Mixing
Sure Thing Arbitrage
$begingroup$
Consider the following model with assets $S_1, S_2$ and three states, and suppose that $r = 10%$
begin{array}{|c|c|c|c|}
hline
n&S_n(0)& S_n(1,omega_1) & S_n(1,omega_2) & S_n(1,omega_3) \ hline
1&4 & 5&6 &3\ hline
2&11 &12 & 9&7\ hline
end{array}
For clarity, time 1 means one year.
Show that there is a sure-thing
arbitrage.
So, what I have done is I tried to find risk-neutral probabilities and I got $Bbb Q = (frac{10}{11},frac{-3}{11},frac{4}{11})$ which has a negative probability so a risk-neutral probability measure does not exist which means we may not be able to replicate the claims. But I am not quite sure how to construct a portfolio for this question.
A friend suggested a portfolio consisting of 11 units of asset 1 and 4 units of asset 2 and then clearly there is arbitrage between the 2. I am just not quite sure about the reasoning for this.
Note, a portfolio $H$ is a sure-thing arbitrage if value $V_0(H) = 0$ and, for every ω ∈ Ω, $V_T (H, ω) > 0$
finance economics actuarial-science
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
$endgroup$
add a comment |
$begingroup$
Consider the following model with assets $S_1, S_2$ and three states, and suppose that $r = 10%$
begin{array}{|c|c|c|c|}
hline
n&S_n(0)& S_n(1,omega_1) & S_n(1,omega_2) & S_n(1,omega_3) \ hline
1&4 & 5&6 &3\ hline
2&11 &12 & 9&7\ hline
end{array}
For clarity, time 1 means one year.
Show that there is a sure-thing
arbitrage.
So, what I have done is I tried to find risk-neutral probabilities and I got $Bbb Q = (frac{10}{11},frac{-3}{11},frac{4}{11})$ which has a negative probability so a risk-neutral probability measure does not exist which means we may not be able to replicate the claims. But I am not quite sure how to construct a portfolio for this question.
A friend suggested a portfolio consisting of 11 units of asset 1 and 4 units of asset 2 and then clearly there is arbitrage between the 2. I am just not quite sure about the reasoning for this.
Note, a portfolio $H$ is a sure-thing arbitrage if value $V_0(H) = 0$ and, for every ω ∈ Ω, $V_T (H, ω) > 0$
finance economics actuarial-science
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
$endgroup$
add a comment |
$begingroup$
Consider the following model with assets $S_1, S_2$ and three states, and suppose that $r = 10%$
begin{array}{|c|c|c|c|}
hline
n&S_n(0)& S_n(1,omega_1) & S_n(1,omega_2) & S_n(1,omega_3) \ hline
1&4 & 5&6 &3\ hline
2&11 &12 & 9&7\ hline
end{array}
For clarity, time 1 means one year.
Show that there is a sure-thing
arbitrage.
So, what I have done is I tried to find risk-neutral probabilities and I got $Bbb Q = (frac{10}{11},frac{-3}{11},frac{4}{11})$ which has a negative probability so a risk-neutral probability measure does not exist which means we may not be able to replicate the claims. But I am not quite sure how to construct a portfolio for this question.
A friend suggested a portfolio consisting of 11 units of asset 1 and 4 units of asset 2 and then clearly there is arbitrage between the 2. I am just not quite sure about the reasoning for this.
Note, a portfolio $H$ is a sure-thing arbitrage if value $V_0(H) = 0$ and, for every ω ∈ Ω, $V_T (H, ω) > 0$
finance economics actuarial-science
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
$endgroup$
Consider the following model with assets $S_1, S_2$ and three states, and suppose that $r = 10%$
begin{array}{|c|c|c|c|}
hline
n&S_n(0)& S_n(1,omega_1) & S_n(1,omega_2) & S_n(1,omega_3) \ hline
1&4 & 5&6 &3\ hline
2&11 &12 & 9&7\ hline
end{array}
For clarity, time 1 means one year.
Show that there is a sure-thing
arbitrage.
So, what I have done is I tried to find risk-neutral probabilities and I got $Bbb Q = (frac{10}{11},frac{-3}{11},frac{4}{11})$ which has a negative probability so a risk-neutral probability measure does not exist which means we may not be able to replicate the claims. But I am not quite sure how to construct a portfolio for this question.
