Download FIN 472 Fixed-Income Analysis Term

FIN 472
Fixed-Income Analysis
Term-Structure Modeling
Professor Robert B.H. Hauswald
Kogod School of Business, AU
Term-Structure Models
• Two purposes
– predicting future yield curves: investing
– modelling current yield curves: pricing
• Two model classes
– equilibrium models
– “arbitrage free”
• One principle: there is not free lunch
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Term Structure of Interest Rates
Interest Rate (%)
• Normal upward
sloping
12
• Inverted
10
8
• Level
6
• Humped
Common Term Structure Shapes
Upward Sloping
Inverted
Level
Humped
4
2
0
1
3
5
10
30
Years
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Introduction to Stochastic Processes
• Interpret the following expression:
dr = µ dt + σ dz
• We are modeling the stochastic process r where r is
the level of interest rates
• The change in r is composed of two parts:
– A drift term which is non-random
– A stochastic or random term that has variance σ 2
– Both terms are proportional to the time interval
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Enhancements to the Process
• In general, there is no reason to believe that
the drift and variance terms are constant
• An Ito process generalizes a Brownian
motion by allowing the drift and variance to
be functions of the level of the variable and
time
dr = µ ( r , t ) dt + σ (r,t ) dz
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Single Factor Models
• The single factor is typically
– the short term interest rate for discrete models
– the instantaneous short term rate for continuous
time models
• Entire term structure is based on the short
term rate: realistic?
• For every short term interest rate there is one,
and only one, corresponding term structure
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Arbitrage Free Models
• Designed to be exactly consistent with current
term structure of interest rates
– current term structure is an input
• Useful for valuing interest rate contingent
securities: pricing derivatives
• Requires frequent recalibration to use model over
any length of time
• Difficult to use for time series modeling
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Matching the Term Structure of
Volatility
• Short rates more volatile than long ones
– term structure of volatility
• Add other time-varying parameters to the
short rate process
– to give it enough degrees of freedom to also
match the term structure of volatilities exactly
• Drawback: the resulting future volatility
term structure is often quite different
– from the initially observed one
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Interest Rate Models
• Interest rate model should produce
– values and distributions of interest rates that are
– consistent with information in the market
• Bond market has information embedded in it
– Values: TVM from today’s spot curve
– Joint distributions: volatilities and correlations
from today’s volatility surface
• Model calibration to extract this information
– The more market information, the more “correct” our
descriptions of interest rate uncertainty in the future
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Key Market Information: Sources
• Forward rates
– Measure of central tendency through time
– Implied from today’s spot curve
– Relatively simple aspect of the model
• Volatility surface
– Measure of variance through time
– Derivatives on interest rates have market prices, thus,
implied volatilities
– Model should reflect these implied volatilities
– By far the more challenging aspect of the model
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Key Market Information: Statistics
Volatility Surface
Forward Curve
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From Market Information to a Model
• Assume the following functional form:
rate change
drift
volatility
– drift and volatility are related to the two sources of
market information
– both need to be calibrated to market prices
• Drift: zero (market) prices from today’s spot curve
• Volatility: market prices of caps and swaptions
– such derivatives are calls and puts on interest rates
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Interest Rate Models
• Three ways to classify these models:
– By time of invention: three generations over past 35 years
– By number of modeled rates: one vs all
– By arbitrage property: equilibrium vs no-arbitrage
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First Generation:
Equilibrium Models
• Originated from simple models of the economy
• Examples:
– Vasicek:
– Cox-Ingersoll-Ross:
• Important features:
–
–
–
–
