Chapter 13
Forward Rate Modeling
This chapter is concerned with interest rate modeling, in which the mean
reversion property plays an important role. We consider the main short rate
models (Vasicek, CIR, CEV, affine models) and the computation of fixed
income products, such as bond prices, in such models. Next we consider the
modeling of forward rates in the HJM and BGM models, as well as in twofactor models.
13.1 Short Term Models and Mean Reversion
Vasicek Model
The first model to capture the mean reversion property of interest rates, a
property not possessed by geometric Brownian motion, is the Vasicek [Vaš77]
model, which is based on the Ornstein-Uhlenbeck process. Here, the short
term interest rate process (rt )t∈R+ solves the equation
drt = (a − brt )dt + σdBt ,
(13.1)
where a, σ ∈ R, b > 0, and (Bt )t∈R+ is a standard Brownian motion, with
solution
wt
a
rt = r0 e −bt + (1 − e −bt ) + σ
e −b(t−s) dBs , t ∈ R+ .
(13.2)
0
b
The probability distribution of rt is Gaussian at all times t, with mean
a
IE[rt ] = r0 e −bt + (1 − e −bt ),
b
and variance
Var[rt ] = σ 2
wt
0
( e −b(t−s) )2 ds = σ 2
wt
0
e −2bs ds =
405
σ2
(1 − e −2bt ),
2b
t ∈ R+ ,
N. Privault
i.e.
rt ' N
a
σ2
r0 e −bt + (1 − e −bt ), (1 − e −2bt ) ,
b
2b
t > 0.
In large time t with b > 0 we have
lim IE[rt ] =
t→∞
a
b
and
lim Var[rt ] =
t→∞
σ2
,
2b
and this distribution converges to the Gaussian N (a/b, σ 2 /(2b)) distribution,
which is also the invariant (or stationary) distribution of (rt )t∈R+ , and the
process tends to revert to its long term mean a/b = limt→∞ IE[rt ].
Figure 13.1 presents a random simulation of t 7−→ rt in the Vasicek model
with r0 = 3%, and shows the mean reverting property of the process with
respect to a/b = 2.5%.
8
7
rt (%)
6
5
4
3
a/b
2
1
0
-1
-2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Fig. 13.1: Graph of the Vasicek short rate t 7→ rt with a = 2.5%, b = 1, and σ = 0.1.
As can be checked from the simulation of Figure 13.1 the value of rt in the
Vasicek model may become negative due to its Gaussian distribution. Although real interest rates can sometimes fall below zero, this can be regarded
as a potential drawback of the Vasicek model.
Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) [CIR85] model brings a solution to the positivity problem encountered with the Vasicek model, by the use the nonlinear
stochastic differential equation
√
drt = β(α − rt )dt + σ rt dBt ,
α > 0,
β > 0.
The probability distribution of rt at time t > 0 admits the noncentral Chi
square probability density function given by
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Forward Rate Modeling
ft (x)
αβ/σ2 −1/2
2β
2β(x + r0 e −βt )
x
= 2
I2αβ/σ2 −1
exp − 2
−βt
−βt
−βt
σ (1 − e
)
σ (1 − e
)
r0 e
(13.3)
!
p
4β r0 x e −βt
,
σ 2 (1 − e −βt )
x > 0, where
Iλ (z) :=
∞
z λ X
2
k=0
(z 2 /4)k
,
k!Γ (λ + k + 1)
z ∈ R,
is the modified Bessel function of the first kind, cf. Corollary 24 in [AL05].
Note that ft (x) is not defined at x = 0 if αβ/σ 2 − 1/2 < 0, i.e. σ 2 > 2αβ, in
which case the probability distribution of rt admits a point mass at x = 0.
On the other hand, rt remains almost surely strictly positive under the Feller
condition 2αβ > σ 2 , cf. the study of the associated probability density in
Lemma 4 of [Fel51].
Figure 13.2 presents a random simulation of t 7−→ rt in the CIR model in
the case σ 2 > 2αβ, in which the process is mean reverting with respect to
α = 2.5% and has a nonzero probability of hitting 0.
8
7
rt (%)
6
5
4
3
α=2.5
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Fig. 13.2: Graph of the CIR short rate t 7→ rt with α = 2.5%, β = 1, and σ = 1.3.
In large time t, using the asymptotics
Iλ (z) 'z→0
z λ
1
,
Γ (λ + 1) 2
the density (13.3) becomes the Gamma density
f (x) = lim ft (x) =
t→∞
"
1
Γ (2αβ/σ 2 )
2β
σ2
2αβ/σ2
2
2
x−1+2αβ/σ e −2βx/σ ,
x > 0.
(13.4)
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N. Privault
with shape parameter 2αβ/σ 2 and scale parameter σ 2 /(2β), which is also the
invariant distribution of rt .
Other classical mean reverting models include the Courtadon (1982) model
drt = β(α − rt )dt + σrt dBt ,
where α, β, σ are nonnegative, and the exponential Vasicek model
drt = rt (η − a log rt )dt + σrt dBt ,
where a, η, σ > 0, cf. Exercises 4.14 and 4.15.
Constant Elasticity of Variance (CEV)
Constant Elasticity of Variance models are designed to take into account
nonconstant volatilities that can vary as a power of the underlying asset.
The Marsh-Rosenfeld (1983) model
−(1−γ)
drt = (βrt
γ/2
+ αrt )dt + σrt
dBt
(13.5)
where α, β, σ, γ are constants and β is the variance (or diffusion) elasticity
coefficient, covers most of the CEV models. Denoting by v(r) := σrγ/2 the
diffusion coefficient in (13.5), constant elasticity refers to the constant ratio
dv(r)/v(r)
rv 0 (r)
d log v(r)
d log rγ/2
γ
=
=
=
=
dr/r
v(r)
d log r
d log r
2
between the relative change dv(r)/v(r) in the variance v(r) and the relative
change dr/r in r.
For γ = 1 this is the CIR model, and for β = 0 we get the standard CEV
model
γ/2
drt = αrt dt + σrt dBt .
If γ = 2 this yields the Dothan [Dot78] model
drt = αrt dt + σrt dBt ,
which is a version of geometric Brownian motion used for short term interest
rate modeling.
Time-dependent affine Models
The class of short rate interest rate models admits a number of generalizations
that can be found in the references quoted in the introduction of this chapter,
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Forward Rate Modeling
among which is the class of affine models of the form
p
drt = (η(t) + λ(t)rt )dt + δ(t) + γ(t)rt dBt .
(13.6)
Such models are called affine because the associated zero-coupon bonds can
be priced using an affine PDE of the type (13.16) below, as will be seen after
Proposition 13.2.
Affine models also include the Ho-Lee model
drt = θ(t)dt + σdBt ,
where θ(t) is a deterministic function of time, as an extension of the Merton
model drt = θdt + σdBt , and the Hull-White model [HW90], cf. Section 13.1,
drt = (θ(t) − α(t)rt )dt + σ(t)dBt
which is a time-dependent extension of the Vasicek model.
13.2 Calibration of the Vasicek model
The Vasicek equation (13.1), i.e.
drt = (a − brt )dt + σdBt
can be discretized according to a discrete-time sequence (tk )k=0,1,...,n as
rtk+1 − rtk = (a − brtk )∆t + σZk ,
k ∈ N,
where ∆t := tk+1 −tk and (Zk )k>0 is a Gaussian white noise with variance ∆t,
i.e. a sequence of independent, centered and identically distributed N (0, ∆t)
Gaussian random variables.
We find
rtk+1 = rtk + (a − brtk )∆t + σZk = a∆t + (1 − b∆t)rtk + σZk ,
k ∈ N.
Based on a set (r̃tk )k=0,...,n of market data we can minimize the residual
n−1
X
r̃tk+1 − a∆t − (1 − b∆t)r̃tk
2
k=0
over a and b using Ordinary Least Square (OLS) regression. For this compute
n−1
2
∂ X
r̃tk+1 − a∆t − (1 − b∆t)r̃tk
∂a
k=0
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N. Privault
= −2∆t −an∆t +
n−1
X
!
r̃tk+1 − (1 − b∆t)r̃tk
k=0
= 0,
and
n−1
2
∂ X
r̃tk+1 − a∆t − (1 − b∆t)r̃tk
∂b
k=0
= ∆t
n−1
X
r̃tk −a∆t + r̃tk+1 − (1 − b∆t)r̃tk
k=0
= ∆t
n−1
X
r̃tk
r̃tk+1 − (1 − b∆t)r̃tk +
k=0
n−1
1X
r̃tl+1 − (1 − b∆t)r̃tl
n
!
l=0
= 0.
This leads to an estimate the parameters a and b respectively as the empirical
mean and covariance of (r̃tk )k=0,1,...,n , i.e.

n−1
n−1
X

1X
1



r̃tk+1 − (1 − b̂∆t)
r̃tk
â∆t =


n
n


k=0
k=0







and







n−1
n−1
n−1

X
X

1X


r̃
r̃
r̃tl+1
r̃
−
t
t
t

k
k
k+1

n
k=0
l=0
1 − b̂∆t = k=0
n−1
n−1
n−1

X
X

1X


r̃tl
r̃tk r̃tk −
r̃tk



n

k=0
k=0 ! l=0

!

