INTEREST-RATE PROCESSES, YIELDS AND BOND VALUATION ∗

8.5 INTEREST-RATE PROCESSES, YIELDS AND BOND VALUATION ∗

Bonds, derivative securities and most economic time series depend intimately on the interest-rate process. It is therefore not surprising that much effort has been devoted to constructing models that can replicate and predict reliably the evolution of interest rates. There are, of course, a number of such models, each expressing some economic rationale for the evolution of interest rates. So far we have mostly assumed known risk-free interest rates. In fact, these risk-free (dis- counting) interest rates vary over time following some stochastic process and as a function of the discount period applied. Generally, and mostly for convenience, an interest-rate process {r(t), t ≥ 0} is represented by an Ito stochastic differential equation:

dr = µ(r, t) dt + σ (r, t) dw

where µ and σ are the drift and the diffusion function of the process, which may or may not be stationary. Table 8.8 summarizes a number of interest rates models. Note that while Merton’s model is nonstationary (letting the

Table 8.8

Author

Diffusion Stationary Merton (1973)

Drift

no Cox (1975)

0 σ r 3/2 yes Vasicek (1977)

yes Dothan (1978)

β (α − r)

yes Brennan–Schwartz (1979)

yes Courtadon (1982)

β r [α − ln(r)]

yes March–Rosenfeld (1983)

β (α − r)

σ r δ/ + βr 2 yes Cox–Ingersoll–Ross (1985)

α r −(1−δ)

σ r 1/2 yes Chan et al. (1992)

β (α − r)

σ r λ yes Constantinidis (1992)

β (α − r)

σ+γr yes Duffie–Kan (1996)

α + βr + γ r 2

(α − r)

σ+γr yes

FIXED INCOME , BONDS AND INTEREST RATES

diffusion-volatility be time-variant), other models have attempted to model this diffusion coefficient. Of course, to the extent that such a coefficient can be mod- elled appropriately, the technical difficulties encountered when the coefficients are time-variant can be avoided and the model parameters estimated (even though with difficulty, since these are mostly nonlinear stochastic differential equations). Further, note that the greater part of these interest rate models are of the ‘mean reversion’ type. In other words, over time short-term interest rates are pulled back to some long-run average level. Thus when the short rate is larger than the average long rate, the drift coefficient is negative and vice versa. Black and Karasinski (1991) (see also Sandmann and Sonderman, 1993) have also suggested that interest models can be modelled as well as a lognormal process. Explicitly, let the annual effective interest rate be given by the nonstationary lognormal model:

dr a (t)

r a (t) (0) = r a, 0

= β(t) dt + σ (t) dW ; r a

and consider the continuously compounded rate R(t) = ln (1 + r a (t)). An appli- cation of Ito’s Lemma to this transformation yields also a diffusion process:

dR(t) = (1 − e −R(t) ) θ

(t) − 2 (1 − e −R(t) )σ dt + σ dW (t)

Another model suggested, and covering a broad range of distributional assumptions, includes the following (Hogan and Weintraub, 1993):

dR(t) = R(t) θ (t) − a ln R(t) + σ 2 dt + R(t)σ dW (t)

The valuation of a bond when interest rates are stochastic is difficult because we cannot replicate the bond value by a risk-free rate. In other words, when rates are stochastic there is no unique way to price the bond. Mathematically this means that there are ‘many’ martingales we can use for pricing the bond and determine its yield (the integral of the spot-rate process). The problem we are faced with is, therefore, to determine a procedure which we can use to select the ‘appropriate martingale’ which can replicate observed bond prices. Specifically, say that the interest-rate model is defined by a stochastic process which is a function of a

If this were the case, the theoretical price of a zero-coupon bond paying $1 at time T is:

B Th

∗ exp  − r

 =E ∗

253 that these bond prices can be observed at time zero for a whole set of future

INTEREST - RATE PROCESSES , YIELDS AND BOND VALUATION

times T and denote these observed values by, B obs (0, T ). In order to determine would minimize in some manner some function of the ‘error’

B =B obs (0, T ) − B Th

There are several alternatives to doing so, as well as numerous mathematical tech- niques we can apply to solving this problem. This is essentially a computational problem (see, for example, Nelson and Siegel, 1987; Wets et al., 2002; Kortanek and Medvedev, 2001; Kortanek, 2003; Delbaen and Lorimier, 1992; Filipovic, 1999, 2000, 2001).

