## Theorem 2.4

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

### Theorem 2.4

Hi, I have a question on the proof of theorem 2.4.

Theorem 2.4 If P and ! are riskless zero-coupon bonds with the same maturity time T, then they are of equal value at all previous times.

Proof: Suppose both bonds P and Q are guarranteed to be worth exactly 1 pound at time T. then Q is worth as much as P in all possible worlds at time T, so Q is worth at least as much as P in all possible worlds at all previous times. By symmetry, we conclude that P is also worth as much as Q., and thus P and Q have the same price in all possible worlds at all times.

I don't understand the sentence highlighted in red. Why must Q be worth as much as P in all possible worlds at time T? Can't Q be worth as much as P in some world states and P be worth as much as Q in some world state? just like in theorem 2.5?

Theorem 2.4 is a stronger assumption then theorem 2.5, Theorems 2.5 states that P is stronger than Q in some state and Q is stronger than P in some state.
weihong

Posts: 3
Joined: Thu Jun 13, 2013 5:33 am

### Re: Theorem 2.4

at time T, P and Q are both worth 1.

If at t <T Q is worth more than P, then sell Q and buy P, wait to time T, and you have an arbitrage.
mj

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

### Re: Theorem 2.4

But if that's the arguement, we can say that theorem 2.5 is a redundant theorem.

Thorem 2.5: If two portfolios P and Q are of equal value today and if at some future time T, P is worth more than Q in some world states, then Q is worth more than P in some world states.

Since P and Q are of equal value, I can argue that in some future time T, P is worth the same as Q. Else, let's say P is worth more than Q in the future, I can construct an arbitrade by buying 1 P now and shorting 1 Q now. and selling them at time T. it will lead me into a riskless profit. So theorem 2.5 will not happen at all?
weihong

Posts: 3
Joined: Thu Jun 13, 2013 5:33 am

### Re: Theorem 2.4

After reading again, I think the keyword to the theorem is some possible world state.I think I am getting it already, just have to read again and again. Thank you.
weihong

Posts: 3
Joined: Thu Jun 13, 2013 5:33 am