n−1
n−1
n−1

X

1X
1X



r̃
−
r̃
r̃
−
r̃
tk
tl
tk+1
tl+1


n
n


l=0
l=0

= k=0
.
!2


n−1
n−1

X

1X


r̃tk −
r̃tk


n
k=0
k=0
This also yields
σ 2 ∆t = Var[σZk ] = Var r̃tk+1 − (1 − b∆t)r̃tk − a∆t ,
k ∈ N,
hence σ can be estimated as
σ̂ 2 ∆t =
n−1
2
1 X
r̃tk+1 − r̃tk (1 − b̂∆t) − â∆t .
n
k=0
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Forward Rate Modeling
Defining r̂tk := rtk − a/b, k ∈ N, we have
r̂tk+1 = rtk+1 − a/b
= rtk − a/b + (a − brtk )∆t + σZk
= rtk − a/b − b(rtk − a/b)∆t + σZk
= r̂tk − br̂tk ∆t + σZk
= (1 − b∆t)r̂tk + σZk ,
k ∈ N.
In other words, the sequence (r̂tk )k∈∈N is modeled according to an autoregressive AR(1) time series in which the current state Xn of the system is
expressed as the linear combination
Xn := σZn + α1 Xn−1 ,
n > 1,
(13.7)
which can be solved recursively as the series
Xn = σZn + α1 (σZn−1 + α1 Xn−2 ) = · · · = σ
∞
X
α1k Zn−k ,
k=0
which converges when |α1 | < 1, i.e. |1 − b∆t| < 1.
Note that the variance of Xn is given by
"∞
#
X
Var[Xn ] = σ 2 Var
α1k Zn−k
k=0
= σ ∆t
2
= σ 2 ∆t
∞
X
k=0
∞
X
α12k
(1 − b∆t)2k
k=0
2
σ ∆t
1 − (1 − b∆t)2
σ 2 ∆t
=
2b∆t − b2 (∆t)2
σ2
'
,
2b
=
which is the expected variance of the Vasicek process in the stationary regime.
"
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library(quantmod)
getSymbols("^TNX",from="2012-01-01",to="2016-01-01",src="yahoo")
rate=Ad(`TNX`)
chartSeries(rate,up.col="blue",theme="white")
n = sum(!is.na(rate))
The next Figure 13.3 displays the yield of the 10 Year Treasury Note on the
Chicago Board Options Exchange (CBOE). Treasury notes usually have a
maturity between one and 10 years, whereas treasury bonds have maturities
beyong 10 years)
rate
[2012−01−03/2015−12−31]
Last 2.269
3.0
2.5
2.0
1.5
Jan 03
2012
Jul 02
2012
Jan 02
2013
Jul 01
2013
Jan 02
2014
Jul 01
2014
Jan 02
2015
Jul 01
2015
Dec 31
2015
Fig. 13.3: CBOE 10 Year Treasury Note yield (TNX).
ratek=as.vector(rate)
ratekplus1 <- c(ratek[-1],0)
b <- (sum(ratek*ratekplus1) - sum(ratek)*sum(ratekplus1)/n)/(sum(ratek*ratek) - sum(ratek)*sum(
ratek)/n)
a <- sum(ratekplus1)/n-b*sum(ratek)/n
sigma <- sqrt(sum((ratekplus1-b*ratek-a)^2)/n)
The next code is generating Vasicek random samples according to the AR(1)
time series (13.7).
for (i in 1:100) {
ar.sim<-arima.sim(model=list(ar=c(b)),n.start=100,n)
y=ratek[1]+a/b+sigma*ar.sim
time <- as.POSIXct(time(TNX), format = "%Y-%m-%d")
yield <- xts(x = y, order.by = time)
chartSeries(yield,up.col="blue",theme="white")
Sys.sleep(0.5)
}
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Forward Rate Modeling
yield
[2012−01−03 08:00:00/2015−12−31 08:00:00]
Last 1.82260342989168
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Jan 03
2012
Jul 02
2012
Jan 02
2013
Jul 01
2013
Jan 02
2014
Jul 01
2014
Jan 02
2015
Jul 01
2015
Dec 31
2015
Fig. 13.4: Calibrated Vasicek samples.
13.3 Zero-Coupon and Coupon Bonds
A zero-coupon bond is a contract priced P (t, T ) at time t < T to deliver
P (T, T ) = $1 at time T . In addition to its value at maturity, a bond may
yield a periodic coupon payment at regular time intervals until the maturity
date.
Fig. 13.5: Five dollar Louisiana bond of 1875 with 7.5% biannual coupons.
The computation of the arbitrage price P0 (t, T ) of a zero-coupon bond based
on an underlying short term interest rate process (rt )t∈R+ is a basic and
important issue in interest rate modeling.
Constant short rate
In case the short term interest rate is a constant rt = r, t ∈ R+ , a standard
arbitrage argument shows that the price P (t, T ) of the bond is given by
P (t, T ) = e −r(T −t) ,
"
0 6 t 6 T.
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N. Privault
Indeed, if P (t, T ) > e −r(T −t) we could issue a bond at the price P (t, T ) and
invest this amount at the compounded risk free rate r, which would yield
P (t, T ) e r(T −t) > 1 at time T .
On the other hand, if P (t, T ) < e −r(T −t) we could borrow P (t, T ) at the rate
r to buy a bond priced P (t, T ). At maturity time T we would receive $1 and
refund only P (t, T ) e r(T −t) < 1.
Deterministic short rates
Similarly to the above, when the short term interest rate process (rt )t∈R+ is
a deterministic function of time, a similar argument shows that
P (t, T ) = e −
rT
t
rs ds
,
0 6 t 6 T.
(13.8)
Stochastic short rates
In case (rt )t∈R+ is an Ft -adapted random process the formula (13.8) is no
longer valid as it relies on future information, and we replace it with
i
h rT
P (t, T ) = IE∗ e − t rs ds Ft ,
0 6 t 6 T,
(13.9)
under a risk-neutral measure P∗ . It is natural to write P (t, T ) as a conditional
expectation under a martingale measure, as the use of conditional expectation
wT
helps to “filter out” the future information past time t contained in
rs ds.
t
The expression (13.9)
makes sense as the “best possible estimate” of the
rT
future quantity e − t rs ds in mean square sense, given information known up
to time t.
Coupon bonds
Pricing bonds with non-zero coupon is not difficult since in general the
amount and periodicity of coupons are deterministic.∗ In the case of a constant, continuous-time coupon yield at the rate c > 0, another application of
the above absence of arbitrage argument shows that the price Pc (t, T ) of the
coupon bond is given by
Pc (t, T ) = e c(T −t) P0 (t, T ),
0 6 t 6 T,
see also Figure 13.9 below. In the sequel we will mostly consider zero-coupon
bonds priced as P (t, T ) = P0 (t, T ), 0 6 t 6 T .
∗
However, coupon default cannot be excluded.
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Martingale property of discounted bond prices
The following proposition shows that Assumption (A) of Chapter 12 is satisfied, in other words, the bond price process t 7−→ P (t, T ) can be used as a
numéraire.
Proposition 13.1. The discounted bond price process
t 7−→ P̃ (t, T ) := e −
rt
0
rs ds
P (t, T )
∗
is a martingale under P .
Proof. By (13.9) we have
e−
rt
0
rs ds
rt
i
h rT
IE∗ e − t rs ds Ft
h rt
i
rT
= IE∗ e − 0 rs ds e − t rs ds Ft
i
h rT
= IE∗ e − 0 rs ds Ft ,
P (t, T ) = e −
0
rs ds
and this suffices to conclude since by the “tower property” (17.37) of conditional expectations, any process (Xt )t∈R+ of the form t 7−→ Xt := IE∗ [F | Ft ],
F ∈ L1 (Ω), is a martingale, cf. Relation (6.1).
Path integrals
In physics, the Feynman path integral
ψ(y, t) :=
w
x(0)=x,x(t)=y
Dx(·) exp
i
S(x(·))
~
where ~ is the Planck constant and S(x(·)) is the action
w t 1
wt
L(x(s), ẋ(s), s)ds =
S(x(·)) =
m(ẋ(s))2 − V (x(s)) ds
0
0
2
N 2
X
(x(ti ) − x(ti−1 ))
'
− V (x(ti−1 )) ∆ti ,
2
2(ti − ti−1 )
i=1
solves the Schrödinger equation
i~
∂ψ
~2 ∂ 2 ψ
(x, t) = −
(x, t) + V (x(t))ψ(x, t).
∂t
2m ∂x2
After the Wick rotation t 7→ −it, the function
φ(y, t) :=
"
w
1
Dx(·) exp − S(x(·))
x(0)=x,x(t)=y
~
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N. Privault
where S(x(·)) is the action
S(x(·)) :=
wt
0
L(x(s), ẋ(s), s)ds =
w t 1
N X
(x(ti ) − x(ti−1 ))2
'
2(ti − ti−1 )2
i=1
0
2
m(ẋ(s))2 + V (x(s)) ds
+ V (x(ti−1 )) ∆ti ,
solves the heat equation
~
∂φ
~2 ∂ 2 φ
(x, t) = −
(x, t) + V (x(t))φ(x, t).
∂t
2m ∂x2
Given the action
S(x(·)) =
'
w t 1
0
2
m(ẋ(s))2 + V (x(s)) ds
N X
(x(ti ) − x(ti−1 ))2
i=1
2(ti − ti−1
)2
+ V (x(ti−1 )) ∆ti ,
we can rewrite the Euclidean path integral as
w
1
Dx(·) exp − S(x(·))
φ(y, t) =
x(0)=x,x(t)=y
~
!
N
N
w
X
1
(x(ti ) − x(ti−1 ))2
1X
=
Dx(·) exp −
V (x(ti−1 ))
−
x(0)=x,x(t)=y
2~ i=1
2∆ti
~ i=1
w
t
1
V (Bs )ds B0 = x, Bt = y .
= IE∗ exp −
~ 0
This type of path integral computation
w
t
φ(y, t) = IE∗ exp − V (Bs )ds B0 = x, Bt = y .
0
(13.10)
is particularly useful for bond pricing, as (13.10) can be interpreted as the
price of a bond with short term interest rate process (rt )t∈R+ := (V (Bt )))t∈R+
conditionally to the value of the endpoint Bt = y, cf. (13.31) below. It can also
be useful for exotic option pricing, cf. Chapter 10, and for risk management.
The path integral (13.10) can be estimated either by closed-form expressions
using Partial Differential Equations (PDEs) or probability densities, by approximations such as (conditional) Moment matching, or by Monte Carlo
estimation, from the paths of a Brownian bridge as shown in Figure 13.6.
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Fig. 13.6: Brownian bridge.
Bond pricing PDE
We assume from now on that the underlying short rate process is solution to
the stochastic differential equation
drt = µ(t, rt )dt + σ(t, rt )dBt
(13.11)
where (Bt )t∈R+ is a standard Brownian motion under P .
∗
Since all solutions of stochastic differential equations such as (13.11) have
the Markov property, cf. e.g. Theorem V-32 of [Pro04], the arbitrage price
P (t, T ) can be rewritten as a function F (t, rt ) of rt , i.e.
i
i
h rT
h rT
P (t, T ) = IE∗ e − t rs ds Ft = IE∗ e − t rs ds rt = F (t, rt ),
and depends on rt only instead of depending on all information available in
Ft up to time t, meaning that the pricing problem can now be formulated as
a search for the function F (t, x).
Proposition 13.2. (Bond pricing PDE). The bond pricing PDE for P (t, T ) =
F (t, rt ) is written as
xF (t, x) =
∂F
1
∂2F
∂F
(t, x) + µ(t, x)
(t, x) + σ 2 (t, x) 2 (t, x), (13.12)
∂t
∂x
2
∂x
t ∈ R+ , x ∈ R, subject to the terminal condition
F (T, x) = 1,
"
x ∈ R.
(13.13)
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Proof. By Itô’s formula we have
rt
rt
rt
d e − 0 rs ds P (t, T ) = −rt e − 0 rs ds P (t, T )dt + e − 0 rs ds dP (t, T )
rt
rt
dF (t, rt )
∂F
= −rt e − 0 rs ds F (t, rt )dt + e − 0 rs ds
(t, rt )(µ(t, rt )dt + σ(t, rt )dBt )
∂x
2
rt
∂
F
∂F
1
σ 2 (t, rt ) 2 (t, rt ) +
(t, rt ) dt
+ e − 0 rs ds
2
∂x
∂t
r
∂F
− 0t rs ds
= e
σ(t, rt )
(t, rt )dBt
∂x
rt
∂F
1
∂2F
∂F
+ e − 0 rs ds −rt F (t, rt ) + µ(t, rt )
(t, rt ) + σ 2 (t, rt ) 2 (t, rt ) +
(t, rt ) dt.
∂x
2
∂x
∂t
(13.14)
= −rt e −
0
rs ds
F (t, rt )dt + e −
0
rs ds
rt
rt
rt
Given that t 7−→ e − 0 rs ds P (t, T ) is a martingale, the above expression
(13.14) should only contain terms in dBt (cf. Corollary II-1, page 72 of
[Pro04]), and all terms in dt should vanish inside (13.14). This leads to the
identities


rt F (t, rt )



∂F
1
∂2F
∂F


 = µ(t, rt )
(t, rt ) + σ 2 (t, rt ) 2 (t, rt ) +
(t, rt )
∂x
2
∂x
∂t




rt

rt


d e − 0 rs ds P (t, T ) = e − 0 rs ds σ(t, rt ) ∂F (t, rt )dBt ,
∂x
(13.15a)
(13.15b)
which recover (13.12) . Condition (13.13) is due to the fact that P (T, T ) = $1.
In the case of an interest rate process modeled by (13.6) we have
p
µ(t, x) = η(t) + λ(t)x
and
σ(t, x) = δ(t) + γ(t)x,
hence (13.12) yields the (time dependent) affine PDE
xF (t, x) =
∂F
∂F
1
∂2F
(t, x) + (η(t) + λ(t)x)
(t, x) + (δ(t) + γ(t)x) 2 (t, x),
∂t
∂x
2
∂x
(13.16)
t ∈ R+ , x ∈ R. By (13.15b), the above proposition also shows that
rt
rt
1
dP (t, T )
=
d e 0 rs ds e − 0 rs ds P (t, T )
P (t, T )
P (t, T )
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Forward Rate Modeling
rt
rt
1
rt P (t, T )dt + e 0 rs ds d e − 0 rs ds P (t, T )
P (t, T )
rt
rt
1
= rt dt +
e 0 rs ds d e − 0 rs ds P (t, T )
P (t, T )
1
∂F
= rt dt +
(t, rt )σ(t, rt )dBt
F (t, rt ) ∂x
∂ log F
(t, rt )dBt .
(13.17)
= rt dt + σ(t, rt )
∂x
=
In the Vasicek case
drt = (a − brt )dt + σdWt ,
the bond price takes the form
F (t, rt ) = P (t, T ) = e A(T −t)+rt C(T −t) ,
where A(·) and C(·) are functions of time, cf. (13.21) below, and (13.17)
yields
dP (t, T )
σ
= rt dt − (1 − e −b(T −t) )dWt ,
(13.18)
P (t, T )
b
since F (t, x) = e A(T −t)+xC(T −t) .
Note that more generally, all affine short rate models as defined in Relation (13.6), including the Vasicek model, will yield a bond pricing formula of
the form
P (t, T ) = e A(T −t)+rt C(T −t) ,
cf. e.g. § 3.2.4. of [BM06].
Probabilistic solution of the Vasicek PDE
Next, we solve the PDE (13.12), written in the Vasicek model
drt = (a − brt )dt + σdBt
with µ(t, x) = a − bx and σ(t, x) = σ, as

