scholarly journals OPTIMAL EXERCISE OF AN EXECUTIVE STOCK OPTION BY AN INSIDER

2011 ◽  
Vol 14 (01) ◽  
pp. 83-106 ◽  
Author(s):  
MICHAEL MONOYIOS ◽  
ANDREW NG
Keyword(s):  
Stock Price ◽  
Stock Option ◽  
Stopping Times ◽  
Inside Information ◽  
Executive Stock Option ◽  
Early Exercise ◽  
Noisy Information ◽  
Exercise Boundary ◽  
Inhomogeneous Diffusion ◽  
The Value Function

We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information. The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. We establish conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward, and derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not. Hence we show that early exercise may arise due to the agent having inside information on the future stock price.

THE VALUE OF BEING LUCKY: OPTION BACKDATING AND NONDIVERSIFIABLE RISK

2021 ◽  
pp. 2150023
Author(s):  
VICKY HENDERSON ◽  
JIA SUN ◽  
A. ELIZABETH WHALLEY
Keyword(s):  
Risk Aversion ◽  
Stock Prices ◽  
Stock Price ◽  
Stock Option ◽  
Risk Averse ◽  
Executive Stock Option ◽  
Early Exercise ◽  
Lookback Option ◽  
Utility Indifference ◽  
Option Compensation

The practice of executives influencing their option compensation by setting a grant date retrospectively is known as backdating. Since executive stock options are usually granted at-the-money, selecting an advantageous grant date to coincide with a low stock price will be valuable to an executive. Empirical evidence shows that backdating of executive stock option grants was prevalent, particularly at firms with highly volatile stock prices. Executives who have the opportunity to backdate should take this into account in their valuation. We quantify the value to a risk averse executive of a lucky option grant with strike chosen to coincide with the lowest stock price of the month. We show the ex ante gain to risk averse executives from the ability to backdate increases with both risk aversion and with volatility, and is significant in magnitude. Our model involves valuing the embedded partial American lookback option in a utility indifference setting with key features of risk aversion, inability to diversify and early exercise.

scholarly journals Optimal Exercise Boundary of American Fractional Lookback Option in a Mixed Jump-Diffusion Fractional Brownian Motion Environment

2017 ◽  
Vol 2017 ◽  
pp. 1-17 ◽  
Author(s):  
Zhaoqiang Yang
Keyword(s):  
Brownian Motion ◽  
Stock Price ◽  
Jump Diffusion ◽  
Fundamental Solutions ◽  
Early Exercise ◽  
Optimal Exercise Boundary ◽  
Lookback Options ◽  
Exercise Price ◽  
Lookback Option ◽  
Exercise Boundary

A new framework for pricing the American fractional lookback option is developed in the case where the stock price follows a mixed jump-diffusion fraction Brownian motion. By using Itô formula and Wick-Itô-Skorohod integral a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

scholarly journals Executive stock options with a rebate: Valuation formula

2006 ◽  
Vol 3 (4) ◽  
pp. 76-79
Author(s):  
P.W.A. Dayananda
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Stock Price ◽  
Stock Options ◽  
Stock Option ◽  
Executive Stock Options ◽  
Exercise Time ◽  
Executive Stock Option ◽  
Exercise Period ◽  
Over Time

We examine the valuation of executive stock option award where there is a rebate at exercise. The rebate depends on the performance of the stock of the corporation over time the period concerned; in particular we consider the situation where the executive can purchase the stock at exercise time at discount proportional to the minimum value of the stock price over the exercise period. Valuation formulae are provided both when assessment is done in discrete time as well as in continuous time. Some numerical illustrations are also presented

THE BRITISH KNOCK-OUT PUT OPTION

2015 ◽  
Vol 18 (02) ◽  
pp. 1550008 ◽  
Author(s):  
LULUWAH AL-FAGIH
Keyword(s):  
Stock Price ◽  
Financial Analysis ◽  
Closed Form Expression ◽  
Form Expression ◽  
Knock Out ◽  
Early Exercise ◽  
New Class ◽  
Put Option ◽  
Exercise Boundary ◽  
Symmetry Relations

