25857 Interest Rate Modelling - WordPress.com · 2014-11-07 · InterestRateModelling Introduction Introduction The interest rate derivative models developed in Chapter 23 took as

Interest Rate Modelling

25857 Interest Rate Modelling

UTS Business SchoolUniversity of Technology Sydney

Chapter 25. The Heath-Jarrow-Morton FrameworkAugust 5, 2014

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Chapter 25. The Heath-Jarrow-Morton Framework1 Introduction2 The Basic Structure3 The Arbitrage Pricing of Bonds4 Arbitrage Pricing of Bond Options5 Forward Risk Adjusted Measure6 Reduction to Markovian Form7 Some Special Models

The Hull-White Extended Vasicek ModelThe General Spot Rate ModelThe Forward Rate Dependent Volatility ModelInterpreting the Subsidiary Variable (t)The Term Structure of Interest RatesPricing European Bond Options

8 Heath-Jarrow-Morton Multi–Factor Models9 Relating Heath-Jarrow-Morton to Hull-White Two-Factor Models10 The Covariance Structure Implied by the Heath-Jarrow-Morton Model

The Covariance Structure of the Forward Rate11 Appendix 2 / 121


Introduction

Introduction

The interest rate derivative models developed in Chapter 23took as their starting point the dynamics of the instantaneousspot rate of interest.

1 The market price of interest rate risk appears in the pricingrelationships.

2 It is possible to remove this dependence of the models onpreference related quantities by expressing terms involving themarket price of interest rate risk in terms of market observedquantities such as the currently observed yield curve andvolatilities of traded interest rate dependent instruments.

3 This can be tedious and for some model specifications may becomputationally intensive.

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Introduction

Introduction

An alternative interest rate modelling approach is theHeath-Jarrow-Morton approach.

1 Starts from the dynamics of the forward rate and requires thespecification of the initial term structure and the volatility ofthe associated forward rate.

2 The dynamics of the spot interest rate are then developedfrom those of the forward rate.

3 The spot interest rate is also an important economic variablewhose assessment determines the evolution of the bond prices.

The major shortcoming of the Heath-Jarrow-Morton approachis that the spot rate dynamics are not path independent (i.e.it is non-Markovian).

1 Increases computational complexity.

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Introduction

Introduction

The key unobserved input to this approach to term structuremodelling is the aforementioned volatility of the forward rates.

1 Many of the forms of the volatility functions reported in theliterature have been chosen for analytical convenience ratherthan on the basis of empirical evidence.

The non-Markovian feature also makes difficult the expressionfor prices of term-structure contingent claims in terms ofpartial differential equations.

1 In the Heath-Jarrow-Morton approach these prices areexpressed as expectation operators, under the equivalentmartingale measure, of appropriate payoffs.

2 It is important to be able to do so in order to apply to theevaluation of interest rate sensitive contingent claims manyuseful computational techniques.

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Introduction

Introduction

The notation used in the original Heath-Jarrow-Morton paperallows for a very general dependence of the forward ratevolatility functions on path dependent quantities.

1 We shall assume in this chapter that the path dependence ofthe forward rate volatility functions arises from dependence onthe instantaneous spot interest rate and/or a set of discretefixed-tenor forward rates.

2 With such a specification, the instantaneous spot rate processin the Heath-Jarrow-Morton framework can be expressed interms of a finite dimensional Markovian system.

3 The dimension of the resultant system of stochastic differentialequations is dependent on the exact form of the volatilityfunction.

4 Usually includes variables that at first sight seem not to bereadily observable.

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Introduction

Introduction

The transformation to the Markovian form also allows easiercomparison with other approaches.

1 This is important in the sense that it allows us to easilyreconcile many of the alternative approaches to the modellingof the term structure of interest rates.

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The Basic Structure

The Basic Structure

The starting point of the Heath-Jarrow-Morton model of theterm structure of interest rates is the stochastic integralequation for the forward rate (for 0 ≤ t ≤ T )

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

∫ t

0�(v, T, !(v))dv+

∫ t

0�(v, T, !(v))dW (v),

(1)

1 where �(v, T, !(v)) and �(v, T, !(v)) are the instantaneousdrift and the volatility function at time v of the forward ratef(v, T ).

2 The instantaneous drift �(v, T, !(v)) and volatility function�(v, T, !(v)) of the forward rate f(v, T ) could depend through! on path dependent quantities, such as the instantaneousspot rate and/or a set of discrete tenor forward rates.

3 The noise term dW (v) is the increment of a standard Wienerprocess generated by a probability measure ℙ.

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The Basic Structure

The Basic Structure

The s.i.e. (1), may alternatively be expressed as the s.d.e.

df(t, T ) = �(t, T, !(t))dt + �(t, T, !(t))dW (t). (2)

It follows from equation (1) that the instantaneous spot rater(t)(≡ f(t, t)) is given by

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

∫ t

0�(v, t, !(v))dv +

∫ t

0�(v, t, !(v))dW (v).

(3)

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The Basic Structure

The Basic Structure

The corresponding s.d.e for r(t) (see equation (22.39)) is

dr = �r(t)dt+ �(t, t, !(t))dW (t), (4)

where

�r(t) = f2(0, t)+�(t, t, !(t))+

∫ t

0

�2(v, t, !(v))dv+

∫ t

0

�2(v, t, !(v))dW (v).

(5)

We recall that the bond price at time t is related to theforward rate by

P (t, T ) = exp

(−

∫ T

t

f(t, s)ds

), 0 ≤ t ≤ T. (6)

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The Basic Structure

The Basic Structure

By the use of Fubini’s theorem for stochastic integrals andapplication of Ito’s lemma (see Section 22.6) the bond pricesatisfies

dP (t, T ) = [r(t) + b(t, T )]P (t, T )dt+ �B(t, T )P (t, T )dW (t), (7)

where

�B(t, T ) ≡ −

∫ T

t

�(t, s, !(t))ds, (8)

and

b(t, T ) ≡ −

∫ T

t

�(t, s, !(t))ds +1

2�2B(t, T ). (9)

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The Basic Structure

The Basic Structure

A quantity of interest is the money market account

A(t) = exp

(∫ t

0r(y)dy

), (10)

This quantity may be used to define the relative bond price

Z(t, T ) =P (t, T )

A(t), (0 ≤ t ≤ T ). (11)

The fact that dA = r(t)A(t)dt and application of the rule forthe quotient of two diffusions (see Section 6.6) yields theresult that the relative bond price satisfies

dZ(t, T ) = b(t, T )Z(t, T )dt+ �B(t, T )Z(t, T )dW (t). (12)

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The Arbitrage Pricing of Bonds


Bonds can be priced using exactly the same hedging portfolioas was used in Chapter 23

1 The instantaneous excess bond return, risk adjusted by itsvolatility must equal the market price of interest rate risk; seeequation (23.9).

The relevant bond dynamics in the current context are givenby (7), so that (23.9) becomes

[r(t) + b(t, T )]− r(t)

�B(t, T )=

market price of

interest rate risk≡ −�(t), (13)

which simplifies to

b(t, T ) + �(t)�B(t, T ) = 0. (14)

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Using expressions (8) and (9) this last equation may bewritten explicitly as

∫ T

t

�(t, s, !(t))ds−1

2

(∫ T

t

�(t, s, !(t))ds

)2

+�(t)

∫ T

t

�(t, s, !(t))ds = 0.

Keeping t fixed and differentiating with respect to maturity T ,the above equation reduces to

�(t, T, !(t))−

(∫T

t

�(t, s, !(t))ds

)�(t, T, !(t))+�(t)�(t, T, !(t)) = 0,

which may be rearranged to

�(t, T, !(t)) = −�(t, T, !(t))

[�(t)−

∫ T

t

�(t, s, !(t))ds

].

