Daniel Dufresne

The Distribution of a Perpetuity, with Applications to Risk Theory and Pension Funding

Abstract If Vk is the discount factor for the kth period, then Z = Σ k⩾1V 1...Vk Ck is the discounted value of a perpetuity paying Ck at time k. In some cases Z is also the limiting distribution of St =Vt (St-1 +Ct-1 ). This paper 1. reviews the literature concerning Z and {St } 2. considers continuous-time counterparts of Z and S, at the same time deriving the distribution of ∫ exp(-γt-σWt )1(0, ∞) (t)dt when W is Brownian motion; 3. gives applications to risk theory and pension funding.

Accelerated simulation for pricing Asian options

When pricing options via Monte Carlo simulations, precision can be improved either by performing longer simulations, or by reducing the variance of the estimators. Two methods for variance reduction are combined: the control variable and the change of measure (or likelihood) methods. We specifically consider Asian options, and show that a change of measure can very significantly improve the precision when the option is deeply out of the money, which is the harder estimation problem. We also show that the simulation method itself can be used to find the best change of measure. This is done by incorporating an updating rule, based on an estimate of the gradient of the variance. The paper includes simulation results.

The log-normal approximation in financial and other computations

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

Stability of pension systems when rates of return are random

Abstract Consider a funded pension plan, and suppose actuarial gains or losses are amortized over a fixed number of years. The paper aims at assessing how contributions (C) and fund levels (F) are affected when the rates of return of the planss assets form an i.i.d. sequence of random variables. This is achieved by calculating the mean and variance of Ct and Ft for t ⩽ ∞.

Weak convergence of random growth processes with applications to insurance

Abstract The paper is concerned with assets accumulating or discounting processes, and their weak convergence when payments are made more and more frequently during each time period. The results are applied to the calculation of the moments of actuarial functions (annuities-certain, life annuities and life insurances). The relationship with random population growth is also briefly discussed.

Stochastic Life Annuities

Abstract This paper gives analytic approximations for the distribution of a stochastic life annuity. It is assumed that returns follow a geometric Brownian motion. The distribution of the stochastic annuity may be used to answer questions such as “What is the probability that an amount F is sufficient to fund a pension with annual amount y to a pensioner aged x?” The main idea is to approximate the future lifetime distribution with a combination of exponentials, and then apply a known formula (due to Marc Yor) related to the integral of geometric Brownian motion. The approximations are very accurate in the cases studied.

Bessel Processes and Asian Options

The goal of this chapter is to give a concise account of the connection between Bessel processes and the integral of geometric Brownian motion. The latter appears in the pricing of Asian options. Bessel processes are defined and some of their properties are given. The known expressions for the probability density function of the integral of geometric Brownian motion are stated, and other related results are given, in particular the Geman and Yor (1993) Laplace transform for Asian option prices.

Pension Funding with Moving Average Rates of Return

In the context of the model of pension funding introduced by Dufresne in 1986, explicit expressions are found for the first two moments of fund level and total contributions, when (1) actuarial gains and losses are amortized over N years, and (2) arithmetic rates of return on assets form a moving average process. The results are obtained via a Markovian representation for the bilinear process obtained for the actuarial losses. One conclusion is that the dependence between successive rates of return may have very significant effects on the financial results obtained.

The Beta Product Distribution with Complex Parameters

The family consisting of the distributions of products of two independent beta variables is extended to include cases where some of the parameters are not positive but negative or complex. This “beta product” distribution is expressible as a Meijer G function. An example (from risk theory) where such a distribution arises is given: an infinite sum of products of independent random variables is shown to have a distribution that is the product convolution of a complex-parameter beta product and an independent exponential. The distribution of the infinite sum is a new explicit solution of the stochastic equation X = (in law) B(X + C). Characterizations of some G distributions are also proved.

Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion

The main result is a two-dimensional identity in law. Let (B t ,L t ) and (β t ,λ t ) be two independent pairs of a linear Brownian motion with its local time at 0. Let A t =∫ 0 t exp(2B s )ds. Then, for fixed t, the pair (sinh(B t ),sinh(L t )) has the same law as (β(A t ),exp(-B t )λ(A t )), and also as (exp(-B t )β(A t ),λ(A t )). This result is an extension of an identity in distribution due to Bougerol that concerned the first components of each pair. Some other related identities are also considered.