a general approach to hedging options

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A GENERAL APPROACH TO HEDGING OPTIONS:

APPLICATIONS TO BARRIER AND PARTIAL BARRIER

OPTIONS

Hans-Peter Bermin

Department of Economics, Lund University

In this paper we consider a Black and Scholes economy and show how the Malliavin calculus

approach can be extended to cover hedging of any square integrable contingent claim. As an

application we derive the replicating portfolios of some barrier and partial barrier options.

KEY WORDS: contingent claims, hedging, barrier options, Malliavin calculus

1. INTRODUCTION

In this paper we consider a Black and Scholes economy and derive a general expression

for the self-financing portfolio that generates any square integrable contingent claim.

The expression is an extension of the results in Karatzas and Ocone (1991), which says

that the number of units to be held at time t in the stock S (with volatility r) is given by

the formula

erTtr1St1EQDtGjFt;

provided that the contingent claim G (maturing at T) is square integrable and that theMalliavin derivative, DtGsatisfies certain conditions. Based on the results in Watanabe

(1984) and U stu nel (1995), we show that the above formula is valid under the weaker

condition that G only has to be square integrable. However, for this to work the

Malliavin derivative, DtGmust be interpreted as a generalized stochastic processthat

is, as a composite of a distribution and a stochastic process. Nevertheless, things turn

out to be tractable because the conditional expectation of the Malliavin derivative

EQDtGjFt is an ordinary stochastic process. For a similar extension in the case of awhite noise driven market, see Aase et al. (2000).

When the contingent claim is a functional of the stock price, we also show that the

required Malliavin derivative can be derived from the classical chain rule. In manycases the formal expression can be evaluated analytically. However, when this is not the

case, Monte Carlo simulations can be implemented to derive the replicating portfolio.

We summarize the theory behind the extension of the formula and focus on the

applicability of the Malliavin calculus.

The author thanks Arturo Kohatsu-Higa, Ali Su leyman U stu nel, and the referee for useful andvaluable comments. Financial support from the Wallander foundation is gratefully acknowledged.

Manuscript received September 1998; final revision received August 2000.

Address correspondence to the author at the Department of Economics, Lund University, P.O. Box 7082,

S-220 07 Lund, Sweden; e-mail: [email protected].

Mathematical Finance, Vol. 12, No. 3 (July 2002), 199218

2002 Blackwell Publishing Inc., 350 Main St., Malden, MA 02148, USA, and 108 Cowley Road, Oxford,

OX4 1JF, UK.

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To motivate the study we consider barrier and partial barrier options. It is well-

known that these contingent claims can be replicated by using the traditional

D-hedging approach, while the Malliavin calculus approach as in Karatzas and

Ocone (1991) is not general enough to handle this situation due to the discontinuity

of the payoff functions. However, by using the extended Malliavin calculus approach

we show how we can find the replicating portfolios of barrier and partial barrieroptions in such a way that they agree with the results from the traditional D-hedging

approach.

By definition, the payoff of a barrier option is equal to the payoff of a standard

option provided that the barrier option is alive at maturity and zero otherwise.

Moreover, we say that the barrier option is alive if the stock stays in some

predefined region. These options are of course less expensive than standard options

as they can become worthless at any time before expiration. Depending on how the

alive region is defined, a lot of slightly different options can be constructed, such as

up-and-out calls/puts, down-and-out calls/puts, up-and-in calls/puts, and down-and-

in calls/puts. A large number of papers have been written on pricing barrier options,hence we will just mention some well-known works. The first published papers go

back to Merton (1973) and Goldman, Sosin, and Shepp (1979). Thereafter

contributions have been made by Conze and Viswanathan (1991) and Reiner and

Rubinstein (1991) among others. For an investor, though, a barrier option may in

some cases have some drawbacks because there is a positive probability that the

option will become worthless at any time before expiration. Therefore, so-called

partial barrier options were introduced in Heynen and Kat (1994; see also Carr

1995). The main feature of these options is that the barrier is only active over a part

of the options lifetime.

This paper is organized as follows. Section 2 summarizes the Black and Scholesframework and states what the replicating strategy looks like in the case of the

D-hedging approach and the Malliavin calculus approach as in Karatzas and

Ocone (1991). In Section 3 we give a short introduction to Malliavin calculus, and

in Section 4 we show why the standard Malliavin calculus approach cannot be

used when the contingent claim is discontinuous. In Section 5 we extend the

Malliavin calculus approach to be valid for any square integrable contingent claim,

and in Section 6 we show how the extended Malliavin calculus approach can be

used to derive the replicating portfolios of some barrier and partial barrier

options. Finally, in Section 7 we present some extensions and summarize the

paper.

2. THE PRELIMINARIES

We fix a finite time interval 0; T and we let fVt : t2 0; Tg be a Wiener process onthe complete probability space X;F; Q. We also denote by F fFt : 0 t Tg theQ-augmentation of the natural filtration generated by V. Now, let us assume that we

are living in a BlackScholes environment, and thus the basic market consists of two

assets: one locally risk-free asset of price B and one stock of price S. The

interpretation of the locally risk-free asset is as usual that of a bank account where

money grows at the short interest rate r. Under the unique equivalent martingale

measure Q the evolution of the asset prices is modeled by the (stochastic) differential

equations

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dBt rBtdtB0 1;

&2:1

dSt rStdt rStdVtS0 S0:

&2:2

From now on we assume that r;r, and S0 are positive constants. The basic market is freefrom arbitrage and complete in the sense that every FT-measurable contingent claimG2 L1X is attainable by a self-financing portfolio.1 In this paper, however, we will focuson contingent claims G2 L2X. The discounted value process Vh=B of the replicatingportfolio h is then a Q-martingale, and the fair price of Gcan thus be expressed as

Vht erTtEQGjFt:2:3Although it is simple to derive the arbitrage-free price of the contingent claim G,

equation (2.3) does not tell us anything about how we can find the self-financing

portfolio h.

