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.

This paper develops a variance reduction technique for Monte Carlo simulations of path-dependent options driven by high-dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables which maximizes the product of probability and payoff---the most important path. The directions used for stratified sampling are optimal for a quadratic approximation to the integrand or payoff function. Indeed, under differentiability assumptions our importance sampling method eliminates variability due to the linear part of the payoff function, and stratification eliminates much of the variability due to the quadratic part of the payoff. The two parts of the method are linked because the asymptotically optimal drift vector frequently provides a particularly effective direction for stratification. We illustrate the use of the method with path-dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of their payoffs. KEY WORDS: Monte Carlo methods, variance reduction, large deviations, Laplace principle 1. INTRODUCTION This paper develops a variance reduction technique for Monte Carlo simulations driven by high-dimensional Gaussian vectors, with particular emphasis on the pricing of pathdependent options. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to...

Simulation has proved to be a valuable tool for estimating security prices for which simple closed form solutions do not exist. In this paper we present two direct methods, a pathwise method and a likelihood ratio method, for estimating derivatives of security prices using simulation. With the direct methods, the information from a single simulation can be used to estimate multiple derivatives along with a security’s price. The main advantage of the direct methods over re-simulation is increased computational speed. Another advantage is that the direct methods give unbiased estimates of derivatives, whereas the estimates obtained by resimulation are biased. Computational results are given for both direct methods and comparisons are made to the standard method of re-simulation to estimate derivatives. The methods are illustrated for a path independent model (European options), a path dependent model (Asian options), and a model with multiple state variables (options with stochastic volatility).

. The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this law is characterized by its integral moments. When the process is asymptotically ff-stable, we prove that t \Gamma1=ff log A t converges in law, as t !1, to the supremum of an ff-stable L'evy process ; in particular, if E [ 1 ] ? 0, then ff = 1 and (1=t) log A t converges almost surely to E [ 1 ]. Eventually, we use Girsanov's transform to give the explicit behavior of E \Theta (a +A t ()) \Gamma1 as t ! 1, where a is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random L'evy environment. 1. Introduction We first describe three different sources of interest for exponential functionals of Brownian mot...

This paper proposes a methodology for the valuation of contingent securities. In particular, it establishes how the characteristic function (of the future uncertainty) is basis augmenting and spans the payoff universe of most, if not all, derivative assets. In one specific application, from the characteristic function of the state-price density, it is possible to analytically price options on any arbitrary transformation of the underlying uncertainty. By differentiating (or translating) the characteristic function, limitless pricing and/or spanning opportunities can be designed. As made lucid via example contingent claims, by exploiting the unifying spanning concept, the valuation approach affords substantial analytical tractability. The strength and versatility of the methodology is inherent when valuing (1) Average-interest options; (2) Correlation options; and (3) Discretely-monitored knock-out options. For each option-like security, the characteristic function is strikingly simple (although the corresponding density is unmanageable/indeterminate). This article provides the economic foundations for valuing derivative securities.

I survey and assess the development of continuous-time methods in finance during the last 30 years. The subperiod 1969 to 1980 saw a dizzying pace of development with seminal ideas in derivatives securities pricing, term structure theory, asset pricing, and optimal consumption and portfolio choices. During the period 1981 to 1999 the theory has been extended and modified to better explain empirical regularities in various subfields of finance. This latter subperiod has seen significant progress in econometric theory, computational and estimation methods to test and implement continuous-time models. Capital market frictions and bargaining issues are being increasingly incorporated in continuous-time theory. THE ROOTS OF MODERN CONTINUOUS-TIME METHODS in finance can be traced back to the seminal contributions of Merton ~1969, 1971, 1973b! in the late 1960s and early 1970s. Merton ~1969! pioneered the use of continuous-time modeling in financial economics by formulating the intertemporal consumption and portfolio choice problem of an investor in a stochastic dynamic programming setting.

Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial Ornstein--Uhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial Ornstein--Uhlenbeck processes, i. e., CIR processes. As a natural extension we study squared Bessel processes and squared Ornstein--Uhlenbeck processes with negative dimensions or negative starting points and derive their properties. Keywords: First hitting times; CIR processes; Bessel processes; radial Ornstein-- Uhlenbeck processes; Bessel processes with negative dimensions 1 Introduction Bessel processes have come to play a distinguished role in financial mathematics for at least two reasons, which have a lot to do with the models being usually considered. One of these models is the Cox--I...

. In this paper, arithmetic average Asian options are studied. It is observed that the Asian option is a special case of the option on a traded account. The price of the Asian option is characterized by a simple one-dimensional partial dierential equation which could be applied to both continuous and discrete average Asian option. The article also provides numerical implementation of the pricing equation. The implementation is fast and accurate even for low volatility and/or short maturity cases. Key words: Asian options, Options on a traded account, Brownian motion, xed strike, oating strike. 1 Introduction Asian options are securities with payo which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. A number of approximations that produce closed form expressions have appeared, se...

In this article we present new results for the pricing of arithmetic Asian options within a Black-Scholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple PDE for the price of arithmetic Asians options. In the case of European average strike options, a proper choice of numeraire reduces the dimension of this PDE to one, leading to a PDE similar to the one derived by Rogers and Shi. We solve this PDE, finding a Laplacetransform representation for the price of average strike options, both seasoned and unseasoned. This extends the results of Geman and Yor, who discussed the case of average price options. Next we use symmetry arguments to show that prices of average strike and average price options can be expressed in terms of each other. Finally we show, again using symmetries, that plain vanilla options on stocks paying known cash dividends are c...

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