In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced in [D] to computing the negative moments of Yor’s accumulation processes for which functional recursion rules are given. Stressing the role of Theta functions, this paper now solves these recursion rules and expresses these negative moments as linear combinations of certain Theta integrals. Using the Jacobi transformation formula, very rapidly and very stably convergent series for them are derived. In this way a computable series for Black–Scholes price of the Asian option results which is numerically illustrated. Moreover, the Laguerre series approach of [D] is made rigorous, and extensions and modifications are discussed. The key for this is the analysis of the integrability and growth properties of the Asia density in [Y], basic problems which seem to be addressed here for the first time. 1. Introduction: Asian

Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the so-called Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman-Watson 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 Hartman-Watson 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.

The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At: = ∫ t exp{Zs}ds, t ≥ 0, where {Zs: s ≥ 0} is a one-dimensional

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].

In the n-period binomial tree model, we provide fast algorithms to compute very accurate lower and upper bounds on the value of a European-style Asian option. These algorithms are inspired by the continuous-time analysis of Rogers and Shi [12]. Specifically we consider lower bounds that are given by grouping stock-price paths in the tree according to the value of a certain random variable Z, and treating all paths in a group as having the same arithmetic stockprice average, namely the average of the arithmetic average over the group. For a specific Z , Rogers and Shi show analytically that the error in the lower bound is small. However they are only able to estimate the lower and upper bounds numerically as integrals in continuous time. Indeed, for their choice of Z these bounds are difficult to compute exactly in the binomial tree model. We show how to choose Z so that the bounds can be computed exactly in time proportional to n 4 by forward induction. Moreover, our bounds are stric...

Introduction Many insurance problems, both for premium calculation as well as for the determination of provisions, can be cast into the form of the evaluation of the distribution of the quantity A t = Z t 0 fl(ø )e \Gammax(ø ) dø; (1) where x(ø) denotes a stochastic process and y(ø) is a deterministic function. This important quantity has been studied by several authors in the actuarial literature as well as in the theory of stochastic processes (see [1-7]). In case x(ø) denotes a stochastic process starting in zero, hence x(0) = 0, the interpretation of (1) is clear. Indeed, (1) can be written as A t = Z t 0 fl 0 (ø) expf\

We derive an integral representation for the fundamental solution of the Kolmogorov forward equation: ft = −((1+µx)f)x + (ν x 2 f)xx associated with the Shiryaev process X solving: dXt = (1+µXt) dt + σXt dBt where µ ∈ IR, ν = σ 2 /2> 0 and B is a standard Brownian motion. The method of proof is based upon deriving and inverting a Laplace transform. Basic properties of X needed in the proof are reviewed. 1.

Abstract. Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (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 self-adjoint operators. We also extend this scheme to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree. 1.

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ’disorder ’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation. 1.

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space H n. This enables us to find asymptotic properties of the kernel. Our starting point is the formula for its Fourier transform. When n = 3, 4 or 6 we give an explicit formula for the Poisson kernel itself. In the general case we give various asymptotics and show convergence to the Poisson kernel of H n.