The models considered in this chapter take the form of the following one-factor SDE:
(1) wheref and g are suitably chosen functions, θ is determined by the mar- ket, and ρ can be chosen by the user of the model or dictated by the market. We will show that θ is the drift of the short rate and ρ is the ten- dency to an equilibrium short rate. The term σ is the local volatility of the short rate. The term arises from a normally distributed Wiener process, since ε ∼ N(0,1), where N(0,1) is the normal distribu- tion with mean 0 and standard deviation of 1. This means that the term σ(r(t),t)dz has an average or expected value of 0.
Equation (1) has two components. The first component is the expected or average change in rates over a small period of time, dt. This is the com- ponent where certain characteristics of interest rates, such as mean rever- sion, are incorporated. The second component is the unknown or the risk term since it contains the random term. This term dictates the distribution characteristics of interest rates. Depending on the model, interest rates are either normally or lognormally distributed.
The Ho-Lee Model
In the HL model or process f(r) = r, g(r) = 0, and ρ = 0 in equation (1).
The HL process is, therefore, given by
dr = θdt + σdz (2)
Sincez is a normally distributed Wiener process, we say the HL process is a normal process for the short rate. The solution to equation (2), assumingr(0) = r0 is given by
(3a) where the integral involving σ is a stochastic integral. If θ is constant this can be expressed as
(3b) Equation (3b) shows that the HL process models an interest rate that can change proportionally with time t through the constant of propor-
df r t( ( )) = [θ( ) ρt + ( )t g r t( ( ))]dt+σ(r t( ),t)dz
dz = ε dt
r t( ) r0 θds
0 t
∫ σdz
0 t
∫
+ +
=
r t( ) r0 θt σdz
0 t
∫
+ +
=
A Review of No Arbitrage Interest Rate Models 43
tionality, θ, and a random disturbance determined by σ. That is, the larger θ is in magnitude the larger the average change in the short rate over time. This is why θ is called the “drift in the short rate.” Also, the smallerθ is the larger the influence of the random disturbance. The short rate can be negative in the HL process. This is a shortcoming of the model. Hull shows that θ is related to the slope of the term structure.14
To obtain a numerical approximation for equation (2) we approxi- mate equation (2) by using equations (3a) and (3b). Letting tk = kτ and rk≈r(kτ) gives
or
(4) where ∆zk is a numerical (discrete) approximation to dz. Since
, we can further approximate equation (4) by
(5) where εk is a random number given by a normal distribution N(0,1).
Equation (5) is the form of the expression that is used for rk+1 to build the HL binomial tree.
We first consider the solution to equation (5) without the stochastic term when θ is constant. Equation (5) under these requirements is
(6a) and the solution is given by
(6b) where c and δ are constants. In particular, c = r0 and δ = θτ. It is seen from this last equation that the mean short rate in the HL process increases or decreases at a constant rate θ over time depending on the sign of θ. As a matter of fact, equation (6b) shows that the short rate grows without bound if θ > 0 and decreases without bound (i.e.
becomes very negative) if θ < 0.
14J. Hull, Options, Futures, and Other Derivatives, Fourth Edition (Saddle River, NJ: Prentice Hall, 2000).
rk+1–rk = θkτ σ+ k∆zk
rk+1 = rk+θkτ σ+ k∆zk
dz = ε dt
rk+1 = rk+θkτ σ+ kεk τ
rk+1 = rk+τθ
rk = c+kδ
3-Buetow/Sochacki Page 43 Thursday, August 29, 2002 10:01 AM
The Hull-White Model
In the HW model or process f(r) = r, g(r) = r, and ρ = −φ. Therefore, the stochastic process for the HW model for the short rate is
dr = (θ − φr)dt + σdz (7) The short rate process in the HW model is seen to be normal as in the HL process. We consider the case where the parameters θ and φ are con- stant over time. Note that if φ = 0 the HL process reduces to the HW process. (The HW process will, therefore, be similar to the HL process if φ is close to 0.) We will see that the introduction of φ in the HW model is an attempt to incorporate mean reversion and to correct for the uncon- trolled growth (or decline) in the HL model shown later in this chapter.
