1 Modelling Term Structures MGT 821/ECON 873 Modelling Term Structures.
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Transcript of 1 Modelling Term Structures MGT 821/ECON 873 Modelling Term Structures.
2
Term Structure Models
Black’s model is concerned with describing the probability distribution of a single variable at a single point in time
A term structure model describes the evolution of the whole yield curve
3
The Zero Curve
The process for the instantaneous short rate, r, in the traditional risk-neutral world defines the process for the whole zero curve in this world
If P(t, T ) is the price at time t of a zero-coupon bond maturing at time T
where is the average r between times t and T
P t T E e r T t( , ) ( )
r
4
Equilibrium Models
Rendleman & Bartter:
Vasicek:
Cox, Ingersoll, & Ross (CIR):
dr r dt r dz
dr a b r dt dz
dr a b r dt r dz
( )
( )
5
Mean Reversion Mean Reversion
Interestrate
HIGH interest rate has negative trend
LOW interest rate has positive trend
ReversionLevel
6
Alternative Term Alternative Term StructuresStructuresin Vasicek & CIR in Vasicek & CIR
Zero Rate
Maturity
Zero Rate
Maturity
Zero Rate
Maturity
8
Equilibrium vs No-Arbitrage Models
In an equilibrium model today’s term structure is an output
In a no-arbitrage model today’s term structure is an input
9
Developing No-Arbitrage Model for r
A model for r can be made to fit the initial term structure by including a function of time in the drift
10
Ho-Lee Model
dr = (t)dt + dz Many analytic results for bond prices and
option prices Interest rates normally distributed One volatility parameter, All forward rates have the same standard
deviation
Diagrammatic Diagrammatic Representation of Ho-LeeRepresentation of Ho-Lee
11
Short Rate
r
r
r
rTime
12
Hull-White Model
dr = [(t ) – ar ]dt + dz Many analytic results for bond prices and option
prices Two volatility parameters, a and Interest rates normally distributed Standard deviation of a forward rate is a
declining function of its maturity
14
Black-Karasinski Model
Future value of r is lognormal Very little analytic tractability
dztdtrtatrd )()ln()()()ln(
15
Options on Zero-Coupon Bonds In Vasicek and Hull-White model, price of call maturing at T on a
bond lasting to s is
LP(0,s)N(h)-KP(0,T)N(h-P) Price of put is
KP(0,T)N(-h+P)-LP(0,s)N(h)
where
TTsσ
KL
a
ee
aKTP
sLPh
P
aTTsa
PP
P
)( Lee-HoFor
price. strike theis and principal theis
2
11
2),0(
),0(ln
1 2)(
16
Options on Coupon Bearing Bonds
In a one-factor model a European option on a coupon-bearing bond can be expressed as a portfolio of options on zero-coupon bonds.
We first calculate the critical interest rate at the option maturity for which the coupon-bearing bond price equals the strike price at maturity
The strike price for each zero-coupon bond is set equal to its value when the interest rate equals this critical value
17
Interest Rate Trees vs Stock Price Trees
The variable at each node in an interest rate tree is the t-period rate
Interest rate trees work similarly to stock price trees except that the discount rate used varies from node to node
18
Two-Step Tree ExampleTwo-Step Tree Example
Payoff after 2 years is MAX[100(r – 0.11), 0]pu=0.25; pm=0.5; pd=0.25; Time step=1yr
10%0.35**
12% 1.11*
10% 0.23
8% 0.00
14% 3
12% 1
10% 0
8% 0
6% 0 *: (0.25×3 + 0.50×1 + 0.25×0)e–0.12×1
**: (0.25×1.11 + 0.50×0.23 +0.25×0)e–0.10×1
19
Alternative Branching Alternative Branching Processes in a Trinomial TreeProcesses in a Trinomial Tree
(a) (b) (c)
20
Procedure for Building Tree
dr = [(t ) – ar ]dt + dz
1.Assume (t ) = 0 and r (0) = 02.Draw a trinomial tree for r to match the
mean and standard deviation of the process for r
3.Determine (t ) one step at a time so that the tree matches the initial term structure
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Building the First Tree for the t rate R
Set vertical spacing:
Change branching when jmax nodes from middle where jmax is smallest integer greater than 0.184/(at)
Choose probabilities on branches so that mean change in R is -aRt and S.D. of change is
tR 3
t
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The First TreeThe First Tree
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
R 0.000% 1.732% 0.000% -1.732% 3.464% 1.732% 0.000% -1.732% -3.464%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
24
Shifting Nodes
Work forward through tree Remember Qij the value of a derivative providing
a $1 payoff at node j at time it Shift nodes at time it by i so that the (i+1)t
bond is correctly priced
25
The Final TreeThe Final Tree
A
B
C
D
E
F
G
H
I
Node A B C D E F G H I
R 3.824% 6.937% 5.205% 3.473% 9.716% 7.984% 6.252% 4.520% 2.788%
p u 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
p m 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
p d 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
26
Extensions
The tree building procedure can be extended to cover more general models of the form:
dƒ(r ) = [(t ) – a ƒ(r )]dt + dz
We set x=f(r) and proceed similarly to before
27
Calibration to determine a and
The volatility parameters a and (perhaps functions of time) are chosen so that the model fits the prices of actively traded instruments such as caps and European swap options as closely as possible
We minimize a function of the form
where Ui is the market price of the ith calibrating instrument, Vi is the model price of the ith calibrating instrument and P is a function that penalizes big changes or curvature in a and
n
iii PVU
1
2)(
HJM Model: NotationHJM Model: Notation
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P(t,T ): price at time t of a discount bond with principal of $1 maturing at T
Wt : vector of past and present values of interest rates and bond prices at time t that are relevant for determining bond price volatilities at that time
v(t,T,Wt ): volatility of P(t,T)
Notation continuedNotation continued
29
ƒ(t,T1,T2): forward rate as seen at t for the period between T1 and T2
F(t,T): instantaneous forward rate as seen at t for a contract maturing at T
r(t): short-term risk-free interest rate at t
dz(t): Wiener process driving term structure movements
30
Modeling Bond Prices
) for
process a get we approach Letting for
process the determine to lemma sIto' use can we
Because
all for
providing function any choose can We
t,TF
TT),Tf(t,T
TT
TtPTtP),Tf(t,T
tttv
v
tdzTtPTtvdtTtPtrTtdP
t
t
(
.
