Fixed Income
Eng.Kelvin
fixed_income.zipSlides FIS5.pdf
Outline
1 Fixed Income Derivatives The Forward-Risk Adjusted Measure
2 Example
Dr Lara Cathcart () 2015 2 / 28
The problem
Consider a fixed-income derivative with a single payo↵ at time T which depends on the term-structure. In particular, we will look at options on zero-coupon bonds. For a call option on a zero-coupon bond maturing at time T
1
, the time T payo↵ and hence the value of the derivative is given by
V
T
= max(P(T, T 1
) � K, 0) (1)
Dr Lara Cathcart () 2015 3 / 28
The problem
By the no-arbitrage theorem, the price today (t=0)is
V
0
= EQ 0
[e� R
T
0
rsds
V
T
] (2)
where the expectation is taken under the risk-neutral distribution (also called the Q measure). Thus the price depends on the stochastic process for the short rate and the contractual specification of the security (i.e how the payo↵ is linked to the term structure).
Dr Lara Cathcart () 2015 4 / 28
The problem
The price V 0
in equation (2) is given by the expectation of the product of two dependent random variables, and calculating this expectation is often quite di�cult. The purpose of this note is presenting a change-of measure technique which considerably simplifies the evaluation of V
0
.
Dr Lara Cathcart () 2015 5 / 28
The problem
Specifically we are going to calculate V 0
as
V
0
= P(0, T)EQ T
0
(V T
) (3)
where QT is a new probability measure (distribution), the so-called forward-risk adjusted measure. This technique was introduced in the fixed-income literature by Jamishidian (1991).
Dr Lara Cathcart () 2015 6 / 28
Model setup and notation
Our term-structure is a general one-factor HJM model see Heath, Jarrow and Morton (1992). Under the Q-measure, forwards rates are governed by
df (t, T) = ��(t, T)� P
(t, T)dt + �(t, T)dW Q t
(4)
where
� P
(t, T) = � Z
T
t
�(t, u)du (5)
Dr Lara Cathcart () 2015 7 / 28
The problem
Bond prices evolve according to the SDE
dP(t, T) = r t
P(t, T)dt + � P
(t, T)P(t, T)dW Q t
(6)
so � P
(t, T) is the time t volatility of the zero maturing at time T.
Dr Lara Cathcart () 2015 8 / 28
The Forward-Risk Adjusted Measure
The price of derivative security follows the SDE
dV
t
= r t
V
t
dt + � V
(t)V t
dW
Q
t
(7)
This means that, under the risk-neutral distribution, the expected rate of return equals the short rate (just like any other security), and the return volatility is � V
(t). So far neither V t
nor � V
(t) are known, but this is not essential for the following arguments. In fact, the only thing that matters is that the process has the form (7) since this facilitates pricing by the forward-risk adjusted measure.
Dr Lara Cathcart () 2015 9 / 28
The Forward-Risk Adjusted Measure
We begin by defining the deflated price process
F
t
⌘ V t
/P(t, T) (8)
for t 2 [0, T]. We can interpret F t
as the price of V t
in units of the T-maturity bond price (i.e., as a relative price).
Dr Lara Cathcart () 2015 10 / 28
The Forward-Risk Adjusted Measure
Using Ito’s lemma, it can be shown that
dF
t
= � P
(� P
� � V
)F t
dt + (� V
� � P
)F t
dW
Q
t
(9)
where � P
and � V
are shorthand notation for � P
(t, T) and � V
(t), respectively.
Dr Lara Cathcart () 2015 11 / 28
The Forward-Risk Adjusted Measure
Furthermore, we define a new probability measure QT , such that
W
Q
T
t
= W Q t
� Z
t
0
� P
(u, T)du t 2 [0, T] (10)
is a Brownian motion under QT .
Dr Lara Cathcart () 2015 12 / 28
The Forward-Risk Adjusted Measure
In a di↵erential form the relationship between W QT t
and W Q t
is
dW
Q
T
t
= dW Q t
� � P
(t, T)dt = dW Q t
� � P
dt (11)
The new probability measure is known as the forward-risk adjusted measure. It is very important to note that there is a di↵erent measure for each T (payo↵ date).
Dr Lara Cathcart () 2015 13 / 28
The Forward-Risk Adjusted Measure
If we substitute (11) into (9), we obtain the dynamics of F t
under the new probability measure QT . Straightforward calculations give
dF
t
= �� P
(� V
� � P
)F t
dt + (� V
� � P
)F t
(dW QT t
+ � P
dt) = (� V
� � P
)F t
dW
Q
T
t
(12) as the two terms with dt cancel out.
Dr Lara Cathcart () 2015 14 / 28
The Forward-Risk Adjusted Measure
Thus under the QT , the drift is zero and F t
is a martingale. The new probability measure was defined in order to obtain this result since the martingale property implies that
F
t
= EQ T
t
(F T
) (13)
Dr Lara Cathcart () 2015 15 / 28
The Forward-Risk Adjusted Measure
Moreover, by definition P(T, T) = 1 so at maturity we have F T
= V T
, and using (13), the current (t = 0) price of the derivative security can now be calculated as
V
0
= P(0, T)F 0
= P(0, T)EQ T
0
(F T
) = P(0, T)EQ T
0
(V T
) (14)
Dr Lara Cathcart () 2015 16 / 28
The Forward-Risk Adjusted Measure
Which is P(0, T) times the expected payo↵ under QT . Generally, the latter calculation is a lot simpler than the direct evaluation of the expectation under Q as in equation (2) above. With (14) at hand, the only remaining task is determining the distribution of the payo↵ under the forward-risk adjusted measure.
Dr Lara Cathcart () 2015 17 / 28
The Forward-Risk Adjusted Measure
We conclude by noting that f (t, T) is a martingale under QT . To see this, substitute (11) into the forward-rate SDE (4)
df (t, T) = ��(t, T)� P
(t, T)dt + �(t, T)(dW Q T
t
+ � P
(t, T)dt) = �(t, T)dW Q T
t
(15) This property turns out to be very useful when pricing fixed income derivative.
Dr Lara Cathcart () 2015 18 / 28
Outline
1 Fixed Income Derivatives The Forward-Risk Adjusted Measure
2 Example
Dr Lara Cathcart () 2015 19 / 28
Example
For concreteness, we use the extended Vasicek model which is a special case of the one-factor HJM model with
�(t, T) = �e�(T�t) (16)
and
� P
(t, T) = � Z
T
t
�(t, u)du = � e
�(T�t) � 1
(17)
The extended Vasicek model is a Markovian HJM model, but the following pricing formula for bond options (27) do not depend on the Markov property.
Dr Lara Cathcart () 2015 20 / 28
Example
We consider the fixed-income derivative to be a call option on a zero-coupon bond maturing at time T
1
. The option expires (matures) at time T < T 1
with the following payo↵:
C
T
= max(P(T, T 1
) � K, 0) (18)
where K is the strike (exercise) price of the option.
Dr Lara Cathcart () 2015 21 / 28
Example
In order to price this security, we need the distribution of C T
under the forward-risk adjusted measure. Since C
T
only depends on P(T, T 1
) and since P(T, T) = 1, we can calculate the expectation of C
T
from the distribution of the relative price,
F(t, T, T 1
) = P(t, T 1
)/P(t, T) (19)
which is also the forward price of the T 1
-maturity bond for delivery at time T.
Dr Lara Cathcart () 2015 22 / 28
Example
Using previous results, the SDE for F(t, T, T 1
) under QT is given by
dF(t, T, T 1
) = (� P
(t, T 1
) � � P
(t, T))F(t, T, T 1
)dW Q T
t
(20)
⌘ � F
(t, T, T 1
)F(t, T, T 1
)dW Q T
t
Since bond prices are always strictly positive the logarithm of F(t, T, T 1
) is well defined, and a simple application of Ito’s lemma gives
Dr Lara Cathcart () 2015 23 / 28
Example
d log F(t, T, T 1
) = �1/2�2 F
(t, T, T 1
)dt + �(t, T, T 1
)dW Q T
t
(21)
Dr Lara Cathcart () 2015 24 / 28
Example
After integrating from t = 0 to t = T we have
log F(T, T, T 1
) = log P(T, T 1
) = log F(0, T, T 1
) � 1/2 Z
T
0
� 2
F
(t, T, T 1
)dt +
Z T
0
� F
(t, T, T 1
)dW
Q
T
t
(22)
The first equality in (22) follows because P(T, T) = 1. moreover, if � F
(t, T, T 1
) is deterministic, i.e if the model is gaussian, it follows from (22) that log P(T, T
1
) is conditionally normally distributed with variance
w
2
F
(T, T 1
) =
Z T
0
�2 F
(t, T, T 1
)dt (23)
and mean
µ F
(T, T 1
) = log F(0, T, T 1
) � 1/2 Z
T
0
�2 F
(t, T, T 1
)dt (24)
= log F(0, T, T 1
) � 1/2w2F(T, T 1
)
Dr Lara Cathcart () 2015 25 / 28
Example
For the extended Vasicek model, � F
(t, T, T 1
)is given by
� F
(t, T, T 1
) =
�
(e
�(T 1
�t) � e�(T�t)) (25)
=
�
e
�(T�t) (e
�(T 1
�T) � 1)
the variance w2 f
(T, T 1
) can be calculated as
w
2
F
(T, T 1
) =
�2
2 (e
�(T 1
�T) � 1)2 Z
T
0
e
�2(T�t) dt (26)
= (
e
�(T 1
�T) � 1
)
2
.(� 2
1 � e�2T
2 )
= B
2
(T
1
� T).VarQ T
0
(r
T
)
Since the last parenthesis in the second line can be recognized as the conditional variance of r
T
, see equation (4) in Jamishidian (1989). The function B(⌧) is the factor loading for the Vasicek model.
Dr Lara Cathcart () 2015 26 / 28
Example
Finally, the price of the call option, denoted C(T, K), is given by
C(T, K) = P(0, T)ET 0
(C T
) (27)
= P(0, T)
Z 1
log K
(ex � K) 1
p 2⇡w
F
e
�(x�µ F
)
2
/2w2 F
dx
= P(0, T)
Z 1
log K
e
x
1 p 2⇡w
F
e
�(x�µ F
)
2
/2w2 F
dx
� P(0, T)K Z 1
log K
1 p 2⇡w
F
e
�(x�µ F
)
2
/2w2 F
dx
= P(0, T 1
)N(d 1
) � P(0, T)KN(d 2
)
where N(.) is the cumulative normal distribution function, w F
is shorthand notation for w(T, T
1
)
Dr Lara Cathcart () 2015 27 / 28
Example
and
d
2
= (log P(0, T
1
)
P(0, T) � log K �
1
2 w
2
F
)/w F
(28)
d
1
= d 2
+ w F
(29)
The calculation is completely analogous to the Black-Scholes model for call options on stock prices, so we skip the intermediate steps leading to the final expression for C(T, K) in equation (27).
Dr Lara Cathcart () 2015 28 / 28
- Fixed Income Derivatives
- The Forward-Risk Adjusted Measure
- Example
Slides FIS4.pdf
Outline
1 Introduction
2 Vasicek model with time-dependent drift
3 The Heath, Jarrow and Morton model
Dr Lara Cathcart () 2012 2 / 74
Introduction
Arbitrage-free models are defined as models which fit the initial yield curve exactly. We consider two types of models, namely equilibrium-style models with time-dependent parameters, also called calibrated models, and the Heath, Jarrow and Morton (HJM) modelling framework. In both cases we focus on one-factor versions of the requisite models.
Dr Lara Cathcart () 2012 3 / 74
One factor models and calibration
The wide-spread popularity of one-factor equilibrium models, such as the Vasicek model, stem from their simplicity.
Dr Lara Cathcart () 2012 4 / 74
One factor models and calibration
At each date, today and in the future, the entire yield curve is a function of a single state variable, the short rate. This feature is very useful when implementing numerical solutions, e.g., when constructing binomial trees to approximate the continuous-time model.
Dr Lara Cathcart () 2012 5 / 74
One factor models and calibration
However, equilibrium models do not fit the current yield curve exactly, and this tends to limit their e↵ectiveness for pricing fixed income derivatives. By introducing time dependent parameters in the model, we can match the current yield curve, while retaining the overall simplicity of the term-structure model.
Dr Lara Cathcart () 2012 6 / 74
One factor models and calibration
This approach is advocated by, especially, Hull and White (1990) who extend the Vasicek and CIR models with time-dependent parameters. Sometimes, these models are referred to as the extended Vasicek and CIR models.
Dr Lara Cathcart () 2012 7 / 74
Outline
1 Introduction
2 Vasicek model with time-dependent drift
3 The Heath, Jarrow and Morton model
Dr Lara Cathcart () 2012 8 / 74
Vasicek model with time-dependent drift
Under the risk neutral measure, the short rate in the ordinary Vasicek (1977) model evolves according to the SDE
dr
t
= (✓ � r t
)dt + �dW Q t
(1)
where ✓ = µ � ��/ is the risk-neutral mean.
Dr Lara Cathcart () 2012 9 / 74
Vasicek model with time-dependent drift
Prices of fixed-income derivatives only depend on the distribution of r t
under the risk neutral measure, so in the following we do not care about the process under the original probability measure (and the true drift).
Dr Lara Cathcart () 2012 10 / 74
Vasicek model with time-dependent drift
In general, the model in equation (1) will not fit the current t = 0 yield curve exactly. Therefore, we augment the risk-neutral process with a time-dependent mean ✓(t),
dr
t
= (✓(t) � r t
)dt + �dW Q t
(2)
Dr Lara Cathcart () 2012 11 / 74
Vasicek model with time-dependent drift
The solution to the SDE (2) can be written as
r
t
= e�tr 0
+
Z t
0
e
�(t�s)✓(s)ds + �
Z t
0
e
�(t�s) dW
Q
s
(3)
Dr Lara Cathcart () 2012 12 / 74
Vasicek model with time-dependent drift
If we define m(t) and x t
by
m(t) = e�tr o
+
Z t
0
e
�(t�s)✓(s)ds (4)
and
dx
t
= �x t
dt + �dW t
with x
0
= 0 (5)
respectively, we can write r t
as
r
t
= m(t) + x t
(6)
Dr Lara Cathcart () 2012 13 / 74
Vasicek model with time-dependent drift
Of course representations (3) and (6) are equivalent, but the calibration is more straightforward in the latter case, so we use (6) in the sequel.
