European compound options written on perpetual american options

Perpetual American warrants have been traded on the stock exchanges or over the counter at least since 1929, as is emphasized in several of the “mostread” 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 article, the formulas for perpetual American options, written on dividend-paying stocks, are expressed in a more intuitive way by defining the “distance to exercise.” After proving that perpetual American options follow a geometric Brownian motion, I show the put-call parity by using perpetual first-touch digitals. Finally, I present formulas for European compound options written on perpetual American options. These formulas use the results by Rubinstein and Reiner for barrier options. I highlight that these authors “implicitly” derived the value of finite-maturity first-touch digitals, which generalize the McKean-Samuelson-Merton results for perpetual American options.