitrary payoff at expiry can always be priced in terms of more

elementary European (i.e. path independent) options. Of course, this is something

that may always be done, provided that the payoff function of the option in question is

non-pathologic, which may lead to non-uniqueness of solutions of the BlackScholes

partial differential equation (BS-PDE). In practice, however, this is a small restriction

on the class of payoffs to which our result applies, which includes all polynomial

payoffs of the underlying stock price with affine exercise conditions. We also allow the

possibility of exponential time-varying boundaries, which are normally referred to in

the literature as curved, as first defined and analysed in the seminal paper of Kunitomoand Ikeda (1992).

498 P. Buchen and O. Konstandatos


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As an application and demonstration of our central result, we will reproduce the

standard double-barrier calls and puts of the literature in this work. The results for the

non-time-varying barrier levels, the so-called flat barrier case, are readily recovered by

setting the appropriate exponential parameters to zero.

In contrast with the probabilistic approaches of Kunitomo and Ikeda (1992),

Rubinstein and Reiner (1991), Rich (1994) and Geman and Yor (1996), our approach

is to extend the results of Buchen (2001), which employed the method of images approach

for the PDE formulation of the barrier problem. This approach actually relies upon the

exploitation of the symmetries of solutions inherent in the BS-PDE. It is somewhat

reminiscent of the well-known method of images of electromagnetics, in which solutions

to electrostatic problems are found by exploiting appropriately placed image charges so

as to obtain the correct boundary conditions along a given physical boundary.

However, in contrast to physical boundaries, we work directly with the BS equation

and consider boundaries in the stock-price space. The utility of our approach stems

from being able to directly treat double-barrier-type problems in the original variables,thus obviating the need of transforming to the Heat equation, with the application of

traditional Greens function techniques and then back-transforming, as is usually

done (e.g. Wilmott et al., 1995). It is worth pointing out that it is quite possible to

develop the equivalent of the central result of this article in terms of the fundamental

solution (or Greens function) of the Heat equation, in the special case of flat

(i.e. constant) barrier levels, as treated in Konstandatos (2003, 2008). Furthermore,

it is also possible to convert any single exponential barrier problem into an equivalent

flat barrier problem, through inclusion of a dividend yield equal to the growth rate of

the barrier. However, when treating the case of double exponential barriers, it is not

possible to use the same trick. By working in the original BS variables, we exploit extrasymmetries of the BS-PDE, which permits exponential time-varying boundaries. Thus,

we derive our pricing methodology as a consequence of the axioms of the BS-image

operator, in a logically consistent fashion.

We will extend the results of Buchen (2001)1 and Konstandatos (2003)2 by introdu-

cing the image function for exponentially time-varying barriers and we will demon-

strate how to exploit such functions of the underlying asset and appropriately placed

image functions of the stock price to construct the solution for the arbitrary double-

knock-out barrier with exponential boundaries. In fact, we also obtain the general

method for pricing single-barrier options with time-varying exponential boundaries as

a special case.

Our method has the great advantage in that it reduces any knock-out double-barrier

problem (and in fact any knock-out barrier problem) to the evaluation of a single path-

independent European option, obviating the need for complicated probabilistic argu-

ments and the calculation of an expectation with respect to a joint density of the

underlying asset with the running maximum and minimum of the asset price. In fact,

application of our technique is especially simple when we are able to decompose the

option payoff into an affine function of the underlying, multiplied by an indicator (or

exercise) function, which gives rise to so-called asset and bond binaries (see Buchen

(2004), also Konstandatos (2003), for a complete treatment, and more recently Bermin

et al. (2008)). Solutions for the path-independent option may be written down by

inspection. The solution for the general double-barrier problem will then follow as asuperposition of a doubly infinite sum of the path-independent option, such that the

A New Approach to Pricing Double-Barrier Options 499


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appropriate boundary and expiry conditions for the double-barrier problem are satis-

fied. Our method effectively exploits the properties of the image function for the BS-

PDE, as introduced in Buchen (2001) for flat barriers, which is a mapping of solutions to

solutions of the BS equation with specific properties.

