Discounted optimal stopping problems for the maximum process

Abstract

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.

Optimal Stopping of the Maximum Process

“…[9], [10], [19], [20], and [24]) In this case the exercise payoff reads as f (x, s) = (s − kx), where k ∈ R + is a known exogenously given constant.…”

“…Under this setting, problem (5.1) can be solved explicitly (see [19] and [24]): Proposition 5.2. Note that γ 1 < 0 and, since μ < r, we have γ 2 > 1.…”

Recent Advances in Applied Probability

“…The result is extended in [Pedersen, 2000a] to Lookback options with fixed and floating strike. The result is extended in [Pedersen, 2000a] to Lookback options with fixed and floating strike.…”

Solving the dual Russian option problem by using change‐of‐measure arguments

“…Guo (2001) presented an explicit solution of the perpetual Russian option problem for a diffusion-type model with switching coefficients. Pedersen (2000) and Guo and Shepp (2001) obtained closed-form solutions to the problems of pricing of more complicated perpetual American lookback options with payoffs depending on the running maxima processes of the underlying geometric Brownian motions. Shepp, Shiryaev, and Sulem (2002) proposed a barrier version of the Russian option in the same model, where the decision about stopping should be taken before the price process reaches a certain positive level.…”