25857 Interest Rate Modelling - WordPress.com · Interest Rate Modelling Introduction Introduction In this chapter we survey models of interest rate derivatives which take the instantaneous

Interest Rate Modelling

25857 Interest Rate Modelling

UTS Business SchoolUniversity of Technology Sydney

Chapter 23. Interest Rate Derivatives - One Factor Spot RateModels

May 22, 2014

1 / 116


Chapter 23. Interest Rate Derivatives - One Factor Spot

Rate Models

1 Introduction

2 Arbitrage Models of the Term Structure

3 The Martingale Representation

4 Some Specific Term Structure ModelsThe Vasicek ModelThe Hull-White ModelThe Cox-Ingersoll-Ross (CIR) ModelCalculation of the Bond Price from the Expectation Operator

5 Pricing Bond Options

6 Solving the Option Pricing EquationThe Hull White ModelThe CIR Model

7 Rendering Spot Rate Models Preference Free-Calibration to the Currently Observed

2 / 116


Introduction

Introduction

This chapter; the problem of pricing options on interest ratederivative securities.

The essential feature of this problem is that we need to takeaccount of the stochastic nature of interest rates.

Chapter 19 illustrated one approach to this problem, namelymodelling the price of pure discount bonds as a stochasticprocess and making this one of the stochastic factors uponwhich the value of the option depends.

The general approach is due to Merton (973b).

3 / 116


Introduction

Introduction

There are however a number of practical difficulties in attempting toimplement this approach.

∼ In particular it requires specification of the average expectedreturn variance over the time interval to maturity, togetherwith the covariance between return and the instantaneousshort term rate.

∼ It is not clear in practice how best to estimate these variancesand covariances.

Nevertheless, Merton’s approach has guided the development ofmany of the subsequent interest rate option models.A characteristic of the stock option model is that there isone basic approach

∼ to which can be added embellishments to account for differentstochastic processes for the underlying asset (e.g. ajump-diffusion process) or to account for different boundaryconditions (e.g. European or American options).

4 / 116


Introduction

Introduction

For interest rate contingent claims however there does notseem to be one basic approach but rather a range ofalternative approaches.

These differ according to what is taken as the underlyingfactor, which is usually one of the instantaneous spot interestrate, the bond price or the forward rate.

Further, some models are presented in a discrete timeframework and some in a continuous time framework.

An important distinction between alternative approachesis whether the initial term structure (i.e. the currentlyobserved yield curve) is itself to be modelled or to betaken as given.

∼ This modelling choice will determine whether the resultingmodels involve the market price of interest rate risk.

5 / 116


Introduction

Introduction

In this chapter we survey models of interest ratederivatives which take the instantaneous spot rate ofinterest as the underlying factor.

The by-now familiar continuous hedging argument is extendedso as to model the term structure of interest rates and fromthis other interest rate derivative securities.

∼ This basic approach is due to Vasicek (1977) and hence weshall often refer to it as the Vasicek approach.

6 / 116


Arbitrage Models of the Term Structure


We consider the perspective of an investor standing at time 0and observing various market rates that enable him/her tocompute the initial forward f(0, T ) (and so the initial bondprice curve P (0, T )) for any maturity T .

This investor wishes to price at any time t (< T ) a puredefault-free discount bond that pays $1 at time T .

The investor seeks the arbitrage-free bond price that does notallow the possibility of riskless arbitrage opportunities betweenbonds of differing maturities.

Furthermore the investor wishes the bond price so obtained tobe consistent with the currently observed initial bond pricecurve.

7 / 116




Fig.1 illustrates the time line for the bond-pricing problem.

b b b0 t T

$1

P (0, T ) observed

oo

oP (t, T ) =?

Figure 1: The time line for the bond-pricing problem

8 / 116




Initially, we assume that the price of a default free bondis a function of only the current short term rate ofinterest and time.

∼ Thus we write P (r(t), t, T ) to denote the price at time t of adiscount bond maturing at time T , having maturity value of$1, when the current instantaneous spot rate of interest isr(t), (which is assumed to be riskless in the sense that moneyinvested at this rate will always be paid back) i.e.,

P (r(T ), T, T ) = 1. (1)

We assume the short term rate follows the diffusion process

dr = �r(r, t)dt+ �r(r, t)dz. (2)

9 / 116




By Ito’s lemma

dP

P= �P (r, t, T )dt + �P (r, t, T )dz, (3)

∼ where

�P (r, t, T ) =1

P

(∂P

∂t+ �r

∂P

∂r+

1

2�2r

∂2P

∂r2

), (4)

�P (r, t, T ) =�rP

∂P

∂r. (5)

Consider an investor who at time t invests $1 in a hedgeportfolio containing two default free bonds maturing attimes T1 and T2 respectively; held in the dollar amounts Q1

and Q2.

10 / 116




Using Pi to denote the price of the bond maturing at time Ti,we can write

dollar return on thehedge portfolio

}= Q1

dP1

P1+Q2

dP2

P2

= (Q1�P1 +Q2�P2)dt+ (Q1�P1 +Q2�P2)dz,(6)

∼ where �Pi, �Pi

denote respectively the expected return andstandard deviation of the bond of maturity Ti (i = 1, 2).

This return can be made certain by choosing the amounts Q1,Q2, so that

Q1

Q2= −�P2

�P1

. (7)

11 / 116




Thus from (6) the dollar return on the now riskless hedgeportfolio is

(Q1�P1 +Q2�P2)dt.

Absence of riskless arbitrage ⇒ this return must be theinstantaneous spot rate of interest r.

Given that the original investment is $1 (i.e. Q1 +Q2 = 1)then this last condition states that

(Q1�P1 +Q2�P2)dt = 1 ⋅ rdt.

12 / 116




Rearranging we obtain

Q1(�P1 − r) +Q2(�P2 − r) = 0,

∼ which when combined with (7) yields the condition forno-riskless arbitrage between bonds of any two maturities,namely

�P1 − r

�P1

=�P2 − r

�P2

. (8)

13 / 116




Since the maturity dates T1, T2 were arbitrary, the ratio

�P (r, t, T ) − r(t)

�P (r, t, T )

must be independent of maturity T .

Let �(r, t) denote the common value of this ratio for bonds ofan arbitrary maturity T . Thus

�P (r, t, T ) − r(t)

�P (r, t, T )= �(r, t). (9)

The quantity � can be interpreted as the market price ofinterest rate risk per unit of bond return volatility.

14 / 116




Thus eqn. (9) asserts that

∼ in equilibrium bonds are priced so that instantaneousbond returns equal the instantaneous risk free rate ofinterest plus a risk premium equal to the market price ofinterest rate risk times instantaneous bond returnvolatility.

Substitution from (4) of the expressions for �P (t, T ) and�P (t, T ) results in the p.d.e. for the bond price,

∂P

∂t+ (�r − ��r)

∂P

∂r+

1

2�2r∂2P

∂r2− rP = 0, (10)

∼ which must be solved subject to the boundary condition

P (r(T ), T, T ) = 1. (11)

15 / 116




To solve (10), either analytically or numerically, we need tospecify the drift �r and diffusion �r as well as form ofthe market price of risk term �(r, t). One commonassumption is that this latter term is constant.

