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Chapter 4
Pricing of Zero-Coupon Bonds
In this chapter we describe the basics of bond pricing in the absence of
arbitrage opportunities. Explicit calculations are carried out for the Vasicek
model, using both the probabilistic and PDE approaches. The definition
of zero-coupon bounds will be used in Chapter 5 in order to construct the
forward rate processes.
4.1 Definition and Basic Properties
A zero-coupon bond is a contract priced P0(t, T) at time t < T to deliver
P0(T, T) = $1 at time T. The computation of the arbitrage price P0(t, T)
of a zero-coupon bond based on an underlying short term interest rate pro-
cess (rt)tR+ is a basic and important issue in interest rate modeling.
We may distinguish three different situations:
a) The short rate is a deterministic constant r > 0.
In this case, P0(t, T) should satisfy the equation
er(Tt)P0(t, T) = P0(T, T) = 1,
which leads to
P0(t, T) = er(Tt), 0 t T.
b) The short rate is a time-dependent and deterministicfunction (rt)tR+ .
In this case, an argument similar to the above shows that
P0(t, T) = e
T
trsds, 0 t T. (4.1)
39
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40 An Elementary Introduction to Stochastic Interest Rate Modeling
c) The short rate is a stochastic process (rt)tR+ .
In this case, formula (4.1) no longer makes sense because the priceP0(t, T), being set at time t, can depend only on information known up
to time t. This is in contradiction with (4.1) in which P0(t, T) depends
on the future values of rs for s [t, T].
In the remaining of this chapter we focus on the stochastic case (c). The
pricing of the bond P0(t, T) will follow the following steps, previously used
in the case of Black-Scholes pricing.
Pricing bonds with non-zero coupon is not difficult in the case of a deter-
ministic continuous-time coupon yield at rate c > 0. In this case the price
Pc(t, T) of the coupon bound is given by
Pc(t, T) = ec(Tt)P0(t, T), 0 t T.
In the sequel we will only consider zero-coupon bonds, and let P(t, T) =
P0(t, T), 0 t T.
4.2 Absence of Arbitrage and the Markov Property
Given previous experience with Black-Scholes pricing in Proposition 2.2, it
seems natural to write P(t, T) as a conditional expectation under a mar-
tingale measure. On the other hand and with respect to point (c) above,
the use of conditional expectation appears natural in this framework since
it can help us filter out the future information past time t contained in(4.1). Thus we postulate that
P(t, T) = IEQ
e
T
trsds
Ft (4.2)under some martingale (also called risk-neutral) measure Q yet to be de-
termined. Expression (4.2) makes sense as the best possible estimate of
the future quantity eT
trsds given information known up to time t.
Assume from now on that the underlying short rate process is solution tothe stochastic differential equation
drt = (t, rt)dt + (t, rt)dBt (4.3)
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Pricing of Zero-Coupon Bonds 41
where (Bt)tR+ is a standard Brownian motion under P. Recall that for
example in the Vasicek model we have
(t, x) = a bx and (t, x) = .
Consider a probability measure Q equivalent to P and given by its density
dQ
dP= e
0KsdBs
12
0|Ks|
2ds
where (Ks)sR+ is an adapted process satisfying the Novikov integrability
condition (2.9). By the Girsanov Theorem 2.1 it is known that
Bt := Bt +t0
Ksds
is a standard Brownian motion under Q, thus (4.3) can be rewritten as
drt = (t, rt)dt + (t, rt)dBt
where
(t, rt) := (t, rt) (t, rt)Kt.
The process Kt, which is called the market price of risk, needs to bespecified, usually via statistical estimation based on market data.
In the sequel we will assume for simplicity that Kt = 0; in other terms we
assume that P is the martingale measure used by the market.
The Markov property states that the future after time t of a Markov process
(Xs)sR+ depends only on its present state t and not on the whole history
of the process up to time t. It can be stated as follows using conditionalexpectations:
IE[f(Xt1 , . . . , X tn) | Ft] = IE[f(Xt1 , . . . , X tn) | Xt]
for all times t1, . . . , tn greater than t and all sufficiently integrable function
f on Rn, see Appendix A for details.