(15)

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Equation (15) is the forward rate drift restriction that wasfirst reported by Heath-Jarrow-Morton

1 If the bond market is free of riskless arbitrage opportunitiesthen the forward rate drift, the forward rate volatility and themarket price of interest rate risk must be tied together asshown by this equation

2 Heath-Jarrow-Morton show that in fact this condition is bothnecessary and sufficient for the absence of riskless arbitrageopportunities

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Up to this point Heath-Jarrow-Morton have not doneanything conceptually different from the standard arbitrageapproach of Chapter 23

1 In the Heath-Jarrow-Morton approach equation (14) is used ina different way.

In the standard arbitrage approach, equation (14) becomes apartial differential equation for the bond price as a function ofthe assumed driving state variable (usually the instantaneousspot rate).

In the Heath-Jarrow-Morton approach, the condition (14)becomes the forward rate drift restriction that is used toconveniently express the bond price dynamics under anequivalent probability measure.

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By use of (14), the stochastic differential equations (7) and(12) for P (t, T ) and Z(t, T ) respectively become

dP (t, T ) = [r(t)− �(t)�B(t, T )]P (t, T )dt + �B(t, T )P (t, T )dW (t),(16)

dZ(t, T ) = −�(t)�B(t, T )Z(t, T )dt + �B(t, T )Z(t, T )dW (t).(17)

At the same time, by substituting (15) into (3) the stochasticintegral equation for r(t) becomes

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

∫ t

0�(v, t, !(v))

∫ t

v

�(v, s, !(v))dsdv

−

∫ t

0�(v, t, !(v))�(v)dv +

∫ t

0�(v, t, !(v))dW (v).

(18)17 / 121




The key advance in the Heath-Jarrow-Morton approach isthat equations (16)–(18), can be written in terms of adifferent Wiener process.We define a new Wiener process W (t) under ℙ by

W (t) =W (t)−

∫ t

0�(s)ds, or (19)

dW (t) = dW (t)− �(t)dt. (20)

Then equations (16)–(18) become

dP (t, T ) = r(t)P (t, T )dt+ �B(t, T )P (t, T )dW (t), (21)

dZ(t, T ) = �B(t, T )Z(t, T )dW (t), (22)

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

∫ t

0

�(v, t, !(v))

∫ t

v

�(v, s, !(v))dsdv+

∫ t

0

�(v, t, !(v))dW (v).

(23)18 / 121




Alternatively, (23) can be expressed as the s.d.e.

dr =

[f2(0, t) +

∂

∂t

(∫ t

0�(v, t, !(v))

∫ t

v

�(v, s, !(v))dsdv

)

+

∫ t

0

∂�(v, t, !(v))

∂tdW (v)

]dt+ �(t, t, !(t))dW (t).

(24)

1 It is at times convenient to deal with the ln of the bond priceB(t, T ) ≡ lnP (t, T )

2 This quantity, by Ito’s lemma, satisfies (under ℙ)

dB(t, T ) = [r(t) −1

2�2B(t, T )]dt+ �B(t, T )dW (t). (25)

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The arbitrage free s.i.e. for the forward rate under ℙ can bewritten,

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

∫ t

0

�(v, T, !(v))

∫ T

v

�(v, s, !(v))dsdv

+

∫ t

0

�(v, T, !(v))dW (v), (26)

and the corresponding s.d.e. as

df(t, T ) = �(t, T, !(t))

∫ T

t

�(t, s, !(t))dsdt+�(t, T, !(t))dW (t).

(27)

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The essential characteristic of the reformulated stochasticdifferential and integral equations is that the empiricallyawkward market price of risk term, �(t), is eliminated fromexplicit consideration

1 From Girsanov’s theorem we obtain the expression for theRadon-Nikodym derivative

dℙ

dℙ= exp

(−1

2

∫ t

0

�2(s)ds+

∫ t

0

�(s)dW (s)

). (28)

2 If we write Et to denote mathematical expectation withrespect to the equivalent probability measure (i.e. the one

associated with dW (t)) then from equation (22)

Et[dZ(t, T )] = 0.

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This last equation implies that Z(t, T ) is a martingale underℙ, i.e.

Z(t, T ) = Et(Z(T, T )),

or, in terms of the bond price

P (t, T ) = Et

[A(t)

A(T )

]= Et

[exp

(−

∫ T

t

r(y)dy

)]. (29)

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Equation (29) is the fundamental bond pricing equation of theHeath-Jarrow-Morton framework

1 The quantity exp(−∫T

tr(y)dy

)should be interpreted as the

stochastic discount factor under ℙ2 The actual implementation of (29) will depend on the form

chosen from the forward rate volatility function3 Closed form analytical expressions for the bond price may be

obtained with appropriate assumptions on the volatilityfunction

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Arbitrage Pricing of Bond Options


Suppose we wish to price at time t an option on the bond, forexample a European call option on the bond, with the optionmaturing at Tc.

1 Under the risk-neutral measure ℙ we can discount the payoff atTc back to t using the stochastic discount factor

exp

(−

∫ Tc

t

r(s)ds

). (30)

Multiplying the payoff by the discount factor we find thatunder one realisation of the spot-rate process under ℙ theoption value at t is given by

exp

(−

∫ Tc

t

r(s)ds

)max [P (Tc, T )−X, 0] . (31)

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The value of the option, C(t, Tc), is then obtained by takingthe expectation of this quantity under the risk-neutralmeasure ℙ (i.e. forming Et)

C(t, Tc) = Et

[exp

(−

∫ Tc

t

r(s)ds

)max[P (Tc, T )−X, 0]

].

(32)

In general, if we have some spot interest rate contingent claimwith payoff at t = Tc given by H(r(Tc), Tc) then its value at tis given by

U(t, Tc) = Et

[exp

(−

∫ Tc

t

r(s)ds

)H(r(Tc), Tc)

]. (33)

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In order to obtain a pricing partial differential equation forU(t, Tc) we need to obtain the Kolmogorov backwardequation associated with equation (24).

1 It is difficult to do this in general because of the non-Markovian

term

∫t

0

∂�

∂tdW (v) that appears in the drift in equation (24).

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Why not mirror the argument of Section 23.6 and use thehedging argument approach to derive the bond option pricingformula?

1 In Section 23.6 there was one underlying factor, r(t), drivingthe uncertainty of the market.

2 In Section 24.1 there were two underlying factors, r(t) andℎ(t), driving the uncertainty of the market and we would writethe option price as C(r, ℎ, t).

3 Application of Ito’s Lemma gave us the dynamics. In bothcases the dynamics of the hedging portfolio could be obtained.

Here we have not so far been so precise about the factorsupon which the volatility function �(t, T, ⋅) depends.

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In order to mirror the hedging argument approach we need tobe more specific about the dynamics of these underlying ratesso that we could then obtain the option price dynamics byapplying Ito’s lemma.

The expressions (29) for the bond price and (33) for theinterest rate derivative hold for quite general specifications ofthe volatility function.

If we want to implement these expressions, using for examplestochastic simulation, then we would need to specify thedynamics (under ℙ) of all stochastic factors entering into thespecification of �(t, T, ⋅).

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Forward Risk Adjusted Measure


We saw in equation (33) that the value of a spot interest ratecontingent claim at time t can be written

U(t, Tc) = Et

[exp

(−

∫ Tc

t

r(s)ds

)H(r(Tc), Tc)

], (34)

1 where H(r(Tc), Tc) denotes the payoff on the claim at time Tc.

Suppose P (t, Tc) represents the price at time t of a purediscount bond maturing at time Tc

1 Then by equation (29),

P (t, Tc) = Et

[exp

(−

∫Tc

t

r(s)ds

)]. (35)

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We can use the results of Chapter 20 to express the value ofthe interest rate contingent claim, using P (t, Tc) as thenumeraire

By forming the quantity Y = U/P we would obtain

U(t, Tc) = P (t, Tc)E∗

t

[U(Tc, Tc)

P (Tc, Tc)

]. (36)

But in the current notation U(Tc, Tc) = H(r(Tc), Tc) andP (Tc, Tc) = 1, hence

U(t, Tc) = P (t, Tc)E∗

t [H(r(Tc), Tc)] . (37)

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The advantage of (37) over (34) is that the stochastic

discounting term exp(−∫ Tctr(s)ds

)is replaced by the

non-stochastic discounting term P (t, Tc).