∼ To formally derive this result, involves some very particularassumptions about how the capital market operates. Theseconditions are discussed briefly in the next subsection.

∼ To give a proper theoretical basis to the choice of �(r, t) itwould be necessary to construct a dynamic general equilibriummodel and relate �(r, t) to investor preferences. This is theapproach adopted by Cox, Ingersoll, and Ross (1985b).

16 / 116


The Martingale Representation


Just as in the case of the stock option model we are able toobtain a martingale representation of the pricingrelationship;

We note from the no riskless arbitrage condition (9) that

�P (r, t) = r + ��P (r, t) (12)

Substitution of (12) into (3) as well as the expression for �Pfrom (4) into (3) yields the stochastic bond price dynamicsunder the condition of no-riskless arbitrage viz.

dP

P= (r + ��P (r, t, T )) dt+ �P (r, t, T )dz. (13)

17 / 116




Following the reasoning used in Chapter 10, define a modifiedWiener process z(t) by

z(t) = z(t) +

∫ t

0�(s)ds. (14)

Under the historical measure ℙ, z(t) is not a standardWiener process (i.e. E(z(t) ∕= 0 where E is the expectationoperation under ℙ) but by an application of Girsanov’stheorem we can obtain an equivalent measure ℙ under whichz(t) is a standard Wiener process (i.e. E(z(t)) = 0 where E isthe expectation operation under ℙ).

Thus in terms of z(t) the s.d.e. (13) for P under the measureℙ becomes

dP

P= rdt+ �P (r, t, T )dz. (15)

18 / 116




However unlike in the stock option situation, the spotrate r is here stochastic,

∼ so we need to define the money market account as

A(t) = e∫

t

0r(s)ds. (16)

∼ It is a simple matter to demonstrate that

dA = rAdt. (17)

19 / 116




We then define the bond price in units of the money marketaccount,1

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

A(t), (18)

∼ By Ito’s lemma;dZ

Z= �P (r, t, T )dz. (19)

1Recall that by the rules of stochastic calculus

dZ

Z=dP

P− dA

A− dP

P⋅ dAA

+

(dA

A

)2

20 / 116




Thus eqn. (19) implies Z(r, t, T ) is a martingale under ℙ,i.e.

Z(r, t, T ) = Et[Z(r(T ), T, T )], (20)

∼ which in terms of the original bond price can be expressed as

P (r, t, T ) = Et

[A(t)

A(T )P (r(T ), T, T )

],

∼ or since P (r(T ), T, T ) = 1, more simply as

P (r, t, T ) = Et

[e−

∫

T

tr(s)ds

]. (21)

21 / 116




To derive the interest rate dynamics under ℙ use (14) toreplace dz by (dz − �(t)dt) in eqn. (2);

dr = (�r − ��r)dt+ �rdz. (22)

An application of the Feynman-Kac formula2 (in particularProposition 8.2) to (21) and (22) would take us back to thep.d.e. (10).

2Make the identificationsx → r, v(t, r) → P (r, t, T ), � = −1, f [s, x(s)] → r(s).

22 / 116




Thus, just as in the stock option situation, we have tworepresentations of the bond price,

∼ the p.d.e. (10)∼ and the expectation operator (21) under the interest rate

dynamics (22).

To use these representations we need to specify thefunction �r,�r and also the functional form for themarket price of interest rate risk �(r, t). This we do in thefollowing subsection for specific term structure models.

23 / 116




Before leaving this subsection we wish to emphasize thediscounted cash flow interpretation of the representation (21).

The factor exp(−∫ Tt r(s)ds) discounts back to t the dollar

received at T , for one particular path followed by r(s).

∼ Since r(s) is stochastic this quantity is in fact a stochasticdiscount factor.

∼ To obtain the discounted value at t of the $1 received at T weneed to average over the range of possible paths followed byr(s) under the measure ℙ.

∼ This is effectively what the Et does; Fig.2 illustrates this idea.

24 / 116




It is also of interest to contrast the bond price expression (21)with the corresponding expression in eqn (22.11) for a worldof certainty, and we see how this is generalised in a naturalway to the world of uncertainty.

We thus have a complete analogy with the stock option pricederivation of Chapter 6 and Chapter 7 with the exceptionthat the pricing relationships here involve the marketprice of interest rate risk �.

But from our discussion in Chapter 10 this is to be expectedsince the underlying factor, the spot interest rate r, is not atraded factor.

25 / 116




b

time

r

t T0

Figure 2: Typical paths for the r process over [t, T ]. Equation (21)

averages the quantity e−∫

T

tr(s)ds over many such paths under the ℙ

measure26 / 116


Some Specific Term Structure Models


A variety of term structure models are obtained

∼ by specifying different forms for �r(r, t) and �r(r, t) in theinterest rate process, eqn. (2),

∼ and/or different forms for the market price of risk term.

27 / 116



The Vasicek Model

The Vasicek Model

The Vasicek Vasicek (1977) model holds a special place in theinterest rate term structure literature as it was the earliestmodel. Its basic assumptions are to take

�r(r, t) = �( − r) and �r(r, t) = �, (23)

∼ with � > 0 and � > 0 are constant, and also assume aconstant market price of interest rate risk �.

Setting � = � − ��, the bond pricing p.d.e. (10) becomes

∂P

∂t+ (� − �r)

∂P

∂r+

1

2�2∂2P

∂r2− rP = 0. (24)

28 / 116



The Vasicek Model

The Vasicek Model

To solve this p.d.e. we seek functions a(t, T ) and b(t, T ) suchthat 3

P (t, T ) = e−a(t,T )−b(t,T )r(t) . (25)

In order that the boundary condition (11) be satisfied for allpossible r(t) it must be the case that a(t, T ) and b(t, T )satisfy

a(T, T ) = 0 and b(T, T ) = 0. (26)

3One motivation for the functional form (25) is that when r isconstant, we have P (t, T ) = e−r(T−t), and (25) is an obviousgeneralisation of this relation.

29 / 116



The Vasicek Model

The Vasicek Model

From (25) we note that4

∂P

∂r= −bP, ∂2P

∂r2= b2P and

∂P

∂t= (−at − btr)P. (27)

Substituting these relations into (24) and gathering terms inpowers of r(r0 = 1) we obtain

[−at − b� +

1

2�2b2

]+ [−bt + �b− 1]r = 0. (28)

If (28) is to hold for all t and all r it must be the case thateach bracket is separately equal to zero.

4We employ the notation at =∂∂ta(t, T ), bt =

∂∂tb(t, T ).

30 / 116



The Vasicek Model

The Vasicek Model

Thus we obtain for a and b the ordinary differential eqns.

bt = �b− 1, (29)

∼ and

at = −b� + 1

2�2b2, (30)

which must be solved subject to the boundary conditions (26).

31 / 116



The Vasicek Model

The Vasicek Model

From (29) we obtain5

b(t, T ) =

[1− e−�(T−t)

]

�. (31)

Substituting (31) into (30), integrating from t to T and, using theboundary condition a(T, T ) = 0 we find that

a(t, T ) =

∫ T

t

[�b(s, T )− 1

2�2b(s, T )2

]ds, (32)

5Eqn. (29) can be re-arranged into

d

dt

(

b(t, T )e−�t

)

= −e−�t

and integrating t to T we obtain

b(T, T )e−�T

− b(t, T )e−�t

= −

∫

T

t

e−�s

ds = −(e−�t

− e−�T

)/�.

