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 cash-flows 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 cashflows (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
∆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
∆ 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)
∆rt ≡ rt +∆t − rt = µ ( t , ∆t , rt ) + σ ( t , ∆t , rt )ε t
• Specific case
∆rt ≡ rt +∆t − rt = µ∆t + σ ∆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 = µdt + σdWt
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
∆ ln r0 ≡ ln r∆t − ln r0 = µ∆t + σε 0 ∆t
(
⇒ ln ru − ln rl = 2σ ∆t or ru = rl exp 2σ ∆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 ∆t = 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 + y2 100 + y2 Pu = and Pd = 1 + ru 1 + rl • Then, taking expectations at time 0, we find an equation that can be solved for ru and rl
100 + y2 100 + y2 + 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 ) dWt
• 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
dWt = 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
1 2 1 2
Brennan-Schwartz (1980)
1
Cox-Ingersoll-Ross (1985)
0.5
1
Dothan (1978) Merton (1973)
Pearson-Sun (1994)
Vasicek (1977)
1
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