We will make use of the following fundamental property, cf e.g. Theorem V-
32 of [Protter (2005)].
Property 4.1. All solutions of stochastic differential equations such as
(4.3) have the Markov property.
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42 An Elementary Introduction to Stochastic Interest Rate Modeling
As a consequence, the arbitrage price P(t, T) satisfies
P(t, T) = IEQ eT
trsdsFt
= IEQ
eT
trsds
rt ,and depends on rt only instead of depending on all information available
in Ft up to time t. As such, it becomes a function F(t, rt) of rt:
P(t, T) = F(t, rt),
meaning that the pricing problem can now be formulated as a search for
the function F(t, x).
4.3 Absence of Arbitrage and the Martingale Property
Our goal is now to apply Itos calculus to F(t, rt) = P(t, T) in order to
derive a PDE satisfied by F(t, x). From Itos formula Theorem 1.8 we have
d
et
0rsdsP(t, T)
= rte
t
0rsdsP(t, T)dt + e
t
0rsdsdP(t, T)
= rte
t
0rsds
F(t, rt)dt + e
t
0rsds
dF(t, rt)= rte
t
0rsdsF(t, rt)dt + e
t
0rsds
F
x(t, rt)((t, rt)dt + (t, rt)dBt)
+et
0rsds
1
22(t, rt)
2F
x2(t, rt)dt +
F
t(t, rt)dt
= et
0rsds(t, rt)
F
x(t, rt)dBt
+et
0rsdsrtF(t, rt) + (t, rt)
F
x(t, rt)
+1
22(t, rt)
2F
x2(t, rt) +
F
t(t, rt)
dt. (4.4)
Next, notice that we have
et
0rsdsP(t, T) = e
t
0rsds IEQ
e
T
trsds
Ft= IEQ
e
t
0rsdse
T
trsds
Ft
= IEQ
e
T
0rsds
Ft
hence
t et
0rsdsP(t, T)
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Pricing of Zero-Coupon Bonds 43
is a martingale (see Appendix A) since for any 0 < u < t we have:
IEQ et
0rsdsP(t, T)Fu = IEQ IEQ e
T
0rsdsFt Fu
= IEQ
eT
0rsds
Fu= IEQ
e
u
0rsdse
T
ursds
Fu= e
u
0rsds IEQ
e
T
ursds
Fu= e
u
0rsdsP(u, T).
As a consequence, (cf. again Corollary 1, p. 72 of [Protter (2005)]), the
above expression (4.4) ofd
et
0rsdsP(t, T)
should contain terms in dBt only, meaning that all terms in dt should vanish
inside (4.4). This leads to the identity
rtF(t, rt) + (t, rt)F
x(t, rt) +
1
22(t, rt)
2F
x2(t, rt) +
F
t(t, rt) = 0,
which can be rewritten as in the next proposition.
Proposition 4.1. The bond pricing PDE for P(t, T) = F(t, rt) is writtenas
xF(t, x) = (t, x)F
x(t, x) +
1
22(t, x)
2F
x2(t, x) +
F
t(t, x), (4.5)
subject to the terminal condition
F(T, x) = 1. (4.6)
Condition (4.6) is due to the fact that P(T, T) = $1. On the other hand,
e
t
0
rsds
P(t, T)t[0,T] and (P(t, T))t[0,T]
respectively satisfy the stochastic differential equations
d
et
0rsdsP(t, T)
= e
t
0rsds(t, rt)
F
x(t, rt)dBt
and
dP(t, T) = P(t, T)rtdt + (t, rt)F
x(t, rt)dBt,
i.e.
dP(t, T)P(t, T) = rtdt + (t, rt)P(t, T) Fx (t, r
t)dBt
= rtdt + (t, rt)log F
x(t, rt)dBt.
