Optimal Upper and Lower Bounds
For the following exercise:
Give optimal upper and lower bounds on the price today for a product that pays a function of the spot price, $ S $ , of a non-dividend paying stock one year from now, there are no interest rates and the spot is $ 100 $ , when the pay-off is $ 0 $ below $ 80 $ , increases linearly from $ 0 $ at $ 80 $ to $ 20 $ at $ 120 $ and then it is constant at $ 20 $ above $ 120 $
The answer is supposed to be $ 0 $ and $ 100/6 $
But I am not understanding how the upper bound is defined.
you have to find $ \alpha $ and $ \beta $ so that
$$ \alpha S_1 + \beta $$ is greater than or equal to the pay-off everywhere. Any such values gives an upper bound of $$ \alpha S_0 + \beta $$ Now try to find the smallest value of that. I am guessing that the answer is $ \alpha = 1/6 $ and $ \beta=0. $