chapitre 6 interest rate

Published on June 2016 | Categories: Types, Presentations | Downloads: 33 | Comments: 0 | Views: 239
of 29
Download PDF   Embed   Report

cours de taux

Comments

Content

6

Interest rate models
Interest rate models are mainly used to price and hedge bonds and bond
options. Hitherto, there has not been any reference model equivalent to the
Black-Scholes model for stock options. In this chapter, we will present the main
features of interest rate modelling (following essentially Artzner and Delbaen
(1989)), study three particular models and see how they are used in practice.

1
1.1

Introduction
Instantaneous interest rate

In most of the models that we have already studied, the interest rate was assumed to be constant. In the real world, it is observed that the loan interest
rate depends both on the date t of the loan emission and on the date T of the
maturity of the loan.
Someone borrowing one dollar at time t, until maturity T , will have to pay
back an amount F (t, T ) at time T , which is equivalent to an average interest
rate R(t, T ) given by the equality
F (t, T ) = e(T −t)R(t,T ) .
If we consider the future as certain, i.e. if we assume that all interest rates
(R(t, T ))t≤T are known, then, in an arbitrage-free world, the function F must
satisfy
∀t < u < s F (t, s) = F (t, u)F (u, s).
(1)
Indeed, it is easy to derive arbitrage schemes when this equality does not hold.
From this relationship and the equality F (t, t) = 1, it follows that, if F is
smooth, there exists a function r(t) s.t.
Z
∀t < T F (t, T ) = exp{

T

r(s)ds}

t

and consequently
1
R(t, T ) =
T −t

1

Z

T

r(s)ds.
t

The function r(s) is called instantaneous interest rate. In fact, the former
equation could be easily seen: If we set
0

r(s) = −F1 (s, s),
then we could find the partial difference of t in equation (1) with the observation
that F is smooth:
0
0
F1 (t, s) = F1 (t, u)F (u, s).
Since F is continuous, we let u = t, and so
0

0

F1 (t, t) =
that is

F1 (t, s)
(∀t < s),
F (t, s)

0

0

F (t, s)
F (t, T )
r(t) = − 1
= 1
(∀t < T )
F (t, s)
F (t, T )

and the result follows.
In an uncertain world, this rationale does not hold any more. At time t,
the future interest rates R(u, T ) for T > u > t are not known. Nevertheless,
intuitively, it makes sense to believe that there should be some relationships
between different rates; the aim of modelling is to determine them, and three
such models will be introduced later.

1.2

Zero-coupon bond

When we say zero-coupon bond, we mean a security paying 1 dollar at a
maturity date T and it could be understood as a short-term-treasury which
generates no dividend, meanwhile its maturity don’t need to be less than one
year. And, coupon-bearing bond is naturally defined as a security paying 1
dollar at the maturity date T and at some certain times before T , it pays some
money according to the coupon. Then, how should such derivatives be priced?
Let’s see an example where the interest rate is a const.
e.g. If the interest rate is r, a two-year zero-coupon bond with nominal
value 100 dollars, would worth 100e−2r dollars now. The value of a two-year
coupon-bearing bond whose nominal value is 100 dollars and with yearly coupon
rate 0.1 (In America, the coupon is payed every half year) would worth
3
X
i=1

100 ×

0.1 −0.5i×0.05
e
+ 105e−2×0.05 .
2

So, we see a coupon-bearing bond could be split into some zero-coupon
bonds, and we will only consider the latter one in the following text first.
We note P (t, T ) the value of zero-coupon bond at time t. Obviously we have
P (T, T ) = 1. If the interest rate is now correlated both to the emission date
and maturity date, we would see from the former subsection that
P (t, T ) = e−
2

RT
t

r(s)ds

.

(2)

1.3

Further discussion

However, in an risky world, the instantaneous interest rate r(t) is no longer
only a function of t, but rather a stochastic process. So on the filtered spaced
(Ω, F, P, (Ft )0≤t≤T ), where (Ft ) is the natural filtration of a standard B.M.
(Wt )0≤t≤T and FT = F, we define the instantaneous interest rate (r(t))0≤t≤T
RT
as an adapted process satisfying 0 |r(t)|dt < ∞ a.s.. It’s also called riskless
interest rate. Therefore, the asset, whose price at time t given by
Rt

St0 = e

0

r(s)ds

is called riskless asset. Nevertheless, it’s indeed risky, and just when compared
with the even more risky asset, zero-coupon bond with maturity less or equal
to the horizon T , can it be considered as riskless.
And for each instant u ≤ T , we define an adapted process (P (t, u))0≤t≤u ,
satisfying P (u, u) = 1 giving the price of the zero-coupon bond with maturity u as a function of time.
We’ve mentioned that the zero-coupon bond is very risky. Why? Before
1980’, the interest rate is relatively stable, and the investors tend to purchase
bond on the emission day and encash them on the maturity day, so the risk of
holding bonds only embodies in the credit risk, or whether the bond could be
reimbursed in time. However, after 1980’, accompanying the great changes occurred in the economic environment, the interest rate fluctuates more frequently
and more acutely, and people now would buy or sell the bond at any time when
they think there are great profits. So not only the credit risk, but also, and at
an even higher proportion, the change of interest rate constitute the great risk
for holding a bond.

2
2.1

Modelling principles
Bond price

In Chapter 4, we first assumed the investors’ demand of the percentile income
from the stock is uncorrelated to the price of the stock, and derived the following
stock price equation:
dSt = St (udt + σdWt ),
where u and σ are constants. It is natural to generalize the Black-Scholes model
into the Black-Scholes model with time-dependent parameters:
½
dSt0 = r(t)St0 dt
dSt = St (u(t)dt + σ(t)dWt ).
Furthermore, we could consider the parameters as stochastic processes:
½
dSt0 = r(t)St0 dt
dPt,T = Pt,T (u(t)dt + σ(t)dWt ).
3

If u(t) and σ(t) bear some good forms, then the latter equation would represents
the price of the zero-coupon bond, which is one of the focuses of this chapter.
However, the relationship between the bond price and the two parameters could
no longer be easily sensed as in the Black-Scholes. So, we will first make the
following assumption and then try to derive the equation for bond price.
(H) There is a probability P* equivalent to P, under which, for
all real-valued u ∈ [0, T ], the process (P˜ (t, u))0≤t≤u defined by
P˜ (t, u) = e−

Rt
0

r(s)ds

P (t, u)

is a martingale.
Remark:
In Chapter 1, we have characterized the absence of arbitrage opportunities
by the existence of an equivalent probability under which discounted asset prices
are martingales. We have also seen such a probability exists in the Black-Scholes
model. So we introduce this probability into this continuous-time model.
To be more strictly, we say for continuous processes the existence of an equivalent probability, under which the discounted bond price is a martingale, still
ensures no arbitrage, but the no arbitrage property for general integrands is no
longer sufficient for the existence of an equivalent local martingale measure[10].
However, we may summaries the present state of the art [10] to claim that under most of the conditions the assumption above is tenable, and therefore, it’s
reasonable for us to set it here:
Numerous authors have presented versions of the fundamental theory of asset pricing under various assumptions and degrees of generality, replacing no arbitrage with slightly different notions, such
as no approximate arbitrage, no free lunch with bounded risk, no
free lunch with vanishing risk, no asymptotic arbitrage and so on,
to reestablish the theorm[2]. The case when the time set is finite is
completely settled in Dalang et al. (1989) and the the use of simple
or even elementary integrands is no restriction at all.
For the case of discrete but infinite time sets, the problem is solved
in Schachermayer (1993). The case of continuous and bounded
processes in continuous time, is solved in Delbaen (1992) and so
on.
One should read [3] and [10] for more details.
Now, under this assumption there are many interesting consequences. Indeed, the martingale property under P ∗ leads to, using equality P (u, u) = 1,
P˜ (t, u) = E ∗ (P˜ (u, u)|Ft ) = E ∗ (e−

4

Ru
0

r(s)ds

|Ft )

and, eliminating the discounting,
P (t, u) = E ∗ (e−

Ru
t

r(s)ds

|Ft )

(3)

This equality, which could be compared to formula (2), shows that the prices
P (t, u) only depend on the behavior of the processes (r(s))0≤t≤T under the
probability P ∗ .
We should pay attention to the hypothesis we made on the filtration (Ft )0≤t≤T ,
which allows us to express the density of the probability P ∗ with respect to P .
In fact, if T is infinite, this may not hold any more [16]. We denote by Lt this
density. For any non negative random variable X, we have
Z
Z


E (X) =
XdP (ω) =
XLT dP = E(XLT ),




and, if Xis Ft -measurable, note Lt = E(LT |Ft ), we have
E ∗ (X) = E(XLT ) = E(E(XLT |Ft )) = E(XE(LT |Ft )) = E(XLt ).