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44 An Elementary Introduction to Stochastic Interest Rate Modeling
4.4 PDE Solution: Probabilistic Method
Our goal is now to solve the PDE (4.5) by direct computation of the con-ditional expectation
P(t, T) = IEQ
e
T
trsds
Ft. (4.7)
We will assume that the short rate (rt)tR+ has the expression
rt = g(t) +
t0
h(t, s)dBs,
where g(t) and h(t, s) are deterministic functions, which is the case in par-ticular in the [Vasicek (1977)] model. Letting u t = max(u, t), using the
fact that Wiener integrals are Gaussian random variables (Proposition 1.3),
and the Gaussian characteristic function (12.2) and Property (a) of condi-
tional expectations, cf. Appendix A, we have
P(t, T) = IEQ
e
T
trsds
Ft
= IEQ
e
T
t(g(s)+
s
0h(s,u)dBu)ds
Ft
= eT
tg(s)ds IEQ
e
T
t
s
0h(s,u)dBuds
Ft
= eT
tg(s)ds IEQ
e
T
0
T
uth(s,u)dsdBu
Ft
= eT
tg(s)dse
t
0
T
uth(s,u)dsdBu IEQ
e
T
t
T
uth(s,u)dsdBu
Ft
= eT
tg(s)dse
t
0
T
th(s,u)dsdBu IEQ
e
T
t
T
uh(s,u)dsdBu
Ft
= eT
tg(s)dse
t
0
T
th(s,u)dsdBu IEQ e
T
t
T
uh(s,u)dsdBu
= e
T
tg(s)dse
t
0
T
th(s,u)dsdBue
12
T
t (T
uh(s,u)ds)
2du.
Recall that in the [Vasicek (1977)] model, i.e. when the short rate process
is solution of
drt = (a brt)dt + dBt,
and the market price of risk is Kt = 0, we have the explicit solution, cf.
Exercise 1.3 and Exercise 3.1:
rt = r0ebt +
a
b(1 ebt) +
t
0
eb(ts)dBs, (4.8)
hence the above calculation yields
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Pricing of Zero-Coupon Bonds 45
P(t, T) = IEQ
e
T
trsds
Ft
= e
T
t(r0e
bs+ ab(1ebs))ds
e
t
0 T
teb(su)
dsdBu
e2
2
T
t(T
ueb(su)
ds)2du
= eT
t(r0e
bs+ ab(1ebs))dse
b(1eb(Tt))
t
0eb(tu)
dBu
e2
2
T
te2bu
ebu
ebT
b
2du
= ert
b(1eb(Tt))+ 1
b(1eb(Tt))(r0e
bt+ ab(1ebt))
e
T
t(r0e
bs+ ab(1ebs))ds+
2
2
T
te2bu
ebu
ebT
b
2du
= eC(Tt)rt+A(Tt),
where
C(T t) = 1
b(1 eb(Tt)),
and
A(T t) =1
b(1 eb(Tt))(r0e
bt +a
b(1 ebt))
Tt
(r0ebs +
a
b(1 ebs))ds
+2
2
Tt
e2buebu ebT
b
2du
=1
b(1 eb(Tt))(r0e
bt +a
b(1 ebt))
r0
b(ebt ebT)
a
b(T t) +
a
b2(ebt ebT)
+2
2b2
Tt
1 + e2b(Tu) 2eb(Tu)
du
= ab2
(1 eb(Tt))(1 ebt) ab
(T t) + ab2
(ebt ebT)
+2
2b2(T t) +
2
2b2e2bT
Tt
e2budu2
b2ebT
Tt
ebudu
=a
b2(1 eb(Tt)) +
2 2ab
2b2(T t)
+2
4b3(1 e2b(Tt))
2
b3(1 eb(Tt))
= 4ab 32
4b3+
2 2ab2b2
(T t)
+2 ab
b3eb(Tt)
2
4b3e2b(Tt).
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46 An Elementary Introduction to Stochastic Interest Rate Modeling
See Exercise 4.5 for another way to calculate P(t, T) in the [Vasicek (1977)]
model.