In order to derive the replicating portfolio h we can use either the well-knownD-hedging approach or the Malliavin calculus approach as in Karatzas and Ocone

(1991). Suppose that we choose the D-hedging approach. Then if there exists a

deterministic function f 2 C1;2 such that ft; St erTtEQGjFt, we can apply theIto formula to ft; St and use the definition of a self-financing portfolio to obtain therepresentation of the replicating portfolio h h0; h1,

h0t ertVht h1tSt;2:4h1t fst; St:2:5

Here h

0

t denotes the number of units to be held at time t in the locally risk-free assetB; and h1t denotes the number of units to be held in the stock S at time t. In someapplications, such as the Asian and Lookback options, there exists no such function f.

In these cases, however, we can usually introduce a supplementary state variable Zt;which for instance can be the running average or the running maximum of the stock

price, such that the price of the option can be expressed as ft; St; Zt. Givendifferentiability conditions on the function f, we can again apply the Ito formula to the

price process of the option and identify the replicating portfolio. The more complex the

contingent claim G is though, in the sense that the more state variables we need to use,

the harder it is to verify the necessary differentiability conditions on the function f.

This becomes a particularly difficult task when we have no analytical solution for the

price of the contingent claim. On the other hand, if we consider the Malliavin calculus

approach, as in Karatzas and Ocone (1991), we get the representation

h0t ertVht h1tSt;2:6h1t erTtr1St1EQDtGjFt:2:7

Here DtG denotes the Malliavin derivative of the contingent claim G, which we will

come back to later. The usual restriction in order for the Malliavin calculus approach

to work is basically that the payoff G is continuous with respect to movements in the

stock price S.

1 A portfolio h h0;h1 is said to be self-financing if the corresponding value functionVht h0tBt h1tSt has the stochastic differential dVht h0tdBt h1tdSt:

GENERAL APPROACH TO HEDGING OPTIONS 201


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By looking at the various price formulas for barrier and partial barrier options we

see that there actually exists a function f such that ft; St erTtEQGjFt if theoption is alive at time t. Hence, as long as the option is alive we can use the traditional

D-hedge, whereas if the option is dead there is nothing to hedge. Moreover, since the

payoff of a barrier or partial barrier option is discontinuous we cannot use the

Malliavin calculus approach.In the next sections, however, we will show how the Malliavin calculus approach can

be generalized to cover discontinuous contingent claims as well.

3. THE MALLIAVIN CALCULUS APPROACH

The Malliavin calculus can be introduced in a number of slightly different ways; see, for

example, Watanabe (1984), Karatzas and Ocone (1991), Nualart (1995a), U stu nel

(1995), or ksendal (1996). Here we basically follow Watanabe and U stu nel. We say

that a stochastic variable F : X

!R belongs to the set

Pif F is in the form

Fx fVt1;x; . . . ; Vtn;x; where the deterministic function f : Rn ! R is apolynomial. We notice that the set Pis dense in LpX for p 1. Next, we define theCameronMartin space H according to

H c : 0; T ! R : ct Zt

0

_ccsds; jcj2H ZT

0

_cc2sds < 1& '

;

and identify our probability space X;F; Q with C00; T;BC00; T; l such thatVt;x xt for all t2 0; T. Here C00;T denotes the Wiener spacethat is, thespace of all continuous real-valued functions x on 0; T such that x0 0, BC00; Tdenotes the corresponding Borel r-algebra and l denotes the unique Wiener measure.

With this setup we can define the directional derivative of a stochastic variable in all the

directions in the CameronMartin space.2

Definition 3.1. The stochastic variable F 2 Phas a directional derivative DcFx atthe point x 2 X in all the directions c 2 H defined by

DcFx dd

Fx c0

:

Moreover, as a consequence of the coordinate mapping process Vt;x xt itfollows that V

ti;x

c

x

ti

c

ti

for all ti2

0; T, and as F

2 Pwe see that the

directional derivative also can be expressed as

DcFx Xni1

@f

@xiVt1;x; . . . ; Vtn;xcti:

remark 3.1. Let us fix s 2 0; T, then it follows from the above definition that thedirectional derivative of the Wiener process is given by DcVs;x cs.

2 The fact that we only define the directional derivative in the directions in the CameronMartin space

is because of the nice property that if the stochastic variables F and G are equal almost surely, then as a

consequence of the Girsanov theorem F c G c for all c 2 H:

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From Definition 3.1 we notice that the map c ! DcFx is continuous for all x 2 Xand consequently there exists a stochastic variable rFx with values in the CameronMartin space H such that DcFx rFx; cH :

RT0

drFdt

tctdt: Moreover, sincerFx is an H-valued stochastic variable, the map ! rFt;x is absolutelycontinuous with respect to the Lebesgue measure on

0; T

. Now, we let the Malliavin

derivative DtFx denote the RadonNikodym derivative ofrFx with respect to theLebesgue measure such that

DcFx ZT

0

DtFx _cctdt:

If we identify this expression with Definition 3.1 we have the following result, which in

many cases is taken directly as a definition.

