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.










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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.










    share|cite|improve this question
























      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.










      share|cite|improve this question













      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






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      asked 6 hours ago









      Rebellos

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