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Models the dynamics of short rate over time
Coefficients are constant
Limited number of factors and parameters
Do not fit spot curve (i.e., allows arbitrage in market)
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© Robert B.H. Hauswald
Second Generation:
No-Arbitrage Models
• Simple generalization of equilibrium models
• Examples:
– Ho-Lee:
– Hull-White:
• Important features:
–
–
–
–
Models the dynamics of short rate over time
Coefficients vary with time
Limited number of factors, but many parameters
Fit spot curve but sometimes not ATM and OTM (=
IV) volatility
surface
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Model Calibration
• Instruments needed
– Zeros: extract time value of money
– Caps and European swaptions:
extract expected volatility
• Procedure
– Start with prior day’s parameters
and price the instruments
– Iteratively adjust parameters to
minimize difference between model
and market prices
– When difference minimized, result
is a set of model parameters
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Third Generation:
Forward Rate Models
• Absence of arbitrage directly imposed
– Heath-Jarrow-Morton:
– Libor Market Model:
• Important features:
– Models the forward rates through time
rather than the short rate
– Can have many factors and parameters
– Fit spot curve and ATM volatility surface
well
– Cannot capture volatility skew and smile
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Model Pricing
• Once calibrated, model can price most fixed-income
securities: two methods
• Tree or lattice: Discretization of the model allows for
efficient pricing
• Monte Carlo simulation
– Simulate many “paths” (e.g., evolutions) of interest rates
through time and calculate a price on each path
– The average price across paths converges with enough paths
– Necessary for mortgage securities and forward rate models
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Model Pricing: Outputs
Monte Carlo simulation
Tree/Lattice
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Sample Application: Hedging
• Derivatives used to hedge residual (long-short) risk
– Swaps, caps, and swaptions
• Metrics needed for hedging: derived from TSIR models
– Effective Duration (D): 1st order sensitivity to interest rates
– Effective Convexity (C): 2nd order sensitivity to interest rates
– Volatility Duration (V): 1st order sensitivity to volatility
• Methods of hedging
– Full hedge: Create portfolio with zero D, C, and V
– Partial hedge: Create portfolio with small D, C, and V
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Building a TSIR Model
• Implementing TSIR models: consistency
– with current actual term structure
– with the term structure of volatility
• Well founded model: absence of arbitrage
– weaker requirement than equilibrium models:
useful for what?
• Two fundamental model classes: illustration
– Ho and Lee: tractable but rates can be negative
– BDT: more involved but avoids negative rates
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Three related interest rates
There are 3 rates/prices that are inter-related. From one,
the other two can be derived.
ZERO COUPON
BOND PRICES P(t,T)
different maturities T
SPOT RATES
continuously compounded
r(t,T) = ln[1/P(t,T)]/(T-t)
T
or r ( t,T) =
∫ f (t,u)du
t
T−t
discrete compounding
1+r(t,T) = [1/P(t,T)]1/(T-t)
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P(T,T)=1
FORWARD RATES
continuously compounded
f ( t, T ) = −
∂ ln P ( t, T )
∂T
 T

P ( t,T ) = exp −∫ f ( t,u ) du 
 t

discrete compounding
or
1 + f ( t, T, T + 1) =
1 + r(t, T + 1)
1 + r(t, T)
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Example of Black-Derman-Toy (BDT) model
Most one-factor interest rate models concentrate on
modelling the short rate.
Define change in the (natural) log of the short rate as
∆ ln r(t) = ln r(t+∆) - ln r(t) .
Example of a simple version of Black-Derman-Toy model
that is a one-factor interest rate model. For simplicity we
shall just describe model as being expressed under
martingale equivalent measures, i.e. a pseudo probability P*.
∆ ln r(t) = [a(t) - b(t) ln r(t)] ∆ + σ(t) ∆W(t)
where ∆W(t) is normally distributed r.v. N(0, ∆) [P*].
Then mean EP* [∆ ln r(t)] = [a(t) - b(t) ln r(t)] ∆ .
Std. dev. {varP* [∆ ln r(t)]}1/2 = σ(t) ∆1/2 . This is the
(conditional on lnr(t) ) volatility of ln r(t+∆).
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Risk-Neutral Probabilities = 1/2
Thus,
1
  a ( t ) − b ( t ) ln r ( t )  ∆ + σ ( t ) ∆ with probability 2
∆ ln r(t) = 
  a ( t ) − b ( t ) ln r ( t )  ∆ − σ ( t ) ∆ with probability 12
.
