1 5. Options on bonds (February 29) Introduction The purpose of this chapter is to give some complements of the interest theory given in the Shreve book "Stochastic Calculus for Finance II, Continuous- Time Models". In particular, we want to draw attention to the simple Vasic- Hull-White short rate model since it allows many explicit computations. In the end of the chapter we will also derive the call price of a bond in the HJM approach to the problem. Throughout this chapter, if possible, we will use the same notation as in the Shreve book and it is as follows. A T -bond or zero coupon bond with maturity date T pays its holder the amount 1 at the date T and its price at time t is denoted by B(t; T ): The yield between times t and T is dened to be Y (t; T )= 1 T t ln B(t; T );t<T or, equivalently, B(t; T )= e Y (t;T )(T t) : We assume the limit lim T !t + Y (t; T ) exists and denote it by Y (t; t) or R(t): Here R(t) is called the short rate at time t: The yield curve at time t is dened by the equation y = Y (t; T );T t: Assuming B(t; T ) smooth as a function of T , the instantaneous forward rate with maturity T; contracted at time t; is dened by f (t; T )= @ ln B(t; T ) @T :
Transcript
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5. Options on bonds(February 29)
Introduction
The purpose of this chapter is to give some complements of the interest theorygiven in the Shreve book "Stochastic Calculus for Finance II, Continuous-Time Models". In particular, we want to draw attention to the simple Vasiµc-Hull-White short rate model since it allows many explicit computations. Inthe end of the chapter we will also derive the call price of a bond in the HJMapproach to the problem.Throughout this chapter, if possible, we will use the same notation as in
the Shreve book and it is as follows.A T -bond or zero coupon bond with maturity date T pays its holder the
amount 1 at the date T and its price at time t is denoted by B(t; T ): Theyield between times t and T is de�ned to be
Y (t; T ) = � 1
T � t lnB(t; T ); t < T
or, equivalently,B(t; T ) = e�Y (t;T )(T�t):
We assume the limitlimT!t+
Y (t; T )
exists and denote it by Y (t; t) or R(t): Here R(t) is called the short rate attime t:The yield curve at time t is de�ned by the equation
y = Y (t; T ); T � t:
Assuming B(t; T ) smooth as a function of T , the instantaneous forwardrate with maturity T; contracted at time t; is de�ned by
f(t; T ) = �@ lnB(t; T )@T
:
2
Since B(t; t) = 1 integration from t to T yields
B(t; T ) = e�R Tt f(t;u)du:
Thus
Y (t; T ) =1
T � t
Z T
t
f(t; u)du; t < T
andY (t; t) = f(t; t):
Suppose we stand at time t and have a worthless portfolio with short oneT -bond and long B(t;T )
B(t;T+�)(T + �)-bonds. Then we must pay the amount 1
at time T since the T -bond matures and we receive the amount B(t;T )B(t;T+�)
attime T + � since the (T + �)-bonds mature. Note that
f(t; T ) = lim�!0+
ln(B(t; T )=B(t; T + �))
�
which explains why we call f(t; T ) the instantaneously forward rate withmaturity T; contracted at time t:The money market account at time t equals
M(t) = eR t0 R(s)ds:
Note thatdM(t) = R(t)M(t)dt:
Investing in the money account may be seen as a self-�nancing rolling overstrategy, which at time t consists of (t+ dt)-bonds.
