chapitre 6 interest rate

8/9/2019 chapitre 6 interest rate

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6

Interest rate models

Interest rate models are mainly used to price and hedge bonds and bondoptions. Hitherto, there has not been any reference model equivalent to theBlack-Scholes model for stock options. In this chapter, we will present the mainfeatures of interest rate modelling (following essentially Artzner and Delbaen

(1989)), study three particular models and see how they are used in practice.

1 Introduction

1.1 Instantaneous interest rate

In most of the models that we have already studied, the interest rate was as-sumed to be constant. In the real world, it is observed that the loan interestrate depends both on the date t of the loan emission and on the date Tof thematurity of the loan.

Someone borrowing one dollar at time t, until maturity T, will have to payback an amount F(t, T) at time T, which is equivalent to an average interest

rateR(t, T) given by the equality

F(t, T) = e(Tt)R(t,T).

If we consider the future as certain, i.e. if we assume that all interest rates(R(t, T))tTare known, then, in an arbitrage-free world, the function F mustsatisfy

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 issmooth, there exists a function r(t) s.t.

t < T F (t, T) = exp{ Tt

r(s)ds}

and consequently

R(t, T) = 1

T t Tt

r(s)ds.

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The function r(s) is called instantaneous interest rate. In fact, the former

equation could be easily seen: If we setr(s) =F1(s, s),

then we could find the partial difference oft in equation (1) with the observationthat F is smooth:

F

1(t, s) = F

1(t, u)F(u, s).

SinceF is continuous, we let u = t, and so

F

1(t, t) = F

1(t, s)

F(t, s) (t < s),

that is

r(t) =F

1(t, s)

F(t, s) =

F

1(t, T)

F(t, T) (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 relationshipsbetween different rates; the aim of modelling is to determine them, and threesuch models will be introduced later.

1.2 Zero-coupon bond

When we say zero-coupon bond, we mean a security paying 1 dollar at amaturity date T and it could be understood as a short-term-treasury which

generates no dividend, meanwhile its maturity dont need to be less than oneyear. And, coupon-bearing bond is naturally defined as a security paying 1dollar at the maturity dateTand at some certain times before T, it pays somemoney according to the coupon. Then, how should such derivatives be priced?Lets 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 nominalvalue 100 dollars, would worth 100e2r dollars now. The value of a two-yearcoupon-bearing bond whose nominal value is 100 dollars and with yearly couponrate 0.1 (In America, the coupon is payed every half year) would worth

3i=1

100 0.12

e0.5i0.05 + 105e20.05.

So, we see a coupon-bearing bond could be split into some zero-couponbonds, 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 haveP(T, T) = 1. If the interest rate is now correlated both to the emission dateand maturity date, we would see from the former subsection that

P(t, T) = eTt r(s)ds. (2)

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1.3 Further discussion

However, in an risky world, the instantaneous interest rate r(t) is no longeronly a function oft, but rather a stochastic process. So on the filtered spaced(, F, P, (Ft)0tT), where (Ft) is the natural filtration of a standard B.M.(Wt)0tTand FT =F, we define the instantaneous interest rate (r(t))0tTas an adapted process satisfying

T0 |r(t)|dt


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Ifu(t) and(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 couldno longer be easily sensed as in the Black-Scholes. So, we will first make thefollowing assumption and then try to derive the equation for bond price.

(H) There is a probability P* equivalent to P, under which, forall real-valued u[0, T], the process (P(t, u))0tu defined by

P(t, u) =et0r(s)dsP(t, u)

is a martingale.

Remark:

In Chapter 1, we have characterized the absence of arbitrage opportunitiesby the existence of an equivalent probability under which discounted asset pricesare martingales. We have also seen such a probability exists in the Black-Scholesmodel. So we introduce this probability into this continuous-time model.

To be more strictly, we say for continuous processes the existence of an equiv-alent probability, under which the discounted bond price is a martingale, stillensures no arbitrage, but the no arbitrage property for general integrands is nolonger 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 un-der most of the conditions the assumption above is tenable, and therefore, itsreasonable for us to set it here:

Numerous authors have presented versions of the fundamental the-ory of asset pricing under various assumptions and degrees of gen-erality, replacing no arbitrage with slightly different notions, suchas no approximate arbitrage, no free lunch with bounded risk, nofree lunch with vanishing risk, no asymptotic arbitrage and so on,to reestablish the theorm[2]. The case when the time set is finite iscompletely settled in Dalang et al. (1989) and the the use of simpleor even elementary integrands is no restriction at all.For the case of discrete but infinite time sets, the problem is solvedin Schachermayer (1993). The case of continuous and boundedprocesses in continuous time, is solved in Delbaen (1992) and soon.One should read [3] and [10] for more details.

Now, under this assumption there are many interesting consequences. In-deed, the martingale property under P leads to, using equality P(u, u) = 1,

P(t, u) = E(P(u, u)|Ft) = E(eu0 r(s)ds|Ft)

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and, eliminating the discounting,

P(t, u) =E(eut r(s)ds|Ft) (3)

This equality, which could be compared to formula (2), shows that the pricesP(t, u) only depend on the behavior of the processes (r(s))0tT under theprobabilityP.

