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The price of a traded security follows a geometric Brownian motion with drift 0.06 and volatility 0.4. Its current price is 40. A brokerage firm is offering, at cost C, an investment that will pay 100 at the end of 1 year either if the price of the security at 6 months is at least 42 or if the price of the security at 1 year is at least 5 percent above its price at 6 months. That is, the payoff occurs if either S(0.5) ≥ 42 or S(1) > 1.05 S(0.5). The continuously compounded interest rate is 0.06.
(a) If this investment is not to give rise to an arbitrage, what is C?
(b) What is the probability the investment makes money for its buyer?
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