Following the economic rationale introduced by Peskir & Samee (2011, 2013) we present a new class of barrier options within the British payoff mechanism where the holder enjoys the early exercise feature of American type options whereupon his payoff (deliverable immediately) is the best prediction of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Should the option holder believe the true drift of the stock price to be unfavorable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. In this paper, we focus on the knock-out put option with an up barrier. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results, we perform a financial analysis of the British knock-out put option. We spot some of the trends previously seen in Peskir & Samee (2011) but observe some behavior unique to the knock-out case. Finally, we derive the British put-call and up-down symmetry relations which express the arbitrage-free price and the rational exercise boundary of the British down-and-out call option in terms of the arbitrage-free price and the rational exercise boundary of the British up-and-out put option.

scholarly journals Efficient valuation and exercise boundary of American fractional lookback option in a mixed jump-diffusion model

2017 ◽  
Vol 04 (02n03) ◽  
pp. 1750033
Author(s):  
Zhaoqiang Yang
Keyword(s):  
Stock Price ◽  
Jump Diffusion ◽  
Fundamental Solutions ◽  
Market Pricing ◽  
Early Exercise ◽  
Lookback Options ◽  
Exercise Price ◽  
Lookback Option ◽  
Exercise Boundary ◽  
Jump Diffusion Model

This study presents an efficient method for pricing the American fractional lookback option in the case where the stock price follows a mixed jump diffusion fraction Brownian motion. By using It ô formula and Wick–It ô–Skorohod integral, a new market pricing model is built. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given. Numerical simulation illustrates some notable features of American fractional lookback options.

scholarly journals Executive Stock Option Exercises and Inside Information

2001 ◽  
Vol 74 (4) ◽  
pp. 513-534 ◽  
Author(s):  
Jennifer N. Carpenter ◽  
Barbara Remmers
Keyword(s):  
Stock Option ◽  
Inside Information ◽  
Executive Stock Option

A note on the nonlinear Volterra integral equation for the early exercise boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Shashiashvili ◽  
Besarion Dochviri ◽  
Giorgi Lominashvili
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Volterra Integral Equation ◽  
Stock Price ◽  
Current Time ◽  
Nonlinear Volterra Integral Equation ◽  
Early Exercise ◽  
Exercise Boundary ◽  
Early Exercise Boundary ◽  
American Put

AbstractIn this paper, we study the nonlinear Volterra integral equation satisfied by the early exercise boundary of the American put option in the one-dimensional diffusion model for a stock price with constant interest rate and constant dividend yield and with a local volatility depending on the current time t and the current stock price S. In the classical Black–Sholes model for a stock price, Theorem 4.3 of [S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 1991, 2, 1–14] states that if the family of integral equations (parametrized by the variable S) holds for all {S\leq b(t)} with a candidate function {b(t)}, then this {b(t)} must coincide with the American put early exercise boundary {c(t)}. We generalize Peskir’s result [G. Peskir, On the American option problem, Math. Finance 15 2005, 1, 169–181] to state that if the candidate function {b(t)} satisfies one particular integral equation (which corresponds to the upper limit {S=b(t)}), then all other integral equations (corresponding to S, {S\leq b(t)}) will be automatically satisfied by the same function {b(t)}.

A NEW STOPPING PROBLEM AND THE CRITICAL EXERCISE PRICE FOR AMERICAN FRACTIONAL LOOKBACK OPTION IN A SPECIAL MIXED JUMP-DIFFUSION MODEL

2018 ◽  
Vol 34 (1) ◽  
pp. 27-52
Author(s):  
Zhaoqiang Yang
Keyword(s):  
Stock Price ◽  
Jump Diffusion ◽  
Fundamental Solutions ◽  
Market Pricing ◽  
Early Exercise ◽  
Exercise Price ◽  
Lookback Option ◽  
Exercise Boundary ◽  
Critical Boundary ◽  
Jump Diffusion Model

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.

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