We should pay attention to the hypothesis we made on the filtration (Ft)0tT,which allows us to express the density of the probability P with respect to P.In fact, ifT is infinite, this may not hold any more [16]. We denote byLt thisdensity. For any non negative random variable X, we have

E(X) =

XdP() =

XLTdP =E(XLT),

and, ifXisFt-measurable, note Lt = E(LT|Ft), we haveE(X) =E(XLT) =E(E(XLT|Ft)) = E(XE(LT|Ft)) = E(XLt). (4)

Thus the random variable Lt is the density ofP restricted toFt with respect

toP.

Lemma1 Let (Mt)0tT 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

=1 T= (inf{t[0, T]|Mt= 0}) T,it follows that 1 is a stopping time and also is 1

. Then using the optional

sampling theorem, we getE(MT|F) = M,

E[1{=T}E(MT|F)] = E[1{=T}M],and

E[1{=T}E(MT|F)] =E(1{=T}MT),E[1{=T}M{}] =E[1{=T}M{}+ 1{=T}M{}] =E[M{}] =E[M{T}].

So from E[MT(1 1=T)] = 0, P(1 1T 0) = 1 and P(MT >0) = 1, weget 1 1=T = 0, a.s., and consequently P(t[0, T], Mt > 0) = 1.

Proposition6.1.1 There is an adapted process (q(t))0tT such that, for all

t[0, T],Lt = exp(

t0

q(s)dBs12

t0

q(s)2ds) a.s. (5)

Proof: The processes (Lt)0tT is a martingale relative to (Ft), which is thenatural filtration of the B.M. (Wt). It follows from the generalized martingale

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representation theorem that there exists an adapted process ((Ht)), satisfying

T0 H2t dt 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 I toformula to the log function and gain

d(log Lt) = 1

LtdLt 1

2

1

L2t(dLt)

2 = 1

LtHtdWt1

2

1

L2tH2t ds a.s.

log Lt = t0

HsLs

12

t0

H2sL2s

a.s.

Write q(t) = HtLt , we finally get

Lt = exp(

t0

q(s)dBs12

t0

q(s)2ds) a.s.

Corollary6.1.2 The price at time t of the zero-coupon bond of maturity utcan be expressed as

P(t, u) =E(exp( ut

r(s)ds +

ut

q(s)dWs 12

ut

q(s)2ds)|Ft). (6)

Proof: A Ft,E[1AE(X|Ft)] = E(1AX) = E(1AXLT)

= E[E(1AXLT|Ft)] = E[1AE(XLT|FtLtLt

] =E(1AE(XLT|Ft)

Lt),

E(X|Ft) = E(XLT|Ft)Lt

. (7)

So, we see

P(t, u) = E(exp( u0

r(s)ds|Ft))

= E(eut r(s)dsLT|Ft)/Lt= the right of (6).

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Proposition6.1.3 For each maturity u, there is an adapted process (ut)0tT

such that, on [0, u]dP(t, u)

P(t, u) =

r(t) utq(t)

dt + utdWt. (8)

Proof: Since the process (P(t, u))0tu is a martingale under P, for anyt > s, A Fs, we have (note Pt = P(t, u) below)

E(1APtLt) = E(1APt) = E[E(1APt|Fs)]

= E[1AE(Pt|Fs)] = E(1APs) =E(1APsLs),

which is to say (P(t, u)Lt)0tu is a martingale under P. With the same pro-cedure as in Proposition 6.1.1, we see there exists an adapted process ut, such

that u0(ut)2


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Remark6.1.4When we compare the risk asset (8) with the riskless asset dS0t =

r(t)S

0

t dt, it is easy to see its the term in dWt which makes the bond riskier.Furthermore, the termr(t) utq(t) corresponds intuitively to the average yieldof the bond at time t, since the increments of B.M. have zero expectation, andthe termutq(t) is the difference between the average yield of bond and theriskless rate, hence the interpretation ofq(t) as a risk premium. With theobservation in Proposition 6.1.1, through the Girsanov theorem, we have underthe probability P,

Wt = Wt t0

q(s)ds

is a standard B.M. and equation (8) becomes

dP(t, u)

P(t, u) =r(t)dt + utd

Wt.

So we say P is a risk neutral probability and ut would be interpreted asvolatility.

2.2 Bond options

To make things clearer, lets first consider a European option with maturityonthe zero-coupon bond with maturity equal to the horizon T. If it is a call withstrike price K, the value of the option at time is obviously (P(, T) K)+and it seems reasonable to hedge this call with a portfolio riskless asset andzero-coupon bond with maturity T. A strategy is then defined by an adaptedprocess ((H0t , Ht))0tT with values in R2, H0t representing the quantity of

riskless asset and Ht the risky asset held in the portfolio at time t. The valueof the portfolio at time t is then given by

Vt = H0t S

0t + HtP(t, T) = H

0t e

t0 r(s)ds + HtP(t, T).