1 The value of this method depends on how easy (or difficult) itis to calculate E

∗

t in (37).2 The measure associated with the E

∗

toperation, which we shall

denote as ℙ∗ is known as the T –forward measure.3 Under ℙ∗ the forward rate at time t is the expectation of the

instantaneous spot rate at T i.e.

f(t, T ) = E∗

t [r(T )]. (38)

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To see this result recall that

P (t, T ) = Et

[exp

(−

∫ T

t

r(s)ds

)].

Differentiating this last equation with respect to T we obtain

∂

∂TP (t, T ) = Et

[∂

∂Texp

(−

∫ T

t

r(s)ds

)]

= Et

[− exp

(−

∫ T

t

r(s)ds

)⋅∂

∂T

∫ T

t

r(s)ds

]

= −Et

[exp

(−

∫ T

t

r(s)ds

)⋅ r(T )

]

= −P (t, T )E∗

t [r(T )]. (39)

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The last line was obtained by applying (37) withH(r(T ), T ) = r(T ).Rearranging the last result we obtain

−∂

∂TlnP (t, T ) = E

∗

t [r(T )]. (40)

However f(t, T ) = − ∂∂T

lnP (t, T ) hence we have establishedthe result in equation (38).For some later applications we need to clarify what form theRadon-Nikodym derivative assumes for the forwardrisk-adjusted measure. We simply identify VT with P (T, T )and V0 with P (0, T ) in equation (20.20) and theRadon-Nikodym derivative becomes

�(0, T ) =P (T, T )

A(T )P (0, T )=

1

A(T )P (0, T ). (41)

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Reduction to Markovian Form


The difficulty in implementing and estimatingHeath-Jarrow-Morton models arises from the non-Markoviannoise term in the stochastic integral equation (23) for r(t).

1 This manifests itself in the third component of the drift termof the stochastic differential equation (24)

2 This component depends on the history of the noise processfrom time 0 to current time t

Depending upon the specification of the volatility function thesecond component of the drift term could also depend on thepath history up to time t.

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In this section we consider a class of functional forms of�(t, T, ⋅) that allow the non-Markovian representation of r(t)and P (t, T ) to be reduced to a finite dimensional Markoviansystem of stochastic differential equations

1 We investigate volatility functions of the forward rate whichhave the general form of

�(t, T, !(t)) = Q(t, T )G(!(t)), 0 ≤ t ≤ T, (42)

Where G is an appropriately well-behaved function

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A useful representation for Q(t, T ) would be

Q(t, T ) = Pn(T − t)e−�(T−t), (43)

Where Pn(u) is the polynomial

Pn(u) = a0 + a1u+ . . .+ anun.

This form would allow the term structure of the volatility toexhibit humps as observed in implied forward rate volatilitiesfrom cap prices.

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Recall the discussion of Section 22.7 when we consideredforward rate volatility functions of the form

�(t, T, !(t)) = �e−�(T−t)G(!(t)), (44)

for � > 0 and � constant

This structure includes forward rate volatilities for a numberof important cases in the literature, including

∙ � > 0, G(!(t)) = 1 leads to a version of the extended Vasicekmodel of Hull-White,

∙ � > 0, G(!(t)) = g(r(t)) leads to the generalised spot ratemodel of Ritchken and Sankarasubramanian (995a).

∙ � > 0, G(!(t)) =√r(t) leads to an extended version of the

CIR model,∙ � > 0, G(!(t)) = g(r(t), f(t, �)) leads to a version of the

model of Chiarella and Kwon (999b).

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Our aim is to express the equations (24) and (27) as aMarkovian system of stochastic differential equations

By considering the drift term of the stochastic differentialequation (27), under the volatility specification of (44), weobtain

�(t, T, !(t))

∫ T

t

�(t, s, !(t))ds

= �2G2(!(t))e−�(T−t)

∫ T

t

e−�(s−t)ds

= �2G2(!(t))e−�(T−t)

(e−�(T−t) − 1

)

−�

= �2(t, T, !(t))

(e�(T−t) − 1

)

�.

38 / 121




Thus, we express the s.d.e. (27) for the forward rate as

df(t, T ) =

[�2(t, T, !(t))

e�(T−t) − 1

�

]dt+�(t, T, !(t))dW (t).

(45)

39 / 121




We also consider the s.d.e. (24) for r(t) and in particular thefirst integral term

∂

∂t

(∫ t

0�(v, t, !(v))

∫ t

v

�(v, s, !(v))dsdv

)

=

∫ t

0

[�2(v, t, !(v))

∫ t

v

�(v, s, !(v))ds + �2(v, t, !(v))

]dv,

(46)

where �2 represents the partial derivative of � with respect toits second argument.

40 / 121




Given the expression for � in (44), we have that,

�2(v, t, !(v)) = −��(v, t, !(v)). (47)

Thus, the right hand side of (46) reduces to

∫t

0

[−��(v, t, !(v))

∫t

v

�(v, s, !(v))ds + �2(v, t, !(v))

]dv. (48)

By using (47) the other two terms of (24) can be expressed as

[∫ t

0

�2(v, t, !(v))dW (v)

]dt+ �(t, t, !(t))dW (t)

=

[−�

∫t

0

�(v, t, !(v))dW (v)

]dt+ �(t, t, !(t))dW (t).

41 / 121




Thus, the s.d.e. (24) becomes

dr =

[f2(0, t)− �

∫ t

0

�(v, t, !(v))

∫ t

v

�(v, s, !(v))dsdv

+

∫t

0

�2(v, t, !(v))dv − �

∫t

0

�(v, t, !(v))dW (v)

]dt

+ �(t, t, !(t))dW (t).

(49)

We note from the s.i.e. (23) that

r(t) − f(0, t) =

∫t

0

�(v, t, !(v))

∫t

v

�(v, s, !(v))dsdv

+

∫t

0

�(v, t, !(v))dW (v).

(50)

42 / 121




Then by using (50), the stochastic differential equation (49)for the spot rate is simplified to

dr = [f2(0, t) + �f(0, t) + (t)− �r(t)] dt+�(t, t, !(t))dW (t),(51)

where we define the subsidiary variable (t) as

(t) =

∫ t

0�2(v, t, !(v))dv. (52)

(t) plays a central role in allowing us to transform the originalnon-Markovian dynamics to Markovian form.Similar subsidiary variables appear in the reduction to Markovianforms of Cheyette (1992), Bhar and Chiarella (997a),Inui and Kijima (1998) and Chiarella and Kwon (999b). (t) may be interpreted as a variable summarising the path historyof the forward rate volatility.

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Some Special Models

Some Special Models

At this point the stochastic differential equation (51) is stillnon-Markovian because the integral in the drift term involvesthe history of the path dependent forward rate volatility.

To proceed any further we need to consider specific functionalforms for G(!(t)) in the volatility specifications (44).

44 / 121


Some Special Models

The Hull-White Extended Vasicek Model


If G(!(t)) = 1 then the subsidiary variable (52) becomes atime function, i.e.

(t) =

∫ t

0�2(v, t, !(v))dv =

∫ t

0�2e−2�(t−v)dv =

�2

2�(1−e−2�t).

Setting

�(t) = f2(0, t) + �f(0, t) +�2

2�(1− e−2�t), (53)

the s.d.e. (51) for r(t) becomes

dr = [�(t)− �r(t)]dt+ �dW (t), (54)

which is a sought Markovian representation.

45 / 121


Some Special Models



Clearly this is the extended Vasicek model with the long runmean allowed to be time varying.

Furthermore, note that the expression we have obtained for�(t) is the same as the one we obtained in Section 23.7 whenwe worked directly from the expression for the bond priceobtained from the continuous arbitrage approach - whichtakes the spot rate process as the driving dynamics.

It is also worth pointing out that by setting � = 0 we obtainthe continuous time specification of the Ho-Lee model.

46 / 121


Some Special Models



We have already seen how to price European options in thisframework in Section 23.7.

To price American options in this framework, the optionpricing equation

1

2�2∂2C

∂r2+ [�(t)− �r]

∂C

∂r+∂C

∂t− rC = 0, (55)

must be solved subject to the boundary conditions for anAmerican option.