Probability (+) : 1/2 and (-) : 1/2 are martingale equivalent
probabilities (not the real or empirical probabilities) that
provides equilibrium prices that do not yield arbitrage
opportunities.
ln rt,H=
lnr ( t) + a( t) − b( t) lnr( t) ∆+σ( t) ∆ withprobability 12
ln r(t)
ln rt,L=
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lnr ( t) + a( t) −b( t) lnr ( t) ∆−σ( t) ∆ withprobability 12
Term-Structure Models Comparison
Note that
ln rt,H - ln rt,L = 2 σ(t) ∆1/2 . Therefore,
rt,H = rt,L × e 2 σ(t) √∆
If ∆ = 1 (yr), then at each t :
rt,H = rt,L × e 2 σ(t) .
Or,
r(t,H) = r(t,L) × e 2 σ(t) .
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Lattice
r(5,HHHHH)
BINOMIAL TREE
r(3,HHH)
r(5,HHHHL)
r(2,HH)
r(1,H)
r(5,HHHLL)
r(3,HHL)
SHORT
RATES
r(0)
r(2,HL)
r(5,HHLLL)
r(3,HLL)
r(1,L)
r(2,LL)
r(5,HLLLL)
r(3,LLL
)
NODE:
LLL
0
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∆
2
3
r(5,LLLLL)
4
5…i
STEPS
TIME = i × ∆
The short rates on the tree are also ∆-period forward rates.
Term-Structure Models Comparison © Robert B.H. Hauswald
t= 0
t= 1
25
t = 2 (end period 2∆)
r(2,HH)
r2e4σ(2)√∆
½
r(1,H)
r1e2σ(1) √∆
½
r(0)
r(2,HL)
½
½
r0
½
r2e2σ(2) √∆
r(1,L)
r1
½
r(2,LL)
r2
Information required to construct the above tree at t=0:
zero prices
annualized volatility (std. dev.) at time t=i∆
P(0,1∆)
P(0,2∆)
P(0,3∆)
P(0,4∆)
σ(1)
σ(2)
σ(3)
σ(4)
…
…
P(0,T)
σ(T)
first interval ∆
2nd interval 2∆
3rd interval 3∆
4th interval 4∆
…
Tth interval T∆
It is important to recognize that the tree is constructed at
each time point t for all information available at t. At the next
t+1, a new tree is constructed based on new information at
t+1. The trees may change over time.
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Data
Alternative Information required to construct the above tree at t=0:
par bond prices
annualized volatility (std. dev.) at time t=i∆
[ytm=coupon][maturity,yrs]
100 [3.5%][1]
100 [4.2%][2]
100 [4.7%][3]
100 [5.2%][4]
σ(1) =10%
σ(2) =10%
σ(3) =10%
σ(4) =10%
…
…
first interval ∆
2nd interval 2∆
3rd interval 3∆
4th interval 4∆
…
100 [y(T)% p.a.][T] σ(T)
Tth interval T∆
The above par bond prices are obtained from on-the-run
Treasury bond issues (coupon bonds). On-the-run issues
(providing benchmark YC or rates) are typically par value
bonds.
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Suppose ∆ = 1 year interval. For simplicity of exposition of the
following calibration method (calibrating or fixing parameters of
tree to initial term structure), assume annual coupon.
t=0
t=1
t = 2 (end period 2)
r(2,HH)
½
r2e4σ(2)
r(1,H)
r1e2σ(1)
½
½
½
r(0)
r0
½
r(2,HL)
r2e2σ(2)
r(1,L)
r1
½
r(2,LL)
r2
Object is to find the 1-period spot rates (short rates) on the tree
i.e. r(0), {r(1,H), r(1,L)}, {r(2,HH), r(2,HL), r(2,LL)}, ….
so that (1) the rates do not admit arbitrage opportunities, and (2) the
rates are consistent with the zero coupon bond (or else the par
value bond) prices at t=0. Note that the above 1-period spot rates
are conditional rates depending on which state or node occurs.
Actual spot rate will take only one of the values at each t. Also, note
that rt = r(t,LLL…L). So r2 = r(2,LL). r(2,HH) > r(2,HL) > r(2,LL). So
the short rates at each t are increasing as we move up.
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Step 1: Calibrating the Lattice
Step One
Find at t=0 node: r(0).
r(0) = ln(1/P(0,1)) . or approx by [(1/P(0,1)-1] .
