The integral of geometric Brownian motion

Dufresne, Daniel

doi:10.1239/aap/999187905

Cited by 83 publications

(93 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The study of the law of A (ρ) T was initiated by Dufresne (1990), where the integral over the whole time axis is considered, and by Yor (1992). Their work was continued in Alili and Gruet (1997), Carr andSchröder (2004), Donati-Martin, Ghomrasni, andYor (2001), Dufresne (2000Dufresne ( , 2001Dufresne ( , 2004Dufresne ( , 2005, Matsumoto and Yor (2003, 2005a, 2005b, Salminen andYor (2005), Schröder (2003), and Yor (2001). The law of A (ρ) T has many interesting applications, for instance, in the theory of Asian options (see Geman and Yor 1993;Rogers and Shi 1995;Fu, Madan, and Wang 1998;Dufresne 2000Dufresne , 2005Donati-Martin et al 2001;Yor 2001;Carr and Schröder 2004).…”

Section: Introduction: Distribution Densities In Stochastic Volatilitmentioning

confidence: 99%

Asymptotic Behavior of Distribution Densities in Models With Stochastic Volatility. I

Gulisashvili

Stein

2010

Mathematical Finance

View full text Add to dashboard Cite

We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of a geometric Brownian motion and the density of the stock price process in the Hull-White model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the price of an Asian option are obtained.

show abstract

Section: Introduction: Distribution Densities In Stochastic Volatilitmentioning

confidence: 99%

Asymptotic Behavior of Distribution Densities in Models With Stochastic Volatility. I

Gulisashvili

Stein

2010

Mathematical Finance

View full text Add to dashboard Cite

show abstract

“…This shows that the minimum of F (ρ) is reached for that value of ρ > 1 for which y 1 = π 2 . From (9) we get that this is ρ = π 2 , and at this point we have F (π/2) = 3π 2 8 . We can obtain also the asymptotics of F (ρ) for very small and very large arguments.…”

Section: 1mentioning

confidence: 86%

“…There are several saddle points along the imaginary axis. We are interested in the saddle point at ξ = iy 1 with 0 < y 1 ≤ π the solution of (9). At this point the second derivative of h is h ′′ (iy 1 ) = 1 + ρ cosh y 1 > 0.…”

Section: Asymptotic Expansion Of θ(ρ/T T) As T →mentioning

confidence: 99%

“…For this reason considerable effort has been devoted to applying analytical methods to simplify Yor's formula. For particular cases µ = 0, µ = 1 simpler expressions as single integrals have been obtained for the density of A (µ) t by Comtet et al [7] and Dufresne [9]. See the review article by Matsumoto and Yor [19] for an overview of the literature.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Matching the quark model to the 1∕N[sub c] expansion

Pirjol¹,

Schat²,

Armstrong³

et al. 2011

AIP Conference Proceedings

View full text Add to dashboard Cite

The Hartman-Watson distribution with density fr(t) = 1 I 0 (r) θ(r, t) with r > 0 is a probability distribution defined on t ≥ 0 which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral θ(r, t) which is difficult to evaluate numerically for small t → 0. Using saddle point methods, we obtain the first two terms of the t → 0 expansion of θ(ρ/t, t) at fixed ρ > 0. As an application we derive, under an uniformity assumption in ρ, the leading asymptotics of the density of the time average of the geometric Brownian motion as t → 0. This has the form P( 1 t t 0 e 2(Bs +µs) ds ∈ da) = (2πt) − 1 2 g(a, µ)e − 1 t J (a) (1 + O(t)), with an exponent J(a) which reproduces the known result obtained previously using Large Deviations theory.

show abstract

“…where n is any nonnegative integer. Dealing with other values of µ as well, we put the above-mentioned reduction in Proposition 1.2 below, which recovers in part Theorem 4.2 of [4] by Dufresne. For every µ ∈ R, we denote by H µ the Hermite function of degree µ and recall its integral representation when µ > −1:…”

Section: Introductionmentioning

confidence: 99%

Untitled

HARIYA¹

2005

AUDIOLOGY JAPAN

View full text Add to dashboard Cite

This paper concerns the density of the Hartman-Watson law. Yor (1980) obtained an integral formula that gives a closed-form expression of the Hartman-Watson density. In this paper, based on Yor's formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a Dufresne's result (2001) that remarkably simplifies representations for the laws of exponential additive functionals of Brownian motion. We also apply that simplification to the law of some additive functional of Bessel process.

show abstract

The integral of geometric Brownian motion

Cited by 83 publications

References 12 publications

Asymptotic Behavior of Distribution Densities in Models With Stochastic Volatility. I

Asymptotic Behavior of Distribution Densities in Models With Stochastic Volatility. I

Matching the quark model to the 1∕N[sub c] expansion

Untitled

Contact Info

Product

Resources

About