Prove that $overline{pi}(C) = S_0^1$ if $text{ess} sup S_1^1 = infty$ and $text{ess}inf S_1^1 = 0$
Exercise :
We consider a financial market of one period $(Omega,mathcal{F},mathbb{P},S^0,S^1)$, where the sample space $Omega$ is finite and the $sigma$-algebra $mathcal{F} = 2^Omega$. Furthermore, $S^0$ is the risk-free asset with initial value $S_0^0 = 1$ at time $t=0$ and interest rate $r>-1$ (which means that $S_1^0 = 1+r)$ and $S^1$ is the risky asset with initial value $S_0^1 >0$ at time $t=0$ and value $S_1^1$ at time $t=1$ which is a random variable. Furthermore, we consider a buying right with payout $C=(S_1^1 - K)^+$ with exercise value $K>0$ and maturity time $T=1$. Let $overline{pi}(C)$ be the no-arbitrage upper bound for $C$. Show that :
$$overline{pi}(C) leq S_0^1$$
and that the equality holds
$$overline{pi}(C) = S_0^1$$
if it is also assumed that $text{ess} sup S_1^1 = infty$ and $text{ess}inf S_1^1 = 0$.
Attempt :
We know, that :
$$overline{pi}(C) = sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{C}{1+r}bigg]=sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg]$$
But, note that :
$$mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg] leq mathbb{E}_mathbb{Q}bigg[frac{S_1^1}{1+r}bigg] = frac{mathbb{E}_mathbb{Q}(S_1^1)}{1+r} = S_0^1 $$
Thus, indeed it is :
$$overline{pi}(C) leq S_0^1$$
Question : I would like to request some help proving the equality. I am really not very familiar with essential infimum and supremum and I would really appreciate a thorough elaboration or explanation over the specific example.
probability-theory discrete-mathematics stochastic-processes martingales finance
asked Nov 20 at 18:01
Rebellos
14.4k31245
add a comment |
Exercise :
We consider a financial market of one period $(Omega,mathcal{F},mathbb{P},S^0,S^1)$, where the sample space $Omega$ is finite and the $sigma$-algebra $mathcal{F} = 2^Omega$. Furthermore, $S^0$ is the risk-free asset with initial value $S_0^0 = 1$ at time $t=0$ and interest rate $r>-1$ (which means that $S_1^0 = 1+r)$ and $S^1$ is the risky asset with initial value $S_0^1 >0$ at time $t=0$ and value $S_1^1$ at time $t=1$ which is a random variable. Furthermore, we consider a buying right with payout $C=(S_1^1 - K)^+$ with exercise value $K>0$ and maturity time $T=1$. Let $overline{pi}(C)$ be the no-arbitrage upper bound for $C$. Show that :
$$overline{pi}(C) leq S_0^1$$
and that the equality holds
$$overline{pi}(C) = S_0^1$$
if it is also assumed that $text{ess} sup S_1^1 = infty$ and $text{ess}inf S_1^1 = 0$.
Attempt :
We know, that :
$$overline{pi}(C) = sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{C}{1+r}bigg]=sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg]$$
But, note that :
$$mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg] leq mathbb{E}_mathbb{Q}bigg[frac{S_1^1}{1+r}bigg] = frac{mathbb{E}_mathbb{Q}(S_1^1)}{1+r} = S_0^1 $$
Thus, indeed it is :
$$overline{pi}(C) leq S_0^1$$
Question : I would like to request some help proving the equality. I am really not very familiar with essential infimum and supremum and I would really appreciate a thorough elaboration or explanation over the specific example.
