Maximum of $frac{1}{1+r}mathbb{E}_{mathbb{Q}_v}big[(K-S_1^1)^+big]$
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Given that $-1<b<0<h$ and $rin (b,h)$, $S_1^1 = S_0^1 cdot (1+R)$ where $S_0^1 >0$ is a positive constant and $R(b) = b, R(0) = 0$ and $R(h) = h$ while $mathbb{Q}(b) = u, mathbb{Q}(0) = v, mathbb{Q}(h) = w$. Also, it is proven (correctly) that :
$$v in (0,1-r/h), quad u = frac{h(1-v)-r}{h-b} quad text{and} quad w = frac{r-(1-v)b}{h-b} $$
How would one define the maximum value for the function
$$frac{1}{1+r}mathbb{E}_{mathbb{Q}_v}big[(K-S_1^1)^+big]$$
with respect to all the probability measures $mathbb{Q}_v$ with $v in (0,1-r/h)$ where $K in (s_0(1+b),s_0)$ ?
Note, that $mathbb{Q}$, in general in this exercise, is an equivalent martingale measure.
I can only see a minimum value, using Jensen's Inequality.
Any tips or elaborations for the maximum will be greatly appreciated.
probability-theory stochastic-processes martingales expected-value
asked 6 hours ago
Rebellos
11.5k21040
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Given that $-1<b<0<h$ and $rin (b,h)$, $S_1^1 = S_0^1 cdot (1+R)$ where $S_0^1 >0$ is a positive constant and $R(b) = b, R(0) = 0$ and $R(h) = h$ while $mathbb{Q}(b) = u, mathbb{Q}(0) = v, mathbb{Q}(h) = w$. Also, it is proven (correctly) that :
$$v in (0,1-r/h), quad u = frac{h(1-v)-r}{h-b} quad text{and} quad w = frac{r-(1-v)b}{h-b} $$
How would one define the maximum value for the function
$$frac{1}{1+r}mathbb{E}_{mathbb{Q}_v}big[(K-S_1^1)^+big]$$
with respect to all the probability measures $mathbb{Q}_v$ with $v in (0,1-r/h)$ where $K in (s_0(1+b),s_0)$ ?
Note, that $mathbb{Q}$, in general in this exercise, is an equivalent martingale measure.
I can only see a minimum value, using Jensen's Inequality.
Any tips or elaborations for the maximum will be greatly appreciated.
probability-theory stochastic-processes martingales expected-value
asked 6 hours ago
Rebellos
11.5k21040
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given that $-1<b<0<h$ and $rin (b,h)$, $S_1^1 = S_0^1 cdot (1+R)$ where $S_0^1 >0$ is a positive constant and $R(b) = b, R(0) = 0$ and $R(h) = h$ while $mathbb{Q}(b) = u, mathbb{Q}(0) = v, mathbb{Q}(h) = w$. Also, it is proven (correctly) that :
$$v in (0,1-r/h), quad u = frac{h(1-v)-r}{h-b} quad text{and} quad w = frac{r-(1-v)b}{h-b} $$
How would one define the maximum value for the function
$$frac{1}{1+r}mathbb{E}_{mathbb{Q}_v}big[(K-S_1^1)^+big]$$
with respect to all the probability measures $mathbb{Q}_v$ with $v in (0,1-r/h)$ where $K in (s_0(1+b),s_0)$ ?
Note, that $mathbb{Q}$, in general in this exercise, is an equivalent martingale measure.
I can only see a minimum value, using Jensen's Inequality.
Any tips or elaborations for the maximum will be greatly appreciated.
probability-theory stochastic-processes martingales expected-value
asked 6 hours ago
Rebellos
11.5k21040
Given that $-1<b<0<h$ and $rin (b,h)$, $S_1^1 = S_0^1 cdot (1+R)$ where $S_0^1 >0$ is a positive constant and $R(b) = b, R(0) = 0$ and $R(h) = h$ while $mathbb{Q}(b) = u, mathbb{Q}(0) = v, mathbb{Q}(h) = w$. Also, it is proven (correctly) that :
$$v in (0,1-r/h), quad u = frac{h(1-v)-r}{h-b} quad text{and} quad w = frac{r-(1-v)b}{h-b} $$
How would one define the maximum value for the function
$$frac{1}{1+r}mathbb{E}_{mathbb{Q}_v}big[(K-S_1^1)^+big]$$
with respect to all the probability measures $mathbb{Q}_v$ with $v in (0,1-r/h)$ where $K in (s_0(1+b),s_0)$ ?
Note, that $mathbb{Q}$, in general in this exercise, is an equivalent martingale measure.
I can only see a minimum value, using Jensen's Inequality.
Any tips or elaborations for the maximum will be greatly appreciated.
probability-theory stochastic-processes martingales expected-value
probability-theory stochastic-processes martingales expected-value
asked 6 hours ago
Rebellos
11.5k21040
asked 6 hours ago
Rebellos
11.5k21040
asked 6 hours ago
Rebellos
11.5k21040
asked 6 hours ago
Rebellos
11.5k21040
asked 6 hours ago
Rebellos
11.5k21040
11.5k21040
add a comment |
add a comment |
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