5.1 Derivation of T-bond dynamics in short rate models
Recall that (t; R(t)) is the initial point on the graph of the yield curve
y = Y (t; T ); T � t:
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Needless to say it is not very likely that every point on this graph is adeterministic function of (t; R(t)): However, this is the starting point in socalled short rate models, which will be our next concern.Assume the short rate is given by the equation
where W is a real-valued standard Brownian motion and �(t; r) and �(t; r)are deterministic functions of (r; t): Furthermore, it will be assumed thatB(t; T ) = F (t; R(t); T ), where F (t; r; T ) is a deterministic function of (t; r; T ):Sometimes it is natural to write F T (t; R(t)) instead of F (t; R(t); T ):Our next aim is to derive a di¤erential equation satis�ed by F T using
an appropriate form of �-hedging. To this end we try to hedge the T -bondwith the money market account and the U -bond, where U is any �xed timewith T < U � �T ; and consider a portfolio with long one T -bond and short� U -bonds. The portfolio value at time t equals
�(t) = F (t; R(t); T )��F (t; R(t); U)
and by Itô�s lemma the noisy part of the di¤erential
d�(t) =
�@F T
@tdt+
@F T
@rdR +
�2
2
@2F T
@r2dt
���
�@FU
@tdt+
@FU
@rdR +
�2
2
@2FU
@r2dt
�disappears if
� =@FT
@r@FU
@r
:
With this choice the portfolio is risk free and, as usual assuming no arbitrages,we get
d�(t) = R(t)�(t)dt = �(t)dM(t)
M(t):
Hence �@F T
@tdt+
�2
2
@2F T
@r2dt
���
�@FU
@tdt+
�2
2
@2FU
@r2dt
�= R(t) fF (t; R(t); T )��F (t; R(t); U)g dt
or@FT
@t+ �2
2@2FT
@r2�RF T
@FT
@r
=@FU
@t+ �2
2@2FU
@r2�RFU
@FU
@r
4
where both sides are independent of T and U: Therefore, we introduce
�(t) = �(t; R(t)) =@FT
@t+ �2
2@2FT
@r2+ �@F
T
@r�RF T
� @FT
@r
and obtain@F T
@t(t; R(t)) +
�2(t; R(t))
2
@2F T
@r2(t; R(t))
+(�(t; R(t))� �(t; R(t))�(t; R(t))@FT
@r(t; R(t))�R(t)F T (t; R(t)) = 0
and, in addition, we have
F T (T;R(T )) = 1:
This context leads us to the equation
@F T
@t(t; r)+
�2(t; r)
2
@2F T
@r2(t; r)+(�(t; r)��(t; r)�(t; r))@F
T
@r(t; r)�rF T (t; r) = 0
for t < T; r 2 R and with the terminal condition
F T (T; r) = 1:
To �nd a solution in probabilistic terms we assume � satis�es the Novikovcondition and de�ne
~W (t) =W (t) +
Z t
0
�(s; R(s))ds; 0 � t � T
where ~W is a standard Brownian motion under the measure ~P = Z�P: Then
dR(t) = (�(t; R(t))� �(t; R(t))�(t; R(t)))dt+ �(t; R(t))d ~W (t); 0 � t � T
and the Feynman-Kac connection yields:
F (t; r; T ) = ~Ehe�
R Tt R(s)ds j R(t) = r
i:
The above process � is known as the market price of risk (for a discussionon this terminology, see T. Björk, Arbitrage Theory in Continuous Time,Oxford Univ. Press (1998), 246-247).
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We end the section by giving a list of some popular short rate models (ifa parameter depends on time this is written explicitly):
(1) Vasiµcek:dR = (b� aR)dt+ �d ~W
(2) Cox-Ingersoll-Ross:
dR = a(b�R)dt+ �pRd ~W
(3) Brennan-Schwartz
dR = a(b�R) + �Rd ~W
(4) Dothan:dR = aRdt+ �Rd ~W
(5) Black-Derman-Toy:
dR = #(t)Rdt+ �(t)Rd ~W
(6) Ho-Lee:dR = #(t)dt+ �d ~W
(7) Hull-White (extended Vasiµcek):
dR = (#(t)� a(t)R)dt+ �(t)d ~W
(8) Hull-White (extended Cox-Ingersoll-Ross)
dR = (#(t)� a(t)R)dt+ �(t)pRd ~W
Below we will often consider a special case of the Hull-White (extendedVasiµcek) model and assume the short rate dynamics is given by the equation
dR = (#(t)� aR)dt+ �d ~W:
Here a and � are constants and #(t); 0 � t � �T ; a deterministic function.For short this model is called the Vasiµcek-Hull-White short rate model.
Exercises
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1. FindP [R(t) < 0 j R(0)]
in the Vasiµceks-Hull-White short rate model.