Or alternately, using (par) bonds, 100 = 103.5/(1+r(0)), or r(0) = 3.5%.
This is illustrated by putting the promised cashflows in the relevant nodal
boxes. r(0) is shown to apply across the time from t=0 (end of period 0 or
start of period 1) to t=1 (end of period 1).
100+3.5
½
100
r(0)
½
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Step 2: Calibrating the Lattice
Step Two
Find at t=1 nodes: r(1,H) and r(1,L). Substitute given σ(1) = 10% p.a. into
the expression involving short rate in the boxes. Note r(0) found in step 1
is substituted at node 0.
100+4.2
½
4.2
r1e2 x 0.1
½
½
½
100
3.5%
½
100+4.2
4.2
r1
½
100+4.2
P.V. cashflows at node 1H is [104.2/(1+r1e0.2) + 4.2].
P.V. cashflows at node 1L is [104.2/(1+r1) + 4.2].
solve r1 in the following equation:
100 = {½ × [104.2/(1+r1e0.2)+4.2] +
½×
[104.2/(1+r1)+4.2]}/1.035
By trial and error, r1= r(1,L) = 4.4448%.
So, r(1,H) = 4.4448% × e0.2 = 5.4289%.
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Step 3: Calibrating the Lattice
Step Three
Find at t=2 nodes: r(2,HH), r(2,HL), and r(2,LL). Substitute given σ(2) =
10% p.a. into the expression involving short rate in the boxes. Note r(1,H)
, r(1,L) found in step 2 are substituted at t=1 nodes.
100+4.7
4.7
r2e4 x 0.1
½
4.7
100+4.7
5.4289%
½
½
½
100
3.5%
½
4.7
2 x 0.1
r2e
100+4.7
4.7
4.4448%
½
4.7
r2
100+4.7
P.V. cashflows at node 2HH is [104.7/(1+r2e0.4) + 4.7].
P.V. cashflows at node 2HL is [104.7/(1+r2e0.2) + 4.7].
P.V. cashflows at node 2LL is [104.7/(1+r2) + 4.7].
bringing PV of
cashflows to the
present is backward
induction process
P.V. cashflows at node 1H is
{½ × [104.7/(1+r2e0.4)+4.7] + ½ × [104.7/(1+r2e0.2)+4.7]}/ 1.054289 +4.7
P.V. cashflows at node 1L is
{½ × [104.7/(1+r2e0.2)+4.7] + ½ × [104.7/(1+r2)+4.7]}/ 1.044448 + 4.7
solve r2 in the following equation:
100 = {½ × [{½ × [104.7/(1+r2e0.4)+4.7] + ½ × [104.7/(1+r2e0.2)+4.7]}/ 1.054289
+4.7] + ½ × [{½ × [104.7/(1+r2e0.2)+4.7] + ½ × [104.7/(1+r2)+4.7]}/ 1.044448 +
4.7]}/1.035
By trial and error, r2= r(2,LL) = 4.6958%. So, r(2,HL) = 4.6958% × e0.2 =
5.7354% , and r(2,HH) = 4.6958% × e0.4 = 7.0053%.
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Steps 4, 5: Calibrating the Lattice
Step Four
Find at t=3 nodes: r(3,HHH), r(3,HHL), r(3,HLL) and r(3,LLL). Substitute
given σ(3) = 10% p.a. into the expression involving short rate in the
boxes. Note r(2,HH), r(2,HL), and r(2,LL) found in step 3 are substituted
at t=2 nodes.
at
at
at
at
node
node
node
node
3HHH is [105.2/(1+r3e0.6) + 5.2]=PV(3HHH)
3HHH is [105.2/(1+r3e0.4) + 5.2]=PV(3HHL)
3HHH is [105.2/(1+r3e0.2) + 5.2]=PV(3HLL)
3LLL is [105.2/(1+r3) + 5.2]=PV(3LLL)
P.V.
P.V.
P.V.
P.V.
cashflows
cashflows
cashflows
cashflows
P.V.
P.V.
P.V.
.
P.V.
