libor market model presentation
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LIBOR Market ModelFrom Zero 3o Hero# - | @Stephan_Chang | [email protected]
Intro.
01.
Intro.
01.
Feynman - Kac
Formula I0o’s Lemma
Ho-Lee (1986) Hull-Whi0e (1994)
Vasicek (1977) CIR (1985)
P(t,T)rt
dP(t,T)
P(t,T)
I0o’s Lemma
Risk-free Portfolio
dF(t,T,U)
F(t,T,U)
Cap
Floor
Swaption
F(t,T,U)f(t,T)
rtf(t,T)
F(t,T,U)
f(t,T) = �� lnP(t,T)
�T
BGM’s
BGM’s LIBOR Ra(e, Swap ra(e
Caplet
Floorlet Swaption
Intro.
01.# Mer(on (1970)
# Vasicek (1977)
# CIR (1985)
# Ho-Lee (1986)
# Hull-Whi(e (1990, 1994)
# Black-Derman-Toy (1994)
# Ho-Lee ex(ension (1986)
# Hull-Whi(e ex(ension (1990)
# Health Jarrow & Mor(on (1991)
BGM’s LIBOR Ra(e, Swap ra(e
Caplet
Floorlet Swaption
Intro.
01.
1973 1995 Black & Scholes Formula
Black & Scholes Formula
Cap Floor Swaption BS-Like 1
2
Intro.
01.3
Payoff
Payoff
C(T) = max(ST � K,0)
P(T) = max(K� ST,0)
T
Payoff
Payoff
Capletk = max(Lk(t) � K,0)
Floorletk = max(K � Lk(t),0)
k
Model
02.
Model
02.
B A
A B $1 A
P(t,T1)
P(t,T1)-
$1 � [1 + (T2 � T1)L(t,T1,T2)]
t T1 T2
$1 $1 � [1 + (T2 � T1)L(t,T1,T2)]
1 + (T2 � T1)L(t,T1,T2)
�[1 + (T2 � T1)L(t,T1,T2)]P(t,T2)+
+
$1- �[1 + (T2 � T1)L(t,T1,T2)]
LIBOR Forward Ra0e
P(t,T2)
Model
02. LIBOR Forward Ra0e
P(t,T1) = P(t,T2)[1 + (T2 � T1)L(t,T1,T2)]
P(t,T2)L(t,T1,T) =P(t,T1) � P(t,T2)
�(T1,T2)
1 + (T2 � T1)L(t,T1,T) =P(t,T1)
P(t,T2)
1 + x LIBOR Ra0e = [ ]= exp[
� T2
T1
r(t,u)du]
HJM f
Model
02.
N(d1) S KN(d2)[ ]C = SN(d1) � Ke�rtN(d2)
Portfolio
+ 1
�(T1,T2) P(t,T1)1
�(T1,T2)P(t,T2)
Price of tradable Asset
P(t,T2)L(t,T1,T2) =P(t,T1) � P(t,T2)
�(T1,T2)
LIBOR Forward Ra0e
Model
02.
= E(Xn|Xn�1) = g(Xn�1)
= E(Xn|Xn�1,Xn�2, ...,X1)
E(Xn|Fn�1)
&
E(Xn|Fn�1) = Xn�1
Spot
Forward
Other Risk-Adjus0ed
measure
measure
measure
QTQ
Qs[ ]Bt
P(t,T)
S
Model
02.
1P(t,T2)
� P(t,T1) � P(t,T2)
�(T1,T2)QT2
P(t,T2)L(t,T1,T2)
P(t,T2)QT2
P(t,T2)L(t,T1,T2) =P(t,T1) � P(t,T2)
�(T1,T2)
LIBOR Ra0e
dL2(t,T1,T2) = �(t,T1,T2)L2(t,T1,T2)dwT2t
dLk(t,Tk�1,Tk) = �(t,Tk�1,Tk)Lk(t,Tk�1,Tk)dwkt( )
[ ]
LIBOR Forward Ra0e
Intro.
02.
