Main Problem

FIN551 Homework 2

March 28, 2022

1. Start with the definition of the “discounted payoff at a time t of a Payer Interest Rate Swap” given on page 14,

β

X D(t,Ti)Nτi(L(Ti−1,Ti)−K).

i=α+1

Find its price as of time t. In other words, express it as function of only zero-coupon bonds P (t, .), which are all known market quantities at time t. Do it in 2 ways:

(a) Using the argument presented on pages 11 and 12 (about the simply- compounded forward interest rate.)

(b) Using Change of Measure toolkit (hint: see section 2.5 on p.38, T- forward measure.)

2. Proveequation2.28Fll(t,ti−1,ti,τi,N,X)=ZBC(t,ti−1,ti,Ni′,N). Namely, show that the floorlet price can be expressed as the price of a call on a zero-coupon bond. Why is this useful?

3. Describe the meaning behind Proposition 2.3.1. (p.33) in plain English without writing down any mathematical formulas.

4. Use FACT ONE (change of numeraire toolkit) on page 29 to prove that any asset S on average grows at risk free under measure Q. In other words, show that RN drift of dS is Strt. (HINT: simply fill in the missing steps in the proof on page 29 by rigorously applying Ito’s lemma and using 2 mathematical facts provided in the lecture notes on page 4).

Notes

1. All references pertain to the second edition of Brigo and Mercurio’s text- book.

2. Please, submit your assignment on Canvas before the due time.

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