TY - JOUR

T1 - A note on first-passage times of continuously time-changed Brownian motion

AU - Hieber, Peter

AU - Scherer, Matthias

PY - 2012/1

Y1 - 2012/1

N2 - The probability of a Brownian motion with drift to remain between two constant barriers (for some period of time) is known explicitly. In mathematical finance, this and related results are required, for example, for the pricing of single-barrier and double-barrier options in a Black-Scholes framework. One popular possibility to generalize the Black-Scholes model is to introduce a stochastic time scale. This equips the modelled returns with desirable stylized facts such as volatility clusters and jumps. For continuous time transformations, independent of the Brownian motion, we show that analytical results for the double-barrier problem can be obtained via the Laplace transform of the time change. The result is a very efficient power series representation for the resulting exit probabilities. We discuss possible specifications of the time change based on integrated intensities of shot-noise type and of basic affine process type.

AB - The probability of a Brownian motion with drift to remain between two constant barriers (for some period of time) is known explicitly. In mathematical finance, this and related results are required, for example, for the pricing of single-barrier and double-barrier options in a Black-Scholes framework. One popular possibility to generalize the Black-Scholes model is to introduce a stochastic time scale. This equips the modelled returns with desirable stylized facts such as volatility clusters and jumps. For continuous time transformations, independent of the Brownian motion, we show that analytical results for the double-barrier problem can be obtained via the Laplace transform of the time change. The result is a very efficient power series representation for the resulting exit probabilities. We discuss possible specifications of the time change based on integrated intensities of shot-noise type and of basic affine process type.

KW - Barrier option

KW - Double-barrier problem

KW - First-exit time

KW - First-passage time

KW - Time-changed Brownian motion

UR - http://www.scopus.com/inward/record.url?scp=80054741831&partnerID=8YFLogxK

U2 - 10.1016/j.spl.2011.09.018

DO - 10.1016/j.spl.2011.09.018

M3 - Article

AN - SCOPUS:80054741831

SN - 0167-7152

VL - 82

SP - 165

EP - 172

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

IS - 1

ER -