The American Put and European Options Near Expiry, Under Levy Processes

A general class of truncated Lévy processes is introduced, and possible ways of fitting parameters of the constructed family of truncated Lévy processes to data are discussed. For a market of a riskless bond and a stock whose log-price follows a truncated Lévy process, TLP-analogs of the Black–Scholes equation, the Black–Scholes formula, the Dynkin derivative and the Leland's model are obtained, a locally risk-minimizing portfolio is constructed, and an optimal exercise price for a perpetual American put is computed.

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Some remarks on first passage of Lévy processes, the American put and pasting principles

The Annals of Applied Probability ◽

10.1214/105051605000000377 ◽

2005 ◽

Vol 15 (3) ◽

pp. 2062-2080 ◽

Cited By ~ 149

Author(s):

L. Alili ◽

A. E. Kyprianou

Keyword(s):

Lévy Processes ◽

Levy Processes ◽

First Passage ◽

American Put

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PRICING OF THE AMERICAN PUT UNDER LÉVY PROCESSES

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024904002463 ◽

2004 ◽

Vol 07 (03) ◽

pp. 303-335 ◽

Cited By ~ 102

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.

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