47 / 121


Some Special Models

The General Spot Rate Model


If G(!(t)) is a function of the spot interest rate r(t), i.e.G(!(t)) = g(r(t)) we need to separately handle thenon-Markovian term appearing in the drift of equation (51)

The subsidiary variable (52) now becomes

(t) =

∫ t

0�2e−2�(t−v)g2(r(v))dv. (56)

By differentiating (56) we have that

d = [�2g2(r(t))− 2� (t)]dt. (57)

48 / 121


Some Special Models



We are now dealing with the two-dimensional Markoviansystem

dr = [f2(0, t) + �f(0, t) + (t) − �r(t)]dt+ � g(r(t))dW (t),

d = [�2g2(r(t))− 2� (t)]dt.

(58)

The representation (58) was obtained byRitchken and Sankarasubramanian (995a).

49 / 121


Some Special Models


If we define the partial differential operator K by

K =1

2�2g2(r)

∂2

∂r2+[f2(0, t)+�f(0, t)+ −�r]

∂

∂r+[�2g2(r)−2� ]

∂

∂

then the Kolmogorov equation for the transition probabilitydensity � is

K� +∂�

∂t= 0.

50 / 121


Some Special Models



Derivative instruments are priced according to the partialdifferential equation

KV +∂V

∂t− rV = 0,

(note V = P for bond price, V = C for option price,) subjectto appropriate boundary conditions, e.g. V (r, T, T ) = 1, forbonds, V (r, Tc, T ) = max [0, P (r, Tc, T )−K], for Europeancall options, etc.

51 / 121


Some Special Models



To evaluate American options we need to employ numericalmethods

1 Chiarella and El-Hassan (1998) have found the method of linesto be very effective in this context

Note that in the special case of g(r(t)) =√r(t), we are

dealing with the extended CIR model

dr = [f2(0, t) + �f(0, t) + (t)− �r(t)]dt+ �√r(t)dW (t),

d = [�2r(t)− 2� (t)]dt.

(59)

52 / 121


Some Special Models

The Forward Rate Dependent Volatility Model


We further generalise the form of the volatility function toinclude forward interest rates

1 G(!(t)) can be a function of the spot interest rate, r(t), andof the forward interest rate, f(t, �) of a fixed maturity � , sothat G(!(t)) = g(r(t), f(t, �)).

2 For example, f(t, �) could be some long forward rate

The intuition behind such a specification is that not only thespot interest rate but also a fixed maturity forward interestrate influence the evolution of the term structure

53 / 121


Some Special Models



The particular forward rate to be used may depend on theapplication under consideration

1 This approach may be considered to be equivalent in somesense to the Brennan and Schwartz (1979) model where ashort rate and a long rate are used to explain the evolution ofthe term structure

2 We need to determine the additional state variables necessaryto make the system Markovian although with a higherdimension

54 / 121


Some Special Models



The associated subsidiary variable (52) is given by

(t) =

∫ t

0�2e−2�(t−v)g2(r(v), f(v, �))dv. (60)

By differentiating (60), we obtain the s.d.e. for (t) as

d = [�2(t, t, !(t)) − 2� (t)]dt

= [�2g2(r(t), f(t, �)) − 2� (t)]dt (61)

55 / 121


Some Special Models



We have now reduced the non-Markovian stochastic dynamicsto a three dimensional Markovian stochastic dynamical systemconsisting of

the s.d.e. for the spot rate r(t) (recall (51))

dr = [f2(0, t) + �f(0, t) + (t)− �r(t)] dt+�g(r(t), f(t, � ))dW (t),(62)

the s.d.e. for the discrete forward rate f(t, � ) (recall (45)),

df(t, � ) = �2(t, �, !(t))

(e�(�−t) − 1)

�dt+ �(t, �, !(t))dW (t),

= �2g2(r(t), f(t, � ))e−2�(�−t) (e

�(�−t)− 1)

�dt

+ �e−�(�−t)

g(r(t), f(t, � ))dW (t), (63)

and the s.d.e (61) for (t).

56 / 121


Some Special Models



Finally we recall that the dynamics of the forward rate to anymaturity T , f(t, T ) is given by (45) and so are determinedonce r(t) and f(t, �) are determined.

1 These latter quantities are driven by the three stochasticdifferential equations (61), (62) and (63) which together formthe Markovian representation.

2 The price of any derivative instrument would then have todepend on r(t) and f(t, �).

3 Thus a bond of maturity T would have a price at time tdenoted by P (t, T, r(t), f(t, �)), and this price is also driven bythe three-dimensional Markovian stochastic differentialequation system referred to above.

57 / 121


Some Special Models

Interpreting the Subsidiary Variable (t)

Interpreting the Subsidiary Variable

It would perhaps be more satisfying to relate (t) to themarket rates r(t) and f(t, �).

1 Indeed it turns out that such a relationship does exist for theforward rate volatility function assumed in equation (44) forthe generalised case of G(!(t)) = g(r(t), f(t, �)).

58 / 121


Some Special Models



Proposition 1The subsidiary integrated square variance quantity (t)defined in equation (60) is related to the rates r(t) andf(t, �) via

(t) = �� (t, � ) [r (t)− f (0, t)]−�e−�(t−� )� (t, � ) [f (t, �)− f (0, �)](64)

where �(t, �) ≡e−�t

e−�� − e−�t.

For the proof of the Proposition see Appendix 25.1.

59 / 121


Some Special Models



An important consequence of Proposition 1 is that it allows usto reduce by one the dimension of the stochastic dynamicsystem (61), (62) and (63) to the two-dimensional oneconsisting of the s.d.e.s (62) and (63) with (t) being definedby (64).

This reduction in dimension is quite significant if we seek to solvefor derivative prices in this framework by use of p.d.e.s or latticebased methods since then we need only deal with two rather thanthree spatial variables in the partial differential operator.

The reduction is less significant, though still useful, whenusing Monte-Carlo simulation

Since Monte Carlo simulation requires the simulation of the oneWiener increment, dW (t).The generation of (t) by equation (64) rather than discretisingequation (61) should lead to some computational efficiency.

60 / 121


Some Special Models



A consequence of Proposition 1 is that we are able to expressthe forward rate to any maturity T in terms of the two ratesr(t) and f(t, �).

Proposition 2The forward rate f(t, T ) to any maturity T is given by

f(t, T )− f(0, T ) =

− e−2�(T−�) �(�, t)

�(T, t)[f(t, � )− f(0, � )] + e

−2�(T−t) �(t, � )

�(T, � )[r(t)− f(0, t)] .

(65)

For the proof of Proposition 2 see Appendix 25.2.

61 / 121


Some Special Models

The Term Structure of Interest Rates


We recall the Heath-Jarrow-Morton approach of defining amoney market account (10) and showing that the relativebond price

Z(t, T ) =P (t, T )

A(t),

is a martingale.

So that the bond price can be written

P (t, T ) = Et

[A(t)

A(T )

]= Et

[exp

(−

∫ T

t

r(y)dy

)]. (66)

62 / 121


Some Special Models



Here, Et is the expectation taken with respect to theprobability distribution generated by the stochastic differentialsystem (62) and (63).

We use� (r(t∗), f(t∗, �) ∣ r(t), f(t, �)) ,

to denote the transition probability density function between tand t∗ (t ≤ t∗).

This quantity satisfies the Kolmogorov backward p.d.e,

K� +∂�

∂t= 0,

whereK is the infinitesimal generator of the diffusion process for f(t, � )r(t) driven by the stochastic differential equations (62) and (63).

63 / 121


Some Special Models



It turns out that K is given by (see Appendix 25.3),

K� ≡ �21

(e�(�−t) − 1

)

�

∂�

∂f+ [f2(0, t) + �f(0, t) + − �r]

∂�

∂r

+1

2�21∂2�

∂f2+

1

2�2r∂2�

∂r2+ �1�r

∂2�

∂f∂r,

(67)

where �1(t) = �(t, �, !(t)) and �r(t) = �(t, t, !(t))

64 / 121


Some Special Models



By application of the Feynman-Kac formula to (66) we findthat the bond price P (t, T, r, f) satisfies the p.d.e.

