Hi! Could someone please explain how to go about answering question 17 of tut 3? I looked at the memo but I dont understand how they got there. Please help! Thank you.

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The easiest way might be to use diagrams to derive bounds. Its useful to draw each payoff profile of each component of the portfolio below one another with proper scaling. This can help you get a handle on how the overall graph changes at certain points. Then you can aggregate the individual graphs by adding their gradients to get the graph you see in figure 1.

If you look at the graph in figure 1, you can see 3 separate lines with 3 separate gradients.Try and map out these lines even further. For example, beyond the point 2K you can see the line has gradient 1 and if you draw out the line back to when \(S_{T}\) is 0 then you can see that the profile always lies below or on the line you've drawn out. This line is \(2K + S_{T}\).

For the lower bound, it can be sufficient to use the straight line 2K but we always provide the greatest possible lower bound as it is more informative. Usually you should find a line with equal gradient to the payoff profile as it tends to infinity (since the lines will be parallel). The line tending to infinity is after the point 2K, and it has a gradient of one. We can thus choose lines such as \(S_T\) or \(K+S_T\). However we want the greatest lower bound and so we see that the straight line 2K is above \(K+S_T\) before the point K and below it from this point on wards, hence we take the max of these 2 lines.

Hope this helps.