5. Barrier options belong to one of four main categories. They can be up-and-out, downand-out, up-and-in, or down-and-in. In each case, there is a specified “barrier,” and when the underlying asset price down or up-crosses this barrier, the option either expires automatically (the “out” case) or comes into life automatically (the “in” case). Consider a European-style up-and-out call written on a stock with a current price of 100 and a volatility of 30%. The stock pays no dividends and follows a geometric price process. The risk-free interest rate is 6% and the option matures in 200 days. The strike price is 110. Finally, the barrier is 120. If the before-maturity stock price exceeds 120, the option automatically expires.

(a) Determine the relevant u and d and the corresponding probability.

(b) Value a call with the same characteristics but without the barrier property.

(c) Value the up-and-out call.

(d) Which option is cheaper?