∂P

∂t+KP − rP = 0, (68)

subject to the terminal condition P (T, T, r, f) = 1, and theboundary conditions

P (t, T,∞, f) = 0, (f ≥ 0),

P (t, T, r,∞) = 0, (r ≥ 0).

The further boundary conditions P (t, T, 0, f) and P (t, T, r, 0)may be obtained by an extrapolation procedure. Note that insubsequent discussion we set

D(t) ≡ f2(0, t) + �f(0, t).

65 / 121


Some Special Models



A consequence of Proposition 2 is that it turns out to bepossible to obtain an analytical expression for the bond price.

Proposition 3The price of bonds driven by the Markovian stochasticdifferential equation system (62) and (63) can be expressed as

P (t, T, r, f) =P (0, T )

P (0, t)exp

[−�(t, T ) (r(t)− f(0, t))−

1

2�2(t, T ) (t)

].

(69)

where �(t, T ) =1

�

(1− e−�(T−t)

), and (t) is defined in

equation (64).

For the proof of Proposition 3 see Appendix 25.3.

66 / 121


Some Special Models



The bond pricing equation (69) has precisely the same form asthe one derived by (995a).

1 Assumed a form for the volatility function in equation (42)with G(!(t)) = g(r(t)) which is independent of the forwardrate f(t, �)

In fact the results in Proposition 2 and Proposition 3 can beconsiderably generalised.

(69) holds in precisely the same form even when the forwardrate volatility depends on a set of discrete forward ratesf(t, �1), f(t, �2), . . . , f(t, �r) wheret ≤ �1 < �2 < ⋅ ⋅ ⋅ < �r ≤ T .

Under these different specifications the history variable (t)will evolve differently but the functional relationship remainsthe same.

67 / 121


Some Special Models

Pricing European Bond Options


Consider an option written on the bond of maturity T . Wesuppose the option matures at time Tc(< T ) and its priceC(t, T, r, f) satisfies the p.d.e.

∂C

∂t+KC − rC = 0, (0 ≤ t ≤ Tc). (70)

If we are dealing with a European call option with strike priceE then the terminal condition for (70) is

C(Tc, T, r, f) = [P (Tc, T, r, f) −E]+ .

The boundary conditions at infinity are

C(t, T,∞, f) = 0, f ≥ 0,

C(t, T, r,∞) = 0, r ≥ 0.

68 / 121


Some Special Models



We recall that the bond prices at option maturity for anygiven values of r(Tc), f(Tc, �) can be obtained directly fromequation (69) without the need to solve the bond pricingp.d.e. (68)

An alternative approach to pricing the European option is touse the result (also derived by Heath-Jarrow-Morton) that

C(t, T, r, f) = Et

[exp

(−

∫Tc

t

r(y)dy

)[P (Tc, T, r(Tc), f(Tc, �)) − E]

+

]

(71)

The expectation in equation (71) could be approximated bysimulating an appropriate number of times the stochasticdifferential equation system (62) and (63) from t to Tc.

69 / 121


Heath-Jarrow-Morton Multi–Factor Models


We focussed on the case where only one noise factor wasimpinging on the forward rate curve.The Heath-Jarrow-Morton framework allows for the possibilityof multiple noise sources, i.e. the general situation is

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

∫ t

0�(v, T, ⋅)dv+

n∑

i=1

∫ t

0�i(v, T, ⋅)dWi(v),

(72)where the n noise terms dWi are the increments ofindependent Wiener processes and the �i(t, T, ⋅) are thevolatility functions associated with each noise term.Setting T = t in (72) we have

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

∫ t

0�(v, t, ⋅)dv +

n∑

i=1

∫ t

0�i(v, t, ⋅)dWi(v).

(73)70 / 121




Substituting (72) into (6) and following the same procedurethat yielded (7) we find that the s.d.e. for the bond pricebecomes

dP (t, T ) = [r(t)+b(t, T, ⋅)]P (t, T )dt+

n∑

i=1

ai(t, T, ⋅)P (t, T )dWi(t),

(74)where

ai(t, T, ⋅) = −

∫ T

t

�i(t, v, ⋅)dv, (75)

and

b(t, T, ⋅) = −

∫ T

t

�(t, v, ⋅)dv +1

2

n∑

i=1

a2i (t, T, ⋅). (76)

71 / 121




The process for the relative bond price Z(t, T ) = P (t,T )A(t) , is

easily found to be

dZ(t, T ) = b(t, T, ⋅)Z(t, T )dt +n∑

i=1

ai(t, T, ⋅)Z(t, T )dWi(t).

(77)

By forming a portfolio of bonds of (n+ 1) different maturitiesand holding these in proportions that ensure the existence ofno riskless arbitrage opportunities results in the condition

[r(t) + b(t, T, ⋅)]− r(t) = −

n∑

i=1

�i(t)ai(t, T, ⋅), (78)

where �i(t) is the market price of risk associated with the itℎ

noise factor.72 / 121




Equation (78) simplifies to

b(t, T, ⋅) +n∑

i=1

�i(t)ai(t, T, ⋅) = 0, (79)

which is the multifactor analogue of the forward rate driftrestriction (14).

By use of (75) equation (79) reads

∫ T

t

�(t, v, ⋅)dv =

n∑

i=1

[1

2a2i (t, T, ⋅) + �i(t)ai(t, T, ⋅)

]. (80)

73 / 121




Differentiating this last equation with respect to maturity Twe find that

�(t, T, ⋅) = −

n∑

i=1

�i(t, T, ⋅)

[�i(t)−

∫ T

t

�i(t, v, ⋅)dv

], (81)

which is the forward rate drift restriction in the multi-factorcase.

74 / 121




Substituting (81) into (73), (74) and (77) the s.d.e. for r(t),P (t, T ) and Z(t, T ) become respectively

dr =

[

f2(0, t) +∂

∂t

n∑

i=1

∫

t

0

�i(v, t, ⋅)

∫

t

v

�i(v, y, ⋅)dydv +n∑

i=1

∫

t

0

∂�i

∂t(v, t, ⋅)dWi(v)

−

n∑

i=1

�i(t)�i(t, t, ⋅) −

n∑

i=1

∫

t

0

�i(v)∂�i

∂t(v, t, ⋅)dv

]

dt +n∑

i=1

�i(t, t, ⋅)dWi(t)

dP (t, T ) =

[

r(t) −

n∑

i=1

�i(t)ai(t, T, ⋅)

]

P (t, T )dt +n∑

i=1

ai(t, T, ⋅)P (t, T )dWi(t),

dZ(t, T ) = −

n∑

i=1

�i(t)ai(t, T, ⋅)Z(t, T )dt +n∑

i=1

ai(t, T, ⋅)Z(t, T )dWi(t).

75 / 121




We then form the new set of processes

Wi(t) =Wi(t)−

∫ t

0�i(s)ds, (i = 1, 2, ⋅ ⋅ ⋅ , n).

By use of Girsanov’s theorem these become Wiener processes

under the equivalent measure ℙ. Thus the forgoing set ofequations become

dP (t, T ) = r(t)P (t, T )dt+

n∑

i=1

ai(t, T, ⋅)P (t, T )dWi(t), (82)

dZ(t, T ) =n∑

i=1

ai(t, T, ⋅)Z(t, T )dWi(t), (83)

dr =

[f2(0, t) +

∂

∂t

n∑

i=1

∫ t

0

�i(v, t, ⋅)

∫ t

v

�i(v, y, ⋅)dydv

+

n∑

i=1

∫ t

0

∂�i

∂t(v, t, ⋅)dWi(v)

]dt+

n∑

i=1

�i(t, t, ⋅)dWi(t). (84)76 / 121




Under ℙ the relative bond price Z(t, T ) is a martingale, sothat

Z(t, T ) = Et [Z(T, T )] , (85)

which in terms of the bond price becomes

P (t, T ) = Et

[exp

(−

∫ T

t

r(y)dy

)], (86)

which of course is the same as (29).

77 / 121




The difference here is the Et is generated by (84) which in its

turn is driven by the n independent noise terms Wi(t).

