The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
The joint distribution of excursion and hitting times of the
Brownian motion with Application to Parisian Option Pricing
You You Zhang (Joint work with Angelos Dassios)
[email protected]
27th May 2014
1 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Motivation
Motivation
Definition
The MinParisianHit Option is triggered either when the age of the excursion
above L reaches time d or a barrier B > L is hit by the underlying price process
S. The MaxParisianHit Option is triggered when both the barrier B is hit and
the excursion age exceeds duration d above L.
2
(r − σ2 )t+σWt
Let the stock price process (St )t≥0 = S0 e
follow a geometric
t≥0
Brownian motion and Q denote the equivalent martingale measure and define
S
gL,t
= sup{s ≤ t : Ss = L}
S
dL,t
= inf{s ≥ t : Ss = L}
S
τd+ (S) = inf{t ≥ 0|1St >L (t − gL,t
) ≥ d}
HB (S) = inf{t ≥ 0|St = B}
2 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Motivation
Motivation
The fair price of a MinParisianHit Up-and-In Call option with payoff (ST − K )+
can be written in terms of hitting times in the following way using Girsanov
theorem
minPHCiu (x, T , K , L, d, r ) = e −rT EQ (ST − K )+ 1min{τ + (S),HB (S)}≤T
d
−(r + 12 m2 )T
σZT
+ mZT
=e
EP (S0 e
− K) e
1min{τ + (Z ),Hb (Z )}≤T
d
Z ∞
−(r + 12 m2 )T
(S0 e σz − K )e mz P ZT ∈ dz, min{τd+ (Z ), Hb (Z )} ≤ T
=e
1
σ
ln SK
0
where (Zt )t≥0 = (Wt + mt)t≥0 is a P-Brownian motion and
m=
+
1
σ
r−
σ2
2
!
τd (Z ) = inf{t ≥ 0|1Zt >l (t − gl,t ) ≥ d}
l =
Z
1
L
ln
σ
S0
gl,t = sup{u ≤ t|Zu = l}
b=
1
B
ln
σ
S0
Hb (Z ) = inf{t ≥ 0|Zt = b}
3 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Motivation
Motivation
(Dassios and Wu. Perturbed Brownian motion and its application to Parisian option pricing. (2))
4 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Martingale Problem
Let W ,µ = µt + Wt be the perturbed drifted Brownian motion around 0. We
define a three state semi-Markov process for b > 0

?

if Wt,µ > b
1
Xt = 1
if 0 < Wt,µ < b


2
if Wt,µ < 0
Let Ut = t − ḡt denote the time elapsed in current state 2 or state 1 and 1?
combined (state 1? is an absorbing state). (Xt , Ut ) becomes a Markov process.
Hence, Xt is a three state semi-Markov process.
We consider a bounded function fi : R2 → R, i = 1∗ , 1, 2, defined as fXt (Ut , t).
The generator A is an operator making
Z t
fXt (Ut , t) −
AfXs (Us , s)ds
0
a martingale. Hence, solving Af = 0 provides us with martingales of the form
fXt (Ut , t).
5 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Martingale Problem
We have
∂f1
∂f1
+
+ λ12 (u) (f2 (0, t) − f1 (u, t)) + λ11? (u) Ae −βt h(u) − f1 (u, t)
∂t
∂u
∂f2
∂f2
Af2 (u, t) =
+
+ λ21 (u) (f1 (0, t) − f2 (u, t))
∂t
∂u
Af1 (u, t) =
Since we are not interested in what happens after the absorbing state 1∗ has
been reached, we do not define Af1∗ , the generator starting from state 1∗ .
We define the times for d, b > 0
Hb = inf{t ≥ 0|Wt,µ = b}
ḡt = sup{s ≤ t|Ws,µ = 0}
τd,+ = inf{t > 0|1Wt,µ >0 (t − ḡt ) > d}
6 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Martingale Problem
Definition
Given a function F (β), the inverse Laplace transform of F , denoted by
L−1 {F (β)}, is the function f whose Laplace transform is F , i.e.
Z ∞
e −βt f (t)dt = F (β)
f (t) = L−1
{F
(β)}|
⇐⇒
L
{f
(t)}(β)
:=
t
t
β
0
Note that we consider the inverse Laplace transform with respect to the
transformation variable β at the evaluation point t.