A friend suggested a portfolio consisting of 11 units of asset 1 and 4 units of asset 2 and then clearly there is arbitrage between the 2. I am just not quite sure about the reasoning for this.
Note, a portfolio $H$ is a sure-thing arbitrage if value $V_0(H) = 0$ and, for every ω ∈ Ω, $V_T (H, ω) > 0$
finance economics actuarial-science
finance economics actuarial-science
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
edited Jan 30 at 17:19
ʎpoqou
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
asked Jan 29 at 13:14
ʎpoqouʎpoqou
3581211
3581211
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Take a long position of $11$ units of $A_1$ and a short position of $4$ units of $A_2$. That costs $0$ initially. Verify that this portfolio has positive value in each of the three scenarios.
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and one can verify that, even with this cost, the position is still a perfect arbitrage. The only close one is the third scenario and in that the portfolio is worth $33-28=5>4.4$.
answered Jan 29 at 13:21
lulululu
43.3k25080
$endgroup$
$begingroup$
I guess this is the part that I dont quite understand. When you go long on 11 units of $A_1$ and short 4 units of $A_2$ do we not "lose" the $A_2$ from the portfolio? I thought of long as something remaining in the portfolio and short as something leaving the portfolio. But I guess that is wrong. I suppose at time 1 I should find that for state $omega_1$ the value of the portfolio is $11*5-4*12>0$. Could you clarify how I should think of the portfolio and long and short positions on assets in the portfolio?
$endgroup$
– ʎpoqou
Jan 29 at 13:31
$begingroup$
To go short an asset means that you have sold an asset you do not posses. Here, we short the asset in the first period and make delivery in the second. If the asset has dropped in price over the period, we make money. If the price has gone up we lose money.
$endgroup$
– lulu
Jan 29 at 13:33
$begingroup$
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and you can verify that, even with this cost, the position is still a perfect arbitrage.
$endgroup$
– lulu
Jan 29 at 13:37
$begingroup$
So taking a position at time 0 kind of means locking in what we will do with the asset at time 1?
$endgroup$
– ʎpoqou
Jan 29 at 13:37
1
$begingroup$
Yes. The person who held $A_1$ seeks to get the riskless return on their asset. In real life, borrowing stocks for short sales can be a tricky business.
$endgroup$
– lulu
Jan 29 at 13:41
|
show 3 more comments
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1 Answer
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1 Answer
1
active
oldest
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active
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votes
$begingroup$
Take a long position of $11$ units of $A_1$ and a short position of $4$ units of $A_2$. That costs $0$ initially. Verify that this portfolio has positive value in each of the three scenarios.
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and one can verify that, even with this cost, the position is still a perfect arbitrage. The only close one is the third scenario and in that the portfolio is worth $33-28=5>4.4$.
answered Jan 29 at 13:21
lulululu
43.3k25080
$endgroup$
$begingroup$
I guess this is the part that I dont quite understand. When you go long on 11 units of $A_1$ and short 4 units of $A_2$ do we not "lose" the $A_2$ from the portfolio? I thought of long as something remaining in the portfolio and short as something leaving the portfolio. But I guess that is wrong. I suppose at time 1 I should find that for state $omega_1$ the value of the portfolio is $11*5-4*12>0$. Could you clarify how I should think of the portfolio and long and short positions on assets in the portfolio?
$endgroup$
– ʎpoqou
Jan 29 at 13:31
$begingroup$
To go short an asset means that you have sold an asset you do not posses. Here, we short the asset in the first period and make delivery in the second. If the asset has dropped in price over the period, we make money. If the price has gone up we lose money.
$endgroup$
– lulu
Jan 29 at 13:33
$begingroup$
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and you can verify that, even with this cost, the position is still a perfect arbitrage.
$endgroup$
– lulu
Jan 29 at 13:37
$begingroup$
So taking a position at time 0 kind of means locking in what we will do with the asset at time 1?
$endgroup$
– ʎpoqou
Jan 29 at 13:37
1
$begingroup$
Yes. The person who held $A_1$ seeks to get the riskless return on their asset. In real life, borrowing stocks for short sales can be a tricky business.