5.2 The T -bond in the Vasiµcek-Hull-White short rate model
Theorem 5.2.1. In the Vasiµcek-Hull-White short rate model,
Furthermore it is suitable to choose the function c(t) such that
dc(t) = (#(t)� ac(t))dt
and c(0) = 0: Thus
c(t) = e�atZ t
0
eas#(s)ds:
Next we solve the equation
dX(t) = �aX(t)dt+ �d ~W (t)
with the initial condition X(0) = R(0) and have
X(t) = e�atR(0) + �e�atZ t
0
easd ~W (s):
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The Gaussian process X = (X(t))t�0 has the expectation and covariancefunction
~E [X(t)] = e�atR(0)
and
Cõv(X(s); X(t)) = �2e�a(s+t) ~E�(
Z s
0
eaud ~W (u)
Z t
0
eaud ~W (u))
�
= �2e�a(s+t)Z min(s;t)
0
e2audu =�2
2ae�a(s+t)(e2amin(s;t) � 1);
respectively.Set
Y =
Z T
0
X(t)dt
and note that Y is Gaussian with expectation
~E [Y ] =
Z T
0
~E [X(t)] dt
=
Z T
0
e�atR(0)dt =R(0)
a(1� e�aT )
and variance
Vãr(Y ) = Cõv(Z T
0
X(s)ds;
Z T
0
X(t)dt)
=
Z T
0
Z T
0
Cõv(X(s); X(t))dsdt =�2
2a
Z T
0
Z T
0
e�a(s+t)(e2amin(s;t) � 1)dsdt
=�2
2a3(2aT � 3 + 4e�aT � e�2aT ):
Hence~Ehe�
R T0 R(t)dt
i= e�
R T0 c(t)dt ~E
�e�Y
�=
= e�R T0 e�at(
R t0 e
au#(u)du)dte�R(0)a(1�e�aT )+ �2
4a3(2aT�3+4e�aT�e�2aT )
= eA(0;T )�C(0;T )R(0):
This proves the special case t = 0 and from this, the general case is immedi-ate, which completes the proof of the theorem.
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Recall thatB(t; T ) = e�
R Tt f(t;u)du
and denote by B�(t; T ) the market price of the T -bond at time t: In a realmarket B�(t; T ) is quoted only for �nitely many values on T and the functionB�(t; T ); 0 � t � �T ; has been obtained with the aid of suitable interpolations.Moreover, assume
B�(t; T ) = e�R Tt f�(t;u)du:
In the next step we want to show that the function #(t); 0 � t � �T ; canbe chosen to get a perfect �t of the yield curve at time 0, that is
B(0; T ) = B�(0; T ) if 0 � T � �T
oreA(0;T )�C(0;T )R(0) = e�
R T0 f�(0;u)du if 0 � T � �T :
To this end we must choose the function #(t) so that
A0T (0; T )� C 0T (0; T )R(0) = �f �(0; T ) if 0 � T � �T :
FromC(t; T ) =
1
a(1� e�a(T�t))
it follows thatC 0T (t; T ) = e
�a(T�t)
and since
A(0; T ) =
Z T
0
�1
2�2C2(u; T )� #(u)C(u; T )
�du
we get
A0T (0; T ) =
Z T
0
��2C(u; T )C 0T (u; T )� #(u)C 0T (u; T )
du
=
Z T
0
��21
a(1� e�a(T�u))e�a(T�u) � #(u)e�a(T�u)
�du:
Accordingly from this
f �(0; T ) = C 0T (0; T )R(0)� A0T (0; T )
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= e�aTR(0) +
Z T
0
#(u)e�a(T�u)du� �2
2a2(1� e�aT )2:
To solve for #(t) we assume f �(0; T ) is smooth function of T and di¤er-entiate with respect to T to get
f �0T (0; T ) = �ae�aTR(0) + #(T )� aZ T
0
#(u)e�a(T�u)ds� �2
a(1� e�aT )e�aT :
Now
f �0T (0; T ) + af�(0; T ) = #(T )� �
2
a(1� e�aT ))e�aT � �
2
2a(1� e�aT )2
or
f �0T (0; T ) + af�(0; T ) = #(T )� �
2
2a(1� e�2aT ):
Thus the choice
#(T ) = f �0T (0; T ) + af�(0; T ) +
�2
2a(1� e�2aT ):
implies thatB(0; T ) = B�(0; T ) if 0 � T � �T :
Exercises
1. Show that
Cor(lnB(t; T ); lnB(t; U)) = 1; t < T � U
in the Vasiµcek-Hull-White short rate model.
