Consider an American call option with a continuously changing strike price X() where dX()/d Define the following

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Consider an American call option with a continuously changing strike price X(τ) where dX(τ)/dτ

and C(S, t; X(t))  max(S  X(t), 0) - C(S, T; X (0)) = max(SX (0), 0).Define the following new set of variables:

un m || S X(t) and F(, T) = C (S, T; X(T)) X (T)

Show that the governing equation for the price of the above American call is given by 

OF T 22 220 = where n(t) = r + ar ditions become F a + n(t). a F a - - n(t)F, 1 ax and r is the riskless

Show that if X(τ) ≥ X(0)e−rτ , then it is never optimal to exercise the American call prematurely. In such a case, show that the value of the above American call is the same as that of a European call with a fixed strike price X(0) (Merton, 1973, Chap. 1).

Show that when the time dependent function η(τ) satisfies the condition ∫τ0 η(s) ds ≥ 0, it is then never optimal to exercise the American call prematurely.

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