Proving d < e^r < u, for Binomial Model

[sophieactrainee]

Hi there,

I am looking at Solution for Q7 of Sep 21 Paper A, where you are asked to prove that d < e^r < u, using proof by contradiction.

They break it up into two cases:
Case 1: e^r <= d < u
Case 2: d < u <= e^r.

However, in the notes they just look at case where e^r < d < u

Is it okay to just use the proof in notes, rather than going through all 2 cases in the ASET solutions?

I.e. would this proof be ok, instead of ASET solution?

Prove d < e^r < u:

Proof:

Assume otherwise, i.e. e^r < d < u

Then we could borrow £1*S0 of cash, and buy £1*S0 of stock

At time 0 this would be net cost of 0 , i.e. -1*S0 + 1*S0 = 0

Now at time 1 this would be worth

-e^4r S0+ S0*u = S0 ( u – e^r) > 0 since u > e^r

OR

-e^4r S0+ S0*d = S0 ( d – e^r) > 0 since d > e^r

So in either case they would get a positive payoff => abitrage opportunity => d < e^r < u holds.




Thanks

 

[John Potter]

You should probably judge this by the number of marks being awarded but I would lean towards quickly rattling off both sides of the proof, especially as the second half is going to be copy/paste of the first half with a couple of buy/sell tweaks...

The point is that the guaranteed return on the cash MUST be sandwiched by the possible returns on the shares, otherwise you just buy "cheap" and sell "expensive", which leads to arbitrage. You formalise this (as you have done) but how long is it then going to take to copy/paste what you've already done?

eg. Let's say e^r < d, this means shares guaranteed to do better than cash, so buy share and borrow cash etc. etc.

Then, copy/paste this and change some words...

Let's say e^r > u, this means shares guaranteed to do worse than cash, so short-sell share and lend cash etc. etc.

More generally, if you've a proof to do where both sides need demonstrating but it's really just the same idea on the other side, (put-call parity for example), if there aren't many marks available, you could write "Similarly, it must be the case that e^r > u."

Honestly, what's in the Examiners' report on stuff like this simply depends on which parallel universe we're in on results day - judge it from the number of marks, I would,

John