Eliminating the stochastic term in equation (7) gives the ordinary differential equation
dr = (θ − φr)dt (8)
whose solution is given by
(9)
where
(10) Ifφ > 0 we see from equation (9) that
Therefore, for positive mean reversion (φ > 0) the HW process will con- verge to the short rate, à. Due to this, the term à is called the “target”
or “long run mean rate.” For negative mean reversion (φ < 0), the short rate grows exponentially over time.
Factoringφ in equation (7) leads to dr = φ(à −r)dt + σdz and eliminating the stocastic term leads to
r t( ) θ φ---+ce–φt
=
c r0 θ φ--- –
=
r t( )
tlim→∞ θ φ--- à
= =
A Review of No Arbitrage Interest Rate Models 45
dr = φ(à −r)dt
We see that if r > à then dr is negative and r will decrease and if r < à thendr is positive and r will increase. That is, rwill approach the target rateà. The larger φ is the faster this approach to the target rate à. This is why φ is called the “mean reversion” or “mean reversion rate.” It reg- ulates how fast the target rate is reached. However, it does not eliminate the negative rates that can occur in the HL process.
Since the target rate à is equal to θ/φ, we can solve for the drift, θ, or the mean reversion, φ. That is,
θ = àφ (11)
or
(12) It is seen from equations (11) and (12) that there is a strong rela- tionship between the drift and mean reversion that can be used to reach any desired target rate. How large the mean reversion should be is an important financial question. Equations (11) and (12) can be used to set target rates. Equations (9) and (10) allow one to determine how long it takes to reach the target rate.
Approximating equation (7) gives us
(13) If θ and φ are constant and we eliminate the stochastic term then the solution to equation (13) has the form
To determine α, β, and γ we substitute this form for rk into equation (13) under these conditions and obtain that β = (1 − φτ),γ = θ/φ = à, and α = r0− à. Therefore,
(14) Note that if 0 < φτ < 2 then −1 < 1 − φτ < 1 and
φ θ
à---
=
rk+1 = rk+(θk–φkrk)τ σ+ kεk τ
rk = αβk+γ
rk α(1–φτ)k θ φ--- +
=
3-Buetow/Sochacki Page 45 Thursday, August 29, 2002 10:01 AM
which is the same result we obtained from equation (9) for the HW SDE.
The condition 0 < φτ < 2 is easily maintained in modeling the short rate.
The Kalotay-Williams-Fabozzi Model
For the KWF process f(r) = ln(r),g(r) = 0, and ρ = 0 in equation (1). This leads to the differential process
d ln(r) = θdt + σdz (15a) This model is directly analogous to the HL model. If u = ln r then we obtain the HL process (equation(2)) for u
(15b) Because u follows a normal process, ln(r) follows a normal process and so r follows a lognormal process. Since u follows the same process as the HL and HW models, u can become negative, but u= ln(r) and r = eu ensuring r is always positive. Therefore, the KWF model eliminates the problems of negative short rates that occurred in the HL and HW models.
Eliminating the stochastic term in equation (15) we obtain d ln(r) = θ(t)dt
and
du = θ(t)dt From equation (3a) we have
sinceu(0) = ln r(0) = ln r0,
Taking the exponential of both sides gives us rk
klim→∞ θ φ--- à
= =
du = θdt+σdz
lnr t( ) u u( )0 θ( )s ds
0
∫t
+
= =
lnr t( ) lnr( )0 θ( )s ds
0
∫t
+
=
A Review of No Arbitrage Interest Rate Models 47
(16) showing that r(t) > 0 since r(0) > 0. Therefore, if θ(t) > 0 the short rate in the KWF process grows without bound and if θ(t) < 0 the short rate in the KWF process decays to 0.
From equation (5) for the HL process the discrete approximation to equation (15b) is
(17a) and the exponential of this equation gives the discrete approximation to equation (15a):
(17b) From equation (17b) and equation (16) we see that the numerical approximation to equation (15a) has similar properties to the solution to the HL SDE. That is, if θ(t) > 0 the short rate grows without bound and if θ(t) < 0 the short rate decays to 0.