Now as the same way in Chapter 4, we make two definitions:

Definition6.1.5* A self-financing strategyis defined by a pair of adaptedprocesses (H0t)0tTand (Ht)0tT, which satisfies1.T0 |H0t |dt +

T0

(Ht)2dt


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thathe0 r(s)ds is square-integrable underP. Then there exists an admissible

strategy whose value at time is equal to h. The value at timet of such astrategy is given byVt= E

e t r(s)dshFt.Proof:1. The method is the same as in Chapter 4. We first observe that if there exitsan admissible strategy ((H0t , Ht)0tTwhich satisfies the conditions set forthin the proposition, we obtain, using the self-financing condition, the formula ofVt, the integration by parts formula and Remark 6.1.4,

Vt = d(VtS0t

) = 1

S0tdVt Vt

(S0t)2

dS0t

= H0t

S0t

dS0t +Ht

S0t

dP(t, T)

H0t S

0t

(S0t)

2dS0t

H0t P(t, T)

(S0t)

2 dS0t

= Ht

S0tdP(t, T) HtP(t, T)

(S0t)2

dS0t

= Ht

1

S0t[P(t, T)r(t)dt + P(t, T)Tt dWt]

P(t, T)

(S0t)2

S0t r(t)dt

=

HtS0t

P(t, T)Tt dWt = HtP(t, T)Tt d Bt.

We deduce, bearing in mind that supt[0,T] Vt is square-integrable under P,

E[

T

0

[HtP(t, T)Tt ]

2dt] =E[

T

0

HtP(t, T)Tt dWt]

2 =E(

T

0

dVt)2


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+1

2 1

X22

dX1dX2

1

X22

dX2dX1+ 0(dX1)2 +

2X1X3

2

(dX2)2

=

AtLt

dWt VtLtL2t

q(t)LtdWt

+1

2

1

L2tAtq(t)Ltdt 1

L2tAtq(t)Ltdt + 2

VtLtL3t

q(t)2L2tdt

=

AtLt

Vtq(t)

dWt

AtLt

Vtq(t)

q(t)dt.

LetJt = AtLt

Vtq(t). Then dVt = JtdWt Jtq(t)dt= JtdWt.So,

V

= V0

+

0

Js

dWs

,

or

he0 r(s)ds =E

he

0 r(s)ds

+

0

JsdWs.

3. Now, we could move to construct the strategy. Set

Ht= Jt

P(t, T)Tt, and H0t =E

he0 r(s)ds|Ft

Jt

Tt

for t . We check easily that ((H0t , Ht))0t defines an admissible strategywhose value at time 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 probabilityequivalent to P under which (P(t, T))0tT is a martingale if and only if theprocess (Tt )= 0,dtdPalmost everywhere. This condition, slightly weaker thanthe hypothesis of Proposition 6.1.6, is exactly what is needed to hedge optionswith bonds of maturity T, which is not surprising when one keeps in mind thecharacterization of complete markets we gave in Chapter 1.

Proposition6.1.8 P is the unique probability equivalent to P under which(P(t, T))0tT is a martingale if and only if the process (Tt )= 0, dtdP almosteverywhere.

Proof: From Proposition 6.1.3, we see when Tt = 0, Tt and q(t) areuniquely determined by the processes (P(t, T))0tT and (rt)0tT. And fromthe martingale representation theorem, the uniqueness ofq(t) and the unique-ness of P are equivalent. So from mP((u, )|Tt = 0) = 0, it is easy toconclude the uniqueness ofP. Let S ={(u, )|Tt = 0}. Under the assumption mP(S) > 0, wewill construct another probability equivalent to P and not equal to P, which

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contradict to the original hypothesis. we would do so in three steps.

1.

T

t =

T

t + q(t), m P(S)> 0, for any >0, letft =

Tt , qt = qt+ 1s, and

ft = ft+ 1s.

Define

= inf{t| exp ( t0

fudBu 12

t0

f2udu)> 2 exp(

t0

fudBu 12

t0

f2udu) , or

exp (

t0

qudBu 12

t0

q2udu)> 2 exp(

t0

qudBu12

t0

q2udu)} T

andqt = q f or t , qt = q for t > ,ft= f f or t , ft= f for t > .

Then for each t we have

exp(

t0

fudBu 12

t0

f2udu)2exp( t0

fudBu12

t0

f2udu)

as well as the same inequality with f andf respectively replaced by qandq.2. Fromfu qu= fuqu and equation (9),

P(t, T)

= P(0, T) exp[

t0

Tu dBu+

(ru 1

2(tu)

2 qufu+ q2u)du]

= P(0, T) exp[

t0

(fu qu)dBu 12

t0

(fu qu)2du + T0

(ru qufu+ q2u)du]

= P(0, T) exp[ t0

(fuqu)dBu 12 t0

(fuqu)2du + t0

(ruqufu+ q2u)du].

