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On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 98 (9 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Levy Integrals and the Stationarity of generalised OrnsteinUhlenbeck processes
"... The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t ..."
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Cited by 21 (9 self)
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The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t
Bessel processes, the integral of geometric Brownian motion, and Asian options
 Theor. Probab. Appl
, 2004
"... Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been stu ..."
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Cited by 7 (0 self)
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Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the HartmanWatson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the HartmanWatson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.
Asymptotic results for sample autocovariance functions and extremes of integrated generalized OrnsteinUhlenbeck processes
, 2008
"... We consider a positive stationary generalized OrnsteinUhlenbeck process Vt = e −ξt ( ∫ t e ξs− dηs + V0 for t ≥ 0, 0 ∫ k k−1 and the increments of the integrated generalized OrnsteinUhlenbeck process Ik = Vt − dLt, k ∈ N, where (ξt,ηt,Lt)t≥0 is a threedimensional Lévy process independent of the s ..."
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Cited by 6 (1 self)
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We consider a positive stationary generalized OrnsteinUhlenbeck process Vt = e −ξt ( ∫ t e ξs− dηs + V0 for t ≥ 0, 0 ∫ k k−1 and the increments of the integrated generalized OrnsteinUhlenbeck process Ik = Vt − dLt, k ∈ N, where (ξt,ηt,Lt)t≥0 is a threedimensional Lévy process independent of the starting random variable V0. The genOU model is a continuous time version of a stochastic recurrence equation. Hence, our models include, in particular, continuous time versions of ARCH(1) and GARCH(1,1) processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of (Vt)t≥0 and (Ik)k∈N. Furthermore, we present a central limit result for (Ik)k∈N. Regular variation and point process convergence play a crucial role in establishing the statistics of (Vt)t≥0 and (Ik)k∈N. The theory can be applied to the COGARCH(1,1) and the Nelson diffusion model.
On the integral of geometric Brownian motion
 Adv. Appl. Prob
, 2003
"... Abstract. This paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is t ..."
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Cited by 5 (0 self)
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Abstract. This paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is the main methodological contribution of the paper. The one of our integrals has a similar structure to that obtained in [Y], while the other is in terms of Hermite functions as those of [Du01]. Performing or not performing a certain Girsanov transformation is identified as the source of these two forms of the laws. For exponents equal to 1 our results specialize to those obtained in [Y], but for exponents equal to minus 1 they give representations for the laws which are markedly different from those obtained in [Du01].
Laplace Transforms for Integrals of Markov Processes
 Markov Processes and Functional Analysis
, 2004
"... Abstract. Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the OrnsteinUhlenbeck, the CoxIngerssolRoss (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractabili ..."
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Cited by 3 (1 self)
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Abstract. Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the OrnsteinUhlenbeck, the CoxIngerssolRoss (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which Laplace transforms for integrals of the diffusion processes and transitional probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain selfadjoint operators. We also extend this scheme to a class of finitestate Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree. 1.
Extremes of ContinuousTime Processes
"... In this paper we present a review on the extremal behavior of stationary continuoustime processes with emphasis on generalized OrnsteinUhlenbeck processes. We restrict our attention to heavytailed models like heavytailed OrnsteinUhlenbeck processes or continuoustime GARCH processes. The survey ..."
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Cited by 2 (0 self)
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In this paper we present a review on the extremal behavior of stationary continuoustime processes with emphasis on generalized OrnsteinUhlenbeck processes. We restrict our attention to heavytailed models like heavytailed OrnsteinUhlenbeck processes or continuoustime GARCH processes. The survey includes the tail behavior of the stationary distribution, the tail behavior of the sample maximum and the asymptotic behavior of sample maxima of our models. 1
Bessel processes, the integral of geometric Brownian motion, and Asian options
"... This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of ..."
Abstract
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This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes us ing the HartmanWatson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the HartmanWatson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.
Applied Probability Trust (5 September 2001) ON THE EQUIVALENCE OF FLOATING AND FIXEDSTRIKE ASIAN OPTIONS
"... There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating and fixedstrike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very use ..."
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There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating and fixedstrike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation and in this case, the result allows the use of more established fixedstrike pricing methods to price floatingstrike Asian options.