The Black-Karasinski Model
In the BK model we set f(r) = ln r,ρ = −φ, and g(r) = ln r in equation (1) to obtain the SDE
d ln r = (θ − φ lnr)dt + σdz (18a) We now work with equation (18a) using equation (7) for the HW pro- cess in a manner similar to how we used results from the HL process to develop the KWF process. If we let u = ln r in equation (18a) we obtain du = (θ − φu)dt + σdz (18b) which is the HW process for u. Again, note that u has all the same prop- erties as r in the HW model. Since r = eu in the BK process, r > 0. This is the advantage the BK model has over the HW model. Therefore, we see that the BK process is an extension of the KWF process as the HW pro- cess is an extension of the HL process. The main difference is the BK is a lognormal extension of the lognormal KWF process. As a matter of fact, ifφ = 0 the BK process reduces to the KWF process. Black and Karasinski introducedφ to control the growth of the short rate in the KWF process.
r t( ) r0e θ
( )s ds 0
∫t
=
uk+1 = uk+θkτ σ+ kεk τ
rk+1 = rkeθkτ σ+ kεk τ
3-Buetow/Sochacki Page 47 Thursday, August 29, 2002 10:01 AM
From equation (9) we have
and after taking exponentials
(19) Forφ < 0 we see that r grows without bound and that for φ > 0
The target rate for the BK process is the exponential of the target rate for the HW process.
As in the HW process, from equation (19) (or equations (9) and (10)) we see that
(20) in the BK process. The closer the initial rate is to the target rate the faster the BK process converges to the target rate. From equations (19) and (20) we see that if the initial short rate is the target rate then r(t) = à for all t in the BK process which is analogous to the HW process.
Given the target rate à we can solve for the drift or the mean rever- sion similarly to equations (11) and (12) in the HW model. We have
θ =φ ln à (21)
and
(22) We discretize u = ln r in equation (18b) just as we did for the HW SDEs and then let r = eu. This is analogous to how we used the HL discrete process to get the KWF discrete process. The equations corresponding to equation (13) are
u t( ) θ φ---+ce–φt
=
r t( ) eu t( ) e
θφ ---+ce–φt
= =
r t( )
tlim→∞ e
θφ
--- à
= =
c lnr0 θ φ--- –
=
φ θ
lnà ---
=
A Review of No Arbitrage Interest Rate Models 49
(23a) or after taking the exponential of both sides of equation (23a)
(23b) For constant θ and φ (similarly to equation (14)), the solution to equa- tion (23b) after eliminating the stochastic term is
(24) Note from equation (24) that
for 0 < φτ < 2. This is similar to the result we obtained from equation (14) for the HW SDEs.
The Black-Derman-Toy Model
The Black-Derman-Toy (BDT) model is a lognormal model with mean reversion, but the mean reversion is endogenous to the model. The mean reversion in the BDT model is determined by market conditions.
The equation describing the interest rate dynamics in the BDT model hasf(r) = ln r and g(r) = ln r in equation (1) as in the BK model. There- fore, the short rate in the BDT model follows the lognormal process
d ln r + [θ(t) + ρ(t) ln r]dt + σ(t)dz
However, in the BDT model giving us
(25a)
Making the substitution u = ln r leads to
(25b) uk+1 = uk+(θk–φkuk)τ σ+ kεk τ
rk+1 = rke(θk–ϕklnrk)τ σ+ kεk τ
rk e
α(1–φr)k θ φ--- +
=
rk
klim→∞ e
θφ
--- à
= =
ρ( )t d
dt---lnσ( )t σ'( )t
σ( )t ---
= =
dlnr θ( ) σt '( )t σ( )t ---lnr
+
dt+σ( )t dz
=
du θ( ) σt '( )t σ( )t ---u
+
dt+σ( )t dz
=
3-Buetow/Sochacki Page 49 Thursday, August 29, 2002 10:01 AM
Notice the similarity in equations (25) and the equations (18) of the BK model. We expect
to behave similarly to −φ(t) in the BK model. This expression should give mean reversion in the short rate when it is negative. That is, we expect that if (implying σ(t) is decreasing) then the BDT model will give mean reversion. On the other hand, when (implying σ(t) is increasing) the short rates in the BDT model will grow with no mean reversion. If σ(t) is constant in the BDT model, then so ρ = 0 and equation (25a) becomes the KWF model (equation (15)). Therefore, we will only study the case of varying local volatility for the BDT model.