2. FindlimT!1
Y (t; T )
in the Vasiµcek short rate model.
3. Find f(t; T ) in the Vasiµcek-Hull-White short rate model.
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4. (Vaµcisek short rate model) A derivative of European type pays theamount
Y = max(0;1
T
Z T
0
R(s)ds�R)
at time of maturity T: Find �Y (0).
5.3 Calls on the U-bond in the Vasiµcek-Hull-White short ratemodel
Let 0 � T < U � �T : A European call on the U -bond with time of maturityT and strike K pays out the amount (B(T; U) �K)+ to its holder at timeT: The price of this call at time t equals
call(t;K; T; U) = ~Ehe�
R Tt R(s)dsmax(0; eA(T;U)�C(T;U)R(T ) �K) j R(t)
iwhich is in principle simple to compute explicitely as the random vector(R TtR(s)ds;R(T )) possesses a bivariate normal distribution. However, the
computations are heavy and below we prefer another metod.
Theorem 5.3.1. Suppose T < U . In the Vasiµcek-Hull-White short ratemodel
call(t;K; T; U) = B(t; U)�(d)�B(t; T )K�(d� ��)where
d =1
��ln
B(t; U)
KB(t; T )+1
2��
and
�� =�
a(1� e�a(U�T ))
r1
2a(1� e�2a(T�t)):
PROOF. It is natural to try to hedge the call with the aid of the U -bondand the T -bond. To this end, choose the T -bond as a numéraire and de�ne
for an appropriate non-anticipating process (t; R(t)); 0 � t � T . Thus weare back in a Black-Scholes like model with vanishing interest rate.Let v(t; R(t)) denote the price of the call at time t 2 [0; T ] in the original
numéraire and de�ne
w(t; S(t)) =v(t; R(t))
B(t; T ):
Now at time t consider a portfolio with long one call and short� U -bonds.The portfolio value �(t) in the new numéraire at time t equals
�(t) = w(t; S(t))��S(t)
and by the Itô lemma
d�(t) =@w
@t(t; S(t))dt+
@w
@s(t; S(t))dS(t) +
1
2
@2w
@s2(t; S(t))(dS(t))2 ��dS(t)
or, equivalently,
d�(t) =@w
@t(t; S(t))dt+
@w
@s(t; S(t))dS(t)+
�2(C(t; U)� C(t; T ))2S(t)22
@2w
@s2(t; S(t))dt
��dS(t):Choosing
� =@w
@s(t; S(t))
the in�nitesimal return d�(t) does not contain the noise d ~W (t) and we set(as usual),
d�(t) = �(t)dM0(t)
M0(t)= 0
or, stated otherwise,
@w
@t(t; S(t)) +
�2(C(t; U)� C(t; T ))2S(t)22
@2w
@s2(t; S(t)) = 0:
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This equation holds if
@w
@t(t; s) +
�2(C(t; U)� C(t; T ))2s22
@2w
@s2(t; s) = 0
and since w(T; S(T )) = max(0; S(T )�K) we are led to the terminal condition
w(T; s) = max(0; s�K):
Now we use Example 4.3.1 (with r = 0) and have
w(t; s) = s�(d(s))�K�(d(s)� ��)
whered(s) =
1
�0lns
K+1
2�0
and
�0 = �
sZ T
t
(C(u; U)� C(u; T ))2du:
Thusv(t; R(t)) = B(t; T )w(t; S(t))
= B(t; U)�(d(S(t))�B(t; T )K�(d(S(t))� ��):Finally using the formula
C(t; T ) =1
a(1� e�a(T�t))
we get�0 = ��
which proves the theorem.
