Brownian Excursions and Parisian Barrier Options

Brownian Excursions and Parisian Barrier Options

Marc Chesney,

HEC, Departement Finance et Economie, 1, rue de la liberation,

78351 Jouy en Josas Cedex, France Monique Jeanblanc-Picque, Equipe d'analyse et probabilites,

Universite d'Evry Val d'Essonne, Boulevard des coquibus, 91025 Evry Cedex, France

Marc Yor,

Laboratoire de probabilites, Tour 56, 3-ieme etage, Universite P.M.C., 4, Place Jussieu,

75252 Paris Cedex, France

July 95, Revised version : December 95 To appear in Adv. Appl. Proba. March 1997

Abstract: A new kind of option, called hereafter a Parisian barrier option, is studied in this paper. This option is the following variant of the so-called barrier option : a down-and- out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecied level and remains constantly below this level for a time interval longer than a xed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

Keywords: Brownian Excursions, Brownian meander, Barrier Options.

AMS classication : 60 G 46, 60 H 60.

1

1 Introduction

The payo of a standard European option only depends on the price of the un- derlying asset at the maturity date. In the barrier (or knock-out) option case, the payo does not only depend on the nal price of the underlying asset but also on whether or not the underlying asset has reached a barrier price during the life span of the option.

The more volatile the underlying value is, the less sense the concept of knock-out option makes. It is acceptable to lose the right to exercise a barrier option only if the probability that the standard option will be in the money at maturity is pretty low. This is not the case if the volatility is high when the option is lost (in this case the value of the option is pretty small). Actually, after a critical level of the volatility, the European knock-out option has a value which is a monotonic decreasing function of the volatility.

In this paper, we dene a new kind of option which we shall call a Parisian barrier option. For conciseness, we shall often drop the term \barrier" in the sequel. A Parisian option is close to a barrier option, studied in 20], the dierence lying in the fact that the owner doesn't lose the option if the value of the underlying value reaches the knock-out level, but only if it remains long enough under (or above) this level.

With a Parisian option, the joint probability to lose the right to exercise and to have at maturity an underlying value which is higher (resp. smaller) than the strike price for a call (resp. for a put) can be monitored.

The Parisian option combines the advantage of the knock-out option, i.e., to reduce the cost, with the advantage of standard options, i.e., to keep the right to exercise longer.

The referee also pointed out another possible advantage of Parisian options we quote : \as far as barrier down-and-out options are concerned, an inuential agent in the market who has written such options and sees the price approaching that limit could try to push the price below this limit, even momentarily. This will make the options worthless, so even if the agent loses some money while doing so, he might be compensated by the elimination of liabilities. However, in the case of Parisian barrier options, he will have to make sure that the price stays below that level for some time this might prove more dicult or more expensive. This makes the market fair in that it protects the holder of such options from deliberate action taken by the writer."

The paper is organised as follows: in section II, we introduce the notation and dene the assumptions. In section III, we obtain an upper and a lower bound for the value of the Parisian option. In section IV, we attempt to value these options, whereas in section V, we give precise formulae for the Laplace transforms of their values. In section VI, we prove the put-call parity. We study cumulative Parisian options in section VII. Technically, in pricing such options, we will rely upon ex-

2

cursion theory, the needed results from which are presented in the appendix.

The main computation made in this paper is an illustration of the following general \principle" : to estimate the usually complicated function

(tK)^def= E11ft Tg(t^;K)⁺]

for (tt 0) a Brownian functional, and T a stopping time, we compute the Laplace transform of (K) :

Z ₁

0 dte^{; t} (tK) =Ee^{; T Z}₀¹ due^{; u}(T⁺u^;K)⁺]

and, in many instances, the right-hand side may be evaluated with the help of the strong Markov property. Two previous applications of this \principle" have been made for the computation on one hand of Asian options 10], on the other hand of double barrier options 11].

Acknowledgements:

We are grateful to Glenn Kentwell and John Cornwall from Banker Trust Australia, for their comments and useful suggestions, to the anonymous referee for constructive remarks and to Laurent Gauthier for helpful discussions. We are of course responsible for any possibly remaining error.