By the estimations in step 1,

E[exp(

T0

qudBu 12

T0

q2udu)] = 1,

exp( T0

rudu) exp[

T0

qudBu12

T0

q2udu]

=P(0, T) exp(

T0

fudBu 12

T0

f2udu)L2.

3. Let Lt= exp(t0 qudBu 12 t0 q2udu), we haveP(t, T) exp(

t0

rudu)Lt = P(0, T) exp(

t0

fudBu 12

t0

f2udu).

Hence P(t, T) = exp( t0

rudu)Lt is a P-martingale, Ltexp(T0

rudu) L2,E[LT] = 1 and (Lt)tT is a P-martingale. DefineP by dP

dP =LT, then its

existence is a contradiction with the uniqueness assumption.

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3 Term structure of interest rate

From above, we see in order to price the zero-coupon bond and the correspondingoption, we need to know the law of (r(t)), and this lead to some conceptionscorrelated 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 actualinvest. And we have mentioned before that now the risk of holding zero-couponbond is partly caused by the manner of people buying and selling bonds at anytime they think profitable. So there exists great correlation between the yieldand the change of the interest rate and this justifies the necessary of interestrate modelling.

If we plot the yield of different bonds with diverse maturity, but the same riskand mobility, this curve is called yield curve; it is the set of{Yt+st }0sTt,and it is also called Term structure of interest rate.

The price of zero-coupon bond{P(t, t+s)}0sTt is called Term struc-ture of zero-coupon bond price.

Spot interest rateis defined as the n-year interest rate which is calculatedfrom today, while forward interest rate f(t, s), t s represents the antici-pated interest rate at time t for time s.{f(t, s)}tsT is named as the time tterm 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 justtry to find the price of the zero-coupon bond above, and would only investigatethe law of interest rate below.

3.2 Three traditional theories[21]

Before 1980, there are three main theories explaining the term structure ofinterest rate.

I. The expectation theory is first proposed in 1896, and developed in1949. It is one of the oldest such theories and it is based on the following fiveassumptions: 1. All the investors are aiming for maximizing profit; 2. All theinvestors think all the bonds with different maturities can be totally superseded

by some others; 3. There is no charge for holding and transaction; 4. Noarbitrage exists; 5. Most of the investors can make accurate expectations andwould behave according to them. Thus the n-year spot interest rate would equalto the average of the 1-year spot interest rate and the expected 1-year forwardinterest rate in the futuren 1 years, that is

i(n)t=it+ i

et+1+ i

et+2+ . . . + i

et+(n1)

n .

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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 expectationtheory. In this theory, they think that some factors, such as the longer thematurity of the bond is, the risk of the volatility of the value of invested moneyis, should be considered too. So, the borrower should provide the investor somerisk premium to insure that the latter would like to buy long-term bonds. So,letk(n)t be the n-year bonds mobility compensation, we have

i(n)t =it+ it+1e + i

et+2+ . . . + i

et+(n1)

n + k(n)t.

III. The market segmentation theory, however, doubt the 2nd as-sumption in the expectation theory. It takes bonds with different maturity ascompletely independent markets and think one bonds interest should be totallydecided by its own demand and provision.

In practice, according to the explanation ability of the three theories to someempirical fact, the liquidity preference theory is more popular.

4 Some classical models

In the last section, three traditional models are introduced, and in the followinganother three modern ones would be discussed. The Vasicek model is highlywelcomed by the practitioners for its simplicity, while the CIR model wouldalways generates non-negative interest rates. And the HJM model provide for-mulae that depend on the parameters of the dynamics of interest rates underP.

4.1 The Vasicek model[22]

In this model, we assume that the process r (t) satisfies

dr(t) =a(b r(t))dt + dWt (10)wherea, b, are non-negative constants. We also assume that the process q(t)is a constantq(t) =, with R. Then

dr(t) = a(b r(t))dt + d Wt (11)whereb= b a and Wt= Wt+ t.

4.1.1 Feature of the solution

LetXt= r(t) b. Then Xt is a solution of the stochastic differential equationdXt=aXtdt + dWt (12)

which means that (Xt) is an O-U process. So from the result of Chapter 3, weknow

r(t) = r(0)eat + b(1 eat) + eat t0

easdWs (13)

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and thatr(t) follows a normal law whose mean is given by E(r(t)) = r(0)eat +

b()1eat

and variance by V ar(r(t)) = 2

e2at

E(t0e2asds) = 21e2at2a .Whent tends to infinity,r(t) converges toN(b,

2

2a). Although this model is very

simple, it is still not satisfactory from a practical point of view since P(r(t) 0.

4.1.2 Preliminary calculation ofP(t, T)

Based on the equations (3) and (11), underP we have

P(t, T) =E(eTt r(s)ds|Ft) = E(e

Tt (Xs+b

)ds|Ft) (14)

whereXt is the solution to the equation

dXt =aXtdt + d Wt. (15)Again using the conclusions in Chapter 3, Remark 3.5.11, we gain

P(t, T) =eb(Tt)E(e

Tt0 X

ys ds)|y=Xt (16)

whereXxt is the unique solution of equation 15 which satisfies Xx0 =x.