∂F
∂F
σ2 ∂ 2 F

 xF (t, x) =
(t, x) + (a − bx)
(t, x) +
(t, x),
∂t
∂x
2 ∂x2


F (T, x) = 1,
by a direct computation of the conditional expectation
i
h rT
P (t, T ) = F (t, rt ) = IE∗ e − t rs ds Ft ,
"
(13.19)
(13.20)
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N. Privault
in the Vasicek [Vaš77] model
drt = (a − brt )dt + σdBt ,
i.e. when the short rate (rt )t∈R+ has the expression
rt = g(t) +
where
and
wt
0
wt
a
h(t, s)dBs = r0 e −bt + (1 − e −bt ) + σ
e −b(t−s) dBs ,
0
b
a
g(t) := r0 e −bt + (1 − e −bt ),
b
h(t, s) := σ e −b(t−s) ,
t ∈ R+ ,
0 6 s 6 t,
are deterministic functions.
Letting u∨t := max(u, t), using the fact that Wiener integrals are Gaussian
random variables and the Gaussian moment generating function, we have
i
h rT
P (t, T ) = IE∗ e − t rs ds Ft
i
h rT
rs
= IE∗ e − t (g(s)+ 0 h(s,u)dBu )ds Ft
w
i
h rT rs
T
= exp −
g(s)ds IE∗ e − t 0 h(s,u)dBu ds Ft
t
w
i
h rT rT
T
= exp −
g(s)ds IE∗ e − 0 u∨t h(s,u)dsdBu Ft
t
w
i
h rT rT
wtwT
T
= exp −
g(s)ds −
h(s, u)dsdBu IE∗ e − t u∨t h(s,u)dsdBu Ft
t
0 u∨t
w
i
h rT rT
wtwT
T
= exp −
g(s)ds −
h(s, u)dsdBu IE∗ e − t u h(s,u)dsdBu Ft
t
0 t
w
h rT rT
i
wtwT
T
h(s, u)dsdBu IE∗ e − t u h(s,u)dsdBu
= exp −
g(s)ds −
t
0 t
2 !
wT
wtwT
1wT wT
h(s, u)ds du
= exp −
g(s)ds −
h(s, u)dsdBu +
u
t
0 t
2 t
w
w
w
T
T
t
a
= exp −
(r0 e −bs + (1 − e −bs ))ds − σ
e −b(s−u) dsdBu
t
0 t
b
w
2 !
2 wT
T
σ
× exp
e −b(s−u) ds du
u
2 t
w
wt
T
a
σ
= exp −
(r0 e −bs + (1 − e −bs ))ds − (1 − e −b(T −t) )
e −b(t−u) dBu
t
0
b
b
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Forward Rate Modeling
× exp
2 !
σ 2 w T 2bu e −bu − e −bT
e
du
2 t
b
rt
1
a
= exp − (1 − e −b(T −t) ) + (1 − e −b(T −t) ) r0 e −bt + (1 − e −bt )
b
b
b
−bu
2 !
wT 2 wT
a
σ
e
− e −bT
r0 e −bs + (1 − e −bs ) ds +
e 2bu
du
× exp −
t
b
2 t
b
= e A(T −t)+rt C(T −t) ,
where
(13.21)
1
C(T − t) := − (1 − e −b(T −t) ),
b
(13.22)
and
4ab − 3σ 2 σ 2 − 2ab
σ 2 − ab −b(T −t) σ 2 −2b(T −t)
+
(T − t) +
e
− 3e
,
3
2
4b
2b
b3
4b
(13.23)
which admits (13.22)-(13.23) for solution.
A(T − t) :=
Analytical solution of the Vasicek PDE
In order to solve the PDE (13.19) analytically, we may look for a solution of
the form
F (t, x) = e A(T −t)+xC(T −t) ,
(13.24)
where A(·) and C(·) are functions to be determined under the conditions
A(0) = 0 and C(0) = 0. Substituting (13.24) into the PDE (13.12) with the
Vasicek coefficients µ(t, x) = (a − bx) and σ(t, x) = σ shows that
x e A(T −t)+xC(T −t) = −(A0 (T − t) − xC 0 (T − t)) e A(T −t)+xC(T −t)
+(a − bx)C(T − t) e A(T −t)+xC(T −t)
1
+ σ 2 C 2 (T − t) e A(T −t)+xC(T −t) ,
2
i.e.
1
x = −A0 (T − t) + xC 0 (T − t) + (a − bx)C(T − t) + σ 2 C 2 (T − t).
2
By identification of terms for x = 0 and x 6= 0, this yields the system of
Riccati differential equations

