Assume the risk-free bond B_t and the (non-dividend paying) stock S_t follow the dynamics of the Black & Scholes model (with interest rate r, stock drift and volatility ).

(a)Consider the American binary option with maturity T and payoff at exercise time

(S)=1(whenS>=K)(S)=0(whenS<K) equal to

Write the free boundary problem for the option. What is the exercise boundary? Explain why.

(b)Consider the perpetual American binary option with the same payoff at exercise time. Write the corresponding free boundary problem.

(c)Find the price of the perpetual binary option by solving the problem in part (b).