1 If we were to use simulation to directly evaluate (86) we wouldneed to simulate n independent sequences of normal randomvariables in order to simulate a path for r(t).

2 Furthermore if the �i(t, T ) depend on other factors, such asdiscrete tenor forward rates, then the processes for these(under ℙ) would have to be simulated as well.

78 / 121


Relating Heath-Jarrow-Morton to Hull-White Two-Factor Models

Relating Heath-Jarrow-Morton to Hull-White Two-Factor

Models

The Hull-White extended Vasicek model can be derived (farmore simply) in the Heath-Jarrow-Morton framework once anappropriate form for the volatility function is chosen.

The Hull-White two-factor model can be obtained as a specialcase of the multi-factor Heath-Jarrow-Morton model. Recallthe Hull-White two-factor model

dr = [�(t) + ℎ(t)− ar(t)]dt+ 1dz1, (87)

where the additional term ℎ(t) in the drift satisfies

dℎ = −b ℎ(t)dt+ 2dz2. (88)

Here z1, z2 are correlated Wiener processes, i.e.

E[dz1 dz2] = �dt.

79 / 121




Models

We may reexpress (87) and (88) in terms of the independentWiener processes w1, w2 as

dr = [�(t)+ℎ(t)−ar(t)]dt+ 1√

1− �2 dw1+ 1� dw2, (89)

dℎ = −b ℎ(t) dt+ 2dw2. (90)

We consider a two-factor Heath-Jarrow-Morton model withvolatility specifications

�i(t, T ) = �ie−�i(T−t), (91)

where �i and �i are constants, for i = 1, 2

80 / 121




Models

Accordingly, the forward rate dynamics are expressed as

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

∫ t

0

�(v, T )dv +

∫ t

0

�1(v, T )dW1(v) +

∫ t

0

�2(v, T )dW2(v),

which under the risk-neutral measure becomes

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

2∑

i=1

∫ t

0

�i(v, T )

∫ T

v

�i(v, y)dy dv +

2∑

i=1

∫ t

0

�i(v, T )dWi(v).

81 / 121




Models

The spot rate process under the risk-neutral measure satisfies

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

2∑

i=1

∫ t

0

�i(v, t)

∫ t

v

�i(v, y)dy dv +

2∑

i=1

∫ t

0

�i(v, t)dWi(v),

(92)

or

dr =

[f2(0, t) +

∂

∂t

2∑

i=1

∫t

0

�i(v, t)

∫t

v

�i(v, y)dy dv

+

2∑

i=1

∫t

0

∂�i∂t

(v, t)dWi(v)

]dt+

2∑

i=1

�i(t, t)dWi(t).

(93)

82 / 121




Models

The volatility functions (91) have the property

∂�i(t, T )

∂T= −�i�i(t, T ).

Thus the s.d.e. (93) for the spot rate becomes

dr =

[f2(0, t) +

∂

∂t

2∑

i=1

Si(t)−

2∑

i=1

�ixi(t)

]dt+

2∑

i=1

�idWi(t), (94)

where we set

Si(t) = �2i

∫ t

0e−�i(t−v)

∫ t

v

e−�i(y−v)dy dv,

and for i = 1, 2,

xi(t) =

∫ t

0�ie

−�i(t−v)dWi(v).

83 / 121




Models

Using techniques introduced in Section 25.6 we can show thatthe xi(t) satisfy the s.d.e.s

dxi = �ie−�i(t−t)dWi(t)−

∫ t

0�i�ie

−�i(t−v)dWi(v)dt

= −�ixi(t)dt+ �idWi(t). (95)

The system (94), (95) for r(t), x1(t), and x2(t) is theMarkovian system that generates the probability distribution

for (E)t.

84 / 121




Models

To obtain the link with the Hull-White two-factor model notethat with the volatility functions (91) the stochastic integralequation for r(t) becomes

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

2∑

i=1

Si(t) + x1(t) + x2(t). (96)

This last equation may be used to eliminate x1(t) in (94), thus

x1(t) = r(t)− f(0, t)−2∑

i=1

Si(t)− x2(t),

85 / 121




Models

Upon substitution into (94) yields

dr = [f2(0, t)+�1f(0, t)+S(t)+(�1−�2)x2(t)−�1r(t)]dt+2∑

i=1

�idWi(t),

(97)

where we have defined

S(t) =2∑

i=1

[∂

∂tSi(t) + �1Si(t)

], (98)

and x2(t) is driven by equation (95) with i = 2, viz

dx2 = −�2x2(t)dt+ �2dW2(t). (99)

86 / 121




Models

To fully obtain the correspondence with the Hull-Whitetwo-factor model in equations (89), (90) set

�(t) = f2(0, t) + �1f(0, t) + S(t),

a = �1,

b = �2, (100)

so that we are now dealing with the system

dr = [�(t) + (a− b)x2(t)− ar(t)]dt+ �1dW1(t) + �2dW2(t),(101)

dx2 = −b x2(t)dt+ �2dW2(t). (102)

87 / 121




Models

If we setℎ(t) = (a− b)x2(t)

then equation (102) becomes

dℎ = −bℎ(t)dt+ (a− b)�2dW2(t). (103)

88 / 121




Models

The system (101) and (103) for r(t) and ℎ(t) is equivalent tothe Hull-White two-factor system (89), (90) if we set

�1 = 1√

1− �2, �2 = 1� and (a− b)�2 = 2. (104)

The parameters of the Hull-White two-factor model may berelated to the parameters of the two-factorHeath-Jarrow-Morton model via

� =�2√�21 + �22

, 1 =√�21 + �22 , 2 = (�1−�2)�2, b = �2.

(105)

The biggest advantage is that the �(t) function in thes.d.e. (101) for r(t) is automatically calibrated to the initiallyobserved forward curve f(0, t).

89 / 121


The Covariance Structure Implied by the Heath-Jarrow-Morton Model

The Covariance Structure Implied by the

Heath-Jarrow-Morton Model

Another important issue is the analysis of the statisticalproperties of the evolution of the forward rates and yieldsunder the jump-diffusion framework.

One factor models allow for only parallel shifts of the yieldcurve, so bond prices and forward rates of all maturities areperfectly correlated.

Multi dimensional models, on the other hand, impose acorrelation structure between forward rates of differentmaturities which shows an exponentially decaying behavior.

We seek to understand the effect on the forward ratecorrelation structure implied by the assumptions concerning inthe forward rate dynamics.

90 / 121



The Covariance Structure of the Forward Rate Changes

Under the risk neutral measure, the changes of the forwardrate follow the dynamics

df(t, T ) =

n∑

i=1

�i(t, T, f(t))�i(t, T, f(t))dt+

n∑

i=1

�i(t, T, f(t))dWi(t).

(106)

Thus

E0[df(t, T )] =

n∑

i=1

�i(t, T, f(t))�i(t, T, f(t))dt. (107)

91 / 121




Denote T1 and T2 two different maturities, then thecovariance of the changes on the forward rate is calculated as

cov0[df(t, T1), df(t, T2)]

= E0[(df(t, T1)− E0[df(t, T1)])(df(t, T2)− E0[df(t, T2)])]

= E0

[n∑

i=1

�i(t, T1, f(t))dWi(t) ⋅n∑

i=1

�i(t, T2, f(t))dWi(t)

].

92 / 121




From the independence of the Wiener increments it readilyfollows that

cov0[df(t, T1), df(t, T2)] =

n∑

i=1

�i(t, T1, f(t))�i(t, T2, f(t))dt,

(108)

The variance of the forward rate changes df(t, Tℎ) (ℎ = 1, 2)as

var0[df(t, Tℎ)] =

n∑

i=1

�2i (t, Tℎ, f(t))dt. (109)

93 / 121




The correlation coefficient between the forward rates changesdf(t, T1) and df(t, T2) is then evaluated as

�(t, T1, T2) =cov0[df(t, T1), df(t, T2)]√

var0[df(t, T1)]√var0[df(t, T2)]

, (110)

where cov0[df(t, T1), df(t, T2)] and var0[df(t, Tℎ)], (ℎ = 1, 2)are defined above.