Dr Lara Cathcart () 2012 14 / 74
Vasicek model with time-dependent drift
Absence of arbitrage implies that the price of a zero-coupon bond is given by the risk-neutral expectation:
P(t, T) = EQ t
h e
� R
T
t
r
s
ds
i (7)
= e� R
T
t
m(s)ds.EQ t
h e
� R
T
t
x
s
ds
i
= exp
� Z
T
t
m(s)ds
! . exp[A(T � t) + B(T � t)x
t
]
Dr Lara Cathcart () 2012 15 / 74
Vasicek model with time-dependent drift
Where
B(⌧) = e
�⌧ � 1
(8)
A(⌧) = 1
2 �2 Z ⌧
0
B
2(s)ds (9)
= 1
2
⇣�
⌘ 2
1 � e�2⌧ � 4(1 � e�⌧ )
2 + ⌧
�
Note that P(t, T) in (7) is written as the product of a deterministic factor and the bond price in an ordinary Vasicek model with zero mean (under the risk neutral measure i.e under Q).
Dr Lara Cathcart () 2012 16 / 74
Vasicek model with time-dependent drift
Calibration of the time-dependent parameters: We want to fit the initial yield curve, represented by the discount function d(T). That is,
P(0, T) = exp
" � Z
T
0
m(s)ds + A(T)
# = d(T) (10)
since x 0
= 0 by the normalization above.
Dr Lara Cathcart () 2012 17 / 74
Vasicek model with time-dependent drift
Calibration of the time-dependent parameters: From (10) we get Z
T
0
m(s)ds = � log d(T) + A(T) (11)
and after di↵erentiating with respect to T on both sides of the equation we arrive at
m(T) = �d log d(T)
dT
+ dA(T)
dT
= f (0, T) + 1
2 �2B2(T) (12)
Dr Lara Cathcart () 2012 18 / 74
Vasicek model with time-dependent drift
Calibration of the time-dependent parameters: This means that m(T) is obtained from the initial forward curve, f (0, T). The time invariant parameters, and �, must, of course, be specified prior to this calculation. In principle, and � can be backed out from market prices of, e.g., interest rate caps, but such calculations are outside the scope of this lecture.
Dr Lara Cathcart () 2012 19 / 74
Vasicek model with time-dependent drift
Calibration of the time-dependent parameters: In order to determine ✓(t), first note that the derivative of m(t) is given by:
m
0(t) = �e�tr 0
+ ✓(t) � Z
t
0
e
�(t�s)✓(s)ds (13)
= ✓(t) � m(t)
Dr Lara Cathcart () 2012 20 / 74
Vasicek model with time-dependent drift
Calibration of the time-dependent parameters: Using this result, and the definition of m(t) in (12) above, we have
✓(t) = m(t) + m0(t) (14)
= f (0, t) + 1
2 �2B2(t) +
@f (0, t)
@t + �2B(t)B0(t)
= f (0, t) + @f (0, t)
@t + �(t)
where
�(t) = 1
2 �2B2(t) + �2B(t)B0(t) (15)
= �2
2 (1 � e�2t)
(the proof that the first of (15) simplifies to the second line is somewhat lengthy and tedious, so it is left out here).
Dr Lara Cathcart () 2012 21 / 74
Vasicek model with time-dependent drift
Calibration of the time-dependent parameters: Finally, we can write the SDE for r
t
in the following way:
dr
t
=
✓ (f (0, t) � r
t
) + @f (0, t)
@t + �(t)
◆ dt + �dW Q
t
(16)
which illustrates how the time-dependent parameters of the SDE are obtained from the initial yield curve, or rather forward curve f (0, t).
Dr Lara Cathcart () 2012 22 / 74
Vasicek model with time-dependent drift Distribution of future bond prices:
The purpose of calibrating the drift ✓(t) to the initial yield curve is, of course, pricing fixed-income derivatives at time t = 0.
Dr Lara Cathcart () 2012 23 / 74
Vasicek model with time-dependent drift
Distribution of future bond prices: Consider a claim with a single payo↵, depending on the term structure at time t, for example a call option on a zero-coupon bond maturing at time T, with exercise (strike) price K. Here, the uncertain payo↵ is given by
C(r t
) = max[P(r t
, t, T) � K, 0] (17)
and the current value (price) of the claim is
V
0
= EQ 0
h e
� R
t
0
r
s
ds
C(r t
) i
(18)
Dr Lara Cathcart () 2012 24 / 74
Vasicek model with time-dependent drift
Distribution of future bond prices: Methods for calculating the price (18) in closed form (if possible), or implementing appropriate numerical procedures, will not be addressed during this course. For our present purposes, it su�ces to note that we need the time t = 0 distribution of the future bond price, P(t,T) which again depends solely on r
t
(besides the deterministic parameters) because of the one-factor assumption.
Dr Lara Cathcart () 2012 25 / 74
Vasicek model with time-dependent drift
Distribution of future bond prices: From (7), the bond price at time t is given by
P(t, T) = exp
" � Z
T
t
m(s)ds + A(T � t) + B(T � t)(r t
� m(t)) #
(19)
where m(s), t s T, is obtained from the calibration to the initial term structure.
Dr Lara Cathcart () 2012 26 / 74
Vasicek model with time-dependent drift
Distribution of future bond prices: It is important to understand that we are looking at the distribution of the future bond price, given the current (t = 0) information. Once we observe P(t, T), we can re-calibrate the function m(s) for s � t, and the new function will generally di↵er from the one obtained from f (0, t). This is the inherent inconsistency of the calibration approach, see Tuckman (1995) chapter 9 for further discussion. However, we are only interested in prices of contingent claims at time t = 0, which leads us to ignore the problem.
Dr Lara Cathcart () 2012 27 / 74
Vasicek model with time-dependent drift
Distribution of future bond prices: In the following, we rewrite this expression to something that is easier to interpret. First, note that the price of the T-maturity bond is given by:
P(0, T)
P(0, t) =
exp[� R T
0
m(s)ds + A(T)]
exp[� R t
0
m(s)ds + A(t)] (20)
= exp[� Z
T
t
m(s)ds + A(T) � A(t)]
Dr Lara Cathcart () 2012 28 / 74
Vasicek model with time-dependent drift
Distribution of future bond prices: Second, using (20) allows us to write (19) as
P(t, T) = P(0, T)
P(0, t) exp[A⇤(t, T) + B(T � t)(r
t
� m(t))] (21)
where A⇤ = A(T � t) + A(t) � A(T). After many lengthy calculations, we can obtain the following formula for A⇤(t, T):
A
⇤(t, T) = � 1
2 B
2(T � t)�(t) + 1
2 �2B(T � t)B2(t) (22)
Dr Lara Cathcart () 2012 29 / 74
Vasicek model with time-dependent drift
Distribution of future bond prices: Finally, since m(t) = f (0, t) + 1
2
�2B(t), we get
P(t, T) = P(0, T)
P(0, t) exp
� 1
2 B
2(T � t)�(t) + B(T � t)(r t
� f (0, t)) �
(23)
which only involves the current forward curve, f (0, t).
Dr Lara Cathcart () 2012 30 / 74
Vasicek model with time-dependent drift
Calibration in other cases: In the above model, only the drift parameter ✓(t) is time-dependent, whereas � and are still time-invariant (constant) parameters. This suggests the following extension of the model,
dr
t
= (t)(✓(t) � r t
)dt + �(t)dW Q t
(24)
or an equivalent generalization of the CIR model
dr
t
= (t)(✓(t) � r t
)dt + �(t) p r
t
dW
Q
t
(25)
Dr Lara Cathcart () 2012 31 / 74
Vasicek model with time-dependent drift
Calibration in other cases: One advantage of letting �(t) or (t) be time-dependent is that the model can match the current volatility structure, in addition to the current yield curve. Not surprisingly, this additional generality comes at a cost, namely that the calibration of the time dependent parameters becomes much more complex.
Dr Lara Cathcart () 2012 32 / 74
Vasicek model with time-dependent drift
Calibration in other cases: In particular, there is no longer a simple relationship between ✓(t) and the initial forward curve f (0, t), as in the equation (14) above. We refer the interested reader to Hull and White (1990) who analyze the models (24) and (25) and discuss di↵erent approaches for calibration of time-dependent parameters.
Dr Lara Cathcart () 2012 33 / 74
Outline
1 Introduction
2 Vasicek model with time-dependent drift
3 The Heath, Jarrow and Morton model
Dr Lara Cathcart () 2012 34 / 74
The Heath, Jarrow and Morton model
The Heath, Jarrow and Morton (HJM) (1992) framework is similar to the calibration approach in the sense that we can match an arbitrary initial yield curve exactly.
Dr Lara Cathcart () 2012 35 / 74
The Heath, Jarrow and Morton model
With calibrated models, the starting point is a SDE for r t
with time-dependent parameters, and the yield curve and volatility structure are determined endogenously from the time dependent parameters.
Dr Lara Cathcart () 2012 36 / 74
The Heath, Jarrow and Morton model
In HJM models, on the other hand, the initial forward curve and volatility structure are specified direclty (exogenously). We use the no-arbitrage assumption to derive restrictions on the future movements of the forward curve, which facilitates using the HJM model for pricing fixed-income derivatives.
Dr Lara Cathcart () 2012 37 / 74
A general one-factor HJM model
Under the true probability measure, the forward curve f (t, T) evolves according to
df (t, T) = ↵(t, T)dt + �(t, T)dW t
for all T � t (26)
where �(t, T) is the forward-rate volatility (volatility sturcture), and ↵(t, T) the drift of the forward curve.
Dr Lara Cathcart () 2012 38 / 74
The Heath, Jarrow and Morton model
Note that all maturities are a↵ected by the same Brownian motion W t
, that is changes in the forward curve are perfectly correlated (one factor model).
Dr Lara Cathcart () 2012 39 / 74
The Heath, Jarrow and Morton model
The short rate r t
is implicitly defined as the forward rate with T = t, that is
r
t
= f (t, t) (27)
Dr Lara Cathcart () 2012 40 / 74
The Heath, Jarrow and Morton model
From the definition of instantaneous forward rates, the logarithm of bond prices are given by the expression
log P(t, T) = � Z
T
t
f (t, u)du (28)
Dr Lara Cathcart () 2012 41 / 74
The Heath, Jarrow and Morton model
In order to price fixed-income derivatives, we need the distribution of f (t, T), and hence bond prices, under the risk-neutral measure. This involve imposing conditions on the forward rate dynamics, so they are consistent with no arbitrage opportunities.
Dr Lara Cathcart () 2012 42 / 74
The Heath, Jarrow and Morton model
We begin by determining the stochastic di↵erential equation for (log) bond prices under the true probability measure
d log P(t, T) = f (t, t)dt � Z
T
t
df (t, u)du
= r t
dt � Z
T
t
(↵(t, u)dt + �(t, u))dW t
)du
=
r
t
� Z
T
t
↵(t, u)du
! dt �
Z T
t
�(t, u)du
! dW
t
(29)
Dr Lara Cathcart () 2012 43 / 74
The Heath, Jarrow and Morton model
An application of Ito’s lemma provides the SDE for P(t,T),
dP(t, T) = µ p
(t, T)P(t, T)dt + �(t, T)P(t, T)dW t
(30)
where
� p
(t, T) = � Z
T
t
�(t, u)du (31)
µ p
(t, T) = r t
� Z
T
t
↵(t, u)du + 1
2 � p
(t, T)2 (32)
Dr Lara Cathcart () 2012 44 / 74
The Heath, Jarrow and Morton model
Since all bond prices are driven by the same Brownian motion (perfect correlation) absence of arbitrage implies the following restriction
µ p
(t, T) = r t
+ �(t)� p
(t, T) for all T (33)
where �(t) is the market price of risk at time t.
Dr Lara Cathcart () 2012 45 / 74
The Heath, Jarrow and Morton model
Using equations (31) and (32), the no arbitrage condition can be written as
Z T
t
↵(t, u)du = �(t)
Z T
t
�(t, u)du + 1
2
Z T
t
�(t, u)du
! 2
(34)
Dr Lara Cathcart () 2012 46 / 74
The Heath, Jarrow and Morton model
If we di↵erentiate with respect to T on both sides of the equation we get
↵(t, T) = �(t)�(t, T) + �(t, T)
Z T
t
�(t, u)du (35)
which means that the forward-rate drift must be a function of the volatility structure and the market price of risk �(t).
Dr Lara Cathcart () 2012 47 / 74
The Heath, Jarrow and Morton model
Thus, under the true probability measure, the SDE for forward rates is given by
df (t, T) =
" �(t)�(t, T) + �(t, T)
Z T
t
�(t, u)du
# dt + �(t, T)dW
t
(36)
Dr Lara Cathcart () 2012 48 / 74
The Heath, Jarrow and Morton model
At this stage two things are worth emphasizing. First, the SDE is still specified under the true original probability measure. Second, the drift of f(t,T) depends on �(t), that is investor preferences.
Dr Lara Cathcart () 2012 49 / 74
The Heath, Jarrow and Morton model
If W Q t
is a Brownian motion under the Q measure (risk neutral measure) we have the following relationship to the true probability measure,
dW
Q
t
= dW t
+ �(t)dt (37)
Dr Lara Cathcart () 2012 50 / 74
The Heath, Jarrow and Morton model
After substituting this into (36) we obtain the SDE under the Q measure (risk-neutral distribution),
df (t, T) =
" �(t, T)
Z T
t
�(t, u)du
# dt + �(t, T)dW Q
t
(38)
= ��(t, T)� p
(t, T)dt + �(t, T)dW Q t
where the second line follows from (31).
