scholarly journals PRICING OF THE AMERICAN PUT UNDER LÉVY PROCESSES

2004 ◽  
Vol 07 (03) ◽  
pp. 303-335 ◽  
Author(s):  
S. Z. Levendorskiǐ
Keyword(s):  
Stock Returns ◽  
Lévy Processes ◽  
Diffusion Processes ◽  
Method Of Lines ◽  
Levy Processes ◽  
Strike Price ◽  
Early Exercise ◽  
Exercise Boundary ◽  
Early Exercise Boundary ◽  
American Put

We consider the American put with finite time horizon T, assuming that, under an EMM chosen by the market, the stock returns follow a regular Lévy process of exponential type. We formulate the free boundary value problem for the price of the American put, and develop the non-Gaussian analog of the method of lines and Carr's randomization method used in the Gaussian option pricing theory. The result is the (discretized) early exercise boundary and prices of the American put for all strikes and maturities from 0 to T. In the case of exponential jump-diffusion processes, a simple efficient pricing scheme is constructed. We show that for many classes of Lévy processes, the early exercise boundary is separated from the strike price by a non-vanishing margin on the interval [0, T), and that as the riskless rate vanishes, the optimal exercise price goes to zero uniformly over the interval [0, T), which is in the stark contrast with the Gaussian case.

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)}.

scholarly journals COMPARISON OF NUMERICAL AND ANALYTICAL APPROXIMATIONS OF THE EARLY EXERCISE BOUNDARY OF AMERICAN PUT OPTIONS

2010 ◽  
Vol 51 (4) ◽  
pp. 430-448 ◽  
Author(s):  
M. LAUKO ◽  
D. ŠEVČOVIČ
Keyword(s):  
Numerical Scheme ◽  
Analytical Approximation ◽  
Approximation Formula ◽  
Put Options ◽  
Early Exercise ◽  
Analytical Approximations ◽  
American Put Options ◽  
Exercise Boundary ◽  
Early Exercise Boundary ◽  
American Put

AbstractWe present qualitative and quantitative comparisons of various analytical and numerical approximation methods for calculating a position of the early exercise boundary of American put options paying zero dividends. We analyse the asymptotic behaviour of these methods close to expiration, and introduce a new numerical scheme for computing the early exercise boundary. Our local iterative numerical scheme is based on a solution to a nonlinear integral equation. We compare numerical results obtained by the new method to those of the projected successive over-relaxation method and the analytical approximation formula recently derived by Zhu [‘A new analytical approximation formula for the optimal exercise boundary of American put options’, Int. J. Theor. Appl. Finance9 (2006) 1141–1177].

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.

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