(4)

Thus the random variable Lt is the density of P ∗ restricted to Ft with respect
to P .
Lemma1 Let (Mt )0≤t≤T be a continuous martingale, such that for any t ∈
[0, T ], P (Mt > 0) = 1. We have P (∀t ∈ [0, T ], Mt > 0) = 1.
Proof: Let
τ = τ1 ∧ T = (inf{t ∈ [0, T ]|Mt = 0}) ∧ T,
it follows that τ1 is a stopping time and also is τ1 ∧ τ . Then using the optional
sampling theorem, we get
E(MT |Fτ ) = Mτ ,
E[1{τ =T } E(MT |Fτ )] = E[1{τ =T } Mτ ],
and
E[1{τ =T } E(MT |Fτ )] = E(1{τ =T } MT ),
E[1{τ =T } M{τ } ] = E[1{τ =T } M{τ } + 1{τ 6=T } M{τ } ] = E[M{τ } ] = E[M{T } ].
So from E[MT (1 − 1τ =T )] = 0, P (1 − 1τ −T ≥ 0) = 1 and P (MT > 0) = 1, we
get 1 − 1τ =T = 0, a.s., and consequently P (∀t ∈ [0, T ], Mt > 0) = 1.
Proposition6.1.1 There is an adapted process (q(t))0≤t≤T such that, for all
t ∈ [0, T ],
Z
Z t
1 t
q(s)2 ds) a.s.
(5)
Lt = exp(
q(s)dBs −
2 0
0
Proof: The processes (Lt )0≤t≤T is a martingale relative to (Ft ), which is the
natural filtration of the B.M. (Wt ). It follows from the generalized martingale

5

representation theorem that there exists an adapted process ((Ht )), satisfying
RT 2
Ht dt < +∞ a.s. and for all t ∈ [0, T ]
0
Z

t

Lt = L0 +

Hs dWs a.s.
0

For LT is the density of the probability P ∗ with respect to P , we have E(LT ) =
1 = L0 , and P (LT > 0) = 1. So
P (Lt > 0) = P (E(LT |Ft ) > 0) = 1, ∀t ∈ [0, T ].
For (Ft ) is right continuous and complete, Mt has a right continuous version.
Using Lemma1, we know
P (∀t ∈ [0, T ], Lt > 0) = 1.
Therefore, we can apply the Itˆ
o formula to the log function and gain
d(log Lt ) =

Write q(t) =

Ht
Lt ,

1 1
1
1 1 2
1
dLt −
(dLt )2 =
Ht dWt −
H ds a.s.
Lt
2 L2t
Lt
2 L2t t
Z t
Z
Hs
1 t Hs2

log Lt =
a.s.

2 0 L2s
0 Ls

we finally get
Z

t

Lt = exp(
0

1
q(s)dBs −
2

Z

t

q(s)2 ds) a.s.

0

Corollary6.1.2 The price at time t of the zero-coupon bond of maturity u ≥ t
can be expressed as
Z
Z u
Z u
1 u
q(s)2 ds)|Ft ).
(6)
P (t, u) = E(exp(−
r(s)ds +
q(s)dWs −
2 t
t
t
Proof: ∵ ∀A ∈ Ft ,
E ∗ [1A E ∗ (X|Ft )] = E ∗ (1A X) = E(1A XLT )
Lt
E(XLT |Ft )
= E[E(1A XLT |Ft )] = E[1A E(XLT |Ft · ] = E ∗ (1A
),
Lt
Lt
∴ E ∗ (X|Ft ) =

E(XLT |Ft )
.
Lt

So, we see
Z
P (t, u) = E ∗ (exp(−
= E(e−

Ru
t

u

r(s)ds|Ft ))
0

r(s)ds

LT |Ft )/Lt = the right of (6).
6

(7)

Proposition6.1.3 For each maturity u, there is an adapted process (σtu )0≤t≤T
such that, on [0, u]
¢
dP (t, u) ¡
= r(t) − σtu q(t) dt + σtu dWt .
P (t, u)

(8)

Proof: Since the process (P˜ (t, u))0≤t≤u is a martingale under P ∗ , for any
t > s, A ∈ Fs , we have (note P˜t = P˜ (t, u) below)
E(1A P˜t Lt ) = E ∗ (1A P˜t ) = E ∗ [E ∗ (1A P˜t |Fs )]
= E ∗ [1A E ∗ (P˜t |Fs )] = E ∗ (1A P˜s ) = E(1A P˜s Ls ),
which is to say (P˜ (t, u)Lt )0≤t≤u is a martingale under P . With the same prou
cedure
R uas uin 2Proposition 6.1.1, we see there exists an adapted process βt , such
that 0 (βt ) < ∞,
Z t
P˜ Lt = P˜ (0, u)L0 +
βtu dWs ,
0

d(log P˜t Lt ) =
Write θtu =

βtu

,
P˜t Lt

βtu

P˜t Lt

dWs −

1 (βtu )2
.
2 (P˜t Lt )2

it follows
Rt

P˜ (t, u)Lt = P˜ (0, u)e

0

θsu dWs − 21

Rt
0

(θsu )2 ds

.

Hence, using the explicit expression of Lt and getting rid of the discounting
factor
R

t
P (t, u) = P˜ (t, u)e 0 r(s)ds
Z t
Z t
¡ u
¢
= P˜ (0, u) exp(
r(s)ds +
θs − q(s) dBs
0
0
Z t
¡
¢
1

(θsu )2 − q(s)2 ds)
2 0
Z t
Z t
¡ u
¢
= P (0, u) exp(
r(s)ds +
θs − q(s) dBs
0
0
Z
¢
1 t¡ u 2
(θs ) − q(s)2 ds)

2 0

Applying the Itˆ
o formula with the exponential function, we get
dP (t, u)/P (t, u)
¡
¢
¢
¢2


= r(t)dt + θtu − q(t) dWt − (θtu )2 − q(t)2 dt + θtu − q(t) dt

2
¡
¢
¢
= r(t) + q(t)2 − θtu q(t) dt + θtu − q(t) dWt ,
which gives the equality (8) with σtu = θtu − q(t).
7

(9)

Remark6.1.4 When we compare the risk asset (8) with the riskless asset dSt0 =
r(t)St0 dt, it is easy to see it’s the term in dWt which makes the bond riskier.
Furthermore, the term r(t) − σtu q(t) corresponds intuitively to the average yield
of the bond at time t, since the increments of B.M. have zero expectation, and
the term −σtu q(t) is the difference between the average yield of bond and the
riskless rate, hence the interpretation of −q(t) as a ’risk premium’. With the
observation in Proposition 6.1.1, through the Girsanov theorem, we have under
the probability P ∗ ,
Z
t

˜ t = Wt −
W

q(s)ds
0

is a standard B.M. and equation (8) becomes
dP (t, u)
˜ t.
= r(t)dt + σtu dW
P (t, u)
So we say P ∗ is a ’risk neutral’ probability and σtu would be interpreted as
volatility.

2.2

Bond options

To make things clearer, let’s first consider a European option with maturity θ on
the zero-coupon bond with maturity equal to the horizon T . If it is a call with
strike price K, the value of the option at time θ is obviously (P (θ, T ) − K)+
and it seems reasonable to hedge this call with a portfolio riskless asset and
zero-coupon bond with maturity T . A strategy is then defined by an adapted
process ((Ht0 , Ht ))0≤t≤T with values in R2 , Ht0 representing the quantity of
riskless asset and Ht the risky asset held in the portfolio at time t. The value
of the portfolio at time t is then given by
Rt

Vt = Ht0 St0 + Ht P (t, T ) = Ht0 e

0

r(s)ds

+ Ht P (t, T ).

Now as the same way in Chapter 4, we make two definitions:
Definition6.1.5* A self-financing strategy is defined by a pair of adapted
processes (Ht0 )0≤t≤T and (Ht )0≤t≤T , which satisfies
RT
RT
1. 0 |Ht0 |dt + 0 (Ht )2 dt < +∞ a.s.
Rt
Rt
2.Ht0 St0 + Ht St = H00 S00 + H0 S0 + 0 Hu0 dSu0 + 0 Hu dSu a.s. for all t ∈ [0, T ].
Definition6.1.5 A strategy φ = ((Ht0 , Ht ))0≤t≤T is admissible if it is selffinancing and if the discounted value V˜t (φ) = Ht0 + Ht P˜ (t, T ) of the corresponding portfolio is, for all t, non-negative and if supt∈[0,T ] Vet is square-integrable
under P ∗ .
Proposition6.1.6 We assume sup0≤t≤T |r(t)| < ∞ a.s and σtT 6= 0 a.s for
all t ∈ [0, θ], let θ < T and let h be an Fθ -measurable random variable, such
8



that he− 0 r(s)ds is square-integrable under P ∗ . Then there exists an admissible
strategy whose value at time θ is equal to h. The value at time t ≤ θ of such a
strategy is given by
¯ ¢
¡ Rθ
Vt = E ∗ e− t r(s)ds h¯Ft .
Proof:
1. The method is the same as in Chapter 4. We first observe that if there exits
an admissible strategy ((Ht0 , Ht )0≤t≤T which satisfies the conditions set forth
in the proposition, we obtain, using the self-financing condition, the formula of
Vt , the integration by parts formula and Remark 6.1.4,
V˜t

Vt
1
Vt
) = 0 dVt − 0 2 dSt0
St0
St
(St )
Ht0 0 Ht
H 0S0
H 0 P (t, T )
dSt + 0 dP (t, T ) − t0 2t dSt0 − t 0 2 dSt0
0
St
St
(St )
(St )
Ht
Ht P (t, T ) 0
dP (t, T ) −
dSt
St0
(St0 )2
¾
½
P (t, T ) 0
1
T ˜
[P
(t,
T
)r(t)dt
+
P
(t,
T

d
W
]

S
r(t)dt
Ht
t
t
St0
(St0 )2 t
Ht
˜ t = Ht Pe(t, T )σ T dB
et .
P (t, T )σtT dW
t
St0

= d(
=
=
=
=

We deduce, bearing in mind that supt∈[0,T ] V˜t is square-integrable under P ∗ ,
Z
E[
0

T

Z
[Ht P˜ (t, T )σtT ]2 dt] = E[

0

T

Z
˜ t ]2 = E(
Ht P˜ (t, T )σtT dW

T

dV˜t )2 < ∞,

0

which meas (V˜t )is a P ∗ martingale. Furthermore,
V˜t = E ∗ (V˜θ |Ft )
Rt

⇒ Vt = St0 V˜t = e

0

r(s)ds



E ∗ (e

0

r(s)ds



h|Ft ) = E ∗ (e

t

r(s)ds

h|Ft ).

2. In order to find an admissible strategy having the same value at any
time,

r(s)ds
0
we first make the following observation. Since we don’t know whether he
˜ t 0 s, t ≤ θ (we only know it is in the σis in the σ-algebra generated by the W
algebra Fθ which can be bigger), so we could not use the Itˆ
o formula under P ∗
directly. Rather, from the proof in Proposition 6.1.3, we know immediately that
V˜t Lt is P -martingale. So under the probability P and in the σ-algebra Ft , we
apply the Itˆ
o formula for V˜t Lt : there exists an adapted process At , such that
˜
dVt = At dWt . Let X1 = V˜t Lt , X2 = Lt , using the results in Proposition 6.1.1,
we find

µ
X1
˜
dVt = d
X2
µ

dX1
X1
=
− 2 dX2
X2
X2
9

·
¸
1
2X1
1 1
2
2
+
dX
dX

dX
dX
+
0(dX
)
+
(dX
)
1
2
2
1
1
2
2 X22
X22
X23
Ã
!
At
V˜t Lt
=
dWt − 2 q(t)Lt dWt
Lt
Lt
"
#
1
1
1
V˜t Lt
2 2
+ − 2 At q(t)Lt dt − 2 At q(t)Lt dt + 2 3 q(t) Lt dt
2
Lt
Lt
Lt
·
¸
·
¸
At
At
=
− V˜t q(t) dWt −
− V˜t q(t) q(t)dt.
Lt
Lt
Let Jt =
So,

At
Lt

˜ t.
− V˜t q(t). Then dV˜t = Jt dWt − Jt q(t)dt = Jt dW
Z
V˜θ = V˜0 +

θ

˜ s,
Js dW

0

or



he

0

r(s)ds

³ Rθ
´ Z
= E ∗ he 0 r(s)ds +

θ

˜ s.
Js dW

0

3. Now, we could move to construct the strategy. Set
³
´
R
Jt
Jt
0

− 0θ r(s)ds
,
and
H
=
E
he
|F
− T
Ht =
t
t
σt
P˜ (t, T )σtT
for t ≤ θ. We check easily that ((Ht0 , Ht ))0≤t≤θ defines an admissible strategy
whose value at time θ is equal to h.
Remark6.1.7 We have not investigated the uniqueness of the probability P ∗
and it is not clear that the risk process (q(t)) is defined without ambiguity.
Actually, we will prove (due to [3]) below that P ∗ is the unique probability
equivalent to P under which (P˜ (t, T ))0≤t≤T is a martingale if and only if the
process (σtT ) 6= 0, dtdP almost everywhere. This condition, slightly weaker than
the hypothesis of Proposition 6.1.6, is exactly what is needed to hedge options
with bonds of maturity T , which is not surprising when one keeps in mind the
characterization of complete markets we gave in Chapter 1.
Proposition6.1.8 P ∗ is the unique probability equivalent to P under which
(P˜ (t, T ))0≤t≤T is a martingale if and only if the process (σtT ) 6= 0, dtdP almost
everywhere.
Proof: ”⇐” From Proposition 6.1.3, we see when σtT 6= 0, σtT and q(t) are
uniquely determined by the processes (P (t, T ))0≤t≤T and (rt )0≤t≤T . And from
the martingale representation theorem, the uniqueness of q(t) and the uniqueness of P ∗ are equivalent. So from m ⊗ P ((u, ω)|σtT = 0) = 0, it is easy to
conclude the uniqueness of P ∗ .
”⇒” Let S = {(u, ω)|σtT = 0}. Under the assumption m ⊗ P (S) > 0, we
will construct another probability equivalent to P and not equal to P ∗ , which
10

contradict to the original hypothesis. we would do so in three steps.
1. ∵ θtT = σtT + q(t), m ⊗ P (S) > 0, for any α > 0, let
ft = θtT , qˆt = qt + α1s , and fˆt = ft + α1s .
Define

Z t
Z t
Z
Z
1 t 2
1 t ˆ2
fu du) > 2 exp(
fu dBu −
f du) , or
(
fˆu dBu −
2 0
2 0 u
0
0
Z t
Z
Z t
Z
1 t 2
1 t 2
exp (
qˆu dBu −
qˆu du) > 2 exp(
qu dBu −
q du)} ∧ T
2 0
2 0 u
0
0

τ = inf{t| exp

and
q˜t = qˆ f or t ≤ τ,
f˜t = fˆ f or t ≤ τ,

q˜t = q f or t > τ,
f˜t = f f or t > τ.

Then for each t we have
Z
Z t
Z
Z t
1 t 2
1 t ˜2
fu du) ≤ 2 exp(
fu dBu −
f du)
exp(
f˜u dBu −
2 0
2 0 u
0
0
as well as the same inequality with f˜ and f respectively replaced by q˜ and q.
2. From fu − qu = f˜u − q˜u and equation (9),
P (t, T )

Z

t

Z

1
(ru − (σut )2 − qu fu + qu2 )du]
2
0
Z t
Z t
Z T
1
2
= P (0, T ) exp[ (fu − qu )dBu −
(fu − qu ) du +
(ru − qu fu + qu2 )du]
2 0
0
0
Z t
Z
Z t
1 t ˜
= P (0, T ) exp[ (f˜u − q˜u )dBu −
(fu − q˜u )2 du +
(ru − q˜u f˜u + q˜u2 )du].
2 0
0
0
σuT dBu

+

By the estimations in step 1,
Z
E[exp(

T

= P (0, T ) exp[

Z

0

q˜u dBu −

Z
0

T

q˜u2 du)] = 1,

Z
1 T 2
exp(−
ru du) exp[
q˜u dBu −
q˜ du]
2 0 u
0
0
Z T
Z
1 T ˜2
= P (0, T ) exp(
f˜u dBu −
fu du) ∈ L2 .
2
0
0
Rt 2
Rt
1
˜ t = exp( q˜u dBu −
˜u du), we have
3. Let L
2 0 q
0
Z t
Z t
Z
1 t ˜2
˜
˜
fu dBu −
ru du)Lt = P (0, T ) exp(
P (t, T ) exp(−
f du).
2 0 u
0
0
Rt
R
˜ t is a P-martingale, L
˜ t exp(− T ru du) ∈ L2 ,
Hence P (t, T ) = exp(− 0 ru du)L
0
˜ T ] = 1 and (L
˜ t )t≤T is a P-martingale. Define P ∗∗ by dP ∗∗ = L
˜ T , then its
E[L
dP
existence is a contradiction with the uniqueness assumption.
T

Z

1
2

T

11

3

Term structure of interest rate

From above, we see in order to price the zero-coupon bond and the corresponding
option, we need to know the law of (r(t)), and this lead to some conceptions
correlated to term structure of interest rate.

3.1

Definitions for term structure

When we say yield, we mean the the ratio of actual earnings to the actual
invest. And we have mentioned before that now the risk of holding zero-coupon
bond is partly caused by the manner of people buying and selling bonds at any
time they think profitable. So there exists great correlation between the yield
and the change of the interest rate and this justifies the necessary of interest
rate modelling.
If we plot the yield of different bonds with diverse maturity, but the same risk
and mobility, this curve is called yield curve; it is the set of {Ytt+s }0≤s≤T −t ,
and it is also called Term structure of interest rate.
The price of zero-coupon bond {P (t, t + s)}0≤s≤T −t is called Term structure of zero-coupon bond price.
Spot interest rate is defined as the n-year interest rate which is calculated
from today, while forward interest rate f (t, s), t ≤ s represents the anticipated interest rate at time t for time s. {f (t, s)}t≤s≤T is named as the time t
term structure of forward interest rates.
We have mentioned three term structures here, and from the equations
Y (t, s) = −

log P (t, s)
,
s−t

f (t, s) = −

∂ log P (t, s)
,
∂s

the equivalence of them is obvious. And it is based on this reason that we just
try to find the price of the zero-coupon bond above, and would only investigate
the law of interest rate below.

3.2

Three traditional theories[21]

Before 1980’, there are three main theories explaining the term structure of
interest rate.
I. The expectation theory is first proposed in 1896, and developed in
1949. It is one of the oldest such theories and it is based on the following five
assumptions: 1. All the investors are aiming for maximizing profit; 2. All the
investors think all the bonds with different maturities can be totally superseded
by some others; 3. There is no charge for holding and transaction; 4. No
arbitrage exists; 5. Most of the investors can make accurate expectations and
would behave according to them. Thus the n-year spot interest rate would equal
to the average of the 1-year spot interest rate and the expected 1-year forward
interest rate in the future n − 1 years, that is
i(n)t =

it + iet+1 + iet+2 + . . . + iet+(n−1)
n
12

.

II. The liquidity preference theory, raised by J.R. hicks in 1993 and
further suggested by J.M. Cullbertson in 1957, is a revision to the expectation
theory. In this theory, they think that some factors, such as the longer the
maturity of the bond is, the risk of the volatility of the value of invested money
is, should be considered too. So, the borrower should provide the investor some
risk premium to insure that the latter would like to buy long-term bonds. So,
let k(n)t be the n-year bond’s mobility compensation, we have
i(n)t =

it + it+1 e + iet+2 + . . . + iet+(n−1)

+ k(n)t .
n
III. The market segmentation theory, however, doubt the 2nd assumption in the expectation theory. It takes bonds with different maturity as
completely independent markets and think one bond’s interest should be totally
decided by its own demand and provision.
In practice, according to the explanation ability of the three theories to some
empirical fact, the liquidity preference theory is more popular.

4

Some classical models

In the last section, three traditional models are introduced, and in the following
another three modern ones would be discussed. The Vasicek model is highly
welcomed by the practitioners for its simplicity, while the CIR model would
always generates non-negative interest rates. And the HJM model provide formulae that depend on the parameters of the dynamics of interest rates under
P.

4.1

The Vasicek model[22]

In this model, we assume that the process r(t) satisfies
dr(t) = a(b − r(t))dt + σdWt

(10)

where a, b, σ are non-negative constants. We also assume that the process q(t)
is a constant q(t) = −λ, with λ ∈ R. Then
˜t
dr(t) = a(b∗ − r(t))dt + σdW


where b = b −
4.1.1

λσ
a

(11)

˜ t = Wt + λt.
and W

Feature of the solution

Let Xt = r(t) − b. Then Xt is a solution of the stochastic differential equation
dXt = −aXt dt + σdWt

(12)

which means that (Xt ) is an O-U process. So from the result of Chapter 3, we
know
Z t
−at
−at
−at
r(t) = r(0)e
eas dWs
+ b(1 − e ) + σe
(13)
0

13

and that r(t) follows a normal law whose mean is given by E(r(t)) = r(0)e−at +
Rt
−2at
b()1 − e−at and variance by V ar(r(t)) = σ 2 e−2at E( 0 e2as ds) = σ 2 1−e2a .
2
When t tends to infinity, r(t) converges to N (b, σ2a ). Although this model is very
simple, it is still not satisfactory from a practical point of view since P (r(t) <
0) > 0.
4.1.2

Preliminary calculation of P (t, T )

Based on the equations (3) and (11), under P ∗ we have
P (t, T ) = E ∗ (e−

RT
t

r(s)ds

|Ft ) = E ∗ (e−

RT
t

(Xs∗ +b∗ )ds

|Ft )

(14)

where Xt∗ is the solution to the equation
˜ t.
dXt = −aXt dt + σdW

(15)

Again using the conclusions in Chapter 3, Remark 3.5.11, we gain
P (t, T ) = e−b



(T −t)

E ∗ (e−

R T −t
0

Xsy ds

)|y=Xt∗

(16)

where Xtx is the unique solution of equation 15 which satisfies X0x = x.
4.1.3

Calculate F (θ, x)


x

In order to calculate P (t, T ) completely, now we try to find F (θ, x) = E ∗ (e− 0 Xs ds ).

We know that Xt is a Gaussian process with continuous paths, so 0 Xsx ds is
a normal random variable, since the integral is the limit of Riemann sum of
Gaussian components. Thus, from the expression of the Laplace transform of a
Gaussian 1
Z
Z θ
¡ θ x ¢¢
¡ R
¢
¡
1
∗ − 0θ Xsx ds

x
Xs ds
E e
= exp − E (
Xs ds) + V ar
2
0
0
¡ ¢
From equalityE ∗ Xsx = xe−as and the Fubini theorem, we deduce
Z
E (


0

θ

Z
Xsx ds)

Z

θ

=

E



0

(Xsx )ds

θ

=

xe−as ds = x

0

1 − e−aθ
.
a

Again, using the Fubini theorem, we have
Z
V ar(
Z
= E [


0
1 If

θ

Xsx ds)

0
θZ θ
0

Z
=E (


Z

θ

Xsx ds

0

−E



θ

0

Xsx ds)2

(Xux − E ∗ Xux )(Xtx − E ∗ Xtx )dtdu]

Y ∼ N (µ, σ 2 ), then E ∗ (e−Y ) =

R

e−y √ 1

2πσ

14

− y−µ
2

e



dy = e

σ2
2

−µ

.

Z

θ

Z

θ

=
0

Z

0

θ

Z

θ

=
0

Z

0

θ

Z

Z

θ

0

θ

Z

0

=
=
4.1.4

Z

u

Z
˜ s )(σe−at
eas dW

Z
σ 2 e−a(u+t) E ∗ (

t

˜ β)
eaβ dW

0
u∧t

˜ s )2 dtdu
eas dW

0

θ

=
σ2
=
2a
Z
= 2

Z
E ∗ (σe−au
0

=
0

E ∗ [(Xux − E ∗ Xux )(Xtx − E ∗ Xtx )]dtdu

0
θ

Z
σ 2 e−a(u+t) E ∗ (
Z

u∧t

e2as ds)dtdu

0
θ

e−a(t+u) (e2a(t∧u) − 1)dtdu

0
0
θZ θ 2

σ −au at
e
(e − e−at )dudt
0
t 2a
σ2
3
2
1
(θ −
+ e−aθ e−2aθ )
2
a
2a a
2a
σ2 θ σ2
σ2
−aθ
− 3 (1 − e
) − 3 (1 − e−aθ )2
2
a
a
2a

bond pricing

Let θ = T − t, R∞ = b∗ − σ 2 /(2a2 )and
R(θ, r) = R∞ −

σ2
1
[(R∞ − r)(1 − e−aθ ) − 2 (1 − e−aθ )2 ].

4a

Then R∞ = limθ→∞ R(θ, r). Combining the above result and equation (16), we
know
P (t, T )
©
ª
1 − e−aθ
σ2 θ
σ2
σ2
= exp − b∗ θ − x
+ 2 − 3 (1 − e−aθ ) − 2 (1 − e−aθ )2
a
2a
2a
4a
2
2
©
ª
σ
1
σ
σ2
= exp − θ(b∗ − 2 ) + [(b∗ − 2 − r)(1 − e−aθ ) − 2 (1 − e−aθ )2 ]
2a
a
2a
4a
= e−θ·R(θ,r(t))
The yield R∞ can be interpreted as a long-term rate; note that it does not
depend on the ’instantaneous spot rate’ r. This last property is considered as
an imperfection of the model by practitioners.
4.1.5

option pricing

Now we have the formula for the price of bond, so based on the Proposition
6.1.6, we can calculate the price of the option. It is easy to verify the hypothesis
of Proposition 6.1.6 holds; and if the option is exercised, or P (θ, T ) > K, then
we have
r(θ) < r∗
15

where
r∗ = R∞ (1 −

σ2
a(T − θ)
a log K
)

.
(1 − e−a(T −θ) ) −
4a2
1 − e−a(T −θ)
1 − e−a(T −θ)

C0 = E ∗ [e−







= E (e




= E (e

= E ∗ (e−


0


0

RT
0



r(s)ds

0

r(s)ds
r(s)ds

(P (θ, T ) − K)+ ]



P (θ, T )1r(θ)<r∗ ) − KE ∗ (e




E (e

r(s)ds

RT
θ

r(s)ds

0

r(s)ds

1r(θ)<r∗ )


|Fθ )1r(θ)<r∗ ) − KE ∗ (e


1r(θ)<r∗ ) − KE ∗ (e

0

r(s)ds

0

r(s)ds

1r(θ)<r∗ )

1r(θ)<r∗ )

Let


RT

dP2
e− 0 r(s)ds
dP1
e−XT
e− 0 r(s)ds
e−Xθ
, and
.
= ∗ −XT =
= ∗ −X =


dP
E (e
)
P (0, T )
dP
E (e θ )
P (0, θ)
From the the mean of XT (or Xθ ), and r(θ) and their correlation coefficient,
which could be calculated as above, and the Laplace transform of r(θ) under
P1 (or P2 ), 2 we have
= P (0, T )P1 (r(θ) < r∗ ) − KP (0, θ)P2 (r(θ < r∗ ))
= P (0, T )N (h) − KP (0, θ)N (h − σp )
q
σp
P (0,T )
σ
1−e−2aθ
−a(T −θ)
log KP
+
,
σ
=
[1

e
]
.
p
(0,θ)
2
a
2a

C0

where h =
4.1.6

1
σp

Estimation of parameters

In practice, parameters must be estimated and a value for r must be chosen. For
r we will choose a short rate (for example, the overnight rate); then we will fit the
parameters b, a, σ by statistical methods, such as MLE, robust statistics, etc., to
the historical data of the instantaneous rate. Finally λ will be determined from
market data by inverting the Vasicek formula [19]. What practitioners really do
is to determine the parameters, including r, by fitting the Vasicek formula on
market data. The parameters of the models shown below could be estimated
in the same way. Reader are referred to [1], [6] and [5] for further information
concerning the test of the parameters.

4.2

The Cox-Ingersoll-Ross model

In 1981, Cox, Ingersoll and Ross forwarded the idea of analyzing term structure
of interest rate using general equilibrium method [7], and in 1985, they proposed
2 Let

(X, Y )be a Gaussian vector with values in R2 under probability P , with correlation

coefficient r and mean and variance ai , σi2 , i = 1, 2. If
exp[

2
σ2

2

˜
dP
dP

=

e−λX
,
E(e−λX )

b2 + (2λrσ1 σ2 − a2 )b], that is to say Y ∼ N ∼ (a2 − 2λrσ1 σ2 , σ22 ).

16

then E ∼ (e−bY ) =

in another two papers [8] [9] the model which is later named as CIR model. In
this model the behavior of the instantaneous rate follows the following equation3 :
p
(17)
dr(t) = (a − br(t))dt + σ r(t)dWt
with
pσ and a non-negative, b ∈ R, and the process (q(t)) being equal to q(t) =
−α r(t), with α ∈ R.
4.2.1

Properties of the solution

Theorem 6.2.3 We suppose that (Wt ) is a standard B.M. defined on [0, ∞[.
For any real number x ≥ 0, there is a unique continuous, adapted process (Xt ),
taking values in R+ , satisfying X0 = x and
p
dXt = (a − bXt )dt + σ Xt dWt on[0, ∞[.
(18)
Proof: From the H¨older property of the square root function, and the theorems
given in [14], it is easy to see the existence and uniqueness of the function
p
(19)
dXt = (a − bXt )dt + σ 2 Xt ∨ 0dWt
with X0 = x ≥ 0. So the only work remains here is to proof the non-negative
property of the solution.

1. If a = 0, equation (18) becomes dXt = −bXt dt + σ 2 Xt ∨ 0dWt . When
X0 = 0, Xt ≡ 0 is obviously a solution and based on the uniqueness of solution,
˜ t = Xt+τ is
we have Xt ≡ 0 a.s.. When X0 ≥ 0, set τ = inf{t; Xt = 0}, then X
˜ 0 = 0 on the space (Ω
˜ = {ω; τ (ω < ∞)}, F˜ = F|
˜ ˜ , P˜ =
a solution of (19) with X

˜
˜
P (·|Ω)). So, X ≡ 0, a.s. on Ω. And we get Xt ≡ Xt∧τ a.s. This implies Xt ≥ 0
a.s.
2. If a > 0, then for any ε > 0, such that bε + a > 0, let τε = inf{t; Xt = −ε}.
Assume that P (τ−ε < ∞) > 0. Then, with probability one, if we take any
r < σ−ε such that X(t) < 0 if t ∈ (r, τ−ε ), we have dXt = (a − bXt )dt on this
interval and hence t 7→ Xt is increasing on this interval, which is impossible
considering X0 ≥ 0 and the definition of τ−ε . This completes the proof.
The theorem tells us that in CIR model, the interest rate would never becomes negative. However, would it be zero at some time point and if so, what
is the probability? The following proposition gives the answer.
Proposition6.2.4
1. If a ≥ σ 2 /2, we have P (τ0x = ∞) = 1, for all x > 0.
2. If 0 ≥ a < σ 2 /2 and b ≥ 0, we have P (τ0x < ∞) = 1, for all x > 0.
3. If 0 ≥ a < σ 2 /2 and b < 0, we have P (τ0x < ∞) ∈]0, 1[, for all x > 0.
3 This is a Continuous state branching process with immigration (CBI), and readers are
referred to [9] to see why such equation is suggested.

17

Proof:
x
1. For x, M > 0, let τM
= inf{t ≥ 0|Xtx = M }, and
Z x
2
2
s(x) =
e2by/σ y −2a/σ dy, x ∈ (0, ∞).
1
x
Then τM
is a stopping time, and

d2 s
dx2

2

= e2bx/σ x−2a/σ

2

2
−1
),
σ 2 (b − ax

so we have

σ 2 d2 s
ds
x 2 + (a − bx)
= 0.
2 dx
dx
x
2. For 0 < ε < x < M , set τ = τε,M
= τεx ∧ τ M x , then for any t > 0,

ds(Xt∧τ ) = σ

2bXt∧τ
σ2

or

2a

(Xt∧τ )− σ2 σ

Z
x
s(Xt∧τ
) = s(x) +

t∧τ

0

p
Xt∧τ dWt ,

s0 (Xsx )σ

p

Xsx dWs .

So,
x
E[s(Xt∧τ
) − s(x)]2
Z
Z
x
2
x
=
[s(Xt∧τ ) − s(x)] dP (w) +
[s(Xt∧τ
) − s(x)]2 dP (w)
t≤τ
τ <t
Z

[s(M ) − s(x)]2 + [s(ε) − s(x)]2 dP (w)
t≤τ
Z
+
[s(M ) − s(x)]2 + [s(ε) − s(x)]2 dP (w)
τ <t

= 2[s(M ) − s(x)]2 + 2[s(ε) − s(x)]2 < ∞,
and


Z
> E[

t∧τ

0

s0 (Xsx )σ

p

Z t
Xsx dWs ]2 = E[
1s≤τ s0 (Xsx )2 σ 2 Xsx ds]
0

Z t
> AE[
1s≤τ ds] → AE(τ ), when t → ∞.
0

Therefore, τ < ∞ a.s.
p
R t∧τ
0
3. Since s (x) is upper bounded and continuous, 0 s0 (Xsx )σ Xsx dWs is a
martingale. With τ < ∞ a.s., so,
Z t∧τ
p
s(x) = E(s(x)) = E[s(Xt∧τ −
s0 (Xsx )σ Xsx dWs )] = E[s(Xt∧τ )]
0
Z
Z
=
s(Xt∧τ )dP (w) +
s(Xt∧τ )dP (w)
x
τεx <τM

=

s(ε)P (τεx

<

x
τM
)

+

x <τ x
τM
ε
x
s(M )P (τM
<

18

τεx ).

4. When a ≥

a2
2 ,

it is obvious that
Z 1
2
2
s(ε) = −
e2by/σ y −2a/σ dy → −∞, when ε → 0.
ε

For fixed 0 < x < M , let ε ↓ 0 in the last equation in step 3, we see P(τ0x <
x
τM
) = 0, for all M > 0. Since for ∀w, ∀K, ∃BK , such that τBK > K, we have
τBK → ∞, τ0 > τBK a.s. That is P (τ0x = ∞) = 1.
2
5.When 0 ≤ a < σ2 , s(0) = limx→0 s(x) exists. So we have
x
x
s(x) = s(0)P(τ0x < τM
) + s(M )P(τ0x > τM
).

If b ≥ 0, we have limx→∞ s(x) = ∞, so P(τ0 < ∞) = 1.
If b < 0, we have limx→∞ s(x) exists, so P(τ0 < ∞) ∈ (0, 1).
4.2.2

Pricing

The following
proposition, which enables us to characterize the joint law of
Rt
(Xtx , 0 Xsx ds), is the key to any pricing within the CIR model.
Proposition6.2.5 For any non-negative λ and µ, we have
Rt x
¢
¡
x
E(e−λXt e−µ 0 Xs ds ) = exp(−aφλ,µ (t) exp − xψλ,µ (t))
where the function φλ,µ and ψλ,µ are given by
φλ,µ = −

¡
¢
2
2γet(γ+b)/2
log 2 γt
2
γt
σ
σ λ(e − 1) + γ − b + e (γ + b)

and
ψλ,µ =

λ(γ + b + eγt (γ − b)) + 2µ(eγt − 1)
σ 2 λ(eγt − 1) + γ − b + eγt (γ + b)

p
with γ = b2 + 2σ 2 µ .
Proof:
1. First, from L´evy theorem, equivalence of the martingale problem and the
weak solution of the corresponding equation, and the good property of the
Bessel diffusion4 [14], we can assert that this expectation can be written as
exp{−aφ(t) − xψ(t)}.
4 Let



i = 1, 2, Y3 = Y1 + Y2 , and
Z ts
B3 (t) =
0

Then W3 is a B.M. and


dYi (t) = 2(Yi (t) ∨ 0)1/2 dWi (t) + αi dt
,
Yi (0) = yi ∈ [0, ∞)
Y1 (s)
dW1 (s) +
Y1 (s) + Y2 (s)

Z

t
0

s
Y2 (s)
dW2 (s).
Y1 (s) + Y2 (s)

dY3 (t) = 2(Y3 (t) ∨ 0)1/2 dW3 (t) + (α1 + α2 )dt
.
Y3 (0) = y1 + y3

19

2. Define
x

F (t, x) = E(eλXt e−u

Rt
0

Xsx ds

).

(20)

It is natural to look for F as a solution of the problem
½
∂F
σ2 ∂ 2 F
∂F
∂t = 2 x ∂x2 + (a − bx) ∂x − uxF
−λx
F (0, x) = e
.

(21)

Indeed, let F be one of the solutions for (21), and suppose it has bounded
derivatives, which could be justified by the finally result. Set
Mt = e−u

Rt
0

Xsx ds

F (T − t, Xtx ),

then from
dMt

= −uXtx e−u

Rt
0

Xsx ds

F (T − t, Xtx )dt − e−u

Rt
0

Xsx ds ∂F (T

− t, Xtx )
dt
∂t

p
− t, Xtx )
dt[(a − bXtx )dt + σ Xtx dWt ]
∂x
2 x
2
x
R
−u 0t Xsx ds ∂ F (T − t, Xt ) σ Xt
dt
+e
2
∂x
2
Rt x
∂F
∂F
σ2 x ∂ 2 F
= e−u 0 Xs ds [−uxF −
+ (a − bx)
+
]|x=Xtx dt
∂t
∂x
2 ∂x2
R t x ∂F p
+e−u 0 Xs ds
σ Xtx dWt
∂x
R t x ∂F p
σ Xtx dWt ,
= e−u 0 Xs ds
∂x
+e−u

Rt
0

Xsx ds ∂F (T

we know Mt is a martingale, and therefore E(Mt ) = M0 , or
M0 = F (T, X0x ) = E[e−u

RT
0

Xsx ds

x

F (0, XTx )] = E(e−λXT e−u

Rt
0

Xsx ds

).

So F in (20) is really a solution to (21).
3. From step 1 and 2, we have
(
2
0
−ψ (t) = σ2 ψ 2 (t) + bψ(t) − u
0
φ (t) = ψ(t)
with√φ(0) = 0, ψ(0) = λ. The first equation above is a Raccati equation, and
−b+ b2 +2uσ 2
is one of its special solutions. Following the standard method, we
σ2
can find the expressions for ψ and φ, which are the same as desired and thus
the proof is finished.
Now with above proposition, let u = 1 λ = 0, we have bond price
¡ RT
¢
P (0, T ) = E ∗ e 0 r(s)ds = e−aφ(T )−r(0)ψ(T )
20

(22)

where the function φ and ψ are given by the following formula


φ(t) = −



¡
¢
2γ ∗ et(γ +b )/2
2
log ∗
∗t
2

γ


σ
γ − b + e (γ + b )

and



ψ(t) =

2(eγ t − 1)

γ − b∗ + eγ ∗ t (γ ∗ + b∗ )

p
(b∗ )2 + 2σ 2 . The price at time t is given by
¡
¢
P (t, T ) = exp − aφ(T − t) − r(t)ψ(T − t) .

with b∗ = b + σα and γ ∗ =

Again, if we apply it with u = 0, we obtain the Laplace transform of Xtx
!2a/σ2
Ã
!
λbe−bt
b
exp −x σ2
) =
σ2
−bt ) + b
−bt ) + b
2 λ(1 − e
2 λ(1 − e
µ
¶2a/σ2
µ

1
λLζ
=
exp −
2λL + 1
2λζ + 1
Ã

−λXtx

E(e

2

with L = σ4b (1 − e−bt ) and ζ = 4xb/(σ 2 (ebt−1 )). With these notations, the
Laplace transform of Xtx /L is given by the function g4a/σ2 ,ζ , where ζ is defined
by
1
λζ
gδ,ζ (λ) =
exp(−
).
2λ + 1
(2λ + 1)δ/2
By statistics knowledge, we see this is the Laplace transform of the non-central
chi-square law 5 with δ degrees of freedom and parameter ζ.
So, if we let
dP1
e−
=
dP ∗


0



r(s)ds

P (θ, T )
dP2
e− 0 r(s)ds
, and
=
,
P (0, T )
dP ∗
P (0, θ)

by the Laplace transform of

r(θ)
L1 (

or

r(θ)
L2 )

under P1 ( or P2 ), where


L1 =

σ2
eγ θ − 1
∗θ

γ
2
2 γ (e
+ 1) + (σ ψ(T − θ) + b∗ )(eγ ∗ θ − 1)

and



L2 =
5 The

σ2
eγ θ − 1
,
∗θ

γ
2 γ (e
+ 1) + b∗ (eγ ∗ θ − 1)

density of this law is given by fδ,ζ =


e−zeta/2 −x/2 δ/4−1/2
e
x
Iδ/2−1 ( xζ),
2ζ δ/4−1/2

f or x >

0, where Iv is the first-order modified Bessel function with index v, defined by Iv (x) =
P
(x/2)2n
( x2 )v ∞
n=0 n!Γ(v+n+1) .

21

is a non-central chi-squared law with
to ζ1 ( or ζ2 ), with

4a
σ2

degrees of freedom and parameter equal


ζ1 =

8r(0)γ ∗2 eγ θ
∗θ
∗θ
2
γ

γ
σ (e
− 1)[γ (e
+ 1) + (σ 2 ψ(T − θ) + b∗ )(eγ ∗ θ − 1)]

and



ζ2 =

γ∗θ

σ 2 (e

8r(0)γ ∗2 eγ θ
.
− 1)[γ ∗ (eγ ∗ θ + 1) + b∗ (eγ ∗ θ − 1)

Now we could price a European call option with maturity θ and exercise
price K, on a zero-coupon bond with maturity T . We can show the hypothesis
of Proposition 6.16. holds; if the call is exercised,
P (θ, T ) > K ⇒ r(θ) < r∗ = −

aφ(T − θ) + log K
,
ψ(T − θ)

so the call price at time 0 is given by
C0

= E ∗ [e−




= E [e


0


0



r(s)ds

(P (θ, T ) − K)+ ]

r(s)ds

(e−aφ(T −θ)−r(θ)ψ(T −θ) − K)+ ]



= E ∗ (e− 0 r(s)ds P (θ, T )1r(θ<r∗ ) ) − KE ∗ (e− 0 r(s)ds 1r(θ<r∗ )
= P (0, T )P1 (r(θ) < r∗ ) − KP (0, θ)P2 (r(θ) < r∗ )
r∗
r∗
= P (0, T )F4a/σ2 ,ζ1 ( ) − KP (0, θ)F4a/σ2 ,ζ2 ( ).
L1
L2
4.2.3

Affine term structure

The bond price of two models above could all be written as
P (t, s) = exp[A(t, s) − B(t, s)r(t)],
where in Vasicek,
B(t, s) =

[B(t, s) − st][a2 b − σ 2 /2] σ 2 B(t, s)2
1 − exp[−a(s − t)]
, A(t, s) =

,
a
a2
4a

and in CIR,
B(t, s) = ψ(s − t), and A(t, s) = −aφ(s − t).
So the yield curve
Y (t, s) =

log P (t, s)
1
=
[B(t, s)r(t) − A(t, s)]
t−s
s−t

is a linear function of r(t), and therefore they are thought to be affine. This
is their main drawback, unable taking the whole yield curve observed on the
market into account in the price structure. 6
6 To

know more about the general affine term structure, one may read [11] and [18].

22

Furthermore, under those two special models, we are able to find out the explicit expression for the volatility of the bond price process as shown in equation
(8). In Vasicek, q(t) = −λ, b∗ = b − λσ
a ,
dP (t, s)
=
P (t, s)

1
[A(t, s)t + σ 2 B(t, s)2 − abB(t, s)]
2
+ [aB(t, s) − B(t, s)t ]r(t)}dt − σB(t, s)t dWt ,
{

where A(t, s)t = dA(t, s)/dt, and it can be checked that
aB(t, s) − B(t, s)t = 1 − e−a(s−t) + e−a(s−t) = 1,
1
λσ −a(s−t)−1
A(t, s)t − σ 2 B(t, s)2 − abB(t, s) =
(e
) = σB(t, s)q(t).
2
a
So, p
σts = −σB(t, s), and θts p= −σB(t, s) − λ. And in CIR, since q(t) =
−α r(t), b∗ = b + σα, γ ∗ = (b∗ )2 + 2σ 2 ,
dP (t, s)
/P (t, s) =
dt

1
[bB(t, s) − B(t, s)t + B(t, s)2 σ 2 ]r(t)
2
p
+ [A(t, s)t − aB(t, s)]}dt − σB(t, s) r(t)dWt .
{

So we just have to do some simple calculation to verify
1
bB(t, s) − B(t, s)t + B(t, s)2 σ 2 = 1 − ασB(t, s),
2
A(s, t)t − aB(s, t) = 0,
p
and it is clear that σts = −σB(t, s) r(t).7

4.3

Heath-Jarrow-Morton model

Some authors have resorted to a two-dimensional analysis to improve the models
in terms of discrepancies between short and long rates, cf [4]. These more
complex models do not lead to explicit formulae and require the solution of
partial differential equations. More recently, Ho and Lee [13] have proposed
a discrete-time model describing the behavior of the whole yield curve, which
is said to be the inception of local equilibrium method. It avoids the twostep procedure: inversion of the term structure and value of the contingent
claims, which is necessary in Vasicek, etc. The continuous-time model we present
now is based on the same idea and has been introduced by Heath,Jarrow and
Morton [12]. Unlike the Ho and Lee model, however, they impose the exogenous
stochastic structure upon forward rates, and not the zero coupon bond prices.
They also think this change in perspective facilitates the mathematical analysis
and it should also facilitate the empirical estimation of the model.
7 Certainly, it can be seen that in CIR, the final expression of the volatility is not as simple
as in Vasicek for direct use.

23

4.3.1

HJM model and elementary properties

Let

Z
P (t, u) = exp(−

u

f (t, s)ds)
t

where (f (t, s)), t ≤ s, is an adapted process already defined as the forward
interest rates. Moreover, it is natural to set f (t, t) = r(t) and constrain the
map (t, s) 7→ f (t, s), defined for t ≤ s, to be continuous. Then the next step
of the modelling consists in assuming that, for each maturity u, the process
(f (t, u))0≤t≤u satisfies an equation of the following form:
Z t
Z t
f (t, u) = f (0, u) +
α(v, u)dv +
σ(f (v, u))dWv
(23)
0

0

the process (α(t, u))0≤t≤u being adapted, the map (t, u) 7→ α(t, u) being continuous and σ being a continuous map from R into R (σ could depend on time
as well).
Then we have to make sure that this model is compatible with the hypothesis
(H). This gives some conditions on the coefficients α and σ of the model. To find
them, we derive the
R udifferential dP (t, u)/P (t, u) and we compare it to equation
(8). Set Xt = − t f (t, s)ds, we have P (t, u) = eXt , from equation (23), we
have
Z u
¡
¢
Xt =
− f (s, s) + f (s, s) − f (t, s) ds
t
Z u
Z u Z s
Z u Z s
¡
¢
¡
¢
= −
f (s, s)ds +
α(v, s)dv ds +
σ(f (v, s))dWv ds
Zt u
Zt u Zt u
Zt u Zt u
¡
¢
¡
¢
= −
f (s, s)ds +
α(v, s)ds dv +
σ(f (v, s))ds dWv
t
t
v
t
v
Z t
Z t Z u
Z t Z u
¡
¢
¡
¢
= X0 +
f (s, s)ds −
α(v, s)ds dv −
σ(f (v, s))ds dWv
0

0

v

0

v

The integrals in the above equation could commute would be justified by noting
!
Z u ÃNX
−1
σ(f (vi , s))1{s≤u} (Wti +1 − Wti ) ds
t

=

i=0

N
−1 µZ u
X
i=0

t


σ(f (vi , s)ds) 1{v≥t} (Wti +1 − Wti )

and both sides converge under L2 to the desired result. We then have
Z
Z u
¢
¡
¢
¡ u
σ(f (t, s))ds dWt
dXt = f (t, t) −
α(t, s)ds dt −
t

t

and by the Itˆ
o formula
1
dP (t, u)/P (t, u) = dXt + d < X, X >t
2
24

=

¡

Z
f (t, t) −

u

α(t, s)ds +
t


2

Z

u

σ(f (t, s))ds

¢2 ¢

dt −

¡

Z

t

u

¢
σ(f (t, s))ds dWt

t

If the hypothesis (H) holds, we must have, from proposition 6.1.3 and equality
f (t, t) = r(t),
Z
σtu q(t)
with σtu = −
Z
t

Ru

u

t

u

=
t

1
α(t, s)ds −
2

µZ

¶2

u

σ(f (t, s))ds
t

σ(f (t, s))ds. So

1
α(t, s)ds =
2

µZ

¶2

u

σ(f (t, s))ds
t

Z
− q(t)

u

σ(f (t, s))ds
t

and, differentiating with respect to u,
µZ
α(t, u) = σ(f (t, u))

u


σ(f (t, s))ds − q(t) .

t

Then equation (23) becomes
µZ
df (t, u) = σ(f (t, u))

u


˜ t.
σ(f (t, s))ds dt + σ(f (t, u))dW

(24)

t

Heath, Jarrow and Morton have also proved the following fine theorem [12].
Theorem 6.2.6 If the function σ is Lipchitz and bounded,8 for any continuous
function φ from [0, T ] to R+ , there exists a unique continuous process with
two indices (f (t, u))0≤t≤u≤T , such that for all u, the process (f (t, u))0≤t≤u is
adapted and satisfies (24), with f (0, u) = φ(u).
We see that, for any continuous process (q(t)), it is then possible to build a
model of the form (23), that a solution of (24) and set
µZ u

α(t, u) = σ(f (t, u))
σ(f (t, s))ds − q(t) .
t

The striking feature of this model is that the law of forward rates under P ∗ only
depends on the function σ, shown in equation (24). It follows that the price of
the options only depends on the function σ.
4.3.2

Option pricing

We will price the option here only when σ is a constant.
Firstly, it is easy to see the hypothesis of Proposition 6.1.6 holds, and
˜t
f (t, u) = f (0, u) + σ 2 t(u − t/2) + σ W
8 This

boundedness condition on σ is essential since, for σ(x) = x, there is no solution [20].