σ2 2

 A0 (s) = aC(s) +
C (s)
2

 0
C (s) = 1 + bC(s),
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N. Privault
which can be solved to recover the above value of P (t, T ) = F (t, rt ).
Vasicek Bond Price Simulations
In this section we consider again the Vasicek model, in which the short rate
(rt )t∈R+ is solution to (13.1). Figure 13.7 presents a random simulation of
t 7−→ P (t, T ) in the same Vasicek model. The graph of the corresponding
deterministic zero coupon bond price obtained for a = b = σ = 0 is also
shown on the Figure 13.7.
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0
5
10
15
20
Fig. 13.7: Graphs of t 7→ F (t, rt ) = P (t, T ) vs t 7→ e −r0 (T −t) .
Figure 13.8 presents a random simulation of t 7−→ P (t, T ) for a (non-zero)
coupon bond with price Pc (t, T ) = e c(T −t) P (t, T ), and coupon rate c > 0,
0 6 t 6 T.
108.00
106.00
104.00
102.00
100.00
0
5
10
15
20
Fig. 13.8: Graph of t 7→ F (t, rt ) = P (t, T ) for a bond with a 2.3% coupon.
The simulation of Figure 13.8 can be compared to the coupon bond market
data of Figure 13.9 below.
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Forward Rate Modeling
Fig. 13.9: Bond price graph with maturity 01/18/08 and coupon rate 6.25%.
See Exercise 13.3 for a bond pricing formula in the CIR model.
Zero coupon bond price and yield data
The following zero coupon bond price data was downloaded at EMMA from
the Municipal Securities Rulemaking Board.
ORANGE CNTY CALIF PENSION OBLIG CAP APPREC-TAXABLEREF-SER A (CA)
CUSIP: 68428LBB9
Dated Date: 06/12/1996 (June 12, 1996)
Maturity Date: 09/01/2016 (September 1st, 2016)
Interest Rate: 0.0 %
Principal Amount at Issuance: $26,056,000
Initial Offering Price: 19.465
library(quantmod)
bondprice <- read.table("bond_data_R.txt",col.names = c("Date","HighPrice","LowPrice","
HighYield","LowYield","Count","Amount"))
head(bondprice)
time <- as.POSIXct(bondprice$Date, format = "%Y-%m-%d")
price <- xts(x = bondprice$HighPrice, order.by = time)
yield <- xts(x = bondprice$HighYield, order.by = time)
chartSeries(price,up.col="blue",theme="white")
chartSeries(yield,up.col="blue",theme="white")
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1
2
3
4
5
6
Date
HighPrice LowPrice
2016-01-13 99.082
98.982
2015-12-29 99.183
99.183
2015-12-21 97.952
97.952
2015-12-17 99.141
98.550
2015-12-07 98.770
98.770
2015-12-04 98.363
98.118
HighYield LowYield Count
1.666
1.501
2
1.250
1.250
1
3.014
3.014
1
2.123
1.251
5
1.714
1.714
2
2.628
2.280
2
price
Amount
20000
10000
10000
610000
10000
10000
[2005−01−26/2016−01−13]
100
Last 99.082
90
80
70
60
50
Jan 26
2005
Aug 03
2006
May 13
2008
Feb 08
2010
Mar 01
2011
Nov 09
2012
Dec 04
2014
Fig. 13.10: Orange Cnty Calif bond prices.
The next Figure 13.11 plots the bond yield y(t, T ) defined as
y(t, T ) = −
log P (t, T )
,
T −t
or P (t, T ) = e −(T −t)y(t,T ) ,
yield
0 6 t 6 T.
[2005−01−26/2016−01−13]
Last 1.666
8
6
4
2
Jan 26
2005
Aug 03
2006
May 13
2008
Feb 08
2010
Mar 01
2011
Nov 09
2012
Dec 04
2014
Fig. 13.11: Orange Cnty Calif bond yields.
Bond pricing in the Dothan model
In the Dothan [Dot78] model, the short term interest rate process (rt )t∈R+ is
modeled according to a geometric Brownian motion
drt = µrt dt + σrt dBt ,
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(13.25)
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Forward Rate Modeling
where the volatility σ > 0 and the drift µ ∈ R are constant parameters and
(Bt )t∈R+ is a standard Brownian motion. In this model the short term interest rate rt remains always positive, while the proportional volatility term σrt
accounts for the sensitivity of the volatility of interest rate changes to the
level of the rate rt .
On the other hand, the Dothan model is the only lognormal short rate
model that allows for an analytical formula for the zero coupon bond price
i
h rT
P (t, T ) = IE∗ e − t rs ds Ft ,
0 6 t 6 T.
For convenience of notation we let p = 1 − 2µ/σ 2 and rewrite (13.25) as
drt = (1 − p)
with solution
σ2
rt dt + σrt dBt ,
2
rt = r0 e σBt −pσ
2
t/2
,
t ∈ R+ ,
(13.26)
where pσ/2 identifies to the market price of risk. By the Markov property of
(rt )t∈R+ , the bond price P (t, T ) is a function F (t, rt ) of rt and time t ∈ [0, T ]:
i
h rT
P (t, T ) = F (t, rt ) = IE∗ e − t rs ds rt ,
0 6 t 6 T.
(13.27)
By computation of the conditional expectation (13.27) using (10.6) we easily
obtain the following result, cf. Proposition 1.2 of [PP11], where the function
θ(v, t) is defined in (10.4).
Proposition 13.3. The zero-coupon bond price P (t, T ) = F (t, rt ) is given
for all p ∈ R by
F (t, x)
= e −σ
2 2
p (T −t)/8
w∞w∞
0
0
1 + z2
e −ux exp −2
σ2 u
(13.28)
! 4z σ 2 (T − t) du dz
θ
,
,
σ2 u
4
u z p+1
x > 0.
Proof. By Proposition 10.1, cf. [Yor92], Proposition 2, the probability distriw T −t
2
bution of the time integral
e σBs −pσ s/2 ds is given by
0
P
w
T −t
=
"
0
w∞
−∞
2
e σBs −pσ s/2 ds ∈ dy
w
t
2
P
e σBs −pσ s/2 ds ∈ dy, Bt − pσt/2 ∈ dz
0
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N. Privault
σz/2 2 σ w ∞ −pσz/2−p2 σ2 t/8
1 + e σz
4e
σ t dy
e
exp −2
θ
,
dz
2
2 −∞
σ y
σ2 y
4
y
w
2
2
∞
2 2
1+z
4z σ (T − t)
dz dy
= e −p σ (T −t)/8
exp −2 2
θ
,
,
0
σ y
σ2 y
4
z p+1 y
=
y > 0,
where the exchange of integrals is justified by the Fubini theorem and the
nonnegativity of integrands. Hence by (10.6) and (13.26) we find
F (t, rt ) = P (t, T )
w
T
= IE∗ exp −
rs ds Ft
t
wT
2
= IE∗ exp −rt
e σ(Bs −Bt )−σ p(s−t)/2 ds Ft
t
wT
2
= IE∗ exp −x
e σ(Bs −Bt )−σ p(s−t)/2 ds
t
w T −t
2
= IE∗ exp −x
e σBs −σ ps/2 ds
0
=
w∞
0
= e
e
−rt y
P
w
T −t
0
−p2 σ 2 (T −t)/8
w∞
0
x=rt
x=rt
ds ∈ dy
w∞
4z σ 2 (T − t)
dz dy
1 + z2
θ
,
.
e −rt y
exp −2 2
0
σ y
σ2 y
4
z p+1 y
e
σBs −pσ 2 s/2
The zero-coupon bond price P (t, T ) = F (t, rt ) in the Dothan model can also
be written for all p ∈ R as
√ p
(2x)p/2 w ∞ −σ2 (p2 +u2 )t/8
u 2
8x
+
i
K
du
F (t, x) =
ue
sinh(πu)
−
Γ
iu
2π 2 σ p 0
2
2
σ
√
∞
(2x)p/2 X 2(p − 2k)+ σ2 k(k−p)t/2
8x
+
e
Kp−2k
, x > 0, t > 0,
σp
k!(p − k)!
σ
k=0
cf. Corollary 2.2 of [PP10], see also [PU13] for numerical computations. Zerocoupon bond prices in the Dothan model can also be computed by the conditional expression
w
w
w
T
∞
T
IE exp −
rt dt
=
IE exp −
rt dt rT = z dP(rT = z),
0
0
0
where rT has the lognormal distribution
dP(rT = z) = dP(r0 eσBT −pσ
2
T /2
(13.29)
2
2
2
1
= z) = √
e−(pσ T /2+log(z/r0 )) /(2σ T ) .
z 2πσ 2 T
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Forward Rate Modeling
In Proposition 13.4 we note that the conditional Laplace transform
w
T
IE exp −
rt dt rT = z
0
cf. (13.10) above, can be computed in a closed integral form using a closed
form integral integral expression based on the modified Bessel function of the
second kind
zζ w ∞
z2
du
Kζ (z) := ζ+1
exp −u −
,
ζ ∈ R, z ∈ C,
(13.30)
0
2
4u uζ+1
cf. e.g. [Wat95] page 183, provided that the real part R(z 2 ) of z 2 ∈ C is
positive.
Proposition 13.4. [PY16], Proposition 4.1. Taking r0 = 1, for all λ, z > 0
we have
r
2
wT
4 e −σ T /8
λ
(13.31)
IE exp −λ
rs ds rT = z = 3/2 2
0
π σ p(z) T
√ p
√
w∞
K1
8λ 1 + 2 z cosh ξ + z/σ
4πξ
2(π 2 −ξ 2 )/(σ 2 T )
p
dξ.
×
e
sin
sinh(ξ)
√
0
σ2 T
1 + 2 z cosh ξ + z
Note however that (13.31) fails for small values of T , and for this reason
the integral can be estimated by a gamma approximation, cf. (13.32) below.
Under the Gamma approximation we can approximate the conditional bond
price on the Dothan short rate rt as
wT
−ν(z)
,
IE exp −λ
rt dt rT = z ' (1 + λθ(z))
0
where the parameters ν(z) and θ(z) are determined by conditional moment
fitting to a gamma distribution, cf. [PY16], which yields
w
wT
∞
−ν(z)
(1 + λθ(z))
dP(rT = z).
(13.32)
IE exp −λ
rs ds
'
0
0
Figures 13.12 shows that the stratified gamma approximation (13.32) matches
the Monte Carlo estimate, while the use of the integral expressions (13.29)
and (13.31) leads to numerical instabilities.
Related computations for yield options in the CIR model can also be found
in [PP17].
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N. Privault
1
stratified gamma
Monte Carlo
integral expression
0.8
F(x,t)
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
8
9
T=10
t
Fig. 13.12: Approximation of Dothan bond prices t 7→ F (t, x) with σ = 0.3 and T = 10.
13.4 Forward Rates
A forward interest rate contract (or Forward Rate Agreement, FRA) gives its
holder the possibility to lock an interest rate denoted by f (t, T, S) at present
time t for a loan to be delivered over a future period of time [T, S], with
t 6 T 6 S. The rate f (t, T, S) is called a forward interest rate. When T = t,
the spot forward rate f (t, t, T ) is also called the yield.
Figure 13.13 presents a typical yield curve on the LIBOR (London Interbank
Offered Rate) market with t =07 May 2003.
5
Forward interest rate
4.5
4
3.5
3
2.5
2
0
5
10
15
20
25
30
years
TimeSerieNb
AsOfDate
2D
1W
1M
2M
3M
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
11Y
12Y
13Y
14Y
15Y
20Y
25Y
30Y
505
7­mai­03
2,55
2,53
2,56
2,52
2,48
2,34
2,49
2,79
3,07
3,31
3,52
3,71
3,88
4,02
4,14
4,23
4,33
4,4
4,47
4,54
4,74
4,83
4,86
Fig. 13.13: Forward rate graph T 7−→ f (t, t, T ).
Maturity transformation, i.e., the ability to transform short term borrowing
(debt with short maturities, such as deposits) into long term lending (credits
with very long maturities, such as loans), is among the roles of banks. Profitability is then dependent on the difference between long rates and short
rates.
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Forward Rate Modeling
Another example of market data is given in the next Figure 13.14, in which
the red and blue curves refer respectively to July 21 and 22 of year 2011.
Fig. 13.14: Market example of yield curves, cf. (13.35).
Forward rates from bond prices
Let us determine the arbitrage or “fair” value of the forward interest rate
f (t, T, S) by implementing the Forward Rate Agreement using the instruments available in the market, which are bonds priced at P (t, T ) for various
maturity dates T > t.
The loan can be realized using the available instruments (here, bonds) on the
market, by proceeding in two steps:
1) At time t, borrow the amount P (t, S) by issuing (or short selling) one
bond with maturity S, which means refunding $1 at time S.
2) Since the money is only needed at time T , the rational investor will
invest the amount P (t, S) over the period [t, T ] by buying a (possibly fractional) quantity P (t, S)/P (t, T ) of a bond with maturity T priced P (t, T )
at time t. This will yield the amount
$1 ×
P (t, S)
P (t, T )
at time T > 0.
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As a consequence, the investor will actually receive P (t, S)/P (t, T ) at time
T , to refund $1 at time S.
The corresponding forward rate f (t, T, S) is then given by the relation
P (t, S)
exp ((S − T )f (t, T, S)) = $1,
P (t, T )
0 6 t 6 T 6 S,
(13.33)
where we used exponential compounding, which leads to the following definition (13.34).
Definition 13.5. The forward rate f (t, T, S) at time t for a loan on [T, S]
is given by
f (t, T, S) =
log P (t, T ) − log P (t, S)
.
S−T
(13.34)
The spot forward rate f (t, t, T ) coincides with the yield given by
f (t, t, T ) = −
log P (t, T )
,
T −t
or P (t, T ) = e −(T −t)f (t,t,T ) ,
0 6 t 6 T.
(13.35)
The instantaneous forward rate f (t, T ) = f (t, T, T ) is defined by taking the
limit of f (t, T, S) as S & T , i.e.
f (t, T ) : = lim f (t, T, S)
S&T
log P (t, S) − log P (t, T )
S−T
log P (t, T + ε) − log P (t, T )
= − lim
ε&0
ε
∂ log P (t, T )
=−
∂T
1
∂P (t, T )
=−
.
P (t, T ) ∂T
= − lim
S&T
(13.36)
The above equation (13.36) can be viewed as a differential equation to be
solved for log P (t, T ) under the initial condition P (T, T ) = 1, which yields
the following proposition.
Proposition 13.6. We have
w
T
P (t, T ) = exp −
f (t, s)ds ,