2 Denitions

2.1 Notation

Let us briey describe what is an excursion at (or away from) the levelLfor an It^o process St, i.e., a process of the form dSt =b(tSt)dt+a(tSt)dWt. For details, see the Appendix and Revuz-Yor 22], ch XII.

LetSt be an It^o process and callL the level of the excursion we consider. We use the notation

g_SLt = sup^fst^jSs =L^g d_SLt= inf^fst^jSs =L^g (we make the usual convention : sup() = 0, and inf() = +¹).

The trajectory of S between g_SLt and d_SLt is the excursion of S at the level L, which straddles time t. The variables g_SLt and d_SLt are the left and the right ends of the excursion. If S remains below L during this excursion, i.e., if St < L, the excursion is said to be belowL.

The length (or life duration) of the excursion which straddles t is d_SLt^;g_SLt. We are interested here with t^;g_SLt, which is the age of the excursion.

In the context of a Parisian option, St is the price of the underlying asset. We suppose that

dSt=St(dt+dBt) S⁰ =x (1) 3

where is a non-negative constant and B a Brownian motion.

We will denote byr the interest rate and by the dividend rate if the underlying is a stock (otherwise, for a currency, will play the role of the foreign interest rate). We assume that both r and are constant.

The maturity of the option is denoted byT and the strike price is K.

2.2 Parisian out option

a. Down-and-out option

The owner of the option loses the option if the un- derlying asset price St reaches the level L and remains constantlybelow this level for a time interval longer than a xed number D, called the option window. If not, the owner will receive a payo (ST), with (x) = (x^;K)⁺ for a call (resp.

(K^;x)⁺ for a put) whereK is a xed value. For a down-and-out barrier option, the case of interest is when the initial price of the underlying asset is greater than the barrier this is no more the case for a down-and-out Parisian option.

b. Up-and-out option

The option is lost by its owner if there is an excursion above the level L which is older than D.

2.3 Parisian in option

a. Down-and-in option:

The owner of a down-and-in option receives the pay- o only if there is an excursion below the level L which is older than D. The down-and-in option (resp. down-and-out) refers to an option which appears (resp.

disappears) when there is an excursion which lasts long enough below the level L.

b. Up-and-in option:

The owner of the option receives the payo only if there is an excursionabove the level L which lasts longer than D.

We assume no rebate in our study we could easily extend our computations to that case.

3 Upper and Lower Bounds of a Parisian down- out option

As presented in the introduction, the Parisian option combines dierent charac- teristics of the knock-out option, and of the standard option. More precisely, let us consider the two following limiting cases for down-and-out options:

St > L and D T ^;t. In this case, the probability to have an excursion below L, between t and T , of length at least equal to D is zero.

4

The values of the Parisian call and put are the standard European options given by the Black and Scholes formula 3].

St> Land D = 0.

In this case, as soon as the underlying value reaches the excursion level L, the option is lost. The option becomes a knock-out option and its value is well known 20].

Between those two limiting cases, whenD decreases fromT^;tto zero, the option value decreases, because the probability to lose the right to exercise increases. We thus have the following formulae for the call and put cases:

a. Call

Ifx > L :

8>

>>

>>

><

>>

>>

>>

:

C_do(xT) xe^;T_N(d1)^;Ke^;rT_N(d2) C_do(xT) xe^;T_N(d1)^;Ke^;rT_N(d2)

;xx L

_;2

N(y) + Ke^;Te^;rT x L

_2;2

N(y^;pT)

(2)

b. Put

Ifx > L :

8>

>>

>>

><

>>

>>

>>

:

P_do(xT) Ke^;rT_N(;d²)^;xe^;T_N(;d¹) P_do(xT) Ke^;rT_N(;d²)^;xe^;T_N(;d¹)

; Ke^;rTe^;T x L

_2;2

N(^;y+^pT) +xx L

_;2

N(^;y) with (3)

d¹ = 1^pT

ln(x

K^{) + (}r^;)T +²T 2

d² =d^1;pT and ^N the distribution function of the normalized gaussian variable,

_{= 12 +} r

² y= 1^pT ^ln( L²

xK ^+pT)

Notice that for each D belonging to 0T], a Parisian option can be dened. The two limit casesD= 0 andD=T correspond respectively to the knock-out option and to the standard option.