This article is structured as follows. The remainder of this section will complete the

background, outlining the basic idea of image functions in the flat barrier case, which

underlies the work in Buchen (2001) and Konstandatos (2003, 2008). In Section 2, we

extend the idea of an image function to incorporate time-varying exponential barrier

levels. In Section 3, we prove the central result of this article, which we call the method

of images for double exponential barrier options. In Section 4, we apply our method to

derive prices for the standard call and put double-barrier options, and finally in

Section 5, we make some remarks regarding the computation of the doubly infinite

sums obtained, offering an explanation for the rapid rate of convergence. We also

present a double-barrier version of the put-call parity relation, and the parity relation

satisfied by the knock-out and knock-in double-barrier options, which allows thepricing of the corresponding knock-in versions of the options under consideration.

Section 6 contains a brief conclusion. We provide a proof of asymptotic convergence

for our main result in the Appendix.

We shall be working throughout with the relative time variable T- t, where t isthe current calendar time and Tthe future expiry date of the option. Thus, . 0 refersto the time remaining to expiry and expiry occurs at 0. It is clear that terminalvalue problems are then converted into equivalent initial value problems. Ifx denotes

the current asset price, then all European-style derivative prices V(x, ) satisfy theBS-PDE.

Definition 1.1. The BS operator L is defined by

LVx; @V@

rV rx @V@x

122x2

@2V

@x2: (1)

The corresponding BS-PDE is defined to be LV 0.European derivative prices satisfy the BS-PDE in the unrestricted asset price domain

x. 0 for all . 0 with a specified initial value V(x, 0) f(x). The functionf(x) is calledthe derivatives payoff function. Such initial value problems are generally easy to solve.

For example, a solution can be written down using the FeynmanKac formula

Vx; er EQffXTjXt xg: (2)

It is now well known (see Harrison and Pliska, 1981) that Equation (2) gives the

arbitrage free price of the derivative if and only if the conditional expectation is taken

with respect to the equivalent martingale measure Q, under which XT has the

representation

XT x expfr 122 BQ g; (3)

where BQ is a Q-Brownian motion. That is, XT/x is the exponential of a Gaussianrandom variable and is hence log normally distributed.



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Pricing barrier options is of course more complex than solving simple initial value

problems because barrier problems must also satisfy boundary conditions: indeed they

belong to the class of initial boundary value (IBV) problems. In particular, the IBV

problem for the double-barrier option considered in this article is expressed by the PDE:

Problem 1.2.

LV 0; > 0; a < x < bV fx; 0; A < x < BV 0; x a; x b;

8 a, and the method of images solution isthen given by Vx; Vax; IaVax; .

2. Images with Exponential Barriers

We seek an extension of Equation (4) and Lemma 1.5 to the case of an exponential time-

varying barrier of the form b()

Be. The required extension is given by Lemma 2.1.

Lemma 2.1. IfV(x, ) satisfies the BS-PDELV 0 with initial value V(x, 0) f(x),then

V x; IbVx;

b

x

qV

b2x

;

; (6)

with q 2(r + )/2 - 1, solves the BS-PDE LV 0 with initial value V x; 0

B=xqfB2=x.Note that this lemma does reduce to that of Lemma 1.5 for the flat barrier case when

0.While it is possible, but rather tedious, to prove this lemma by direct substitution ofEquation (6) into the BS-PDE, it is more instructive to use symmetry arguments

described by the next result.

Lemma 2.2. Ifur(x, ) solves the BS-PDE for constant risk-free rate r, then

vr;sx; esursxes;

solves the BS-PDE for any real s. Furthermore, if the initial value ur(x, 0) is indepen-

dent ofr, then vr,s(x, ) ; ur(x, ) is an identity for all . 0 and real s.

We omit the proof, which is elementary. However, as shown below, Lemma 2.2 canbe used to prove the important result of Lemma 2.1.



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Proof of Lemma 2.1. Let wr(x, ) be any solution of the BS-PDE with initial valuewr(x, 0) independent ofr. Then, let us apply Lemma 2.2 to the BS solution

urx; IBwrx; w rx; :

Clearly, the initial value ur(x, 0) will depend on r, since the factor (B/x)q depends on r

through q 2r/2 - 1. However, we may conclude that vr;x; e w rxe; satisfies the BS-PDE for any real . But, as w

rx; B=xqwrB2=x; , we obtain

vr;x; e Be

x

qwr

B2e

x;

Be

x

q ewrye; ; y Be2x

Be

x

q wry; ; by Lemma 2:2

bx

qwr

b2

x;

:

where b b() Be. This completes the proof. &

Remark 2.3. Clearly, the image solution IbVx; with b Be satisfies all theproperties listed in Remark 1.4. Furthermore, the method of images described in

Theorem 1.6 will also still be valid, taking care to interpret the initial value Vbx; 0 fxIx < B correctly.

