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EXERCISE

We consider a financial market of one period with two stocks $(S_0,S_1)$.
We further consider a purchase option with repayment $C = (S_1^{1} - K)^+$
with a value of $K> 0$ and maturing time $T = 1$. Let $underline{π}(C)$,$overline{π}(C)$ the lower and upper no-arbitrage barriers for the value of $C$.

a)Show that the following inequality applies
$$underline{π}(C)geq bigg(S_1^{1} - dfrac{k}{1+r}bigg)^+$$

Please also show that the following equality applies
$$underline{π}(C)= bigg(S_1^{1} - dfrac {k}{1+r}bigg)^+$$

if it is further assumed that:
$$P[S_1^{1}=S_0^{1} cdot(1+r)]> 0$$

Hint : Use the probability measure $$Q[·]: = P [·|S_1^{1} =S_0^{1}cdot (1+r)]$$

b) Also show that the following inequality applies
$$underline {π}(C)leq S_0^{1}$$

and that the following equality applies
$$overline{π}(C)=S_0^{1}$$

assuming that $rm ess sup S_1^{1}=infty $ and that $rm ess inf S_1^{1}=0$.

ATTEMPT

a)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,$$underline{π}(C)geq inf_{Q in P}E_Qbigg[dfrac{c}{1+r}bigg]=inf_{Q in P}E_Qbigg[dfrac{S_1^{1}-k}{1+r}bigg]=inf_{Q in P}dfrac{1} {1+r}E_Q[S_1^{1}-k]$$

UPDATE

Ι think that i have to use put-call-parity equation:

$$C_t-P_t=S_t-kB(t,T)$$
with $B(t,T)=dfrac{S_0^{0}} {S_0^{1}}=dfrac{1}{1+r}$

Then we rearrange the equation so that C is on the left hand side and use P_t>0 (using 1/(1+r) for the discount factor).So,we have $$C_t=P_t+S_t-k dfrac{1}{1+r}$$

Can anyone help me from here with a thorough solution so to find the question a)?

I don't know if this is right but i want to know if I am in the right way to solve the problem.

Also,I would like to ask if $min=inf$ in this problem

For question b) i know that i have to take this theorem

and this remark but i don't know even how to start!

I would really appreciate any hints/thorough solution because it's the first problem that I have to solve and I don't have any experience in this type of exercise and generally it's my first try in the field of financial mathematics

Thanks, in advance!

$mathbf{(a)}$ From the extension of the Theorem hinted, the lower no-arbitrage bound is :
$$underline{pi}(C) = inf_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{C}{1+r}bigg]=inf_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg]$$

The Jenses inequality in context of probability theory, says thati f $X$ is a random variable and $varphi$ is a convex function, then it is : $varphibig(mathbb{E}[X]big) leq mathbb{E}[varphi(X)]$. The function $f(x) = (x-K)^+, ; K>0$ is convex, thus :

$$mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg] geqfrac{1}{1+r}big[mathbb{E}_mathbb{Q}(S_1^1 - K)big]^+ $$
$$= frac{1}{1+r}big[mathbb{E}_mathbb{Q}(S_1^1) - mathbb{E}_mathbb{Q}(K)big]^+= frac{1}{1+r}bigg[S_0^1(1+r) - Kbigg]^+$$
$$implies$$
$$underline{pi}(C) geq bigg(S_0^1 - frac{K}{1+r}bigg)^+$$
where the probability measure $mathbb{Q}[cdot] := mathbb{P}[cdot | S_1^1 = S_0^1(1+r)]$ was used.

Now if $mathbb{P}[S_1^1 = S_0^1(1+r)] >0$, this means that at some point it will be $S_1^1 = S_0^1(1+r)$ and thus :

$$mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg] = mathbb{E}_mathbb{Q}bigg[frac{(S_0^1(1+r) - K)^+}{1+r}bigg] = mathbb{E}_mathbb{Q}bigg[bigg(S_0^1 - frac{K}{1+r}bigg)^+bigg]$$
which means that the equality will hold, since $S_0^1$ is just a constant $>0$ :
$$underline{pi}(C) = bigg(S_0^1 - frac{K}{1+r}bigg)^+$$

For part $mathbf{(b)}$, again from the extension of the Theorem hinted, we have :
$$overline{pi}(C) = sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{C}{1+r}bigg]=sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg]$$

But, note that simply it is :

$$mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg] leq mathbb{E}_mathbb{Q}bigg[frac{S_1^1}{1+r}bigg] = frac{mathbb{E}_mathbb{Q}(S_1^1)}{1+r} = S_0^1 $$
$$implies$$
$$overline{pi}(C) leq S_0^1 $$
where the probability measure $mathbb{Q}[cdot] := mathbb{P}[cdot | S_1^1 = S_0^1(1+r)]$ was used.

Can you figure out a similar approach for the last part now ?

(a) Use put-call parity and the fact that the put price must be +ve. (b) Call price cannot exceed the value of the stock, as otherwise buying the stock and selling the call would generate an arbitrage profit.