Definition 3.2. The Malliavin derivative of a stochastic variable F 2 P is thestochastic process

fDtF : t

2 0; T

ggiven by

DtFx Xni1

@f

@xiVt1;x; . . . ; Vtn;x1tti :

We note that the Malliavin derivative is well defined almost everywhere dt dQ.

remark 3.2. Let us fix s 2 0; T, then it follows from the above definition that theMalliavin derivative of the Wiener process is given by DtVs;x 1ts.

More generally we can define the k-times iterated Malliavin derivative Dkt1;...;tkFx Dt1Dt2

DtkF

x

such that Dkt1;...;tkF is defined almost everywhere dt

k

dQ. It turns out

that it is natural to introduce the norm k kk;p on the set Paccording to

kFkk;p EQjFjp Xkj1

EQ Djt1;...;tjF

pL20;Tj

!" #1=p;3:1

for p> 1 and k 2 N [ f0g. Here we set kFkp0;p EQjFjp. Now, as the Malliavinderivative is a closable operator (see Nualart 1995a), we define by Dk;p the Banach

space which is the closure ofPunder k kk;p. For future applications we also define thecomplete, countably normed, vector space D1 as the intersection D1 \k1\p>1Dk;p.3Hence, we have the inclusions D1 & Dk;p & Dj;q whenever k j and p q.

remark 3.3. Let us fix s 2 0; T. The solution Ss s0 expr 12r2s rVs tothe stochastic differential equation (2.2) belongs to the Banach space D1;2 and

DtSs rSs1ts. In order to prove this standard result, we approximate the solutionSs by a sequence in Pand use Definition 3:2. together with the closability of theMalliavin derivative. Moreover, we deduce for future applications that Ss 2 D1 sinceSs 2 LpX for all p:

What makes Malliavin calculus interesting in mathematical finance is the Clark

Ocone formula. We present the formula as a theorem and refer to Nualart (1995a) for a

complete proof.

3 Although D1 is not a normed space it is a countably normed spacethat is, a metric space where themetric is constructed from the countably many norms k kk;p.



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Theorem 3.1 (The ClarkOcone formula). Let the FT-measurable stochastic variableG belong to the space D1;2. Then

G EQG ZT

0

EQDtGjFtdVt:

This theorem is a generalization of the Ito representation theorem, see ksendal

(1998), in the sense that it gives an explicit expression for the integrand. Moreover,

since the discounted value process Vh=B of a self-financing portfolio h is aQ-martingale, the Ito formula tells us that

VhT erTVh0 ZT

0

erTth1trStdVt:3:2

Hence, in order to find the replicating portfolio (i.e., the self-financing portfolio h such

that VhT G a.s.) we can identify the coefficients in the ClarkOcone formula and(3.2):

Vh0 erTEQG;h1t erTtr1St1EQDtGjFt:

&

Note that the initial amount Vh0 required to replicate the stochastic variable(contingent claim) G is just our previously defined unique price according to (2.3).

This explains equations (2.6) and (2.7) since Vht h0tBt h1tSt by defini-tion.

In order to explicitly derive the replicating portfolio h of a contingent claim G2 D1;2,we need to calculate the Malliavin derivative of G. This is fairly simple if the contingent

claim G is a Lipschitz function of a stochastic vector process belonging to D1;2 asproved in Nualart (1995a).

Proposition 3.1. Let u : Rn ! R be a function such thatjux uyj Kjx yj;

for any x;y 2 Rn and some constant K. Suppose that F F1; . . . ;Fn is a stochasticvector whose components belong to the space D1;2 and suppose that the law of F is

absolutely continuous with respect to the Lebesgue measure on Rn. Then uF 2 D1;2 and

DtuF Xni1

@u@xi

FDtFi:

In order to see the implications of this proposition, we derive the replicating

portfolio of a standard call option.

example 3.1. Let Pt denote the time t price of a standard call optionthat is, acontingent claim with payoff function ST K for some strike price K. Sincex K is a Lipschitz function for all x and ST has a density function, it followsfrom Remark 3.3 and Proposition 3.1 that ST K 2 D1;2 with

DtST K 1fST > KgrST:Consequently, the replicating portfolio is given by

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h1t erTtr1S1tEQ1fST > KgrSTjFt erTtS1tEQST K 1fST > KgKjFt S1tPt S1terTtKQST > KjFt:

The Malliavin calculus approach is both elegant and straightforward to use for

contingent claims in the space D1;2.