94 / 121




We assume the volatility functions are of the form

�i(s, t) = �0ie−��i(t−s), i = 1, . . . , n, (111)

where the �0i, ��i are constant

Then the covariance between df(t, T1) and df(t, T2) iscalculated as

cov0[df(t, T1), df(t, T2)] =

n∑

i=1

�20ie−��i(T1+T2−2t), (112)

95 / 121




The correlation coefficient between the forward rates changesdf(t, T1) and df(t, T2) is evaluated as

�(t, T1, T2) =

∑ni=1 �

20ie

−��i(T1+T2−2t)

√var0[df(t, T1)]

√var0[df(t, T2)]

, (113)

where the variance of the forward rate changes df(t, Tℎ)(ℎ = 1, 2) is

var0[df(t, Tℎ)] =n∑

i=1

�20ie−2��i(Tℎ−t). (114)

96 / 121



The Covariance Structure of the Forward Rate


The forward rate under the risk neutral measure is given by

f(t, T ) = f(0, T ) +n∑

i=1

∫ t

0�i(s, T, f(s))�i(s, T, f(s))ds

+

n∑

i=1

∫ t

0�i(s, T, f(s))dWi(s).

(115)

Thus

E0[f(t, T )] = f(0, T ) +n∑

i=1

∫ t

0�i(s, T, f(s))�i(s, T, f(s))ds.

(116)

97 / 121





Denote T1 and T2 two maturities then the covariance of theforward rates f(t, T1) and f(t, T2) is calculated as

cov0[f(t, T1), f(t, T2)]

= E0[(f(t, T1)− E0[f(t, T1)])(f(t, T2)− E0[f(t, T2)])]

= E0

[n∑

i=1

∫ t

0�i(s, T1, f(s))dWi(s) ⋅

n∑

i=1

∫ t

0�i(s, T2, f(s))dWi(s)

].

(117)

98 / 121





Using the result

E0[

∫ t

0�i(s, T1)dWi(s)

∫ t

0�i(s, T2)dWi(s)] =

∫ t

0�i(s, T1)�i(s, T2)ds,

The covariance is given by

cov0[f(t, T1), f(t, T2)] =n∑

i=1

∫ t

0�i(s, T1)�i(s, T2)ds. (118)

99 / 121





Considering again the volatility functions of the form

�i(s, t) = �0ie−��i(t−s), i = 1, . . . , n, (119)

Then

∫ t

0�i(s, T1)�i(s, T2)ds =

�20ie−��i(T1+T2)

2��i(e2��it − 1). (120)

100 / 121





The covariance between the forward rates f(t, T1) andf(t, T2) becomes

cov0[f(t, T1), f(t, T2)] =

n∑

i=1

�20ie−��i(T1+T2)

2��i(e2��it − 1),

(121)And the correlation coefficient �(t, T1, T2) between theforward rates f(t, T1) and f(t, T2) is evaluated as

∑ni=1

�20ie

−��i(T1+T2)

2��i(e2��it − 1)

√var0[f(t, T1)]

√var0[f(t, T2)]

, (122)

where the variance of the forward rate f(t, Tℎ) (ℎ = 1, 2) is

var0[f(t, Tℎ)] =

n∑

i=1

�20ie−2��iTℎ

2��i(e2��it − 1). (123)

101 / 121


Appendix

Proof of Proposition 25.1

Recall that r(t) satisfies the stochastic integral equation (23) and f(t, � )satisfies the stochastic integral equation (26) with T set equal to � .

We assume the forward rate volatility specifications

� (v, t, !(v)) = �e−�(t−v)

g (r(v), f(v, � ))

and set

�∗ (v, t, !(v)) = � (v, t, !(v))

∫ t

v

� (v, s, !(v)) ds

= �2e−�(t−v)

g (r(v), f(v, � ))

∫ t

v

e−�(s−v)

g (r(v), f(v, � )) ds

= �2g2 (r(v), f(v, � )) e−�(t−v)

(1− e−�(t−v)

�

).

102 / 121


Appendix


Note that the first integral term in equation (23) can be written

∫ t

0

�∗ (v, t, !(v)) dv = �

2

∫ t

0

g2 (r(v), f(v, � )) e−�(t−v)

(1− e−�(t−v)

)

�dv

=e−�t�2

�

∫ t

0

g2(r(v), f(v, � ))e�vdv −

e−2�t�2

�

∫ t

0

g2(r(v),

≡e−�t

�I(t;�)−

e−2�t

�I(t; 2�).

Next note that the second integral in equation (23) may be written as

∫ t

0

� (v, t, !(v))dW (v) = �

∫ t

0

e−�(t−v)

g (r(v), f(v, � ))dW (v)

= �e−�t

∫ t

0

g (r(v), f(v, � )) e�vdW (v)

≡ e−�t

J(t;�).

103 / 121


Appendix


Similarly the first integral term in equation (26) can be written

∫ t

0

�∗ (v, �, !(v)) dv = �

2

∫ t

0

g2 (r(v), f(v, � )) e−�(�−v)

(1− e−�(�−v)

)

�dv

=e−��

�I(t;�)−

e−2��

�I(t; 2�).

The second integral term in equation (26) may be similarly treated, sothat

∫ t

0

� (v, �, !(v))dW (v) = �

∫ t

0

e−�(�−v)

g (r(v), f(v, � ))dW (v)

= �e−��

∫ t

0

e�vg (r(v), f(v, � ))dW (v)

≡ e−��

J(t;�).

104 / 121


Appendix


We may thus write the stochastic integral equations for r(t) and f(t, � )in terms of the integrals I(t;�), I(t; 2�) and J(t;�) as

r(t) = f(0, t) +e−�t

�I(t;�)−

e−2�t

�I(t; 2�) + e

−�tJ(t; �), (124)

f(t, � ) = f(0, � ) +e−��

�I(t;�)−

e−2��

�I(t; 2�) + e

−��J(t; �). (125)

We note that equations (124) and (125) can be re-expressed as

r(t)− f(0, t) +e−2�t

�I(t; 2�) = e

−�t

[I(t;�)

�+ J(t;�)

],

f(t, � )− f(0, � ) +e−2��

�I(t; 2�) = e

−��

[I(t;�)

�+ J(t;�)

].

105 / 121


Appendix


We may combine the above equations to express I(t; 2�) as a function ofr(t) and f(t, � ), i.e.,

I(t; 2�) =�e��

e−�t − e−��[f(t, � )− f(0, � )]−

�e�t

e−�t − e−��[r(t)− f(0, t)]

(126)

Finally we note that

(t) =

∫ t

0

�2 (v, t, !(v)) dv

= �2

∫ t

0

e−2�(t−v)

g2 (r(v), f(v, � )) dv

= �2e−2�t

∫ t

0

e2�v

g2 (r(v), f(v, � )) dv

= e−2�t

I(t; 2�).

106 / 121


Appendix


Thus we finally have

(t) = ��(t, � ) [r(t)− f(0, t)]− �e−�(t−�)

�(t, � ) [f(t, � )− f(0, � )] .(127)

where we set

�(t, � ) ≡e−�t

e−�� − e−�t.

107 / 121


Appendix


It is readily verified that the manipulations that led to equation (125) ofAppendix 25.1 are equally valid for t set to a general maturity T . Thus(126) holds for t set to T , i.e.,

I(t; 2�) =�e�T

e−�t − e−�T[f(t, T )− f(0, T )]

−�e�t

e−�t − e−�T[r(t)− f(0, t)]

= e2�t (t).

Substituting the expression for (t) we find that

I(t; 2�) = �e2�t

(�(t, � )[r(t)− f(0, t)]− e

−�(t−�)�(t, � )[f(t, � )− f(0, � )]

)

= �

(e�T

e−�t − e−�T[f(t, T )− f(0, T )]−

e�t

e−�t − e−�T[r(t)− f(0, t)]

)

108 / 121


Appendix


On rearranging

e�T

e−�t − e−�T[f(t, T )− f(0, T )]

=e�t

e−�t − e−�T[r(t)− f(0, t)] + e

2�t�(t, � )[r(t)− f(0, t)]

− e2�te−�(t−�)

�(t, � )[f(t, � )− f(0, � )],

from which

f(t, T )− f(0, T ) = [r(t)− f(0, t)]

(e�t

e�T+e2�t(e−�t − e−�T )

e�T�(t, � )

)

−e2�te−�(t−T )

e�T�(t, � )(e−�t − e

−�T )[f(t, � )− f(0, � )].