The Nelson and Siegel approach is applied by many banks and consists in estimating the zero-coupon yield curve by fitting for all available bonds data in a sector credit combination the yield curve:

i ) are the model parameters. The Roger Wets approach (www.episolutions.com) is based upon a Taylor series approximation of the discount function in integral form. It is based on an approximation, and in this sense it shares properties with purely spline methods. Kortanek and Medvedev (2001), however, use a dynamical systems approach for modelling the term structure of interest rates based on a stochastic linear differential equation by constructing perturbation functions on either the unobservable spot interest rate or its integral (the yield) as unknown functions. Functional parameters are then estimated by minimizing a norm of the error comparing computed yields against observed yields over an observation period, in contrast to using the expectation operator for a stochastic process. When applied to a future period, the solved-for spot-rate function becomes the forecast of the unobservable function, while its integral approximates the yield function to the desired accuracy.

Some prevalent methods for computing (extracting) the zeros, curve-fitting procedures, equating the yield curve to observed data in the central bank include, among others: in Canada using the Svensson procedure and David Bolder (Bank of Canada); in Finland the Nelson–Siegel procedure; in France, the Nelson–Siegel, Svensson procedures; in Japan and the USA the banks use smoothing splines etc. (see Kortanek and Medvedev, 2001; Filipovic, 1999, 2000, 2001). Explicit solutions can be found for selected models, as we shall see below when a number of examples are solved. In particular, we shall show that approaches based on the optimal control of selected models can also be used.

FIXED INCOME , BONDS AND INTEREST RATES

8.5.1 The Vasicek interest-rate model

The Vasicek model has attracted much attention and is used in many theoretical and empirical studies. Its validity is of course, subject to empirical verification. An analytical study of the Vasicek model is straightforward since it is a classical model used in stochastic analysis (also called the Ornstein–Uhlenbeck process, as we saw in Chapter 4). In Vasicek’s model the interest-rate change fluctuates around a long-run rate, α. This fluctuation is subjected to random and normal

perturbations of mean zero and variance σ 2 dt however. dr = β(α − r) dt + σ dw

This model’s solution at time t when the interest rate is r (t) is r (u; t):

−β(u−t)

(u; t) = α + e −β(u−τ ) (r (t) − α) + σ e dw(τ )

determining a number of martingales (or bond prices) that obey the model above, namely bond prices at time t = 0 can theoretically equal the following:

B th (0, T ; α, β, σ ) = E ∗ exp  − r (u; α, β, σ ) du 

In this simple case, interest rates have a normal distribution with a known mean and variance (volatility) evolution. Therefore

r (u,α, β, σ ) du

has also a normal probability distribution with mean and variance given by: m (r (0), T ) = αT + (1 − e −βT

3 (4 e −βT −e −2βT 2β + 2βT − 3) In these equations the variance is independent of the interest rate while the mean

v (r (0), T ) = v(T ) =

is a linear function of the interest which we write by: m

(1 − e −βT ) (r (0), T ) = α T−

(1 − e −βT )

+ r(0)

β This property is called an affine structure and is of course computationally de-

sirable for it will allow a simpler calculation of the desired martingale. Thus, the

255 theoretical zero-coupon bond price paying $1 T periods hence can be written by:

INTEREST - RATE PROCESSES , YIELDS AND BOND VALUATION

B th (0, T ; α, β, σ ) = E exp  r (u, α, β, σ ) du 

=e −m(r(0),T )+v(T )/2

(4 e 3 −βT −e −2βT + 2βT − 3);

Now assume that a continuous series of bond values are observed and given by

B obs (0, T ) which we write for convenience by, B obs −R T (0, T ) = e T . Without loss of generality we can consider the yield error term given by:

T =R T − (A(T ) − r 0 D (T ))

and thus select the parameters (i.e. select the martingale) that is closest in some sense to observed values. For example, a least squares solution of n observed bond values yields the following optimization problem:

α,β,σ Min

i =1

When the model has time-varying parameters, the problem we faced above turns out to have an infinite number of unknown parameters and therefore the yield curve estimation problem we considered above might be grossly underspecified. Explicitly, let the interest rate model be defined by:

dr (t) = β [α(t) − r(t)] dt + σ dw

The theoretical bond value has still an affine structure and therefore we can write:

