pricing of zero-coupon bonds

7/29/2019 Pricing of Zero-Coupon Bonds

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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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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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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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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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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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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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-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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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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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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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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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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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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(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).

pricing of zero-coupon bonds

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