3. Method of Images for Double Barriers

Recall that the single-barrier option can be priced in terms of a related initial

value problem and its image with respect to the barrier level. We show that

the same principle applies to double-barrier options as well. However, there isone important difference. Whereas the single-barrier case requires only a single

image, the double-barrier problem requires a doubly infinite sequence of

images: one infinite sequence for each barrier. We state the main result of this

article in Theorem 3.1 and then describe a number of non-trivial analytical

steps to prove it.

Theorem 3.1. (Method of Images for Double Exponential Barriers). Let U(x,)solve the (unrestricted) problem for the BS-PDE LU 0, with initial valueUx; 0 fxIA < x < B. Then, the unique arbitrage free solution of Problem1.2 for a() , x , b() can be expressed entirely in terms of U(x, ). It is givenexplicitly by the doubly infinite sum



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Vx; X1

n1npnfx=aqnU2nx; a=xpn Ua22n=x; g; (7)

where

pn nq n 1q; qn nq q (8)

ba

B

A

e: (9)

Remark 3.2. Equation (7) is the main result of this article. It provides a representa-

tion for the price of any double-knock-out (exponential) barrier option with an

arbitrary payoff function f(x). This representation is closed as long as we can solvethe related initial value problem for U(x, ). This will certainly be the case for all simplepayoffs including those for calls and puts. We shall demonstrate later that our

representation yields a result that is slightly different to the double-barrier option

prices first presented by Kunitomo and Ikeda (1992). We further show in the Appendix

that this doubly infinite series is at the least asymptotically convergent for all bounded

payoff functions f(x).

We shall now derive a number of intermediate results that have a direct bearing on

the proof of Theorem 3.1. As a notational convenience, denote any sequence of

consecutive image operations:Ia;Ib;Ic;Id, for example, by the symbol

Idcba IdIcIbIa:

With this notation, our first lemma needed for the proof of Theorem 3.1 is the

following.

Lemma 3.3. Let Kba denote the doubly infinite sequence of image operators

Kba I Ia Iba Iaba Ibaba . . .

Ib

Iab

Ibab

Iabab

. . .

(10)

Then the solution of Problem 1.2 is given by

Vx; KbaUx; ; (11)

where U(x,) solves the IV problem described in Theorem 3.1.

Proof. Since solutions of the BS-PDE with well-behaved payoff functions f(x) both

exist and are unique, it is only necessary to show that Equation (11) satisfies the BS-

PDE; the boundary conditions at x a(), b() and the initial value V(x, 0) f(x) inA, x, B.



10/20

First, KbaU satisfies the BS-PDE because U does and by Lemma 2.1 so does anysequence of images of U. The boundary conditions follow from the factorizations of

Kba written below.

Kba I IaI Ib Iba Ibab Ibaba . . . I IaHba say

Kba I IbI Ia Iab Iaba Iabab . . . I IbHab say:

Hence V I IaHbaU vanishes at x a() by property (2) of Remark 1.4.Similarly, V

I

Ib

HabU vanishes at x

b().

It remains to show that V(x, 0) f(x) in the range A, x, B. This follows from theobservation that

Vx; 0 KbaUx; 0 Ux; 0 image sequence fUx; 0g:

Now for A , x , B, U(x, 0) f(x), by the definition of U and so by property (3) ofRemark 1.4, any image sequence of U(x, 0) will evaluate U at points outside the

interval (A, B) where U(, 0) vanishes. This completes the proof. &

Remark 3.4. The intuition behind the operatorK

b

ashould not escape the reader.

We subtractIaU from Uto satisfy the BC at x a; then we add a termIbaU to satisfythe BC at x b; but now the BC is no longer satisfied at x a, so we subtract afurther term IabaU. We continue this process indefinitely to obtain the first infinitesequence of images of U. The second infinite sequence starting with IbU follows asimilar pattern, first satisfying the BC at x b, then x a, back to x b and so on.

We now present a lemma that shows how an arbitrary sequence of three images with

respect to exponential price levels can be written in terms of a single image.

