FINANCIAL MATHEMATICS ACTIVITIES
|February 1, 2015|
Financial Mathematics Seminars - October 25, 2000
Marcel Rindisbacher, Joseph L. Rotman School of Management, University of Toronto
This talk and associated article extend the standard continuous time financial market model pioneered by Samuelson (1969) and Merton (1971) to allow for insider information. We prove that if the investment horizon of an insider ends after his initial information advantage has disappeared, an insider has arbitrage opportunities if and only if the anticipative information is so informative that it contains zero-probability events given initial public information. When it ends before or anticipative information does not contain such events we derive expressions for optimal consumption and portfolio policies, which allow to analyze, how the anticipative information affects optimal strategies of ins iders. Optimal insider policies are shown not to be fully revealing. Individually, anticipative information is of no value and therefore does not affect the optimal behavior of insiders if and only if it is independent from public information. We show that arbitrage opportunities allow to replicate arbitrary consumption streams such that Merton's consumption-investment problem with general convex von Neumann-Morgenstern preferences has no solution whenever investment horizons are longer than resolution times of signals. If the true insider signal is perturbed by independent noise this problem can be avoided. But since in this case non-insiders will never learn the anticipative information we argue that this is not appropriate to capture important features of insider information. We also show that the valuation of contingent claims measurable with respect to the public information by arbitrage is invariant to insider information if it does not allow for arbitrage opportunities. In contrast contingent claims have no value for insiders with anticipative information generated by signals with continuous distributions.
S. David Promislow, Department of Mathematics & Statistics, York University
A life annuity is a contract which promises a stream of payments at fixed times and of fixed amounts, with the provision that the purchaser must be alive at the time of each payment in order to collect. The pricing of such contracts involves an assessment of both future interest rates and future mortality experience. This talk will discuss a joint project with M. Milevsky, which considers the problem of valuing options on these mortality-contingent claims. A typical product of this type would give the option holder the right to purchase a life annuity at some future date, at a price which is guaranteed now. Although these do not appear to be sold directly at the present time, many U.S. insurance companies offer this type of option as an additional benefit to holders of their tax sheltered savings plans. The valuation of annuity options requires a somewhat different approach towards mortality measurement than the traditional actuarial technique. In order to model the uncertainty in future mortality, one must view the force of mortality ( hazard rate) as a stochastic process, rather than a fixed function of time. We show that under certain natural assumptions, both the mortality and interest rate risk can be hedged, and the option to annuitize can be priced by constructing a replicating portfolio involving insurance, annuities, and default-free bonds. Both discrete time and continuous time models will be discussed.