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Statistics and Probability Letters 82 (2012) 165172

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Statistics and Probability Letters

journal homepage: www.elsevier.com/locate/stapro

A note on first-passage times of continuously time-changedBrownian motionPeter Hieber 1, Matthias Scherer Lehrstuhl fr Finanzmathematik, Technische Universitt Mnchen, Parkring 11, 85748 Garching-Hochbrck, Germany

a r t i c l e i n f o

Article history:Received 22 July 2011Received in revised form 23 September2011Accepted 23 September 2011Available online 1 October 2011

Keywords:Double-barrier problemFirst-exit timeFirst-passage timeTime-changed Brownian motionBarrier option

a b s t r a c t

The probability of a Brownian motion with drift to remain between two constantbarriers (for some period of time) is known explicitly. In mathematical finance, this andrelated results are required, for example, for the pricing of single-barrier and double-barrier options in a BlackScholes framework. One popular possibility to generalize theBlackScholes model is to introduce a stochastic time scale. This equips the modelledreturns with desirable stylized facts such as volatility clusters and jumps. For continuoustime transformations, independent of the Brownianmotion,we show that analytical resultsfor the double-barrier problem can be obtained via the Laplace transform of the timechange. The result is a very efficient power series representation for the resulting exitprobabilities. We discuss possible specifications of the time change based on integratedintensities of shot-noise type and of basic affine process type.

2011 Elsevier B.V. All rights reserved.

1. Introduction

One-sided and two-sided exit problems for stochastic processes are classical problems with various applications inmathematical finance; see, for example, Darling and Siegert (1953), Black and Cox (1976), Kunitomo and Ikeda (1992),Geman and Yor (1996), Bertoin (1998), Lin (1999), Kyprianou (2000), Pelsser (2000), Rogers (2000), Sepp (2004), and Hurd(2009). For instance, first-passage time problems appear when barrier options are to be priced or when default probabilitiesin structural models are to be computed. For most stochastic processes, however, a closed-form solution to the variousproblem specifications is unknown; an exception is the case of Brownian motion with drift, which is sufficient to pricesingle-barrier and double-barrier options in the seminal BlackScholesmodel. Thismodel, however, is often criticized for itssimplicity. Consequently, various extensions have been proposed. One popular way of generalizing the BlackScholesmodelis to replace the calendar time by some suitable increasing stochastic process2. Related techniques can also be applied tocredit risk models; see, for example, Kammer (2007), Hurd (2009). In the present model, we show how several results canbe generalized to the situation of a continuous3 time shift. What is required is the Laplace transform of the time change,which is known for several popular specifications. We explicitly discuss models where the time change is constructed as anintegrated intensity. This includes, for example, models where the intensity is a basic affine process or a shot-noise process.

Corresponding author. Tel.: +49 89 289 17402; fax: +49 89 289 17407.E-mail addresses: [email protected] (P. Hieber), [email protected], [email protected] (M. Scherer).

1 Tel.: +49 89 289 17414; fax: +49 89 289 17407.2 Clark (1973) gave the followingmotivation: the different evolutions of price series on different days is due to the fact that information is available to traders

at a varying rate. On days when no new information is available, trading is slow, and the price process evolves slowly. On days when new information violates oldexpectations, trading is brisk, and the price process evolves much faster. Geman et al. (2000) show that a time change represents a measure of activity in theeconomy.3 To stress the limitations of this technique, we discuss why subordination with jumps must be treated differently.

0167-7152/$ see front matter 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2011.09.018

166 P. Hieber, M. Scherer / Statistics and Probability Letters 82 (2012) 165172

All results are verified using Monte Carlo techniques. Also note that we present an alternative approach for some single-barrier problems that are currently solved via Fourier inversion.

2. Notation and problem formulation

Throughout, we work on the probability space (,F , P), supporting all required stochastic processes. We consider aBrownian motion B = {Bt}t0 with drift R, volatility > 0, and initial value B0 = 0, satisfying the stochasticdifferential equation (sde) dBt = dt + dWt , where W = {Wt}t0 is a standard Brownian motion. Assume that thereare two constant barriers b < 0 < a, and define

Tab := inf {t 0 : Bt (b, a)} , (1)as well as T+ab := {Tab | Tab = Ta,} and Tab := {Tab | Tab = T,b}. If the lower barrier b is hit first, the first-exit time isTab = Tab; if the upper barrier a is hit first, the first-exit time is Tab = T+ab.

