FIXED-INCOME SECURITIES
Chapter 12
Modeling the Yield Curve
Dynamics
Outline
•
•
•
•
•
Motivation
Interest Rate Trees
Single-Factor Continuous-Time Models
Multi-Factor Continuous-Time Models
Arbitrage Models
Motivation
Why do we Care?
•
Pricing (and hedging) of fixed-income securities that pay cashflows which are known with certainty at the initial date (e.g.,
plain vanilla bonds
– Boils down to a computation of the sum of these cash-flows, discounted at a
suitable rate
– Challenge for the bond portfolio manager is therefore limited to being able
to have access to a robust methodology for extracting implied zero-coupon
prices from market prices (see Chapter 4)
•
Pricing and hedging fixed-income securities that pay uncertain
cash-flows (e.g., options on bonds)
– It requires not only the knowledge on discount factors at the present date,
but also some kind of understanding of how these discount factors (i.e., the
term structure of pure discount rates) are going to evolve over time
– In particular, one needs to account for potential correlations between the
discount factor and the promised payoff, volatility of the payoff, etc.
– Some dynamic model of the term structure of interest rates is therefore
needed
Interest Rate Tree
Binomial Model
• General binomial model
– Given current level of short-term rate r, next-period short rate, can
take on only two possible values: an upper value ru and a lower
value rl, with equal probability 0.5
– In period 2, the short-term interest rate can take on four possible
values: ruu, rul, rlu, rll
– More generally, in period n, the short-term interest rate can take on
n
2 values => very time-consuming and computationally inefficient
• Recombining trees
– Means that an upward-downward sequence leads to the same
result as a downward-upward sequence
– For example, rul = rlu
– Only (n+1) different values at period n
Interest Rate Tree
Binomial Model
Period 0
Period 1
Period 2
Period 3
Period 4
ruuuu
ruuu
ruu
ru
r
ruuul
ruul
rul
rl
ruull
rull
rll
rulll
rlll
rllll
Recombining tree
…
Interest Rate Tree
Analytical Formulation
• We may write down the binomial process as
Δ rt ≡ rt +1 − rt = σε t
where εt are independent variables taking on values (+1,-1) with proba (1/2,1/2)
• Problem is rates can take on negative values with
positive probability
• Fix that problem by working with logs
Δ ln rt ≡ ln rt +1 − ln rt = σε t
⇒ rt +1
⎛ u = exp (σ ) ⎞
⎟⎟
= rt × exp (σε t ) = rt × ⎜⎜
⎝ d = exp (− σ )⎠
with probability (1/2,1/2)
Interest Rate Tree
Analytical Formulation
• More general models (could be written on log rates)
Δ rt ≡ rt + Δt − rt = μ (t , Δ t , rt ) + σ (t , Δ t , rt )ε t
• Specific case
Δ rt ≡ rt + Δt − rt = μ Δ t + σ Δ t ε t
– Focus and state- and time-independent models
– Square-root of time law is consistent with the absence of serial
correlation (independent increments feature in the random walk)
• Continuous-time limit (Merton (1973))
drt ≡ rt + dt − rt = μ dt + σ dW t
Interest Rate Tree
Calibration
• Calibration of the model is performed so as to make
model consistent with the current term structure
• We have at date 0
Δ ln r0 ≡ ln rΔt − ln r0 = μ Δ t + σε 0 Δ t
(
⇒ ln ru − ln rl = 2σ Δ t or ru = rl exp 2σ Δ t
)
• We take as given an estimate for σ, the current par
yield curve yt, and we iteratively find the values ru, rl,
ruu, rul, rlu, rll, etc., consistent with the input data
Interest Rate Tree
Calibration – Con’t
• Consider a 2 period tree with Δt = 1 for simplicity
• The price one year from now of the 2-year par
Treasury bond can take two values: a value Pu
associated with ru, and a value Pl, associated with rl
100 + y 2
100 + y 2
and Pd =
Pu =
1 + ru
1 + rl
• Then, taking expectations at time 0, we find an
equation that can be solved for ru and rl
100 + y 2
⎛ 100 + y 2
⎞
+ y2
+ y2 ⎟
⎜
1 + rl
1 ⎜ 1 + rl exp (2σ )
⎟
100 =
+
⎟
2⎜
1 + y1
1 + y1
⎜
⎟
⎝
⎠
Interest Rate Tree
Calibration – Time for an Example!
• Consider current Treasury bond par yield curve: y1 =
4%, y2 = 4.30%
• We want to calibrate a binomial interest rate tree,
assuming a volatility of 1% for the one-year interest
rate
• We have
100 + 4 . 3
⎛ 100 + 4 . 3
⎞
+ 4 .3
+ 4 .3 ⎟
⎜
⎧ rl = 4 . 57 %
1 + rl
1 ⎜ 1 + rl exp (. 02 )
⎟
⇒ ⎨
100 =
+
⎟
2⎜
1 + 4%
1 + 4%
⎩ ru = 4 . 66 %
⎜
⎟
⎝
⎠
Single-Factor Continuous-Time Models
General Formulation
• General expression for a single-factor continuoustime model
drt = μ (t , rr )dt + σ (t , rr )dW t
• The term W denotes a Brownian motion, which a
process with independent normally distributed
increments
–
–
–
–
dW represents the instantaneous change.