4.1.3 Calculate F(, x)

In order to calculate P(t, T) completely, now we try to find F(, x) = E(e0 X

xs ds).

We know that Xt is a Gaussian process with continuous paths, so

0 Xxs 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 aGaussian 1

E

e0 X

xs ds

= exp E(

0

Xxs ds) +1

2V ar

0

Xxs ds

From equalityE

Xxs

= xeas and the Fubini theorem, we deduce

E( 0

Xxs ds) =

0

E(Xxs)ds= 0

xeasds= x1 ea

a .

Again, using the Fubini theorem, we have

V ar( 0

Xxs ds) = E(

0

Xxs ds E 0

Xxs ds)2

= E[ 0

0

(Xxu EXxu)(Xxt EXxt)dtdu]

1If Y N(, 2), then E(eY) =

ey 12

e y

22 dy= e2

2 .

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=

0

0

E[(Xxu

EXxu)(Xxt

EXxt)]dtdu

=

0

0

E(eau u0

easdWs)(eat

t0

eadW)

=

0

0

2ea(u+t)E( ut0

easdWs)2dtdu

=

0

0

2ea(u+t)E( ut0

e2asds)dtdu

= 2

2a

0

0

ea(t+u)(e2a(tu) 1)dtdu

= 2

0

t

2

2aeau(eat eat)dudt

= 2

a2( 3

2a+

2

aea

1

2ae2a)

= 2

a2

2

a3(1 ea)

2

2a3(1 ea)2

4.1.4 bond pricing

Let = T t, R = b 2/(2a2)and

R(, r) =R 1a

[(R r)(1 ea) 2

4a2(1 ea)2].

ThenR = lim R(, r). Combining the above result and equation (16), we

know

P(t, T)

= exp b x 1 ea

a +

2

2a2

2

2a3(1 ea)

2

4a2(1 ea)2

= exp (b 2

2a2) +

1

a[(b

2

2a2 r)(1 ea)

2

4a2(1 ea)2]

= eR(,r(t))

The yield R can be interpreted as a long-term rate; note that it does notdepend on the instantaneous spot rate r. This last property is considered asan 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 Proposition6.1.6, we can calculate the price of the option. It is easy to verify the hypothesisof Proposition 6.1.6 holds; and if the option is exercised, or P(, T)> K, thenwe have

r()< r

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where

r = R(1 a(T )1 ea(T) )

2

4a2(1 ea(T)) a log K

1 ea(T) .

C0= E[e

0 r(s)ds(P(, T) K)+]

= E(e0 r(s)dsP(, T)1r()


17/29

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 equation

3

:

dr(t) = (a br(t))dt +

r(t)dWt (17)

with and a non-negative, bR, and the process (q(t)) being equal to q(t) =

r(t), with 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 x0, there is a unique continuous, adapted process (Xt),taking values in R+, satisfying X0= x and

dXt = (a

bXt)dt + XtdWt on[0, [. (18)Proof: From the Holder property of the square root function, and the theoremsgiven in [14], it is easy to see the existence and uniqueness of the function

dXt= (a bXt)dt +

2Xt 0dWt (19)

with X0 = x 0. So the only work remains here is to proof the non-negativeproperty of the solution.1. If a = 0, equation (18) becomes dXt =bXtdt+

2Xt 0dWt. When

X0= 0,Xt0 is obviously a solution and based on the uniqueness of solution,we have Xt0 a.s.. When X00, set = inf{t; Xt = 0}, then Xt =Xt+ isa solution of (19) with X0= 0 on the space ( ={; ( 0, then for any >0, such that b+a >0, let = inf{t; Xt =}.Assume that P( 0. Then, with probability one, if we take anyr < such that X(t)< 0 ift(r, ), we have dXt = (a bXt)dt on thisinterval and hence t Xt is increasing on this interval, which is impossibleconsideringX00 and the definition of. This completes the proof.

The theorem tells us that in CIR model, the interest rate would never be-comes negative. However, would it be zero at some time point and if so, whatis the probability? The following proposition gives the answer.

Proposition6.2.41. Ifa2/2, we have P(x0 =) = 1, for all x >0.2. If 0a < 2/2 and b0, we have P(x0 0.3. If 0a < 2/2 and b


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Proof:

1. For x, M >0, let

x

M= inf{t0|Xx

t =M}, ands(x) =

x1

e2by/2

y2a/2

dy,x(0, ).

ThenxMis a stopping time, and d2sdx2 =e

2bx/2x2a/2 22 (b ax1), so we have

2

2 x

d2s

dx2+ (a bx) ds

dx = 0.

2. For 0< < x < M, set =x,M =x Mx, then for any t >0,

ds(Xt) = 2bXt

2 (Xt) 2a2

XtdWt,

or

s(Xxt) =s(x) + t0

s(Xxs)

Xxs dWs.