4. LIMITATIONS OF THE MALLIAVIN CALCULUS APPROACH

In Section 2 we stated that any contingent claim G2 L2X could be replicated bya self-financing portfolio. However, the ClarkOcone formula works only for those

contingent claims that belong to the space D1;2 & D0;2 L2X: Hence theMalliavin calculus approach is indeed more restrictive than the D-hedging

approach in some cases. In order to see this, let us start with the followingsimple example.

example 4.1. Let us fix the FT-measurable contingent claim G 1fST Kg.Then

Vht erTtEQGjFt erTtQST KjFt erTtFSTK;where FST denotes the Ft-conditional cumulative distribution function of ST. FromAppendix A it follows that FSTK is in the form /ln KSt; t for some deterministicfunction /: Hence /s

ln

K

S

t

; t

S1

t

K/K

ln

K

S

t

; t

S1

t

KfS

T

K

; where

fST denotes the corresponding Ft-conditional density function of ST. It is easilyverified that the deterministic function ft;s erTt/lnK

s; t is of class C1;2; and

consequently we can use the D-hedging approach to identify the replicating portfolio

h1t according to 2:5 as

h1t fst; St erTt/s lnK

St

; t

erTtS1tKfSTK:

Now let us see what happens if we use the Malliavin calculus approach in a

similar way. If we assume that A is any FT-measurable set such that 1fAg 2 D1;2,then from Proposition 3.1 with u

x

x2 on

0; 1

; it follows that

Dt1fAg 21fAgDt1fAg: Hence Dt1fAg 0 on the complement to A, andDt1fAg 2Dt1fAg on the set A. Thus, the only solution on the set A is givenby Dt1fAg 0, and thereby we get that Dt1fAg 0 almost everywhere. By usingthe ClarkOcone formula we see that in this case 1fAg EQ1fAg QA;consequently it follows that

QA 6 1fAg ! 1fAg j2D1;2:If we return to Example 4.1 and put A fST Kg we see that

1fST Kg j2D1;2, although 1fST Kg 2 L2X. As a consequence we cannotuse the Malliavin calculus approach to identify the replicating portfolio h. This is

rather disturbing because the Ito representation theorem (see ksendal 1998) tells us

that there exists a unique F-adapted process fwt : t2 0; Tg such thatG EQG

RT0wtdVt; with



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EQ

ZT0

w2tdt !

< 1;4:1

for any FT-measurable stochastic variable G2 L2X.In the next section, however, we will see that the problem of calculating the process w

may be solved by regarding the Malliavin derivative in the sense of distributions.

5. EXTENSIONS OF THE CLARKOCONE FORMULA

In this section we show that the ClarkOcone formula is valid for any FT-measurablestochastic variable in L2X, and thereafter we show that the Malliavin calculusapproach to derive the replicating portfolio of a contingent claim as in Karatzas and

Ocone (1991) can be extended in a similar way. At first glance, the extension of the

ClarkOcone formula seems rather innocent since the space D1;2 is a dense subspace of

L2

X

. This is nevertheless not the case. The problem we are facing is that the Malliavin

derivative cannot be defined in the usual way for stochastic variables in L2X. Hence,what first needs to be done is to extend this definition and thereafter characterize the

new space for which the ClarkOcone formula is valid. U stu nel (1995) shows that a

ClarkOcone formula can be derived for elements in the dual space ofD1, which is amuch larger space than L2X. However, this extension is made in a distributional senseand therefore we have no guarantee a priori that the formula actually make sense in the

usual way when restricted to L2X. We show that this is indeed the case. In order toprove our statement, we briefly summarize the results of U stu nel and also refer to

Watanabe (1984).

The preceding arguments will be based on the Wiener chaos expansion, which we

state as a theorem and refer to, for example, ksendal (1996) for a complete proof.

Theorem 5.1 (Wiener chaos expansion). Let F be an FT-measurable stochasticvariable such that F 2 L2X. Then there exists a sequence ffng1n0 of deterministic

functions fn 2 ^LL2Rn such that

F X1n0

Infn EQF X1n1

Infn:

Here ^LL2Rn denotes the space of symmetric square integrable functions on 0; Tn andInfn is the iterated Ito integral Infn RT0 RT0 fnt1; . . . ; tndVt1 dVtn:Moreover, for future simplicity we also denote by JnF the orthogonal projection ofthe stochastic variable F on the nth Wiener chaos; that is, JnF Infn.

We see that the Wiener chaos expansion is closely related to the Ito representation

theorem and the ClarkOcone formula. Motivated by this observation we follow

Watanabe (1984) and introduce, similar to (3.1), the norm jk k jk;p on the set Paccording to

jkFkjk;p X1

n

0

1 nk2JnF

LpX

; F 2 P;

for p> 1 and k2 R. From the Wiener chaos expansion we see directly thatjkFkj0;p kFkL2X kFk0;p: Hence, the norms jk k jk;p and k kk;p are equivalent for

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p> 1 and k 0. However, more surprisingly, it can be shown by means of the Meyerinequalities and the multiplier theorem (see, e.g., U stu nel 1995) that the norms jk k jk;pand k kk;p are equivalent whenever p> 1 and k2 N. Moreover, the coupleP; jk k jk;p is a Banach space, and consequently we extend our previous definitionofDk;p to be the closure of

Punder

jk k jk;p such that the following inclusions are valid

Db;p & Da;p & D0;p LpX & Da;p & Db;p[ [ [ [ [Db;q & Da;q & D0;q LqX & Da;q & Db;q

whenever 1 < p< q and 0 < a < b. Watanabe (1984) shows that the dual of the spaceDb;p, denoted D

0b;p, is given by Db;q with

1q 1

p 1, where the elements ofDb;q are to

be interpreted as generalized stochastic variablesthat is, as composites of distribu-

tions and stochastic processes. We set D1 : \k2R\p>1Dk;p such that this space isdefined just as before. The dual of the complete countably normed vector space D1 isnow given by

