Elias S. W. Shiu

On the Time Value of Ruin

Abstract This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ult...

Optimal Dividends: Analysis with Brownian Motion

Abstract In the absence of dividends, the surplus of a company is modeled by a Wiener process (or Brownian motion) with positive drift. Now dividends are paid according to a barrier strategy: Whenever the (modified) surplus attains the level b, the “overflow” is paid as dividends to shareholders. An explicit expression for the moment-generating function of the time of ruin is given. Let D denote the sum of the discounted dividends until ruin. Explicit expressions for the expectation and the moment-generating function of D are given; furthermore, the limiting distribution of D is determined when the variance parameter of the surplus process tends toward infinity. It is shown that the sum of the (undiscounted) dividends until ruin is a compound geometric random variable with exponentially distributed summands. The optimal level b* is the value of b for which the expectation of D is maximal. It is shown that b* is an increasing function of the variance parameter; as the variance parameter tends toward infinity, b* tends toward the ratio of the drift parameter and the valuation force of interest, which can be interpreted as the present value of a perpetuity. The leverage ratio is the expectation of D divided by the initial surplus invested; it is observed that this leverage ratio is a decreasing function of the initial surplus. For b = b*, the expectation of D, considered as a function of the initial surplus, has the properties of a risk-averse utility function, as long as the initial surplus is less than b*. The expected utility of D is calculated for quadratic and exponential utility functions. In the appendix, the original discrete model of De Finetti (1957) is explained and a probabilistic identity is derived.

The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin

Abstract We examine the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transform, which can naturally be interpreted as discounting. We show that, as a function of the initial surplus, the joint density satisfies a certain renewal equation. We generalize Dicksons (1992) formula, which expresses the joint distribution of the surplus immediately before ruin and the deficit at ruin in terms of the probability of ultimate ruin.

Actuarial bridges to dynamic hedging and option pricing

Abstract 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.

Risk Theory with the Gamma Process

The aggregate claims process is modelled by a process with independent, stationary and nonnegative increments. Such a process is either compound Poisson or else a process with an infinite number of claims in each time interval, for example a gamma process. It is shown how classical risk theory, and in particular ruin theory, can be adapted to this model. A detailed analysis is given for the gamma process, for which tabulated values of the probability of ruin are provided.

The Probability of Eventual Ruin in the Compound Binomial Model

This paper derives several formulas for the probability of eventual ruin in a discrete-time model. In this model, the number of claims process is assumed to be binomial. The claim amounts, premium rate and initial surplus are assumed to be integer-valued.

Discounted probabilities and ruin theory in the compound binomial model

Abstract The aggregate claims are modeled as a compound binomial process, and the individual claim amounts are integer-valued. We study f(x, y; u), the “discounted” probability of ruin for an initial surplus u, such that the surplus just before ruin is x and the deficit at ruin is y. This function can be used to calculate the expected present value of a penalty that is due at ruin, and, if it is interpreted as a probability generating function, to obtain certain information about the time of ruin. An explicit formula for f(x, y; 0) is derived. Then it is shown how f(x, y; u) can be expressed in terms of f(x, y; 0) and an auxiliary function h(u) that is the solution of a certain recursive equation and is independent of x and y. As an application, we use the asymptotic expansion of h(u) to obtain an asymptotic formula for f(x, y; u). In this model, certain results can be obtained more easily than in the compound Poisson model and provide additional insight. For the case u=0, expressions for the expected present value of a payment of 1 at ruin and the expected time of ruin (given that ruin occurs) are obtained. A discrete version of Dickson’s formula is provided.

Martingale Approach to Pricing Perpetual American Options

The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the risk-neutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skip-free. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skip-free. Under the classical assumption that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual down-and-out call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the relationship between Samuelsons high contact condition and the first order condition for optimality.

From ruin theory to pricing reset guarantees and perpetual put options

Abstract We examine the joint distribution of the time of ruin, the surplus immediately before ruin, the deficit at ruin, and the cause of ruin. The time of ruin is analyzed in terms of its Laplace transform, which can naturally be interpreted as discounting. We present two financial applications – the pricing of reset guarantees for a mutual fund or an equity-indexed annuity, and the pricing of a perpetual American put option. In both cases, the logarithm of the price of the underlying asset is modeled as a shifted compound Poisson process. Hence the asset price process has downward discontinuities, with the times and amounts of the drops being random.

Maximizing Dividends without Bankruptcy

Consider the classical compound Poisson model of risk theory, in which dividends are paid to the shareholders according to a barrier strategy. Let b * be the level of the barrier that maximizes the expectation of the discounted dividends until ruin. This paper is inspired by Dickson and Waters (2004). They point out that the shareholders should be liable to cover the deficit at ruin. Thus, they consider b 0 , the level of the barrier that maximizes the expectation of the difference between the discounted dividends until ruin and the discounted deficit at ruin. In this paper, b * and b 0 are compared, when the claim amount distribution is exponential or a combination of exponentials.