So,

E[s(Xxt) s(x)]2

=

t

[s(Xxt) s(x)]2dP(w) +


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4. When a a22

, it is obvious that

s() = 1

e2by/2

y2a/2

dy , when 0.

For fixed 0 < x < M, let 0 in the last equation in step 3, we see P(x0 0. Since forw, K, BK, such that BK > K, we haveBK , 0> BK a.s. That is P(x0 =) = 1.5.When 0a < 2

2 , s(0) = limx0 s(x) exists. So we have

s(x) =s(0)P(x0 < xM) + s(M)P(

x0 >

xM).

Ifb0, we have limx s(x) =, so P(0


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2. Define

F(t, x) = E(eXxt eut0 X xs ds). (20)

It is natural to look for F as a solution of the problem F

t = 2

2x

2Fx2 + (a bx)Fx uxF

F(0, x) = ex. (21)

Indeed, let F be one of the solutions for (21), and suppose it has boundedderivatives, which could be justified by the finally result. Set

Mt = eu t0 Xxs dsF(T t, Xxt),

then from

dMt = uXxteut0 X

xs dsF(T t, Xxt)dt eu

t0 X

xs ds

F(T t, Xxt)t

dt

+eut0 X

xs ds

F(T t, Xxt)x

dt[(a bXxt)dt +

XxtdWt]

+eut0X xs ds

2F(T t, Xxt)x2

2Xxt2

dt

= eut0 X

xs ds[uxF F

t + (a bx) F

x +

2x

2

2F

x2]|x=Xxtdt

+eut0 X

xs ds

F

x

XxtdWt

= eut0 X

xs ds

F

xXxtdWt,

we knowMt is a martingale, and therefore E(Mt) = M0, or

M0= F(T, Xx0 ) =E[e

u T0 Xxs dsF(0, XxT)] = E(e

XxTeut0X xs ds).

So Fin (20) is really a solution to (21).3. From step 1 and 2, we have

(t) = 22

2(t) + b(t) u

(t) =(t)

with (0) = 0, (0) = . The first equation above is a Raccati equation, and

b+b2+2u22 is one of its special solutions. Following the standard method, wecan find the expressions for and , which are the same as desired and thusthe proof is finished.

Now with above proposition, let u = 1= 0, we have bond price

P(0, T) = E

eT0 r(s)ds

= ea(T)r(0)(T) (22)

20


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where the function and are given by the following formula

(t) =22

log 2et(+b)/2

b+ et(+ b)

and

(t) = 2(e

t 1) b+ et(+ b)

withb = b + and =

(b)2 + 22. The price at time t is given by

P(t, T) = exp a(T t) r(t)(T t).

Again, if we apply it with u = 0, we obtain the Laplace transform ofXxt

E(eXxt ) =

b2

2(1 ebt) + b

2a/2exp

x bebt2

2(1 ebt) + b

=

1

2L + 1

2a/2exp

L

2+ 1

with L =

2

4b(1ebt) and = 4xb/(2(ebt1)). With these notations, the

Laplace transform ofXxt/L is given by the function g4a/2,, where is definedby

g,() = 1

(2 + 1)/2exp(

2 + 1).

By statistics knowledge, we see this is the Laplace transform of the non-central

chi-square law 5 with degrees of freedom and parameter .So, if we let

dP1dP

=e

0 r(s)dsP(, T)

P(0, T) , and

dP2dP

=e

0 r(s)ds

P(0, ) ,

by the Laplace transform of r()L1 (or r()L2

) under P1( or P2), where

L1=2

2

e 1

(e + 1) + (2(T ) + b)(e 1)and

L2=

2

2 e

1(e + 1) + b(e 1) ,

5The density of this law is given by f, = ezeta/2

2/41/2ex/2x/41/2I/21(

x), for x >

0, where Iv is the first-order modified Bessel function with index v, defined by Iv(x) =

(x2 )v

n=0(x/2)2n

n!(v+n+1) .

21


22/29

is a non-central chi-squared law with 4a2 degrees of freedom and parameter equal

to1( or 2), with

1= 8r(0)2e

2(e 1)[(e + 1) + (2(T ) + b)(e 1)]and

2= 8r(0)2e

2(e 1)[(e + 1) + b(e 1) .

Now we could price a European call option with maturity and exercisepriceK, on a zero-coupon bond with maturity T. We can show the hypothesisof Proposition 6.16. holds; if the call is exercised,

P(, T)> K

r()< r=

a(T ) + log K

(T ) ,

so the call price at time 0 is given by

C0 = E[e

0 r(s)ds(P(, T) K)+]

= E[e0 r(s)ds(ea(T)r()(T) K)+]

= E(e0 r(s)dsP(, T)1r(


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Furthermore, under those two special models, we are able to find out the ex-

plicit expression for the volatility of the bond price process as shown in equation(8). In Vasicek, q(t) =, b = b a ,

dP(t, s)

P(t, s) = { [A(t, s)t+1

22B(t, s)2 abB(t, s)]

+ [aB(t, s) B(t, s)t]r(t)}dt B(t, s)tdWt,

whereA(t, s)t= dA(t, s)/dt, and it can be checked that

aB(t, s) B(t, s)t = 1 ea(st) + ea(st) = 1,

A(t, s)t 12

2B(t, s)2 abB(t, s) = a

(ea(st)1) =B(t, s)q(t).