A European put on the U -bond with strike K and time of maturity Tpays the amount (K�B(T; U))+ to its holder at time T and its price at timet is denoted by put(t;K; T; U): As
In the Vasiµcek-Hull-White short rate model we thus have the following putprice formula
put(t;K; T; U) = KB(t; T )�(�p � d)�B(t; U)�(�d):
Example 5.3.1. (Vasiµcek model) In this example we assume that #(t) = bis constant and want to �nd the time 0 price �Y (0) of a derivative paying theamount Y = R(T )K at time of maturity T; where K is a positive number.First note that
B(0; T ) = eA(T )�C(T )R(0)
whereC(T ) =
1
a(1� e�aT )
and
A(T ) =(C(T )� T )(ab� �2
2)
a2� �
2C2(T )
4a
The price formula
�Y (0) = ~Ehe�
R T0 R(s)dsR(T )K
iyields
�Y (0) = �K@
@TB(0; T )
whereB(0; T ) = ~E
he�
R T0 R(s)ds
i:
Hence
�Y (0) = �KB(0; T )((e�aT � 1)(ab� �2
2)
a2� �
2C(T )
2ae�aT � e�aTR(0)
)
= KB(0; T )
(C(T )(ab� �2
2)
a+�2C(T )
2ae�aT + e�aTR(0)
)
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= KB(0; T )
�C(T )(b� �
2
2C(T )) + e�aTR(0)
�:
Exercices
1. (Vasiµcek model, that is #(t) = b is constant) Let T and K be positiveconstants. Find the time zero price of a derivative paying the amountY at time T where
Y =
�1 if R(T ) � K0 if R(T ) < K:
5.4. Calls on a �xed coupon bond in the Vasiµcek-Hull-Whiteshort rate model
Suppose T0 < T1 < T2 < ::: < Tn; c1; :::; cn > 0 and N > 0: A �xed couponbond with emission date T0 pays the owner the amount ci at the coupon dateTi for each i = 1; :::; n. In addition, the owner obtains the face value N attime Tn: De�ning
ai =
�ci; i = 1; :::; n� 1cn +N; i = n
the value Bc(t) of the bond at time t 2 [T0; T1[ equals
Bc(t) =nXi=1
aiB(t; Ti):
Next consider a European call on the coupon bond with strike K andmaturity T 2 [T0; Tn] n fT0; T1; :::; Tng :We want to �nd the call price v(t) attime t in the Vasiµcek-Hull-White short rate model. Note that
v(T ) = max(0;XTi>T
aiB(T; Ti)�K):
Without loss of generality we may assume T0 < T < T1: First recall that
B(t; U) = B(t; U ;R(t)) = eA(t;U)�C(t;U)R(t)
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where A(t; U) and C(t; U) are deterministic and C(t; U) > 0 if t < U: Lett < T be �xed and choose � such that
K =nXi=1
aiB(T; Ti; �)
which implies that
v(T ) = max(0;
nXi=1
ai(B(T; Ti;R(T ))�B(T; Ti; �))
and
v(T ) =nXi=1
aimax(0; B(T; Ti;R(T ))�B(T; Ti; �)):
Thus
v(t) =nXi=1
aicall(t; B(T; Ti; �); T; Ti):
5.4. Swaptions in the Vasiµcek-Hull-White short rate model
If you borrow the amount 1 over the period [T; T + �] and pay interest at theend of the period you must pay the interest
�L(T; T ) = 1� ( 1
B(T; T + �)� 1)
at time T + � (the notion is in line with the Shreve book p 436). Thus
B(T; T + �) =1
1 + �L(T; T ):
Here L(T; T ) is called the spot Libor at time T (with period length �).A simple swap with principal 1 and swap rate K pays its owner the
amountY = 1� �(L(T; T )�K)
16
at time T + �: We want to �nd the value v(t) of this simple swap at timet � T .First
�L(T; T ) =1
B(T; T + �)� 1
and, hence,
Y =1
B(T; T + �)� ~K;
where~K = 1 + �K:
Note that Y is known already at time T and in a model free from arbitrageswe get
v(T ) = 1� ~KB(T; T + �)):
