17. Pricing
03.
Model for LIBOR (London In0er-Bank Offer Ra0e)
-HJM framework
-Fini0e number, N of time periods
-LIBOR over each period lognormal
- Black’s Formula for caplets satisfied.
BGM model = LMM = LFM
Brace Ga0arek Musiela (1997)
Assumption
- L > 0
- L continuous time
- L follows a lognormal process with de0erministic vol.
Thus, dL(t, Tk 1, Tk) = k(t, Tk 1, Tk)dwk
t
t [0, Ti], i = 1, ...N wk
t : is brownian motion under Qk
23. Pricing
03.
Cap
pf: Trivial ! Caplet Cap
Portfolio
Cap = 𝞢 Caplet
Cap(t, T, N, K) =
i= +1
N P(t, Ti)EQ
[
e
Ti
t
rudu
P(t, Ti 1)
(Ti 1, Ti)(L(t, Ti 1, Ti) K)+
|Ft]
= N
i= +1
P(t, Ti) (Ti 1, Ti)Blackc(K, L(t, T1, T2),
Ti 1
t
2
i (u)du)