$endgroup$
– lulu
Jan 29 at 13:41
|
show 3 more comments
$begingroup$
Take a long position of $11$ units of $A_1$ and a short position of $4$ units of $A_2$. That costs $0$ initially. Verify that this portfolio has positive value in each of the three scenarios.
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and one can verify that, even with this cost, the position is still a perfect arbitrage. The only close one is the third scenario and in that the portfolio is worth $33-28=5>4.4$.
answered Jan 29 at 13:21
lulululu
43.3k25080
$endgroup$
$begingroup$
I guess this is the part that I dont quite understand. When you go long on 11 units of $A_1$ and short 4 units of $A_2$ do we not "lose" the $A_2$ from the portfolio? I thought of long as something remaining in the portfolio and short as something leaving the portfolio. But I guess that is wrong. I suppose at time 1 I should find that for state $omega_1$ the value of the portfolio is $11*5-4*12>0$. Could you clarify how I should think of the portfolio and long and short positions on assets in the portfolio?
$endgroup$
– ʎpoqou
Jan 29 at 13:31
$begingroup$
To go short an asset means that you have sold an asset you do not posses. Here, we short the asset in the first period and make delivery in the second. If the asset has dropped in price over the period, we make money. If the price has gone up we lose money.
$endgroup$
– lulu
Jan 29 at 13:33
$begingroup$
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and you can verify that, even with this cost, the position is still a perfect arbitrage.
$endgroup$
– lulu
Jan 29 at 13:37
$begingroup$
So taking a position at time 0 kind of means locking in what we will do with the asset at time 1?
$endgroup$
– ʎpoqou
Jan 29 at 13:37
1
$begingroup$
Yes. The person who held $A_1$ seeks to get the riskless return on their asset. In real life, borrowing stocks for short sales can be a tricky business.
$endgroup$
– lulu
Jan 29 at 13:41
|
show 3 more comments
$begingroup$
Take a long position of $11$ units of $A_1$ and a short position of $4$ units of $A_2$. That costs $0$ initially. Verify that this portfolio has positive value in each of the three scenarios.
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and one can verify that, even with this cost, the position is still a perfect arbitrage. The only close one is the third scenario and in that the portfolio is worth $33-28=5>4.4$.
answered Jan 29 at 13:21
lulululu
43.3k25080
$endgroup$
Take a long position of $11$ units of $A_1$ and a short position of $4$ units of $A_2$. That costs $0$ initially. Verify that this portfolio has positive value in each of the three scenarios.
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and one can verify that, even with this cost, the position is still a perfect arbitrage. The only close one is the third scenario and in that the portfolio is worth $33-28=5>4.4$.
answered Jan 29 at 13:21
lulululu
43.3k25080
edited Jan 29 at 13:39
answered Jan 29 at 13:21
lulululu
43.3k25080
answered Jan 29 at 13:21
lulululu
43.3k25080
answered Jan 29 at 13:21
lulululu
43.3k25080
43.3k25080
$begingroup$
I guess this is the part that I dont quite understand. When you go long on 11 units of $A_1$ and short 4 units of $A_2$ do we not "lose" the $A_2$ from the portfolio? I thought of long as something remaining in the portfolio and short as something leaving the portfolio. But I guess that is wrong. I suppose at time 1 I should find that for state $omega_1$ the value of the portfolio is $11*5-4*12>0$. Could you clarify how I should think of the portfolio and long and short positions on assets in the portfolio?
$endgroup$
– ʎpoqou
Jan 29 at 13:31
$begingroup$
To go short an asset means that you have sold an asset you do not posses. Here, we short the asset in the first period and make delivery in the second. If the asset has dropped in price over the period, we make money. If the price has gone up we lose money.
$endgroup$
– lulu
Jan 29 at 13:33
$begingroup$
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and you can verify that, even with this cost, the position is still a perfect arbitrage.
$endgroup$
– lulu
Jan 29 at 13:37
$begingroup$
So taking a position at time 0 kind of means locking in what we will do with the asset at time 1?
$endgroup$
– ʎpoqou
Jan 29 at 13:37
1
$begingroup$
Yes. The person who held $A_1$ seeks to get the riskless return on their asset. In real life, borrowing stocks for short sales can be a tricky business.
$endgroup$
– lulu
Jan 29 at 13:41
|
show 3 more comments
$begingroup$
I guess this is the part that I dont quite understand. When you go long on 11 units of $A_1$ and short 4 units of $A_2$ do we not "lose" the $A_2$ from the portfolio? I thought of long as something remaining in the portfolio and short as something leaving the portfolio. But I guess that is wrong. I suppose at time 1 I should find that for state $omega_1$ the value of the portfolio is $11*5-4*12>0$. Could you clarify how I should think of the portfolio and long and short positions on assets in the portfolio?
$endgroup$
– ʎpoqou
Jan 29 at 13:31
$begingroup$
To go short an asset means that you have sold an asset you do not posses. Here, we short the asset in the first period and make delivery in the second. If the asset has dropped in price over the period, we make money. If the price has gone up we lose money.
$endgroup$
– lulu
Jan 29 at 13:33
$begingroup$
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and you can verify that, even with this cost, the position is still a perfect arbitrage.
$endgroup$
– lulu
Jan 29 at 13:37
$begingroup$
So taking a position at time 0 kind of means locking in what we will do with the asset at time 1?
$endgroup$
– ʎpoqou
Jan 29 at 13:37
1
$begingroup$
Yes. The person who held $A_1$ seeks to get the riskless return on their asset. In real life, borrowing stocks for short sales can be a tricky business.
$endgroup$
– lulu
Jan 29 at 13:41
$begingroup$
I guess this is the part that I dont quite understand. When you go long on 11 units of $A_1$ and short 4 units of $A_2$ do we not "lose" the $A_2$ from the portfolio? I thought of long as something remaining in the portfolio and short as something leaving the portfolio. But I guess that is wrong. I suppose at time 1 I should find that for state $omega_1$ the value of the portfolio is $11*5-4*12>0$. Could you clarify how I should think of the portfolio and long and short positions on assets in the portfolio?
$endgroup$
– ʎpoqou
Jan 29 at 13:31
$begingroup$
I guess this is the part that I dont quite understand. When you go long on 11 units of $A_1$ and short 4 units of $A_2$ do we not "lose" the $A_2$ from the portfolio? I thought of long as something remaining in the portfolio and short as something leaving the portfolio. But I guess that is wrong. I suppose at time 1 I should find that for state $omega_1$ the value of the portfolio is $11*5-4*12>0$. Could you clarify how I should think of the portfolio and long and short positions on assets in the portfolio?
$endgroup$
– ʎpoqou
Jan 29 at 13:31
$begingroup$
To go short an asset means that you have sold an asset you do not posses. Here, we short the asset in the first period and make delivery in the second. If the asset has dropped in price over the period, we make money. If the price has gone up we lose money.
$endgroup$
– lulu
Jan 29 at 13:33
$begingroup$
To go short an asset means that you have sold an asset you do not posses. Here, we short the asset in the first period and make delivery in the second. If the asset has dropped in price over the period, we make money. If the price has gone up we lose money.
$endgroup$
– lulu
Jan 29 at 13:33
$begingroup$
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and you can verify that, even with this cost, the position is still a perfect arbitrage.
$endgroup$
– lulu
Jan 29 at 13:37
$begingroup$
Note: it is not clear to me whether or not you are meant to take the cost of the borrow into account. Since rates are $10%$, whoever loaned you $A_1$ should get a $10%$ return on the proceeds from the short sale. That comes to $4.4$ here and you can verify that, even with this cost, the position is still a perfect arbitrage.
$endgroup$
– lulu
Jan 29 at 13:37
$begingroup$
So taking a position at time 0 kind of means locking in what we will do with the asset at time 1?
$endgroup$
– ʎpoqou
Jan 29 at 13:37
$begingroup$
So taking a position at time 0 kind of means locking in what we will do with the asset at time 1?
$endgroup$
– ʎpoqou
Jan 29 at 13:37
1
1
$begingroup$
Yes. The person who held $A_1$ seeks to get the riskless return on their asset. In real life, borrowing stocks for short sales can be a tricky business.
$endgroup$
– lulu
Jan 29 at 13:41
$begingroup$
Yes. The person who held $A_1$ seeks to get the riskless return on their asset. In real life, borrowing stocks for short sales can be a tricky business.
$endgroup$
– lulu
Jan 29 at 13:41
|
show 3 more comments
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