Dr Lara Cathcart () 2012 51 / 74
The Heath, Jarrow and Morton model
Note that the drift of this SDE is independent of �(t), so valuation of fixed-income derivatives is truly preference-free. The reason for this independence is that we are pricing interest-rate derivatives relative to the current yield curve (forward curve), and the yield curve reflects all relevant investor preferences.
Dr Lara Cathcart () 2012 52 / 74
The Heath, Jarrow and Morton model
There are two inputs to a HJM model: the initial forward curve, f (0, T), and the volatility structure, �(t, T).
Dr Lara Cathcart () 2012 53 / 74
The Heath, Jarrow and Morton model
The first is simply the current (t = 0) forward curve, whereas the latter must be specified somehow-either from historical estimates or backed out from prices of interest-rate derivatives (like ”implied volatility” in the Black-Scholes model).
Dr Lara Cathcart () 2012 54 / 74
The Heath, Jarrow and Morton model
In any case, the purpose of using HJM model is pricing derivatives whose payo↵ depend on the future term structure. Therefore, it is convenient to have an expression like (23) for the HJM model.
Dr Lara Cathcart () 2012 55 / 74
The Heath, Jarrow and Morton model
Starting from the initial forward curve, f (0, s), and using the SDE under the risk-neutral measure (38), we have that
log P(t, T) = � Z
T
t
f (t, s)ds
=
Z T
t
✓ �f (0, s) �
Z t
0
df (u, s)
◆ ds
=
Z T
t
✓ �f (0, s) +
Z t
0
�(u, s)� p
(u, s)du � Z
t
0
�(u, s)dW Q u
◆ ds(39)
Dr Lara Cathcart () 2012 56 / 74
The Heath, Jarrow and Morton model
Second, note that
� Z
T
t
f (0, s)ds = � Z
T
0
f (0, s)ds +
Z t
0
f (0, s)ds = log
P(0, T)
P(0, t)
� (40)
Dr Lara Cathcart () 2012 57 / 74
The Heath, Jarrow and Morton model
Finally, by combining (39) and (40), we get
P(t, T) = P(0, T)
P(0, t) exp
✓Z T
t
Z t
0
�(u, s)� p
(u, s)duds � Z
T
t
✓Z t
0
�(u, s)dW Q
u
◆ ds
◆ (41)
Dr Lara Cathcart () 2012 58 / 74
The Heath, Jarrow and Morton model
As in (23), we can express P(t, T) as the forward price multiplied by a random factor. However, contrary to the Vasicek model with time-dependent drift, the random factor involves the entire path of the Brownian motion, W Q
s
, between 0 and t, not just a single random variable, like the short rate at time t.
Dr Lara Cathcart () 2012 59 / 74
The Heath, Jarrow and Morton model
This path dependency can be problematic, especially when implementing numerical approximations to the HJM model. For example, binomial trees cannot be recombining (so the dimension grows exponentially with the number of time steps), and when using Monte Carlo methods, we must simulate the movements of the entire forward curve which is more time consuming than simulating the path of a single state variable.
Dr Lara Cathcart () 2012 60 / 74
Markovian HJM models
Having noted the potential di�culties with path-dependencies in the general HJM model, we turn to a special case of HJM model where the term structure can always be expressed as a function of a finite number of state variables. We obtain this cases, known as Markovian HJM models, by imposing certain restrictions on the volatility structure.
Dr Lara Cathcart () 2012 61 / 74
Markovian HJM models
In any model, bond prices are given by
P(t, T) = EQ t
h e
� R
T
t
r
s
ds
i (42)
so a Markovian HJM model (as defined above) corresponds to a Markovian stochastic process for the short rate under the risk neutral measure. The bond price in (42) is given by a conditional expectation, and if the conditional distribution of r
s
only depends on a finite number of state variables X t
the bond price will be a function of these state variables.
Dr Lara Cathcart () 2012 62 / 74
Markovian HJM models
We start by writing the short rate of the general HJM model in di↵erential form. The short rate is given by
r
t
= f (t, t) = f (0, t) � Z
t
0
�(s, t)� p
(s, t)ds +
Z t
0
�(s, t)dW Q s
(43)
Dr Lara Cathcart () 2012 63 / 74
Markovian HJM models
And the di↵erential form with respect to t is
dr
t
= (
@f (0, t)
@t � �(t, t)�
p
(t, t) � Z
t
0
@�(s, t)
@t � p
(s, t)ds
� Z
t
0
�(s, t) @�
p
(s, t)
@t ds
+
Z t
0
@�(s, t)
@t dW
Q
s
)dt + �(t, t)dW Q
t
(44)
Dr Lara Cathcart () 2012 64 / 74
Markovian HJM models
Since � p
(t, t) = 0 and @� p
(s, t)/@t = ��(s, t), this reduces to
dr
t
=
@f (0, t)
@t � Z
t
0
@�(s, t)
@t � p
(s, t)ds +
Z t
0
@�(s, t)
@t dW
Q
s
+
Z t
0
�2(s, t)ds
� dt
+ �(t, t)dW Q t
(45)
Dr Lara Cathcart () 2012 65 / 74
Markovian HJM models
Note how the drift is path-dependent since it involves the entire path of the Brownian motion W Q
s
between 0 and t. However, if the volatility structure satisfies the restriction
@�(s, t)
@t = �(t)�(s, t) (46)
for some (t), the di↵erential form (45) simplifies to
dr
t
=
" @f (0, t)
@t + (t)
Z t
0
�(s, t)� p
(s, t)ds � (t) Z
t
0
�(s, t)dW Q
s
+
Z t
0
� 2
(s, t)ds
# dt
+ �(t, t)dW Q
t
(47)
Dr Lara Cathcart () 2012 66 / 74
Markovian HJM models
which in light of the identity
r
t
� f (0, t) = � Z
t
0
�(s, t)� p
(s, t)ds +
Z t
0
�(s, t)dW Q s
(48)
can be written as
Dr Lara Cathcart () 2012 67 / 74
Markovian HJM models
dr
t
=
✓ @f (0, t)
@t + (t)(f (0, t) � r
t
) + � t
◆ dt + �(t, t)dW Q
t
(49)
with
� t
=
Z t
0
�2(s, t)ds (50)
The di↵erential form of � t
is
d� t
=
✓ �2(t, t) + 2
Z t
0
�(s, t) @�(s, t)
@t ds
◆ dt (51)
= � �2(t, t) � 2(t)�
t
� dt
and there are no path-dependencies in the drift of either (49) and (51).
Dr Lara Cathcart () 2012 68 / 74
Markovian HJM models
In summary, if the volatility structure �(s, t) obeys the restriction (46), the short rate is governed by a bivariate Markovian SDE consisting of equations (49) and (51), respectively. The mean reversion parameter (t) in (49) is time-varying and the coe�cient is obtained from the volatility structure. Moreover, if �(s, t) is non-stochastic �
t
in (51) reduces to a time-dependent function, and the short-rate dynamics become equivalent to the extended Vasicek process (24) see Hull and White (1993). The Markov restriction on the volatility structure (46) has the form of an ordinary di↵erential equation.
Dr Lara Cathcart () 2012 69 / 74
Markovian HJM models
Specifically,
@�(s, u)/@u
�(s, u) =
@ log �(s, u)
@u = �(u) for u � s (52)
Dr Lara Cathcart () 2012 70 / 74
Markovian HJM models
Integrating from u = s to u = t in (52) yields
log �(s, t) � log �(s, s) = � Z
t
s
(u)du (53)
which can be written �(s, t) = �(s, s)e�
R t
s
(u)du (54)
where �(s, s) is the short-rate volatility at time s, see (49).
Dr Lara Cathcart () 2012 71 / 74
Markovian HJM models
Thus, the volatility structure has to be of the form (54)in order to obtain a Markovian HJM model. Note that �(s, s) can depend on the short rate at at time s, so does not have to be a deterministic function. For example, the following specification can be used
�(s, t) = �r� s
e
�(t�s) (55)
Dr Lara Cathcart () 2012 72 / 74
Markovian HJM models
We now turn to the distribution of future (time t) bond prices. If the volatility structure is of the form (54), it can be shown that (41) simplifies to
P(t, T) = P(0, T)
P(0, t) exp
✓ � 1
2 �2(t, T)�
t
+ �(t, T)[f (0, t) � r t
]
◆ (56)
where
�(t, T) = � � p
(t, T)
�(t, t) =
Z T
t
e
� R
s
t
(u)du ds (57)
and � t
is defined above see Ritchken and Sankarasubramanian (1995) for a proof.
Dr Lara Cathcart () 2012 73 / 74
Markovian HJM models
If (u) is constant, we get a closed-form expression for (57),
�(t, T) =
Z T
t
e
�(s�t) ds =
1 � e�(T�t)
k
(58)
which apart from the sign, is the factor loading in the Vasicek model.
Dr Lara Cathcart () 2012 74 / 74
- Introduction
- Vasicek model with time-dependent drift
- The Heath, Jarrow and Morton model
Slides FIS3.pdf
Outline
1 Assumptions
2 A general framework for multi-factor model
3 The exponential-a�ne class of models
Dr Lara Cathcart () 2015 2 / 46
Assumptions
The bond market is frictionless: no distorting taxes, no transactions costs, no short-sale restrictions, and all bonds are infinitely divisible.
Dr Lara Cathcart () 2015 3 / 46
Assumptions
Investors always prefer more wealth to less, i.e. the marginal utility of wealth is positive at all levels of wealth.
Dr Lara Cathcart () 2015 4 / 46
Assumptions
All bond prices are function of m ⇤ 1 vector of state variables, denoted Xt. Together with the next assumption, this implies that the market prices of risk at time t, �(Xt), are functions of the m state variables.
Dr Lara Cathcart () 2015 5 / 46
Assumptions
The short rate is a known function of Xt, i.e. rt = r(Xt). In most cases, rt is the first element of the vector Xt, that is rt = w
0Xt, where w is a vector with one as the first element and zeros elsewhere.
Dr Lara Cathcart () 2015 6 / 46
Assumptions
The dynamics of the state variables are governed by:
dXt = µ(Xt)dt + �(Xt)dWt (1)
where µ(X) is m ⇤ 1 drift vector and �(X) is an m ⇤ m matrix containing the volatilities coe�cients, and Wt is an m-dimensional Brownian motion.
Dr Lara Cathcart () 2015 7 / 46
Outline
1 Assumptions
2 A general framework for multi-factor model
3 The exponential-a�ne class of models
Dr Lara Cathcart () 2015 8 / 46
Multi-factor models
We start by deriving a stochastic process for bond prices, including an expression for the expected returns on di↵erent bonds. Next we use the absence of arbitrage to impose a restriction on bond returns, which requires assumptions about the market prices of risk. Finally we obtain a PDE for bond prices, as well as the risk-neutral process for the short rate (through the m state variables).
Dr Lara Cathcart () 2015 9 / 46
Multi-factor models
By an appropriate multivariate version of Ito’ s lemma, bond prices can be shown to evolve according to (Note that T is fixed, and t denotes calendar time.)
dP(t, T) = µp(t, T)P(t, T)dt + mX
i=1
�pi(t, T)P(t, T)dWit (2)
Dr Lara Cathcart () 2015 10 / 46
Multi-factor models
µp(t, T)P(t, T) = mX
i=1
@P
@Xi µi(X) +
@P
@t +
1
2
mX
i=1
mX
j=1
@2P
@Xi@Xj �i(X)�j(X)⇢ij (3)
�pi(t, T)P(t, T) = @P
@Xi �i(X) (4)
Note that ⇢ii = 1 in equation (3). The expected return and the bond volatilities depend on the state variables, but to keep the notation manageable, this dependence is suppressed here.
Dr Lara Cathcart () 2015 11 / 46
Multi-factor models
In order to derive the appropriate restriction on µ(t, T) for di↵erent maturity dates T, we construct a portfolio of bonds ✓t consisting of K = m + 1 distinct maturities. The number of bonds with maturity date Ti, i = 1, .....K, is denoted by wi.
Dr Lara Cathcart () 2015 12 / 46
Multi-factor models
The instantaneous changes in the value of this portfolio, ✓t can be written as:
d✓t = KX
k=1
wkdP(t, Tk) (5)
d✓t = [ KX
k=1
wkµp(t, Tk)P(t, Tk)]dt (6)
+ mX
i=1
[ KX
k=1
wk�pi(t, Tk)P(t, Tk)]dWit
where we have interchanged the order of summation between i and k in equation (6).
Dr Lara Cathcart () 2015 13 / 46
Multi-factor models
Since there are more bonds than sources of risk, it must be possible to choose non-zero portfolio weights, wk which make the portfolio locally riskless. This means that the weights must satisfy m restrictions of the form
KX
k=1
wk�pi(t, Tk)P(t, Tk) = 0 (7)
Dr Lara Cathcart () 2015 14 / 46
Multi-factor models
By continuously readjusting the portfolio weights, we can ensure that the price dynamics of the portfolio are always riskless, or deterministic. Absence of arbitrage requires that the expected (and realized) return is equal to the short rate rt otherwise there is a ”free lunch” by either buying or selling the portfolio and taking the opposite in the money markets, both investments are locally riskless.
