EXERCISE
We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$
Let C be a financial derivative of the form $C:=h(S_1^{1})$ where $h\geq0$ is a convex function.Show that the hedging value $\overline π(C)$ of derivative $C$ is given by the formula
$$\overline π(C)=\dfrac{h(β)}{1+r}\cdot \dfrac{(1+r)S_0^{1}-α}{β-α}+\dfrac{α}{1+r}\cdot \dfrac{β-(1+r)S_0^{1}}{β-α}$$
QUESTIONS
We have that : $$α<S_0^{1}\cdot(1+r)<β$$ and $$0<min_{ω \in Ω} S_1^{1}(ω)<S_0^{1}\cdot(1+r)<max_{ω \in Ω} S_1^{1}(ω)$$
We have also that $$α<β\Longrightarrow min_{ω \in Ω} S_1^{1}(ω)<max_{ω \in Ω} S_1^{1}(ω)$$
We have a financial market with no-arbitrage so we have the form: $$π(C)=E_Q\bigg[\dfrac{c}{1+r}\bigg]<\infty$$ for $Q\subset P$
So,I am new in financial Mathematics and i don't have the experience to understand how to proceed with this data!Can anyone help me with this?How can i use the fact that function $h \geq 0$ is a convex function.Did i miss any data from what the exercise gives me?How can i start so to estimate $\overline π(C)$
I would really appreciate any hints/thorough solution because I don't have any experience in this type of exercise.
Thanks, in advance!