4 The valuation of Parisian options

LetQdenote the risk neutral probability. UnderQ, the dynamics ofS is given by dSt=St ((r^;)dt+ dWt) S⁰ =x (4)

5

where (Wtt 0) is a Q-Brownian motion and x >0. It follows that St=x exp^"r^;;2

2

! t+Wt

# :

Let us introduce the following notation m= 1

r^{;; 2} 2

! b = 1 ^ln

L x

where Lis the excursion level.

The price of the asset is

St=xexp((mt+Wt)):

4.1 Parisian out-option

4.1.1 Down-and-out option.

The owner of a down-and-out Parisian option loses it if St reaches L and remains constantly below the level L for a time interval longer thanD.

This is the case if and only if ~H_LD^; (S) T, where ~H_LD^; (S) is the rst time at which the age of an excursion below the level L for the process St is greater (\older") than or equal to D

H~_LD^; (S) = inf^ft 0^j11St<L(t^;g_SLt)D^g (5) We can give a more convenient form to ~H_LD^; (S) by refering to the Brownian motionWt. Actually, ~H_LD^; (S) =H_bD^m;, where

H_bD^m;= inf^ft 0^j11Wt⁺mt<b(t^;g_mbt)D^g (6) where g_mbt is the left extremity of the excursion straddling time t and away from level b for the process Wt+mt:

g_mbt= sup^fu < t^jWu+mu =b^g (7) Relying on Cameron-Martin-Girsanov theorem, we introduce a new probability P, which makes (Zt=Wt+mt0tT) a P-Brownian motion. The price of a down-and-out option with payo (ST) is, in an arbitrage free model,

e^;rTEQ

(ST)11_{Hb D}^m;_>T= exp^;(r+m²

2 )T]EP

11_{Hb D}^; _>T (xeZ^T)e^mZT (8) where

H_bD^; =H_bD^0; = inf^ft0^j11Zt<b(t^;gbt)D^g gbt= sup^fut^jZu =b^g:

6

In many formulae involving a function of the maturity T, as in (8), the dis- counted factor exp^;(r + ^m₂²)T] appears. In order to give concise formulae, we introduce the following notation

?(T) = exp(r+m²

2 )T] (T) (9)

and we shall refer to this quantity as a (rm)-discounted value.

a. Parisian down-and-out call :

We now treat in details the Parisian down- and-out call. Let us denote byC_do(xTKLDr) the value of a Parisian down- and-out call.

This value depends on the parameters x (the value of the underlying asset), T (the maturity) and KLDr. When the parameters KLDr are xed, we use the concise notationC_do(xT). From (8), we have

C_do(xTKLDr) = exp^;(r+m²

2 )T]EP

11_{Hb D}^; _>T (xexp(ZT) ^;K)⁺ exp(mZT) therefore, using notation (9), we obtain the following expression

?C_do(xTKLDr) = EP

11_{Hb D}^; _>T (xexp(ZT) ^;K)⁺ exp(mZT)

b. Parisian down-and-out put :

Using obvious notation, we get the following:

?P_do(xTKLDr) = EP

11_H^;_{b D>T} (K ^;xexp(ZT))⁺ exp(mZT)

4.1.2 Up-and-out options

The (rm)-discounted value of an up-and-out option with payo (ST) is : EP

11_{Hb D}⁺ _>T (xeZ^T) exp(mZT) where

H_bD⁺ = inf^ft0^j11Zt>b(t^;gbt)D^g gbt= sup^fu < t^jZu =b^g

We denote by C_uo(xTKLDr) (resp P_uo(xTKLDr)) the value of an up-and-out call (resp. put).

4.2 Parisian in options

The \in" case follows from the \out" case: for example, let us denote by

?C_di(xTKLDr) = EP

11fH_{b D}^; T^g(xeZ^{T ;}K)⁺ exp(mZT) 7

the (rm)-discounted value of a down-and-in call. Then,

C_do(xTKLDr) = ^BS(xT)^; C_di(xTKLDr) (10) where ^BS(xT) is the Black-Scholes price, i.e.,

BS(xT) = exp^;(r+ m²

2 )T]EP

(xeZ^{T ;}K)⁺ exp(mZT): In the same way, the values of an up-and-out call and of an up-and-in call

C_ui(xTKLDr) = EP

11fH_{b D}⁺ Tg(xeZ^{T ;}K)⁺ exp(mZT) satisfy

C_uo(xTKLDr) = ^BS(xT)^; C_ui(xTKLDr)

4.2.1 Some references

A number of papers 6, 13, 15, 16, 17, 21] and 24] have been devoted to the study of the longest duration of excursions of Brownian motion on the interval 0T], and more generally to the sequences

V¹(T)> V²(T)>...> Vn(T)...

of ranked lengths of excursions during the interval 0T], includingT^;gT. A more complete discussion of the existing literature is presented in 18], where this kind of study is furthered by showing that the law of the normalized sequence

1(V1( )V2( )...Vn( )...)

for certain very special random times , is identical to the law obtained for =T, a constant time this is in particular the case for

=H¹(D) = inf^ft :V¹(t)> D^g

and, more generally for = Hn(D) = inf^ft : Vn(t) > D^g. Such identities in law have originally motivated the writing of 18] and of the present paper. We remark that the scaling property implies that T

Vn(T) ^(law=⁾ Hn(1), an identity which has been exploited by Wendel 24]. It would be possible to construct options based on H²(D) H³(D)... however, we shall not develop this here.

8

4.3 Reduction to the case

^x

⁼

^L

The case x = L (or b = 0) is important, since it is possible to reduce the other cases to it. In this case, we use the following special notation : for excursions below the level L (withx=L orb = 0)

C_o^;(xTKDr) = e^;rTEQ

(ST ^;K)⁺11_H^{^}_{L D}^; (S)>T

or, using the notation (9):

?C_o^;(xTKDr) = EP

(xeZ^{T ;}K)⁺ exp(mZT)11_H0^;_D>T (11) We refer to this case as a zero down-and-out call. For excursions above the level L (zero up-and-out call)

C_o⁺(xTKDr) = e^;rT EQ

(ST ^;K)⁺11_H^~_{L D}⁺ (S⁾>T

(12)

4.3.1 Case

b >0

We show that the b >0 (or L > x) case reduces to the case b = 0. We study the case of call options, the general case follows immediately. Let us denote byTb the rst hitting time of b, related to the Brownian motion (Btt 0)

Tb = inf^ft 0^jBt=b^g and its well known law by b(du) :

b(du) = _pbe^;b22u 2u³ du whose Laplace transform equals

E(exp(^;2

2 Tb)) = exp^;b for0:

a. Down call

In the case b > 0, if H_bD^; T D, then Tb D. Therefore, the (rm)- discounted value of a down-and-out call in the case L > xis

?C_do(xTKLDr) = EP

11_{Hb D}^; _T11TbDxexp((ZT ^;ZTb+b) ^;K]⁺exp(m(ZT ^;ZTb +b)) 9

=^{Z D}

0 b(du)EP

11H~_{b D}^b; T^;uxexp((WT;u+b)) ^;K]⁺exp(m(WT;u+b)) where ~H_bD^b; is the rst time such that an excursion below b, for the Brownian motionWt+b issued from b, is older than D. It follows

?C_do(xTKLDr) =^{Z D}

0 b(du)EP

11_H0^;_{D T};uxexp(ZT^;u)^;K]⁺exp(mZT^;u) therefore, we obtain :

Proposition 1

The (rm)-discounted value of a down-and-out call in the case L > x can be expressed in terms of zero down-and-out calls

?C_do(xTKLDr) =^Z₀^Db(du)^?C_o^;(xT ^;uKDr) where

?C_o^;(xTKDr) = EP(11_H0^;_D>T(xeZ^{T ;}K)⁺ exp(mZT)) and b is the law of Tb.

b. Up call

If b >0 and H_bD⁺ T, then Tb T. Then, we prove the following

Proposition 2

The(rm)-discounted value of an up-and-in call in the caseL > x can be expressed in terms of zero up-and-in calls

?C_ui(xTKLDr) = ^Z₀^T b(du)^?C_i⁺(xT ^;uKDr) where

?C_i⁺(xT) =EP

11fH₀⁺_DT^g(xeZ^{T ;}K)⁺ exp(mZT) corresponds to a zero up-and-in call.

4.3.2 Case

b <0

.

