# 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 $\beta=0.$