## Exercise 2.15 [2nd ed.]

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

### Exercise 2.15 [2nd ed.]

Hello evrebody,

I don't know if I'm posting in the right section then sorry if I'm wrong. I write here on the forum because I don't understand the answer to an exercise on Prof. Joshi's book and hope somebody can help; the exercise is the 2.15:

S (non-dividend stock), exp(rt) (riskless bond).
A contract pays S(t2)-S(t1) at time t2. replicate this contract by trading in the stock and riskless bond.

Now it's clear to me how lock S(t1) and S(t2), that is buying S at t0 (will be S(t2) at t2 and buy S(t1)*exp[-r(t2-t1)] bonds at t1 so these will be S(t1) at t2......the answer to the exercise is to buy today (at t0) S(t0) stocks and S(t0)exp[-r(t2-t1)] stocks. Then sell the second amount of shares and buy the bonds at t1....I don't understand why this procedure will give me S(t2)-S(t1) at t2.....mean why the difference between them?

So excuse me for my terrible eng and thanks everybody for the attention, regards

iacopo
Sparvi3ro

Posts: 3
Joined: Fri Oct 04, 2013 1:09 pm

### Re: Exercise 2.15 [2nd ed.]

you buy 1 stock at t0 and sell at it at t2 to get S(t_2)

you buy e^{-r(t_2-t_1)} stocks at t0 and sell at t1 to get S(t_1)e^{-r(t_2-t_1)} units of cash at t_1. Turn these into bonds to get S(t_1) units of cash at t_2.

Hope this helps.
mj

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

### Re: Exercise 2.15 [2nd ed.]

Thanks professor Joshi.

Ok, this procedure is clear (almost clear, probably I miss something). If I do some gain calculus at the end of the trading I don't find the result:

first operation: buy S(t0) and sell at t2. The gain is obviously S(t2)-S(t0).

second operation: buy S(t0)*e^[-r(t2-t1)] and cash at t1, then buy S(t1)*e^[-r(t2-t1)] bonds and cash at t2.
The gain is (S(t1)-S(t0))*e[-r(t2-t1)] + S(t1)-S(t1)*e^[-r(t2-t1)] = S(t1)-S(t0)*e^[-r(t2-t1)].

Then the total gain is S(t2)+S(t1)-S(t0)*(1+e^[-r(t2-t1)]) != S(t2)-S(t1).

Where is the mistake?...Excuse me if I'm demanding but unlike the other exercises I do not understand this one, although I put effort into this. Thank you for your patience. Kind regards,

Iacopo
Last edited by Sparvi3ro on Wed Oct 09, 2013 4:04 pm, edited 1 time in total.
Sparvi3ro

Posts: 3
Joined: Fri Oct 04, 2013 1:09 pm

### Re: Exercise 2.15 [2nd ed.]

we are thinking in terms of the absolute cost of replication not the values of changes.

if we do it your way:

"first operation: buy S(t0) and sell at t2. The gain is obviously S(t2)-S(t0)."

actually it isn't because you have to fund the cost of buying S at time 0.
mj

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

### Re: Exercise 2.15 [2nd ed.]

Well, if is the absolute value what we are looking for is now clear. Anyway I don't understand "you have to fund the cost", isn't the cost -S(t0)?... I think I don't grasp the concept at all. I will study more in depth or move forward with the reading of the book, probably it will be clear in a couple weeks of hard study. Thanks for the answers Professor, kind regards

iacopo
Sparvi3ro

Posts: 3
Joined: Fri Oct 04, 2013 1:09 pm