D01

[k2R

[p>1

D0k;p

[k2R

[q>1

Dk;q : D1:5:1

The dual space D1 is large enough to contain every composition of Schwartzdistributions and nondegenerate stochastic variables in D1 such as dxST, where dxdenotes the Dirac delta function4 with point mass at x 2 R. Recall that the stockST 2 D1 according to Remark 3.3. We let the term dxST serve as our mainexample of a generalized stochastic variable: note that by itself it has no meaning and

that all the moments of order two and higher are not defined. Still, the (conditional)

expectation ofdxST exists with

EQdxSTjFt :Z1

0

dxsfSTsds fSTx; x 2 0;1;

where as usual fST denotes the Ft-conditional density function of ST.Now we return to the ClarkOcone formula and consider the implications of the

previous definitions. We know that D1 & D1;2, hence the ClarkOcone formula isobviously valid for any element ofD1. Moreover, U stu nel (1995) shows that the mapG! RT

0EQDtGjFtdVt from D1 ! D1 extends as a continuous mapping to

D1 ! D1, which results in the following proposition.

Proposition 5.1. Let G be an element ofD

1. Then we have the representationformula

G EQG ZT

0

EQDtGjFtdVt:

For the proof see U stu nel (1995). The above formula is written in a somewhat sloppy

way since Gis a generalized stochastic variable. Hence the interpretation must be taken

in the sense of distributions, meaning that what looks like an ordinary expectation and

an Ito integral need not be so. However, an explanation for expressing Proposition 5.1

as above is that we are only interested in stochastic variables G that belong to

4 Note that the Dirac delta function is not an ordinary function but a distribution defined such thatRbadxyhydy hx 1fa x bg for any sufficiently smooth function h. Moreover, dx is the formal

derivative of the indicator function 1f > xg.



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L2X & D1; and in this case all the terms in Proposition 5.1 make sense in the usualway. This is proved in the following lemma.

Lemma 5.1. The map G! EQDtGjFt is continuous from L2X ! L20; T X,andR0 EQ

DtG

jFt

dV

t

is an ordinary Ito integral.

Proof. It is clearly sufficient to show that the map is continuous from a dense subset

ofL2X to L20; T X, and here we choose our dense subset to be the linear span ofthe Wick exponentials, or, as they are also called, the martingale exponentials,

exp

ZT0

hsdVs 12

ZT0

h2sds

; h 2 L20; T& '

:

If we denote by ui expR

0hisdVs 12

R0

h2i sds a Wick exponential, then thefollowing three properties hold: (a) EQuiTujT exp

RT

0hishjsds, (b)

EQui

T

jFt

ui

t

, and (c) ui

T

2D1;2 with Dtui

T

hi

t

ui

T

; see, for example,

U stu nel (1995). All that remains is to consider a real-valued linear combination of

Wick exponentials UnT Pni1 ciuiT and to show thatkEQDsUnTjFskL20;TX KkUnTkL2X

for some constant K. By using property (a) we see that

kUnTk2L2X EQXni1

ciuiT 224

35 Xn

i1

Xnj1

cicj exp

ZT0

hishjsds

:

Moreover, from properties (b) and (c) we get that

EQDsUnTjFs EQXni1

cihisuiTjFs" #

Xni1

cihisuis:

Hence, inserting the results and using the Fubini theorem together with property (a)

yield

kEQDsUnTjFsk2L20;TX ZT

0

Xni1

Xnj1

cicjhishjs expZs

0

hithjtdt

ds

Xn

i1Xnj1

cicj expZs

0hithjtdt !

T

s0

Xni1

Xnj1

cicj exp

ZT0

hithjtdt

1

kUnTk2L2X Xni1

ci

2;

which completes the first part of the proof. Finally, by a careful analysis of

Proposition 5.1 in U stu nel (1995) it follows that R

0EQDtGjFt:dVt is an ordinary Ito

integral. h

Although the Malliavin derivative DtG of a square integrable stochastic variable

must be interpreted as a generalized stochastic process (i.e., as a composite of a

208 H. P. BERMIN


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distribution and a stochastic process), the conditional expectation of the Malliavin

derivative is an ordinary stochastic process. Thanks to this smoothing effect we have

the following important result.

Theorem 5.2. Any contingent claim G

2L2

X

can be replicated by the self-financing

portfolio h h0; h1 defined byh0t ertVht h1tSt;h1t erTtr1St1EQDtGjFt:

&

Proof. The result is a direct consequence of (3.2), Proposition 5.1, Lemma 5.1, and

the uniqueness of the Ito integral. h

In order to derive the replicating portfolio using the above theorem, we must

establish how to calculate the Malliavin derivative. If we denote the space ofRn-valued

Schwartz distributions by S0

R

n

, we have the following extension of Proposition 3.1.Corollary 5.1. Let T 2 S0Rn. Suppose that F F1; . . . ;Fn is a stochastic vector

whose components belong to the space D1 and suppose that the law of F is absolutelycontinuous with respect to the Lebesgue measure on Rn. Then the composite TF 2 D1and

DtTF Xni1

@T

@xiFDtFi:

Here @T@xi F shall be interpreted as an element ofD1 for each i 1; . . . ; n.

For the proof we refer to Watanabe (1984) or U stu nel (1995). To see the implications of

the previous extension of the ClarkOcone formula let us consider the following example.

example 5.1. Let us fix the FT-measurable contingent claim G 1fST Kg 2L2X as in Example 4.1. Then by using Corollary 5.1 we can formally calculate thereplicating portfolio according to Theorem 5.2 as

h1t erTtr1S1tEQDt1fST KgjFt

erTtr1S1

t

EQ

dK

S

T

rS

T

jFt

erTtS1tZ1

0

dKssfSTsds

erTtS1tKfSTK;which is precisely the result we obtained in Example 4.1.