L(t,T1,T2)
QT2
dL2(t,T1,T2)
L2(t,T1,T2)
L(t,T1,T2) = µL(t,T1,T2)dt + L(t,T1,T2)�(t,T1,T2)dwt
µL =�L(t,T1,T2)
�T1+ L(t,T1,T2)�(t,T1,T2)�(t,T1,T2) +
(T2 � T1)L2(t,T1,T2)
1 + (T2 � T1)L(t,T1,T2)�(t,T1,T2)|�(t,T1,T2)|2
dL2(t,T1,T2) = �(t,T1,T2)L2(t,T1,T2)dwT2t
Martingale Condition
P
dL2(t,T1,T2)
L2(t,T1,T2)
Model
02.
P
L(t,T1,T2)
QT2
dL2(t,T1,T2)
L2(t,T1,T2)
dL2(t,T1,T2) = �(t,T1,T2)L2(t,T1,T2)dwT2t
QT1
How about QT1
dL2(t,T1,T2)
L2(t,T1,T2)
dL2(t,T1,T2)
L2(t,T1,T2)
dL(t,T1,T2) = L(t,T1,T2)�(t,T1,T2)k�
i=2
(T2 � T1)�k,i�i(t,T1,T2)L(i,T1,T, 2)1 + (T2 � T1)L(t,T1,T2)
dt
+L(t,T1,T2)�(t,T1,T2)dwT11
?
Model
02. BGM’s contribution (1994 - 1997)
1 Rocks SucksQT1QT2
2-
-
-
Pricing
03.
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)dwkt
t � [0,Ti], �i = 1, ...N wkt : is brownian motion under Qk
Pricing
03.
Caplet
Floorlet
Cap
Floor
Portfolio
Portfolio
Cap = 𝞢 Caplet
Floor = 𝞢 Floorlet
FRA
IRS
SwaptionPortfolio Portfolio
IRS= 𝞢 FRA
Pricing
03.Caplet
Cpl(t,T1,T2,N,K) = N � P(t,T2) � �(T1,T2)Blackc(K,L(t,T1,T2), �)
� =
�� T1
t�2(u,T1,T2)du Blackc(K,F, �) = F�(
ln(F/K) + �2
�) � K�(
ln(F/K) � �2
�)
pf:
Cpl = N � P(t,T2) � �(T1,T2) �EQ[e�
� T2t rudu
P(t,T2)(L(t,T1,T2) � K)+|Ft]
N � P(t,T2) � �(T1,T2) �EQT2
[(L(t,T1,T2) � K)+|Ft]=
MartigaleQT2
dQT2
dQ
Pricing
03.EQT2
[(L(t,T1,T2) � K)I{L(t,T1,T2)>K}|Ft]
1 2
2 EQT2[I{L(t,T1,T2)>K}|Ft] = PrT2(L(t,T1,T2) > K|Ft)
= EQT2[L(t,T1,T2)I{L(t,T1,T2)>K}|Ft] �KEQT2
[I{L(t,T1,T2)>K}|Ft]
= PrT2(lnL(t,T1,T2) > lnK|Ft)
= PrT2(z < d2,k) = �(d2,k)
= ...
Pricing
03.1 EQT2
[L(t,T1,T2)I{L(t,T1,T2)>K}|Ft]
= EQT2
[L2(t)e�12
� T1t �2
T2(u)du+
� T1t �T2 (u)dw
T2 (u))I{L(t,T1,T2)>K}|Ft]
dwR(t) = dwT2(t) � �T2(t)dt
= L2(t,T1,T2)ER[I{L(t,T1,T2)>K}|Ft]
= L2(t,T1,T2)PrR(L(t,T1,T2) > K|Ft)
= L2(t,T1,T2)PrR(lnL(t,T1,T2) > lnK|Ft)
= L2(t,T1,T2)�(d1, k)
Pricing
03.EQT2
[(L(t,T1,T2) � K)I{L(t,T1,T2)>K}|Ft]
= EQT2[L(t,T1,T2)I{L(t,T1,T2)>K}|Ft]
1 2
�KEQT2[I{L(t,T1,T2)>K}|Ft]
21= +
= L2(t,T1,T2)�(d1,k) � K�(d2,k)
Pricing
03.Cap
pf: Trivial !
Caplet
CapPortfolio
Cap = 𝞢 Caplet
Cap(t,T,N,K) =��
i=�+1
N � P(t,Ti)EQ[
e� Tit 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�2i (u)du)
Pricing
03.Floorlet Floor
Caplet
Floorlet
Cap
Floor
Portfolio
Portfolio
Cap = 𝞢 Caplet
Floor = 𝞢 Floorlet
Blackc
Blackp
pf: Trivial !