The same method leads to the following formulae :

?C_di(xTKLDr) =^{Z T}

0 b(du)^?C_i^;(xT ^;uKDr) where

?C_i^;(xT) = EP

11fH0^;DT^g(xeZ^{T ;}K)⁺ exp(mZT) :

5 A formula for the valuation

We study, in this section, the valuation of a down-and-in option. The value of a down-and-out option was obtained in (10).

10

5.1 Parisian down-and-in call

Let us denote by ^Ft = (Zss t) the completed ltration of the Brownian motion (Ztt 0). Remark thatH_bD^; is an ^Ft-stopping time. We have

?C_di(xT) = EP

11_{Hb D}^; T EP

e^mZT(xe^{ZT ;}K)^+jF_{Hb D}^; (13) Thus by use of the strong Markov property

?C_di(xT) =EP(11_{Hb D}^; T ^PT;H_{b D}^; (fx)(Z_{Hb D}^; )) (14)

with fx(z) =e^mz(xe^z;K)⁺ (15)

and

Ptf(z) = 1^p2t

Z ₁

;1f(u)exp(^;(u^;z)² 2t ⁾du

the Brownian semi-group acting on f. The same kind of formula applies for a down option with payo (ST).

As recalled in the Appendix, the random variables Z_{Hb D}^; and H_bD^; are inde- pendent. Let us denote the law of Z_{Hb D}^; by(dz).

Consequently, we obtain

?C_di(xT) = ^Z_;1¹ (dz)E(11_{Hb D}^; T^PT^;H_{b D}^; (fx)(z))

= ^{Z 1}

;1dy fx(y)hb(Ty) (16) where

hb(ty) =^Z_;1¹ (dz)EP

0 BB BB

@11_{Hb D}^; _<t exp^;(b^;z^;y)² 2(t^;H_bD^; )

q2(t^;H_bD^; )

1 CC CC

A (17)

Let us remark that hb(ty) = 0 if t < D.

5.1.1 Case

b <0 In this case,

P(Z_{Hb D}^{; 2}dx) = dx

D ⁽b^;x) exp(^;(x^;b)² 2D ^)11xb

11

We establish in the Appendix that, for > 0 the Laplace transform ^hb(y) of hb(y) is

Z ₁

0 dte^{; t}hb(ty) = e^bp2

D^p2 (^p2D)

Z ₁

0 dz z exp(^;z²

2D ^{; j}b^;z^;y^jp2) where

(z) =^{Z 1}

0 dx x exp(^;x²

2 +zx) = 1 +z^p2exp(z²

2 )^N(z): (18) The integral

K D(y^;b)^def= ^{Z 1}

0 dz z exp(^; z²

2D ^{; j}b^;z^;y^jp2) can be easily evaluated.

Ifa =y^;b <0, we obtain, using change of variables

K D(a) = exp(a^p2)1^;2e ^DpD^N(^;p2D)] = exp(a^p2) (^;p2D) Ifa >0, a similar method leads to

K D(a) =e^;ap2 + 2^pD e ^De^{;ap2 N}(_pa

D ^;p2D)^{; N}(^;p2D)] ^;e^ap2 1^{; N}(_pa

D ^+p2D)]

IfD= 0, we obtain that ^hb(y) = eb^p2

p2 e^;jb^;y^jp2, therefore we have the formula for a barrier option (it is easy to invert ^hb and thus to nd the right-hand side of (2)).

Using the explicit expression (15) of fx, we obtain

?C_di(xTKLDr) = ^Z1

(x) dy e^my(xe^{y ;}K)hb(Ty) where (x) = 1^lnK

x^.

Particular case:

K > LIn this case, we need only the value of hb(ty) for y > b and we obtain

^hb(y) = (^;p2D)

(^p2D) e^(2b_p^;y)p2 2

It can easily be proved, since that e^;p2 is a Laplace Transform, that (^;p2D) is a Laplace transform. Therefore, in order to invert ^hb, it suces to invert

(^p12D). This is not easy : see 5] for some computation.

12

Theorem 1

In the case x > L (i.e., b < 0), the (rm)-discounted value of a Parisian down-and-in call with level L is

?C_di(xTKLDr) =^Z1

(x) dy e^my(xe^y;K)hb(Ty) where (x) = 1 ^lnK

x^.