Use of the boundary condition b(T, T ) = 0 and some re-arrangement yields (31).

32 / 116



The Vasicek Model

The Vasicek Model

∼ which reduces to6

6Note that∫ T

t

b(s, T )ds =

∫ T

t

(1− e−�(T−s))

�ds

=

∫ T−t

0

(1− e−�u)

�du =

(T − t)

�+

1

�2(e−�(T−t)

− 1)

and∫ T

t

b2(s, T )ds =

∫ T−t

0

(1− e−�u)2

�2du

=1

�2

[

(T − t) +2

�(e−�(T−t)

− 1)−1

2�(e−2�(T−t)

− 1)

]

=(T − t)

�2−

1

2�3

{

(e−�(T−t)− 1)2 − 2(e−�(T−t)

− 1)}

.

33 / 116



The Vasicek Model

The Vasicek Model

a(t, T ) =

(�

�− �2

2�2

)(T − t) +

(�

�2− �2

2�3

)(e−�(T−t) − 1)

+�2

4�3(e−�(T−t) − 1)2. (33)

The corresponding expression for the yield to maturity is

�(t, T ) =− lnP (t, T )/(T − t) = (a(t, T ) + b(t, T )r(t))/(T − t)

=

(�

�− �2

2�2

)(1 +

e−�(T−t) − 1

�(T − t)

)

+�2

4�3(e−�(T−t) − 1)2

T − t+

(1− e−�(T−t))

T − tr(t). (34)

34 / 116



The Vasicek Model

The Vasicek Model

By letting (T − t) → ∞ we find that the yield at infinitematurity is given by 7

�∞ =�

�− �2

2�2. (35)

One may then express the bond price as

P (r, t, T ) = exp

[(e−�(T−t) − 1)

�(r − �∞)− �∞(T − t)

− �2

4�3(e−�(T−t) − 1)2

]. (36)

7One could use this insight to infer a value for the unknown factor �from the currently observed yield curve, from which one could obtain anestimate of �∞.

35 / 116



The Vasicek Model

The Vasicek Model

The Vasicek model is now of historical interest, but itcontains all the basic ingredients needed to deal with the moresophisticated models. Namely

∼ a technique to solve the pricing p.d.e.∼ and the idea of relating the parameters of the model to

information that can be obtained from the currentlyobserved yield curve.

The last observation also makes evident one of theshortcomings of the Vasicek model. By setting t = 0 in (36)we obtain

P (r0, 0, T ) = exp

[(e−�T − 1)

�(r0 − �∞)− �∞T − �2

4�3(e−�T − 1)2

].

(37)

36 / 116



The Vasicek Model

The Vasicek Model

If � is chosen so as to match the long term yield �∞ then weonly have two parameters, � and �, left to makeexpression (36) consistent with the entire currentlyobserved yield curve P (r0, 0, T ).

∼ Clearly this is impossible as at the most we could choose � and� to fit two points exactly, or alternatively choose them toobtain some sort of least squares fit.

These observations suggest that one possibility to develop amodel that fits the currently observed yield curve is to makeat least one, if not more, of the quantities �, and �time varying.

∼ We would then have at our disposition a whole set of values ofsay � (if it were allowed to be time varying) with which tomatch the theoretical model to the currently observed yieldcurve.

37 / 116



The Vasicek Model

The Vasicek Model

This is the essential insight of the Hull-White model to whichwe turn in the next subsection.

However before turning to the Hull-White model we considerwhat the solution (25) implies for the bond price dynamics.

We know that under ℙ the bond price dynamics are given by(13). However at that point in our development we did nothave an explicit expression for ∂P

∂r .

∼ The solution (25) now enables us to calculate this expression,in fact it is given in (27).

∼ Substituting this expression into (13)

dP

P= (r − ��b(t, T ))dt− �b(t, T )dz. (38)

38 / 116



The Vasicek Model

The Vasicek Model

The last eqn. indicates that the standard deviation of bondreturn is −�b, the minus sign simply indicates that a positiveshock to the interest rate dynamics (i.e. a positive dz) resultsin a negative shock to the bond price dynamics. This ismerely a reflection of the fact that interest rates andbond prices are inversely related.

As we have seen in the discussion of the general case inSection 23.2, eqn. (38) can be transformed, under theequivalent measure ℙ, to

dP

P= rdt− �b(t, T )dz. (39)

∼ The last eqn. leads, as we saw in Section 23.2 to themartingale representation (21).

39 / 116



The Vasicek Model

The Vasicek Model

Furthermore the interest rate dynamics under ℙ becomes

dr = (� − �r)dt+ �dz, (40)

where we recall that � has already been defined as� = � − ��.

These are the dynamics with respect to which theexpectation Et in (21) is to be calculated.

40 / 116



The Hull-White Model


Hull-White (1990) take as the process for the short rate

dr = �(t)( (t)− r)dt+ �(t)dz. (41)

The difference from the Vasicek model being the timedependence of the coefficients �(t), (t) and �(t), themotivation being the points discussed at the conclusion of theprevious subsection.

The bond-pricing p.d.e. (10) now becomes

∂P

∂t+ (�(t)− �(t)r)

∂P

∂r+

1

2�2(t)

∂2P

∂r2− rP = 0, (42)

∼ where we set�(t) = �(t) (t)− ��(t). (43)

41 / 116





The only difference from the p.d.e. (24) being the timedependence of the coefficients �(t),�(t) and �(t).

∼ It seems not unreasonable to attempt again a solution of theform (25). In fact precisely the same manipulations yield forthe time coefficients a(t, T ) and b(t, T ) the ordinarydifferential eqns.

bt = �(t)b − 1, (44)

and

at = −b�(t) + 1

2�(t)2b2, (45)

∼ the only difference being that the two ordinary differentialeqns. we must solve now have time varying coefficients.

42 / 116





If we define

K(t) =

∫ t

0�(s)ds, (46)

∼ then the solution to (44) can be written8

b(t, T ) =

∫ T

t

eK(t)−K(s)ds. (47)

8Note that ddtK(t) = �(t) so that d

dteK(t) = eK(t)�(t). Multiplying across

(44) by e−K(t) and re-arranging we obtain

d

dt(e−K(t)b(t, T )) = −e−K(t).

The result then follows by integrating t to T and following manipulationssimilar to those in footnote 5.

43 / 116





Substituting (47) into (45) and integrating t to T yields

a(t, T ) =

∫ T

tb(s, T )�(s)ds− 1

2

∫ T

t�2(s)b(s, T )2ds. (48)

For general forms of the functions �(t), �(t) and �(t) it maybe necessary to perform numerically the integrations in (47)and (48).

In fact to perform the integrations we would need to also havesome functional form for �, and this would be difficult toobtain.

∼ It turns out that we can instead find from market datathe function �(t) (which contains �) and this is (together with�(t) and �(t)) all we need to use the bond pricing formula.

