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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 $hgeq0$ is a convex function.Show that the overcompensation 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 $$α<βLongrightarrow min_{ω in Ω} S_1^{1}(ω)<max_{ω in Ω} S_1^{1}(ω)$$

We have a financial market with no-arbitrage so we have the form:
$$π(C)=E_Qbigg[dfrac{c}{1+r}bigg]<infty$$ for $Qsubset P$

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!

$hle L$ where $L$ is the linear function joining $(alpha,h(alpha))$ to $(beta,h(beta))$. Then calculate the discounted expected payoff of $L$.