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 THE BRITISH ASSET-OR-NOTHING PUT OPTION

2017 ◽  
Vol 20 (04) ◽  
pp. 1750030
Author(s):  
MIN GAO
Keyword(s):  
Stock Price ◽  
Option Price ◽  
Mathematical Finance ◽  
Closed Form Expression ◽  
Attractive Alternative ◽  
Strike Price ◽  
Early Exercise ◽  
Put Option ◽  
Exercise Boundary ◽  
Price Movements

Following the economic rationale of Peskir & Samee [The British put option, Applied Mathematical Finance 18 (6), 537–563 (2011); The British call option, Quantitative Finance 13 (1), 95–109 (2013)], we present a new class of asset-or-nothing put option where the holder enjoys the early exercise feature of American asset-or-nothing put option whereupon his payoff is the ‘best prediction’ of the European asset-or-nothing put option payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, the option holder finds that if the true drift of the stock price is unfavorable, then he can substitute it with the contract drift and minimize his losses. The key to the British asset-or-nothing put option is the protection feature as not only can the option holder exercise at or above the strike price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favorable he will generally receive high returns. 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. We also analyze the financial meaning of the British asset-or-nothing put option using the results above and show that with the contract drift properly selected, the British asset-or-nothing put option becomes a very attractive alternative to the classic European/American asset-or-nothing put option.

APPROXIMATE SOLUTIONS FOR THE BRITISH PUT OPTION AND ITS OPTIMAL EXERCISE BOUNDARY

2016 ◽  
Vol 57 (3) ◽  
pp. 222-243
Author(s):  
JOANNA GOARD
Keyword(s):  
Stock Price ◽  
Approximate Solutions ◽  
Financial Derivatives ◽  
Put Options ◽  
Early Exercise ◽  
Empirical Results ◽  
Analytic Approximations ◽  
Optimal Exercise Boundary ◽  
Put Option ◽  
Exercise Boundary

British put options are financial derivatives with an early exercise feature whereby on payoff, the holder receives the best prediction of the European put payoff under the hypothesis that the true drift of the stock price is equal to a contract drift. In this paper, we derive simple analytic approximations for the optimal exercise boundary and the option valuation, valid for short expiry times – which is a common feature of most options traded in the market. Empirical results show that the approximations provide accurate results for expiries of at least up to two months.

Double barrier American put option pricing under uncertain volatility model

2021 ◽  
pp. 2150038
Author(s):  
El Kharrazi Zaineb ◽  
Saoud Sahar ◽  
Mahani Zouhir
Keyword(s):  
Single Point ◽  
Option Prices ◽  
Double Barrier ◽  
Early Exercise ◽  
Put Option ◽  
Volatility Model ◽  
Uncertain Volatility ◽  
Exercise Boundary ◽  
Early Exercise Boundary ◽  
American Style

This paper aims to study the asymptotic behavior of double barrier American-style put option prices under an uncertain volatility model, which degenerates to a single point. We give an approximation of the double barrier American-style option prices with a small volatility interval, expressed by the Black–Scholes–Barenblatt equation. Then, we propose a novel representation for the early exercise boundary of American-style double barrier options in terms of the optimal stopping boundary of a single barrier contract.

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.

Sensitivity analysis of the optimal exercise boundary of the American put option

2016 ◽  
Vol 23 (3) ◽  
pp. 429-433
Author(s):  
Nasir Rehman ◽  
Sultan Hussain ◽  
Wasim Ul-Haq
Keyword(s):  
Stock Price ◽  
Obstacle Problem ◽  
One Dimensional ◽  
American Put Option ◽  
Optimal Exercise Boundary ◽  
Put Option ◽  
Dimensional Diffusion ◽  
Exercise Boundary ◽  
American Put ◽  
Continuity Estimate

AbstractWe consider the American put problem in a general one-dimensional diffusion model. The risk-free interest rate is constant, and volatility is assumed to be a function of time and stock price. We use the well-known parabolic obstacle problem and establish the continuity estimate of the optional exercise boundaries of the American put option with respect to the local volatilities, which may be considered as a generalization of the Achdou results [1].

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.

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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