Lemma 1 (Double Exit Problem for a BrownianMotionwith Drift). Consider a Brownianmotion Bt with drift R and volatility > 0. Then

P(T+ab T ) =exp

2b

2

1

exp 2b

2

exp

2a

2

+ exp a 2

KT (b), (2)

P(Tab T ) =1 exp

2a

2

exp

2b

2

exp

2a

2

expb 2

KT (a), (3)

P(Tab T ) = 1exp

b 2

KT (a) exp

a 2

KT (b)

, (4)

where

KNT (k) := 2

(a b)2N

n=1

n(1)n+12

2 2+ 2n22

2(ab)2exp

2

2 2+

2n22

2(a b)2Tsin

nka b

. (5)

Proof. Denote by f +ab (t), fab (t), and fab(t) the corresponding first-passage time densities. Their Laplace transforms fab() :=

0 exp(t)fab(t)dt are derived in Darling and Siegert (1953) as

f +ab () = expa 2

sinh2+2 2 2 bsinh

2+2 2 2

(b a) , f ab () = expb 2

sinh2+2 2 2

a

sinh

2+2 2 2

(b a) .

From Laplace inversion tables (see, for example, Oberhettinger and Badii (1973), p. 295), one obtains

L1f +ab(t) =

2

(a b)2n=1

n(1)n+1 exp2

2 2+

2n22

2(a b)2tsin

nba b

exp

a 2

.

Finally, P(T+ab T ) = T0 f

+ab (t)dt can be obtained by integration. To obtain the given representation in Lemma 1, the

identity sinh(kx) = 2sinh(k)

n=1

(1)n+1nn2+k2 sin(nx) (see, for example, Rottmann (2008), p. 126) has to be used. By setting

x = b/(a b), k = (a b)/( 2), and using that sinh(y) = sinh(y) for all y R, we find

expa 2

2(a b)2

n=1

(1)n+1n2

2 2+ 2n22

2(ab)2sin

nba b

= exp

a 2

sinh b 2

sinh

(ab) 2

=

1 exp 2b

2

exp

2b

2

exp

2a

2

.

P. Hieber, M. Scherer / Statistics and Probability Letters 82 (2012) 165172 167

By symmetry, P(Tab T ) (respectively, f ab ) can be obtained from P(T+ab T ) (respectively, f +ab ) by using the substitutiona b, b a, and . The expression for P(Tab T ) = P(T+ab T ) + P(Tab T ) is given in, for example,Darling and Siegert (1953), p. 6334and in Domin (1996), p. 173.

Using Lemma 1, we now investigate the situation of a time-changed Brownian motion. To do so, let = {t}t0 be a(pathwise) continuous and increasing stochastic process with limtt = P-a.s. and0 = 0. This stochastic time scaleis used to time change B, i.e. we consider the process St := Bt , for t 0. The idea to approach exit problems for the processS is conceptually simple. Conditioned onT , we are back in a situation that was already solved. When we integrate outTto obtain unconditional probabilities, we observe that the required quantities can be interpreted as functions of the Laplacetransform of T , for which we exploit the specific structure of (5). Fortunately, this Laplace transform is known for mostpopular specifications of; see the examples presented below.

Theorem 2 (Double Exit Problem for a Time-Changed Brownian Motion). Consider a time-changed Brownian motion St := Btwith a (pathwise) continuous time change, independent of B. Denote the Laplace transform of T byT (u) := E[exp(uT )],u 0. Then

P(T+ab T ) =exp

2b

2

1

exp 2b

2

exp

2a

2

+ exp a 2

KT (b), (6)

P(Tab T ) =1 exp

2a

2

exp

2b

2

exp

2a

2

expb 2

KT (a), (7)

P(Tab T ) = 1exp

b 2

KT (a) exp

a 2

KT (b)

, (8)

where

KNT (k) := 2

(a b)2N

n=1

n(1)n+12

2 2+ 2n22

2(ab)2T

2

2 2+

2n22

2(a b)2sin

nka b

.