(128)

109 / 121


Appendix


Consider the following:(i)

e�t + e2�t(e−�t − e−�T )

e�T�(t, � ) =

e�t

e�T+e2�t(e−�t − e−�T )e−�t

e�T (e−�� − e−�t)

=e�t(e−�� − e−�t) + e�t(e−�t − e−�T )

e�T (e−�� − e−�t)

=e�t(e−�� − e−�T )

e�T (e−�� − e−�t)

=e2�t

e2�Te−�t

e−�� − e−�te−�� − e−�T

e−�T

= e−2�(T−t) �(t, � )

�(T, � )

110 / 121


Appendix


(ii)

e2�te−�(t−�)

�(t, � )(e−�t − e−�T )

e�T= e

2�t−�t+��(t, � )

(e−�t − e−�T )

e2�T e−�T

=e�te��e−�t(e−�t − e−�T )

e2�T e−�T (e−�� − e−�t)

=e2��

e2�Te−��

−(e−�t − e−�� )

(e−�t − e−�T )

e−�T

= −e−2�(T−�) �(�, t)

�(T, t).

111 / 121


Appendix


Hence equation (128) can be rewritten

f(t, T )− f(0, T ) = e−2�(T−t) �(t, � )

�(T, � )[r(t)− f(0, t)]

− e−2�(T−�) �(�, t)

�(T, t)[f(t, � )− f(0, � )]

where

�(�1, �2) ≡e−��1

e−��2 − e−��1.

We have thus proved Proposition 25.2.

112 / 121


Appendix

Details of the Infinitesimal Generator

We recall the following result from Section5.4 concerning the infinitesimalgenerator of an n dimensional Ito process

In our application we set

X1 ≡ f(t, � ),

a1 ≡ �2(t, �, !(t))

(e�(�−t) − 1)

�,

�11 ≡ �1 ≡ �(t, �, !(t)),

X2 ≡ r(t),

a2 ≡ f2(0, t) + �f(0, t) + (t)− �r(t),

�21 ≡ �r ≡ �(t, t, !(t)).

113 / 121


Appendix

Details of the Infinitesimal Generator

Thus the matrix S assumes the form[�21 �1�r

�1�r �2r

].

Using the foregoing expression for S the expression for the operator K inequation (67) is readily derived.

114 / 121


Appendix


Using the relationship

P (t, T ) = exp

(−

∫ T

t

f(t, s)ds

)

and equation (26) for the forward rate f(t, s) we obtain for the bondprice the expression

P (t, T ) =P (0, T )

P (0, t)exp

[−

(∫ T

t

∫ t

0

�∗(v, s, ⋅)dvds+

∫ T

t

∫ t

0

�(v, s, ⋅)dW (v)ds

)

where

�(v, T, ⋅) = �e−�(T−v)

g(r(v), f(v, � ))

�∗(v, T, ⋅) = �(v, T, ⋅)

∫ T

v

�(v, s, ⋅)ds

= �2g2(r(v), f(v, � ))e−�(T−v)

∫ T

v

e−�(s−v)

ds.

115 / 121


Appendix


Set

I =

∫ T

t

∫ t

0

�∗(v, s, ⋅)dvds+

∫ T

t

∫ t

0

�(v, s, ⋅)dW (v)ds

≡ I1 + I2

=

∫ t

0

∫ T

t

�∗(v, s, ⋅)dsdv +

∫ t

0

∫ T

t

�(v, s, ⋅)dsdW (v),

where we have interchanged the order of integration to obtain the lastequality

116 / 121


Appendix


Next note that∫ T

t

�∗(v, s, ⋅)ds = � (r(v), f(v, � ))

∫ T

t

e−�(s−v)

∫ s

v

e−�(y−v)

� (r(v), f(v, � )) dyds

= � (r(v), f(v, � ))

∫ T

t

e−�(s−v)

{∫ t

v

e−�(y−v)

� (r(v), f(v, � )) dy

+

∫ s

t

e−�(y−v)

� (r(v), f(v, � )) dy

}ds

= �2g2(r(v), f(v, � ))

∫ T

t

e−�(s−v)

ds

∫ t

v

e−�(y−v)

dy

+ �2g2(r(v), f(v, � ))

∫ T

t

e−�(s−v)

∫ s

t

e−�(y−v)

dyds

= �2g2(r(v), f(v, � ))e−�(t−v)

(∫ T

t

e−�(s−t)

ds

)∫ t

v

e−�(y−v)

dy

+ �2g2(r(v), f(v, � ))e−2�(t−v)

∫ T

t

e−�(s−t)

∫ s

t

e−�(y−t)

dyds

= �∗(v, t, ⋅)�(t, T ) + �

2(v, t, ⋅)�(t, T )117 / 121


Appendix


where

�(t, T ) =

∫ T

t

e−�(s−t)

ds =1

�

(1− e

−�(T−t))

�(t, T ) =

∫ T

t

e−�(s−t)

∫ s

t

e−�(y−t)

dyds =1

2�2(t, T )

i.e. we have shown that∫ T

t

�∗(v, s, ⋅)ds = �(t, T )�∗(v, t, ⋅) +

1

2�2(t, T )�2(v, t, ⋅).

118 / 121


Appendix


Returning to the expressions for I1, I2 we can now write

I1 =

∫ t

0

[�(t, T )�∗(v, t, ⋅) +

1

2�2(t, T )�2(v, t, ⋅)

]dv,

and

I2 =

∫ t

0

�(t, T )�(v, t, ⋅)dW (v),

so that

I =1

2�2(t, T )

∫ t

0

�2(v, t, ⋅)dv + �(t, T )

[∫ t

0

�∗(v, t, ⋅)dv +

∫ t

0

�(v, t, ⋅)dW (v)

].

119 / 121


Appendix


However we note from equation (23), for the instantaneous spot rater(t), that

∫ t

0

�∗(v, t, ⋅)dv+

∫ t

0

�(v, t, ⋅)dW (v) = r(t)− f(0, t).

Hence

I =1

2�2(t, T )

∫ t

0

�2(v, t, ⋅)dv + �(t, T ) [r(t)− f(0, t)] .

120 / 121


Appendix


Recalling the definition of the subsidiary stochastic variable (t) we canfinally write

I =1

2�2(t, T ) (t) + �(t, T ) [r(t)− f(0, t)] .

Hence the expression for the bond price may be written as inProposition ??.

121 / 121


References

Bhar, R. and C. Chiarella (1997a).Transformation of Heath-Jarrow-Morton Models to Markovian

Systems.European Journal of Finance 3, 1–26.

Brennan, M. J. and E. S. Schwartz (1979).A Continuous-Time Approach to the Pricing of Bonds.Journal of Banking Finance 3, 135–155.

Cheyette, O. (1992).Term Structure Dynamics and Mortgage Valuation.Journal of Fixed Income 1, 28–41.

Chiarella, C. and N. El-Hassan (1998).Pricing American Interest Rate Options in a

Heath-Jarrow-Morton Framework Using Method of Lines.Quantitative Finance Research Group, School of Finance and

Economics, UTS .121 / 121


Appendix

Chiarella, C. and O. K. Kwon (1999b).Forward Rate Dependent Markovian Transformations of the

Heath-Jarrow-Morton Term Structure Model.Quantitative Finance Research Group, School of Finance and

Economics, UTS (Research Paper 5).

Inui, K. and M. Kijima (1998).A Markovian Framework in Multi-Factor Heath-Jarrow-Morrton

Models.Journal of Financial and Quantitative Analysis 33, 423–440.

Ritchken, P. and L. Sankarasubramanian (1995a).Volatility structures of forward rates and the dynamics of the

term structure.Mathematical Finance 5, 55–72.

121 / 121

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