It is stochastic (uncertain)
It behaves as a normal distribution with zero mean and variance dt
It can be thought of as
dW t = ε t dt
Single-Factor Continuous-Time Models
Popular Models
• All popular fall into the following class
drt = [μ1 + μ 2 rt ]dt + [σ 1 + σ 2 rt ] dWt
α
• Listing of popular models
Model
Ω1 Ω2 α1 α2 ϑ
Brennan-Schwartz (1980)
Φ
Φ
Φ
1
Cox-Ingersoll-Ross (1985) Φ
Φ
Φ
0.5
Φ
1
Dothan (1978)
Merton (1973)
Φ
Φ
Pearson-Sun (1994)
Φ
Φ
Φ
Vasicek (1977)
Φ
Φ
Φ
1
Φ
0.5
1
Single-Factor Continuous-Time Models
What is a good Model?
• A good model is a model that is consistent with reality
• Stylized facts about the dynamics of the terms
structure
– Fact 1: (nominal) interest rates are positive
– Fact 2: interest rates are mean-reverting
– Fact 3: interest rates with different maturities are imperfectly
correlated
– Fact 4: the volatility of interest rates evolves (randomly) in time
• A good model should also be
– Tractable
– Parsimonious
31
/1
29 /90
/
30 6/9
/1 0
1
30 /90
/4
30 /91
/9
28 /91
/2
31 /92
/
31 7/9
/1 2
2
31 /92
/
29 5/9
/1 3
0
31 /93
/3
31 /94
/8
31 /94
/1
30 /95
/
30 6/9
/1 5
1
30 /95
/4
30 /96
/9
28 /96
/2
31 /97
/
31 7/9
/1 7
2
29 /97
/
30 5/9
/1 8
0
31 /98
/3
31 /99
/8
/9
9
DOW JONES Index Value
12000
30
10000
25
8000
20
6000
15
4000
10
2000
5
0
0
Fed Fund Rate (in %)
Single-Factor Continuous-Time Models
Empirical Facts 1, 2 and 4
Single-Factor Continuous-Time Models
Empirical Fact 3
1M
3M
6M
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
1M
1
3M
0.992 1
6M
0.775 0.775 1
1Y
0.354 0.3
2Y
0.214 0.165 0.42
3Y
0.278 0.246 0.484 0.79
4Y
0.26
5Y
0.224 0.179 0.381 0.737 0.879 0.935 0.981 1
6Y
0.216 0.168 0.352 0.704 0.837 0.892 0.953 0.991 1
7Y
0.228 0.182 0.35
8Y
0.241 0.199 0.351 0.614 0.745 0.826 0.892 0.936 0.968 0.992 1
9Y
0.238 0.198 0.339 0.58
9Y
10Y
0.637 1
0.901 1
0.946 1
0.225 0.444 0.754 0.913 0.983 1
0.661 0.792 0.859 0.924 0.969 0.991 1
0.712 0.798 0.866 0.913 0.95
0.981 0.996 1
10Y 0.202 0.158 0.296 0.576 0.705 0.779 0.856 0.915 0.952 0.976 0.985 0.99 1
Daily changes in French swap markets in 1998
Single-Factor Continuous-Time Models
Vasicek Model
• Vasicek (1977) model
drt = a(b − rt )dt + σdWt
• This process exhibit a mean-reverting feature
– The parameter b may be regarded as the equilibrium level of the
short-term interest rate, around which it stochastically evolves
– When r falls far below (above) its long-term value b, the expected
instantaneous variation of r is positive (negative)
– In this case, the short-term rate will tend to move up (down)
– It will move towards its long-term value quickly when it is far from it
and when the parameter a (speed of return to the long-term mean
value) is high
• On the other hand, it is not consistent with facts 1, 3
and 4
Single-Factor Continuous-Time Models
Cox-Ingersoll-Ross Model
• CIR (1985) model
drt = a(b − rt )dt + σ rt dWt
•
•
•
•
This process exhibit a mean-reverting feature
It also prevents interest rates to become negative
It exhibits a stochastic volatility component
On the other hand
– Interest rate and volatility risks are perfectly correlated
– Interest rates of different maturities are perfectly correlated (only
one source of randomness)
Multi-Factor Continuous-Time Models
Some Popular Models
• Fong and Vasicek (1991) model
– Fong and Vasicek (1991) take the short rate and its volatility as two
state variables
– Variance of the short-rate changes is a key element in the pricing of
fixed-income securities, in particular interest rates derivatives
• Longstaff and Schwartz (1992) model
– Longstaff and Schwartz (1992) use the same two state variables,
but with a different specification
– Allows them to get closed-form solution for the price of a discount
bond and a call option on a discount bond
• Chen (1996) and Balduzzi et al. (1996) models
– Chen (1996) and Balduzzi et al. (1996) suggest the use of a threefactor model by adding the short-term average of the short rate
– These three state variables can be assimilated to the three factors
which can be empirically obtained though a PCA of the term
structure dynamics
Arbitrage Models
Calibration of Continuous-Time Factor Models
• In practice, single or multi-factor models are
calibrated in such a way that the models' parameters
are obtained as solutions to a minimization program
of the difference (i.e., the squared spread) between
market prices of reference bonds and theoretical
values generated from the model
• This is the analogue of the calibration of an interest
rate tree
• Then the model is used to price interest rates
derivatives (see Chapters 14 and 15)
• The difference between the derived yield curve and
the observed curve, even though minimized, can not
be entirely eliminated
Arbitrage Models
Some Popular Models
• These models are built to be consistent with currently
observed bond prices, which makes them popular
among practitioners
• First example is Ho and Lee (1986)
– Discrete-time binomial setting
– Discount bond prices are driven by a single source of uncertainty
• Heath, Jarrow and Morton (1990ab,1992) have
generalized this approach by allowing discount bonds
prices to be driven by a multi-dimensional uncertainty
in a continuous-time framework
• Markovian versions of the HJM model are often used
– Translate into recombining trees in discrete-time
– Can be implemented without too much numerical complexity