t
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0 6 t 6 T.
(13.37)
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Forward Rate Modeling
Proof. We check that
log P (t, T ) = log P (t, T ) − log P (t, t) =
w T ∂ log P (t, s)
wT
ds = −
f (t, s)ds.
t
t
∂s
Proposition 13.6 also shows that
f (t, t) =
=
=
=
=
∂ wT
f (t, s)ds|T =t
∂T t
∂
log P (t, T )|T =t
−
∂T
1
∂
−
P (t, T )|T =t
P (t, T ) |T =t ∂T
i
h
rT
∂
−
IE∗ e − t rs ds Ft
∂T
|T =t
i
h
rT
IE∗ rT e − t rs ds Ft
|T =t
= IE∗ [rt | Ft ]
= rt ,
i.e. the short rate rt can be recovered from the instantaneous forward rate
as
rt = f (t, t) = lim f (t, T ).
T &t
As a consequence of (13.33) and (13.37) the forward rate f (t, T, S), 0 6 t 6
T 6 S, can be recovered from (13.34) and the instantaneous forward rate
f (t, s), as:
log P (t, T ) − log P (t, S)
S−T
w
wS
T
1
f (t, s)ds
=−
f (t, s)ds −
t
t
S−T
1 wS
=
f (t, s)ds,
0 6 t 6 T < S.
S−T T
f (t, T, S) =
(13.38)
In particular, the spot forward rate or yield f (t, t, T ) can be written as
f (t, t, T ) = −
"
log P (t, T )
1 wT
=
f (t, s)ds,
T −t
T −t t
0 6 t < T.
(13.39)
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Differentiation with respect to T of the above relation shows that the yield
f (t, t, T ) and the instantaneous forward rate f (t, s) are linked by the relation
wT
1
∂f
1
f (t, s)ds +
(t, t, T ) = −
f (t, T ),
2
∂T
(T − t) t
T −t
0 6 t < T,
from which it follows that
1 wT
∂f
f (t, s)ds + (T − t)
(t, t, T )
T −t t
∂T
∂f
= f (t, t, T ) + (T − t)
(t, t, T ),
0 6 t < T.
∂T
f (t, T ) =
Forward Swap Rates
The first interest rate swap occured in 1981 between IBM and the World
Bank. The vanilla interest rate swap makes it possible to exchange a sequence of variable forward rates f (t, Tk , Tk+1 ), k = 1, 2, . . . , n − 1, against a
fixed rate κ over a time period [T1 , Tn ]. Over the succession of time intervals
[T1 , T2 ), [T2 , T3 ), . . . , [Tn−1 , Tn ] defining a tenor structure, see Section 14.1
for details, the combination of such exchanges will generate a cumulative
discounted cash flow
!
!
n−1
n−1
r Tk+1
r Tk+1
X
X
rs ds
rs ds
(Tk+1 − Tk ) e − t
f (t, Tk , Tk+1 ) −
κ(Tk+1 − Tk ) e − t
k=1
k=1
=
n−1
X
(Tk+1 − Tk ) e
−
r Tk+1
t
rs ds
(f (t, Tk , Tk+1 ) − κ),
k=1
at time t = T0 , in which we used simple (or linear) interest rate compounding.
This cash flow is used to make the contract fair, and it can be priced at time
t as
"n−1
#
r Tk+1
X
rs ds
IE∗
(Tk+1 − Tk ) e − t
(f (t, Tk , Tk+1 ) − κ) Ft
k=1
=
n−1
X
rT
k+1
rs ds (Tk+1 − Tk )(f (t, Tk , Tk+1 ) − κ) IE∗ e − t
Ft
k=1
=
n−1
X
(Tk+1 − Tk )P (t, Tk+1 ) f (t, Tk , Tk+1 ) − κ .
k=1
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Forward Rate Modeling
The swap rate S(t, T1 , Tn ) is by definition the value of the rate κ that makes
the contract fair by making this cash flow vanish. The next Proposition 13.7
makes use of the annuity numéraire
P (t, T1 , Tn ) :=
n−1
X
(Tk+1 − Tk )P (t, Tk+1 ),
0 6 t 6 T1 ,
(13.40)
k=1
which represents the present value at time t of future $1 receipts at times
T1 , T2 , . . . , Tn , weighted by the time intervals Tk+1 − Tk , k = 1, 2, . . . , n − 1.
Proposition 13.7. The LIBOR swap rate S(t, T1 , Tn ) is given by
S(t, T1 , Tn ) =
n−1
X
1
(Tk+1 − Tk )P (t, Tk+1 )f (t, Tk , Tk+1 ).
P (t, T1 , Tn )
k=1
(13.41)
Proof. By definition, S(t, T1 , Tn ) is the fixed rate over [T1 , Tn ] that will
be agreed in exchange for the family of forward rates f (t, Tk , Tk+1 ), k =
1, 2, . . . , n − 1, and it solves
n−1
X
(Tk+1 − Tk )P (t, Tk+1 ) f (t, Tk , Tk+1 ) − S(t, T1 , Tn ) = 0,
(13.42)
k=1
i.e.
0=
n−1
X
(Tk+1 − Tk )P (t, Tk+1 )f (t, Tk , Tk+1 )
k=1
−S(t, T1 , Tn )
n−1
X
(Tk+1 − Tk )P (t, Tk+1 )
k=1
=
n−1
X
(Tk+1 − Tk )P (t, Tk+1 )f (t, Tk , Tk+1 ) − P (t, T1 , Tn )S(t, T1 , Tn ),
k=1
which shows (13.41) by solving for S(t, T1 , Tn ) .
The time intervals (Tk+1 − Tk )k=1,2,...,n−1 in the definition (13.40) of the
annuity numéraire can be replaced by coupon payments (ck+1 )k=1,2,...,n−1
occurring at times (Tk+1 )k=1,2,...,n−1 , in which case the annuity numéraire
becomes
P (t, T1 , Tn ) =
n−1
X
ck+1 P (t, Tk+1 ),
0 6 t 6 T1 ,
(13.43)
k=1
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which represents the value at time t of the future coupon payments discounted
according to the bond prices P (t, Tk+1 )k=1,2,...,n−1 . This expression can also
be used for amortizing swaps in which the value of the notional decreases
over time, or for accreting swaps in which the value of the notional increases
over time.
LIBOR Rates
Recall that the forward rate f (t, T, S), 0 6 t 6 T 6 S, is defined using
exponential compounding, from the relation
f (t, T, S) = −
log P (t, S) − log P (t, T )
.
S−T
(13.44)
In order to compute swaption prices one prefers to use forward rates as defined on the London InterBank Offered Rates (LIBOR) market instead of the
standard forward rates given by (13.44).
The forward LIBOR L(t, T, S) for a loan on [T, S] is defined using linear
compounding, i.e. by replacing (13.44) with the relation
1 + (S − T )L(t, T, S) =
P (t, T )
,
P (t, S)
0 6 t 6 T,
which yields the following definition.
Definition 13.8. The forward LIBOR rate L(t, T, S) at time t for a loan on
[T, S] is given by
L(t, T, S) =
1
S−T
P (t, T )
−1 ,
P (t, S)
0 6 t 6 T < S.
(13.45)
Note that (13.45) above yields the same formula for the (LIBOR) instantaneous forward rate
L(t, T ) : = lim L(t, T, S)
S&T
P (t, S) − P (t, T )
(S − T )P (t, S)
P (t, T + ε) − P (t, T )
= lim
ε&0
εP (t, T + ε)
P (t, T + ε) − P (t, T )
1
lim
=
P (t, T ) ε&0
ε
1
∂P (t, T )
=−
P (t, T ) ∂T
= lim
S&T
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Forward Rate Modeling
=−
∂ log P (t, T )
,
∂T
as (13.36).
In addition, Relation (13.45) shows that the LIBOR rate can be viewed
as a forward price X̂t = Xt /Nt with numéraire Nt = (S − T )P (t, S) and
Xt = P (t, T ) − P (t, S), according to Relation (12.7) of Chapter 12. As a
consequence, from Proposition 12.4, the LIBOR rate (L(t, T, S))t∈[T,S] is a
martingale under the forward measure P̂ defined by
rS
1
dP̂
e − 0 rt dt .
=
dP∗
P (0, S)
LIBOR Swap Rates
The LIBOR swap rate S(t, T1 , Tn ) satisfies the same relation as (13.42) with
the forward rate f (t, Tk , Tk+1 ) replaced with the LIBOR rate L(t, Tk , Tk+1 ),
i.e.
n−1
X
(Tk+1 − Tk )P (t, Tk+1 )(L(t, Tk , Tk+1 ) − S(t, T1 , Tn )) = 0.
k=1
Proposition 13.9. The LIBOR swap rate S(t, T1 , Tn ) is given by
S(t, T1 , Tn ) =
P (t, T1 ) − P (t, Tn )
,
P (t, T1 , Tn )
0 6 t 6 T1 .
(13.46)
Proof. By (13.41), (13.45) and a telescoping sum. we have
n−1
X
1
(Tk+1 − Tk )P (t, Tk+1 )L(t, Tk , Tk+1 )
P (t, T1 , Tn )
k=1
n−1
X
P (t, Tk )
1
P (t, Tk+1 )
−1
=
P (t, T1 , Tn )
P (t, Tk+1 )
S(t, T1 , Tn ) =
k=1
n−1
X
1
=
(P (t, Tk ) − P (t, Tk+1 ))
P (t, T1 , Tn )
k=1
=
P (t, T1 ) − P (t, Tn )
.
P (t, T1 , Tn )
(13.47)
Clearly, a simple expression for the swap rate such as that of Proposition 13.9
cannot be obtained using the standard (i.e. non-LIBOR) rates defined in
(13.44). Similarly, it will not be available for amortizing or accreting swaps
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N. Privault
because the telescoping summation argument does not apply to the expression
(13.43) of the annuity numeraire.
When n = 2, the swap rate S(t, T1 , T2 ) coincides with the forward rate
L(t, T1 , T2 ):
S(t, T1 , T2 ) = L(t, T1 , T2 ),
(13.48)
and the bond prices P (t, T1 ) can be recovered from the forward swap rates
S(t, T1 , Tn ).
Similarly to the case of LIBOR rates, Relation (13.46) shows that the
LIBOR swap rate can be viewed as a forward price with (annuity) numéraire
Nt = P (t, T1 , Tn ) and Xt = P (t, T1 ) − P (t, Tn ). Consequently the LIBOR
swap rate (S(t, T1 , Tn )t∈[T,S] is a martingale under the forward measure P̂
defined from (12.1) by
P (T1 , T1 , Tn ) − r0T1 rt dt
dP̂
=
e
.
dP∗
P (0, T1 , Tn )
Yield curve data
We refer to Chapter III-12 of [Cha14] on the R package “YieldCurve” [Gui15]
for the following code and further details on yield curve and interest rate
modeling using R.
install.packages("YieldCurve")
require(YieldCurve)
data(FedYieldCurve)
first(FedYieldCurve,'3 month')
last(FedYieldCurve,'3 month')
mat.Fed=c(0.25,0.5,1,2,3,5,7,10)
n=50
plot(mat.Fed, FedYieldCurve[n,], type="o",xlab="Maturities structure in years", ylab="Interest rates
values")
title(main=paste("Federal Reserve yield curve observed at",time(FedYieldCurve[n], sep=" ") ))
grid()
The next Figure 13.15 is plotted using this code∗ which is adapted from
http://www.quantmod.com/examples/chartSeries3d/chartSeries3d.alpha.R
∗
Click to open or download.
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Forward Rate Modeling
15%
14%
13%
12%
11%
10%
9%
8%
7%
6%
5%
4%
3%
Jan Jan
19821984 Jan Jan
Jan
1986
Jan
1988
Jan
1990
1992
1994
2%
1%
Jan
1996
Jan
1998
Jan
2000
Jan
2002
Jan
2004
Jan
2006
Jan
2008
Jan
2010
Jan
2012
0% R_10Y
R_7Y
R_5Y
R_3Y
R_2Y
R_1Y
R_6M
R_3M
Jan
2012
Fig. 13.15: Federal Reserve yield curves from 1982 to 2012.
European Central Bank (ECB) data can be similarly obtained.
data(ECBYieldCurve)
first(ECBYieldCurve,'3 month')
last(ECBYieldCurve,'3 month')
mat.ECB<-c(3/12, 0.5, 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)
for (n in 200:400) {
plot(mat.ECB, ECBYieldCurve[n,], type="o",xlab="Maturities structure in years", ylab="Interest
rates values",ylim=c(3.1,5.1))
title(main=paste("European Central Bank yield curve observed at",time(ECBYieldCurve[n], sep=" ")
))
grid()
Sys.sleep(0.5)
}
The next Figure 13.16 represents the output of the above script.
Fig. 13.16: European Central Bank yield curves.∗
∗
The animation works in Acrobat Reader on the entire pdf file.
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Decreasing yield curves can occur when central banks attempts to limit
inflation by tightening interest rates. In the next section we turn to the modeling of the market curves observed in Figure 13.16.
13.5 The HJM Model
From the beginning of this chapter we have started with the modeling of
the short rate (rt )t∈R+ , followed by its consequences on the pricing of bonds
P (t, T ) and on the expressions of the forward rates f (t, T, S) and L(t, T, S).
In this section we choose a different starting point and consider the problem of directly modeling the instantaneous forward rate f (t, T ). The graph
given in Figure 13.17 presents a possible random evolution of a forward interest rate curve using the Musiela convention, i.e. we will write
g(x) = f (t, t + x) = f (t, T ),
under the substitution x = T − t, x > 0, and represent a sample of the
instantaneous forward curve x 7−→ f (t, t + x) for each t ∈ R+ .
Forward rate
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
20
15
10
0
5
10
x
t
5
15
20
0
Fig. 13.17: Stochastic process of forward curves.
In the Heath-Jarrow-Morton (HJM) model, the instantaneous forward rate
f (t, T ) is modeled under P∗ by a stochastic differential equation of the form
dt f (t, T ) = α(t, T )dt + σ(t, T )dBt ,
0 6 t 6 T,
(13.49)
where t 7−→ α(t, T ) and t 7−→ σ(t, T ), 0 6 t 6 T , are allowed to be random (adapted) processes. In the above equation, the date T is fixed and the
differential dt is with respect to t.
Under basic Markovianity assumptions, a HJM model with deterministic
coefficients α(t, T ) and σ(t, T ) will yield a short rate process (rt )t∈R+ of the
form
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Forward Rate Modeling
drt = (a(t) − b(t)rt )dt + σ(t)dBt ,
cf. § 6.6 of [Pri12], which is the Hull-White model [HW90], with explicit
solution
wt
wt
rt
rt
rt
σ(u) e − u b(τ )dτ dBu ,
e − u b(τ )dτ a(u)du +
rt = rs e − s b(τ )dτ +
s
s
0 6 s 6 t.
The HJM Condition
How to “encode” absence of arbitrage in the defining HJM Equation (13.49)
is an important question. Recall that under absence of arbitrage, the bond
price P (t, T ) has been constructed as
w
w
T
T
P (t, T ) = IE∗ exp −
rs ds Ft = exp −
f (t, s)ds , (13.50)
t
t
cf. Proposition 13.6, hence the discounted bond price process is given by
w
w
wT
t
t
t 7−→ exp − rs ds P (t, T ) = exp − rs ds −
f (t, s)ds
(13.51)
0
t
0
is a martingale under P∗ by Proposition 13.1 and Relation (13.37) in Proposition 13.6. This shows that P∗ is a risk-neutral measure, and by the first
fundamental Theorem 5.7 of asset pricing we conclude that the market is
without arbitrage opportunities.
Proposition 13.10. (HJM Condition [HJM92]). Under the condition
α(t, T ) = σ(t, T )
wT
t
σ(t, s)ds,
t ∈ [0, T ],
(13.52)
which is known as the HJM absence of arbitrage condition, the discounted
bond price process (13.51) is a martingale, and the measure P∗ is risk-neutral.
Proof. Consider the spot forward rate, or yield, given from (13.39) as
f (t, t, T ) =
and let
Xt =
wT
t
1 wT
f (t, s)ds,
T −t t
f (t, s)ds = − log P (t, T ),
0 6 t 6 T,
with the relation
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N. Privault
f (t, t, T ) =
1 wT
Xt
f (t, s)ds =
,
T −t t
T −t
0 6 t 6 T,
(13.53)
where the dynamics of t 7−→ f (t, s) is given by (13.49). We note that when
f (t, s) = g(t)h(s) is a smooth function which satisfies the separation of variables property we have the relation
dt
wT
t
g(t)h(s)ds = −g(t)h(t)dt + g 0 (t)
wT
h(s)dsdt,
t
which extends to f (t, s) as
dt
wT
t
f (t, s)ds = −f (t, t)dt +
wT
t
dt f (t, s)ds,
which can be seen as a form of the Leibniz integral rule. Therefore we have
dt Xt = dt
wT
f (t, s)ds
wT
= −f (t, t)dt +
dt f (t, s)ds
t
wT
wT
= −f (t, t)dt +
α(t, s)dsdt +
σ(t, s)dsdBt
t
w
tw
T
T
= −rt dt +
α(t, s)ds dt +
σ(t, s)ds dBt ,
t
t
t
hence we have
|dt Xt |2 =
w
T
t
2
σ(t, s)ds dt.
Hence by Itô’s calculus we have
dt P (t, T ) = dt e −Xt
1 −Xt
e
(dt Xt )2
2
2
w
T
1
= − e −Xt dt Xt + e −Xt
σ(t, s)ds dt
t
2
wT
wT
−Xt
= −e
−rt dt +
α(t, s)dsdt +
σ(t, s)dsdBt
= − e −Xt dt Xt +
t
1
+ e −Xt
2
w
T
t
t
2
σ(t, s)ds dt,
and the discounted bond price satisfies
w
t
dt exp − rs ds P (t, T )
0
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w
w
t
t
= −rt exp − rs ds − Xt dt + exp − rs ds dt P (t, T )
0
0
w
w
t
t
= −rt exp − rs ds − Xt dt − exp − rs ds − Xt dt Xt
0
0
w
w
2
t
T
1
+ exp − rs ds − Xt
σ(t, s)ds dt
0
t
2
w
t
= −rt exp − rs ds − Xt dt
0
w
wT
wT
t
− exp − rs ds − Xt
−rt dt +
α(t, s)dsdt +
σ(t, s)dsdBt
t
0
t
w
w
2
t
T
1
σ(t, s)ds dt
+ exp − rs ds − Xt
0
t
2
w
w
t
T
= − exp − rs ds − Xt
σ(t, s)dsdBt
t
0
2 !
w
w
t
T
1 wT
σ(t, s)ds
dt.
− exp − rs ds − Xt
α(t, s)dsdt −
t
0
t
2
Thus, the discounted bond price process
w
t
t 7−→ exp − rs ds P (t, T )
0
will be a martingale provided that
wT
t
α(t, s)ds −
1
2
w
T
t
2
σ(t, s)ds = 0,
0 6 t 6 T.
(13.54)
Differentiating the above relation with respect to T , we get
α(t, T ) = σ(t, T )
which is in fact equivalent to (13.54).
wT
t
σ(t, s)ds,
13.6 Forward Vasicek Rates
In this section we consider the Vasicek model, in which the short rate process
is the solution (13.2) of (13.1) as illustrated in Figure 13.1.
In the Vasicek model, the forward rate is given by
f (t, T, S) = −
"
log P (t, S) − log P (t, T )
S−T
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rt (C(S − t) − C(T − t)) + A(S − t) − A(T − t))
S−T
σ 2 − 2ab
=−
2
2b
σ2
1
rt
σ 2 − ab
e −b(S−t) − e −b(T −t) − 3 e −2b(S−t) − e −2b(T −t) ,
−
+
3
S−T
b
b
4b
=−
and spot forward rate, or yield, satisfies
log P (t, T )
rt C(T − t) + A(T − t)
=−
T −t
T −t
1
rt
σ 2 − ab
σ2
σ 2 − 2ab
+
+
(1 − e −b(T −t) ) − 3 (1 − e −2b(T −t) ) .
=−
2
3
2b
T −t
b
b
4b
f (t, t, T ) = −
In this model, the forward rate t 7−→ f (t, T, S) can be represented as in
Figure 13.18, with here b/a > r0 .
0.01
f(t,T,S)
0.0095
0.009
0.0085
0.008
0.0075
0.007
0.0065
0.006
0.0055
0.005
0
2
4
6
8
10
t
Fig. 13.18: Forward rate process t 7−→ f (t, T, S).
Note that the forward rate cure t 7−→ f (t, T, S) appears flat for small values
of t, i.e. longer rates are more stable, while shorter rates show higher volatility
or risk. Similar features can be observed in Figure 13.19 for the instantaneous
short rate given by
∂ log P (t, T )
(13.55)
∂T
2
a
σ
2
= rt e −b(T −t) + 1 − e −b(T −t) − 2 1 − e −b(T −t) ,
b
2b
f (t, T ) : = −
from which the relation limT &t f (t, T ) = rt can be easily recovered.
The instantaneous forward rate t 7−→ f (t, T ) can be represented as in Figure 13.19, with here t = 0 and b/a > r0 :
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0.14
f(t,T)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
2
4
6
8
10
12
14
16
18
20
t
Fig. 13.19: Instantaneous forward rate process t 7−→ f (t, T ).
The HJM coefficients in the Vasicek model are in fact deterministic and
taking a = 0 we have
dt f (t, T ) = σ 2 e −b(T −t)
wT
t
e b(t−s) dsdt + σ e −b(T −t) dBt ,
i.e.
α(t, T ) = σ 2 e −b(T −t)
wT
t
e b(t−s) ds = σ 2 e −b(T −t)
1 − e −b(T −t)
,
b
and σ(t, T ) = σ e −b(T −t) , and the HJM condition reads
α(t, T ) = σ 2 e −b(T −t)
wT
t
e b(t−s) ds = σ(t, T )
wT
t
σ(t, s)ds.
(13.56)
Random simulations of the Vasicek instantaneous forward rates are provided
in Figures 13.20 and 13.21.
Fig. 13.20: Forward instantaneous curve (t, x) 7−→ f (t, t + x) in the Vasicek model.∗
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Fig. 13.21: Forward instantaneous curve x 7−→ f (0, x) in the Vasicek model.∗
For x = 0 the first “slice” of this surface is actually the short rate Vasicek
process rt = f (t, t) = f (t, t + 0) which is represented in Figure 13.22 using
another discretization.
0.07
0.065
0.06
0.055
0.05
0.045
0.04
0.035
0.03
0
5
10
15
20
Fig. 13.22: Short term interest rate curve t 7−→ rt in the Vasicek model.
13.7 Modeling Issues
Parametrization of Forward Rates
In the Nelson-Siegel parametrization the instantaneous forward rate curves
are parametrized by 4 coefficients z1 , z2 , z3 , z4 , as
g(x) = z1 + (z2 + z3 x) e −xz4 ,
x > 0.
An example of a graph obtained by the Nelson-Siegel parametrization is given
in Figure 13.23, for z1 = 1, z2 = −10, z3 = 100, z4 = 10.
∗
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4
z1+(z2+xz3)exp(-xz4)
2
0
-2
-4
-6
-8
-10
0
0.2
0.4
0.6
0.8
1
Fig. 13.23: Graph of x 7−→ g(x) in the Nelson-Siegel model.
The Svensson parametrization has the advantage to reproduce two humps instead of one, the location and height of which can be chosen via 6 parameters
z1 , z2 , z3 , z4 , z5 , z6 as
g(x) = z1 + (z2 + z3 x) e −xz4 + z5 x e −xz6 ,
x > 0.
A typical graph of a Svensson parametrization is given in Figure 13.24, for
z1 = 7, z2 = −5, z3 = −100, z4 = 10, z5 = −1/2, z6 = −1.
5
x->z1+(z2+z3*x)*exp(-x*z4)+z5*x*exp(-z6*x)
4.5
4
3.5
3
2.5
2
0
5
10
15
20
25
30
lambda
Fig. 13.24: Graph of x 7−→ g(x) in the Svensson model.
Figure 13.25 presents a fit of the market data of Figure 13.13 using a Svensson
curve.
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5
4.5
4
3.5
3
2.5
Market data
Svensson curve
2
0
5
10
15
years
20
25
30
Fig. 13.25: Comparison of market data vs a Svensson curve.
It can be shown, cf. § 3.5 of [Bjö04b], that the forward yield curves of the
Vasicek model are included neither in the Nelson-Siegel space, nor in the
Svensson space. In addition, the Vasicek yield curves do not appear to correctly model the market forward curves cf. also Figure 13.13 above.
In the Vasicek model we have
∂f
σ2
σ 2 −b(T −t)
(t, T ) = −brt + a −
+
e
e −b(T −t) ,
∂T
b
b
and one can check that the sign of the derivatives of f can only change once
at most. As a consequence, the possible forward curves in the Vasicek model
are limited to one change of “regime” per curve, as illustrated in Figure 13.26
for various values of rt , and in Figure 13.27.
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
20
Fig. 13.26: Graphs of forward rates.
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Forward Rate Modeling
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0
2
20
4
x
15
6
10
8
5
10
t
0
Fig. 13.27: Forward instantaneous curve (t, x) 7−→ f (t, t + x) in the Vasicek model.
One may think of constructing an instantaneous rate process taking values in
the Svensson space, however this type of modelization is not consistent with
absence of arbitrage, and it can be proved that the HJM curves cannot live
in the Nelson-Siegel or Svensson spaces, cf. §3.5 of [Bjö04b].
Another way to deal with the curve fitting problem is to use deterministic
shifts for the fitting of one forward curve, such as the initial curve at t = 0,
cf. e.g. § 8.2 of [Pri12].
Fitting the Nelson-Siegel and Svensson models to yield curve data
Recall that in the Nelson-Siegel parametrization the instantaneous forward
rate curves are parametrized by four coefficients z1 , z2 , z3 , z4 , as
f (t, t + y) = z1 + (z2 + z3 y) e −yz4 ,
y > 0.
(13.57)
Recall taking x = T − t, the yield f (t, t, T ) is given as
1 wT
f (t, s)ds
T −t t
1 wx
=
f (t, t + y)dy
x 0
z2 w x −yz4
z3 w x −yz4
= z1 +
e
dy +
ye
dy
x 0
x 0
1 − e −xz4
1 − e −xz4 + x e −xz4
= z1 + z2
+ z3
.
xz4
xz4
f (t, t, T ) =
The expression (13.57) can be represented in the parametrization
f (t, t + x) = z1 + (z2 + z3 x) e −xz4 = β0 + β1 e −x/λ +
"
β2 −x/λ
xe
,
λ
x > 0,
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cf. [Cha14], with β0 = z1 , β1 = z2 , β2 = z3 /z4 , λ = 1/z4 .
require(YieldCurve)
data(ECBYieldCurve)
mat.ECB<-c(3/12, 0.5, 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)
first(ECBYieldCurve, '1 month')
Nelson.Siegel(first(ECBYieldCurve, '1 month'), mat.ECB)
for (n in seq(from=70, to=290, by=10)) {
ECB.NS <- Nelson.Siegel(ECBYieldCurve[n,], mat.ECB)
ECB.S <- Svensson(ECBYieldCurve[n,], mat.ECB)
ECB.NS.yield.curve <- NSrates(ECB.NS, mat.ECB)
ECB.S.yield.curve <- Srates(ECB.S, mat.ECB,"Spot")
plot(mat.ECB, as.numeric(ECBYieldCurve[n,]), type="o", lty=1, col=1,ylab="Interest rates", xlab=
"Maturity in years", ylim=c(3.2,4.8))
lines(mat.ECB, as.numeric(ECB.NS.yield.curve), type="l", lty=3,col=2,lwd=2)
lines(mat.ECB, as.numeric(ECB.S.yield.curve), type="l", lty=2,col=6,lwd=2)
title(main=paste("ECB yield curve observed at",time(ECBYieldCurve[n], sep=" "),"vs fitted yield
curve"))
legend('bottomright', legend=c("ECB data","Nelson-Siegel","Svensson"),col=c(1,2,6), lty=1, bg='
gray90')
grid()
Sys.sleep(0.5)
}
Fig. 13.28: ECB data vs fitted yield curve.∗
The Correlation Problem and a Two-Factor Model
The correlation problem is another issue of concern when using the affine
models considered so far. Let us compare three bond price simulations with
maturity T1 = 10, T2 = 20, and T3 = 30 based on the same Brownian path,
as given in Figure 13.29. Clearly, the bond prices F (rt , T1 ) = P (t, T1 ) and
∗
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F (rt , T2 ) = P (t, T2 ) with maturities T1 and T2 are linked by the relation
P (t, T2 ) = P (t, T1 ) exp(A(t, T2 ) − A(t, T1 ) + rt (C(t, T2 ) − C(t, T1 ))), (13.58)
meaning that bond prices with different maturities could be deduced from
each other, which is unrealistic.
1
0.9
0.8
0.7
0.6
0.5
0.4
P(t,T1)
P(t,T2)
P(t,T3)
0.3
0
5
10
15
t
20
25
30
Fig. 13.29: Graph of t 7−→ P (t, T1 ).
In affine short rates models, by (13.58), log P (t, T1 ) and log P (t, T2 ) are linked
by the linear relationship
log P (t, T2 ) = log P (t, T1 ) + A(t, T2 ) − A(t, T1 ) + rt (C(t, T2 ) − C(t, T1 ))
log P (t, T1 ) − C(t, T1 )
= log P (t, T1 ) + A(t, T2 ) − A(t, T1 ) + (C(t, T2 ) − C(t, T1 ))
A(t, T1 )
C(t, T2 ) − C(t, T1 )
= 1+
log P (t, T1 )
A(t, T1 )
C(t, T1 )
+A(t, T2 ) − A(t, T1 ) − (C(t, T2 ) − C(t, T1 ))
A(t, T1 )
with constant coefficients, which yields the perfect (anti)correlation
Cor(log P (t, T1 ), log P (t, T2 )) = ±1,
depending on the sign of the coefficient 1 + (C(t, T2 ) − C(t, T1 ))/A(t, T1 ), cf.
§ 8.3 of [Pri12],
A solution to the correlation problem is to consider a two-factor model
based on two control processes (Xt )t∈R+ , (Yt )t∈R+ which are solution of