44 / 116





Consider the bond price dynamics implied by the bond pricingformula (13) with a(t, T ) and b(t, T ) now given by (47) and(48).We follow exactly the corresponding manipulations for theVasicek model that led to eqn. (38), which are not alteredby the fact that �, and � are now time varying.From the general expression (25) for the bond price andequation (4)

�P (t, T ) = −�(t)b(t, T ). (49)

Thus we obtaindP

P= (r − ��(t)b(t, T ))dt − �(t)b(t, T )dz, (50)

Note the time dependence of �(t) and the fact that b(t, T ) isgiven by eqn. (47).

45 / 116





Under the equivalent measure ℙ the bond price dynamics are

dP

P= rdt− �(t)b(t, T )dz, (51)

∼ which leads to the martingale representation (21) as we haveshown in the general case in Section ??.

The interest rate dynamics under which Et is calculated aregiven by (after setting dz = dz − �(t)dt in (41))

dr = (�(t)− �(t)r)dt+ �(t)dz. (52)

46 / 116



The Cox-Ingersoll-Ross (CIR) Model


Cox, Ingersoll, and Ross (1985a)(CIR) consider the interestrate process

dr = �(t)( − r)dt+ �√rdz. (53)

As we discussed in Section 22.3 this process guaranteesnon-negative (or positive if �(t) > �2/2) spot interest ratesample paths.

Using a dynamic general equilibrium framework.

In fact CIR employ a dynamic general equilibrium frameworkto derive the bond pricing equation and under specificassumptions about investor preferences, end up with a marketprice of risk given by (54).

47 / 116





In order to obtain a tractable bond pricing equation weassume that the market price of interest rate risk is a functionof r given by

�(r) = �√r, (54)

where � is a constant.The pricing p.d.e. (10) becomes

1

2�2r

∂2P

∂r2+(�(t) − (�(t)+��)r)

∂P

∂r+∂P

∂t− rP = 0. (55)

Given the very similar structure to the p.d.e. encountered inthe Vasicek and Hull/White models (the only difference is ther in front of the second derivative) we again try a solution ofthe same form viz.

P (t, T ) = e−rb(t,T )−a(t,T ). (56)

48 / 116





Again, the condition P (T, T ) = 1 can only be guaranteed if

b(T, T ) = 0, a(T, T ) = 0. (57)

We note also that here

∂P

∂r= −bP, ∂2P

∂r2= b2P,

∂P

∂t= (−rbt − at)P,

∼ which upon substitution into (55) and re-arrangement of termsyields

[−�(t) b− at]+r

[1

2�2b2 + (�(t) + ��)b − bt − 1

]= 0. (58)

49 / 116





In order that this relation hold for all r and all t it must bethe case that

1

2�2b2 + (�(t) + �)b− bt − 1 = 0, (59)

∼ and

− �(t) b− at = 0. (60)

50 / 116





The difference compared to the solution of the Hull-Whitemodel is the b2 term in the ordinary differential equation (59).

This is in fact the well-known Ricatti ordinary differentialequation whose solution is known. We show in Appendix ??that the solution to (59) is

b(t, T ) =2

�2[1− e−�(T−t)]

[�1e−�(T−t) − �2], (61)

where

�1 = −(�(t) + �)

�2+

�

�2, �2 =

−(�(t) + �)

�2− �

�2,

� =√

(�(t) + �)2 + 2�2.

Equation (61) appears in many different forms in theliterature.

51 / 116





Eqn. (60) may be written

da

dt= −�(t) b(t, T ),

∼ which upon integration from t, T yields (using a(T, T ) = 0)

− a(t, T ) = −�(t) ∫ T

t

b(s, T )ds. (62)

We show in Appendix ?? that (62) integrates to

a(t, T ) =2�(t)

��2

[−� (T − t)

�1− (�1 − �2)

�1�2ln(

�1 − �2e�(T−t)

�1 − �2)

].

(63)

∼ There are also many alternative representations of (63) in theliterature.

52 / 116





We saw in the discussion on the Hull-White model, that inorder to be able to calibrate the model to market data weneeded the additional flexibility required by allowing thecoefficients in (53) to be time varying.

We can adopt exactly the same procedure with the CIR modelso that any or all of the coefficients �,�(t), and � in thepartial differential equation (55) become time varying.

Again we try a solution of the form (56) and eqns. (59) and(60) still emerge as the eqns. determining the coefficients band a. Only now it needs to be borne in mind that thecoefficients �, �(t), � and are time-varying.

53 / 116





We show in the appendix that the functional form (61) is stillvalid for b(t, T ) except that the constant � is replaced by thetime averaged function

�(t, T ) =1

T − t

∫ T

t�(s)ds. (64)

The expression for a(t, T ) can only be left as the integral

a(t, T ) =

∫ T

t�(s) (s)b(s, T )ds, (65)

∼ since the integration would in general be impossible analyticallybecause �(t, T ) could be a quite complicated time function.

54 / 116





From (56) we can readily calculate that

∂P

∂r= −b(t, T )P

∼ hence eqn. (15) for the risk neutral bond price dynamics in theCIR case become

dP

P= rdt− �b(t, T )

√rdz. (66)

The interest rate dynamics under the equivalent measure ℙareobtained by setting dz = dz − �

√rdt in (53) and so are given

bydr = (�(t)− �(t)r)dt+ �

√rdz

where

�(t) = �(t) (t) and �(t) = �(t) + ��.55 / 116



Calculation of the Bond Price from the Expectation Operator

Calculation of the Bond Price from the Expectation

Operator

We have seen in Section 23.4 how to obtain an explicitexpression for the bond price by solving the pricing p.d.e. (10)under various assumptions about �r and �r.

It is also of interest to see how to obtain the same result bystarting from the martingale or expectation operatorexpression (21).

The particular spot interest rate models with which we areworking provide one of the rare instances where we can carryout analytically, both the solution of the p.d.e. and thecalculation of the expectation operator.

The key to carrying out the expectation operation in(21) is to determine the distributional characteristics of∫ Tt r(s)ds under ℙ.

56 / 116





Operator

We shall now show that in the case of the Hull-White modelthis quantity is normally distributed with mean and variancethat we calculate below.∼ We know from Chapter 6 how to calculate the expectation of

the exponential of a normally distributed random variable.

The appropriate interest rate dynamics are given by eqn. (52),which using the quantity K(t) defined by eqn. (46), can bewritten

d(r(t)eK(t)) = eK(t)�(t)dt+ eK(t)�(t)dz.

Integrating from t to s(< T ) and re-arranging we find that

r(s) = r(t)eK(t)−K(s)+

∫ s

t

eK(u)−K(s)�(u)du+

∫ s

t

eK(u)−K(s)�(u)dz(u).

(67)57 / 116





Operator

b b bt s T

Figure 3: The Region of Integration for equation (67)

58 / 116





Operator

Next integrate eqn. (67) from t to T to obtain∫ T

t

r(s)ds =r(t)

∫ T

t

eK(t)−K(s)ds+

∫ T

t

(∫ s

t

eK(u)−K(s)�(u)du

)ds

+

∫ T

t

(∫ s

t

eK(u)−K(s)�(u)dz(u)

)ds.

