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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 $h\geq0$ 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!

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  • $\begingroup$ Can you define the “overcompensation value”? $\endgroup$ – Daneel Olivaw Nov 15 '18 at 13:12
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    $\begingroup$ Hint: use that $h(\lambda x_1 + (1-\lambda)x_2) \leq \lambda h(x_1) + (1-\lambda)h(x_2)$ for some interval $[x_1, x_2]$ and $\lambda \in [0,1]$. Now try to figure out what $\lambda$ and $[x_1, x_2]$ should be to solve your problem. $\endgroup$ – Freelunch Nov 15 '18 at 13:26
  • $\begingroup$ yeah i mean hedging price!I think that it's famous with this name!So my exercise want me to find the hedgin price $\overline π(C)$.Any ideas? $\endgroup$ – Paris K. Patsogiannis Nov 15 '18 at 13:28
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    $\begingroup$ 1. Please get your accounts merged Rebellos. 2. Please don't cross post math.stackexchange.com/questions/2999677. $\endgroup$ – LocalVolatility Nov 15 '18 at 15:40
  • $\begingroup$ @LocalVolatility I Wish i could write,and answer an exercise as Rebellos but unfortunately i can't.He is more familiar with this site!Actually , Rebellos is a friend of mine and we are colleagues! ;) $\endgroup$ – Paris K. Patsogiannis Nov 15 '18 at 16:50
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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.

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  • $\begingroup$ Great answer!Thanks @Freelunch!You helped me a lot!So we have to work with the convex functions and martingale! $\endgroup$ – Paris K. Patsogiannis Nov 15 '18 at 16:52

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