7 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Martingale Problem
Lemma
For the -perturbed process we find the Laplace transform
,+
AE e −βHb h(uHb )1H <τ ,+ + BE e −βτd 1τ ,+ <H b
d
d
b
Rd
Be −βd P̄1 (d) + A 0 e −βw h(w )p11? (w )dw
=
1 − P̃21 (β)P̂12 (β)
(1)
Proof: Assuming f having the form fi (u, t) = e −βt gi (u) and solving Af ≡ 0
with the constraints g1 (d) = B and g2 (∞) = 0 and g1? (u) = Ah(u) we can
solve for g1 which provides us with martingales of the form
Mt := fXt (Ut , t) = e −βt gXt (Ut ). Let T = min{Hb , τd,+ }, then Optional
Sampling suggests
g1 (0) = E(M0 ) = E(MT ∧t ) = E(MT 1T <t ) + E(Mt 1T >t )
= E(MHb 1H <τ ,+ 1Hb <t ) + E(Mτ ,+ 1τ ,+ <H 1τ ,+ <t ) + E(Mt 1T >t )
d
b d
d
b ,+d
−→ E e −βHb g1? (UHb )1H <τ ,+ + E e −βτd g1 (Uτ ,+ )1τ ,+ <H t→∞
b
d
d
d
b
,+
−βHb
−βτd
= AE e
h(uHb )1H <τ ,+ + BE e
1τ ,+ <H b
d
d
b
8 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Single Laplace transform of Hitting and Excursion time
Proposition
The Laplace transform of the hitting and excursion time for drifted Brownian motion
µt + Wt is given by
+
AE e −βHb h(uHb )1H <τ + + BE e −βτd 1τ + <H =
b
b
d
d
(
∞ h
i
X
1
g (k, 0) − e µb g (k + , 0) + µ [f (k, 0) − f (k + 1, 0)] +
2Be −βd
2
k=0
)
r
Z d
∞
(2k+1)2 b 2
2 µb− µ2 w X (2k + 1)2 b 2
−
−βw
2
2w
−1 e
+A
e
dw ×
e
h(w )
πw 3
w
0
k=0
( ∞
)
i −1
X hp
× 2
2β + µ2 f (k, β) + g (k, β)
k=0
where
√
f (k, β) = e
−
r
g (k, β) =
2β+µ2 2kb
N
q
2kb
(2β + µ2 )d − √
d
√
−e
2β+µ2 2kb
N
q
2kb
− (2β + µ2 )d − √
d
2 − (2β+µ2 )d − 2(kb)2
2
d
e
πd
9 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Double Laplace transform of Hitting and Excursion time
case Hb < τd+
Proposition
The double Laplace transform of Hitting and Excursion time of a drifted Brownian
motion where Hb < τd+ is
+
E0 e −βHb −γτd 1H <τ + =
b
d
"
Z
d
0
µ(d − w ) − b
−µ(d − w ) − b
√
√
−N
+
d −w
d −w
q
√
b
2
−γ τ̂d
+
(2γ + µ2 )(d − w ) − √
+ Eµ
) e −( 2γ+µ +µ)b N
0 (e
d −w
!#
q
√
b
2
+ e 2γ+µ −µ)b N − (2γ + µ2 )(d − w ) − √
×
d −w
r
∞ (2k+1)2 b 2
2 µb− µ2 w X (2k + 1)2 b 2
2
×
e
−
1
e − 2w dw ×
πw 3
w
k=0
( ∞
)
i −1
X hp
× 2
2β + µ2 f (k, β) + g (k, β)
e −βw e −γd
1 − e −2µb N
k=0
10 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Proof:
Lemma
E e −βHb h(uHb )1H
+
b <τd
Rd
0
=
e −βw h(w )
q
2
µ2 w
2
e µb− 2
πw 3
∞ hp
P
∞ P
(2k+1)2 b 2
k=0
w
(2k+1)2 b 2
− 1 e − 2w dw
i
2β + µ2 f (k, β) + g (k, β)
k=0
Define h(uHb ) := E e
E e −βHb h(uHb )1H
−γτd+
+
b <τd
|Hb
and the l.h.s becomes
+
= E e −βHb E e −γτd |Hb 1H
+
b <τd
+
= E e −βHb e −γτd 1H
+
b <τd
On the other hand, we have
−γ(Hb +d−uH )
−γ(Hb +H̃0 +τ̂d )
b 1
h(uHb ) = E0 e
1H̃ <d−u Hb
H̃0 >d−uH Hb + E0 e
0
Hb
b
h
i
−γ(d−uH )
b Pb (H̃0 > d − uH ) + Eb e −γ H̃0 1
= e −γHb e
E0 (e −γ τ̂d )
b
H̃ <d−u
0
Hb
11 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
case τd+ < Hb
Proposition
The double Laplace transform of Hitting and Excursion time of a drifted Brownian
motion where τd+ < Hb is
q
√
h
b
2
e −βd e −b( 2γ+µ −µ) N √ − (2γ + µ2 )d −
b
d
d
iX
∞ h
q
√
b
1
2
− e b( 2γ+µ −µ) N − √ − (2γ + µ2 )d
g (k, 0) − e µb g (k + , 0)+
2
d
k=0
) ( ∞ q
i
i
Xh
µ [f (k, 0) − f (k + 1, 0)]
×
2(β + γ) + µ2 f (k, β + γ) + g (k, β + γ) ×
+
E(e −βτd
(
−γHb
1τ + <H ) =
k=0
)−1
µt − b
−µt − b
2µb
√
√
1−N
−e
N
d
d
12 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Double Laplace of Excursion and Hitting time
Proof:
Lemma
e −βd
−βτd+
E e
1τ + <H =
∞ h
P
k=0
b
d
g (k, 0) − e µb g (k + 12 , 0) + µ [f (k, 0) − f (k + 1, 0)]
i
i
∞ hp
P
2β + µ2 f (k, β) + g (k, β)
k=0
We define a new generator starting at time τd+ . State 1? is a killed state.