25

is a solution to equation (24), since under this f , the left and right hand sides
are respectively
Z u
˜ t , and σ(
˜ t = σ2(u − t)dt + σdW
˜ t.
df (t, u) = σ 2 (u − t)dt + σdW
σds) + σdW
t

So, from the definition of P ,
µ

P (0, T )
σ2 θ
˜
exp −σ(T − θ)Wθ −
(T − θ)2
P (0, θ)
T
( Z
)
T
2
2
2
2
σ
θ(T

T
θ
+
θ

θ
)
˜θ −
= exp −
f (0, s)ds − σ(T − θ)W
2
θ
( Z
)
Z θ
T
σ 2 θT (T − θ)
˜
= exp −
f (0, s)ds +
f (0, s)ds − σ(T − θ)Wθ −
2
0
0
( Z
)
T
−σ 2 θ 2
1 2 2
2
˜ θ (T − θ)
= exp −
f (0, s)ds +
(T − θ ) + σ θ (T − θ) − σ W
2
2
θ
( Z
)
T
θ
2
˜
= exp −
[f (0, s) + σ θ(s − ) + σ Wθ ]ds
2
θ
= e−

RT
θ

f (θ,s)ds

= P (θ, T ).

Secondly, using the properties of Winner process, we know the L2 − lim of


P2n −1 ˜
θ
2
˜
˜
˜
[−σ i=0 W
θ ·
2n ] is (−σ 0 Ws ds), (−σ 0 Ws ds + λWθ ) is the L − lim of

P2n −1
n
−W θn i ]} and the last term follows the Gaussian
{ i=0 [λ− σθ
2n (2 −i)][W n θ
2

2 (i+1)

2 3

distribution with mean 0 and variance θλ2 −θ2 σλ+ σ 3θ . So from the knowledge
of Laplace transform of Gaussian, we know
³ Rθ
´ 1
σ2 θ3
˜
˜
).
E ∗ e− 0 Ws ds eλWθ = (θλ2 − θ2 σλ +
2
3
And then, if we let
dP1
e−
=
dP∗


0



r(s)ds

dP2
P (θ, T )
e− 0 r(s)ds
=
, and
,

P (0, T )
dP
P (0, θ)

we have
˜



˜

E 1 (e−λWθ ) = E ∗ [e−λWθ −
˜

= E ∗ [e−λWθ e−


0

so

˜ θ − σθ2
W
2


e

+θσT

θ

way, we know

r(s)ds P (θ, T )

P (0, T )

]

R
2
2 2
˜ θ − σ θT (T −θ)
˜ s ]ds+ θ f (0,s)ds−σ(T −θ)W
[f (0,s)+ σ 2s +σ W
2
0

˜ θ −σ
−(λ+σ(T −θ))W

= E ∗ [e

0


0

˜ s ds
W

2 3
− σ 6θ

]e

σ 2 θT (T −θ)

2

=e

λ2 θ
σθ 2
2 −λ( 2

]

−θσT )

,

is a standard normal distribution under P 1 and in the same

˜ θ + σθ2
W
√ 2
θ

is a a standard normal distribution under P 2 .
26

Finally, if the option is exercised,
˜θ <
P (θ, T ) > K ⇒ W
Let d = (Wθ∗ −
price is
C0

σθ 2
2

= E ∗ [e−
= E ∗ [e−


+ θσT )/ θ =

0


0

Wθ∗

=

σ 2 θT (T −θ)
2

+ log

P (0,θ)K
P (0,T )

−σ(T − θ)


σ θ(T −θ)
2

KP (0,θ)



log P (0,T )

σ θ(T −θ)

r(s)ds

(P (θ, T ) − K)+ ]

r(s)ds

∗ −
P (θ, T )1{W
˜ θ <W ∗ } ] − KE [e

, we know the option


0

r(s)ds

θ

˜θ <
= P (0, T )P1 (W

.

1{W
˜ θ <W ∗ } ]
θ

˜θ <
− KP (0, θ)P2 (W

= P (0, T )N (d) − KP (0, θ)N (d − σ θ(T − θ)).

4.4

Wθ∗ )

Wθ∗ )

Coupon-bearing bond pricing

We have discussed how to convert a coupon-bearing bond pricing problem to
the zero-coupon bond pricing problem when the interest rate is a constant at the
beginning of this chapter. Now, at the end of this chapter, based on the work
of Jamshidian [15], we could give the formula for coupon-bearing bond pricing
under the Vasicek and CIR model, etc., or under the simple affine model which
bear positive coefficient B(t, s). Now, suppose a bond option with maturity T
and exercise price X based on a bond which generates n times coupon after time
T , and the i’th cash flow which happens at time si is ci (1 ≤ i ≤ n, s1 > T ).
Set r∗ be the time T instantaneous interest rate under which the value of the
coupon-bearing bond equals X, and let Xi be the value of the zero-coupon
bond with maturity si at time T when r = r∗ . r∗ could be quickly worked out
using, for example, Newton-Raphson method. Denote the time T price of the
zero-coupon bond with maturity si by P (T, si ), then the yield of this option is
max[0,

n
X

ci P (T, si ) − X].

i=1

Since all rates are increasing function of r, all bond prices are decreasing functions of r, this means that the coupon-bearing bond is worth more than X at
time T and should be exercised if, and only if, r < rX . Furthermore, the zerocoupon bond maturing at time si underlying the coupon-bearing bond worth
more than ci Xi at time T if, and only if, r < rX . So the yield of the option
could now be written as
n
X

ci max[0, P (T, si ) − Xi ],

i=1

which shows that the coupon-bearing bond could be thought as the sum of n
zero-coupon bonds.
27

References
[1] Ait-Sahakia, Y., Testing continuous-time model of the spot interest rate, Review Financial Studies 12 (1996), 721-762.
[2] Ammann, M., Credit Risk Valuation – Methods, Models and Applicatons, Springer.
[3] Artzner, P. and Delbaen, F., The martingale approach, Advances
in applied mathematics 10 (1989), 95-129.
[4] Brennan,M.J. and E.S. Schwartz, The valuation of the American
put option, Journal of Finance, 32, (1977), 449-462.
[5] Chapman, D.A., Pearson, N.D.: Is the short rate drift actually
nonlinear. Journal of Finance, 55, 355-388 (2000).
[6] Songxi Chen, The coefficients of the difusion equations, preprint.
[7] Cox, J.C., J.E. Ingersoll and S.A. Ross, A Re-examination of
traditional hypothesis about the term structure of interest rates,
Journal of finance, (1981).
[8] Cox, J.C., J.E. Ingersoll and S.A. Ross, An intertemporal general
equilibrium model of asset prices, Econometrica, (1985).
[9] Cox, J.C., J.E. Ingersoll and S.A. Ross, A theory of the term
structure of interest rates, Econometrica, 53 (1985), 385-407.
[10] Delbaen, F. and Schachermayer, W., A general version of the
fundamental theorem of asset pricing,Math. Ann. 300, 463-520,
1994.
[11] D. Duffie, D. Filipovi´c, W. Schachermayer, Affine processes and
applications in finance, The Annals of applied probability, (2003),
Vol 13,984-1053.
[12] Heath, D., A. Jarrow and A. Morton, Bond pricing and term
structure of interest rates, Econometrica, Vol. 60, No. 1 (1992),
77-105.
[13] Thomas. Y. Ho, Sang-bing Lee, Term structure movement and
pricing interest rate contingent claims, Journal of finance, (1986).
[14] N. Ikeda, S. Watanabe, Stochastic differential equations and
diffusion processes, Chap 4, North-Holland, 1981.
28

[15] Farshid Jamshidian, An exact bond option Formula, Journal of
finance, Vol. XLIV, NO. 1, (1989).
[16] Karatzas, I. and Shreve, E., Brownian Motion and stochastic
calculus, Springer-Verlag (1988), 193.
[17] Lamberton, D. and Lapeyre, B.,Introduction to stochastic calculus applied to finance, Chapman and Hall, 1996.
[18] Li Zenghu, Catalytic branching processes and affine interest
rate models, preprint.
[19] Longzhen Fan, wei Hu, Financial engineering, Shanghai people
publishing house.
[20] Morton, A.J., Arbitrage and Martingales, Ph.D. thesis, Cornell
University,1989.
[21] Shi Bingchao, et al., Theories and policies about interest rate.
[22] O.A. Vasicek, An equilibrium characterization of the term
structure, Journal of financial economics, 5 (November 1977),
88-177.

29

Sponsor Documents

Or use your account on DocShare.tips

Hide

Forgot your password?

Or register your new account on DocShare.tips

Hide

Lost your password? Please enter your email address. You will receive a link to create a new password.

Back to log-in

Close