Interchanging the order of integration in the second integral andapplying Fubini’s theorem Section 22.4 version III is being usedhere), the last eqn. becomes∫ T

t

r(s)ds =r(t)

∫ T

t

eK(t)−K(s)ds+

∫ T

t

(∫ T

u

eK(u)−K(s)ds

)�(u)du

+

∫ T

t

(∫ T

u

eK(u)−K(s)ds

)�(u)dz(u). (68)

59 / 116





Operator

Using the definition of b(t, T ) at eqn. (47), eqn. (68) becomes∫ T

t

r(s)ds = b(t, T )r(t)+

∫ T

t

b(u, T )�(u)du+

∫ T

t

b(u, T )�(u)dz(u).

(69)

Eqn. (69) implies that∫ T

tr(s)ds is normally distributed

(conditional on information at time t) since the coefficients on theright-hand side are at most time functions or involve realisedstochastic quantities (in this case r(t)).∼ The mean, M(t), and variance V 2(t), are easily calculated to

be

M(t) = b(t, T )r(t) +

∫ T

t

b(u, T )�(u)du, (70)

and

V 2(t) =

∫ T

t

b(u, T )2�2(u)du. (71)60 / 116





Operator

From the above discussion we can assert that (under ℙ)

∫ T

t

r(s)ds ∼ N(M(t), V 2(t)), (72)

and so

−∫ T

t

r(s)ds ∼ N(−M(t), V 2(t)). (73)

61 / 116





Operator

Finally using the results of (iv) in Section 6.3 we obtain the result

P (r, t, T ) = Et[e−

∫

T

tr(s)ds]

= e−M(t)+ 12V

2(t)

= exp

[−b(t, T )r(t)−

∫ T

t

b(u, T )�(u)du

+1

2

∫ T

t

b(u, T )2�2(u)du

]

= exp[−b(t, T )r(t)− a(t, T )], (74)

by making use of the definition of a(t, T ) in eqn. (48).

We see that in eqn. (74) we have recovered the bond pricingformula (25) obtained by solving the p.d.e. (42) .

62 / 116


Pricing Bond Options


We continue to assume that the short-term rate follows theprocess (2). We also assume that there are no risklessarbitrage opportunities in the bond market.

Thus the price of the discount bond of any maturity is stillgiven by the solution to the p.d.e. (10).

Let C(r, t) denote the price at time t of a call option ofmaturity TC written on a bond having maturity T (> TC).

63 / 116




b b bt TC T

bond option

matures

underlying bond

matures

Figure 4: Time Line for The Bond Option Problem

64 / 116




By Ito’s lemmadC

C= �Cdt+ �Cdz, (75)

where

�C =1

C

(∂C

∂t+ �r

∂C

∂r+

1

2�2r∂2C

∂r2

), (76)

�C =�rC

∂C

∂r. (77)

Consider an investor who at time t invests $1 in a hedgeportfolio containing the bond of maturity T held in the dollaramount QP and the option of maturity TC held in thedollar amount QC .∼ The dollar return on this hedge portfolio over time interval dt

is given by65 / 116




dollar return on thehedge portfolio

}= QP

dP

P+QC

dC

C

= (QP�P +QC�C)dt+ (QP�P +QC�C)dz.

The hedge portfolio is rendered riskless by choosing QP ,QC

such that

QP

QC= −�C

�P. (78)

66 / 116




The absence of riskless arbitrage means that the hedgeportfolio can only earn the same return as the original $1invested at the risk-free rate;

(QP�P +QC�C)dt = 1 ⋅ rdt. (79)

Recalling that QP +QC = 1, the conditions (78) and (79)imply

�C − r

�C=�P − r

�P. (80)

But by eqn. (9) we know that in an arbitrage-free bondmarket (�P − r)/�P is equal to the market price of interestrate risk.

67 / 116




Thus we arrive at the no-riskless arbitrage condition betweenthe option and bond markets, viz.

�C(t, s)− r(t)

�C(t, s)=�P (t, s)− r(t)

�P (t, s)= �(r, t). (81)

This has the now familiar interpretation that in the absence ofriskless arbitrage the excess return risk adjusted on both thebond and the option are equal. The common factor to whichthey are equal is the market price of risk of the spot interestrate, the underlying factor.

Eqn. (81) yields the p.d.e. (10) for the bond price P , and forthe option price C, the p.d.e.

∂C

∂t+ (�r − ��r)

∂C

∂r+

1

2�2r∂2C

∂r2− rC = 0, (82)

68 / 116




∼ which in the case of a European call option (with exercise priceE) on the bond must be solved on the time interval 0 < t < TCsubject to the boundary conditions

C(r(TC), TC) = max[0, P (r(TC), TC , T )− E],

C(∞, t) = 0.(83)

∼ The last condition is a consequence of the result that

P (∞, t, T ) = 0,

i.e. the bond value declines to zero as the interest rate becomeslarge.

69 / 116




Note the two-pass structure of the solution process.

∼ We must first solve the partial differential equation (10) withboundary condition (1) for the bond price P (r(s), s, T ) on thetime interval TC ≤ s ≤ T .

∼ The value P (r(TC), TC , T ) is used in the solution of thepartial differential equation (82) (in fact the same partialdifferential equation) via the boundary condn. in (83); this twopass procedure illustrated in Fig.5.

70 / 116




C(r(t), t)

b b bt TC T︸︷︷︸︸︷︷︸

solve option pricing p.d.e

subject to

C(r(TC ), TC ) = ℎ[P (r(TC), TC , T )]

︷︸︸︷

solve bond pricing p.d.e

subject to P (r(T ), T, T ) = 1

P (r(TC), TC , T )︸︷︷︸

Figure 5: The two-pass procedure for solving the bond option pricingproblem

71 / 116




In order to obtain the martingale representation for the optionprice we follow almost identical steps to those we followed inSection 23.3 to obtain the martingale representation for thebond price.

First we observe from the no-arbitrage condition (81) that

�C = r + ��C . (84)

72 / 116




Substituting (84) into (75) the arbitrage free option pricedynamics are given by

dC

C= (r + ��C)dt+ �Cdz.

The last eqn. may in turn be written in terms of z(t) (seeeqn. (14)) as

dC

C= rdt+ �Cdz(t), (85)

∼ where we recall that under the equivalent measure ℙ, thequantity z(t) is a standard Wiener process.

73 / 116




If we set

Y (r, t) =C(r, t)

A(t),

the option price measured in units of the money marketaccount, then from Ito’s lemma

dY

Y=�rC

∂C

∂rdz(t).

The last eqn. implies that Y is a martingale under ℙ,thus

Y (r, t) = Et[Y (r(TC), TC)],

∼ which in terms of the option price itself can be expressed as

C(r, t) = Et[e−

∫ TCt

r(s)dsC(r(TC), TC)]. (86)

74 / 116




If for example we wish to price a European call option on abond then the maturity condition is

C(r(TC), TC) = max[0, P (r(TC ), TC , T )−X].

The interest rate dynamics under ℙ are still given by eqn.(22), viz.

dr = (�r − ��r)dt+ �rdz.

Application of the Feynman-Kac formula to (86) (seeProposition 8.3 will take us back to the option pricing p.d.e.(82).

Recalling the discussion about stochastic discounting under ℙat the end of Section 23.3 we see that equation (86) has anobvious expected (under ℙ) discounted payoff interpretation.