Lemma 3.5. Let a Ae, b Be and c Ce, where (A, B, C) are positiveconstants and (, , ) are any constants. Then

Icba Ic Ib Ia A; B; CId; (12)

where

A; B; C A2=2 B2=2 C2=2

and

d d ac=b De; D AC=B;



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Proof. By definition of the image operator, we have

IcbaVx; c=zqb=yqa=xq V!; ;

wherey a2/x, z b2/y and ! c2/z. Carrying out the algebraic manipulations leads to

IcbaVx; a; b; c dx

qV

d2

x;

a; b; c albmcn AlBmCn elmn

q q q ql q q 2

2

m q q 2

2

n q q 2 2

:

The result now follows from the observation that l + m + n 0. &

Definition 3.6. Define by Hnab for all integer n . 0, the image operator

Hnab IabIab I ab; fn abpairsg

Iabab

ab

and for n 0 set H0ab I, the identity operator.

Lemma 3.7. The 2n-fold image operator Hnab is equivalent to the double image

Hnab Ib Ibn1=an: (13)

Proof. We prove the result by induction. First for n 0, Equation (13) readsH0ab Ib Ib I by property (1) of Remark 1.4. Now assume the result true for nand calculate

Hn1ab Iab Hnab Iab IbIbn1=an Ia Ibn1=an; since Ibb I

Ib IbIaIc; with c bn1

an

C; A; B IbId; with d bca

bn2

an1:

We have used Lemma 3.5 in the last line above. The result now follows from thestraightforward calculation (C, A, B) 1 when



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c Ce; C Bn1

An; n 1 n

q n 1q nq 2=2r 1:

This completes the proof. &

Lemma 3.7 leads to an interesting corollary that gives meaning to the operator Hnabfor all integer n . 0.

Corollary 3.8. For every integer n . 0.

Hnab Hnba (14)Proof. Formally replacing n by -n in Hnab IbIbn1=an gives

Hn

ab Ib Ian

=bn

1

Iba IaIan=bn1; as Iaa I Iba Hn1ba Hnba:

The last line derives from Definition 3.6. &

The next result shows how the operator Kba can be expressed in terms of the multipleimage operators Hnba.

Lemma 3.9.

Kba I IaX1

n1Hnba: (15)

Proof. We start with the factorization ofKba given in the proof of Lemma 3.3.

Kba I IaI Ib Iba Ibab Ibaba Ibabab

I IaX1n0

Hnba IbX1n0

Hnab" #

I Ia X1

n0Hnba Ia X

1

n1Hnab" #;

using I2a I and Iab Hnab Hn1ab

I IaX1n0

Hnba X1

n1Hnba

" #;

using I IaIa I Ia and Hnba Hnab I Ia

X1

n

1

Hnba:

This completes the proof. &



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Corollary 3.10. Since Kba Kab and Hnab Hnba, it follows that we also have theequivalent representations

Kba

I

Ia

P1

n1 Hnab

I

Ia

P1

n1 Hnba

I IbP1

n1Hnba I Ib

P1n1

Hnab: (16)

We now present the proof of our main result.

Proof of Theorem 3.1. Using Lemma 3.3 and Corollaries 3.8 and 3.10 we have the

representation

Vx; Kba Ux;

I Ia X1

n1 Hn

ab Ux;

X1n1

Hnba IaHnabUx; :

We first calculate, with the aid of Lemma 3.7,

Hnba Ux; Ia IcUx; ; c an1

bn

ax

c2

q

c

x

qU

a2x

c2

; n2qq x

a

qqU2nx;

npn xa

qnU2nx; ; (17)

with pn 2q - q nq - (n - 1)q, qn (q - q) n(q - q) and we havedefined () b()/a().Next, if we replace n by -n and define pn pn, we obtain using (17),

Ia HnabUx; Ia Hnba Ux; Ia npn a

x

qn U2nx;

npn ax

qn 2nax

qU

a22n

x;

npn ax

pnU

a22n

x;

; (18)

where we have used

pn 2q pn and q qn pn:

Subtracting the two terms (17) and (18) above gives the result of Theorem 3.1. &



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Remark 3.11. Due to the many symmetries entailed by Corollary 3.10, there exists

equally many different but equivalent representations of Equation (7) stated in

Theorem 3.1.

