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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
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Interest Rate Modelling
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
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Interest Rate Modelling
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).
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Interest Rate Modelling
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).
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Interest Rate Modelling
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.
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Interest Rate Modelling
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.
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Interest Rate Modelling
Arbitrage Models of the Term Structure
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.
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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)
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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.
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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)
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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.
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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)
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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.
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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)
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Interest Rate Modelling
Arbitrage Models of the Term Structure
Arbitrage Models of the Term Structure
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).
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Interest Rate Modelling
The Martingale Representation
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)
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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)
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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)
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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)
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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).
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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.
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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.
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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.
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Interest Rate Modelling
The Martingale Representation
The Martingale Representation
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
Interest Rate Modelling
Some Specific Term Structure Models
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.
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Interest Rate Modelling
Some Specific Term Structure Models
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)
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Interest Rate Modelling
Some Specific Term Structure Models
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.
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Interest Rate Modelling
Some Specific Term Structure Models
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 ).
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Interest Rate Modelling
Some Specific Term Structure Models
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).
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Interest Rate Modelling
Some Specific Term Structure Models
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).
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Interest Rate Modelling
Some Specific Term Structure Models
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)}
.
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Interest Rate Modelling
Some Specific Term Structure Models
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)
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Interest Rate Modelling
Some Specific Term Structure Models
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 �∞.
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Interest Rate Modelling
Some Specific Term Structure Models
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)
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Interest Rate Modelling
Some Specific Term Structure Models
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.
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Interest Rate Modelling
Some Specific Term Structure Models
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)
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Interest Rate Modelling
Some Specific Term Structure Models
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).
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Interest Rate Modelling
Some Specific Term Structure Models
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Hull-White Model
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)
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Interest Rate Modelling
Some Specific Term Structure Models
The Hull-White Model
The Hull-White Model
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Hull-White Model
The Hull-White Model
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Hull-White Model
The Hull-White Model
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Hull-White Model
The Hull-White Model
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).
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Interest Rate Modelling
Some Specific Term Structure Models
The Hull-White Model
The Hull-White Model
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)
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
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).
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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)
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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)
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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)
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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.
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Interest Rate Modelling
Some Specific Term Structure Models
The Cox-Ingersoll-Ross (CIR) Model
The Cox-Ingersoll-Ross (CIR) Model
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
Interest Rate Modelling
Some Specific Term Structure Models
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.
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Interest Rate Modelling
Some Specific Term Structure Models
Calculation of the Bond Price from the Expectation Operator
Calculation of the Bond Price from the Expectation
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
Interest Rate Modelling
Some Specific Term Structure Models
Calculation of the Bond Price from the Expectation Operator
Calculation of the Bond Price from the Expectation
Operator
b b bt s T
Figure 3: The Region of Integration for equation (67)
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Interest Rate Modelling
Some Specific Term Structure Models
Calculation of the Bond Price from the Expectation Operator
Calculation of the Bond Price from the Expectation
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)
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Interest Rate Modelling
Some Specific Term Structure Models
Calculation of the Bond Price from the Expectation Operator
Calculation of the Bond Price from the Expectation
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
Interest Rate Modelling
Some Specific Term Structure Models
Calculation of the Bond Price from the Expectation Operator
Calculation of the Bond Price from the Expectation
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)
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Interest Rate Modelling
Some Specific Term Structure Models
Calculation of the Bond Price from the Expectation Operator
Calculation of the Bond Price from the Expectation
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) .
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Interest Rate Modelling
Pricing Bond Options
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).
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
b b bt TC T
bond option
matures
underlying bond
matures
Figure 4: Time Line for The Bond Option Problem
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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
Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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)
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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)
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
∼ 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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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)
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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)
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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
Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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).
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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
Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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.
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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)
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Interest Rate Modelling
Pricing Bond Options
Pricing Bond Options
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
Interest Rate Modelling
Solving the Option Pricing Equation
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
Interest Rate Modelling
Solving the Option Pricing Equation
The Hull White Model
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
Interest Rate Modelling
Solving the Option Pricing Equation
The Hull White Model
The Hull White Model
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
Interest Rate Modelling
Solving the Option Pricing Equation
The Hull White Model
The Hull White Model
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
Interest Rate Modelling
Solving the Option Pricing Equation
The Hull White Model
The Hull White Model
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
Interest Rate Modelling
Solving the Option Pricing Equation
The Hull White Model
The Hull White Model
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
Interest Rate Modelling
Solving the Option Pricing Equation
The Hull White Model
The Hull White Model
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
Interest Rate Modelling
Solving the Option Pricing Equation
The Hull White Model
The Hull White Model
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.
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Interest Rate Modelling
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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)
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Interest Rate Modelling
Solving the Option Pricing Equation
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).
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Interest Rate Modelling
Solving the Option Pricing Equation
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.
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Interest Rate Modelling
Solving the Option Pricing Equation
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)
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Interest Rate Modelling
Solving the Option Pricing Equation
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 ).
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Interest Rate Modelling
Rendering Spot Rate Models
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)
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Interest Rate Modelling
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Rendering Spot Rate Models
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 ).
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Rendering Spot Rate Models
∼ 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)
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Interest Rate Modelling
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Rendering Spot Rate Models
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).
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Interest Rate Modelling
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Rendering Spot Rate Models
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.
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Interest Rate Modelling
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Rendering Spot Rate Models
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)
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Interest Rate Modelling
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Rendering Spot Rate Models
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)
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Interest Rate Modelling
Rendering Spot Rate Models
Rendering Spot Rate Models
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
Interest Rate Modelling
Rendering Spot Rate Models
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.
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Interest Rate Modelling
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Appendix 1. Solution of the Ordinary Differential Equation (59)
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Interest Rate Modelling
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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),
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Interest Rate Modelling
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Solution of the Ordinary Differential Equation
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.
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Interest Rate Modelling
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Solution of the Ordinary Differential Equation
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),
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Interest Rate Modelling
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Solution of the Ordinary Differential Equation
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)
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Interest Rate Modelling
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Solution of the Ordinary Differential Equation
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.
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Interest Rate Modelling
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Solution of the Ordinary Differential Equation
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
Interest Rate Modelling
Rendering Spot Rate Models
Solution of the Ordinary Differential Equation
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)
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Interest Rate Modelling
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Solution of the Ordinary Differential Equation
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 ).
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Appendix 1. Calculating �(T ) in the Calibration of the H-W Model
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Interest Rate Modelling
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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
)
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Interest Rate Modelling
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Calculating �(T ) in the Calibration of the H-W Mode
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)
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Interest Rate Modelling
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Calculating �(T ) in the Calibration of the H-W Mode
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
)).
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Interest Rate Modelling
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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.
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