Dr Lara Cathcart () 2015 15 / 46
Multi-factor models
Stated otherwise, the expected excess return must be zero,
KX
k=1
wk(P(t, Tk).(µp(t, Tk) � rt) = 0 (8)
Dr Lara Cathcart () 2015 16 / 46
Multi-factor models
We have shown that, if the vector z = [P(t, T 1
)w 1
, ....P(t, TK )wK ] 0 with z 6= 0,
solves the system of equations: 2
66 4
�p1(t, T1) , . . . , �p1(t, TK ) �p2(t, T1) , . . . , �p2(t, TK )
. . . , . . . , . . . �pm(t, T1) , . . . , �pm(t, TK )
3
77 5
2
66 4
P(t, T 1
)w 1
P(t, T 2
)w 2
. . . P(t, TK )wK
3
77 5 ⌘ A1z = 0
Dr Lara Cathcart () 2015 17 / 46
Multi-factor models
the same K ⇤ 1 vector z also solves the larger system:
2
6666 4
�p1(t, T1) , . . . , �p1(t, TK ) �p2(t, T1) , . . . , �p2(t, TK )
. . . , . . . , . . . �pm(t, T1) , . . . , �pm(t, TK )
µp(t, T1) � rt , . . . , µp(t, TK ) � rt
3
7777 5
2
66 4
P(t, T 1
)w 1
P(t, T 2
)w 2
. . . P(t, TK )wK
3
77 5 ⌘ A2z = 0 (9)
Dr Lara Cathcart () 2015 18 / 46
Multi-factor models
Since the above is a homogeneous system of equations and z 6= 0, this only possible if the rank of A
2
is equal to m. Since A 2
has m + 1 rows but rank m, the last row can be written as a linear combination of the other rows.
Dr Lara Cathcart () 2015 19 / 46
Multi-factor models
Moreover, this result does not depend on the specific maturities Tk, so for any T we have
µp(t, T) = rt + mX
i=1
�i(Xt)�pi(t, T) (10)
where �i(Xt) is the market price of risk for the i’th state variable (factor). Note that the risk premia can only depend on Xt and possibly calendar time t, but not on the maturity dates T.
Dr Lara Cathcart () 2015 20 / 46
Multi-factor models
To complete the derivation of bond prices, we substitute (10) into (3). After rearranging terms and using the definition of �pi(t, T) in (4), we get the following PDE for bond prices:
1
2
mX
i=1
mX
j=1
@2P
@Xi@Xj �i(X)�j(X)⇢ij + (11)
mX
i=1
@P
@Xi (µi(X) � �i(X)�i(X)) +
@P
@t � rP = 0
with boundary condition P(T, T) = 1.
Dr Lara Cathcart () 2015 21 / 46
Multi-factor models
As in the one-factor case, we can use the Feynman-Kac theorem to represent the solution of the PDE (11) as the risk neutral expectation
P(t, T) = EQt
h e�
R T t
r(Xs)ds i
(12)
where the expectation is taken under the probability measure corresponding to the drift-adjusted stochastic process (SDE):
dXt = (µ(Xt) � �(Xt)�(Xt)) dt + �(Xt)dW Qt (13)
Dr Lara Cathcart () 2015 22 / 46
Multi-Factor models
Thus far, our discussion of multi-factor models may appear somewhat abstract. For example, we have not made any attempts to interpret the state vector, Xt, except for being the driving force of changes in the yield curve.
Dr Lara Cathcart () 2015 23 / 46
Multi-Factor models
However, once we have specified a stochastic process for the state variables and made assumptions about the risk premia, we can solve the bond-pricing equation and determine the functional relationship between P(t, T) and Xt, for any maturity date T. Given m di↵erent bond prices (i.e points on the yield curve), we can invert the bond-pricing equation and express the m state variables in terms of m zero-coupon yields. This approach is useful in practical implementation of the models, but the interpretation of the state variables may not be straightforward.
Dr Lara Cathcart () 2015 24 / 46
Multi-Factor models
The last problem suggests that we should use observable state variables when building term-structure models. Using macroeconomic variables is an interesting idea, and inflation an obvious candidate for any nominal term-structure model, but there are several problems. In particular, macroeconomic variables are observed relatively infrequently (monthly, at most), and the data quality is often rather poor due to measurement problems.
Dr Lara Cathcart () 2015 25 / 46
Outline
1 Assumptions
2 A general framework for multi-factor model
3 The exponential-a�ne class of models
Dr Lara Cathcart () 2015 26 / 46
A�ne Multi-Factor Models
In the following we consider a few examples from a class of models called exponential-a�ne models, where a general analytical solution is available. For some parametric specifications, we can obtain a closed-form expression like in the Vasicek model, but in the worst case we will have to solve a system of ordinary di↵erential equations numerically, and this can be done very e�ciently with the Runga-kutta method.
Dr Lara Cathcart () 2015 27 / 46
Examples of a�ne multi-factor models
For models with multiple factors, there are a lot of di↵erent specifications and we cannot provide an exhaustive list. Instead we will o↵er only a few examples from the multi-factor literature.
Dr Lara Cathcart () 2015 28 / 46
Gaussian central tendency model
This model has been proposed by among others Beaghole-Tenney (1991) and Jegadeesh-Pennacchi (1996). The short rate is governed by the two-factor Gaussian process
drt = 1(µt � rt)dt + �1dW1t (14)
dµt = 2(✓ � µt)dt + �2dW2t (15)
and the two Brownian motions may be correlated with correlation coe�cients ⇢.
Dr Lara Cathcart () 2015 29 / 46
Gaussian central tendency model
The market prices of risks are specified as constant � 1
and � 2
. The central tendency models generalizes the Vasicek model by letting the short rate revert towards a time varying (stochastic) mean which is governed by a separate process. Sometimes this feature is referred to as a ”double decay” model.
Dr Lara Cathcart () 2015 30 / 46
Gaussian central tendency model
The PDE is given by:
1
2
@2P
@r2 �2 1
+ 1
2
@2P
@µ2 �2 2
+ @2P
@r@µ ⇢�
1
� 2
+ @P
@r [
1
(µ � r) � � 1
� 1
] + (16)
@P
@µ [
2
(✓ � µ) � � 2
� 2
] + @P
@t � rP = 0
subject to the boundary condition P(T, T) = 1.
Dr Lara Cathcart () 2015 31 / 46
Gaussian central tendency model
It is straightforward to verify that this model is exponential-a�ne (since the model is Gaussian). Therefore,
P(t, T) = exp[A(⌧) + B 1
(⌧)rt + B2(⌧)µt] (17)
If we substitute the requisite partial derivatives into (16) and divide by P on both sides of the equation we get,
1
2 B2 1
(⌧)�2 1
+ 1
2 B2 2
(⌧)�2 2
+ B 1
(⌧)B 2
(⌧)⇢� 1
� 2
+ B 1
(⌧)[ 1
(µ � r) � � 1
� 1
] + (18)
B 2
(⌧)[ 2
(✓ � µ) � � 2
� 2
] � A0(⌧) � B0 1
(⌧)r � B0 2
(⌧)µ � r = 0
Dr Lara Cathcart () 2015 32 / 46
Gaussian central tendency model
Since (18) must hold for all values of r and µ, we obtain the following ODE systems after collecting terms:
B0 1
(⌧) = � 1
B 1
(⌧) � 1 (19) B0 2
(⌧) = 1
B 1
(⌧) � 2
B 2
(⌧) (20)
A0(⌧) = 1
2 �2 1
B2 1
(⌧) + 1
2 �2 2
B2 2
(⌧) + ⇢� 1
� 2
B 1
(⌧)B 2
(⌧) (21)
�� 1
� 1
B 1
(⌧) + ( 2
✓ � � 2
� 2
)B 2
(⌧)
with boundary (initial) conditions B 1
(0) = 0, B 2
(0) = 0, and A(0) = 0 as P(T, T) = 1, for all rt and µt.
Dr Lara Cathcart () 2015 33 / 46
Gaussian central tendency model
It is possible to solve the entire ODE system in closed form. We start by concentrating on B
1
(⌧) and B 2
(⌧).
Dr Lara Cathcart () 2015 34 / 46
Gaussian central tendency model
First, note that the ODE defining B 1
(⌧) is exactly the same as in the Vasicek model. This means that
B 1
(⌧) = e�1⌧ � 1
1
(22)
Dr Lara Cathcart () 2015 35 / 46
Gaussian central tendency model
If we substitute (22) into (20), we get another linear ODE, which can be solved by the same technique we used in the one factor Vasicek model.
B 2
(⌧) = e�2⌧ � 1
2
� e�1⌧ � e�2⌧
1
� 2
(23)
Dr Lara Cathcart () 2015 36 / 46
Gaussian central tendency model
Finally, we can substitute (22) and (23) into (21) and A(⌧) can be calculated by ordinary integration. The expression for A(⌧) is rather lengthy.
Dr Lara Cathcart () 2015 37 / 46
Fong-Vasicek Stochastic Volatility model
Fong-Vasicek (1991) propose another extension of the Vasicek model where the Ornstein-Uhlenbeck process is augmented with a stochastic volatility factor:
drt = 1(µ � rt)dt + p VtdW1t (24)
dVt = 2(↵ � Vt)dt + ⌘ p VtdW2t (25)
The correlation coe�cient between the two Brownian motions is denoted ⇢.
Dr Lara Cathcart () 2015 38 / 46
Fong-Vasicek Stochastic Volatility model
Fong-Vasicek (1991) specify the market prices of risk as
�i(.) = �i p V (26)
since this is the only specification which preserve the a�ne property.
Dr Lara Cathcart () 2015 39 / 46
Fong-Vasicek Stochastic Volatility model
With these assumptions we derive the PDE subject to the appropriate boundary condition and guess a solution of the form
P(t, T) = exp[A(⌧) + B 1
(⌧)r + B 2
(⌧)V ] (27)
After substituting the requisite partial derivatives of (27) into the PDE and collecting terms, we get an ODE system defining the function A(⌧), B
1
(⌧) and B 2
(⌧). The solution for B 1
(⌧) is the same as in the Vasicek model.
Dr Lara Cathcart () 2015 40 / 46
Fong-Vasicek Stochastic Volatility model
Closed-form expressions for B 2
(⌧) and A(⌧) are presented in Selby-Strickland (1995). The espressions are quite complicated involving (infinite order series expansion) so it might be worthwhile to consider solving the ODEs numerically instead.
Dr Lara Cathcart () 2015 41 / 46
Multi-factor CIR model
Neither the Gaussian central tendency model nor the Fong-Vasicek stochastic volatilty model restrict the short rate to be non-negative.
Dr Lara Cathcart () 2015 42 / 46
Multi-factor CIR model
A popular multi-factor model with this property is the m factor CIR model which is obtained by adding m independent square root processes.
drt = mX
i=1
yit (28)
dyit = i(µi � yit)dt + �i p yitdWit (29)
and the market price of risk for the i’th factor is specified as in the one-factor CIR model,
�i(.) = (�i/�i) p yit (30)
Dr Lara Cathcart () 2015 43 / 46
Multi-factor CIR model
In this case the easiest way to derive an expression for bond prices is using the Feynman-kac formula
P(t, T) = Et h e�
R T t (
Pm i=1 yis)ds
i (31)
where yit evolves according to
dyit = (i(µi � yit) � �iyit)dt + �i p yitdW
Q it (32)
Dr Lara Cathcart () 2015 44 / 46
Multi-factor CIR model
By interchanging the order of integration and summation equation (31) can rewritten as
P(t, T) = EQt
h e�
R T t (
Pm i=1 yis)ds
i (33)
P(t, T) = EQt
" mY
i=1
e� R
T t
yisds
# (34)
P(t, T) = mY
i=1
EQt
h e�
R T t
yis ds i
(35)
P(t, T) = mY
i=1
Pi(t, T) (36)
Dr Lara Cathcart () 2015 45 / 46
Multi-factor CIR model
Where Pi(t, T) is the price formula for a one-factor CIR model with parameters i, µi, �i and �i, as well as ”short rate” yit. Note that the third line follows because of independence between the m square-root processes.
Dr Lara Cathcart () 2015 46 / 46
- Assumptions
- A general framework for multi-factor model
- The exponential-affine class of models
Slides FIS1.pdf
Outline
1 Introduction
2 Interest Rate Derivatives
3 Black’s Model
4 Introduction to Interest Rate Models
Dr Lara Cathcart () 2015 2 / 75
Yield Curve Basics: Definitions
Zero Coupon Bonds
Time to Maturity
Yield to Maturity or Continuously Compounded Spot Interest Rate
Short Rate
Accumulator Account
Forward Rate
No Arbitrage Relationships
Dr Lara Cathcart () 2015 3 / 75
Zero Coupon Bonds
A T maturity zero coupon bond (pure discount bond) is a contract that guarantees its holder the payment of one unit of currency at time T with no intermediate payment. The contract value at time t < T is denoted by P(t, T). Clearly P(T, T) = 1 for all T. If we are at time t, a zero coupon bond for the maturity T is a contract that establishes the present value of one unit of currency to be paid at time T.
Dr Lara Cathcart () 2015 4 / 75
Time to Maturity
The time to maturity is the amount of time in years from the present time t to the maturity time T, T > t. The definition T � t make sense as long as t and T are real numbers associated to two time instants.
Dr Lara Cathcart () 2015 5 / 75
Time to Maturity Continued
If t and T denote two dates expressed as day—month—year, say D1 = (d1, m1, y1) and D2 = (d2, m2, y2) we need to define the amount of time between the two dates in terms of the number of days between them. This choice however is not unique and the market evaluates the time between t and T in di↵erent ways (according to the relevant market convention).
Dr Lara Cathcart () 2015 6 / 75
Yield to Maturity
The continuously compounded spot interest rate prevailing at time for the maturity T is denoted by R(t, T) and is the constant rate at which an investment of P(t, T) units of currency at time t accrues continuously to yield a unit amount of currency at maturity T.
