A New York firm is offering a new financial instrument called a happy call. It has a

Question:

A New York firm is offering a new financial instrument called a "happy call." It has a payoff function at time $T$ equal to $\max (.5 S, S-K)$, where $S$ is the price of a stock and $K$ is a fixed strike price. You always get something with a happy call. Let $P$ be the price of the stock at time $t=0$ and let $C_{1}$ and $C_{2}$ be the prices of ordinary calls with strike prices $K$ and $2 K$, respectively. The fair price of the happy call is of the form

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