4. Double-Barrier Calls and Puts

Consider now double-barrier calls and puts with strike price ksatisfying A, k, B. If

k,A, calls (puts) are always (never) exercised at expiry, while ifk.B, calls (puts) are

never (always) exercised at expiry. For double-barrier call options, it is only necessary

to determine UC(x, ), which solves the BS-PDE initial value problem, LUC 0 withpayoff function

UCx; 0 x kIA < X< B

x kx > k x kx > B Ckx; 0; k Ckx; 0; B:

The first term above is clearly the payoff of a standard European call option of strike

price k; the second term that of gap call option of strike price kand exercise price B

(the gap is equal to B- k). Both these options are easy to price using standard methods

(e.g. using Equation (2)) and have the following expressions, for . 0, in terms of thestandard GaussianN d

Ckx; ; k xN dx=k; kerN d0x=k; (19)

Ckx; ; B xN dx=B; kerN d0x=B; ; (20)

where

d;d0y; logy r 12

2

ffiffiffi

p : (21)

Thus,

UC

x; Ckx; ; k Ckx; ; B (22)is the function Uwe must use in Equation (7) for double-barrier call options.

A similar argument applies for put options with payoff function

UPx; 0 k xIA < X < B k xIx < k x kIx < A Pkx; 0; k Pkx; 0; A:

The present values are then,



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Pkx; ; k xNdx=k; kerNd0x=k; (23)

Pk

x; ; A

x

Nd

x=B;

ker

Nd0

x=A;

: (24)

This leads to an expression equal to the difference between a standard European put

option and a gap put option (of gap k- A),

UPx; Pkx; ; k Pkx; ; A; (25)

for the function Uto be used in Equation (7) for double-barrier put options.

Remark 4.1. Since, in each case, UC and UP contain a sum of four Gaussians,

Equation (7) contains a doubly-infinite sum of eight such Gaussians. This structure isalso evident in the solution obtained by Kunitomo and Ikeda (1992). After careful

comparison, we find our representation is indeed identical to theirs, provided a small

adjustment to their reported formulae is made. In particular, their Equations (3.2) for

the double-barrier call price C(t) and (3.8) for the double-barrier put price P(t) are

correct only for the single time t 0. Our corresponding formulae are however correctfor all times t , T.

5. Computations

Apart from some minor differences in some of the entries, we have been able to

reproduce the double-barrier call price tables of Kunitomo and Ikeda (1992), parts

of which we include here for convenience using parameter values equivalent to those

given in their paper:

x 1000; 1=12; k 1000; r 0:05; 0:2:

The three columns labelled (a), (b) and (c), respectively, correspond to the following

choices for the exponential parameters, mirroring the choices in Kunitomo and Ikeda

(1992): The call option values that we obtain are reproduced in Table 1. The corre-sponding put prices that we obtain, not published in Kunitomo and Ikeda (1992), are

reproduced in Table 2.

The first column of prices corresponds to (, ) (-0.1,0.1) when the barriers arediverging relative to calendar time. The second and third columns, respectively,

correspond to the flat barrier case (, ) (0,0) and the converging barriers case(, ) (0.1, -0.1).

Table 1 agrees with Table 3.1 of Kunitomo and Ikeda (1992), who noted through

numerical experimentation that very few terms of the doubly infinite series are

required for good convergence. We also observe the rapid rate of convergence of the

sum. In fact, in many cases only the n 1 terms beyond the dominant n 0 term arenecessary to achieve numerical convergence.



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We offer below an explanation of this phenomenon.

Figure 1 shows the functions U(x, ) for European calls and puts that are used in

Equation (7) to price their corresponding double-barrier options. The parameters arethose of Kunitomo and Ikeda (1992) and are given explicitly below.

The central observation is the fact that both functions decay rapidly to zero in the

limits x ! 0 and x ! 1. A careful asymptotic analysis shows that at both ends of theasset price range,

Table 1. Call option prices.

(, )

A B (0.1,-

0.1) (0,0) (-

0.1,0.1)

0 1 25.1207 25.1207 25.1207400 1600 25.1207 25.1207 25.1207500 1500 25.1207 25.1207 25.1207600 1400 25.1207 25.1207 25.1207700 1300 25.1196 25.1187 25.1170800 1200 24.8809 24.7568 24.5790850 1150 23.2123 22.5367 21.6872900 1100 16.1748 14.4023 12.5033930 1070 8.5259 6.6861 4.9622950 1050 3.3923 2.1462 1.1731

Table 2. Put option prices.

