## Exercise 2.18

This forum is to discuss the book "the concepts and practice of mathematical finance" by Mark Joshi.

### Exercise 2.18

While constructing super-replicating portfolio in part (iii), why don't we consider also the portfolio

/alpha = 0, /beta = 0.75 ?

It does super-replicate, doesn't it? Because it then gives better upper bound =0.75 for first two cases, than given in the answers (0.8 and 0.9).
Sorry for possible confusion.
lime

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Joined: Tue Jul 30, 2013 7:24 am

### Re: Exercise 2.18

i think you're right.
mj

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

### Re: Exercise 2.18

* super-replicating portofolio's for (iii)
There exists a more optimal super-replicating portfolio than portfolio
alpha = 0.5 and beta = 0.5,
Portfolio alpha = 0.5 and beta = 0.25 will always lead to a lower cost and is super-replicating.
The upper-bounds for this portfolio are: 0.65, 0.7, 0.525 and 0.6 respectively. Portfolio alpha = 1 and beta = 0 leads in all 4 cases to a higher cost.
Also note that the portfolio suggested above alpha = 0 and beta = 0.75 will lead to higher bounds, than portfolio alpha = 0.5 and beta = 0.25.
G_CAM

Posts: 10
Joined: Mon Aug 12, 2013 12:46 pm

### Re: Exercise 2.18

Ex. 2.18 sub-replication (iv).
As opposed to a sub-replication solution propsed for (iv). (5/3,-5/6) cannot be a sub-replication solution.
One also needs to take the point (0.6, 0) into account. As we have a 0.5 digital call struck at 0.6
(5/3)*0.6-5/6 = 1/6 and exceeds 0, therefore is (5/3,-5/6) is not sub-replicating. Taking (0.6,0) into account (5/2,-1.5) is also sub-replicating solution.
For the 4 cases we have then as optimal values:
0.5 (either solution (5/2, -1.5) or (5,-3.5))
1 solution (5,-3.5)
0.15 solution (5/2,-1.5)
0.25 solution (5/2,-1.5)
G_CAM

Posts: 10
Joined: Mon Aug 12, 2013 12:46 pm