Accordingly from this v(t) = B(t; T )� ~KB(t; T + �):To de�ne more involved swaps, to begin with let
T0 < T1 < ::: < Tn
and� = Ti � Ti�1; i = 1; :::; n:
If you borrow the amount 1 over the period [T0; Tn] and for each �xed i 2f1; :::; ng agree to pay interest for the period [Ti�1; Ti] at the end of thisperiod the corresponding amount equals
�Li�1 =1
B(Ti�1; Ti)� 1
where Li�1 = L(Ti�1; Ti�1): Recall that a simple swap with principal 1 andswap rate K over the period [Ti�1; Ti] pays its owner the amount
Yi = �(Li�1 �K)
at time Ti. Thus at time t � T0 the value of this simple swap equals
�Yi(t) = B(t; Ti�1)� ~KB(t; Ti):
Now consider a so called swap with principal 1 and swap rate K whichpays out the amount Yi at time Ti for every i = 1; :::; n: The value v(t) ofthis swap at time t � T0 must be
swap(t;K) =nXi=1
(B(t; Ti�1)� ~KB(t; Ti))
17
=nXi=1
(B(t; Ti�1)� (1 + �K)B(t; Ti))
= B(t; T0)�B(t; Tn)� �KnXi=1
B(t; Ti):
If the swap is written at time t the swap rate K(t) is chosen so that the swapis of zero value: Thus
K(t) =B(t; T0)�B(t; Tn)�Pn
i=1B(t; Ti):
From now on suppose t � T � T0: A swaption with the swap rate K paysthe amount
Y = max(0; B(T; T0)�B(T; Tn)� �KnXi=1
B(T; Ti))
at maturity T: In the special case T = T0
Y = max(0; 1�B(T; Tn)� �RnXi=1
B(T; Ti)):
In the Vasiµcek-Hull-White short rate model the value of this swaptionbefore time T can be treated as a European put on a �xed coupon bond andits price is simple to compute using the same trick as in the previous section.
5.6 Caps in the Vasiµcek-Hull-White short rate model
A caplet with cap rate K pays at maturity T + � the amount
Y = 1� �max(L(T; T )�K; 0)
where we assume a unit nominal amount. With notation as above,
Y = max(0;1
B(T; T + �)� ~K):
18
If v(t) denotes the price of the derivative at time t � T we, in particular,have
v(T ) = max(0; 1� ~KB(T; T + �))
orv(T ) = ~Kmax(0;
1~K�B(T;K + �)):
Accordingly from this the caplet is equivalent to a number of European puts,which we know how to price expicitly in the Vasiµcek-Hull-White short ratemodel.A so called cap with a unit nominal amount and cap rate K is de�ned as
follows. LetT0 < T1 < ::: < Tn
and� = Ti � Ti�1; i = 1; :::; n:
The cap pays its owner the amount
1� �max(L(Ti�1; Ti�1)�K; 0)
at time Ti for every i 2 f1; :::; ng : It follows from the above that the cap hasan explicit price in every model where each caplet possesses a closed formprice formula.
5.8 HJM; a method based on forward rates
Let �T be a �xed future point of time and let (W (t))0�t� �T be an n-dimensionalstandard Brownian motion in the time interval
�0; �T
�: Set F(t) = �(W (s);
s � t); 0 � t � �T :The Heath-Jarrow-Morton approach to the bond market starts with the
equationsdf(t; T ) = �(t; T )dt+ �(t; T )dW (t); 0 � t � T
where �T = (�(t; T ))0�t�T and �T = (�(t; T ))0�t�T ; are progressively mea-surable for every 0 � T � �T : Here �(t; T ) is an 1� n matrice for every t. Inaddition, we assume
f(0; T ) = f �(0; T ); 0 � T � �T
19
whereB�(0; T ) = e�
R T0 f�(0;u)du
and B�(0; T ) denotes the market price of the T -bond at time 0 (this so calledmarket price is an interpolation from true market prices). Since
B(0; T ) = e�R T0 f(0;u)du
we have a perfect �t of the yield curve, that is
