Valuation of American Call Options

RALPH VILLIGER provides an analytical solution for American call options assuming proportional dividends.

The purpose of this paper is to provide an analytical solution for American call options assuming proportional dividends. Proportional dividends are more realistic for long-term options than absolute dividends and the formula does not have the flaws known from absolute dividend formulae.

The holder of an American call option has the right but not the obligation to buy the underlying share at a fixed price, the strike, any time until expiry of the option. In the case of no dividends it would not make any sense to execute an American call option prematurely because the option could always be sold at a higher price than the inner value of the option, the difference
being the time value. Nevertheless an early execution can be advisable if the share pays dividends. At payment the share price decreases by the dividend. If the dividend is higher than the time value of the option after the payment then the holder should execute the option.

Roll (1977), Geske (1979, 1981), and Whaley (1981) provided an analytical solution for the case of known absolute dividends. As a shortcoming we can only assume the ex-dividend share price (i.e. the share price net of the discounted dividend payment) to follow a geometric Brownian motion. For options with long maturities, and consequently several dividend payments, this corresponds to a considerable reduction of the actual volatility, because only the ex-dividend share price is assumed to fluctuate. Several authors tried to adjust the volatility (Chriss (1997) or Haug and Haug (1998)). In an extreme case, assuming the dividend discount model (i.e. the share price equals the sum of all discounted dividend payments) the share price for a perpetual American call wouldn’t even fluctuate anymore.

Of course this is an inadmissible exaggeration of the Roll-GeskeWhaley model; its authors only assumed the one dividend payment case as a realistic scenario. Nevertheless the example of a perpetual call exhibits the problem of the ex-dividend share price. As a second drawback we repeat Geske’s doubt (1979) that dividends are difficult to forecast in such precise terms.

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