1. Consider a perpetual American put option V (S), which satisfics the Cauchy-Euler problem dV dS2 dV dS dV dS where S > 0 is the spot price. E > 0 is the strike, S > 0 is the optimal exercise boundary, > 0 is the constant volatility, r > 0 is the constant interest rate and D is the dividend yield. By assuming a solution of the form V (S) = Srn show that Show that one of the roots of this equation, m, is always negative. Obtain the option price V (S) and hence deduce that for S>S* Finally, show that 1. Consider a perpetual American put option V (S), which satisfics the Cauchy-Euler problem dV dS2 dV dS dV dS where S > 0 is the spot price. E > 0 is the strike, S > 0 is the optimal exercise boundary, > 0 is the constant volatility, r > 0 is the constant interest rate and D is the dividend yield. By assuming a solution of the form V (S) = Srn show that Show that one of the roots of this equation, m, is always negative. Obtain the option price V (S) and hence deduce that for S>S* Finally, show that