A New Stopping Problem and the Critical Exercise Price for American Fractional Lookback Option in a Special Mixed Jump-Diffusion Model

Abstract

A new stopping problem and the critical exercise price of American fractional lookback option are developed in the case where the stock price follows a special mixed jump diffusion fractional Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built, and the fundamental solutions of stochastic parabolic partial differential equations are deduced under the condition of Merton assumptions. With an optimal stopping problem and the exercise boundary, the explicit integral representation of early exercise premium and the critical exercise price are also derived. Numerical simulation illustrates the asymptotic behavior of this critical boundary.

The reliability of geometric Brownian motion forecasts of S&P500 index values

“…Halperin and Dixon (2020) "quantum equilibrium-disequilibrium" model requires a nonlinear two-parametric extension of the classical GBM model. In cases where stock prices follow a special mixed jump-diffusion fractional Brownian motion, Yang (2020) developed a new stopping problem and the critical exercise price of American fractional lookback options.…”

Option to survive or surrender: carbon asset management and optimization in thermal power enterprises from China

“…To describe the strategy and actions during the carbon asset operation, Markov decision process is applied to simulate the decision-making as in [2] for energy storage system, [3] for management of greenhouses, [4] for optimizing energy conversion and [5] for micro-grid power optimal control. Paralleling to the approach of Markov decision process, optimal stopping model is also usually used to characterize the timing of decision and operation, especially for those financial environments with continuous dynamic of price process, see [6] for asset trading strategy and [7], [8], [9] for option pricing. The former has its advantage of describing the decision-making for all points in the time horizon, while the later seeks one or limited several (multiple optimal stopping times) timings for some specific action and it handles both discrete and continuous environment.…”