In order to continue with barrier contracts we must be able to control the whole

trajectory of the stock. This motivates the introduction of the following stochastic

variables:

MSt1;t2 supt2t1;t2

St and mSt1;t2 inft2

t1;t2St;

for 0 t1 t2 T. Now, let us consider the following example of the simplest existingbarrier contract.



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write down this generalization of Proposition 3.1 for square integrable stochastic

variables.

Corollary 5.3. Suppose that F F1; . . . ;Fn is a stochastic vector whosecomponents belong to the space D1;2 and suppose that the law of F is absolutely

continuous with respect to the Lebesgue measure on Rn. Let u : Rn ! R be a piecewiseLipschitz function such that uF 2 L2X then DtuF 2 D1;2L20; T and

DtuF Xni1

@u

@xiFDtFi:

Here @u@xi F shall be interpreted as an element ofD1;2 for each i 1; . . . ; n:

Proof. Since u is a piecewise Lipschitz function and D1;2 is large enough tocontain variables such as dF, F 2 D1;2, we can differentiate u in the sense ofdistributions. h

example 5.3. Let us fix the FT-measurable contingent claim G 1fMS0;T Hg as inExample 5.2. Then by using corollaries 5.2 and 5.3 we can formally calculate the

replicating portfolio according to Theorem 5.2 as

h1t erTtr1S1tEQDtf1fMS0;T HggjFt erTtr1S1tEQdHMS0;TMS0;T1fMS0;t MSt;TgjFt erTtr1S1tEQdHMSt;TMSt;T1fMS0;t MSt;TgjFt

er

T

tr

1

S1

tZ1

0dHmm1fM

S0;t mgfMSt;Tmdm

erTtr1S1tHfMSt;TH1fMS0;t Hg;

which again is the same result as in the D-hedging approach according to Example 5.2.

Now we are ready to use the extended Malliavin calculus approach to find the

replicating portfolios of some barrier contracts.

6. HEDGING BARRIER AND PARTIAL BARRIER OPTIONS

In this section, we derive the self-financing portfolios that generate the square

integrable payoff functions of barrier and partial barrier options by using the

extended Malliavin calculus approach. Actually we will only consider partial barrier

options since these contingent claims might be seen as generalizations of standard

barrier options. Though not presented the results can thereafter easily be verified to

equal the replicating portfolios obtained by using the D-hedging approach. The

prices of the options have already been derived in the paper by Heynen and Kat

(1994), so according to (2.3) we already know the initial amount of money needed

to replicate these contingent claims. In fact we will only derive the number of units

to be held in the stock S, since we know from Theorem 5.2 that the number of

units to be held in the bank account B will then be implicitly determined.Henceforth, we consider the times 0 s T and interpret time 0 as today, time sas the monitoring time, and time T as the maturity of the options. Moreover, we



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will denote by Nx the cumulative distribution function of a standard normalstochastic variable and by Nx;y;q the cumulative distribution function of abivariate standard normal stochastic variable with correlation q. Let us also

introduce the variables

d1 ln

S0=K

llT

rffiffiffi

Tp ; d2 d1 r ffiffiffiffiTp ; d01 d1 2 lnH=S0rffiffiffiTp ; d02 d2 2 lnH=S0rffiffiffiTpe1 lnH=S0llsrffiffisp ; e2 e1 r ffiffiffisp ; e01 e1 2 lnH=S0rffiffisp ; e02 e2 2 lnH=S0rffiffisp ;

where l r 12r2 and ll r 1

2r2:

For both standard and partial barrier options it is possible to distinguish between

knock-out and knock-in options. However, since the sum of a knock-out option and a

knock-in option by definition is equal to a standard option, we will only consider the

knock-out options. In order to facilitate the reading, though, we will just call them out

options. Finally we introduce the variables

g 1 if call option

1 if put option& ; m 1 if up-and-out option1 if down-and-out option&in order to keep track of all the relevant combinations.

There are four possible combinations of out barrier options, namely the up-and-out

call (UOC), the up-and-out put (UOP), the down-and-out call (DOC), and the down-

and-out put (DOP). According to (2.3) the time t prices of the partial out barrier

options, PPOt, may formally be written asP

POt erTtEQST K1fMS0;s HgjFt if UOC and H K;P

POt erTtEQK ST1fMS0;s HgjFt if UOP and H K;PPOt erTtEQST K1fmS0;s HgjFt if DOC and H K;P

POt erTtEQK ST1fmS0;s HgjFt if DOP and H K:Note that by setting the monitoring time s T, we obtain the standard out barrieroptions. Hence, if we know how to price and hedge partial out barrier options we also

know how to price and hedge standard out barrier options.

