Out performance certificates (also called "sprint certificates", "accelerator certificates", or "speeders") are offered to investors by many European banks as a way of investing in a company's stock. The initial investment equals the company's stock price, S0. If the stock price goes up between time 0 and time T, the investor gains k times the increase at time T where k is a constant greater than 1.0. However, the stock price used to calculate the gain at time T is capped at some maximum level M. If the stock price goes down the investor's loss is equal to the decrease. The investor does not receive dividends.

(a) Show that the outperformance certificate is a package.

(b) Calculate using DerivaGem the value of a one-year outperformance certificate when the stock price is 50 euros, k = 1.5, M = 70 euros, the risk-free rate is 5%, and the stock price volatility is 25%. Dividends equal to 0.5 euros are expected in 2 months, 5 month, 8 months, and 11 months.