Note that more generally, all affine short rate models as defined in Rela-
tion (3.1), including the Vasicek model, will yield a bond pricing formula
of the form
P(t, T) = eA(Tt)+C(Tt)rt ,
cf. e.g. 3.2.4. of [Brigo and Mercurio (2006)].
4.5 PDE Solution: Analytical Method
In this section we still assume that the underlying short rate process is
the Vasicek process solution of (4.3). In order to solve the PDE (4.5)
analytically we look for a solution of the form
F(t, x) = eA(Tt)+xC(Tt), (4.9)
where A and Care functions to be determined under the conditions A(0) =
0 and C(0) = 0. Plugging (4.9) into the PDE (4.5) yields the system of
Riccati and linear differential equations
A(s) = aC(s) 2
2C2(s)
C(s) = bC(s) + 1,
which can be solved to recover
A(s) =4ab 32
4b3+ s
2 2ab
2b2+
2 ab
b3ebs
2
4b3e2bs
and
C(s) = 1
b(1 ebs).
As a verification we easily check that C(s) and A(s) given above do satisfy
bC(s) + 1 = ebs = C(s),
and
aC(s) +2C2(s)
2
= a
b
(1 ebs) +2
2b2
(1 ebs)2
=2 2ab
2b2
2 ab
b2ebs +
2
2b2e2bs
= A(s).
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Pricing of Zero-Coupon Bonds 47
-0.5
0
0.5
1
1.5
2
0 5 10 15 20
Fig. 4.1 Graph of t Bt.
4.6 Numerical Simulations
Given the Brownian path represented in Figure 4.1, Figure 4.2 presents the
corresponding random simulation of t rt in the Vasicek model withr0 = a/b = 5%, i.e. the reverting property of the process is with respect to
its initial value r0 = 5%. Note that the interest rate in Figure 4.2 becomes
negative for a short period of time, which is unusual for interest rates but
may nevertheless happen [Bass (October 7, 2007)].
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0 5 10 15 20
Fig. 4.2 Graph of t rt.
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48 An Elementary Introduction to Stochastic Interest Rate Modeling
Figure 4.3 presents a random simulation of t P(t, T) in the same Va-
sicek model. The graph of the corresponding deterministic bond price ob-
tained for a = b = = 0 is also shown on the same Figure 4.3.
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20
Fig. 4.3 Graphs of t P(t, T) and t er0(Tt).
Figure 4.4 presents a random simulation oft P(t, T) for a coupon bondwith price Pc(t, T) = e
c(Tt)P(t, T), 0 t T.
100.00
102.00
104.00
106.00
108.00
0 5 10 15 20
Fig. 4.4 Graph of t P(t, T) for a bond with a 2.3% coupon.
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Pricing of Zero-Coupon Bonds 49
Finally we consider the graphs of the functions A and C in Figures 4.5 and
4.6 respectively.
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 5 10 15 20
Fig. 4.5 Graph of t A(T t).
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 5 10 15 20
Fig. 4.6 Graph of t C(T t).
The solution of the pricing PDE, which can be useful for calibration pur-
poses, is represented in Figure 4.7.
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50 An Elementary Introduction to Stochastic Interest Rate Modeling
00.20.40.60.81
00.02
0.040.06
0.080.1
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
t
x
Fig. 4.7 Graph of (x, t) exp(A(T t) + xC(T t)).
4.7 Exercises
Exercise 4.1. Consider a short term interest rate process (rt)tR+ in a
Ho-Lee model with constant coefficients:
drt = dt + dWt,and let P(t, T) will denote the arbitrage price of a zero-coupon bond in this
model:
P(t, T) = IEP
exp
Tt
rsds
Ft
, 0 t T. (4.10)
(1) State the bond pricing PDE satisfied by the function F(t, x) defined
via
F(t, x) = IEP
exp
T
t
rsds
rt = x
, 0 t T.
(2) Compute the arbitrage price F(t, rt) = P(t, T) from its expression
(4.10) as a conditional expectation.