(1)

 dXt = µ1 (t, Xt )dt + σ1 (t, Xt )dBt ,
(13.59)

 dY = µ (t, Y )dt + σ (t, Y )dB (2) ,
t
2
t
2
t
t
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(1)
(2)
where (Bt )t∈R+ , (Bt )t∈R+ have correlated Brownian motion with
(2)
Cov(Bs(1) , Bt ) = ρ min(s, t),
and
(1)
dBt
(2)
· dBt
(13.60)
s, t ∈ R+ ,
= ρdt,
(13.61)
for some correlation parameter ρ ∈ [−1, 1]. In practice, (B )t∈R+ and
(B (2) )t∈R+ can be constructed from two independent Brownian motions
(W (1) )t∈R+ and (W (2) )t∈R+ , by letting

(1)
(1)

 Bt = Wt ,
(1)

 B (2) = ρW (1) + p1 − ρ2 W (2) ,
t
t
t
t ∈ R+ ,
and Relations (13.60) and (13.61) are easily satisfied from this construction.
In two-factor models one chooses to build the short term interest rate rt via
rt := Xt + Yt ,
t ∈ R+ .
By the previous standard arbitrage arguments we define the price of a bond
with maturity T as
w
T
P (t, T ) : = IE∗ exp −
rs ds Ft
t
w
T
= IE∗ exp −
rs ds Xt , Yt
t
w
T
∗
= IE exp −
(Xs + Ys )ds Xt , Yt
t
= F (t, Xt , Yt ),
(13.62)
since the couple (Xt , Yt )t∈R+ is Markovian. Applying the Itô formula with
two variables to
w
T
t 7−→ F (t, Xt , Yt ) = P (t, T ) = IE∗ exp −
rs ds Ft ,
t
and using the fact that the discounted process
w
rt
T
t 7−→ e − 0 rs ds P (t, T ) = IE∗ exp −
rs ds Ft
0
is an Ft -martingale under P , we can derive a PDE
∗
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Forward Rate Modeling
∂F
∂F
(t, x, y) + µ2 (t, y)
(t, x, y)
∂x
∂y
1
1
∂2F
∂2F
+ σ12 (t, x) 2 (t, x, y) + σ22 (t, y) 2 (t, x, y)
2
∂x
2
∂y
∂2F
∂F
+ρσ1 (t, x)σ2 (t, y)
(t, x, y) +
(t, Xt , Yt ) = 0,
(13.63)
∂x∂y
∂t
−(x + y)F (t, x, y) + µ1 (t, x)
on R2 for the bond price P (t, T ). In the Vasicek model

(1)