So, st = B(t, s), and st = B(t, s) . And in CIR, since q(t) =r(t), b = b + , = (b)2 + 22,dP(t, s)

dt /P(t, s) = { [bB(t, s) B(t, s)t+ 1

2B(t, s)22]r(t)

+ [A(t, s)t aB(t, s)]}dt B(t, s)

r(t)dWt.

So we just have to do some simple calculation to verify

bB(t, s) B(t, s)t+ 12

B(t, s)22 = 1 B(t, s),

A(s, t)t aB(s, t) = 0,and it is clear that st =B(t, s)r(t).74.3 Heath-Jarrow-Morton model

Some authors have resorted to a two-dimensional analysis to improve the modelsin terms of discrepancies between short and long rates, cf [4]. These morecomplex models do not lead to explicit formulae and require the solution ofpartial differential equations. More recently, Ho and Lee [13] have proposeda discrete-time model describing the behavior of the whole yield curve, whichis said to be the inception of local equilibrium method. It avoids the two-step procedure: inversion of the term structure and value of the contingentclaims, 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 andMorton [12]. Unlike the Ho and Lee model, however, they impose the exogenousstochastic structure upon forward rates, and not the zero coupon bond prices.They also think this change in perspective facilitates the mathematical analysisand it should also facilitate the empirical estimation of the model.

7Certainly, it can be seen that in CIR, the final expression of the volatility is not as simpleas in Vasicek for direct use.

23


24/29

4.3.1 HJM model and elementary properties

LetP(t, u) = exp(

ut

f(t, s)ds)

where (f(t, s)), t s, is an adapted process already defined as the forwardinterest rates. Moreover, it is natural to set f(t, t) = r(t) and constrain themap (t, s) f(t, s), defined for t s, to be continuous. Then the next stepof the modelling consists in assuming that, for each maturity u, the process(f(t, u))0tu satisfies an equation of the following form:

f(t, u) =f(0, u) +

t0

(v, u)dv+

t0

(f(v, u))dWv (23)

the process ((t, u))0t

u being adapted, the map (t, u)

(t, u) being con-

tinuous and being a continuous map from R into R( could depend on timeas well).

Then we have to make sure that this model is compatible with the hypothesis(H). This gives some conditions on the coefficients and of the model. To findthem, we derive the differential dP(t, u)/P(t, u) and we compare it to equation(8). Set Xt =

ut

f(t, s)ds, we have P(t, u) = eXt , from equation (23), wehave

Xt =

ut

f(s, s) + f(s, s) f(t, s)ds=

ut

f(s, s)ds +

ut

st

(v, s)dv

ds +

ut

st

(f(v, s))dWv

ds

= ut

f(s, s)ds + ut u

v(v, s)dsdv+ u

t u

v(f(v, s))dsdWv

= X0+

t0

f(s, s)ds t0

uv

(v, s)ds

dv t0

uv

(f(v, s))ds

dWv

The integrals in the above equation could commute would be justified by noting ut

N1i=0

(f(vi, s))1{su}(Wti+1 Wti)

ds

=N1i=0

ut

(f(vi, s)ds)

1{vt}(Wti+1 Wti)

and both sides converge under L2 to the desired result. We then have

dXt=

f(t, t) ut

(t, s)ds

dt ut

(f(t, s))ds

dWt

and by the Ito formula

dP(t, u)/P(t, u) = dXt+1

2d < X,X >t

24


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= f(t, t) u

t

(t, s)ds +1

2 u

t

(f(t, s))ds2

dt u

t

(f(t, s))dsdWtIf the hypothesis (H) holds, we must have, from proposition 6.1.3 and equalityf(t, t) =r(t),

utq(t) =

ut

(t, s)ds 12

ut

(f(t, s))ds

2withut =

ut (f(t, s))ds. So u

t

(t, s)ds=1

2

ut

(f(t, s))ds

2 q(t)

ut

(f(t, s))ds

and, differentiating with respect to u,

(t, u) = (f(t, u))

ut

(f(t, s))ds q(t)

.

Then equation (23) becomes

df(t, u) =(f(t, u))

ut

(f(t, s))ds

dt + (f(t, u))dWt. (24)

Heath, Jarrow and Morton have also proved the following fine theorem [12].

Theorem 6.2.6 If the function is Lipchitz and bounded,8 for any continuousfunction from [0, T] to R+, there exists a unique continuous process with

two indices (f(t, u))0tuT, such that for all u, the process (f(t, u))0tu isadapted and satisfies (24), with f(0, u) =(u).