Obviously, the price of a partial out barrier option will depend on whether time t

is less or greater than the monitoring time s. If the option is alive at time s then the

price is equal to the price of a standard option, which we already know how to

price and hedge. Let us therefore assume that t

s. From the results in Heynen and

Kat (1994) we know that the time 0 price of the partial out barrier options is givenby

PPO0 g S0 N gd1; me1;gm

ffiffiffiffis

T

r H

S0

2ll=r2N gd

01; me

01;gm

ffiffiffiffis

T

r " #(

erTK N gd2; me2;gmffiffiffiffis

T

r H

S0

2l=r2N gd

02; me

02;gm

ffiffiffiffis

T

r " #);

and according to (2.3) this is just the initial amount of money we need to replicate the

options. In order to find the replicating strategies let us consider the partial up-and-out

call. From Theorem 5.2 it follows that the replicating portfolio must consist of

h1t erTtr1S1tEQDtST K1fMS0;s HgjFt

212 H. P. BERMIN


15/20

units of the stock Sat time twhenever t s. This expression is evaluated in Appendix B.By repeating the procedure for the other possibilities we get for the general case when

t s that

h

1

t g N gln

StK llT trffiffiffiffiffiffiffiffiffiffiffi

T tp ; m ln St

H lls trffiffiffiffiffiffiffiffiffiffis tp ;gmq0@ 1A8 s we notice from thedefinition of the payoff functions that the hedging strategy is equal to the hedging

strategy of a standard option provided that the partial out barrier option is alive at

time s. Moreover, setting s T gives the replicating portfolios for the standard outbarrier options.

7. EXTENSIONS AND SUMMARY

In this paper we show how the Malliavin calculus approach to hedging contingent

claims can be extended to any square integrable payoff function, or more precisely we

show that the ClarkOcone formula can be extended to square integrable random

variables. From a practical point of view the extension is interesting because there are a

lot of contingent claims, such as barrier options, that do not belong to the subspace

D1;2 & L2X. It is also clear that our results still hold when we consider more generaldynamics of the stock price. For instance, if we define the stock price S as the solution

to the stochastic differential equation:

dSt rStdt rStdVtS0 S0;

&

and assume that the functional form of r is such that rx > 0 for x > 0; zero is anunattainable boundary, and the solution is nonexploding, then Theorem 5.2 still holds

after the obvious modification of the replicating portfolio

h1t erTtrSt1EQDtGjFt:7:1



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Moreover, under these conditions it is known (see Nualart 1995a) that the stock price has

a density which simplifies the calculation of the Malliavin derivativeDtGconsiderably. In

the case where the analytical form of the density is known, the replicating portfolio can

usually be calculated in closed form. However, when this is not possible we use equation

(7.1) as a basis for Monte Carlo simulation. If the contingent claim G

2D1;2 then (7.1) is

often directly amenable for simulations; however, if G is only in L2X the situation ismore complicated because the Malliavin derivative of G will in general include Dirac

delta functions which if approximated with deterministic functions will generate a high

variance for the estimated value of the replicating portfolio. A way to overcome this

problem was presented in Fournie et al. (1999). Their approach, which was based on the

integration by parts formula of Malliavin calculus, was to reexpress (7.1) in the form

h1t erTtrSt1EQGHjFt;for some stochastic variable H. In that paper explicit forms for the stochastic

variable H were derived for standard and Asian options, after which Monte Carlo

simulations were carried out. Recently, the integration by parts technique has alsobeen applied to lookback and barrier options in papers by Gobet and Kohatsu-Higa

(2001) and Bernis, Gobet, and Kohatsu-Higa (2002), confirming the efficiency of the

method.

If we compare the Malliavin calculus approach with the well-known D-hedging

approach, we find an interesting difference. While we only need the contingent claim to

be square integrable in order to derive a formal expression for the replicating portfolio

with the Malliavin calculus approach, the standard D-hedging approach requires

differentiability conditions at any point in time. For instance, if the time t price of the

contingent claim G can be expressed in the form ft; St; Zt; with Z being someadditional state variable of bounded variation, and f is of class C

1;2;1

, then theD-hedging approach implies that

h1t fst; St; Zt;ft;s;z erTtEQGjSt s; Zt z:

&

However, by using (7.1) as a starting point it is possible to show that the D-hedging

formula holds under the weaker condition that ft;s;z only is Lipschitz continuous ins; provided that the pair S; Z has a joint density (see Bermin 2002 for details). Thisresult is of particular interest when we depart from the standard BlackScholes setup

with a constant volatility and thus are forced to use Monte Carlo simulations, since the

integration by parts technique of Malliavin calculus in general requires a D-hedgingformula to start with.

As an application of our general results we derive the replicating portfolios for some

barrier and partial barrier options. It should be pointed out however that in the

standard BlackScholes model where the joint density of the stock price and its running

maximum (minimum) is analytically known and closed-form solutions for the option

prices are available, it is usually easier to apply the D-hedging formula directly rather

than to use the extended Malliavin calculus approach.

APPENDIX AIn this appendix we present the different cumulative distribution functions that we will

use. These joint cumulative (conditional) distribution functions are all consequences of

214 H. P. BERMIN


17/20

the reflection principle (see, e.g., Karatzas and Shreve 1996), and can be derived from

the results in Heynen and Kat (1994) and Carr (1995).