The function hb(ty) is characterized by its Laplace transform

^hb(y) = e^bp2

D^p2 (^p2D)

Z ₁

0 dz z exp(^; z²

2D ^{; j}b^;z^;y^jp2) where (z) is dened in (18).

In the case K > L, the Laplace transform of hb(ty) for y > (x) is h^b(y) = (^;p2D)

(^p2D) e^(2b_p^;y)p2 2 :

Remark:

In each case, it is possible to compute the Laplace transform of the

\Delta" term, i.e., the derivative with respect toxof the value of the option. This Laplace transform has a somewhat complicated form. For example, in the case K > L, the Laplace transform of !(T) = @C_di

@x ⁽xT) is

^!() = ^Z₀¹e^{; t}!(t)dt= (^;p2D)

(^p2D) _p e^2bp2 2 m+^;p2

K x

m+^;p2 where =+^m₂² +r.

5.1.2 Case

b >0

In the case b >0, the Laplace transform of hb(:y) is complicated. We obtain

Z ₁

0 dte^{; t}hb(ty) = E^{Z 1}

;1(dz)e^{; Hb D; p1}₂ ^exp(;jy^;z^jp2) Using the results of the Appendix,

Z ₁

0 dte^{; t}hb(ty) = (19) D^p2 (1 ^p2D)

Z ₁

0 dz z exp(^; z²

2D^{; j}y^;b+z^jp2) ^{Z D}

0 b(dx)e^{; x} + e^{; D}

2^pD

Z b

;1 dzexp(^; z²

2D^);exp;(z^;2b)² 2D

e^;jy^;z^jp2 13

The second term of the right member of (19) is the Laplace transform of g(y) where

g(ty) = 112^pt>Dt^;D

Z b

;1exp(y^;z)²

2(t^;D) exp(^; z²

2D^);exp(;(z^;2b)² 2D ⁾

dz :

Particular case

Ify > b, the rst term on the right member of (19) is equal to(20) (^;p2D)

(^p2D) e^;(_p^y;b)p2 2

Z D

0 b(dx)e^{; x} (21) This term is the product of four Laplace transforms, however the inverse of ₍^p1_{2 D)} is not identied.

Theorem 2

In the case x < L (i.e., b > 0), the (rm)-discounted value of a Parisian down-and-in call with level L is

?C_di(xTKLDr) =^Z1_(x) dy e^my(xe^y;K)hb(Ty) where (x) = 1 ^lnK

x^.

The function hb(ty) is characterized by its Laplace transform ^hb(y) =

1

D^p2 (^p2D)

Z ₁

0 dz z exp(^;z²

2D^;jy^;b+z^jp2) ^{Z D}

0 b(dz)e^{; z}+^g(ty) where g is dened in (20).

6 Parisian put

We establish a put-call parity in the same manner as Grabbe 12]. The (rm)- discounted value of a Parisian down-and-out put is

?P_do(xTKLDr) = EP

11_{Hb D}^; _>T (K^;xe^ZT)⁺ exp(mZT) The right side is equal to

xK EP

11_H^;_{b D>T} (e^;ZT

x ^; K^{1 )+ exp((}+m)ZT)^!

and this expression can be reduced, by introducing the Brownian motion Wt =

;Zt. Thus, the hitting time H_bD^; related to Z is the stopping time H;⁺bD related toW, and it follows that

?P0^d(xT) =xK EP

11_H;⁺_{b D>T}(e^WT

x ^; K^{1)+ exp(;(}+m)WT)^! 14

Writing the terms (+m) and ^m₂²+r in an explicit form with (r) as parameters, it is now easy to check that

P_do(xTKLDr) =xKC_uo(1xT ¹KLD¹ r)

For other Parisian options, the same parity relation holds. For example, for a Parisian up-and-out put,

P_uo(xTKLDr) =xKC_uo(1xT ¹KLD¹ r)

7 Cumulative Parisian options

In this section, we dene and study an option which is lost by its owner when the time spent by the underlying asset below the level L is greater than D.

More precisely, let ;^;_T^L = ^R0^T 11StLdt and ;⁺_T^L = ^R0^T 11St Ldt. The value of the option is

C_c(xTKLDr) =e^;rT EQ((ST ^;K)⁺11_;T ^L _D): The problem reduces to the computation of

?C_c(xTKLDr) = EP(11_A^b_{T D}(xe^{ZT ;}K)⁺exp(mZT)) where A^b_T^;=^R0^T 11Ztbdt, andA^b_T⁺=^R0^T 11Zt bdt.