We see that, for any continuous process (q(t)), it is then possible to build amodel of the form (23), that a solution of (24) and set

(t, u) = (f(t, u))

ut

(f(t, s))ds q(t)

.

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

4.3.2 Option pricing

We will price the option here only when is a constant.Firstly, it is easy to see the hypothesis of Proposition 6.1.6 holds, and

f(t, u) = f(0, u) + 2t(u t/2) + Wt8This boundedness condition on is essential since, for (x) =x, there is no solution [20].

25


26/29

is a solution to equation (24), since under this f, the left and right hand sides

are respectively

df(t, u) = 2(u t)dt + d Wt, and ( ut

ds) + d Wt = 2(u t)dt + d Wt.

So, from the definition ofP,

P(0, T)

P(0, ) exp

(T )W

2

T (T )2

= exp

T

f(0, s)ds (T )W2(T2 T + 2 2)

2

= exp

T

0

f(0, s)ds +

0

f(0, s)ds (T )W2T(T )

2 = exp

T

f(0, s)ds +2

2 (T2 2) +1

222(T ) W(T )

= exp

T

[f(0, s) + 2(s 2

) + W]ds

= e

T f(,s)ds =P(, T).

Secondly, using the properties of Winner process, we know the L2 lim of[2n1i=0 W

2 2n

] is ( 0

Wsds), (0

Wsds+ W) is the L2 lim of{

2n1i=0 [ 2n (2ni)][W 2n(i+1) W 2n i]} and the last term follows the Gaussian

distribution with mean 0 and variance22 + 2

3

3 . So from the knowledgeof Laplace transform of Gaussian, we know

E

e0 Wsdse

W

=1

2(2 2 +

23

3 ).

And then, if we let

dP1dP

=e

0 r(s)dsP(, T)

P(0, T) , and

dP2dP

= e

0 r(s)ds

P(0, ) ,

we have

E1(eW ) = E[eW0 r(s)ds

P(, T)

P(0, T)]

= E[eWe0[f(0,s)+

2s2

2 +Ws]ds+

0 f(0,s)ds(T)W

2T(T)2 ]

= E[e(+(T))We0 Wsds]e

23

6 2T(T)

2 =e22 (

2

2 T),

soW 22 +T

is a standard normal distribution under P1 and in the same

way, we knowW+

2

2

is a a standard normal distribution underP2.

26


27/29

Finally, if the option is exercised,

P(, T)> K W < W =2T(T)

2 + log P(0,)KP(0,T)

(T ) .

Let d = (W 2

2 +T)/

=

(T)2

logKP(0,)P(0,T)

(T) , we know the option

price is

C0 = E[e

0 r(s)ds(P(, T) K)+]

= E[e0 r(s)dsP(, T)1{W


28/29

References

[1] Ait-Sahakia, Y., Testing continuous-time model of the spot in-terest rate, Review Financial Studies12 (1996), 721-762.

[2] Ammann, M.,Credit Risk Valuation Methods, Models and Ap-plicatons, Springer.

[3] Artzner, P. and Delbaen, F., The martingale approach,Advancesin applied mathematics10 (1989), 95-129.

[4] Brennan,M.J. and E.S. Schwartz, The valuation of the Americanput option, Journal of Finance, 32, (1977), 449-462.

[5] Chapman, D.A., Pearson, N.D.: Is the short rate drift actuallynonlinear. 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 oftraditional 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 generalequilibrium model of asset prices, Econometrica, (1985).

[9] Cox, J.C., J.E. Ingersoll and S.A. Ross, A theory of the termstructure of interest rates, Econometrica, 53 (1985), 385-407.

[10] Delbaen, F. and Schachermayer, W., A general version of thefundamental theorem of asset pricing,Math. Ann. 300, 463-520,1994.

[11] D. Duffie, D. Filipovic, W. Schachermayer, Affine processes andapplications in finance,The Annals of applied probability, (2003),Vol 13,984-1053.

[12] Heath, D., A. Jarrow and A. Morton, Bond pricing and termstructure of interest rates, Econometrica, Vol. 60, No. 1 (1992),

77-105.

[13] Thomas. Y. Ho, Sang-bing Lee, Term structure movement andpricing interest rate contingent claims, Journal of finance, (1986).

[14] N. Ikeda, S. Watanabe, Stochastic differential equations anddiffusion processes, Chap 4, North-Holland, 1981.

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[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 cal-culus applied to finance, Chapman and Hall, 1996.

[18] Li Zenghu, Catalytic branching processes and affine interestrate models, preprint.

[19] Longzhen Fan, wei Hu, Financial engineering, Shanghai peoplepublishing house.

[20] Morton, A.J., Arbitrage and Martingales, Ph.D. thesis, CornellUniversity,1989.

[21] Shi Bingchao, et al., Theories and policies about interest rate.

[22] O.A. Vasicek, An equilibrium characterization of the termstructure, Journal of financial economics, 5 (November 1977),88-177.

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