(, )

A B (0.1,-0.1) (0,0) (-0.1,0.1)

0 1 20.9627 20.9627 20.9627400 1600 20.9627 20.9627 20.9627500 1500 20.9627 20.9627 20.9627600 1400 20.9627 20.9627 20.9627

700 1300 20.9627 20.9627 20.9627800 1200 20.9518 20.9440 20.9312850 1150 20.5242 20.3205 20.0401900 1100 16.0030 14.7652 13.3584930 1070 8.8902 7.2223 5.5842950 1050 3.5324 2.3039 1.3080

Parameter values

k r a b

0.25 1000 0.05 0.20 850 1100 -0.10 0.10



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limx!f0;1g

Ux; Od=d; d j log xj=ffiffiffip ;where z 1ffiffiffiffi2p e

12z2 is the Gaussian density function.

The shaded regions in Figure 1 show the interior of the current barrier levels. Thetwo sets of dots show the image points yn (b/a)2nx (lower sequence) and zn (b/a)2na2/x (upper sequence) for -1 n 1 and for the asset price x 1000. We see thateven for n 2, the image points (off-scale in Figure 1) lie well into the tails of theU-functions for both calls and puts. Higher order images lie even further into the tails.

It is therefore not surprising that good accuracy is obtained using only first-order

images, corresponding to the first two terms away from the n 0 term.

Remark 5.1. We complete this section with two observations.

(1) Contrary to claims made in the literature, there does exist a double-barrier put

call parity relation. It can be succinctly stated as

CDBx; ; k PDBx; ; k FDBx; ; k; (26)

where the last term on the right of Equation (26) is the price of double-barrier

forward, i.e. a double barrier with expiry payoff given by f(x) (x - k).(2) Once we have priced double-barrier knock-out options, it is a straightforward matter

to price their corresponding knock-in versions. Let VEUx; denote the price of astandard European derivative with payoff f(x), absent any barriers. Then, in an

obvious notation, the double-barrier knock-in, knock-out parity relation is simply

VKIx; VKOx; VEUx;: (27)

500 1000 1500

0

1

2

3

4

5

6

78

9

10

Stock price x

Call option U

0

0

1

1

1

1

Put option U

Stock price x

0

0

1

1

1

1

Figure 1. Functions U(x, ) for European calls and puts used in Equation (7) to price theircorresponding double-barrier options.



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6. Conclusion

We have presented a new and general method for pricing double barrier options with

arbitrary payoff, where we allow exponential time-varying boundaries for the

knock-out barrier levels. The difficulty in pricing such options lies in the fact thatthey are path dependent, and traditional approaches have relied upon the calculation

of an expectation with respect to a joint density of the stock price with the maximum

and minimum processes of the underlying. Such approaches require the use of the

reflection principle of Brownian motion to determine the required density. Our

method reduces the problem of pricing any double knock-out barrier option to that

of pricing an equivalent European (i.e. path independent) option, which generally is

much easier to accomplish. Our central theorem (Equation (7)) demonstrates this for

any given payoff, not just for the traditional call and put payoffs. The reader may

appreciate the fact that our representation of the double-barrier knock-out price is not

limited to affine or call-type payoff functions of the underlying asset price, but is quite

general, provided the option payoffs are restricted to non-pathological types. Our

central theorem represents a major conceptual and computational simplification of the

problem on pricing options with double-barrier features, by obviating the need for

complicated transition density integrations. The reader may also note that we were

able to reproduce the results for the standard calls and puts of the literature without

recourse to any complicated calculation and we were able to write down the results

essentially by inspection.

In this work we have also demonstrated that our approach agrees, up to minor

correction, with the published results of Kunitomo and Ikeda (1992) for the double-

barrier knock-out call. We have included part of the tables of the double-barrier

knock-out call option prices presented by the above authors and have supplementedthis by including the corresponding double-barrier knock-out put option prices for the

given parameters. We have presented an explanation for the rapid rate of convergence

of the doubly infinite sum, demonstrating that the sequence of multiple images

inherent in the structure of the double-barrier solution adds terms that lie progres-

sively further into tails of the normal density function, demonstrating that their

contributions will decay rapidly. In fact, this rapid convergence will also be observed

for any affine or power payoff in the stock price, given the fact that the normal

distribution function will always appear in the pricing of the associated European

option of our methodology.

Our approach gives great scope for further work. One immediate application is toprice partial-time double-barrier options with exponential boundaries, which gener-

alizes the work of Heynen and Kat (1994) in the single-barrier case. Such results have

not appeared in the literature yet. Another area of extension is in the case of multi-asset

barrier and double-barrier options, where the authors have done much work.