7.1 Case

^b

^{= 0}

Forb = 0, the law of the pair (ZTA⁺_T) may be obtained from the equality in law (ZTgTA⁺_gT) = (^pT ^;gT m¹gTgTU) where m¹ is the Brownian meander at time 1 (see Appendix), U is uniform on 01], is a symmetric Bernoulli variable on^f;1+1^g, the variables (m1gTU) being independent.

We denote by T(x)dx the sub-probabilityP(ZT ²dxA⁺_T > D). It follows that, since ^fZT ²dxA⁺_T^g=^fZT ²dxA⁺_gT^gfor x <0,

T(x) = ^jx^j 2

Z T

D _q s^;D

s³(T ^;s)³ exp(^; x²

2(T ^;s))ds x <0: Forx >0, we use the decomposition

fZT ²dxA⁺_T > D^g=^fZT ²dx T^;gT > D^gfZT ²dx D > T^;gT > D^;A⁺_gT^g 15

and it follows (see the Appendix for the law of (ZTgT)) that, for x >0 : T(x) = x

2

Z T^;D

0

exp(^; x² 2(T ^;s))

qs(T ^;s)³ ds+x(T ^;D) 2

Z T T;D

exp(^;_2(T^x2_;s))

qs³(T ^;s)³ ds:

The (rm)-discounted price of a cumulative Parisian zero down call (b= 0) is

?C_c⁺(xTKxDr) = ^{Z 1}

;1dy(xe^y;K)⁺exp(my)T(y):

Another proof of the pricing formula of cumulative options was proposed to us by the referee, involving some results on -quantiles (see, e.g., 1, 7, 9]). Dene the -quantile of ^fZt 0t T^g as

M(T) = inf^fx:A^x_T^;> T^g Then, in order to compute

E11_A^b_T^;_>Dxe^{ZT ;}K⁺expmZT

we note that ^fA^b_T^; > D^g = ^fM(^D_TT) < b^g. The decomposition in Dassios 7]

and its extension in 9] state that (M(D

T T⁾ZT)^(law= ( sup⁾

0sDZs+ inf_0sTZ~s ZD+ ~ZT^;D)

where ~Zt is an independent copy of Zt. From this identity, a discrete time version of which goes back to Wendel 23] and Port 19], one can calculate the joint density of M(^D_TT) andZT and thus obtain an expression for the value of the option.

7.2 Down-and-out case,

^{b <}

⁰ .

As in the case of the Parisian down-and-out call with b <0, we establish that

?C_c^;(xTKLDr) =^{Z T}

0 b(du)^?C_c⁺(xT ^;uKxDr)

7.3 Parity relations

Since A^;_T +A⁺_T =T, we have the parity relation :

C_c^;(xTKLDr) +C_c⁺(xTKLDr) =^BS(xK) and P_c⁺(xTKLDr) =xKC_c^;(1xT ¹KLDr)

16

7.4 Some references

The law of the pair (ZTA⁺_T) is studied in 1, 9, 25, 26]. The law of the triple (ZTLTA⁺_T) whereLT is the local time at 0 is studied more completely in 14].

8 Appendix

8.1 Some denitions and notation

8.1.1 Excursions intervals

Let (Ztt0) be a standard Brownian motion. For eacht >0, dene

gt = sup^fs^js tZs= 0^g (22) dt = inf^fs^jstZs= 0^g (23) We have almost surely gt< t < dt.

The interval (gtdt) is the interval of the excursion¹ which straddles timet. For u in this interval, sgnZu remains constant. The law of the pair (gtZt) is

11st ^jx^j

2^qs(t^;s)³ exp^; x²

2(t^;s)dsdx (24)

More generally, we can dene, for b²IR,

gbt = sup^fs^js tZs=b^g (25) dbt = inf^fs^js tZs=b^g (26) Let H_bD^; = inf^ft : 11Ztb(t^;gbt) D^g be the rst time at which the age of an excursion below b is greater than or equal to D. In the same way, we dene H_bD⁺ = inf^ft : 11Zt b(t^;gbt) D^g to be the rst time at which the age of an excursion above b is greater than or equal to D. We are interested in the laws of the pairs (Z_{Hb D}^; H_bD^; ) and (Z_{Hb D}⁺ H_bD⁺ ) which are described below, in 8.3.