R(t, T) = � ln P(t, T)
T � t (1)
Dr Lara Cathcart () 2015 7 / 75
Yield to Maturity Continued
The continuously compounded interest rate is therefore a constant rate that is consistent with zero coupon bond prices in that
exp(R(t, T)(T � t))P(t, T) = 1 (2)
and from which we can express the bond price in terms of the continuously compounded rate R
P(t, T) = exp �(R(t, T)(T � t)) (3)
Dr Lara Cathcart () 2015 8 / 75
Yield to Maturity Continued
Given a set of pure discount bonds P(t, T), t < T the term structure of interest rates is the set of yield to maturity or continuously compounded spot interest rate R(t, T). (The term structure of interest rate is a plot at time t of continuously compounded spot interest rate for all maturities T)
Dr Lara Cathcart () 2015 9 / 75
Short Rate
The short rate r t
⌘ r(t) is the rate on instantaneous borrowing or lending.
Dr Lara Cathcart () 2015 10 / 75
Accumulator account
A sum of 1 invested in the short rate at time zero and continuously rolled over is called the accumulator account or the money market account. Its value p
t
at time t is
p
t
= exp(
Z t
0 r
s
ds) (4)
A money market account represents a risk-free investment, where profit is accrued continuously at the risk-free rate (short rate) prevailing in the market at every instant.
Dr Lara Cathcart () 2015 11 / 75
Note
The instantaneous short rate is a theoretical entity which does not exist in real life. A short Libor rate, such as a three-month rate, is often used as a surrogate for the short rate, but it must not be confused with the true instantaneous short rate, in some markets the overnight rate can be used as a surrogate, but in others it may be an extremely volatile rate reflecting short term supply and demand that does not reflect the underlying levels of interest rate.
Dr Lara Cathcart () 2015 12 / 75
Forward Rate
Forward rates are rates it is possible to lock into today for borrowing or lending in the future. They are characterized by three time instants, namely the time t at which the rate is considered, its expiry T1 and its maturity T2. Their values can be derived directly from pure discount bond prices. Define f
t
(T1, T2) to be continuously compounded forward rate available at time t for borrowing at time T1, and repaying at time T2.
Dr Lara Cathcart () 2015 13 / 75
No arbitrage Relationship
e
R(t,T2)(T2�t) = eR(t,T1)(T1�t)eft(T1,T2)(T2�T1) (5)
so that
f
t
(T1, T2) = 1
T2 � T1 [R(t, T2)(T2 � t) � R(t, T1)(T1 � t)] (6)
or
f
t
(T1, T2) = 1
(T2 � T1) ln
P(t, T1)
P(t, T2) (7)
otherwise arbitrage would be possible.
Dr Lara Cathcart () 2015 14 / 75
Note
The instantaneous forward rate of maturity T is
f
t
(T) = f t
(T, T) = lim T2!T ft(T, T2) (8)
It is the rate obtainable at time t for instantaneous borrowing at time T. (Intuitively, the instantaneous forward rate f
t
(T) is a forward interest rate at time t whose maturity is very close to its expiry.)
Dr Lara Cathcart () 2015 15 / 75
Note Continued
1 It is assumed that all rates are riskless rates, so that there is no default risk 2 Spot rates are forward rates for immediate delivery,
R(t, T) = f t
(t, T) = lim T1!tft(T1, T) and rt = ft(t)
3 Investing at a spot rate is equivalent to rolling over at the appropriate instantaneous forward rates
e
R(t,T)(T�t) = e R
T
t
f
t
(s)ds (9)
4 Instantaneous forward rates can be read directly o↵ the yield curve,
f
t
(T) = � @(lnP(t, T))
@T = R(t, T) + (T � t)
@R(t, T)
@T (10)
Dr Lara Cathcart () 2015 16 / 75
Note Continued
Instantaneous forward rates are fundamental quantities in the theory of interest. For example the HJM model use the instantaneous forward rates as fundamental quantities to be modelled
Dr Lara Cathcart () 2015 17 / 75
Simple Interest Rate Instruments
1 LIBOR 2 Deposit 3 Accrual Factor 4 Discount Factor 5 Forward Rate Agreements, FRAs 6 Interest Rate Swaps
Dr Lara Cathcart () 2015 18 / 75
LIBOR
Libor sometimes called spot Libor is the London Interbank O↵er Rate. As the name implies it is the rate of interest that one London Bank will o↵er to pay on a deposit by another. There will in general be a di↵erent LIBOR for each of the standard deposit maturities.
Dr Lara Cathcart () 2015 19 / 75
Deposit
A deposit is an agreement between two parties in which one pays the other a cash amount an in return receives this money back at some pre-agreed future date, with a pre-agreed additional payment of interest. This interest is of course proportional to the amount of cash initially deposited.
Dr Lara Cathcart () 2015 20 / 75
Deposit Continued
A deposit is made until a fixed date referenced in the retail market as a fixed term deposit. Deposits are available for a range of maturities or terms, only a small number of maturities none greater than a year are quoted as standard. The amount of interest paid at the end of the period is naturally larger the longer the period. The way this interest is quoted is via an accrual factor and an interest rate.
Dr Lara Cathcart () 2015 21 / 75
Accrual Factor
The total amount of interest rate that the depositing bank will receive is calculated by multiplying the LIBOR by the amount of time as a proportion of a year, for which this money has been on deposit.This amount of time is called accrual factor or day count fraction. Accrual factors are calculated by dividing the number of days in the period by the number of days in year. Di↵erent markets use di↵erent conventions.
Dr Lara Cathcart () 2015 22 / 75
Discount Factors
Suppose we are at time t. For any time T � t there is a discount factor denoted by �(t
T
), defined to be the value at time t of a zero coupon bond ZCB paying a unit amount at time T. Note that for all t �(t
t
) = 1 since a unit cash flow now is worth one unit.
Dr Lara Cathcart () 2015 23 / 75
Discount Factors
We usually assume that the initial discount curve �(0 T
), T � 0 if today is time zero, is known.
Dr Lara Cathcart () 2015 24 / 75
LIBOR Continued
Let L(t, ⌧) define the Libor rate at time t, of tenor ⌧ where the tenor of an interest rate is its maturity period, the period from the point of investment to the time that interest is paid. If one lends at Libor then one pays 1 at time t. At time t + ⌧ one receives back 1 + L(t, ⌧)↵
L
(t, t + ⌧), where ↵ L
(t, t + ⌧) is the day count fraction for L. ↵
L
(t, t + ⌧) is the proportion of L paid out at time t + ⌧, and is computed as a fraction of a year. For simplicity we set ↵(t, t + ⌧) ⌘ ⌧, a constant.
Dr Lara Cathcart () 2015 25 / 75
LIBOR Continued
To avoid arbitrage we must have
P(t, t + ⌧) = 1
1 + L(t, ⌧)↵ L
(t, ⌧) (11)
Dr Lara Cathcart () 2015 26 / 75
Forward Rate Agreements (FRAs)
Forward rate agreements are the market equivalents of the theoretical forward rates defined above. Let F = F0(t, ⌧) denotes the FRA rate agreed at time 0 for time t, tenor ⌧. The rate F is fixed so that the premium at time 0 is 0.
Dr Lara Cathcart () 2015 27 / 75
FRA Continued
At time t the holder of the FRA receives
c = (L(t, ⌧) � F) ⇥ ↵
L
(t, ⌧)
1 + L(t, ⌧)↵ L
(t, ⌧) (12)
This quantity can be positive or negative, it represents the present value (at time t) of the di↵erence between borrowing at Libor of tenor ⌧ at time t and borrowing at the FRA rate F. It is as if one locked in at time 0, to a borrowing rate of F at time t, tenor ⌧, since the FRA pays o↵ the di↵erence what one actually has to borrow at L(t, ⌧) and the FRA rate. One can buy or sell FRAs, equivalent to future borrowing or lending at the FRA rate.
Dr Lara Cathcart () 2015 28 / 75
FRA Continued
To avoid arbitrage the FRA rate F must be related to market LIBOR rates.
1 + L(0, t + ⌧)↵ L
(0, t + ⌧) = (1 + L(0, t)↵ L
(0, t))(1 + F0(t, ⌧)↵(t, ⌧)) (13)
where for example t = ⌧ = three months, L(0, t + ⌧) is a six month Libor rate and L(0, t) is a three-month Libor rate. If equation (13) is not satisfied an arbitrage is possible. One lends at the more expensive rates, hedging by borrowing at the cheaper rates.
Dr Lara Cathcart () 2015 29 / 75
FRA Continued
Since 1/(1 + L(0, t)↵ L
(0, t)) = P(0, t) the FRA rate is also given by
F0(t, ⌧) = 1
⌧
✓ P(0, t)
P(0, t + ⌧) � 1
◆ (14)
Dr Lara Cathcart () 2015 30 / 75
Interest Rate Swaps
A swap rate is the coupon rate,S, on fixed rate borrowing with equal present value to floating rate borrowing over the same period. The present value of borrowing a principal of 1 at a fixed rate S with coupons paid at times t
i
i = 1, ..., n is
Pv(fixed leg) = nX
i=1
S↵ s
(t i�1, ti)�(ti) + �(tn) (15)
Where ↵ s
is the relevant day count fraction and �(t) is the discount factor for cash flows occurring at time t.
Dr Lara Cathcart () 2015 31 / 75
Interest Rate Swaps Continued
The present value of the floating rate leg (the stream of floating cashflows) of the swap is
Pv(floating leg) = nX
i=1
L(t i�1, ti)↵(ti�1, ti)�(ti) + �(tn) (16)
where for simplicity we are assuming that the tenors of the floating and fixed legs are the same. L(t
i�1, ti) is fixed at time ti�1 but the cashflows it determines is paid at time t
i
.
Dr Lara Cathcart () 2015 32 / 75
Interest Rate Swaps Continued
It is possible to reduce equation (16) considerably. Although L(t i�1, ti) is only
fixed at time t i�1 one can find the value at t = 0 of a portfolio generating the
amount L(t i�1, ti)↵L(ti�1, ti)�(ti) at time ti�1.
Dr Lara Cathcart () 2015 33 / 75
Interest Rate Swaps Continued
Now the value of the floating leg simplifies:
Pv(floating leg) = nX
i=1
L(t i�1, ti)↵(ti�1, ti)�(ti) + �(tn)
= nX
i=1
(�(t i�1) � �(ti)) + �(tn)
= �(t0) = 1
1 = nX
i=1
S↵ s
(t i�1, ti)�(ti) + �(tn) (17)
Dr Lara Cathcart () 2015 34 / 75
Interest Rate Swaps Continued
This means that S can be interpreted as the annual coupon rate, payable at time t
i
giving a bond a market price of 1 at t = 0. This is just the par coupon rate, so swap rates are equivalent to par coupon rates. Re-arranging the previous expression equation we can express the time tn swap rates S
tn
in terms of discount factors
S
tn
= 1 � �(t
n
) P
n
i=1 ↵s(ti�1, ti)�(ti) (18)
so from a knowledge of the discount factors one immediately derives swap rates
Dr Lara Cathcart () 2015 35 / 75
Outline
1 Introduction
2 Interest Rate Derivatives
3 Black’s Model
4 Introduction to Interest Rate Models
Dr Lara Cathcart () 2015 36 / 75
Interest rate derivatives
1 Swaptions 2 Caps 3 Floor
Dr Lara Cathcart () 2015 37 / 75
Introduction
Interest rate derivatives are instruments whose payo↵s are dependent in some ways on the level of the interest rate. Interest rate derivatives are more di�cult to value than equity and foreign exchange derivatives.
Dr Lara Cathcart () 2015 38 / 75
Introduction Continued
Firstly, the behavior of an individual interest rate is more complicated than that of a stock price or exchange rate. Secondly, for the valuation of many products, it is necessary to develop a model describing the behavior of the entire yield curve.Thirdly, the volatilities of di↵erent points on the yield curve are di↵erent. Fourthly, interest rates are used for discounting as well as for defining the payo↵ from the derivative.
Dr Lara Cathcart () 2015 39 / 75
Introduction Continued
In the following we examine two of the most popular over-the-counter interest rate derivatives (swaptions, and interest rate caps and floors) and we describe how the lognormal assumption underlying the Black and Scholes model can be used to value these instruments.
Dr Lara Cathcart () 2015 40 / 75
Swaptions
A swaption is an option to enter a swap at a pre-agreed fixed rate, the exercises rate. Swaptions come in many varieties but we shall be concerned only with vanilla swaptions. In fact a vanilla swaption is equivalent to an option on a coupon bond.
Dr Lara Cathcart () 2015 41 / 75
Swaptions Continued
We wish to describe the cashflows generated by swaptions. To do so we extend the notation introduced above. Suppose ↵
S
is the day basis of a swap. At times t i
i = 1, ..., n the swap generates cashflows
p
i
= ↵ s
(t i�1, ti)(L(ti�1, ti) � Lx) (19)
Where L x
is the rate on the fixed leg of the swap. If the swap was created some time in the past,L
x
may no longer equal the current swap rate and the swap will no longer have net zero value.
Dr Lara Cathcart () 2015 42 / 75
Swaptions Continued
Write V = V (t, t0, ↵s, n, LX ) for the time t value of the swap t t0, and let P(t, T) be the time t discount factor for time T. When t = t0, V is zero if Lx has the value
S = S t
(t, t n
) = 1 � P(t, t
n
) P
n
i=1 ↵sP(t, ti) (20)
which is just the current swap rate.
Dr Lara Cathcart () 2015 43 / 75
Swaptions Continued
If t < t0 we have a forward start swap. Cashflows are exchanged starting at a future time rather than immediately. The value of the floating leg is now P(t, t0), and the value of the fixed leg is
P n
i=1 ↵sLxP(t, ti) + P(t, tn), so the value V at time t of the forward start swap is
V (t, t0, ↵s, n, Lx) = P(t, t0) � nX
i=1
↵ s
L
x
P(t, t i
) � P(t, t n
) (21)
Dr Lara Cathcart () 2015 44 / 75
Swaptions Continued
This is zero when L x
= S t
(t0, tn),
S
t
(t0, tn) ⌘ St(t0, tn, ↵s) = P(t, t0) � P(t, tn)P
n
i=1 ↵sP(t, ti) (22)
S
t
(t0, tn) is the forward rate start swap rate at time t on a forward start swap starting at time t0 > t and maturing at time tn, where the tenor ⌧ ⇠ ↵s is understood and has been omitted from the notation.