75 / 116




One of the difficulties with evaluating the expectation in (86)is that one needs the joint distribution of

exp(−∫ TC

tr(s)ds)

andC(r(TC), TC).

Calculation of this joint distribution may be quite difficult.

76 / 116




b b bt TC T

P (t, TC)

P (t, T )

bond prices observed at time t

Figure 6: Using P (t, TC) as numeraire - the forward measure

77 / 116




A simpler calculation may be obtained by using the so-calledforward measure9, which consists in choosing as numeraire abond of maturity TC (see Fig.6).

That is we consider

Y (r, t, TC , T ) =C(r, t, TC , T )

P (r, t, TC ), (87)

9The measure ℙ∗ that we develop below is known as the forward measure

because under this measure the instantaneous forward rate equals the expectedfuture forward rate, as we show in Section (25.5).

78 / 116




The dynamics of Y under ℙ are given by (see Section 6.6 and recallthat under ℙ the dynamics for P and C are given respectively byequations (15) and (85))

dY

Y= −�P (t, TC)(�C − �P (t, TC))dt+ (�C − �P (t, TC))dz.

(88)

Equation (88) may be rearranged to

dY

Y= (�C − �P (t, TC))(dz − �P (t, TC)dt).

Following the discussion in Section 20.1 we can define a new process

z∗(t) = z(t)−∫ t

0

�P (u, TC)du, (89)

and a new measure ℙ∗ such that z∗(t) is a Wiener process under

this measure.79 / 116




Thus the dynamics for Y become

dY

Y= (�C − �P (t, TC))dz

∗,

and it follows that Y is a martingale under ℙ∗.

Using E∗t to denote expectations under ℙ∗, formed at time t, then

Y (r(t), t, TC , T ) = E∗

t

[Y (r(TC), TC , TC , T )

], (90)

Upon use of the definition of Y we obtain

C(r(t), t, TC , T ) = P (r(t), t, TC) E∗

t

[C(r(TC), TC , TC , T )

]. (91)

The difference between the expressions (86) and (91) for the valueof the bond option lies in the way the stochastic discounting is done.

80 / 116




In (86), the stochastic discounting is done along eachstochastic interest rate path from t and TC , and sincethese paths are stochastic this term must appear under theexpectation operator.

In (91) the discounting from t to TC is done using the bondof maturity TC , which is known to the investor at time t andhence this term does not need to appear under theexpectation operator.

It sometimes turns out that the expectation operation in (91)can be calculated explicitly, as we shall see in Section 23.7 forthe Hull-White and CIR models.

81 / 116




If it is necessary to evaluate the expectation in (91) bysimulation then we will need the dynamics for r under ℙ∗.

These are easily obtained by using (89) to replace dz in (22)by dz∗ + �P (t, TC)dt so that

dr =(�r + �r(�P (t, TC)− �)

)dt+ �rdz

∗. (92)

82 / 116




It is also of interest to obtain the dynamics under ℙ∗ for therelative bond price

X(r, t, TC , T ) =P (r, t, T )

P (r, t, TC). (93)

This follows by noting that under ℙ∗ we have

dP (t, T )

P (t, T )=(r + �P (t, T )�P (t, TC)

)dt+ �P (t, T )dz

∗.

Using the results of Section (6.6), we obtain

dX

X=(�P (t, T )− �P (t, TC)

)dz∗. (94)

83 / 116


Solving the Option Pricing Equation


In this section we apply the general spot interest rate pricingframework of Section 23.6 to two special models that yieldclosed form solutions.

First the Hull-White model, which assumes a Gaussian processfor the spot interest rate.

Second the CIR model which assumes a Feller or square rootprocess for the spot interest rate.

In both cases it is convenient to use the bond of optionmaturity as numeraire.

The option pricing formula is basically Black-Scholes in theHull-White case or Black-Scholes like in the CIR case.

84 / 116



The Hull White Model


Recall eqn. (52) that for the Hull-White model the spotinterest rate dynamics under ℙ are given by

dr = (�(t)− �(t)r)dt+ �(t)dz,

∼ where �(t) is defined at eqn. (43).

In this case eqn. (82) becomes

∂C

∂t+ (�(t)− �(t)r)

∂C

∂r+

1

2�2(t)

∂2C

∂r2− rC = 0, (95)

∼ subject to the boundary condition (83).

85 / 116





In turns out that the solution to (95) can be veryelegantly obtained by an application of the change ofnumeraire results of Chapter 20.

∼ Instead of using the money market account as the numeraire,it is more convenient to use the price of the pure discountbond P (r, t, TC) whose maturity date is TC .

From (49) with T = TC the bond return volatility for theHull-White model is

�P (t, TC) = −�(t) b(t, TC) (96)

with b(t, T ) defined by (47).

86 / 116





Consider the specific case of a European call bond option forwhich

C(r(TC), TC , TC , T ) =(P (r, TC , T )− E

)+.

Recalling the definition of the relative bond price, seeequation (93), this payoff may be written

C(r(TC), TC , TC , T ) =(X(TC , TC , T )− E

)+.

Substituting (96) into (94) we find that the dynamics of Xbecome

dX

X= �(t)[b(t, TC )− b(t, T )]dz∗(t). (97)

In terms of the relative bond price X we can express (91) as

87 / 116





C(r, t, T )

P (r, t, TC)= E

∗t [(X(TC , TC , T )− E)+]. (98)

Since the expectation in (98) is with respect to outcomes forthe X variable, the relevant stochastic dynamicsunderlying the probability distn. in the calculation of E∗

t is thes.d.e. (97).

From equation (97) dX/X is normally distributed under ℙ∗,with

E∗t

[dX

X

]= 0, (99)

var∗[dX

X

]= �2(t)[b(t, TC)− b(t, T )]2dt ≡ v2(t)dt. (100)

88 / 116





The calculation of the expectation in (98) with drivingdynamics (97) is simply the Black-Scholes European calloption pricing problem with r = 0, exercise price E andtime varying variance v2(t).Thus

E∗t [(X(TC , TC , T )− E)+] = X(t, TC , T )N (d∗1)−EN (d∗2),

(101)

∼ where

d∗1 =ln(X(t, TC , T )/E) + v2(TC − t)/2

v√TC − t

,

d∗2 = d∗1 − v√TC − t,

v2 =1

TC − t

∫ TC

tv2(s)ds.

89 / 116





Substituting (101) into (98);

C(r, t, T ) = P (r, t, TC )X(t, TC , T )N (d∗1)− EP (r, t, TC )N (d∗2)

= P (r, t, T )N (d∗1)−EP (r, t, TC )N (d∗2)(102)

∼ with d∗1 given by

d∗1 =ln (P (r, t, T )/P (r, t, TC)E) + v2(TC − t)/2

v√TC − t

∼ and d∗2 by

d∗2 = d∗1 − v√TC − t.

By the put-call parity condition, the corresponding put optionprice can similarly be expressed as

U(r, t, T ) = EP (r, t, TC )N (−d∗2)− P (r, t, T )N (−d∗1).90 / 116





The structure of the option pricing formula (102) should becompared with eqn 19.23 in Chapter 19, the one obtained forthe Black-Scholes model with stochastic interest rates.

