Another Look at the Hartman-Watson Distributions
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Another Look at the Hartman-Watson Distributions Jacek Jakubowski1 · Maciej Wi´sniewolski1 Received: 2 October 2018 / Accepted: 9 October 2019 / © The Author(s) 2019
Abstract The article deals with the Hartman-Watson distributions and presents a new approach to them by investigating a special function u. The function u is strictly related to the distribution of the exponential functional of Brownian motion appearing in the mathematical finance framework. The study of the latter leads to new explicit representations for the function u. One of them is through a new parabolic PDE. Integral relations of convolution type between Hartman-Watson distributions and modified Bessel functions are presented. It turns out that u can be represented as an integral convolution of itself and the modified Bessel function K0 . Finally, excursion theory and a subordinator connected to the hyperbolic cosine of Brownian motion are involved in order to obtain yet another representation for u. Possible applications of the resulting explicit formulas are discussed, among others Monte Carlo evaluations of u. Keywords Hartman-Watson distributions · Additive functional of Brownian motion · Asian options · PDE · Excursions of Brownian motion · Le´vy measure · Modified Bessel functions Mathematics Subject Classification (2010) 60G40 · 60G17 · 91G80
1 Introduction The recent paper of Lyasoff [17] has shed a new light on the long studied subject of finding the distribution of an important additive functional of geometric Brownian motion, that is t At = 0 e2Bu du, where (Bt , t ≥ 0) is a real typical Brownian motion, and also on HartmanWatson distributions. At has been the subject of the deep studies of Marc Yor and co-authors in several articles; see e.g. [19, 22, 23] and many others. Such functionals, besides their intrinsic interest, are important for mathematical finance, in particular for pricing financial instruments including the so-called Asian options (see e.g. [7, 18, 23]). They belong to the Maciej Wi´sniewolski
[email protected] Jacek Jakubowski [email protected] 1
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
J. Jakubowski, M. Wi´sniewolski
first generation of exotic options and are especially useful for hedging the risk of oscillations of the asset price at the expiration date of the option (see Wystup [27, Sect. 1.5.4]). The difficulties in pricing Asian options come from the payoff function. The latter is an integral functional of the asset price; for instance, for a call option with t exercise price K > 0 and expiration date t it is given by (At − K)+ , where At = 1t 0 Su du and Su denotes the price of the basic financial asset at u. Clearly, the task of pricing Asian options reduces to the description of the probability distribution of At , which is a well known and difficult problem in financial mathematics (see Geman and Yor [9]). In particular, there are known connections between the distribution of At and the family of Hartman-Watson probability distributions. This pa
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