A happy Call option is a Call option with payoff \(\max (\alpha S, S-K)\). So we always get something with a happy Call! If \(C_{1}\) and \(C_{2}\) are the prices of two call options with strikes \(n K\) and \(m K\), show that the price of a happy Call is given by the formula \[C_{H}=a_{1} C_{1}+a_{2} C_{2}+a_{3} S\] and find the constants \(a_{1}, a_{2}, 2_{3}\).