We extend the method of Esscher transforms to changing probability measures in a certain class of stochastic processes that model security prices. According to the Fundamental Theorem of Asset Pricing, security prices are calculated as expected discounted values with respect to a (or the) equivalent martingale measure. If the measure is unique, it is obtained by the method of Esscher transforms; if not, the risk-neutral Esscher measure provides a unique and transparent answer, which can be justified if there is a representative investor maximizing his expected utility. We construct self-financing replicating portfolios in the (multidimensional) geometric shifted (compound) Poisson process model, in which the classical (multidimensional) geometric Brownian motion model is a limiting case. With the aid of Esscher transforms, changing numeraire is explained concisely. We also show how certain American type options on two stocks (for example, the perpetual Margrabe option) can be priced. Applying the optional sampling theorem to certain martingales (which resemble the exponential martingale in ruin theory), we obtain several explicit pricing formulas without having to deal with differential equations.

Actuarial bridges to dynamic hedging and option pricing

Ruin problems with compounding assets

(1997). Some Probabilistic Aspects of Set Partitions. The American Mathematical Monthly: Vol. 104, No. 3, pp. 201-209.

Some Probabilistic Aspects of Set Partitions

A bibliography on life testing and related topics

The aim of this paper is to analyse two functions that are of general interest in the collective risk theory, namely F, the distribution function of the total amount of claims, and II, the Stop Loss premium. Section 2 presents certain basic formulae. Sections 17-18 present five claim distributions. Corresponding to these claim distributions, the functions F and II were calculated under various assumptions as to the distribution of the number of claims. These calculations were performed on an electronic computer and the numerical method used for this purpose is presented in sections 9, 19 and 20 under the name of the C-method which method has the advantage of furnishing upper and lower limits of the quantities under estimation. The means of these limits, in the following regarded as the “exact” results, are given in Tables 4-20. Sections 11-16 present certain approximation methods. The N-method of section 11 is an Edgeworth expansion, while the G-method given in section 12 is an approximation by a...

Studies in risk theory with numerical illustrations concerning distribution functions and stop loss premiums. Part I

Let the random variable ξ be distributed with distribution function G(ξ) of the continuous, discontinuous or mixed type, and the random variables x ξ be distributed with the distribution functions V ξ(x) with the corresponding characteristic functions , η being a real variable, and i the imaginary unit, and with the generating functions defined by ψξ(− i log z). If the range of x is restricted to the right semi-plane the lower limit of the integral in the definition of ψξ(η) can be replaced by zero.

A review of the collective theory of risk

A. Confluent Hypergeometric Functions The theory of such functions will be recapitulated here following a recently published book (Slater, L. J., Confluent Hypergeometric Functions, Cambr. Univ. Press 1960).

The theory of confluent hypergeometric functions and its application to compound poisson processes

Let the nth moment about zero of a Poisson distribution designated by ±1 with ±1 = t, let i = √−1 and τ a real variable and let z = z(τ) be equal to eτi . Then, ±n,n = 1, 2, ... fulfill the followi...

A note on moments of a poisson probability distribution

1. For the definition of general processes with special regard to those concerned in Collective Risk Theory reference is made to Cramer (Collective Risk Theory, Skandia Jubilee Volume, Stockholm, 1955). Let the independent parameter of such a process be denoted by τ, with the origin at the point of departure of the process and on a scale independent of the number of expected changes of the random function. Denote with p(τ, n)dt the asymptotic expression for the conditional probability of one change in the random function while the parameter passes from τ to τ + dτ: relative to the hypothesis that n changes have occurred, while the parameter passes from 0 to τ. Assume further—unless the contrary is stated—that the probability of more than one change, while the parameter passes from τ to τ + dτ, is of smaller order than dτ.

A note on different models of stochastic processes dealt with in the collective theory of risk

F. Esscher has recently generalized the transformation of the distribution functions defining a Poisson process, which he introduced in 1932 and which Ammeter in 1948 applied to a Polya process. This generalization was first achieved for a positive compound Poisson process with an unconditioned distribution function of the risk (in the sequel called the risk distribution) and with a conditioned distribution relative to the hypothesis that a change has occurred in the random function attached to the process, of the size of such a change (in the sequel called claim distribution), which both are independent of the parameter t. In the following section 2 and throughout this paper the investigation will be restricted in the same way. An extension to cases where the risk distribution and/or the claim distribution depend on t can easily be derived by using the results given by the present author in a report to the congress in London (1964). Also the theory may easily be extended to cases where the chang...

Carl Philipson

Papers

A review of the collective theory of risk

The theory of confluent hypergeometric functions and its application to compound poisson processes

A note on moments of a poisson probability distribution

A note on different models of stochastic processes dealt with in the collective theory of risk

On Esscher transforms of distribution functions defining a compound poisson process for large values of the parameter