 dXt = −aXt dt + σdBt ,

 dY = −bY dt + ηdB (2) ,
t
t
t
this yields the solution F (t, x, y) of (13.63) as
P (t, T ) = F (t, Xt , Yt ) = F1 (t, Xt )F2 (t, Yt ) exp (Uρ (t, T )) ,
(13.64)
where F1 (t, Xt ) and F2 (t, Yt ) are the bond prices associated to Xt and Yt in
the Vasicek model, and
e −a(T −t) − 1
e −b(T −t) − 1
e −(a+b)(T −t) − 1
ση
T −t+
+
−
Uρ (t, T ) := ρ
ab
a
b
a+b
(1)
(2)
is a correlation term which vanishes when (Bt )t∈R+ and (Bt )t∈R+ are independent, i.e. when ρ = 0, cf [BM06], Chapter 4, Appendix A, and § 8.4 of
[Pri12].
Partial differentiation of log P (t, T ) with respect to T leads to the instantaneous forward rate
f (t, T ) = f1 (t, T ) + f2 (t, T ) − ρ
ση
(1 − e −a(T −t) )(1 − e −b(T −t) ),
ab
(13.65)
where f1 (t, T ), f2 (t, T ) are the instantaneous forward rates corresponding to
Xt and Yt respectively, cf. § 8.4 of [Pri12].
An example of a forward rate curve obtained in this way is given in Figure 13.30.
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0.24
0.23
0.22
0.21
0.2
0.19
0.18
0
5
10
15
20
T
25
30
35
40
Fig. 13.30: Graph of forward rates in a two-factor model.
Next in Figure 13.31 we present a graph of the evolution of forward curves
in a two factor model.
0.24
0.235
0.23
0.225
0.22
0.215
1.4
1.2
1
0.8
t
0.6
0.4
0.2
0 0
2
1
3
4
5
6
7
8
x
Fig. 13.31: Random evolution of forward rates in a two-factor model.
13.8 The BGM Model
The models (HJM, affine, etc.) considered in the previous chapter suffer
from various drawbacks such as nonpositivity of interest rates in Vasicek
model, and lack of closed form solutions in more complex models. The BGM
[BGM97] model has the advantage of yielding positive interest rates, and to
permit to derive explicit formulas for the computation of prices for interest
rate derivatives such as caps and swaptions on the LIBOR market.
In the BGM model we consider two bond prices P (t, T1 ), P (t, T2 ) with maturities T1 , T2 and the forward measure
r T2
e − 0 rs ds
dP2
=
,
dP∗2
P (0, T2 )
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with numeraire P (t, T2 ), cf. (12.6). The forward LIBOR rate L(t, T1 , T2 ) is
modeled as a geometric Brownian motion under P2 , i.e.
dL(t, T1 , T2 )
(2)
= γ1 (t)dBt ,
L(t, T1 , T2 )
(13.66)
0 6 t 6 T1 , i = 1, 2, . . . , n − 1, for some deterministic function γ1 (t), with
solution
w
u
1wu
|γ1 |2 (s)ds ,
L(u, T1 , T2 ) = L(t, T1 , T2 ) exp
γ1 (s)dBs(2) −
t
2 t
i.e. for u = T1 ,
L(T1 , T1 , T2 ) = L(t, T1 , T2 ) exp
w
T1
t
γ1 (s)dBs(2) −
1 w T1
|γ1 |2 (s)ds .
2 t
Since L(t, T1 , T2 ) is a geometric Brownian motion under P2 , standard caplets
can be priced at time t ∈ [0, T1 ] from the Black-Scholes formula.
The following graph 13.32 summarizes the notions introduced in this chapter.
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Bond price
P (t, T ) = e −(T −t)f (t,t,T )
2
Bondh price
i
rT
P (t, T ) = IE∗ e − t rs ds | Ft
Short rate1 rt
LIBOR rate3
(t,T )−P (t,S)
L(t, T, S) = P(S−T
)P (t,S)
Forward rate3
)−log P (t,S)
f (t, T, S) = log P (t,TS−T
Bond price
rT
P (t, T ) = e − t f (t,s)ds
Instantaneous forward rate4
P (t,T )
f (t, T ) = L(t, T ) = − ∂ log∂T
Short rate
rt = f (t, t) = f (t, t, t)
Spot forward rate (yield)
rT
f (t, t, T ) = t f (t, s)ds/(T − t)
Instantaneous forward rate4
f (t, T ) = L(t, T ) = limS&T f (t, T, S)
= limS&T L(t, T, S)
1
2
3
4
Can
Can
Can
Can
be
be
be
be
modeled
modeled
modeled
modeled
by Vasiçek and other short rate models
from dP (t, T )/P (t, T ).
in the BGM model
in the HJM model
Fig. 13.32: Roadmap of stochastic interest rate modeling.
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Exercises
Exercise 13.1 Consider a tenor structure {T1 , T2 } and a bond with maturity
T2 and price given at time t ∈ [0, T2 ] by
w
T2
P (t, T2 ) = exp −
f (t, s)ds ,
t ∈ [0, T2 ],
t
where the instantaneous yield curve f (t, s) is parametrized as
f (t, s) = r1 1[0,T1 ] (s) + r2 1[T1 ,T2 ] (s),
s ∈ [t, T2 ].
Find a formula to estimate the values of r1 and r2 from the data of P (0, T2 )
and P (T1 , T2 ).
Same question for when f (t, s) is parametrized as
f (t, s) = r1 s1[0,T1 ] (s) + (r1 T1 + r2 (s − T1 ))1[T1 ,T2 ] (s),
s ∈ [t, T2 ].
Exercise 13.2 Let (Bt )t∈R+ denote a standard Brownian motion started at
0 under the risk-neutral measure P∗ . We consider a short term interest rate
process (rt )t∈R+ in a Ho-Lee model with constant deterministic volatility,
defined by
drt = adt + σdBt ,
where a ∈ R and σ > 0. Let P (t, T ) will denote the arbitrage price of a
zero-coupon bond in this model:
w
T
P (t, T ) = IE∗ exp −
rs ds Ft ,
0 6 t 6 T.
(13.67)
t
a) State the bond pricing PDE satisfied by the function F (t, x) defined via
w
T
F (t, x) := IE∗ exp −
rs ds rt = x ,
0 6 t 6 T.
t
b) Compute the arbitrage price F (t, rt ) = P (t, T ) from its expression (13.67)
as a conditional expectation.
Hint. One may use the integration by parts relation
wT
t
Bs ds = T BT − tBt −
wT
t
sdBs
= (T − t)Bt + T (BT − Bt ) −
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= (T − t)Bt +
wT
t
(T − s)dBs ,
2 2
and the Laplace transform identity IE[ e λX ] = e λ η /2 for X ' N (0, η 2 ).
c) Check that the function F (t, x) computed in question (b) does satisfy the
PDE derived in question (a).
d) Compute the forward rate f (t, T, S) in this model.
From now on we let a = 0.
e) Compute the instantaneous forward rate f (t, T ) in this model.
f) Derive the stochastic equation satisfied by the instantaneous forward rate
f (t, T ).
g) Check that the HJM absence of arbitrage condition is satisfied in this
equation.
Exercise 13.3
Consider the CIR process (rt )t∈R+ solution of
√
drt = −art dt + σ rt dBt ,
where a, σ > 0 are constants (Bt )t∈R+ is a standard Brownian motion started
at 0.
a) Write down the bond pricing PDE for the function F (t, x) given by
w
T
F (t, x) := IE∗ exp −
rs ds rt = x ,
0 6 t 6 T.
t
Hint: Use Itô calculus and the fact that the discounted bond price is a
martingale.
b) Show that the PDE of Question (a) admits a solution of the form
F (t, x) = e A(T −t)+xC(T −t) where the functions A(s) and C(s) satisfy
ordinary differential equations to be also written down together with the
values of A(0) and C(0).
Exercise 13.4 Convertible bonds. Consider an underlying stock price process
(St )t∈R+ given by
(1)
dSt = rSt dt + σSt dBt ,
and a short term interest rate process (rt )t∈R+ given by
(2)
drt = γ(t, rt )dt + η(t, rt )dBt ,
(1)
(2)
where (Bt )t∈R+ and (Bt )t∈R+ are two correlated Brownian motions under
(1)
(2)
the risk-neutral measure P∗ , with dBt · dBt = ρdt. A convertible bond
is a corporate bond that can be exchanged into a quantity α > 0 of the
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underlying company’s stock Sτ at a future time τ , whichever has a higher
value, where α is a conversion rate.
a) Find the payoff of the convertible bond at time τ .
b) Rewrite the convertible bond payoff at time τ as the linear combination
of P (τ, T ) and a call option payoff on Sτ , whose strike price is to be
determined.
c) Write down the corporate bond price at time t ∈ [0, τ ] as a function
C(t, St , rt ) of the underlying asset price and interest rate, using a discounted conditional expectation, and show that the discounted corporate
bond price
rt
e − 0 rs ds C(t, St , rt ),
t ∈ [0, τ ],
is a martingale.
rt
d) Write down d e − 0 rs ds C(t, St , rt ) using the Itô formula and derive the
pricing PDE satisfied by the function C(t, x, y) together with its terminal
condition.
e) Taking the bond price P (t, T ) as a numeraire, price the convertible bond
as a European option with strike price K = 1 on an underlying asset
˜T.
priced Zt := St /P (t, T ), t ∈ [0, τ ] under the forward measure IE
f) Assuming the bond price dynamics dP (t, T ) = rt P (t, T )dt+σB (t)P (t, T )dBt ,
determine the dynamics of the process (Zt )t∈R+ under the forward mea˜T.
sure IE
g) Assuming that (Zt )t∈R+ can be modeled as a geometric Brownian motion,
price the corporate bond option using the Black-Scholes formula.
Exercise 13.5 Given (Bt )t∈R+ a standard Brownian motion, consider a HJM
model given by
dt f (t, T ) =
σ2
T (T 2 − t2 )dt + σT dBt .
2
(13.68)
a) Show that the HJM condition is satisfied by (13.68).
b) Compute f (t, T ) by solving (13.68).
rt
Hint: We have f (t, T ) = f (0, T ) + 0 ds f (s, T ) = · · ·
c) Compute the short rate rt = f (t, t) from the result of Question (b).
d) Show that the short rate rt satisfies a stochastic differential equation of
the form
drt = η(t)dt + (rt − f (0, t))ψ(t)dt + ξ(t)dBt ,
where η(t), ψ(t), ξ(t) are deterministic functions to be determined.
Exercise 13.6 Let (rt )t∈R+ denote a short term interest rate process. For
any T > 0, let P (t, T ) denote the price at time t ∈ [0, T ] of a zero coupon
bond defined by the stochastic differential equation
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dP (t, T )
= rt dt + σtT dBt ,
P (t, T )
0 6 t 6 T,
(13.69)
under the terminal condition P (T, T ) = 1, where (σtT )t∈[0,T ] is an adapted
process. Let the forward measure PT be defined by
P (t, T ) − r t rs ds
dPT ,
0 6 t 6 T.
e 0
IE∗
Ft =
∗
dP
P (0, T )
Recall that
BtT := Bt −
wt
0
σsT ds,
0 6 t 6 T,
is a standard Brownian motion under PT .
a) Solve the stochastic differential equation (13.69).
b) Derive the stochastic differential equation satisfied by the discounted bond
price process
rt
t 7−→ e − 0 rs ds P (t, T ),
0 6 t 6 T,
and show that it is a martingale.
c) Show that
i
h rT
rt
IE∗ e − 0 rs ds Ft = e − 0 rs ds P (t, T ),
0 6 t 6 T.
d) Show that
i
h rT
P (t, T ) = IE∗ e − t rs ds Ft ,
0 6 t 6 T.
e) Compute P (t, S)/P (t, T ), 0 6 t 6 T , show that it is a martingale under
PT and that
w
T
1wT S
P (t, S)
P (T, S) =
exp
(σsS − σsT )dBsT −
(σs − σsT )2 ds .
t
P (t, T )
2 t
f) Assuming that (σtT )t∈[0,T ] and (σtS )t∈[0,S] are deterministic functions,
compute the price
i
i
h rT
h
+ + IE∗ e − t rs ds (P (T, S) − κ) Ft = P (t, T ) IET (P (T, S) − κ) Ft
of a bond option with strike price κ.
Recall that if X is a centered Gaussian random variable with mean mt
and variance vt2 given Ft , we have
2
vt
1
IE[( e X − K)+ | Ft ] = e mt +vt /2 Φ
+ (mt + vt2 /2 − log K)
2
vt
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vt
1
−KΦ − + (mt + vt2 /2 − log K)
2
vt
where Φ(x), x ∈ R, denotes the Gaussian cumulative distribution function.
Exercise 13.7 (Exercise 4.13 continued). Bridge model. Assume that the
price P (t, T ) of a zero coupon bond is modeled as
T
P (t, T ) = e −µ(T −t)+Xt ,
t ∈ [0, T ],
where µ > 0.
a) Show that the terminal condition P (T, T ) = 1 is satisfied.
b) Compute the forward rate
f (t, T, S) = −
1
(log P (t, S) − log P (t, T )).
S−T
c) Compute the instantaneous forward rate
f (t, T ) = − lim
S&T
1
(log P (t, S) − log P (t, T )).
S−T
d) Show that the limit lim f (t, T ) does not exist in L2 (Ω).
T &t
e) Show that P (t, T ) satisfies the stochastic differential equation
dP (t, T )
1
log P (t, T )
= σdBt + σ 2 dt −
dt,
P (t, T )
2
T −t
t ∈ [0, T ].
f) Show, using the results of Exercise 13.6-(d), that
h rT T i
P (t, T ) = IE∗ e − t rs ds Ft ,
where (rtT )t∈[0,T ] is a process to be determined.
g) Compute the conditional density
#
"
P (t, T ) − r t rsT ds
dPT e 0
IE∗
Ft =
∗
dP P (0, T )
of the forward measure PT with respect to P∗ .
h) Show that the process
B̃t := Bt − σt,
0 6 t 6 T,
is a standard Brownian motion under PT .
i) Compute the dynamics of XtS and P (t, S) under PT .
Hint: Show that
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−µ(S − T ) + σ(S − T )
wt
0
1
S−T
dBs =
log P (t, S).
S−s
S−t
j) Compute the bond option price
i
i
h
h rT T
IE∗ e − t rs ds (P (T, S) − K)+ Ft = P (t, T ) IET (P (T, S) − K)+ Ft ,
0 6 t < T < S.
Exercise 13.8 (Exercise 4.16 continued). Write down the bond pricing PDE
for the function
h rT
i
F (t, x) = IE∗ e − t rs ds rt = x
and show that in case α = 0 the corresponding bond price P (t, T ) equals
P (t, T ) = e −B(T −t)rt ,
where
B(x) =
with γ =
p
0 6 t 6 T,
2( e γx − 1)
,
2γ + (β + γ)( e γx − 1)
β 2 + 2σ 2 .
Exercise 13.9 Consider a short rate process (rt )t∈R+ of the form rt = h(t) +
Xt , where h(t) is a deterministic function and (Xt )R+ is a Vasicek process
started at X0 = 0.
a) Compute the price P (0, T ) at time t = 0 of a bond with maturity T , using
h(t) and the function A(T ) defined in (13.23) for the pricing of Vasicek
bonds.
b) Show how the function h(t) can be estimated from the market data of the
initial instantaneous forward rate curve f (0, t).
Exercise 13.10
a) Given two LIBOR spot rates L(t, t, T ) and L(t, t, S), compute the corresponding LIBOR forward rate L(t, T, S).
b) Assuming that L(t, t, T ) = 2%, L(t, t, S) = 2.5% and t = 0, T = 1,
S = 2T = 2, would you buy a LIBOR forward contract over [T, 2T ] with
rate L(0, T, 2T ) if L(T, T, 2T ) remained at L(T, T, 2T ) = L(0, 0, T ) = 2%?
Exercise 13.11 Black-Derman-Toy model. Consider a two-step interest rate
model in which the short term √interest rate r0 on [0, 1] √can turn into two
possible values r1u = r0 e µ∆t+σ ∆t and r1d = r0 e µ∆t−σ ∆t on [1, 2] with
equal probabilities 1/2 at time ∆t = 1 year and σ = 22% per year, and two
zero coupon bonds with prices P (0, 1) and P (0, 2) at time t = 0.
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a)
b)
c)
d)
Write down the value of P (1, 2) using r1u and r1d .
Write down the value of P (0, 2) using r1u , r1d and r0 .
Estimate the value of r0 from the market price P (0, 1) = 91.74.
Estimatethe values of r1u and r1d from the market price P (0, 2) = 83.40.
Exercise 13.12 Consider a yield curve (f (t, t, T ))06t6T and a bond paying
coupons c1 , c2 , . . . , cn at times T1 , T2 , . . . , Tn until maturity Tn , and priced as
n
X
P (t, Tn ) =
ck e −(Tk −t)f (t,t,Tk ) ,
0 6 t 6 T1 ,
k=1
where cn is inclusive of the last coupon payment and the nominal $1 value
of the bond. Let f˜(t, t, Tn ) denote the compounded yield to maturity defined
by equating
P (t, Tn ) =
n
X
˜
ck e −(Tk −t)f (t,t,Tn ) ,
0 6 t 6 T1 ,
(13.70)
k=1
i.e. f˜(t, t, Tn ) solves the equation
F (t, f˜(t, t, Tn )) = P (t, Tn ),
with
F (t, x) :=
n
X
ck e −(Tk −t)x ,
0 6 t 6 T1 ,
0 6 t 6 T1 .
k=1
The bond duration D(t, Tn ) is the relative sensitivity of P (t, Tn ) with respect
to f˜(t, t, Tn ) defined as
D(t, Tn ) := −
∂F
1
(t, f˜(t, t, Tn )),
P (t, Tn ) ∂x
0 6 t 6 T1 .
The bond convexity C(t, Tn ) is defined as
C(t, Tn ) :=
1
∂2F
(t, f˜(t, t, Tn )),
P (t, Tn ) ∂x2
0 6 t 6 T1 .
a) Compute the bond duration in case n = 1.
b) Show that the bond duration D(t, Tn ) can be interpreted as an average of
times to maturity weighted by the respective discounted bond payoffs.
c) Show that the bond convexity C(t, Tn ) satisfies
C(t, Tn ) = (D(t, Tn ))2 + (S(t, Tn ))2 ,
where S(t, Tn ) measures the dispersion of the duration of the bond payoffs
around the portfolio duration D(t, Tn ).
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d) Consider now the zero-coupon yield defined as
fα (t, t, Tn ) := −
1
log P (t, t + α(Tn − t)),
α(Tn − t)
where α ∈ (0, 1). Compute the bond duration associated to the yield
fα (t, t, Tn ) in affine bond pricing models of the form
P (t, T ) = e A(T −t)+rt B(T −t) ,
0 6 t 6 T.
e) [Wu00] Compute the bond duration associated to the yield fα (t, t, Tn ) in
the Vasicek model in which B(T − t) := (1 − e −b(T −t) )/b, 0 6 t 6 T .
Exercise 13.13 Stochastic string model [SCS01]. Consider an instantaneous
forward rate f (t, x) solution of
dt f (t, x) = αx2 dt + σdt B(t, x),
(13.71)
with a flat initial curve f (0, x) = r, where x represents the time to maturity,
and (B(t, x))(t,x)∈R2+ is a standard Brownian sheet with covariance
IE[B(s, x)B(t, y)] = (min(s, t))(min(x, y)),
s, t, x, y ∈ R+ ,
and initial conditions B(t, 0) = B(0, x) = 0 for all t, x ∈ R+ .
a) Solve the equation (13.71) for f (t, x).
b) Compute the short term interest rate rt = f (t, 0).
c) Compute the value at time t ∈ [0, T ] of the bond price
w
T −t
P (t, T ) = exp −
f (t, x)dx
0
with maturity T .
d) Compute the variance IE
r T −t
"
w
2 #
T −t
0
B(t, x)dx
of the centered Gaussian
B(t, x)dx.
random variable 0
e) Compute the expected value IE∗ [P (t, T )].
f) Find the value of α such that the discounted bond price
w T −t
α
e −rt P (t, T ) = exp −rT − t(T − t)3 − σ
B(t, x)dx ,
0
3
t ∈ [0, T ].
satisfies e −rt IE∗ [P (t, T )] = e −rT .
w
T
g) Compute the bond option price IE∗ exp −
rs ds (P (T, S) − K)+
0
by the Black-Scholes formula, knowing that
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IE[(x e m+X −K)+ ] = x e m+
v2
2
Φ(v+(m+log(x/K))/v)−KΦ((m+log(x/K))/v),
when X is a centered Gaussian random variable with mean m = rτ −v 2 /2
and variance v 2 .
Exercise 13.14
(Exercise 13.7 continued).
a) Compute the forward rate
f (t, T, S) = −
1
(log P (t, S) − log P (t, T )).
S−T
b) Compute the instantaneous forward rate
f (t, T ) = − lim
S&T
1
(log P (t, S) − log P (t, T )).
S−T
c) Show that the limit lim f (t, T ) does not exist in L2 (Ω).
T &t
d) Show that P (t, T ) satisfies the stochastic differential equation
1
log P (t, T )
dP (t, T )
= σdBt + σ 2 dt −
dt,
P (t, T )
2
T −t
t ∈ [0, T ].
e) Show, using the results of Exercise 13.6-(c), that
h rT T i
P (t, T ) = IE∗ e − t rs ds Ft ,
where (rtT )t∈[0,T ] is a process to be determined.
f) Compute the conditional density
dPT P (t, T ) − r t rsT ds
IE∗
e 0
Ft =
∗
dP
P (0, T )
of the forward measure PT with respect to P∗ .
g) Show that the process
B̃t := Bt − σt,
0 6 t 6 T,
is a standard Brownian motion under PT .
h) Compute the dynamics of XtS and P (t, S) under PT .
Hint: Show that
−µ(S − T ) + σ(S − T )
wt
0
S−T
1
dBs =
log P (t, S).
S−s
S−t
i) Compute the bond option price
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i
h rT T
IE∗ e − t rs ds (P (T, S) − K)+ Ft = P (t, T ) IET (P (T, S) − K)+ Ft ,
0 6 t < T < S.
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