Hedging Value-Financial Mathematics

EXERCISE

We consider a free from arbitrage financial market $$(Ω,F,P,S_0,S_1)$$ with $$α,where $$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 : $$α and $$0

We have also that $$α<β\Longrightarrow min_{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.

• Can you define the “overcompensation value”? – Daneel Olivaw Nov 15 '18 at 13:12
• 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. – Freelunch Nov 15 '18 at 13:26
• 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? – Paris K. Patsogiannis Nov 15 '18 at 13:28
• 1. Please get your accounts merged Rebellos. 2. Please don't cross post math.stackexchange.com/questions/2999677. – LocalVolatility Nov 15 '18 at 15:40
• @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! ;) – Paris K. Patsogiannis Nov 15 '18 at 16:50

Note that we can write $$S_1(\omega)$$ as a convex combination of $$\alpha$$ and $$\beta$$ with
$$$$S_1(\omega) = \frac{\beta-S_1(\omega)}{\beta-\alpha} \alpha + \frac{S_1(\omega) - \alpha}{\beta-\alpha} \beta$$$$
Since $$h$$ was a convex function then by definition $$$$h(S_1(\omega)) \leq \frac{\beta-S_1(\omega)}{\beta-\alpha} h(\alpha) + \frac{S_1(\omega) - \alpha}{\beta-\alpha} h(\beta)$$$$
Taking the $$\mathbb{Q}$$-expected value, while noting that $$E^\mathbb{Q}[S_1] = (1+r)S_0$$, $$$$E^\mathbb{Q}[h(S_1)] \leq \frac{\beta-(1+r)S_0}{\beta-\alpha} h(\alpha) + \frac{(1+r)S_0 - \alpha}{\beta-\alpha} h(\beta)$$$$ which gives you an upper bound on the price of the derivative.