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6.1.2 Parameters illustrated with VMARKET experiments

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The pararameters of American options are identical to those of the European listed in sect.4.1.2. The payoff, however, is here altered by the possibility of an early exercise and the parametric dependencies will here be illustrated with numerical experiments. Starting with the simplest situation without drift (SpotRate=Dividend=0) the VMARKET applet below shows that the European and the American payoff can sometimes be equal.

VMARKET applet:  press Start/Stop to calculate the price of an American put option in the absence of drifts SpotRate=Dividend=0; compare the payoff with the one obtained for a European option. The black (alt. grey) line shows the present (alt. intrinsic) value of the option V(S,t) for a range of underlying prices 0 < S < 20, as the time runs from the expiry date (T=0) back to three quarter of a year (T-t=0.75) before the expiry date.



Virtual market experiments: volatility
  1. Repeat the experiments comparing the American and European payoff using vanilla Call and also binary options such as VSpread, SuperShr.
  2. Under which circumstance do you get the same American & European payoff?

The experiments with vanilla call and put options show that American and European options have the same value in the absence of drifts; dramatic differences do however appear for super-share and other binary options, for which the terminal payoff $ \Lambda(S)$ is convex.

To study how the American payoff is modified in the presence of drifts, let us perform a second series of experiments setting the volatility to zero and increasing the spot rate and the dividend yield parameters to unrealistically large values.

VMARKET applet:  press Start/Stop to calculate the price of an American call option in the absence of volatility and unrealistically large drifts parameters SpotRate=0.6, Dividend=0.4 -- corresponding to 60% spot rate and 40% dividend yield. The black (alt. grey) line shows the present (alt. intrinsic) value of the option V(S,t) for a range of underlying prices 0 < S < 20, as the time runs from the expiry date (T=0) back to three quarter of a year (T-t=0.75) before the expiry date.



Virtual market experiments: spot rate and dividend yield
  1. Select Put option and compare the American and European exercise styles; explain what you observe.
  2. Switch back to American and Call option, and vary the SpotRate=0-0.6 and the Dividend=0-0.6 to determine under which circumstances the payoff is made of three (rather than two) segments.
  3. Try to list all the contracts where the American and the European exercise style results in the same payoff.

Rather than repeating conclusions similar to those that have been obtained from experiments with European options, we encourage the reader to review sect.4.1.2 and develop an intuition for how the volatility, the spot rate and the dividend yield affects the payoff for both European and American options.

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