One sees that they are identical in structure if one replaces theunderlying traded asset (the stock S) of Chapter 19 with theunderlying traded asset (the bond P ) of the current situation.

91 / 116



The CIR Model

The CIR Model

In the case of the CIR model with the interest rate processgiven by (53), eqn. (82) becomes

∂C

∂t+ [�( − r)− ��r]

∂C

∂r+

1

2�2r

∂2C

∂r2− rC = 0. (103)

92 / 116



The CIR Model

The CIR Model

Eqn. (103) can also be solved by using the change ofnumeraire ideas of Chapter 20.

The derivation follows exactly the same lines as in theprevious subsection, the only difference is that now the bondprice dynamics are given by (66).

∼ As a result the dynamics for X are given by

dX

X= v(t)

√rdW ∗, (104)

wherev(t) = �[b(t, T )− b(t, TC)], (105)

with b(t, T ) given by eqn. (61).

93 / 116



The CIR Model

The CIR Model

The Kolmogorov eqn. associated with eqn. (104) is

1

2v2(t)r

∂2�

∂r2+∂�

∂t= 0. (106)

∼ The p.d.f. arising from eqn. (106) is essentially given byequation 22.19 in Chapter 23 in the limit �→ 0.Integration of the call option payoff with respect to this distn.yields the option price.

94 / 116



The CIR Model

The CIR Model

For instance, if the boundary condition is given by (83) theexpression for the option price turns out to be

C(r, t, TC ;T,K)

= P (r, t, T )�2

(2r∗[�+ +B(TC , T )];

4�

�2,

2�2re�(TC−t)

�+ +B(TC , T )

)

(107)

− EP (r, t, TC )�2

(2r∗[�+ ];

4�

�2,2�2re�(TC−t)

�+

),

(108)

95 / 116



The CIR Model

The CIR Model

∼ where

� ≡ ((�+ �)2 + 2�2)1/2,

� ≡ 2�

�2(e�(TC−t) − 1),

≡ �+ �+ �

�2,

r∗ ≡ 1

B(TC , T )

[log

(A(TC , T )

E

)],

�2(⋅) is the noncentral chi-square distn. function and r∗ is thecritical interest rate below which exercise will occur, namely thatobtained by solving E = P (r∗, TC , T ).

96 / 116


Rendering Spot Rate Models


Consider again the model with interest rate dynamics underthe historical measure ℙ given by (41). We know the bondprice is

P (r, t, T ) = e−a(t,T )−b(t,T )r(t), (109)

∼ where

b(t, T ) =

∫ T

t

eK(t)−K(s)ds, K(t) =

∫ t

0

�(s)ds, (110)

a(t, T ) =

∫ T

t

b(s, T )�(s)ds− 1

2

∫ T

t

�(s)2b(s, T )2ds, (111)

∼ and�(t) = �(t) (t)− �(t)�(t). (112)

97 / 116




In order to use this model we need estimates (from marketdata) for �(t), �(t) and �(t).

Note that �(t) impounds in itself the functions (t) and�(t), which do not therefore need to be separatelyestimated, at least for the purposes of pricing derivativesecurities.

We assume that we already have estimates of �(t) from theprices of interest rate caps using the corresponding optionpricing formula (see Section 23.7), thus it only remains todetermine �(t) and �(t).

We assume that we also have available market information onthe volatility of bonds returns of all maturities at time 0.

We know from eqns. (13) and (109) that the volatility ofbond returns is −�(t)b(t, T ).

98 / 116




∼ Thus we assume �(0)b(0, T ) is given as a function of maturityT . Putting t = 0 in eqn. (83) we have

b(0, T ) =

∫ T

0e−K(s)ds. (113)

Differentiating with respect to maturity T yields

K(T ) = − ln

(∂

∂Tb(0, T )

),

∼ and since K′(t) = �(t), we obtain

�(T ) = − ∂

∂T

[ln

(∂

∂Tb(0, T )

)]. (114)

99 / 116




Next set t = 0 in the bond pricing eqn. so that

P (r0, 0, T ) = e−a(0,T )−b(0,T )r0 . (115)

The function P (r0, 0, T ) would be available from the currentlyobserved yield curve. Consider (115) in the form

a(0, T ) = − lnP (r0, 0, T )− b(0, T )r0. (116)

From the last eqn. a(0, T ) can be considered as known (frommarket data) as a function of T , we shall further assume thatthis function is sufficiently smooth to be at least twicedifferentiable.

Thus our remaining task is to determine the function �(t).

100 / 116




We recall from (111) that

a(0, T ) =

∫ T

0b(s, T )�(s)ds− 1

2

∫ T

0�2(s)b(s, T )2ds. (117)

The second term on the right hand side, perhaps vianumerical integration, will simply be a known function of T .

Thus eqn. (117) constitutes an integral eqn. for theunknown function �.By a process of successive differentiations we find that

�(T ) = e−K(T ) ∂

∂T

(

eK(T ) ∂

∂Ta(0, T )

)

+e−K(T ) ∂

∂T

(

eK(T ) ∂

∂T

(

1

2

∫

T

0�2(s)b(s, T )

2ds

))

.

(118)

Whilst equation (118) involves awkward looking algebraicexpressions, its numerical evaluation would be a routine task.

101 / 116




Consider the case where �, � are constant, so that �(t) is theonly time varying parameter.

Now we simply have K(t) = �t, and hence

b(t, T ) =

∫ T

te�t−�sds

=1

�(1− e�(t−T )), (119)

from which

b(0, T ) =1

�(1− e−�T ). (120)

102 / 116




Furthermore

a(t, T ) =

∫ T

tb(s, T )�(s)ds− �2

2

∫ T

tb2(s, T )ds, (121)

and so

a(0, T ) =

∫ T

0b(s, T )�(s)ds− �2

2

∫ T

0b2(s, T )ds. (122)

Differentiating (122) with respect to T we obtain

∂a(0, T )

∂T=

∫ T

0

�(s)∂

∂T

(

1

�(1− e−�(T−s))

)

ds−�2

2

∂

∂T

(∫ T

0

b2(s, T )ds

)

= e−�T

∫ T

0

�(s)e�sds−�2

2

∂

∂T

(∫ T

0

b2(s, T )ds

)

. (123)

103 / 116




Now, using (119), equation (122) may be written

a(0, T ) =

∫ T

0

1

�(1 − e−�(T−s))�(s)ds− 1

2

∫ T

0

�2b2(s, T )ds (124)

=1

�

∫ T

0

�(s)ds − e−�T

�

∫ T

0

e�s�(s)ds − 1

2

∫ T

0

�2b2(s, T )ds.

Using (124) to eliminate the e−KT∫ T

0 �(s)eKsds term in (123) weobtain

∂a(0, T )

∂T= −�a(0, T ) +

∫

T

0�(s)ds −

��2

2

∫

T

0b2(s, T )ds −

�2

2

∂

∂T

(∫

T

0b2(s, T )ds

)

,

which upon re-arrangement yields

∫

T

0�(s)ds =

∂a(0, T )

∂T+ �a(0, T ) +

�2

2

[

�

∫

T

0b2(s, T )ds +

∂

∂T

(∫

T

0b2(s, T )ds

)]

.