However, we leave all the above for subsequent publications.

Notes

1Where single flat barriers were treated.2Where the flat double-barrier case is treated.



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References

Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables (New York: Dover Publications Inc.).

Bermin, H., Buchen, P. and Konstandatos, O. (2008) Two exotic lookback options. Applied Mathematical

Finance, 15(4), pp. 387402.

Buchen, P. (2001) Image options and the road to barriers. Risk Magazine, 14(9), pp. 127130.

Buchen, P. (2004) The pricing of dual expiry exotics. Quantitative Finance, 4(1), pp. 18.

Buchen, P. and Konstandatos, O. (2005) A new method of pricing lookback options. Mathematical Finance,

15(2), pp. 245259.

Carr, P. (1995) Two extensions to barrier option valuation. Applied Mathematical Finance, 2(3),pp. 173209.

Geman, H. and Yor, M. (1996) Pricing and hedging double-barrier options: a probabilistic approach.

Mathematical Finance, 6(4), pp. 365378.

Goldman, M., Sosin, H. and Gatto, M. (1979) Path dependent options: buy at the low, sell at the high.

Journal of Finance, 34(5), pp. 11111127.

Harrison, J. and Pliska, S. (1981) Martingales and stochastic integrals in the theory of continuous trading.

Stochastic Processes and their Applications, 11, pp. 215260.

Heynen, R. and Kat, H. (1994) Partial barrier options. Journal of Financial Engineering, 3(3/4), pp. 253274.Konstandatos, O. (2003) A new framework for pricing barrier and lookback options. PhD thesis, The

University of Sydney.

Konstandatos, O. (2008) PricingPathDependentOptions: A ComprehensiveFramework, Saabr 1sted.(Applications

of Mathematics) (Saarbrucken, Germany: VDM Verlag Dr. Muller Aktiengesellschaft & Co.).

Kunitomo, N. and Ikeda, M. (1992) Pricing options with curved boundaries. Mathematical Finance, 2,

pp. 276298.

Merton, R. (1973) Theory of rational option pricing. Bell Journal of Economics and Management Science,

4(1), pp. 141183.

Rich, D. (1994) The mathematical foundations of barrier option pricing theory. Advances in Futures and

Operations Research, 7, pp. 267311.

Rubinstein, M. and Reiner, E. (1991) Breaking down the barriers. Risk Magazine, 4(8), pp. 2835.

Wilmott, P., Howison, S. and Dewynne, J. (1995) The Mathematics of Financial Derivatives: A StudentIntroduction (New York: Cambridge University Press).

Appendix A

We prove here that the doubly infinite series given in Equation (7) is asymptotically

convergent, for all bounded payoffs, in the following sense. We sayP1

n1 Sn isasymptotically convergent if for sufficiently large n . N, Sn is asymptotic to Tn andPjnj>N

Tn is convergent.

It is only necessary to prove asymptotic convergence of the first of the two terms in

Equation (7), for once that is done, the second follows mutatis mutandis. By definitionof Ux; , it can be represented in terms of the BS Greens Function on the interval[A, B], through

Ux; ZBA

fyGx=y; dyy

; (A1)

where

Gx=y; er

ffiffiffip logx=y r

12

2 ffiffiffip

(A2)and z 1ffiffiffiffi

2p e

12z2 is the usual standard Gaussian density. Assuming the



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payoff function f(y) is bounded on the interval [A, B], that is, jfyj 0.Appealing to the asymptotics of N(z) for z ! 1 (see Abramowitz and Stegun,

1964) we find the leading term

U2n

x;,

Mer

jzj1

z; zkn: (A4)Let Un npnx=aqn U2nx; denote the nth term of the double infinite series we areconsidering. In light of Equation (A4), we need only show that the term

Tn npn kn

is asymptotically order Oehn2 for some h . 0. The term x=aqn is at most Oecn forsome bounded c . 0, which is dominated by the factor Tn Oehn2 as n ! 1. Beinghigher order, x=aqn can be safely neglected. With our definitions ofpn and k, we calculate

log Tn,2

2log 1

2k2

n2

2 log 2

log n2

2 log 2

logB=An2;

where we have used b=a B=Ae. Hence, for large jnj we have shown

Tn,ehn2 ; h

2 log

2log

B=A

:

Since . 1 and B>A, we have h > 0, completing the proof. &


double barrier with arbitrary payoffs

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