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Über dieses Buch

This monograph contains: - ten papers written by the author, and co-authors, between December 1988 and October 1998 about certain exponential functionals of Brownian motion and related processes, which have been, and still are, of interest, during at least the last decade, to researchers in Mathematical finance; - an introduction to the subject from the view point of Mathematical Finance by H. Geman. The origin of my interest in the study of exponentials of Brownian motion in relation with mathematical finance is the question, first asked to me by S. Jacka in Warwick in December 1988, and later by M. Chesney in Geneva, and H. Geman in Paris, to compute the price of Asian options, i. e. : to give, as much as possible, an explicit expression for: (1) where A~v) = I~ dsexp2(Bs + liS), with (Bs,s::::: 0) a real-valued Brownian motion. Since the exponential process of Brownian motion with drift, usually called: geometric Brownian motion, may be represented as: t ::::: 0, (2) where (Rt), u ::::: 0) denotes a 15-dimensional Bessel process, with 5 = 2(1I+1), it seemed clear that, starting from (2) [which is analogous to Feller's repre­ sentation of a linear diffusion X in terms of Brownian motion, via the scale function and the speed measure of X], it should be possible to compute quan­ tities related to (1), in particular: in hinging on former computations for Bessel processes.

Inhaltsverzeichnis

Frontmatter

0. Functionals of Brownian Motion in Finance and in Insurance

Abstract
In 1900, the mathematician Louis Bachelier proposed in his dissertation “Théorie de la Spéculation” to model the dynamics of stock prices as an arithmetic Brownian motion (the mathematical definition of Brownian motion had not yet been given by N. Wiener) and provided for the first time the exact definition of an option as a financial instrument fully described by its terminal value. In his 1965 paper “Theory of Rational Warrant Pricing”, the economist and Nobel prize winner Paul Samuelson, giving full recognition to Bachelier’s fondamental contribution, transformed the arithmetic Brownian motion into a geometric Brownian motion assumption to account for the fact that stock prices cannot take negative values.
Hélyette Geman

1. On Certain Exponential Functionals of Real-Valued Brownian Motion

J. Appl. Prob. 29 (1992), 202–208
Abstract
Dufresne [1] recently showed that the integral of the exponential of Brownian motion with negative drift is distributed as the reciprocal of a gamma variable. In this paper, it is shown that this result is another formulation of the distribution of last exit times for transient Bessel processes. A bivariate distribution of such integrals of exponentials is obtained explicitly.
Marc Yor

2. On Some Exponential Functionals of Brownian Motion

Adv. Appl. Prob. 24 (1992), 509–531
Abstract
In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T]of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit time distributions and the fixed time case is recovered by inverting Laplace transforms.
Marc Yor

3. Some Relations between Bessel Processes, Asian Options and Confluent Hypergeometric Functions

C.R. Acad. Sci., Paris, Sér. I 314 (1992), 471–474 (with Hélyette Geman)
Abstract
A closed formula is obtained for the Laplace transform of moments of certain exponential functionals of Brownian motion with drift, which give the price of some financial options, so-called Asian options. A second equivalent formula is presented, which is the translation, in this context, of some intertwining properties of Bessel processes or confluent hypergeometric functions.
Marc Yor

4. The Laws of Exponential Functionals of Brownian Motion, Taken at Various Random Times

C.R. Acad. Sci., Paris, Sér. I 314 (1992), 951–956
Abstract
With the help of several different methods, a closed formula is obtained for the laws of the exponential functionals of Brownian motion with drift, taken at certain random times, particularly exponential times, which are assumed to be independent of the Brownian motion.
Marc Yor

5. Bessel Processes, Asian Options, and Perpetuities

Mathematical Finance, Vol. 3, No. 4 (October 1993), 349–375 (with Hélyette Geman)
Abstract
Using Bessel processes, one can solve several open problems involving the integral of an exponential of Brownian motion. This point will be illustrated with three examples. The first one is a formula for the Laplace transform of an Asian option which is “out of the money.” The second example concerns volatility misspecification in portfolio insurance strategies, when the stochastic volatility is represented by the Hull and White model. The third one is the valuation of perpetuities or annuities under stochastic interest rates within the Cox-Ingersoll-Ross framework. Moreover, without using time changes or Bessel processes, but only simple probabilistic methods, we obtain further results about Asian options: the computation of the moments of all orders of an arithmetic average of geometric Brownian motion; the property that, in contrast with most of what has been written so far, the Asian option may be more expensive than the standard option (e.g., options on currencies or oil spreads); and a simple, closed-form expression of the Asian option price when the option is “in the money,” thereby illuminating the impact on the Asian option price of the revealed underlying asset price as time goes by. This formula has an interesting resemblance with the Black-Scholes formula, even though the comparison cannot be carried too far.
Marc Yor

6. Further Results on Exponential Functionals of Brownian Motion

Abstract
Let (Bt,t ≥ 0) denote a real-valued Brownian motion starting from 0, and let v be a real.
Marc Yor

7. From Planar Brownian Windings to Asian Options

Insurance: Mathematics and Economics 13 (1993), 23–34
Abstract
It is shown how results presented in Insurance: Mathematics and Economics 11, no. 4, in several papers by De Schepper, Goovaerts, Delbaen and Kaas, concerning the arithmetic average of the exponential of Brownian motion with drift [which plays an essential role in Asian options, and has also been studied by the author, jointly with H. Geman] are related to computations about winding numbers of planar Brownian motion. Furthermore, in the present paper, Brownian excursion theory is being used in an essential way, and helps to clarify the role of some Bessel functions computations in several formulae.
Marc Yor

8. On Exponential Functionals of Certain Lévy Processes

Stochastics and Stochastic Rep. 47 (1994), 71–101 (with P. Carmona and F. Petit)
Abstract
In this article we generalize the work of M. Yor concerning the law of A T = ∫ 0 T exp (ξ s ) ds where ξ is a Brownian motion with drift and T an independent exponential time, to the case where ξ belongs to a certain class of Lévy processes. Our method hinges on a bijection, introduced by Lamperti, between exponentials of Lévy processes and semi-stable Markov processes.
Marc Yor

9. On Some Exponential-integral Functionals of Bessel Processes

Mathematical Finance, Vol. 3, No.2 (April 1993), 231–240
Abstract
This paper studies the moments of some exponential-integral functionals of Bessel processes, which are of interest in some questions of mathematical finance, including the valuation of perpetuities and Asian options.
Marc Yor

10. Exponential Functionals of Brownian Motion and Disordered Systems

J. Appl. Prob. 35 (1998), 255–271 (with Alain Comtet and Cécile Monthus)
Abstract
The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts, such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.
Marc Yor

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