European Compound Options Written on Perpetual American Options

Posted: 17 Nov 2010

See all articles by Gaia Barone

Gaia Barone

School of Business, National College of Ireland

Date Written: November 1, 2010


Perpetual American warrants have been traded on the stock exchanges or over the counter at least since 1929, as it is emphasized in several of the "most-read" finance books. The first rational model to evaluate perpetual American call options appeared as early as 1965, when McKean (Samuelson’s Appendix) derived a closed-form valuation formula under the now-standard hypothesis of a geometric Brownian motion for the price of the underlying stock. A formula for perpetual American put options was later derived by Merton (1973) for the no-dividend case.

In this paper, I review the formulas for perpetual American call and put options, written on dividend paying stocks, and show that they can be expressed in a more intuitive way by defining the "distance to exercise". Then, by using perpetual first-touch digitals, I derive a put-call parity for perpetual American options. Finally, I present formulas for European compound options written on perpetual American options. These formulas use the results for barrier options obtained by Rubinstein and Reiner. I highlight that these authors "implicitly" derived the value of finite-maturity first-touch digitals, which generalize the McKean-Samuelson-Merton formulas for perpetual American options.

Keywords: perpetual options, compound options, barrier options, first-touch digitals, Greeks, put-call parity, distance to exercise, time to exercise

JEL Classification: G13

Suggested Citation

Barone, Gaia, European Compound Options Written on Perpetual American Options (November 1, 2010). Journal of Derivatives, Vol. 20, No. 3: pp. 61-74, Spring 2013, Available at SSRN: or

Gaia Barone (Contact Author)

School of Business, National College of Ireland ( email )

Mayor Street
Dublin, 1

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