probability-theory discrete-mathematics stochastic-processes martingales finance
asked Nov 20 at 18:01
Rebellos
14.4k31245
add a comment |
Exercise :
We consider a financial market of one period $(Omega,mathcal{F},mathbb{P},S^0,S^1)$, where the sample space $Omega$ is finite and the $sigma$-algebra $mathcal{F} = 2^Omega$. Furthermore, $S^0$ is the risk-free asset with initial value $S_0^0 = 1$ at time $t=0$ and interest rate $r>-1$ (which means that $S_1^0 = 1+r)$ and $S^1$ is the risky asset with initial value $S_0^1 >0$ at time $t=0$ and value $S_1^1$ at time $t=1$ which is a random variable. Furthermore, we consider a buying right with payout $C=(S_1^1 - K)^+$ with exercise value $K>0$ and maturity time $T=1$. Let $overline{pi}(C)$ be the no-arbitrage upper bound for $C$. Show that :
$$overline{pi}(C) leq S_0^1$$
and that the equality holds
$$overline{pi}(C) = S_0^1$$
if it is also assumed that $text{ess} sup S_1^1 = infty$ and $text{ess}inf S_1^1 = 0$.
Attempt :
We know, that :
$$overline{pi}(C) = sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{C}{1+r}bigg]=sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg]$$
But, note that :
$$mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg] leq mathbb{E}_mathbb{Q}bigg[frac{S_1^1}{1+r}bigg] = frac{mathbb{E}_mathbb{Q}(S_1^1)}{1+r} = S_0^1 $$
Thus, indeed it is :
$$overline{pi}(C) leq S_0^1$$
Question : I would like to request some help proving the equality. I am really not very familiar with essential infimum and supremum and I would really appreciate a thorough elaboration or explanation over the specific example.
probability-theory discrete-mathematics stochastic-processes martingales finance
asked Nov 20 at 18:01
Rebellos
14.4k31245
Exercise :
We consider a financial market of one period $(Omega,mathcal{F},mathbb{P},S^0,S^1)$, where the sample space $Omega$ is finite and the $sigma$-algebra $mathcal{F} = 2^Omega$. Furthermore, $S^0$ is the risk-free asset with initial value $S_0^0 = 1$ at time $t=0$ and interest rate $r>-1$ (which means that $S_1^0 = 1+r)$ and $S^1$ is the risky asset with initial value $S_0^1 >0$ at time $t=0$ and value $S_1^1$ at time $t=1$ which is a random variable. Furthermore, we consider a buying right with payout $C=(S_1^1 - K)^+$ with exercise value $K>0$ and maturity time $T=1$. Let $overline{pi}(C)$ be the no-arbitrage upper bound for $C$. Show that :
$$overline{pi}(C) leq S_0^1$$
and that the equality holds
$$overline{pi}(C) = S_0^1$$
if it is also assumed that $text{ess} sup S_1^1 = infty$ and $text{ess}inf S_1^1 = 0$.
Attempt :
We know, that :
$$overline{pi}(C) = sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{C}{1+r}bigg]=sup_{mathbb{Q} in mathcal{P}}mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg]$$
But, note that :
$$mathbb{E}_mathbb{Q}bigg[frac{(S_1^1 - K)^+}{1+r}bigg] leq mathbb{E}_mathbb{Q}bigg[frac{S_1^1}{1+r}bigg] = frac{mathbb{E}_mathbb{Q}(S_1^1)}{1+r} = S_0^1 $$
Thus, indeed it is :
$$overline{pi}(C) leq S_0^1$$
Question : I would like to request some help proving the equality. I am really not very familiar with essential infimum and supremum and I would really appreciate a thorough elaboration or explanation over the specific example.
probability-theory discrete-mathematics stochastic-processes martingales finance
probability-theory discrete-mathematics stochastic-processes martingales finance
asked Nov 20 at 18:01
Rebellos
14.4k31245
asked Nov 20 at 18:01
Rebellos
14.4k31245
asked Nov 20 at 18:01
Rebellos
14.4k31245
asked Nov 20 at 18:01
Rebellos
14.4k31245
asked Nov 20 at 18:01
Rebellos
14.4k31245
14.4k31245
add a comment |
add a comment |
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