Dr Lara Cathcart () 2015 45 / 75
Swaptions Continued
If t n
is clear from the context, we shall abbreviate S t
(t0, tn) to St(t0) Let’s consider a European swaption on a forward start swap with exercise rate L
s
, exercised at time t0. The payo↵ to this swaption at time t0 is
p = max(S t0 � Ls, 0) ⇥ P (23)
where
P = nX
i=1
↵ s
P(t0, ti) (24)
p = max( nX
i=1
(↵ s
P(t0, ti)St0 � ↵sP(t0, ti)Ls), 0) (25)
makes it clear that the payo↵ to a swaption is the di↵erence between the discounted cashflows on the fixed legs of swaps at the current swap rate of S
t0
and the swaption exercise rate L s
.
Dr Lara Cathcart () 2015 46 / 75
Caps
On each date that an interest payment is due a cap pays out the di↵erence between interest payment at a floating rate r
t
, and interest payments at a fixed exercise rate r
x
, when r t
is greater than r x
, on a principal of P.
Dr Lara Cathcart () 2015 47 / 75
Cap Continued
On each exercise date the cap pays out p = ⌧Pmax(0, r t
� r x
), where ⌧ is the tenor of r
t
. For a floating rate borrower, buying a cap e↵ectively caps interest payments at the rate r
x
. The floating rate r t
is usually a Libor rate, L t
of some fixed tenor ⌧. We assume this from now on, writing L
ti
for the Libor value L(t
i
, t i+1). An interest rate cap is a strip of individual interest rate options on rt,
called caplets. There is one caplet for each exercise date.
Dr Lara Cathcart () 2015 48 / 75
Floor
An interest rate floor pays o↵ p = ⌧Pmax(0, r x
� r t
) at each exercise date, ensuring that interest is paid at a rate at least as great as r
x
. A floor is a strip of individual floorlets.
Dr Lara Cathcart () 2015 49 / 75
Caps and Floors Continued
Caps and floors are used by corporates in the management of interest rate risk on floating rate debt. Purchasing a cap allows the floating rate borrower to determine the maximum interest rate they will ever have to pay on the debt. A floating rate borrower who sells a floor generates capital at the cost of ensuring that interest is always paid at a rate at least that of the floor rate.
Dr Lara Cathcart () 2015 50 / 75
Caps and Floors Continued
Equivalence to a portfolio of Bond Options: Suppose a cap is composed of caplets on Libor at time t
i
,i = 1, ..., n. The payo↵ to the caplet on L ti
is made at time t
i+1. The payment of ⌧Pmax(0, Lti � rx) must be discounted back from time ti+1 to time t
i
. The correct discount rate is the Libor rate L(t i
, t i+1) which is known at
time t i
. The value v i
at time t i
of the ith caplet is
v
i
= ⌧Pmax(0, L
ti
� r x
)
1 + ⌧L ti
(26)
= max(0, P � P(1 + ⌧r
x
)
(1 + ⌧L ti
) ) (27)
Dr Lara Cathcart () 2015 51 / 75
Caps and Floors Continued
But P(1 + ⌧r x
)/1 + ⌧L ti
is the value at time t i
of a pure discount bond maturing at time t
i+1 with value P(1 + ⌧rx), so the value vi of the ith caplet at time ti is the value of a put maturing at time t
i
with exercise price P, on a pure discount bond maturing at time t
i+1 with value P(1 + ⌧rx).
Dr Lara Cathcart () 2015 52 / 75
Caps and Floors Continued
Each caplet is equivalent to a bond put option this is important because many of the simpler interest rate models we shall meet later have explicit formulae for bond option values, which means that caps can be priced very easily in those models.
Dr Lara Cathcart () 2015 53 / 75
Outline
1 Introduction
2 Interest Rate Derivatives
3 Black’s Model
4 Introduction to Interest Rate Models
Dr Lara Cathcart () 2015 54 / 75
Black’s Model
Since the Black-Scholes model was first published in 1973, it has become a very popular tool. The model has been extended so that it can be used to value options on foreign exchange, options on indices, and options on futures contract. The model we will is usually referred to as Black’s model because the formulas are similar to those in the model suggested by Fischer Black for valuing options on commodity futures.
Dr Lara Cathcart () 2015 55 / 75
Black’s Model Continued
Consider a European call option on a variable whose value is V . Define
T: Maturity date of the option
F: Forward price of V for a contract with maturity T
F0: Value of F at time zero
X: Strike price of the option
P(t, T): Price at time t of a zero-coupon bond paying $1 at time T
V
T
: Value of V at time T
�: Volatility of F
Dr Lara Cathcart () 2015 56 / 75
Black’s Model Continued
Black’s model calculates the expected payo↵ from the option assuming 1
V
T
has a lognormal distribution with the standard deviation of ln V T
equal to � p T.
2 The expected value of V T
is F0
It then discounts the expected payo↵ at the T-year risk-free rate by multiplying by P(0, T).
Dr Lara Cathcart () 2015 57 / 75
Black’s Model Continued
The payo↵ from the option is max(V T
� X, 0) at time T. The lognormal assumption implies that the expected payo↵ is
E(V T
)N(d1) � XN(d2) (28)
where E(V T
) is the expected value of V T
and
d1 = ln[E(V
T
)/X] + �2T/2
� p T
(29)
d2 = ln[E(V
T
)/X] � �2T/2 � p T
= d1 � � p T (30)
Because we are assuming that E(V T
) = F0 the value of the option is
c = P(0, T)[F0N(d1) � XN(d2)] (31)
Dr Lara Cathcart () 2015 58 / 75
Black’s Model Continued
where
d1 = ln(F0/X) + �
2 T/2
� p T
(32)
d2 = ln(F0/X) � �2T/2
� p T
= d1 � � p T (33)
Dr Lara Cathcart () 2015 59 / 75
Black’s Model Continued
An important feature of Black’s model is that we do not have to assume geometric Brownian motion for the evolution of either V or F. All that we require is that V
T
be lognormal at time T. The parameter, �, is usually referred to as the volatility of F or the forward volatility of V . However, its only role is to define the standard deviation of lnV
T
by means of the relationship
Standard Deviation of lnV
T
= � p T (34)
The volatility parameter does not necessarily say anything about the standard deviation of lnV at times other than time T.
Dr Lara Cathcart () 2015 60 / 75
Black’s Formula for Swaptions
Suppose that t i
� t i�1 = ⌧ is a constant and that ↵s(ti�1,t
i
) = ⌧, the swap has constant tenor ⌧ and final cashflow time t
n
. Let S t
(t0) be the time t forward start swap rate on a forward start swap of tenor ⌧ starting at time t0 with a final cashflow time t
n
.
Dr Lara Cathcart () 2015 61 / 75
Black’s Formula for Swaptions Continued
At time t < t0 the swaption has value v.
v = P(S t
(t0)N(d1) � LsN(d2)) (35)
where P = P
n
i=1 ⌧P(t, ti), N(d) is the standard normal distribution function and
d1 = 1
� p t0 � t
ln S
t
(t0)
L
s
+ 1
2 � p t0 � t (36)
d2 = d1 � � p t0 � t (37)
with � = �(t, t0, ⌧, n, Ls), the Black swaption volatility.
Dr Lara Cathcart () 2015 62 / 75
Black’s Formula for Swaptions
Black’s formula assumes that the forward start swap rate S t
(t0) is the underlying state variable. If S
t
(t0) were log-normal with volatility �, then Black’s formula would be correct. Swaption prices are usually quoted as the volatility �. All other variables in Black’s formula are known form either the specification of the swaption, or form current market prices.
Dr Lara Cathcart () 2015 63 / 75
Black’s Formula for Caps
Consider a cap on Libor of tenor ⌧ with individual caplets maturing at time t
i
= t0 + i⌧, i = 1, ..., n. The payo↵ to the ith caplet at time ti+1 is
p
i
= ⌧max(0, L ti
� r X
) (38)
where L ti
is the market quoted rate at time t i
, r X
is the exercise rate and the notional principal is 1.
Dr Lara Cathcart () 2015 64 / 75
Black’s Formula for Caps Continued
Let c i
be the value at time t of the ith caplet the value of the cap at time t < t0 is c =
P n
i=1 ci. For an individual caplet, maturing at time ti, Black’s formula for the value of the caplet is
c
i
= ⌧P(t, t i+1)(Ft(ti, ti+1)N(d1) � rX N(d2)) (39)
where P(t, t i+1) is the time t value of the time ti+1 maturity pure discount bond,
F
t
(t i
, t i
+ 1) is the market forward rate,
d1 = 1
� i
p t
i
� t ln
F
t
(t i
, t i+1)
r
X
+ 1
2 � i
p t
i
� t (40)
d2 = d1 � �i p t
i
� t (41)
and � i
is the Black’s volatility of the ith caplet.
Dr Lara Cathcart () 2015 65 / 75
Outline
1 Introduction
2 Interest Rate Derivatives
3 Black’s Model
4 Introduction to Interest Rate Models
Dr Lara Cathcart () 2015 66 / 75
Introduction to Interest Rate Models
Any interest rate model has to have two main ingredients if it is to have any hope of practical application. It must provide a statistical description of how the state variables in the model change through time, and it must provide a procedure to price interest rate derivatives from the statistical description.
Dr Lara Cathcart () 2015 67 / 75
Introduction to Interest Rate Models
The mathematical form that the statistical description of an interest rate takes entails firstly, identifying the state variable that the model uses, and secondly, specifying the dynamics of the state variables. In this set of lectures we will look at the three main types of interest rate models.
1 Classical models 2 Arbitrage-free models 3 Market models
Other types of model exist for example random field models and models with jump components, but are beyond the scope of this course.
Dr Lara Cathcart () 2015 68 / 75
Criteria for model Selection
An interest rate model may have strong theoretical properties, unless it actually works it is of relatively little use. Determining whether a model works or not is an ill-defined objective, but there are various desirable properties a model could possess which increase the likelihood of it being used in practice. Firstly, the model should fit market data. Secondly, it should have good dynamical properties, and it should be tractable.
Dr Lara Cathcart () 2015 69 / 75
Fitting market data
Interest rate models can be regarded as methods of interpolation and extrapolation. Given some set of market data, choose the parameters of the model so that the model fits closely to the market data and using these parameters other instruments may now be priced consistently with the market data. In practice it is usually not possible to fit more than a few aspects of the complete set of available market data.
Dr Lara Cathcart () 2015 70 / 75
Fitting market data
One would like to fit 1 the current yield curve 2 current caplet/bond option prices 3 the current volatility structure, for instance the term structure of at the
money bond option implied volatilities.
Dr Lara Cathcart () 2015 71 / 75
Good Dynamics
Current market prices are essentially static features . A model should also be able to fit to some extent the way that prices change through time. This means having the theoretical model distributions of future prices as close as possible to the distributions of actual market prices. Putting this to one side, in practice one may only hope at best to match selected dynamical features. Even then, if the dynamical feature is observed under the objective measure rather than a pricing measure, one may have to estimate the prices of risk for the model, as well as the coe�cient of interest.
Dr Lara Cathcart () 2015 72 / 75
Good Dynamics
The types of dynamical featured that it may be thought desirable to match may include:
1 the dynamics of the short rate, for instance its reversion level and rate, and its volatility;
2 the dynamics of the whole yield curve; 3 the dynamics of selected instruments.
Dynamics may be measured as moments of an historical time series.
Dr Lara Cathcart () 2015 73 / 75
Tractability
To be useful, any model must give computable solutions. It is no accident that some of the most popular interest rate models are those which have either explicit solution for the values of a range of instruments, or else come equipped with good computational methods, such as lattice implementation.
Dr Lara Cathcart () 2015 74 / 75
Tractability
No model has explicit solutions for every type of instrument, so one may ask how tractable a model is for di↵erent classes of instrument does the model:
1 have explicit solutions for simple instruments such as bonds, bond options and caplets?
2 have simple numerical solution methods for more complicated instruments such as swaptions, American options, and path-dependent instruments? In may cases one still has to use the full machinery of finite di↵erence methods or Monte Carlo methods, but even here some models may be easier to compute with than others.
Dr Lara Cathcart () 2015 75 / 75
- Introduction
- Interest Rate Derivatives
- Black's Model
- Introduction to Interest Rate Models
FIS2.pdf
1 A General One-Factor Model
In this section we explain the common mathematical structure of term-structure models with a single factor,
a framework that encompasses the Vasicek 1977 and CIR 1985 models.
We make the following assumptions:
1. The bond market is frictionless: no distorting taxes, no transactions costs, no short-sale restrictions,
and all bonds are infinitely divisible.
2. Investors always prefer more wealth to less, i.e. the marginal utility of wealth is positive at all levels
of wealth.
3. All bond prices, i.e. P(t,T ) for all T > t, depend only on a single state variable (factor): the short
rate rt in addition to t and T . This means that changes in the yield curve at different maturities are
perfectly correlated.
4. The short rate follows the general SDE:
drt = µ(rt)dt + s(rt)dWt (1)
where µ(r) and s(r) are the drift and volatility functions, respectively, and Wt a Brownian motion
(Wiener) process.
It is important to realize that we do not assume that the relationship between P(t,T ) and the short rate rt is
known. On the contrary, the entire purpose of the following is to derive that function endogenously, from
the above assumptions.
In the first step, we determine the stochastic process (SDE) for the bond price P(t,T ) note that T is
fixed, and t denotes calendar time.