Proof. The first-passage times of Brownian motion (St = Bt , i.e.t t) are given in Lemma 1. Then,

P(T+ab T ) = EP(T+ab T | T = T )

=

exp 2b

2

1

exp 2b

2

exp

2a

2

+ exp

a 2

2(a b)2

n=1

n(1)n+12

2 2+ 2n22

2(ab)2E[exp

2

2 2+

2n22

2(a b)2T

]sin

nba b

=exp

2b

2

1

exp 2b

2

exp

2a

2

+ expa 2

2(a b)2

n=1

n(1)n+12

2 2+ 2n2 2

2(ab)2

T2

2 2+

2n2 2

2(a b)2sinnba b

.

Note that the first equality holds for continuous time changes only (see Remark 5). The expressions for P(Tab T ) andP(Tab T ) are obtained analogously.Example 3 (CIR Process). A popular possibility to include a continuous stochastic time change is an integratedCoxIngersollRoss (CIR) process as introduced in, for example, Duffie et al. (2000). The CIR process is given via the sde

dt = ( t)dt + tdWt , 0 > 0, (9)

where , , and are non-negative constants, and {Wt}t0 is a one-dimensional Brownian motion. The Feller condition(see Feller (1951)), 2 > 2, guarantees that the process is almost surely positive. A sample path of t is presented in

4 Note that the expression in Darling and Siegert (1953), p. 633, contains two typos: 2 has to be replaced by and (1)n by (1)n+1 .


Fig. 1 (right). The Laplace transform of the integrated processT := T0 sds is given by (see, for example, Cox et al. (1985);

Dufresne (2001))

CIRT (u) := E[exp

u T0sds

]

=

exp(T/2)cosh(T/2)+

sinh(T/2)

2 2

exp

u0

2 sinh(T/2)cosh(T/2)+

sinh(T/2)

, (10)

where = 2 + 2u 2. Furthermore,E[T ] = 0

+ T + exp(T )

0+

. (11)

The case = 1/2, = 1 is the special case of aHeston-type stochastic volatilitymodel treated in Lipton (2001), p. 492ff. Asa generalization of the CIR process, one can use basic affine processes; see, for example, Duffie et al. (2000). Such processesallow for an additional jump component of the intensity process. Note that the integrated intensity remains continuousin t .

Example 4 (Shot-Noise Process). Consider a shot-noise process5 as in Cox and Isham (1980) and Dassios and Jang (2003):

t = 0 exp(t)+sit

Mi exp(t si), (12)

where 0 > 0 is the initial intensity, > 0 is the exponential decay rate, {si}i=1 are the jump times of a time-homogeneousPoisson process with intensity > 0, and Mi Exp( ) are the jump sizes. A sample path of t is given in Fig. 1 (left). TheLaplace transform of the integrated processT :=

T0 sds is given by

6

snT (u) := Eexpu T0sds

|0

= expu0

1 exp(T ) T +

1+ u

T + 1

ln1+ u

1 exp(T ) . (13)

Furthermore,

E[T ] =0

2

1 exp(T )+ T

. (14)

Remark 5 (Restriction to Continuous Time Transformations). Note that Theorem 2 does not hold for discontinuous timetransformations, for example, for Lvy subordinators. If there is a jump in the time transformation , we do not observethe time-changed Brownian motion on the whole interval [0,T ]. Thus, the true barrier hitting probabilities in the case ofa discontinuous time transformation are always lower than the expressions given in Theorem 2. Hurd (2009) introduces a(slightly modified) first-passage time of the second kind to account for this issue.

3. Implementation

To implement the probabilities given in Lemma 1 and Theorem 2, the infinite sums KT have to be approximated by finitesums KNT . Given a predefined precision > 0, Lemma 6 gives lower bounds for the number of required summands N . Theseare conveniently small.