Eliminating the stochastic term in equation (25) leads to
(26)
Solving this equation for u as we did in the KF and BK models, gives us
or
or
(27) Note that the BDT mean short rate depends on the local volatility. If the local volatility has a decreasing structure, then the first exponential term in equation (27) has a negative exponent and will cause a decrease in the short rate and vice versa if the local volatility has an increasing structure. It is important to note that mean reversion in the BDT model comes from the local volatility structure (i.e., it is endogenous).
σ'( )t σ( )t ---
σ'( )t <0
σ'( )t >0
σ'( )t = 0
dlnr du θ( ) σt '( )t σ( )t ---u
+
dt θ( ) σt '( )t σ( )t ---lnr
+
dt
= = =
u t( ) u( )0 σ( )0
--- θ( )s
σ( )s ---ds
0
∫t
+ σ( )t
=
r t( ) e
log( )r0 σ0
--- θ( )s σ( )s ---ds 0
∫t
+
σ( )t
e
σ( )tlog( )r0 σ0 ---
e
σ( )t θ( )s σ( )s ---ds 0
∫t
= =
r t( ) r0e
σ( ) σt – 0 σ0
---log( )r0
e
σ( )t θ( )s σ( )s ---ds 0
∫t
=
A Review of No Arbitrage Interest Rate Models 51
We now consider numerical solutions to the BDT process. To dis- cretize equation (25a) for the BDT model we start off again by approxi- matingdu in equation (25b) by u to get
(28) The exponential of equation (28) gives us
(29) where
We approximate this term by
That is, we approximate by a discrete approximation to the deriva- tive. We now have
or
(30)
If the random term is 0 equation (30) becomes
(31) uk+1 = uk+(θk+ρkuk)τ σ+ kεk τ
rk+1 = rke[(θk+ρklnrk)τ σ+ kεk τ]
ρk
σk' σk
---
=
σk+1–σk
---τ σk
--- σk'
uk+1 uk θk
σk+1–σk
---τ σk
---uk +
τ σkεk τ
+ +
=
uk+1 σk+1
σk
---uk+θkτ σ+ kεk τ
=
uk+1 σk+1
σk
---uk+θkτ
=
3-Buetow/Sochacki Page 51 Thursday, August 29, 2002 10:01 AM
In particular, if
whereα is a constant then
The exponential of this gives
This equation is interesting because ln r0 < 0. If α > 1 then the first exponential term decreases. When θ < 0 the second exponential term also decreases and the BDT short rate should approach a target rate.
Conversely, when θ > 0 the second exponential term increases. In this case we can approach a target rate or the second term can dominate. If α < 1 then a similar situation arises. Therefore, in order to get meaning- ful numerical results for the BDT short rates we strongly recommend that α be close to 1 and that the term structure of spot rates not have too large a slope.
The analysis of the equations without the stochastic term presented in this section is important. Recall that the characteristics of the random term are such that average influence of this term will be much smaller than the mean term in the SDEs. Consequently, the properties presented within this section will also hold under more general circumstances. The discrete approximations we developed for the models will be used to build the binomial and trinomial models in the next section. Note that we are highlighting the difference across the models and do not cali- brate the models to market information.
For numerical reasons, the BK and HW models are best imple- mented in the trinomial framework. The HL, KWF, and BDT models are more easily implemented in the binomial framework.15 We will discuss
15See G.W. Buetow and J. Sochacki, Binomial Interest Rate Models, AIMR Research Foundation, 2001.
σk+1
σk
--- = α
uk αku0 αjθk j– –1τ
j=0 k–1
∑
+
=
rk r0eα
k–1 ( )lnr0
e
αjθk j– –1τ j=0
k–1
∑
=
A Review of No Arbitrage Interest Rate Models 53
the specifics of this in the next section. For the trinomial framework we use the approach of Hull and White.16