sol2


Total Posts: 3 
Joined: Nov 2018 


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 zerocoupon riskfree 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)/(Tt). 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 


Come on! It should be a simple question for those who know theory... :( Anyone?



pj


Total Posts: 3604 
Joined: Jun 2004 


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 


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: 1360 
Joined: Jun 2005 


@pj
you are heartkinded. and you will suffer

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c) 