B th [t, T ; α(t), β, σ ] = E ∗ exp  − A r (u; α, β, σ ) du  =e (t,T )−r(t)D(t,T )

The integral interest-rate process is still normal with mean and variance leading to:

2 2 1 −β(T −t) A & (t, T ) =

2 (s, T ) − βα(s)D(s, T ) ds; D(t, T ) = β %1 − e

or:

d A(t, T )

= α(t) %1 − e −β(T −t)

%1 − e −β(T −t) & 2 , A (T , T ) = 0

dt

FIXED INCOME , BONDS AND INTEREST RATES

in which α(t), β, σ are unspecified. If we equate this equation to the available bond data we will obviously have far more unknown variables than data points and therefore the yield curve estimate will depend again on the optimization technique we use to generate the best fit parameters β ∗ ,σ ∗ and the function, α ∗ (t). Such problems can be formulated as standard problems in the calculus of variations (or optimal control theory). For example, if we consider the observed prices B obs (t, T ), t ≤ T < ∞, for a specific time, T , and minimize the follow- ing squared error in continuous time, we obtain the following singular control problem:

Min

2 du α (u) = [ A(u, T ) − c(u, T )]

subject to:

d A(u, T ) du

= α(u)a(u, T ) − b(u, T ), A(T, T ) = 0 with

c (u, t) = y obs −β(T −u) (u, T ) + r(u) & ,

β %1 − e

2 %1 − e 2β −β(T −u) and α(u) is the control and A(u, T ) is the state which can be solved by the

a (u, t) = %1 − e

−β(T −u)

& ; b(u, t) =

usual techniques in optimal control. The solution of this problem leads either to

a bang-bang solution, or to a singular solution. Using the deterministic dynamic programming framework, the long-run (estimated) rate is given by solving:

∂ A [α(u)a(u, T ) − b(u, T )] culate α(u), we can proceed by a change of variables and transform the original

= Min α (u) [ A(u, T ) − c(u, T )] + ∂ u

control problem into a linear quadratic control problem which can be solved by the standard optimal control methods. Explicitly, set:

y (u) = [A(u, T ) − c(u, T )] dw(u)

with du = α(u) and z(u) = y(u) − a(u, T )w(u) Thus, the problem is reduced to:

Min

w (u) = [z(u) + a(u, T )w(u)] du

257 subject to: dz(u)

INTEREST - RATE PROCESSES , YIELDS AND BOND VALUATION

˙a(u, T ) = da(u, T )/du and at time T,

= − ˙a(u, T )w(u) − b(u, T ) − c(u, T ); du

z (T ) = −c(T, T ) − a(T, T )w(T )

This is a linear control problem whose objective is quadratic in both the state and the control. As a result, the problem solution of this standard control problem is the linear feedback form:

w (u) = Q(u) + S(u)z(u) or α(t) = w (u) du

The functions Q(u), S(u) can be found by inserting in the problem’s conditions for optimality. This problem is left for self-study, however (see also Tapiero, 2003).

Problem: The cox–ingersoll–Ross (CIR) model

By changing the interest-rate model, we change naturally the results obtained. Cox, Ingersoll and Ross (1985), for example, suggested a model, called the square root process , which has a volatility given as a function of interest rates as well, namely, they assume that:

√ dr = β (α − r) dt + σ r dw

First show that the interest rate process is not normal but its mean and variance are given by:

0 ) = c(t) σ 2 +ξ ; Var(r (t) |r 0 ) = c(t) σ 2 + 4ξ where

σ 2 [exp(βt) − 1] Demonstrate then that this process has an affine structure as well by verifying

that:  T

B (r, t, T ) = E exp  − r (T − u) du  =e ( A(t,T )−r D(t,T )) ; B(r, T , T ) = 1

and at the boundary A(T , T ) = 0, D(T, T ) = 0. Finally, calculate both A(t, T ) and D(t, T ) and formulate the numerical problem which has to be solved in order

to determine the bond yield curve based on available bond prices.

