Abstract
The purpose of this paper is to present a survey of recent developments concerning the distributions of occupation times of Brownian motion and their applications in mathematical finance. The main result is a closed form version for Akahori's generalized arc-sine law which can be exploited for pricing some innovative types of options in the Black & Scholes model. Moreover a straightforward proof for Dassios' representation of the α -quantile of Brownian motion with drift shall be provided.
Highlights
Problems of pricing derivative securities in the traditional Black & Scholes framework are often closely connected to the knowledge of distributions induced by application of measurable functionals to Brownian motion
Occupation times of Brownian motion had been a subject of intensive research in stochastic calculus years ago (e.g. a nice proof of Levy’s famous arc-sine law is presented in Billingsley (1968)) the interest of financial economists and applied mathematicians in questions concerning quantile options has caused a renaissance in this topic [see Akahori (1995), Dassios (1995)]
With ν ∈ gR we introduce the Brownian motion with drift Z defined by Zt = Wt + νt, t ≥ 0
Summary
Problems of pricing derivative securities in the traditional Black & Scholes framework are often closely connected to the knowledge of distributions induced by application of measurable functionals to Brownian motion. In the Appendix miscellaneous remarks and computational aspects of proofs are provided for those readers who are interested in technical details
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