Lemma A.1. Given that 0 t s T. Let H > St and define k lnK=St,h

ln

H=S

t

. Then Q

S

T

K;MSt;s

H

jFt:

is equal to

Nk lT trffiffiffiffiffiffiffiffiffiffiffi

T tp ;h ls trffiffiffiffiffiffiffiffiffiffis tp ;

ffiffiffiffiffiffiffiffiffiffiffis tT t

r

e2hlr2N k 2h lT trffiffiffiffiffiffiffiffiffiffiffi



r :

Lemma A.2. Given that 0 t s T. Let H > St and define k lnK=St,h lnH=St. Then QST > K;MSt;s HjFt is equal to

Nk lT t

rffiffiffiffiffiffiffiffiffiffiffi



r




r :

Lemma A.3. Given that 0 t s T. Let H < St and define k lnK=St,h lnH=St. Then QST K; mSt;s > HjFt is equal to

Nk

l

T

t

r ffiffiffiffiffiffiffiffiffiffiffiT tp ;h

l

s

t

r ffiffiffiffiffiffiffiffiffiffis tp ; ffiffiffiffiffiffiffiffiffiffiffis

t

T tr e2hlr2N k 2h lT t

rffiffiffiffiffiffiffiffiffiffiffi



r :

Lemma A.4. Given that 0 t s T. Let H < St and define k lnK=St,h lnH=St. Then QST > K; mSt;s > HjFt is equal to

Nk lT t

r ffiffiffiffiffiffiffiffiffiffiffiT tp ;h ls t

r ffiffiffiffiffiffiffiffiffiffis tp ; ffiffiffiffiffiffiffiffiffiffiffi

s tT

tr




r :

In order to obtain the corresponding density functions, we need to know how to

differentiate the bivariate normal cumulative distribution function. If we define the

variable WK;H by

WK;H NAK;H;BK;H;q ZAK;H

1

ZBK;H1

uu; v; qdudv;

where A; and B; are some continuously differentiable functions with respect toboth arguments and u; ;q being the density function of a standard bivariate normalstochastic variable with correlation q, it then follows that



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@WK;H@H

uAK;HN BK;H qAK;Hffiffiffiffiffiffiffiffiffiffiffiffiffi1 q2

p

@AK;H@H

u

B

K;H

N

AK;H qBK;Hffiffiffiffiffiffiffiffiffiffiffiffiffi1 q2p

@BK;H@H

:

Here, u denotes the density function of a standard normal stochastic variable.

APPENDIX B

In this appendix we derive the conditional expectation of the Malliavin derivative for

the partial up-and-out call in Section 6. We use the notation dH for the Dirac deltafunction with point mass at H, which is the formal derivative of the indicator function

1

f> H

g. Moreover, we define the RadonNikodym derivative

dQS

dQ ST

EQST on FT;

such that QS is a probability measure absolutely continuous with respect to Q. It is easy

to show that the Girsanov kernel for this transformation is just equal to r, and

consequently the distribution functions in Appendix A taken with respect to QS are

obtained by simply replacing l with ll l r2. For a complete and detailed study ofthese aspects, see Geman, El Karoui, and Rochet (1995). Moreover, for every

stochastic variable X such that EQSjXj < 1 it follows thatEQSTXjFt EQSTjFtEQSXjFt;B:1

see, for example, ksendal (1998) for details.

Proposition B.1. Let the payoffGbe defined by G ST K1fMS0;s Hg: Then,for t s,

DtG rST1fST > K;MSt;s Hg1fMS0;t HgrST KHdHMSt;s1fMS0;t Hg:

Proof. By using the fact that the joint law ofST;MS0;s is absolutely continuouswith respect to the Lebesgue measure on R2 and Corollary 5:3 we get that

DtG 1fST > KgrST1fMS0;s HgST KdHMS0;srMS0;s1fMS0;t MSt;sg

rST1fST > K;MS0;s HgrST KdHMSt;sMSt;s1fMS0;t MSt;sg

rS

T

1

fS

T

> K;MSt;s

H

g1

fMS0;t

H

grST KHdHMSt;s1fMS0;t Hg;

which completes the proof. h

216 H. P. BERMIN


19/20

Our next step is to derive the conditional expectation

EQDtST K1fMS0;s HgjFt; for t s < T:Given that MS0;t H, these formal calculations follow from (B.1):

EQDtGjFt rEQSTjFtQS

ST > K;MSt;s HjFt

rHEQSTjFtEQS1fST > KgdHMSt;sjFtrHKEQ1fST > KgdHMSt;sjFt:

We let Fl

ST;MSt;sand F

ll

ST;MSt;s, denote the joint cumulative Ft-conditional distribution

functions of the pair ST;MSt;s with respect to the probability measures Q and QS,respectively, and introduce the notations

fiST;MSt;ss; m

@2

@s@mFiST;MSt;ss; m

FiSTjMSt;sHs @@mFiST;MSt;ss; mmH

8>: ; i l; llNote that the upper index indicates the relevant probability measure (i.e., if we are to

use l or ll in the formulas that are presented in Appendix A). Now, since

EQ1fST > KgdHMSt;sjFt Z1

0

Zm0

1fs > KgdHmflST;MSt;ss; mds dm

ZH

0

1fs > KgflST;MSt;s

s;Hds

ZH

K f

l

ST;MSt;ss;Hds Fl

STjMSt;sHH Fl

STjMSt;sHK;

and similar for the term EQS1fST > KgdHMSt;sjFt, we finally get thatEQDtGjFt rerTtStQSST > K;MSt;s HjFt

rHerTtSthF

ll

STjMSt;sHH Fll

STjMSt;sHK

irHK Fl

STjMSt;sHH Fl

STjMSt;sHK

h i;

if MS0;t H and zero otherwise. Inserting the formulas from Appendix A yields thedesired results presented in Section 6.

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a general approach to hedging options

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