B(0; T ) = B�(0; T ) all 0 < T � �T :
FromB(t; T ) = e�
R Tt f(t;u)du; 0 � t � T
the Itô lemma yields
dB(t; T ) = B(t; T )d(�Z T
t
f(t; u)du) +1
2B(t; T )(d(�
Z T
t
f(t; u)du))2
where (using a true calculus)
d
Z T
t
f(t; u)du = �f(t; t)dt+Z T
t
(df(t; u))du
= �f(t; t)dt+Z T
t
(�(t; u)dt+ �(t; u)dW (t))du
= (�f(t; t) +Z T
t
�(t; u)du)dt+ (
Z T
t
�(t; u)du)dW (t):
Next we de�ne
��(t; T ) =
Z T
t
�(t; u)du
and
��(t; T ) =
Z T
t
�(t; u)du
and have
�dZ T
t
f(t; u)du = (R(t)� ��(t; T ))dt� ��(t; T )dW (t):
20
Now
dB(t; T ) = B(t; T )
�(R(t)� ��(t; T ) + 1
2j ��(t; T ) j2)dt� ��(t; T )dW (t)
�and, hence,
B(t; T ) = B(0; T )eR t0 (R(s)��
�(s;T ))ds�R t0 �
�(s;T )dW (s)
or, equivalently,
B(t; T )
M(t)= B(0; T )e�
R t0 �
�(s;T )ds�R t0 �
�(s;T )dW (s)
Our next task is to �nd conditions that ensure an equivalent martingalemeasure. To this end assume there exists a progressively measurable Rn-valued and integrable random function � such that
���(t; T ) + ��(t; T )�(t) = �12j ��(t; T ) j2; 0 � t � T � �T
and introduce
~W (t) =W (t) +
Z t
0
�(u)du; 0 � t � �T :
In addition, we assume � satis�es the Novikov condition so that ~P = Z�P isa probability measure under which ~W is an n-dimensional standard Brownianmotion. Now
B(t; T )
M(t)= B(0; T )e�
R t012j��(s;T )j2ds�
R t0 �
�(s;T )d ~W (s)
and
dB(t; T )
M(t)=B(t; T )
M(t)(���(t; T ))d ~W (t):
Thus (B(t;T )M(t)
;F(t))0�t�T is a martingale under ~P for every T � �T :As
���(t; T ) + ��(t; T )�(t) = �12j ��(t; T ) j2
di¤erentiation with respect to T yields the so called HJM no arbitrage con-dition
��(t; T ) + �(t; T )�(t) = ��(t; T )Z T
t
�(t; u)|du:
21
and de�ning
~�(t; T ) = �(t; T )
Z T
t
�(t; u)|du
we getdf(t; T ) = ~�(t; T )dt+ �(t; T )d ~W (t); 0 � t � T:
Example 5.7.1 (Ho-Lee model). Suppose n = 1 and �(t; T ) = �; where� > 0 is a constant. Then
~�(t; T ) = �
Z T
t
�ds = �2(T � t)
anddf(t; T ) = �2(T � t)dt+ �d ~W (t):
Thusf(t; T ) = f(0; T ) + �2t(T � t
2) + � ~W (t)
and
R(t) = f(0; t) + �2t2
2+ � ~W (t):
Example 5.7.2. Suppose n = 1; 0 � �(t; T ) � 1 if 0 � t � T � �T ; andthat the HJM no-arbitrage condition is ful�lled. A �nancial derivative ofEuropean type has the payo¤
Y = R(T ) exp(
Z T
0
R(t)dt)
at time of maturity T: We want to prove that �Y (0) � f(0; T ):To prove this inequality recall the equation
df(t; T ) = �(t; T )��(t; T )dt+ �(t; T )d ~W (t)
where
��(t; T ) =
Z T
t
�(t; u)du:
Note that ��(t; T ) � 0: Moreover,
f(t; T ) = f(0; T ) +
Z t
0
�(s; T )��(s; T )ds+
Z t
0
�(s; T )d ~W (s)
22
and
R(T ) = f(0; T ) +
Z T
0
�(s; T )��(s; T )ds+
Z T
0
�(s; T )d ~W (s):
Thus
�Y (0) = ~E [R(T ) j F0] = f(0; T )+ ~E
�Z T
0
�(s; T )��(s; T )ds j F0�� f(0; T ):
Theorem 5.7.1. Suppose �(t; T ); 0 � t � T � �T is a deterministic functionand suppose for �xed T < U that
inf0�t�T
j ��(t; U)� ��(t; T ) j> 0:
Set
�(t) =
sZ T
t
j ��(u; U)� ��(u; T ) j2 du if 0 � t � T:
A European call on the U-bond with strike K and maturity T has the price
call(t;K; T; U) = B(t; U)�(d1)�KB(t; T )�(d2)
at time t < T; where
d1 =ln B(t;U)
KB(t;T )+ 1
2�2(t)
�(t)
and
d2 =ln B(t;U)
KB(t;T )� 1
2�2(t)
�(t)
If the U-bond with strike K and maturity T have the price put(t;K; T; U)at time t, then
