this paper the value of the option (i.e. the supremum in (1.2)) will be found exactly, and in particular it will be shown that the maximum in (1.2) is finite if and only if r ? ¯ : (1.4) Assuming (1.4), an explicit formula is given for both the maximal expected present value and the optimal stopping rule in (2.4), which is not a fixed time rule but depends heavily on the observed values of X t and S t . We call the financial option described above a "Russian option" for two reasons. First, this name serves to (facetiously) differentiate it from American and European options, which have been extensively studied in financial economics, especially with the new interest in market economics in Russia. Second, our solution of the stopping problem (1.2) is derived by the so-called principle of smooth fit, first enunciated by the great Russian mathematician, A. N. Kolmogorov, cf. [4, 5]. The Russian option is characterized by "reduced regret" because the owner is paid the maximum stock price up to the time of exercise and hence feels less remorse at not having exercised at the maximum. For purposes of comparison and to emphasize the mathematical nature of the contribution here, we conclude the paper by analyzing an optimal stopping problem for the Russian option based on Bachelier's (1900) original linear model of stock price fluctuations, X

One of the predominant modes of accessing information in the World Wide Web consists in surfing from one document to another along hypermedia links. We have studied the dynamics of Web surfing within an economics context by considering that there is value in each page that an individual visits, and that clicking on the next page assumes that the information will continue to have some value. Within this formulation an individual will continue to surf until the expected cost of continuing is perceived to be larger than the expected value of the information to be found in the future. This problem is similar to that of a real option in financial economics. We consider the options viewpoint as a descriptive theory of information foraging by Internet users, and we show how it leads to a kind of “law of surfing ” which has been verified experimentally in several large independent datasets. But the real options perspective, which is by now a well-established field in financial economics, may also provide a rich normative model for designing rational Internet agents.

Some non-linear optimal stopping problems can be solved explicitly by using a common method which is based on time-change. We describe this method and illustrate its use by considering several examples dealing with Brownian motion. In each of these examples we derive explicit formulas for the value function and display the optimal stopping time. The main emphasis of the paper is on the method of proof and its unifying scope.

d the book and do the problems you will have mastered all that you will need to get a good grade. An important class of stochastic processes, or "random functions", X(t), satisfy E[X(t + s)jX(t)] = X(t) (1) which is called the martingale hypothesis; we say that the stochastic process is a martingale. The reason people believe that stock prices follow the martingale law (1) is rather deep: people believe that all knowledge of the future has already been incorporated into the price of the stock so that the price is a "martingale" and no trends remain. That is, the expected value of the price at any time in the future is simply the present value. They believe that arbitrageurs have already correctly found the right value and they have removed all profits; this assumes the stock market is all-knowing and wise and there is no way to obtain any additional information. To make the notion of martingale more precise we need to know some more probability theory. A random variable is a funct

A large class of continuous time optimal stopping problems is shown to have solutions explicitly determined by roots of equations xH(x) = I where H involves Laplace transforms. These results motivate the specification of discrete time optimal stopping problems whose solutions are approximated by solutions to corresponding continuous time problems, making rigorous a procedure sometimes employed in the literature. A fairly self-contained treatment of continuous time optimal stopping is also included, albeit for highly structured situations.

“You got to know when to hold em, know when to fold em, know when to walk away... ”