P.V.
cashflows at node 2HH is {½×PV(3HHH)+½×PV(3HHL)}/1.070053 + 5.2.
cashflows at node 2HL is {½×PV(3HHL)+½×PV(3HLL)}/1.057354 + 5.2.
cashflows at node 2LL is {½×PV(3HLL)+½×PV(3LLL)}/1.046958 + 5.2.
cashflows at node 1H is {½×PV(2HH)+½×PV(2HL)]}/ 1.054289 + 5.2
cashflows at node 1L is {½×PV(2HL)+½×PV(2LL)]}/ 1.044448 + 5.2
solve r3 in the following equation:
100 = {½ × [PV(1H)] + ½ × [PV(1L)]}/1.035
By trial and error, find r3= r(3,LLL).
Hence, r(3,HLL) = r3 e0.2, r(3,HHL) = r3 e0.4 , r(3,HHH) = r3 e0.6 .
Step Five
Find at t=4 nodes: r(4,HHHH), r(4,HHHL), r(4,HHLL), r(4,HLLL), and
r(4,LLLL). Substitute given σ(4) = 10% p.a. into the expression involving
short rate in the boxes. Note r(3,HHH), r(3,HHL), r(3,HLL) and r(3,LLL)
found in step 4 are substituted at t=3 nodes.
Continue …
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3-Period Binomial Lattice
We saw that given the bonds (whether zeros or coupon)
prices at t=0 with maturities up to T, and in addition
specifying volatility σ(t) for t=1,2,3,….,T, we calibrate (i.e.
fix the short rates in each node) the tree up to t=T.
t=0
t=1
t=2
t=3
9.1987%
7.0053%
½
7.5312%
5.4289%
½
½
½
3.5%
5.7354%
6.1660%
½
4.4448%
½
4.6958%
5.0483%
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Role of Calibration
• Calibration not to derive no-arbitrage equilibrium prices
of (riskfree) option-free coupon bonds up to maturity T
– can be priced directly from the zero rates (or zero discount YC) by
bootstrapping the original set of coupon bonds (t=1,…,T).
– One intuition from the above is that therefore the calibration will not affect
such option-free bond prices.
• The calibration however affects the conditional short
rates on the nodes
– depending on the chosen volatility function σ(t).
• The conditional short rates affect the price of a security
whose payoffs or cash-flows are
– contingent on the short rates.
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Earlier,


1
1
1
×
×L ×
P(0, n ) = E P* 

1
+
r
0
1
+
r
1
1
+
r
n
−
1
(
)
(
)
(
)
(
)
(
)
(
)


where r(t), t>0, is a random variable taking conditional values r(t,HH..H),
r(t,HH..L), etc. This zero-coupon bond is not affected by calibration. But if a
European security payoff S is dependent on the last short rate or “state of the world
at t=n”, e.g. S(n) = S[r(n-1)], then the security's price C(0,n) is given by

S  r ( n − 1 ) 
1
1
C(0, n ) = E P* 
×
×L × 
 .
(1 + r ( n − 1 ) ) 
 (1 + r ( 0 ) ) (1 + r (1 ) )
Since S values at nodes at t=n, change with r(n-1), then given P(0,t),
t=0,1,2,….,n, a different calibration based on different volatility will
induce different sets of r(t)'s, and hence different values for C(0,n). In
other words, the information contained in P(0,t), t=0,1,2,….,n, alone is
insufficient to price C(0,n). The interest rate model (or process) is
required with specification of volatility in this case. Then the calibration
and thus the tree is required to price the interest rate option. The latter
would extend to American interest rate contingent claims, hence also
embedded options in bonds.