By Ito’s lemma we get:
dP(t,T ) = µp(t,T )P(t,T )dt + sp(t,T )P(t,T )dWt (2)
where
µp(t,T )P(t,T ) = ∂P ∂r
µ(r)+ ∂P ∂t
+ 1 2
∂2P ∂r2
s2(r) (3)
2
sp(t,T )P(t,T ) = ∂P ∂r
s(r) (4)
In equation (2), µp(t,T ) is the expected instantaneous return of the bond with maturity date T, and
sp(t,T ) is the volatility (standard deviation)of the bond return. Both depend on the short rate, rt , but to
simplify the notation this dependence is suppressed here.
The problem is that equilibrium expected returns µp(t,T ) for different T 0s are unknown, so a general ex-
pression for the bond price P(t,T ) cannot be determined at this stage. The intermediate goal in the following
is developing some form of equilibrium model for the expected returns, µp(t,T ), for all T . Concretely, we
use the principle of no-arbitrage to reduce this problem to specifying a single market price risk parameter.
Suppose we construct a portfolio consisting of w1 bonds with maturity date T1 and w2 bonds with maturity
T2. We require T1 6= T2, but apart from that T1 and T2 can be arbitrary. The value of the resulting portfolio,
at time t, is denoted by
qt = w1P(t,T1)+ w2P(t,T2), (5)
and the value, qt , satisfies the SDE:
dqt = [w1µp(t,T1)P(t,T1)+ w2µp(t,T2)P(t,T2)]dt (6)
+[w1sp(t,T1)P(t,T1)+ w2sp(t,T2)P(t,T2)]dWt
Since there are two bonds, and only one source of risk, it must be possible to choose w1 and w2 such that
w1sp(t,T1)P(t,T1)+ w2sp(t,T2)P(t,T2) = 0 (7)
In general, this requires continuous adjustment of the portfolio (which can be done costlessly since
we have assumed away transactions costs). If w1 and w2 are continuously readjusted according to (7), the
portfolio SDE (6) reduces to
dqt = [w1µp(t,T1)P(t,T1)+ w2µp(t,T2)P(t,T2)]dt (8)
which is deterministic (riskless).
3
A riskless portfolio has a simple dynamic:
dqt = qrt dt (9)
To prevent arbitrage (9) and (8) should be equal i.e the excess returns above the short rate rt must be
zero.
w1(µp(t,T1)� rt)P(t,T1)+ w2(µp(t,T2)� rt)P(t,T2) = 0 (10)
Rearranging the above and simplifying we get:
µp(t,T1)� rt sp(t,T1)
= µp(t,T2)� rt
sp(t,T2) (11)
The fact that the equality in equation (11) holds for any pair of maturities implies that the expressions on
either side of the equals side must be independent of Tj i.e. for any T ,
µp(t,Tj) = rt + l(rt)sp(t,Tj) (12)
j=1,2. Where l(rt) is the so-called market price of risk. Further note that the particular choices of
T1 and T2 play no role in the above derivation so (12) must hold for all T and l(r) must be independent
of T . In summary, we have reduced the problem of determining µp(t,T ), for all possible T, to that of
specifying a single market price parameter l(r), which is at most a function of the short rate r. Of course
this requires an additional assumption about market preferences, and different models (Vasicek,CIR) use
different specification of l(r).
Finally we substitute (12) into (3) and after rearranging terms (and using the definition of (sp(t,T )) we
get the following fundamental partial differential equation (PDE) for the bond price:
1 2
∂2P ∂r2
s2(r)+ ∂P ∂r
[(µ(r)� l(r)s(r)]+ ∂P ∂t
� rP = 0 (13)
with boundary condition P(T,T ) = 1. Analytical solutions to this PDE exist for several one-factor models,
including the Vasicek (1977) and CIR (1985) models.
A general representation of the solution to (13) is furnished by the Feynman-Kac formula:
P(t,T ) = E Qt [e � R T
t rsds] (14)
4
where the expectation is taken under the probability measure (probability distribution) corresponding to the
drift-adjusted stochastic process:
drt = (µ(rt)� l(rt)s(rt))dt + s(rt)dW Q
t (15)
Of course we still need to calculate (14) in order to get a closed-form expression for the bond price
P(t,T ), and in most cases it is actually simpler to solve the PDE directly.
However, equation (14) offers a lot of intuition about the mechanics of arbitrage-free term-structure
models. The current price, P(t,T ), is obtained by discounting the final payment of one unit of account back
to the present (time t), and since the future short-term interest rates are random, we take the expectation,
conditional on the current value of the short rate, rt .
2 Examples of one-factor models
In this section we present the bond-pricing formula, i.e. the solution P(t,T ) of the fundamental PDE for two
different models: Vasicek (1977) and CIR (1985).
2.1 Vasicek (1977) Model
The Vasicek model defines the short rate as a mean-reverting Gaussian process (also known as the Ornstein-
Uhlenbeck process):
dr = k(µ � r)dt + sdW
where k > 0 measures the speed of mean reversion, µ > 0 is the unconditional mean and s > 0 is the
volatility of the short rate. The market price of risk is assumed to be constant l(r) = l.
1. Form a riskless bond portfolio and derive the partial differential equation for the bond price
P(t,T )
The value of the bond P is a function of two independent variables; the value of the short rate r, and
calendar time t (or equivalently time to maturity t = T �t (i.e. dt = �dt). The differential of P(r,t,T )
5
is calculated using Ito’s lemma as follows:
dP(r,t,T ) = Pt dt + Prdr + 1 2
Ptt(dt)2 + 1 2
Pr(dr)2 + Prt drdt
= Pt dt + Prdr + 1 2
Prr(dr)2
Plugging now the expression for the short rate differential, as given by the Vasicek model we get:
dP(r,t,T ) = Pt dt + Pr (k(µ � r)dt + sdW )+ 1 2
Prrs2dt
=
Pt + Prk(µ � r)+
1 2
Prrs2 �
dt + PrsdW (16)
Without any loss of generality, we can assume that the bond price follows a geometric brownian motion with
non-constant drift and volatility terms, i.e.
dP(r,t,T ) = µP(r,t,T )P(r,t,T )dt + sP(r,t,T )P(r,t,T )dW (17)
Equating terms in equations (16) and (17), we have:
µP(r,t,T )P(r,t,T ) = Pt + Prk(µ � r)+ 1 2
Prrs2 (18)
sP(r,t,T )P(r,t,T ) = Prs (19)
Since the short rate is not a tradable asset, let’s construct an arbitrage portfolio q that consists of w1 bonds
maturing at date T1, P(r,t,T1) ⌘ P1 (for ease of notation) and w2 bonds maturing at date T2 6= T1, P(r,t,T2) ⌘
P2:
q = w1P1 + w2P2
) dq = w1dP1 + w2dP2 =
= w1 [µP(r,t,T1)P1dt + sP(r,t,T1)P1dW ]+ w2 [µP(r,t,T2)P2dt + sP(r,t,T2)P2dW ]
= [w1µP(r,t,T1)P1 + w2µP(r,t,T2)P2]dt +[w1sP(r,t,T1)P1 + w2sP(r,t,T2)P2]dW
The term that involves risk is [w1sP(r,t,T1)P1 + w2sP(r,t,T2)P2]dW . This risk can be eliminated by accord-
ingly choosing the weights w1 and w2. In particular, setting this term equal to zero yields the ratio between
6
the weights that needs to be satisfied in order to render q riskless:
[w1sP(r,t,T1)P1 + w2sP(r,t,T2)P2]��dW = 0
, w1 w2
= � sP(r,t,T2)P2 sP(r,t,T1)P1
(20)
Under these circumstances the bond portfolio is riskless and consequently, by the no-arbitrage condition, its
value should grow with the risk free rate (which is obviously equal to the short rate):
dq = rqdt = [w1µP(r,t,T1)P1 + w2µP(r,t,T2)P2]dt
, r [w1P1 + w2P2]��dt = [w1µP(r,t,T1)P1 + w2µP(r,t,T2)P2]��dt
, w1(µP(r,t,T1)� r)P1 + w2(µP(r,t,T2)� r)P2 = 0
, w1 w2
= � (µP(r,t,T2)� r)P2 (µP(r,t,T1)� r)P1
(21)
However, we already know what the ratio w1w2 by the equation (20). Thus, equating equations (20) and (21)
yields:
sP(r,t,T2)P2 sP(r,t,T1)P1
= (µP(r,t,T2)� r)P2 (µP(r,t,T1)� r)P1
(µP(r,t,T1)� r)��P1 sP(r,t,T1)��P1
= (µP(r,t,T2)� r)��P2
sP(r,t,T2)��P2 (µP(r,t,T1)� r)
sP(r,t,T1) =
(µP(r,t,T2)� r) sP(r,t,T2)
The left hand side of the above equation is only a function of T1, whereas the second is only a function of
T2. This implies that the above expression is independent of the maturity and should be the same across all
maturities! As a consequence, we can define the -constant- market price of risk l for any T :
l = µP(r,t,T )� r
sP(r,t,T ) (22)
7
Plugging in the above expression the results for µP(r,t,T ) and sP(r,t,T ) for the Vasicek model, as given by
equations (18) and (19) respectively, we finally come up with the PDE for the bond price:
l = Pt +Pr k(µ�r)+ 12 Prr s
2
P(r,t,T ) � r Pr s
P(r,t,T )
, lPrs = Pt + Prk(µ � r)+ 1 2
Prrs2 � rP
, 1 2
Prrs2 + Pr [k(µ � r)� ls]+ Pt � rP = 0 (23)
with terminal condition at maturity P(T,T ) = 1.
8
Guess that the solution to the above PDE has the so called exponential-affine structure:
P(t,T ) = eA(t)+B(t)r
where t = T � t is the time to bond maturity. Derive the ODE’s that A(t) and B(t) need to satisfy and
solve those ODE’s for A(t) and B(t) subject to appropriate conditions
First we need to compute the derivatives of P that appear in the PDE of equation (23):
Pr = ∂ ∂r
h eA(t)+B(t)r
i =
✓ ∂ ∂r
[A(t)+ B(t)r] ◆
eA(t)+B(t)r = B(t)P
Prr = ∂ ∂r
B(t)P = B(t) ∂ ∂r
P = B2(t)P
Pt = �Pt = � ∂ ∂t
eA(t)+B(t)r = �[A0(t)+ B0(t)r]P
We can now plug in the above results to the PDE:
1 2
B2(t)◆Ps2 + B(t)◆P[k(µ � r)� ls]� [A0(t)+ B0(t)r]◆P � r◆P = 0
� ⇥ B0(t)+ kB(t)+ 1
⇤ r +
1 2
B2(t)s2 + B(t)[kµ � ls]� A0(t) �
= 0
The original PDE and subsequently the above expression should be satisfied for all values of the short rate
r. Given that, the above expression should have zero coefficient for r and zero ”constant” term with respect
to r. Thus, we get the two ODE’s for A(t) and B(t):
B0(t)+ kB(t)+ 1 = 0 (24) 1 2
B2(t)s2 + B(t)[kµ � ls]� A0(t) = 0 (25)
with boundary conditions stemming straightforward from P(T,T ) = 1, i.e. at time to maturity t = T �T = 0:
P(T,T ) = eA(t=0)+B(t=0)r = 1 ) A(0)+ B(0)r = log(1) ) A(0)+ B(0)r = 0
, A(0) = B(0) = 0 (26)
where the above stems again from the fact that it should be satisfied for all values of r.
It now only remains to solve equations (24) and (25) subject to (26). Solving equation (24) for B(t) is
9
straightforward by simply multiplying by the integrating factor ekt and integrating in [0,t]:
B0(t)+ kB(t)+ 1 = 0 , ektB0(t)+ ektkB(t) = �ekt
, d [ektB(t)] = �ekt , Z t
0 d ⇥ eksB0(s)
⇤ = �
Z t
0 eksds
, ektB(t)� e0 B(0) |{z} =0
= �
eks
k
�t
0 , ektB(t) = �
ekt � 1 k
, B(t) = e�kt � 1
k
Given the value of B(t), we can then solve equation (25) for A(t):
A0(t) = 1 2
B2(t)s2 + B(t)[kµ � ls] = 1 2
e�kt � 1
k
�2 s2 +
e�kt � 1 k
[kµ � ls]
, Z t
0 A0(s)ds =
Z t
0
( 1 2
e�ks � 1
k
�2 s2 +
e�ks � 1 k
[kµ � ls] )
ds
, A(t)� A(0) |{z} =0
= s2
2k2
Z t
0
⇥ e�ks � 1
⇤2 ds +
kµ � ls k
Z t
0
⇥ e�ks � 1
⇤ ds
, A(t) = s2
2k2 Z t
0
⇥ e�2ks � 2e�ks + 1
⇤ ds +
kµ � ls k
Z t
0
⇥ e�ks � 1
⇤ ds
= s2
2k2
e�2ks
�2k � 2
e�ks
�k + s
�t
0 +
kµ � ls k
e�ks
�k � s
�t
0
= s2
2k2
e�2kt
�2k � 2
e�kt
�k + t �
1 �2k
+ 2 1 �k
� +
kµ � ls k
e�kt
�k � t �
1 �k
�
= � s2
2k
e�2kt
2k2 � 2
e�kt
k2 �
t k �
1 2k2
+ 2 1 k2
� �
kµ � ls k
2
66 4
e�kt � 1 k| {z }
=B(t)
+t
3
77 5
= � s2
2k
2
666 4
e�2kt � 2e�kt + 1 2k2| {z }
= 12 B 2(t)
+ 2e�kt
2k2 �
1 2k2
� 2 e�kt
k2 �
t k �
1 2k2
+ 2 1 k2
3
777 5 �
kµ � ls k
[B(t)+ t]
10
= � s2
2k
2
666 4
1 2
B2(t)� e�kt � 1
k2| {z } = 1k B(t)
� t k
3
777 5 �
kµ � ls k
[B(t)+ t]
= � s2
2k
1 2
B2(t)� B(t)+ t
k
� �
kµ � ls k
[B(t)+ t]
= [B(t)+ t]
s2
2k2 �
kµ � ls k
� �
s2
4k B2(t)
The last manipulations for A(t) (once the integration is completed) are not mandatory...they simply simplify
the expression.