FIXED INCOME , BONDS AND INTEREST RATES

Problem: The nonstationary Vasicek model

Show for the nonstationary model dr = µ(t)(m(t) − r) dt + σr dw that its solu- tion is:

 r (t) = exp [−A(t)]  y+ µ (s)m(s) exp [−A(s)] 

A (t) = M(t) + σ 2 t/ 2−σ dw and M (t) = µ (s) ds

8.5.2 Stochastic volatility interest-rate models

Cotton, Fouque, Papanicolaou and Sircar (2000) have shown that a single factor model (i.e. with one source of uncertainty) driven by Brownian motion implies perfect correlation between returns on bonds for all maturities T , which is not seen in empirical analysis. They suggest, therefore, that the volatility in the Va- sicek model ought to be stochastic as well. Their derivation, based on a mean reverting model in the short rate, shows an exponential decay in the short-term, (two weeks). This is small compared to bonds with maturities of several years.

Denote the variance in an interest model by V = σ 2 (r, t), then an interest-rate ‘stochastic volatility model’ consists of two stochastic differential equations, with

two sources of risk (w 1 ,w 2 ) which may be correlated or not. An example would be:

√ dr = µ(r, t) dt + V (r, t) dw 1

dV = ν(V, r, t) dt + γ (V, r) dw 2

where the variance V appears in both equations. Hull and White (1988) for ex- ample suggest that we use a square root model with a mean reverting variance model given by:

dr √

= µ dt + V dw λ 1 ; dV = α(β − V ) dt + γ r V dw 2 ,ρ dt = E dw 1 dw r 2 In this case, note that when stock prices increase, volatility increases. Further

when volatility increases, interest rates (or the underlying asset we are modeling) increase as well. Cotton et al. (2000), in contrast, suggested that, in a CIR-type model such as dr = (µ − r) dt + σr γ dW , γ is not equal to a half but rather is equal to one and half and thereby certainly greater than one. The model they suggest turns out:

dr = θ r (µ r − r) dt + α r +β r V dW ∗

dV = θ V (µ V − V ) dt + α V +β V V dZ ∗ where (dW ∗ , dZ ∗ ) are Brownian motion under the pricing measure. Note here

that the volatility is a mean reverting driving process. The advantage in using such a model is that it also leads to an affine structure where the time-dependent

259 coefficients are given by the solutions of differential equations. In this case, esti-

INTEREST - RATE PROCESSES , YIELDS AND BOND VALUATION

mation of the yield curve can be reached, as we have stated above, by the solution of an optimal control problem. In other words, once a theoretical estimate of the bond price is found, and observed bond prices are available, we can calculate the parameters of the model by solving the appropriate optimization problem.

8.5.3 Term structure and interest rates

Interest rates applied for known periods of time, say T , change necessarily over time. In other words, if r (t, T ) is the interest rate applied at t for T , then at t + 1, the relevant rate for this period T would be r (t + 1, T − 1), while the going interest for the same period would be r (t + 1, T ). If these interest rates are not equal, there may be an opportunity for refinancing. As a result, the evolution of interest rates for different maturity dates is important. For example, if a model is constructed for interest rates of maturity T , then we may write:

dr (t, T ) = µ(r, T ) dt + σ (r, T ) dw

The price of a zero-coupon bond is a function of such interest rates and is given by B(t, T ) = exp [−r(t, T )(T − t)] whose differential equation (see the mathe-

matical Appendix to this chapter) ∂ B ∂ B 1 ∂ 2 B 2

0= + t

∂ r [µ(r, T ) − λ(r, t)] + 2 ∂ r 2 ∂ σ (r, T ) − r B

B (r, T , T ) = 1

where the price of risk, a known function of r and time t, is proportional to the returns standard deviation and given by:

1 ∂ B α (r, t, T ) = r + λ(r, t)

The solution of this equation, although cumbersome, can in some cases be deter- mined analytically, and in others it can be solved numerically. For example, if we

2 set (µ(r, T ) − λ(r, t)) = θ; σ 2 (r, T ) = ρ where (θ, ρ) are constant then a solu- tion of the partial differential equation of the bond price (see the Mathematical

Appendix), we obtain an affine structure type:

1 2 3 (r, t, T ) = exp −r(T − t) − θ

2 Set the following equalities: µ(r, T ) − λ(r, t) = k(θ − r); σ 2 (r, T ) = ρ r (which is the CIR model seen earlier) and show that the solution for the bond

price equation is of the following form:

B (r, t, T ) = exp {A(T − t) + r D(T − t)}

A solution for the function A(.) and D(.) can be found by substitution.

FIXED INCOME , BONDS AND INTEREST RATES