A mathematical analysis of the optimal exercise boundary for american put options

We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem, and we derive and rigorously prove high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the boundary: one...

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Bibliographic Details
Published in:SIAM journal on mathematical analysis 2007-01, Vol.38 (5), p.1613-1641
Main Authors: XINFU CHEN, CHADAM, John
Format: Article
Language:English
Subjects:
Accuracy
Applied mathematics
Approximation
Dividends
Exact sciences and technology
Interest rates
Mathematical analysis
Mathematics
Monte Carlo simulation
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Partial differential equations
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Put & call options
Sciences and techniques of general use
Securities prices
Stock prices
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Summary:We study a free boundary problem arising from American put options. In particular we prove existence and uniqueness for this problem, and we derive and rigorously prove high order asymptotic expansions for the early exercise boundary near expiry. We provide four approximations for the boundary: one is explicit and is valid near expiry (weeks); two others are implicit involving inverse functions and are accurate for longer time to expiry (months); the fourth is an ODE initial value problem which is very accurate for all times to expiry, is extremely stable, and hence can be solved instantaneously on any computer. We further provide an ode iterative scheme which can reach its numerical fixed point in five iterations for all time to expiry. We also provide a large time (equivalent to regular expiration times but large interest rate and/or volatility) behavior of the exercise boundary. To demonstrate the accuracy of our approximations, we present the results of a numerical simulation.
ISSN:0036-1410
1095-7154
DOI:10.1137/s0036141003437708