(3) Check that the function F(t, x) computed in Question (2) does satisfy
the PDE derived in Question (1).
Exercise 4.2. (Exercise 3.2 continued). Write down the bond pricing PDEfor the function
F(t, x) = E
eT
trsds
rt = x
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Pricing of Zero-Coupon Bonds 51
and show that in case = 0 the corresponding bond price P(t, T) equals
P(t, T) = eB(Tt)rt , 0 t T,
where
B(x) =2(ex 1)
2+ (+ )(ex 1),
with =
2 + 22.
Exercise 4.3. Let (rt)tR+ 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
dP(t, T)
P(t, T)= rtdt +
Tt dBt, 0 t T, (4.11)
under the terminal condition P(T, T) = 1, where (Tt )t[0,T] is an adapted
process. Let the forward measure PT be defined by
IE
dPTdP
Ft
=P(t, T)
P(0, T)e
t
0rsds, 0 t T.
Recall that
BTt := Bt
t
0
Ts ds, 0 t T,
is a standard Brownian motion under PT.
(1) Solve the stochastic differential equation (4.11).
(2) Derive the stochastic differential equation satisfied by the discounted
bond price process
t et
0rsdsP(t, T), 0 t T,
and show that it is a martingale.(3) Show that
IE
eT
0rsds
Ft = e t0 rsdsP(t, T), 0 t T.(4) Show that
P(t, T) = IE
eT
trsds
Ft , 0 t T.(5) Compute P(t, S)/P(t, T), 0 t T, show that it is a martingale under
PT and that
P(T, S) =P(t, S)
P(t, T)exp
Tt
(Ss Ts )dB
Ts
1
2
Tt
(Ss Ts )
2ds
.
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52 An Elementary Introduction to Stochastic Interest Rate Modeling
Exercise 4.4. (Exercise 1.8 continued). Assume that the price P(t, T) of a
zero coupon bond is modeled as
P(t, T) = e(Tt)+XT
t , t [0, T],
where > 0. Show that the terminal condition P(T, T) = 1 is satisfied.
Problem 4.5. Consider the stochastic differential equation
dXt = bXtdt + dBt, t > 0,
X0 = 0,
(4.12)
where b and are positive parameters and (Bt)tR+ is a standard Brownian
motion under P, generating the filtration (Ft)tR+ . Let the short term
interest rate process (rt)tR+ be given by
rt = r + Xt, t R+,
where r > 0 is a given constant. Recall that from the Markov property, the
arbitrage price
P(t, T) = IEP
expTt
rsds Ft , 0 t T,
of a zero-coupon bond is a function F(t, Xt) = P(t, T) of t and Xt.
(1) Using Itos calculus, derive the PDE satisfied by the function (t, x)
F(t, x).
(2) Solve the stochastic differential equation (4.12).
(3) Show thatt0
Xsds =
b
t0
(eb(ts) 1)dBs
, t > 0.
(4) Show that for all 0 t T,Tt
Xsds =
b
t0
(eb(Ts) eb(ts))dBs +
Tt
(eb(Ts) 1)dBs
.
(5) Show that
IE
Tt
XsdsFt
=
b
t0
(eb(Ts) eb(ts))dBs.
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Pricing of Zero-Coupon Bonds 53
(6) Show that
IET
t XsdsFt
=
Xtb (1 e
b(Tt)
).
(7) Show that
Var
Tt
XsdsFt
=
2
b2
Tt
(eb(Ts) 1)2ds.
(8) What is the distribution of
Tt
Xsds given Ft?
(9) Compute the arbitrage price P(t, T) from its expression (4.10) as aconditional expectation and show that
P(t, T) = eA(t,T)r(Tt)+XtC(t,T),
where C(t, T) = (eb(Tt) 1)/b and
A(t, T) =2
2b2
Tt
(eb(Ts) 1)2ds.
(10) Check explicitly that the function F(t, x) = eA(t,T)+r(Tt)+xC(t,T)
computed in Question (9) does solve the PDE derived in Question (1).