)],(ln[)],(ln[
0),,(
)(),(),,(),()(),(
1221
12
2121
31
The process for The process for FF((tt,,TT))
factor one
than more is there whenhold results Similar
have must we
(),(
write weif that means result This
dτs(t,)ΩT,s(t,)ΩT, m(t,
)dzΩT,s(t,dtΩT,t,mTtdF
tdzTtvdtTtvTtvTtdF
T
t ttt
tt
tTtTt
),
)
)(),,(),,(),,(),(
33
The LIBOR Market Model
The LIBOR market model is a model constructed in terms of the forward rates underlying caplet prices
34
Notation
t k
F t t t
m t t
t F t t
v t P t t t
t t
k
k k k
k k
k k
k k k
: th reset date
forward rate between times and
: index for next reset date at time
volatility of at time
volatility of ( , at time
( ):
( )
( ): ( )
( ): )
:
1
1
35
Volatility Structure
We assume a stationary volatility structure
where the volatility of depends only on
the number of accrual periods between the
next reset date and [i.e., it is a function only
of ]
F t
t
k m t
k
k
( )
( )
36
In Theory the ’s can be determined from Cap Prices
yinductivel
determined be to s the allows This
have must
weprices cap to fit perfect a provides
model the If caplet. the for volatility the is If
when of volatility the as Define i
'
),(
)()(
11
22
1
k
iiikkk
kkk
k
t
tt
itmktF
37
ExampleExample
If Black volatilities for the first three
caplets are 24%, 22%, and 20%, then
0=24.00%
1=19.80%
2=15.23%
38
ExampleExample
n 1 2 3 4 5
n(%) 15.50 18.25 17.91 17.74 17.27
n-1(%) 15.50 20.64 17.21 17.22 15.25
n 6 7 8 9 10
n(%) 16.79 16.30 16.01 15.76 15.54
n-1(%) 14.15 12.98 13.81 13.60 13.40
39
The Process for Fk in a One-Factor LIBOR Market Model
dF F dz
P t t
k k m t k
i
( )
( , ),
The drift depends on the world chosen
In a world that is forward risk -neutral
with respect to the drift is zero1
40
Rolling Forward Risk-Rolling Forward Risk-NeutralityNeutrality
It is often convenient to choose a world that is always FRN wrt a bond maturing at the next reset date. In this case, we can discount from ti+1 to ti at the i rate observed at time ti. The process for Fk is
dF
F
F
Fdt dzk
k
i i i m t k m t
i ij m t
i
k m t
( ) ( )
( )( )1
41
The LIBOR Market Model and HJM
In the limit as the time between resets tends to zero, the LIBOR market model with rolling forward risk neutrality becomes the HJM model in the traditional risk-neutral world
42
Monte Carlo Implementation Monte Carlo Implementation of LMM Model of LMM Model
We assume no change to the drift between
reset dates so that
F t F tF t
Lk j k ji i j i j k j
j j
k j
j k
i
k k j j( ) ( ) exp( )
1
2
1 2
43
Multifactor Versions of LMM
LMM can be extended so that there are several components to the volatility
A factor analysis can be used to determine how the volatility of Fk is split into components
44
Ratchet Caps, Sticky Caps, and Flexi Caps
A plain vanilla cap depends only on one forward rate. Its price is not dependent on the number of factors.
Ratchet caps, sticky caps, and flexi caps depend on the joint distribution of two or more forward rates. Their prices tend to increase with the number of factors
45
Valuing European Options in the LIBOR Market Model
There is an analytic approximation that can be used to value European swap options in the LIBOR market model.
46
Calibrating the LIBOR Market Model In theory the LMM can be exactly calibrated to
cap prices as described earlier In practice we proceed as for short rate models
to minimize a function of the form
where Ui is the market price of the ith calibrating instrument, Vi is the model price of the ith calibrating instrument and P is a function that penalizes big changes or curvature in a and
n
iii PVU
1
2)(
47
Types of Mortgage-Backed Securities (MBSs)
Pass-Through Collateralized Mortgage
Obligation (CMO) Interest Only (IO) Principal Only (PO)
48
Option-Adjusted Spread(OAS)
To calculate the OAS for an interest rate derivative we value it assuming that the initial yield curve is the Treasury curve + a spread
We use an iterative procedure to calculate the spread that makes the derivative’s model price = market price.
This spread is the OAS.