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sol2


Total Posts: 3
Joined: Nov 2018
 
Posted: 2018-11-07 09:06
I don’t know theoretically accurate solution to the following problem that I frequently face valuing irregular floating rate notes. Can somebody help me out?

Problem
Let’s assume that there is a market of zero-coupon risk-free bonds.
Price of a bond at time t with maturity at T and face=1 is P(t,T).
Bond’s interest rate is R(t,T) = (1/P(t,T)-1)/(T-t).
Forward interest rate from T_1 to T_2 at t is F(t,T_1,T_2) = (P(t,T_1)/P(t,T_2) – 1)/(T_2 – T_1).
There is a tradable derivative “f” that pays R(T_1, T_2) (interest rate R fixed at T_1) at T_3>T_2.

Question
What is value of “f” at t=0?

The question is trivial for the case T_3=T_2: f(0) = P(0,T_2)*F(0,T_1,T_2) – “discounted forward rate”. Is it possible to find f(0) for T_3>T_2?

sol2


Total Posts: 3
Joined: Nov 2018
 
Posted: 2018-11-09 08:16
Come on! It should be a simple question for those who know theory... :-( Anyone?

pj


Total Posts: 3415
Joined: Jun 2004
 
Posted: 2018-11-09 09:16
If your product at time T_3 is worth X.
How much will it be worth at T_2?
Hint: Discounting.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

sol2


Total Posts: 3
Joined: Nov 2018
 
Posted: 2018-11-09 13:35
pj, THANK YOU for the answer!!!

So your solution is f(0) = P(0,T_3)*F(0,T_1,T_2). I am not sure that it is right (but some people usually do what you suggest). Can you prove it strictly?

Solution to the “T_3=T_2” case is based on changing probability measure to a measure where numeraire is P(t,T_2), i.e. g(t)/P(t,T_2) is a martingale for any traded instrument g(t). Or you can simply hedge f(t) with bonds P(t,T_1) and P(t,T_2).

But it doesn’t work for the “T_3>T_2” case. It is not that simple, I think.

I hope that people here understand that you can’t just discount however you want and whatever you want…

nikol


Total Posts: 553
Joined: Jun 2005
 
Posted: 2018-11-09 17:32
@pj

you are heart-kinded.
and you will suffer
Evil Grin
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