2.2 Cox-Ingersoll-Ross Model
The famous CIR, or Cox, Ingersoll and Ross, model uses the so-called square-root SDE (process) for the
short rate:
drt = k(µ � rt)dt + s p
rt dWt (27)
CIR specifies the market price of risk as follows: l(r) = l p
r/s, The scaling by s is done only to
simplify the subsequent derivations.
The derivations are considerably more tedious than in the Vasicek case. Once again, the bond-pricing
formula takes the familiar exponential-affine form.
The main advantage over the Vasicek model is that rt is restricted to be nonnegative. However, for
realistic parameters values, there is rarely much difference between the yield curves obtained from the
Vasicek and CIR models, respectively.
2.3 Advantages and Disadvantages of One-factor models
The main advantage of one-factor models is their simplicity as the entire yield curve is a function of just
one state variable. Moreover, this state variable is observable-at least in principle (in practice we use a
short-term interest rate as a proxy). However, there are several problems with one-factor models. First,
the model assumes that changes in the yield curve, and hence bond returns, are perfectly correlated across
maturities, and not surprisingly this assumption is easily contradicted by the empirical evidence. Also the
Vasicek and CIR models can accommodate yield curve that are monotonic increasing or decreasing and
11
humped. An inversely humped yield curve cannot be generated with these models. Moreover with time-
invariant parameters one factor models tend to provide a very poor fit to the actual yield curves observed in
the market. The latter problem can be solved by calibration which is discussed at a later stage. By making
some parameters time-dependent, we obtain a perfect fit to the current (initial) yield curve, and the calibrated
model can only be used to price fixed-income derivatives. If the modeling purpose is identifying bonds that
are mispriced, the calibration approach cannot be used.
12
Coursework.pdf
7
Part III. Pricing (15 marks) Assume that these rates are annual bond versus 1 year LIBOR annual bond.
Maturity Swap Rate
1 Year 6.00%
2 Year 6.25%
3 Year 6.50%
4 Year 6.75%
5 Year 7.00%
1. What is the 5 year swap rate? Please build up the term structure of the forward rates and corresponding zero coupon bond prices.
2. What is the 2x5 swap rate? Here 2x5 swap refers to the 3-year swap, two years forward.
8
Appendix Bootstrap Method Following Wilmott (2007), the bootstrap method for deriving the yield curve from market swap rates is based upon rearranging the formula for the swap rate to solve for the longest discount factor. The time 𝑡 swap rate is
𝑆 = 1 − 𝛿(𝑡 )
∑ 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 ) ,
Where 𝛼 (𝑡 ,𝑡 )is the day count fraction between times 𝑡 and 𝑡 , and 𝛿(𝑡 ) is the discount factor for cash flows occurring at time 𝑡 . Rearranging the above equation,
𝑆 = 1 − 𝛿(𝑡 )
𝛼 (𝑡 ,𝑡 )𝛿(𝑡 ) + ∑ 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 )
⇒ 𝑆 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 ) + 𝑆 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 ) = 1 − 𝛿(𝑡 ) ,
so
𝛿(𝑡 ) = 1 − 𝑆 ∑ 𝛼 (𝑡 ,𝑡 )𝛿(𝑡 )
1 + 𝑆 𝛼 (𝑡 ,𝑡 ) .
If we have available the swap rates for many equally-spaced maturities, we may set 𝛼 (𝑡 ,𝑡 ) = 𝛼 and it simplifies to
𝛿(𝑡 ) = 1 − 𝑆 𝛼 ∑ 𝛿(𝑡 )
1 + 𝑆 𝛼
Then, starting with the one-period swap rate, 𝑆 and noting that 𝛿(𝑡 ) = 𝛿(0) = 1, we have
𝛿(𝑡 ) = 1
1 + 𝑆 𝛼 . The remaining discount factors 𝛿(𝑡 ), 𝑗 = 2,…,𝑛 can then be found sequentially by applying
𝛿 𝑡 = 1 − 𝑆 𝛼 ∑ 𝛿(𝑡 )
1 + 𝑆 𝛼 .
Note that in practice the swap rates we observe in the market may not be equally spaced, nor may they have identical payment frequencies and/or day count bases. Almost certainly the swap contracts will not actually start on the pricing date 𝑡 = 𝑡 . Furthermore, we may wish to introduce elaborate schemes for interpolating between the discrete maturity dates available to us. Such considerations lead to much more complicated bootstrapping algorithms but which are all founded on the principles presented here.
9
Reference D. Brigo, F. Mercurio. Interest Rate Models: Theory and Practice. Springer, 2nd edition, 2006. A. Cairns. Interest Rate Models. Princeton University Press, 2004. F. A. Longstaff and E.S. Schwartz. Interest rate volatility and the term structure: A two-factor general equilibrium model. The Journal of Finance, 47:1259–1282, 1992. F. A. Longstaff and E.S. Schwartz. Implementation of the longstaff-schwartz interest rate model. Journal of Fixed Income, September: 7–14, 1993.11 Riccardo Rebonato. Interest-Rate Option Models. John Wiley and Sons, 2nd edition, 1996. P. Wilmott. Paul Wilmott introduces Quantitative Finance. John Wiley and Sons, 2nd edition, 2007.
FIS6.pdf
1 The LIBOR and Swap Market Models
Before market models were introduced, short-rate models used to be the main choice for pricing and hedging
interest-rate derivatives. Short rate models are still chosen for many applications and are based on modelling
the instantaneous spot interest rate (short rate) via (possibly multi-dimensional) diffusion process. The
diffusion process characterizes the evolution of the complete yield curve in time.
To fix ideas, let us consider the time 0 price of a T2 maturity caplet resetting at time T1 (0 < T1 < T2)
with strike X and a notional amount of 1. Let τ denote the year fraction between T1 and T2. Such a contract
pays out the amount
τmax(L(T1,T2)− X,0) (1)
at time T2, where in general L(u,s) is the LIBOR rate at time u for maturity s.
Again to fix the ideas, let us choose a specific short-rate model and assume we are using the extended
Vasicek model. Such model allows for an analytical formula for forward LIBOR rates F .
F(t,T1,T2) = F(t,T1,T2,rt,θ) (2)
L(T1,T2) = F(T1,T1,T2,rT1,θ) (3)
One can try to price a caplet. To this end one can compute the risk neutral expectation of the payoff.
E Q0
! exp(−
! T2
0 rsds)τmax(F(T1,T1,T2,rT1,θ)− X,0)
" (4)
This too turns out to be feasible and leads to a function
UC(0,T1,T2,X,θ) (5)
On the other hand, the market has been pricing caplets (caps) for years. One possible derivation of
Black’s formula for caplet is based on the following approximation. When pricing the discounted payoff
E Q0
! exp(−
! T2
0 rsds)τmax(L(T1,T2)− X,0)
" = ... (6)
One first assumes the discount factor exp(− " T2
0 rsds) to be deterministic and identifies it with the corre-
2
sponding bond price P(0,T2). Then one factors out the discount factor to obtain
... ≈ P(0,T2)τE Q 0 (max(L(T1,T2)− X,0)) = P(0,T2)τE
Q 0 (max(F(T1,T1,T2)− X,0)) (7)
Now inconsistently with the previous approximation, one goes back to assuming rates to be stochastic,
and models the forward LIBOR rate F(t,T1,T2) as in the classical Black Scholes option pricing setup i.e as
a (driftless) geometric Brownian motion
dF(t,T1,T2) = vF(t,T1,T2)dW Q
t (8)
Where v is the instantaneous volatility assumed here to be constant for simplicity.
Then the expectation
E Q0 (max(F(T1,T1,T2)− X,0)) (9)
can be viewed simply as a T1 maturity call option price with strike X and whose underlying asset has
volatility v.
We therefore obtain
C pl(0,T1,T2,X) = P(0,T2)τE Q 0 (max(F(T1,T1,T2)− X,0)) (10)
= P(0,2 )τ(F(0,T1,T2)N(d1)− X N(d2)) (11)
d1 = log(F/X)+ w2/2
w (12)
d2 = log(F/X)− w2/2
w (13)
Where N(.) is the cumulative normal distribution function.
From the way we just introduced it, this formula seems to be partly based on inconsistencies. However,
with the change of measure set up we encountered in the previous lecture, the formula can be given full
mathematical rigor.
Denote by QT2 the T2 forward risk adjusted measure and by E T2 the corresponding expectation. We then
obtain
3
E Q0
! exp(−
! T2
0 rsds)τmax(L(T1,T2)− X,0)
" = P(0,T2)E T2 (τ(max(L(T1,T2)− X,0)) (14)
Now the last expectation is no longer taken under the risk neutral measure but rather under T2 forward
risk adjusted measure.
Since by definition F(t,T1,T2) can be written as the price of a tradable asset divided by P(t,T2) it needs
follow a martingale under QT2 . Martingale means driftless when dealing with diffusion process.
Therefore
dF(t,T1,T2) = vF(t,T1,T2)dW QT2
t (15)
Notice that the driftless lognormal dynamics above is precisely the dynamics we need in order to recover
exactly Black’s formula without approximation. Following this rigorous approach we indeed obtain Black’s
formula since the process F has the same distribution as in the approximated case above and hence the
expected value has the same value as before. The example just introduced is a simple case of what is known
as ”lognormal forward-LIBOR model”.
If we go back to the short rate model formula UC and ask ourselves whether this formula can be com-
patible with the above reported Black’s market formula.
It is well known that the two formulas are not compatible. In general, no known short-rate model can
lead to Black’s formula for caplets and more generally for caps.
What is done with short rate models in general is after obtaining a perfect fit of the initial term structure,
one also tries to look for parameters that produce caplet prices that are closest to a number of observed
market cap prices. The model is thus calibrated to the cap market and should reproduce well the observed
prices. Still the prices are complicated nonlinear functions of the deterministic parameters, and this renders
the parameters themselves difficult to interpret. On the contrary, the parameter v in the above ”lognormal
forward-LIBOR model” (LFM) for F has a clear interpretation as a lognormal percentage (instantaneous)
volatility and traders feel confident handling such kind of parameters.
When dealing with several caplets involving different forward rates, different structures of instantaneous
volatilities can be employed. For example one can select a different v for each forward rate by assuming each
forward rate to have a constant instantaneous volatility. Moreover, different forward rates can be modelled
as each having different sources W that are instantaneously correlated. Modelling correlation is necessary
for pricing payoffs depending on more than a single rate at a given time.
4
The model we briefly introduced is the market model for half of the interest rate derivatives world, i.e
the cap market. But what happens when dealing with basic products form the other half of this world, such
as swaptions?
Swaptions are priced by a Black-like market formula that is in many respects similar to the cap formula.
This market formula can be given full rigor as in the case of the caps formula. However, doing so involve a
different change of measure (called the forward swap measure)under which the relevant forward swap rate
(rather than a particular forward LIBOR rate) is driftless and lognormal. The obtained model is known as
”lognormal forward swap model” (LSM).
One may wonder whether the two models are distributionally compatible or not. Well the two models
are incompatible. If we adopt the LFM for caps we cannot recover the market formula given by the LSM
for swaptions. There are some recent works investigating the size of the discrepancy between these two
models, results seem to suggest that the difference is not large in most cases. However, the problem remains
of choosing either of the two models for the whole market. Brace, Dun and Barton (1998) suggest to adopt
the LFM as central model for the two markets mainly for its mathematical tractability. Also because the fact
that forward rates are somehow more natural and more representative of the yield curve than swap rates.
We are now left with the problem of finding a way to compute swaption prices with the LFM. In order
to understand the difficulties of this task, let us consider a very simple swaption. Assume we are at time 0.
The underlying interest rate swap starts at T1 and pays T2 and T3. All times are equally spaced by a year
fraction denoted by τ and we take a unit notional amount. The (payer) swaption payoff can be written as
Max (P(T1,T2)τ(F(T1,T1,T2)− K)+ P(T1,T3)τ(F(T2,T2,T3)− k),0) (16)
The key point here is the following, the payoff is not additively separable with respect to the different
rates. As a consequence, when you take expectation of such a payoff the joint distribution of the two rates
F(T1,T1,T2) and F(T2,T2,T3) is involved in the calculation so that the correlation between the two rates has
an impact on the value of the contract. This does not happen with caps. Let’s consider a cap consisting of
the T2 and T3 caplets. The cap payoff as seen from time 0 would be
exp ! − ! T2
0 rudu
" τ(Max(F(T1,T1,T2)− K,0)+ exp
! − ! T3
0 rudu
" τ(Max(F(T2,T2,T3)− K,0) (17)
5
This time, the payoff is additively separated with respect to different rates. Indeed we can compute (17) as
P(0,T2)τE T20 (Max(F(T1,T1,T2)− K,0)+ P(0,T3)τE T3 0 (Max(F(T2,T2,T3)− K,0) (18)
In this last expression we have two expectations each one involving a single rate. The joint distribution of
the two rates F(T1,T1,T2) and F(T2,T2,T3) is not involved, accordingly the correlation between these two
rates does not affect the payoff.
As a consequence of this example it is clear that adequately modelling correlation can be important in
defining a model that can be effectively calibrated to swaption prices. If a short model has to be chosen, it
is better to choose a multifactor model. As far as the LFM is concerned, the solution is usually to assign a
different Brownian motion to each forward rate and to assume such Brownian motions to be instantaneously
correlated. Choosing an instantaneous correlation structure flexible enough to express a large number of
swaptions prices and at the same time parsimonious enough to be tractable is a delicate task. Generally
to evaluate swaptions and other payoffs with the LFM one has usually to resort to Monte Carlo pricing by
simulating the relevant forward LIBOR rates.
6
