Hedging Value-Financial Mathematics

Question

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!

Answer 1

Note that we can write $S_1(\omega)$ as a convex combination of $\alpha$ and $\beta$ with

\begin{equation} S_1(\omega) = \frac{\beta-S_1(\omega)}{\beta-\alpha} \alpha + \frac{S_1(\omega) - \alpha}{\beta-\alpha} \beta \end{equation}

Since $h$ was a convex function then by definition \begin{equation} h(S_1(\omega)) \leq \frac{\beta-S_1(\omega)}{\beta-\alpha} h(\alpha) + \frac{S_1(\omega) - \alpha}{\beta-\alpha} h(\beta) \end{equation}

Taking the $\mathbb{Q}$-expected value, while noting that $E^\mathbb{Q}[S_1] = (1+r)S_0$, \begin{equation} E^\mathbb{Q}[h(S_1)] \leq \frac{\beta-(1+r)S_0}{\beta-\alpha} h(\alpha) + \frac{(1+r)S_0 - \alpha}{\beta-\alpha} h(\beta) \end{equation} which gives you an upper bound on the price of the derivative.