8.1.2 The slow Brownian ltration

We suppose that b = 0. Let us denote by (^Ftt 0) the natural ltration of the Brownian motion Z.

If R is a random variable such that R > 0 a.s., we dene the sigma-eld ^F_R^; of the past up to R as the -algebra generated by the variables R, where is a predictable process.

In particular, we consider the -algebra ^F_g^;_t which is included in ^Ft and is in- creasing with respect to t. Denote by (^F_g⁺_tt 0) the slow Brownian ltration²

Fg⁺t =^F_g^;_t ^_(sgn(Zt)):

1For details, see Chung 4] and e.g., Revuz-Yor 22] p.107 Ex 3.23 and Ch. XII, par 3.

2See Dellacherie-Maisonneuve-Meyer 8] for details and comments.

17

8.2 The Brownian meander

We denote by g =g¹ = sup^fs 1 : Zs = 0^g the left extremity of the excursion which straddles time 1.

The Brownian meander³ is dened as the process

mu = 1^p1^;g^jZg+u(1^;g)^ju1 (27) The process m is a Brownian scaled part of the (normalized) Brownian excursion which straddles time 1.

The process m is independent of ^Fg⁺. The law ofm1 may be deduced from (24) P(m1 2dx) =xexp(^;x²

2 )11^x>0dx (28)

Using Brownian scaling again, we remark that for every t, the process m⁽_u^t) = 1^pt^;gt^jZgt+u(t^;gt)^j u1

is a Brownian meander independent of the -eld ^F_g⁺_t in particular, the law of m^(t) does not depend on t.

8.2.1 The Azema martingale

We now introduce the so-called Az#ema martingale t = (sgnZt)^pt^;gt, which is a remarquable ^F_g⁺_t-martingale. Following 2] closely, we project the^Ft-martingale exp(Zt^{; 2}

2 t) on the ltration ^F_g⁺_t: E(exp(Zt^;2

2 t)^jF_g⁺_t) =Eexp(m⁽1^t)t^;2

2 t)^jF_g⁺_t^! and, from the independence property we just recalled, we get

E(exp(Zt^{; 2}

2 t)^jFg⁺t) = exp(^;2

2 t) (t) where (z) =E(exp(zm1)) = ^Z₀¹xexp(zx^;x²

2 )dx.

3see Chung 4] or Revuz-Yor 22] Ex 3.8 ch. XII.

18

8.3 The law of ⁽

^{Hb D; ZH}b D^;

) 8.3.1 Case

b= 0

In this case, we denoteH_D^;=H0^;D. The variableH_D^;is a^F_g⁺_t, hence a^Ft, stopping time. The process

p1D^jZg_HD^;+uD^ju1^!

is a Brownian meander independent of^Fg⁺_HD^;. In particular 1^pDZ^HD; is distributed as^;m1:

P(Z_HD^{; 2}dx) = ^;x

D ^exp(;x²

2D^)11x<0dx

and the variablesH_D^; and Z_HD^; are independent. For any >0, the local martin- gale ( (^;_t^H_D^;) exp(^;2

2 (t^{^}H_D^;)t0) is bounded. Hence, using the optional stopping theorem at H_D^;, we obtain

E (^;_HD^;)exp(^;2

2 H_D^;)^!= (0) = 1 and the left hand side is equal to (^pD)E(exp(^;2

2 H_D^;)). The formula E(exp(^;2

2H_D^;)) = 1 (^pD) follows. Such formulae go back to Wendel 24].

8.3.2 Case

b <0

This case study may be reduced to the previous one, with the help of the stopping time Tb. Since

H_bD^; =Tb+ ^H0^;D(W) where

H^0^;D(W) = inf^ft0 11Wt⁰(t^;g_t^W)D^glaw= H0^;D

Wt =ZT_b+t^;b g_t^W = sup^fut Wu = 0^g it follows, from the independence ofTb and ^H0^;D(W), that

E(exp^;2

2 H_bD^; ) = E(exp^;2

2 Tb)E(exp^;2

2 H^0^;D(W)) 19