Consider two time points, $t = 0,1$ , and let there be three assets $S^0$ , $S^1$ and $S^2$ ; we write $S_t^i$ for the value of the $i$ th asset at time $t$ (all prices quoted in $\mathcal{L}$ ). While $S_0^0 = 4/3$ and $S_0^1 = S_0^2 = 10/3$ , the value of the assets at time $t = 1$ depend on the temperature at that time in the following way, for some $x,y \in \mathbb{R}$ :

S _ {1} ^ {0} = \left\{ \begin{array}{l l} 1 & \text {if warm} \\ 2 & \text {if cold} \end{array} \right. \quad S _ {1} ^ {1} = \left\{ \begin{array}{l l} 3 & \text {if warm} \\ 4 & \text {if cold} \end{array} \right. \quad \text {and} \quad S _ {1} ^ {2} = \left\{ \begin{array}{l l} x & \text {if warm} \\ y & \text {if cold} \end{array} \right.

It will be either warm or cold and both weather scenarios have a positive probability of occurring. Consider also a financial derivative which ensures the holder a payoff, at time $t = 1$ , according to the following:

X = \left\{ \begin{array}{l l} {\left(S _ {1} ^ {0} + S _ {1} ^ {2}\right) ^ {2}} & \text {if warm} \\ {\left(S _ {1} ^ {1} + S _ {1} ^ {2}\right) ^ {2}} & \text {if cold} \end{array} \right.

Address the following: