Mixing
Risk-neutral probability measures
$begingroup$
Consider a market with $Omega= (omega_1, omega_2, omega_3)$, $r = 0$ and one asset $S$.
Suppose that $S(0) = 2$ and $S$ has claim $bar S = (1, 3, 3)$ at time 1.
Find all the risk-neutral probability measures on $Omega$
What I did for this question is construct a similar table as in Sure Thing Arbitrage. I got a system of solutions which gave
$Bbb Q=(frac{1}{2}, p, frac{1}{2}-p)$ as the set of all possible risk neutral probability measures. But again, I am not sure if I can take the claims at time 1 to be the same at the price/value of the asset at time 1.
finance economics actuarial-science
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
$endgroup$
add a comment |
$begingroup$
Consider a market with $Omega= (omega_1, omega_2, omega_3)$, $r = 0$ and one asset $S$.
Suppose that $S(0) = 2$ and $S$ has claim $bar S = (1, 3, 3)$ at time 1.
Find all the risk-neutral probability measures on $Omega$
What I did for this question is construct a similar table as in Sure Thing Arbitrage. I got a system of solutions which gave
$Bbb Q=(frac{1}{2}, p, frac{1}{2}-p)$ as the set of all possible risk neutral probability measures. But again, I am not sure if I can take the claims at time 1 to be the same at the price/value of the asset at time 1.
finance economics actuarial-science
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
$endgroup$
$begingroup$
A claim represents the payoff of the asset in each state of the world; in this case, the claim tells you how much you will earn given three possible outcomes. Economically speaking, the asset can be thought of a bond that pays well in two states (where the claim is 3) and pays badly in one state (where the claim in 1). Then, you can construct the risk-neutral probabilities by solving the following system of equations.
$endgroup$
– Waie
Jan 30 at 12:33
$begingroup$
So is my thinking for the question correct? As in, I can take the claim for each state to be the value of the asset at that time in that state?
$endgroup$
– ʎpoqou
Jan 30 at 13:34
$begingroup$
I think so; however, when you calculate the simultaneous equations system that is, x+3y+3z=1 and x+y+z=1, you get that x=0.5 and y=0.5-z. So, you will end up with multiple risk-neutral probabilities as y+z=0.5 and so you have that: 0<y<0.5, 0<z<0.5 and y+z=0.5. In fact, let's assume that you choose arbitrarily that y=1/8 and z=3/8 (which satisfies the conditions), then you still have valid risk-neutral probability measures. Multiple risk-neutral probabilities mean incomplete markets.
$endgroup$
– Waie
Jan 30 at 17:26
add a comment |
$begingroup$
Consider a market with $Omega= (omega_1, omega_2, omega_3)$, $r = 0$ and one asset $S$.
Suppose that $S(0) = 2$ and $S$ has claim $bar S = (1, 3, 3)$ at time 1.
Find all the risk-neutral probability measures on $Omega$
What I did for this question is construct a similar table as in Sure Thing Arbitrage. I got a system of solutions which gave
$Bbb Q=(frac{1}{2}, p, frac{1}{2}-p)$ as the set of all possible risk neutral probability measures. But again, I am not sure if I can take the claims at time 1 to be the same at the price/value of the asset at time 1.
finance economics actuarial-science
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
$endgroup$
Consider a market with $Omega= (omega_1, omega_2, omega_3)$, $r = 0$ and one asset $S$.
Suppose that $S(0) = 2$ and $S$ has claim $bar S = (1, 3, 3)$ at time 1.
Find all the risk-neutral probability measures on $Omega$
What I did for this question is construct a similar table as in Sure Thing Arbitrage. I got a system of solutions which gave
$Bbb Q=(frac{1}{2}, p, frac{1}{2}-p)$ as the set of all possible risk neutral probability measures. But again, I am not sure if I can take the claims at time 1 to be the same at the price/value of the asset at time 1.
finance economics actuarial-science
finance economics actuarial-science
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
asked Jan 29 at 13:20
ʎpoqouʎpoqou
3581211
3581211
$begingroup$
A claim represents the payoff of the asset in each state of the world; in this case, the claim tells you how much you will earn given three possible outcomes. Economically speaking, the asset can be thought of a bond that pays well in two states (where the claim is 3) and pays badly in one state (where the claim in 1). Then, you can construct the risk-neutral probabilities by solving the following system of equations.
$endgroup$
– Waie
Jan 30 at 12:33
$begingroup$
So is my thinking for the question correct? As in, I can take the claim for each state to be the value of the asset at that time in that state?
$endgroup$
– ʎpoqou
Jan 30 at 13:34
$begingroup$
I think so; however, when you calculate the simultaneous equations system that is, x+3y+3z=1 and x+y+z=1, you get that x=0.5 and y=0.5-z. So, you will end up with multiple risk-neutral probabilities as y+z=0.5 and so you have that: 0<y<0.5, 0<z<0.5 and y+z=0.5. In fact, let's assume that you choose arbitrarily that y=1/8 and z=3/8 (which satisfies the conditions), then you still have valid risk-neutral probability measures. Multiple risk-neutral probabilities mean incomplete markets.
$endgroup$
– Waie
Jan 30 at 17:26
add a comment |
$begingroup$
A claim represents the payoff of the asset in each state of the world; in this case, the claim tells you how much you will earn given three possible outcomes. Economically speaking, the asset can be thought of a bond that pays well in two states (where the claim is 3) and pays badly in one state (where the claim in 1). Then, you can construct the risk-neutral probabilities by solving the following system of equations.
$endgroup$
– Waie
Jan 30 at 12:33
$begingroup$
So is my thinking for the question correct? As in, I can take the claim for each state to be the value of the asset at that time in that state?
$endgroup$
– ʎpoqou
Jan 30 at 13:34
$begingroup$
I think so; however, when you calculate the simultaneous equations system that is, x+3y+3z=1 and x+y+z=1, you get that x=0.5 and y=0.5-z. So, you will end up with multiple risk-neutral probabilities as y+z=0.5 and so you have that: 0<y<0.5, 0<z<0.5 and y+z=0.5. In fact, let's assume that you choose arbitrarily that y=1/8 and z=3/8 (which satisfies the conditions), then you still have valid risk-neutral probability measures. Multiple risk-neutral probabilities mean incomplete markets.
$endgroup$
– Waie
Jan 30 at 17:26
$begingroup$
A claim represents the payoff of the asset in each state of the world; in this case, the claim tells you how much you will earn given three possible outcomes. Economically speaking, the asset can be thought of a bond that pays well in two states (where the claim is 3) and pays badly in one state (where the claim in 1). Then, you can construct the risk-neutral probabilities by solving the following system of equations.
$endgroup$
– Waie
Jan 30 at 12:33
$begingroup$
A claim represents the payoff of the asset in each state of the world; in this case, the claim tells you how much you will earn given three possible outcomes. Economically speaking, the asset can be thought of a bond that pays well in two states (where the claim is 3) and pays badly in one state (where the claim in 1). Then, you can construct the risk-neutral probabilities by solving the following system of equations.
$endgroup$
– Waie
Jan 30 at 12:33
$begingroup$
So is my thinking for the question correct? As in, I can take the claim for each state to be the value of the asset at that time in that state?
$endgroup$
– ʎpoqou
Jan 30 at 13:34
$begingroup$
So is my thinking for the question correct? As in, I can take the claim for each state to be the value of the asset at that time in that state?
$endgroup$
– ʎpoqou
Jan 30 at 13:34
$begingroup$
I think so; however, when you calculate the simultaneous equations system that is, x+3y+3z=1 and x+y+z=1, you get that x=0.5 and y=0.5-z. So, you will end up with multiple risk-neutral probabilities as y+z=0.5 and so you have that: 0<y<0.5, 0<z<0.5 and y+z=0.5. In fact, let's assume that you choose arbitrarily that y=1/8 and z=3/8 (which satisfies the conditions), then you still have valid risk-neutral probability measures. Multiple risk-neutral probabilities mean incomplete markets.
$endgroup$
– Waie
Jan 30 at 17:26
$begingroup$
I think so; however, when you calculate the simultaneous equations system that is, x+3y+3z=1 and x+y+z=1, you get that x=0.5 and y=0.5-z. So, you will end up with multiple risk-neutral probabilities as y+z=0.5 and so you have that: 0<y<0.5, 0<z<0.5 and y+z=0.5. In fact, let's assume that you choose arbitrarily that y=1/8 and z=3/8 (which satisfies the conditions), then you still have valid risk-neutral probability measures. Multiple risk-neutral probabilities mean incomplete markets.
$endgroup$
– Waie
Jan 30 at 17:26
add a comment |
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$begingroup$
A claim represents the payoff of the asset in each state of the world; in this case, the claim tells you how much you will earn given three possible outcomes. Economically speaking, the asset can be thought of a bond that pays well in two states (where the claim is 3) and pays badly in one state (where the claim in 1). Then, you can construct the risk-neutral probabilities by solving the following system of equations.
$endgroup$
– Waie
Jan 30 at 12:33
$begingroup$
So is my thinking for the question correct? As in, I can take the claim for each state to be the value of the asset at that time in that state?
$endgroup$
– ʎpoqou
Jan 30 at 13:34
$begingroup$
I think so; however, when you calculate the simultaneous equations system that is, x+3y+3z=1 and x+y+z=1, you get that x=0.5 and y=0.5-z. So, you will end up with multiple risk-neutral probabilities as y+z=0.5 and so you have that: 0<y<0.5, 0<z<0.5 and y+z=0.5. In fact, let's assume that you choose arbitrarily that y=1/8 and z=3/8 (which satisfies the conditions), then you still have valid risk-neutral probability measures. Multiple risk-neutral probabilities mean incomplete markets.
$endgroup$
– Waie
Jan 30 at 17:26