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Use the calibrated tree to price a CALLABLE BOND:
4 years to maturity. coupon rate 6.5%. callable (by issuer) after 1
year at flat call price of 100.
t=0
t=1
t=2
t=3
9.1987%
7.0053%
½
7.5312%
5.4289%
½
½
½
3.5%
5.7354%
6.1660%
½
4.4448%
½
4.6958%
5.0483%
P(0,1) = 1/1.035 = 0.96618
P(0,2) = 1/2[1/(1.035*1.054289)]+1/2[1/(1.035*1.044448)]=0.92075
P(0,3) = 1/4[1/(1.035*1.054289*1.070053)]+ 1/4[1/(1.035*1.054289*1.057354)]
+1/4[1/(1.035*1.044448*1.057354)] +1/4[1/(1.035*1.044448*1.046958)]=0.870405
P(0,4) =
1/8[1/(1.035*1.054289*1.070053*1.091987)]+1/8[1/(1.035*1.054289*1.070053*1.075312)]
+1/8[1/(1.035*1.054289*1.057354*1.075312)] +1/8[1/(1.035*1.054289*1.057354*1.06166)]
+1/8[1/(1.035*1.044448*1.057354*1.075312)] +1/8[1/(1.035*1.044448*1.057354*1.06166)]
+1/8[1/(1.035*1.044448*1.046958*1.06166)] +1/8[1/(1.035*1.044448*1.046958*1.050483)]
=0.814276
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price of option-free bond 4yr, 6.5%, =
P(0,4)*106.5+P(0,3)*6.5+P(0,2)*6.5+P(0,1)*6.5=104.643
COMPUTED CALLABLE BOND VALUE IF NOT CALLED
RESET ABOVE TO CALL VALUE 100 IF COMPUTED VALUE >
CALL PRICE (assume firm calls when bond price reaches call price)
1/2(97.925+100)
+ 6.5
discounted by
1.054289
t=0
t=1
t=2
½
½
t=3
97.925
7.0053%
100.032 100
5.4289%
102.899
3.5%
106.5/1.091987
97.529
9.1987%
99.041
7.5312%
½
½
½
1/2 (97.529 + 99.041)
+ 6.5
discounted by
1.070053
100.270 100
5.7354%
100.315 100
6.1660%
101.97 100
4.4448%
½
101.723 100
4.6958%
1/2(100+100)+6.5
discounted by 1.035
callable only after 1
yr
101.382 100
5.0483%
Note how the cashflows are contingent on different r(t)’s over time. It
can change from computed values to contingent call prices, i.e.
min (computed price, call price).
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Model Comparison
• By construction, the two models are equally able to fit the term
structure of interest rates; but
• The models generate important differences in the implied risk
neutral probability distribution of interest rates in the future
– The Ho-Lee model gives non-zero probability to negative
interest rates, and small probability to high interest rates
– The Simple BDT model gives essentially zero probability to
interest rates below 1%, but assigns higher probability to high
interest rates
• These differences are not important for bond prices, as both
models exactly match the term structure of interest rates; but
– they generate important differences for other securities that have
asymmetric payoff structures, such as options
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Risk-Neutral Interest-Rate
Distribution
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Caps
• A cap has multiple potential payoffs determined
by a settlement frequency and a maturity
– portfolio of single payment options
• At each settlement date, if the underlying index is
below the strike rate, no payments are exchanged
– aside from the premium
• If the underlying index exceeds the strike rate, the
seller of the cap must pay:
(Index Rate - Strike Rate) × (Days in settlement period ÷ 360)
× Notional Amount
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Cap Example
• A borrower has issued a floating rate note and is
paying LIBOR quarterly over the next three years
• By purchasing a cap with a 10% strike rate,
–
–
–
–
quarterly settlement frequency,
a maturity of three years,
and a notional amount equal to the amount of the note,
the borrower can hedge its exposure to increasing
LIBOR: insurance kicks in when?
• What happens if LIBOR rises?
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Swap Rates and Cap Prices
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Flat and Forward Volatility
• The Ho-Lee model appears to overprice short term
caps, and underprice long term caps, while the
• Simple BDT model in this case always underprices it
• One possible problem with the model is that the
volatility σ has been mis-measured
– The volatility of interest rates is time varying, and thus we
may be using the wrong level of volatility
• A single value of σ that makes the observed cap price
consistent with the model does not exist
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Summary
• Two approaches:
– estimate parameters, generate tree: Vasicek
– fit spot rate tree to market data, recover parameters
of process: Black-Derman-Toy
• Methodology:
– three up: NFL, current YC consistency, nonnegative spot rate
– one down: YC shapes - requires multi-factor models
• Lattice pricing and hedging
– T-bill options and T-bill futures
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