Shout options

6. Shout options can be structured so that holders of this contract have one op- portunity to \"shout\" or lock in profits. This allows the holders to continue to benefit from positive market movements without the possibility of losing already locked-in profits. Suppose the underlying stock price is S(t),t > 0, the strike price of the shout option is K, and the maturity date of the shout option is 7'. The option holder has an opportunity to \"shout\" at any stopping time 7 before maturity. When he shouts, he locks in the profit S(7) K, i.e., receives the cash S(7) K at maturity for sure. After shouts, the shout option becomes a usual European with the strike price reset to be S(7). In other words, the option holder still benefits from the positive market movements after locking some minimum profit. Therefore, the payoff from the option at maturity is S(7) K + max(S(T) S(1),0) = max(S(T) K, S(7) K). Because an shout time is involved, the shout option can be valued by the same procedure for valuing American option. Consider a 3-month shout option on a stock with a strike price of $50. The stock price is $51, the risk-free rate is 3% per annum, and the volatility is 25% per annum. The stock pays no dividend. Use a three-step binomial tree to calculate the option price