this paper we consider the class of spectrally negative L'evy processes. These are real valued random processes with stationary independent increments which have no positive jumps. Amongst others Emery [11], Suprun [23], Bingham [4] and Bertoin [3] have all considered fluctuation theory for this class of processes. Such processes are often considered in the context of the theories of dams, queues, insurance risk and continuous branching processes; see for example [6, 4, 5, 19]. Following the exposition on two sided exit problems in Bertoin [3] we study first exit from an interval containing the origin for spectrally negative L'evy processes reflected in their supremum (equivalently spectrally positive L'evy processes reflected in their infimum). In particular we derive the joint Laplace transform of the time to first exit and the overshoot. The aforementioned stopping time can be identified in the literature of fluid models as the time to buffer overflow (see for example [1, 13]). Together Universit'e de Pau, e-mail: Florin.Avram@univ-pau.fr y Utrecht University, e-mail: kyprianou@math.uu.nl z Utrecht University, e-mail: pistorius@math.uu.nl 1 with existing results on exit problems we apply our results to certain optimal stopping problems that are now classically associated with mathematical finance. In sections 2 and 3 we introduce notation and discuss and develop existing results concerning exit problems of spectrally negative L'evy processes. In section 4 an expression is derived for the joint Laplace transform of the exit time and exit position of the reflected process from an interval containing the origin. This Laplace transform can be written in terms of scale functions that already appear in the solution to the two sided exit problem. In Section 5 we ou...

Consider the American put and Russian option [33, 34, 17] with the stock price modeled as an exponential Lévy process. We find an explicit expression for the price in the dense class of Lévy processes with phase-type jumps in both directions. The solution rests on the reduction to the first passage time problem for (reflected) Lévy processes and on an explicit solution of the latter in the phase-type case via martingale stopping and Wiener-Hopf factorisation. Also the first passage time problem is studied for a regime switching Lávy process with phase-type jumps. This is achieved by an embedding into a a semi-Markovian regime switching Brownian motion.

A firm whose net earnings are uncertain, and that is subject to the risk of bankruptcy, must choose between paying dividends and retaining earnings in a liquid reserve. Also, different operating strategies imply different combinations of expected return and variance. We model the firm's cash reserve as the difference between the cumulative net earnings and the cumulative dividends. The first is a diffusion (additive), whose drift/volatility pair is chosen dynamically from a finite set, A. The second is an arbitrary nondecreasing process, chosen by the firm. The firm's strategy must be nonclairvoyant. The firm is bankrupt at the first time, T , at which the cash reserve falls to zero (T may be infinite), and the firm's objective is to maximize the expected total discounted dividends from 0 to T , given an initial reserve, x; denote this maximum by V (x). We calculate V explicitly, as a function of the set A and the discount rate. The optimal policy has the form: (1) pay no dividends if ...

A model of an incomplete market with the incorporation of a new notion of "inside information" is posed. The usual assumption that the stock price is Markovian is modified by adjoining a hidden Markov process to the Black-Scholes exponential Brownian motion model for stock fluctuations. The drift and volatility parameters take different values when the hidden Markov process is in different states. For example, it is 0 when there is no subset of the market which has or which believes it has, extra information. However, the hidden process is in state 1 when information is not equally shared by all, and then the behavior of the members in the subset causes increased fluctuations in the stock price. This model

We solve the following three optimal stopping problems for dierent kinds of options, based on the Black-Scholes model of stock uctuations: (i) The perpetual lookback American option for the running maximum of the stock price during the life of the option. This problem is more dicult 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 dierential 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 non-algebraic equation. (iii) A new call option for the option buyer who is risk-averse and gets to choose, a priori, a xed 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 ...

We propose a new call option where the option seller pays the minimum price (in inflated dollars) that the asset has ever traded at during the time period (which may be indefinitely long) between the selling time and the delivery time (to be chosen by the seller). This option is the dual of the put option where the option buyer receives the maximum price (in discounted dollars) that the asset has ever traded at during the time period (which may be indefinitely long) between the buying time and the exercise time (to be chosen by the buyer). Because the settlement payoff is at the minimum (for the call) or the maximum (for the put) there is reduced regret in the sense that it is not necessary for the option holder to worry about missing a good price in the recent past (of course he may regret not holding on longer) since he gets the best price up to the settlement time. We give the exact simple formula for the optimal expected present value (fair price) that can be derived from the opt...

Abstract In this article it is shown that one is able to evaluate the price of perpetual calls, puts, Russian and integral options directly as the Laplace transform of a stopping time of an appropriate diffusion using standard fluctuation theory. This approach is offered in contrast to the approach of optimal stopping through free boundary problems [see volume 39,1 of Theory of Probability and its Applications]. Following ideas in [5], we discuss the Canadization of these options as a method of approximation to their finite time counterparts. Fluctuation theory is again used in this case.

Given a standard Brownian motion (Bt)t 0 and the equation of motion: dXt = vt dt + p 2 dBt

We consider an online auction setting where the seller attempts to sell an item. Bids arrive over time and the seller has to make an instant decision to either accept this bid and close the auction or reject it and move on to the next bid, with the hope of higher gains. What should be the seller’s strategy to maximize gains? Using techniques from convex analysis, we provide an explicit closedform optimal solution (and hence a simple optimum online algorithm) for the seller. Our methodology is attractive to online auction systems that have to make an instant decision, especially when it is not humanly possible to evaluate each bid individually, when the number of bids is large or unknown ahead of time, and when the bidders are unwilling to wait.

Abstract. We examine the early exercise policies and pricing behaviors of one-asset American options with lookback payoff structures. The classes of option models considered include floating strike lookback options, Russian options, fixed strike lookback options and pricing model of dynamic protection fund. For each class of the American lookback options, we analyze the optimal stopping region, in particular the asymptotic behavior at times close to expiration and at infinite time to expiration. The inter-relations between the price functions of these American lookback options are explored. The mathematical technique of analyzing the exercise boundary curves of lookback options at infinitesimally small asset value is also applied to the American two-asset minimum put option model. Key words. Lookback options, American feature, free boundary problems, two-asset minimum put option