104 / 116



Rendering Spot Rate ModelsDifferentiating the last equation with regard to T yields,

�(T ) =∂

∂T

[

∂a(0, T )

∂T+ �a(0, T )

]

+�2

2

∂

∂T

[

�

∫

T

0b2(s, T )ds +

∂

∂T

(∫

T

0b2(s, T )ds

)]

.

With the function �(T ) now at our disposal we can computethe time function a(t, T ) and hence bond prices calibrated tomarket data.

In the case of the CIR model, the steps taken to calibrate themodel to the initial yield curve and cap and swaption data forexample are similar to the Hull-White model. We do notprovide details here.

105 / 116



Appendix 1. Solution of the Ordinary Differential Equation (59)

106 / 116



Solution of the Ordinary Differential Equation

Solving for b(t, T )

Consider the ordinary differential eqn.

db

dt= �0b

2 + �1b− 1 = �0

[b2 +

�1

�0b− 1

�0

]. (125)

∼ The quadratic in the brackets on the RHS can be factorised as

b2 +�1

�0b− 1

�0= (b− �1)(b− �2),

106 / 116




where

�1 = − �1

2�0+

�

2�0,

�2 = − �1

2�0− �

2�0, (126)

� =√�21 + 4�0.

Thus the ordinary differential eqn. (125) can be written

db

dt= �0(b− �1)(b− �2), (127)

∼ or asdb

(b− �1)(b− �2)= �0dt.

107 / 116




With some slight re-arrangement the last eqn. can be written[

1

b− �1− 1

b− �2

]db = �0(�1 − �2)dt = �dt. (128)

Integrating the last eqn. from t to T we obtain[ln

(b− �1b− �2

)]T

t

= �(T − t), (129)

i.e.

ln

(b(T, T )− �1b(T, T )− �2

)− ln

(b(t, T )− �1b(t, T )− �2

)= �(T − t),

∼ which on making use of b(T, T ) = 0 becomes

ln

(b(t, T )− �1b(t, T )− �2

)= ln

(�1�2

)− �(T − t),

108 / 116




i.e.

b(t, T )− �1b(t, T )− �2

= exp

[ln

(�1�2

)− �(T − t)

]

=�1�2e−�(T−t).

Solving the last eqn. for b(t, T ) we obtain

b(t, T ) =�1�2(1− e−�(T−t))

�2 − �1e−�(T−t). (130)

∼ Using the fact that �1�2 = −1/�0 this simplifies slightly to

b(t, T ) =1

�0

[1− e−�(T−t)]

[�1e−�(T−t) − �2]. (131)

109 / 116




Solving for a(t, T )

From equation (62) of Section ??

a(t, T ) = �

∫ T

tb(s, T )ds.

Making the transformation u = �(T − s) we see that

a(t, T ) =+�

�

∫ �(T−t)

0b(T − u

�, T )du.

Substituting the expression (131) for b(t, T ) (and setting� = T − t) we obtain

a(t, T ) = +�

�

2

�2

∫ ��

0

(1− e−u)

(�1e−u − �2)du.

110 / 116




Consider the integral

I =

∫ ��

0

(

1− e−u

�1e−u − �2

)

du

=

∫ ��

0

du

�1e−u − �2−

∫ ��

0

e−udu

�1e−u − �2

=

∫ ��

0

eudu

�1 − �2eu−

∫ ��

0

e−udu

�1e−u − �2

=

[

−1

�2ln(�1 − �2e

u)

]��

0

+

[

1

�1ln(�1e

−u− �2)

]��

0

=(�1 − �2)

�1�2ln(�1 − �2)−

1

�2ln(�1 − �2e

��) +1

�1ln(�1e

−��− �2)

=(�1 − �2)

�1�2ln(�1 − �2)−

1

�2ln(�1 − �2e

��)

+1

�1ln[e−��/(�1 − �2e

�� )]

= −��

�1+

(�1 − �2)

�1�2

{

ln(�1 − �2)− ln(�1 − �2e�� )

}

. 111 / 116




Finally

I = − �

�1� − (�1 − �2)

�1�2ln

(�1 − �2e

��

�1 − �2

),

and so

a(t, T ) =2�

��2

[−�(T − t)

�1− (�1 − �2)

�1�2ln(

�1 − �2e�(T−t)

�1 − �2)

].

Using the fact that �1�2 = −2/�2 and �1 − �2 = 2�/�2 wefinally obtain

a(t, T ) =2�

�2

[−(T − t)

�1+ ln

(�1 − �2e

�(T−t)

�1 − �2

)]. (132)

112 / 116




Allowing For Time Varying Coefficients

The steps leading to (128) remain the same as in the constantcoefficients case, only now �1, �2, � and � become functionsof time. Thus in order to use the same functional form for thesolution we need to define

�(t, T ) =1

T − t

∫ T

t�(s)ds.

Thus integration of (128) will yield (131) with � replaced by�(t, T ).

113 / 116



Appendix 1. Calculating �(T ) in the Calibration of the H-W Model

114 / 116



Calculating �(T ) in the Calibration of the H-W Mode

From (110) we note that

∂b

∂T= eK(t)−K(T ).

Differentiating (117) with respect to T yields

∂

∂Ta(0, T ) = b(T, T )�(T ) +

∫ T

0eK(s)−K(T )�(s)ds

− ∂

∂T

(1

2

∫ T

0�2(s)b(s, T )2ds

)

= e−K(T )

∫ T

0eK(s)�(s)ds− ∂

∂T

(1

2

∫ T

0�2(s)b(s, T )2ds

)

114 / 116




Rearranging

eK(T ) ∂

∂Ta(0, T ) =

∫ T

0eK(s)�(s)ds− eK(T ) ∂

∂T(1

2

∫ T

0�2(s)b(s, T )2ds

).

Differentiating again with respect to T we obtain

∂

∂T

(eK(T ) ∂

∂Ta(0, T )

)= eK(T )�(T ) (133)

− ∂

∂T

(eK(T ) ∂

∂T

(1

2

∫ T

0�2(s)b(s, T )2ds

)). (134)

115 / 116




Rearranging this last equation we obtain

�(T ) = e−K(T ) ∂

∂T

(eK(T ) ∂

∂Ta(0, T )

)

+ e−K(T ) ∂

∂T

(eK(T ) ∂

∂T

(1

2

∫ T

0�2(s)b(s, T )2ds

)).

116 / 116



Cox, J. C., J. E. Ingersoll, and S. A. Ross (1985a).A Theory of the Term Structure of Interest Rates.Econometrica 53(2), 385–406.

Cox, J. C., J. E. Ingersoll, and S. A. Ross (1985b).An Intertemporal General Equilibrium Model of Asset Prices.Econometrica 53(2), 363–384.

Merton, R. C. (1973b).Theory of Rational Option Pricing.Bell Journal of Economics and Management Science 4, 141–183.

Vasicek, O. (1977).An Equilibrium Characterisation of the Term Structure.Journal of Financial Economics 5, 177–188.

116 / 116

Documents

25857 Interest Rate Modelling - WordPress.com · Interest Rate Modelling Introduction Introduction In this chapter we survey models of interest rate derivatives which take the instantaneous