Article
Reference
A correction note on the first passage time of an Ornstein-Uhlenbeck process to a boundary
LEBLANC, Barbara, RENAULT, Olivier, SCAILLET, Olivier
LEBLANC, Barbara, RENAULT, Olivier, SCAILLET, Olivier. A correction note on the first passage time of an Ornstein-Uhlenbeck process to a boundary. Finance and stochastics , 2000, vol. 4, no. 1, p. 109-111
DOI : 10.1007/s007800050007
Available at:
http://archive-ouverte.unige.ch/unige:41811
Disclaimer: layout of this document may differ from the published version.
1 / 1
Finance Stochast. 4, 109–111 (2000)
c Springer-Verlag 2000
A correction note on the first passage time
of an Ornstein-Uhlenbeck process to a boundary
B. Leblanc1, O. Renault2, O. Scaillet2,3,?
1 Banque Nationale de Paris, BFI-MC, 13, rue Lafayette, F-75009 Paris, France (e-mail: bleblanc@bnp.fr)
2 D´epartement des Sciences Economiques, Universit´e Catholique de Louvain, 3 Place Montesquieu, B-1348 Louvain-la-Neuve, Belgique (e-mail: renault@ires.ucl.ac.be)
3 Institut d’Administration et de Gestion, Universit´e Catholique de Louvain, 3 Place Montesquieu, B-1348 Louvain-la-Neuve, Belgique (e-mail: scaillet@ires.ucl.ac.be)
Abstract. This paper provides the derivation of the hitting time density of an Ornstein-Uhlenbeck process to a flat boundary. The derivation relies on a change of measure approach and delivers an explicit formula. This formula is an amended expression of the result given in Leblanc and Scaillet (1998). It corresponds to the formula given by a time substitution approach when the boundary level coincides with the mean of the invariant measure. It can for example be used to price digital up-and-in credit spread options when the logarithm of the credit spread is assumed to follow an Ornstein-Uhlenbeck process.
Key words: Hitting time, Ornstein-Uhlenbeck process, path dependent option JEL classification: E43, G13
Mathematics Subject Classification (1991): 60E10, 60G17, 60J70, 65U05 This paper presents the derivation of the hitting time density of an Ornstein- Uhlenbeck process to a flat boundary. The method is based on a change of probability measure originally proposed by Leblanc and Scaillet (1998). How- ever their derivation in Appendix 4 contains computational errors although the methodology is correct. More specifically we correct the form of the change of measure to switch from an Ornstein-Uhlenbeck to a Brownian motion. A direct application of the derived density can be found in the pricing of European digital up-and-in credit spread options in the setting of Longstaff and Schwartz (1995).
In such a framework the logarithm of the credit spread is assumed to evolve according to an Ornstein-Uhlenbeck process.
?We would like to thank the associate editor M. Schweizer for his comments. The second and third author gratefully acknowledge financial support from Marie Curie Fellowship ERB4001GT970319 and from the Belgian Program on Interuniversity Poles of Attraction (PAI nb. P4/01), respectively.
Manuscript received: February 1999; final version received: April 1999
110 B. Leblanc et al.
Hitting time density of an Ornstein-Uhlenbeck process Consider the process satisfying under the probability measure Q :
drt= (φ−λrt) dt +p
βd ˜Wt, (1)
where ˜Wtis a Q−Brownian motion. This is an Ornstein-Uhlenbeck process start- ing from r0<k . We are interested in getting : Q sup[0,T ]rt ≥k
= Q (Tk≤T ) where Tk= inf{t : rt ≥k},and k represents the boundary level.
To simplify the computations we adopt the useful time and drift changes :
˜rt = rt
β −φλ so that :
d ˜rt =−λ˜˜rtdt + d ˜˜Wt, (2) with ˜λ=λ/β,˜r0= r0−φλ,˜k = k −φλ and ˜T˜k= Tk/β= inf
t : ˜rt ≥˜k .
Proposition 1 If ˜rt is solution of (2), then the distribution of the hitting time of level ˜k by the Ornstein-Uhlenbeck process ˜rt is characterized by the density :
Q ˜T˜k ∈dt
(3)
= ˜k−˜r0
√2π exp λ˜
2 ˜r02−˜k2+ t−( ˜k−˜r0)2coth ˜λt λ˜ sinh ˜λt
!32 .
Proof. It parallels the proof given in Leblanc and Scaillet (1998). We introduce the following change of measure :
Q = exp
−1 2
Z t 0
λ˜2˜ru2du + Z t
0
−λ˜˜rud ˜ru
P
= exp
−λ˜2 2
Z t 0
˜ru2du−λ˜
2 ˜rt2−˜r02−t P.
Then under P , ˜rt = ˜˜Wt −λ˜Rt
0 ˜rudu is a Brownian motion starting from ˜r0. Furthermore :
Q ˜T˜k∈dt
= E˜rP0
exp
−λ˜2 2
Z t 0
˜ru2du−λ˜
2 ˜rt2−˜r02−tT˜˜k = t
P T˜˜k∈dt
= exp λ˜
2 ˜r02−˜k2+ t E˜rP
0
exp
−λ˜2 2
Z t 0
˜ru2du T˜˜k= t
P T˜˜k ∈dt Since ˜ru is conditionally on ˜T˜k = t a Bessel Bridge with dimension 3 starting from x = ˜k−˜r0 and conditioned to be equal to 0 at time t , we can replace
E˜rP0
exp
−λ˜2 2
Z t 0
˜ru2du T˜˜k= t
, by :
First passage time of O-U process 111
ExP
exp
−λ˜2 2
Z t 0
R2udu Rt = 0
,
where Ru is under P a three-dimensional Bessel bridge starting at x . The result follows from formula 2.5 p.18 in Yor (1992) and the well-known first passage time density of a standard Brownian motion. Q.E.D.
When the boundary level is set equal to the marginal mean of the pro- cess, equation (3) can also be obtained via a time substitution approach. In- deed, from Cox and Miller (1965) p. 229 using the time change f (t ) =
exp 2 ˜λt
−1
/2 ˜λ˜r02, we can show that the problem boils down to the first passage time of a Brownian motion Bf (t )to a flat boundary when ˜k = k−φλ = 0.
Besides we may verify that the limit of (3) coincides with the hitting time density of a standard Brownian motion when ˜λgoes to zero.
References
1. Cox, D., Miller, H.: The theory of stochastic processes. London: Methuen 1965
2. Leblanc, B., Scaillet, O.: Path dependent options on yields in the affine term structure model.
Finance and Stochastics 2, 349–367 (1998)
3. Longstaff, F., Schwartz, E.: Valuing credit derivatives. Journal of Fixed Income June, 6–12 (1995)
4. Yor, M.: Some Aspects of Brownian Motion. Basel: Birkh¨auser 1992