Lemma 6 (Truncating KT ). Define the (absolute) computation error of P(Tab T ) by

:=P(Tab T ) 1 expb 2 KNT (a) expa 2 KNT (b)

. (15)5 Note that the presented model is not the most general shot-noise process. It is without difficulty possible to, for example, change the jump size

distribution; see Dassios and Jang (2003).6 This Laplace transform can be obtained from Dassios and Jang (2003) by introducing exponential jump sizes g(u) = 1/(1 + u/ ) in Eq. (2.10),

p. 79.


To obtain a given precision , a lower bound for the summation index N N is obtained as follows.(a) In the case of Brownian motion:

N max1,2(a b)2

2 2T

ln

3 2T4(a b)2

a 2

. (16)

(b) In the case of an integrated CIR process with parameters (, , , 0):

N max1,

(a b)22 + 4 20(1 exp(

2 + 2 2T ))

a 2

+ 2T 2

ln20(1 exp(

2 + 2 2T ))

42(a b)2 + 2 2

.

(17)

(c) In the case of an integrated shot-noise process with parameters (, , , 0):

N max1, 2(a b)2

2 201 exp(T )

ln

3 20

1 exp(T )

4(a b)2 a 2

. (18)

The same lower bounds hold for the (absolute) computation error of P(T+ab T ) and P(Tab T ).Proof. If T (u) J exp(Mu), where J > 0,M > 0 are constants, we get from Eq. (15)

=expb 2 KT (a) KNT (a) expa 2 KT (b) KNT (b)

()expb 2

+ exp

a 2

2(a b)2

n=N+1

n 2n222(ab)2

T

2

2 2+

2n22

2(a b)2

expa 2 4

n=N+1

1nT

2n22

2(a b)2

J expa 2 4

N

1n

exp

M

2 2

2(a b)2 n2

dn

J expa 2 4(a b)23 2M exp

M

2 2

2(a b)2N2

.Thus, we get

N max1,2(a b)2

2 2M

ln

3 2M4(a b)2J

a 2

. (19)

In the Gaussian case (T (u) = exp(uT )), we setM = T , J = 1.Since one can show by a lengthy calculation that g(u) := 1+ u

T + 1

ln1 + u

1 exp(T ) is decreasing for

u [0,)with g(0) = T , we get

snT (u) exp0

1 exp(T )u,

and thusM = 01 exp(T )/, J = 1 if the time change is an integrated shot-noise process.

If T (u) J exp(Mu), where J > 0,M > 0 are constants, we get analogously

()J expa 2 4

N

1nexp

M

2(a b)ndn

J expa 2 4

2(a b)2M

exp

M

2(a b)N .

Then,

N max1,

2(a b)M

ln

2M

42(a b)J

a 2

. (20)


Fig. 1. Sample paths of a shot-noise process (left) and a CIR process (right) are displayed. Whenever t > 1, we approach faster than calendar time.

If the time change is an integrated CIR process, using that cosh(T/2)+/ sinh(T/2) cosh(T/2) 1, exp(T/2) exp(T/2), and / 1, we get

CIRT (u) =

exp(T/2)cosh(T/2)+

sinh(T/2)

2 2

exp

u0

2 sinh(T/2)cosh(T/2)+

sinh(T/2)

exp2T 2

exp

u 0

2 + 2u 21 exp2 + 2u 2T

exp2T 2

exp

u 0

2 + 2 21 exp2 + 2 2T ,

where the last inequality holds for u 1. ThusM = 01 exp(2 + 2 2T )/2 + 2 2, J = exp2T/ 2.

The estimation() is an upper bound for the (absolute) computation error of P(Tab T ) and P(T+ab T ). Thus, the given

lower bounds for N hold for those expressions, too.

4. Numerical examples

The parameters in this section are chosen such that E[T ] = T , i.e. on average our time change T increases as thecalendar time. If we set a = b = 0.2 and want to evaluate P(Tab T ) for a precision = 1e08, we obtain for reasonableparameters (, ) = (0.1, 0.2), (, , , 0) = (0.12, 1.00, 0.05, 1.00), and (, , , 0) = (0.80, 0.80, 0.64, 1.00) lowerbounds of only N 5 for the Brownian motion and the shot-noise-type time change. If the time change is an integrated CIRprocess, our error bound yields N 25. Note, however, that the error bounds in Lemma 6 are rather coarse, and the actualprecision is supposedly much higher.

