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Summary: A Simple Heuristic for Valuing Certain Perpetual American-Type
Securities
John E. Angus and Xiao Hong
December 17, 1997
Abstract. We present a simple, purely probabilistic method for valuing
certain perpetual (i.e. with no expiration) derivative securities which may be
exercised at the holder's option.
Keywords: Perpetual options, martingales, Markov time
1. Introduction and Background
Consider an asset (e.g. a stock) whose share price S at time t 0 is governed by the
stochastic differential equation
dS = (µ - D0)Sdt + SdX (1)
where µ > 0, D0 0, and > 0 are constants, and {X(t), t 0} is a standard
Brownian motion. Here, is referred to as the volatility, and D0 is interpreted as
the constant rate of (continuous) dividend payment. Equation (1) can be solved
explicitly to yield the share price at time t, given the initial price at t = 0 is S0 :
S(t) = S0 exp
³³
µ - D0 - 2
/2
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