Under certain assumptions on the dependence structure of the residual lives of the insured (i.e. their independence, positive association, or negative association), in this paper we establish some laws of large numbers for the convex upper bounds, derived by the technique of comonotonicity, of the present value function of a homogeneous portfolio composed of the whole-life insurance policies.

We solve the following three optimal stopping problems for different kinds of options, based on the Black-Scholes model of stock fluctuations. (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more difficult than the closely related one for the Russian option, and we show that for a class of utility functions the free boundary is governed by a nonlinear ordinary differential equation. (ii) A new type of stock option, for a company, where the company provides a guaranteed minimum as an added incentive in case the market appreciation of the stock is low, thereby making the option more attractive to the employee. We show that the value of this option is given by solving a nonalgebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a fixed constant l as a ‘hedge’ on a possible downturn of the stock price, where the buyer gets the maximum of l and the price at any exercise time. We show that the optimal policy depends on the ratio of x/l, where x is the current stock price.

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.