First-passage time probabilities for single-barrier problems can be obtained by letting a (or b ,respectively). If we choose a such that P(T+ab T ) 0, then P(Tab T ) P(T,b < T ). In the case of Brownian motion, itis a classical result (used in credit risk; see Black and Cox (1976)) that

P(T,b T ) = b TT

+ exp

2b 2

b+ TT

. (21)

For time-changed Brownian motion, Hurd (2009) derived the following equation using Fourier inversion:

P(T,b T ) = 1+ exp(2b/2)

2

+ 1R

[ 0

exp(2b/ 2) exp(ixb) exp(ixb)ix

TixT + x2 2T/2dx] , (22)

whereR[x + iy] = x denotes the real part of the complex number x + iy. Due to the fast convergence of our power seriesexpansion, we get very fast and accurate results that provide a meaningful alternative to the Fourier inversion algorithm


Table 1Comparison of the power series and the Fourier inversion technique for P(T,b 1) of a Brownian motion ( = 0.1, = 0.2) in calendar time (top), timechanged by an integrated CIR process (middle; (, , , 0) = (0.12, 1.00, 0.05, 1.00)), and time changed by an integrated shot-noise process (bottom;(, , , 0) = (0.80, 0.80, 0.64, 1.00)). The Fourier integral in Eq. (22) was evaluated using a trapezoid rule with discretization = 0.01 on the interval[0, 100]. The absolute error of both approaches is given.Brownian motion in calendar time True Power series (a = 3.0, N = 40) Fourier inversion

P(T,b 1) P(T3,b 1) Error P(T,b 1) Errorb = 0.80 7.81e06 7.81e06


the result to single-barrier first-exit problems by setting the barriers appropriately. This provides a reasonable alternativeto the Fourier inversion technique presented in Hurd (2009). Due to the discretization bias, one should avoid simulatingfirst-exit times on a grid.

References

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7395.Domin, M., 1996. First passage time distribution of aWiener process with drift concerning two elastic barriers. Journal of Applied Probability 33, 164175.Duffie, D., Pan, J., Singleton, K., 2000. Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 13431376.Dufresne, D., The integrated square-root process. working paper, University of Montreal, 2001.Feller, W., 1951. Two singular diffusion problems. The Annals of Mathematics 54 (1), 173182.Geman, H., Yor, M., 1996. Pricing and hedging double-barrier options: a probabilistic approach. Mathematical Finance 6 (4), 365378.Geman, H., Madan, D., Yor, M., 2000. Time changes for Lvy processes. Mathematical Finance 11, 7996.Hurd, T., 2009. Credit risk modeling using time-changed Brownian motion. International Journal of Theoretical and Applied Finance 12 (8), 12131230.Kammer, S., A general first-passage-time model for multivariate credit spreads and a note on barrier option pricing. PhD thesis, JustusLiebig Universitt

Gieen, 2007.Kunitomo, N., Ikeda, M., 1992. Pricing options with curved boundaries. Mathematical Finance 2 (4), 275298.Kyprianou, A.E., 2000. Introductory Lectures on Fluctuations of Lvy Processes with Applications. Springer Verlag.Lin, X.S., 1999. Laplace transform and barrier hitting time distribution. Actuarial Research Clearing House 1, 165178.Lipton, A., 2001. Mathematical Methods for Foreign Exchange. World Scientific.Oberhettinger, F., Badii, L., 1973. Tables of Laplace Transforms. Springer Verlag.Pelsser, A., 2000. Pricing double barrier options using Laplace transforms. Finance and Stochastics 4, 95104.Rogers, L., 2000. Evaluating first-passage probabilities for spectrally one-sided Lvy processes. Journal of Applied Probability 37, 11731180.Rottmann, K., 2008. Mathematische Formelsammlung. Spektrum Verlag.Sepp, A., 2004. Analytical pricing of double-barrier options under a double-exponential jump diffusion process: applications of Laplace transform.

International Journal of Theoretical and Applied Finance 7 (2), 151175.

A note on first-passage times of continuously time-changed Brownian motionIntroductionNotation and problem formulationImplementationNumerical examplesConclusionReferences

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