1
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:
3
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
dR(t) = �(t; R(t))dt+ �(t; R(t))dW (t); 0 � t � �T
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,
B(t; T ) = eA(t;T )�C(t;T )R(t)
whereC(t; T ) =
1
a(1� e�a(T�t))
and
A(t; T ) =
Z T
t
�1
2�2C2(u; T )� #(u)C(u; T )
�du:
PROOF. Setting R(t) = X(t) + c(t); we get
dX(t) + dc(t) = (#(t)� aX(t)� ac(t))dt+ �d ~W (t):
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
S(t) =B(t; U)
B(t; T ); 0 � t � T
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and
M0(t) =B(t; T )
B(t; T )= 1; 0 � t � T:
Clearly,S(t) = e(A(t;U)�A(t;T ))�(C(t;U)�C(t;T ))R(t)
anddS(t) = S(t)( (t; R(t))dt� �(C(t; U)� C(t; T ))d ~W (t))
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
B(T; U)�max(0; B(T; U)�K)
= K �max(0; K �B(T; U))
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it follows that
B(t; U)� call(t;K; T; U) = KB(t; T )� put(t;K; T; U):
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)
)
14
= 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)
15
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
B(t; U)� call(t;K; T; U) = KB(t; T )� put(t;K; T; U):
23
PROOF. The proof is very similar to the proof of Theorem 5.3.1. Let theT -bond be numéraire and write
max(0; B(T; U)�K) = B(T; T )max(0; B(T; U)B(T; T )
�K):
Set
S(t) =B(t; U)
B(t; T ); 0 � t � T
andM0(t) = 1; 0 � t � T :
Note that
S(t) = S(0)eR t0 (��
�(u;U)+��(u;T ))du+R t0 (��
�(u;U)+��(u;T ))dW (u)
anddS(t) = S(t)( (t)dt+ (���(t; U) + ��(t; T ))dW (t))
for an appropriate progressively measurable (t); 0 � t � T .We assume the price of the call at time t equals v(t) in the original
numéraire and de�ne
w(t; S(t)) =v(t)
B(t; T ):
Now at the time t 2 [0; T ] consider a portfolio with long one U -bond andshort � U -bonds. The portfolio value �(t) in the new numéraire t equals
�(t) = w(t; S(t))��S(t)
and as usual we choose � such that the noisy part disappears in d�(t) and,moreover,
d�(t) = 0 (= �(t)dM0(t)
M0(t)):
Now by Itô�s lemma
@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)
=@w
@t(t; S(t))dt+
@w
@s(t; S(t))dS(t)+
j ��(t; U)� ��(t; T ) j2 S(t)22
@2w
@s2(t; S(t))dt
��dS(t) = 0:
24
Thus
� =@w
@s(t; S(t))
and@w
@t(t; S(t)) +
j ��(t; U)� ��(t; T ) j2 S(t)22
@2w
@s2(t; S(t)) = 0:
This equation holds if
@w
@t(t; s) +
j ��(t; U)� ��(t; T ) j2 s22
@2w
@s2(t; s) = 0
and, in addition, we insert the terminal condition
w(T; s) = max(0; s�K):
Thus, using Example 4.3.1, we have
w(t; S(t)) = S(t)�(d1)�K�(d2)
andv(t) = B(t; T )w(t; S(t))
= B(t; U)�(d1)�B(t; T )K�(d2):The last part of the theorem follows as the model is free from arbitrage.
Exercises
1. Suppose n = 1 and �(t; T ) = �e�a(T�t); where a; � > 0 are parameters.Find the probability law of (R(t))0�t� �T under ~P?
2. Setd ~P T =
1
B(0; T )e�
R T0 R(s)dsd ~P
and assume
(X(t)
B(t))0�t�T
is a ~P -martingale. Show that
(X(t)
B(t; T ))0�t�T
is a ~P T -martingale.
25
3. Prove thatf(0; T ) = ~ET [R(T )] :
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