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 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.

[1]  H. P. Jr. Mackean,et al.  Appendix : A free boundary problem for the heat equation arising from a problem in mathematical economics , 1965 .

[2]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[3]  P. Carr,et al.  ALTERNATIVE CHARACTERIZATIONS OF AMERICAN PUT OPTIONS , 1992 .

[4]  A. Friedman.  Variational principles and free-boundary problems , 1982 .

[5]  D. Lamberton,et al.  Variational inequalities and the pricing of American options , 1990 .

[6]  J. B. Keller,et al.  American options on assets with dividends near expiry , 2002 .

[7]  P. Wilmott,et al.  Option pricing: Mathematical models and computation , 1994 .

[8]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal Stopping and Free-Boundary Problems , 2006 .

[9]  Xinfu Chen,et al.  CONVEXITY OF THE EXERCISE BOUNDARY OF THE AMERICAN PUT OPTION ON A ZERO DIVIDEND ASSET , 2007 .

[10]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[11]  Rachel Kuske,et al.  Optimal exercise boundary for an American put option , 1998 .

[13]  Erik Ekström.  Convexity of the optimal stopping boundary for the American put option , 2004 .

[14]  J. Hull.  Options, futures, and other derivative securities , 1989 .

[15]  Guy Barles,et al.  CRITICAL STOCK PRICE NEAR EXPIRATION , 1995 .

[16]  Robert Stamicar,et al.  THE EARLY EXERCISE BOUNDARY FOR THE AMERICAN PUT NEAR EXPIRY: NUMERICAL APPROXIMATION , 1999 .

[17]  I. Kim.  The Analytic Valuation of American Options , 1990 .

[18]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[19]  P. Moerbeke.  On optimal stopping and free boundary problems , 1973, Advances in Applied Probability.

[20]  Charles Knessl.  Asymptotic analysis of the American call option with dividends , 2002, European Journal of Applied Mathematics.

[21]  Herb Johnson,et al.  The American Put Option and Its Critical Stock Price , 2000 .

[22]  G. Barone-Adesi,et al.  Efficient Analytic Approximation of American Option Values , 1987 .

[23]  G. Peskir.  ON THE AMERICAN OPTION PROBLEM , 2005 .

[24]  R. Elliott,et al.  Approximations for the values of american options , 1991 .