Af1 (u, t) =
∂f1
∂f1
+
+ λ11? (u) e −γt − f1 (u, t)
∂t
∂u
At time τd+ we are in state 1 with f1 (d, 0) = g1 (d). Solving Af ≡ 0 with constraint
g1 (∞) = 0 we derive g1 (d). As a result we have found a martingale Mt := fXt (Ut , t)
with M0 = f1 (d, 0) = g1 (d). Also, with Tb being the first hitting time of b of our
process starting at τd+ and hence Hb = τd+ + Tb ,
MTb = f1? (UTb , Tb ) = e −γTb
Hence, OST yields g1 (d) = E(e −γTb ) and the double Laplace becomes
+
+
E(e −βτd e −γHb 1τ + <H ) = E(e −βτd 1τ + <H E(e −γHb |τd+ ))
b
d
= E(e
−βτd+
1τ + <H E(e
d
b
d
−γ(τd+ +Tb )
|τd+ ))
b
+
= g1 (d)E(e −(β+γ)τd 1τ + <H )
d
b
13 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Application
Application to MinParisianHit options
u
minPHCi = e
−(r + 1 m2 )T
2
∞
Z
1 ln K
σ
x
(xe
σz
− K )e
mz
+
P ZT ∈ dz, min{τd (Z ), Hb (Z )} ≤ T
Proposition
The joint density of position at maturity and minimum of hitting and excursion time for standard
Brownian motion is
ZT
+
Zb
1
P(ZT ∈ dz, min{τd , Hb } ≤ T ) =
p
t=0 w =−∞
∞
P
"
×
k=−∞
∞
P
w +2kb −
e
d
(2kb)2
e − 2d
2π(T − t)
e
−
(z−w )2
2(T −t)
(w +2kb)2
2d
#
−1
−
×
(2k+1)2 b 2
2d
e−
−1
Lβ {H1 (β)}|t + δ(w −b) Lβ {H2 (β)}|t dw dt
k=−∞
e −βd
H1 (β) =
∞ h√
P
k=0
∞ √
P
i
∞ h
P
g (k, 0) − g (k + 12 , 0)
k=0
i
2βf (k, β) + g (k, β)
,
H2 (β) =
k=0
2βf (k + 21 , β) + g (k + 12 , β)
∞ h√
P
2βf (k, β) + g (k, β)
i
k=0
14 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Application
Proof: Let Z denote a standard Brownian motion and τ := min{τd+ , Hb }.
P(ZT ∈ dz, min{τd+ , Hb } ≤ T ) =
ZT
Zb
P(ZT ∈ dz|τ = t, Zτ ∈ dw )P(τ ∈ dt, Zτ ∈ dw )
t=0 w =−∞
ZT
Zb
=
t=0 w =−∞
ZT
Zb
=
t=0 w =−∞
2
(z−w )
1
−
p
e 2(T −t) dzP(τ ∈ dt, Zτ ∈ dw )
2π(T − t)
2
n
(z−w )
1
−
p
e 2(T −t) dz P(τ ∈ dt, Zτ ∈ dw |Hb < τd+ )P(Hb < τd+ ) +
2π(T − t)
+ P(τ ∈ dt, Zτ ∈ dw |τd+ < Hb )P(τd+ < Hb )
o
15 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Application
case Hb < τd+ :
P(τ ∈ dt, Zτ ∈ dw |Hb < τd+ )P(Hb < τd+ )
= P(ZHb ∈ dw |τ = t, Hb < τd+ )P(τ ∈ dt, Hb < τd+ )
n o +
= δ(w −b) dw L−1
E e −β min{Hb ,τd } 1H <τ +
dt
β
b
case
τd+
t
d
< Hb :
P(τ ∈ dt, Zτ ∈ dw |τd+ < Hb )P(τd+ < Hb )
= P(Zτ + ∈ dw |τ = t, τd+ < Hb )P(τ ∈ dt, τd+ < Hb )
d
= lim P (Zd ∈ dw | inf Zs > 0, sup Zs < b)P(τ ∈ dt, τd+ < Hb )
→0
0<s<d
∞
P
=
k=−∞
∞
P
e
w +2kb −
e
d
(2kb)2
− 2d
w +2kb)2
2d
−e
0<s<d
dw
(2k+1)2 b 2
−
2d
n o +
L−1
E e −βτd 1τ + <H
dt
β
b
d
t
k=−∞
16 / 17
The joint distribution of excursion and hitting times of the Brownian motion with Application to Parisian Option Pricing
Application
References
[1] Marc Chesney, Monique Jeanblanc-Picque, and Marc Yor. Brownian Excursions and Parisian Barrier Options.
Annals of Applied Probability, 29, 1997.
[2] Angelos Dassios and Shanle Wu. Perturbed Brownian motion and its application to Parisian option pricing.
Finance and Stochastics, 14(3):473–494, November 2009.
[3] Angelos Dassios and Shanle Wu. Double-Barrier Parisian options. Journal of Applied Probability, 48(1):1–20,
2011.
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