I would like to solve below exercise but got stuck. Please kindly help me. The exercise is taken from the actuarial exam organized in my country, and my translation of the exercise into English is not perfect, however I hope it's understood.
Suppose that we are given a stock A on the Black-Scholes market. Assume that we are able to buy and sell European call options for A. We denote the price of A at the moment $T$ by $S_T$. At the moment $t=0$ we are building the portfolio in such a way that the pay-off at the moment $t=1$ is the following: $W_{1}=\left\{ \begin{array}{cc} 0 & \text{if }S_{1}\notin \lbrack 50,250] \\ S_{1}-50 & \text{if }S_{1}\in \lbrack 50,100] \\ 50 & \text{if }S_{1}\in \lbrack 100,200] \\ 250-S_{1} & \text{if }S_{1}\in \lbrack 200,250]% \end{array}% \right. $
Calculate the price $C$ of the portfolio at $T=0$ under the assumptions that $r=2\%$, $\sigma=2\%$, $S_0 = 150$.
My work so far: as $W_1=(S_1-50)^+ -(S_1-100)^+ - (S_1 - 200)^+ +(S_1 - 250)^+$, the price $C$ I am looking for is equal(?) to $C=C(50)+C(250)-C(100)-C(200)$, where $C(K)$ denotes the European call option for $A$ with a strike $K$. I suppose that the last equality is wrong and on the right side we have $\infty - \infty$(?) Could you please help? The right answer is approximately 49 - that's all I know.