Modelling of Forward Libor
and Swap Rates
Marek Rutkowski
Faculty of Mathematics and Information Science
Warsaw University of Technology, 00-661 Warszawa, Poland
Contents
1 Introduction
2
2 Modelling of Forward Libor Rates
2.1 Forward and Futures Libor Rates . . . . . . . . . . . . . . . . . . .
2.1.1 Single-period Swaps Settled in Arrears . . . . . . . . . . . .
2.1.2 Single-period Swaps Settled in Advance . . . . . . . . . . .
2.1.3 Eurodollar Futures Contracts . . . . . . . . . . . . . . . . .
2.2 Lognormal Models of Forward Libor Rates . . . . . . . . . . . . . .
2.2.1 Miltersen-Sandmann-Sondermann Approach . . . . . . . . .
2.2.2 Brace-Ga̧tarek-Musiela Approach . . . . . . . . . . . . . . .
2.2.3 Musiela-Rutkowski Approach . . . . . . . . . . . . . . . . .
2.2.4 Jamshidian’s Approach . . . . . . . . . . . . . . . . . . . .
2.3 Dynamics of Libor Rates and Bond Prices . . . . . . . . . . . . . .
2.3.1 Dynamics of L(·, Tj ) under PTj . . . . . . . . . . . . . . . .
2.3.2 Dynamics of FB (·, Tj+1 , Tj ) under PTj . . . . . . . . . . . .
2.4 Caps and Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Market Valuation Formula for Caps and Floors . . . . . . .
2.4.2 Valuation in the Lognormal Model of Forward Libor Rates
2.4.3 Hedging of Caps and Floors . . . . . . . . . . . . . . . . . .
2.4.4 Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . .
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4
4
4
6
7
7
8
8
10
13
15
16
18
19
20
21
23
24
3 Modelling of Forward Swap Rates
3.1 Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Lognormal Model of Forward Swap Rates . . . . . . . . . . . . . .
3.3 Valuation of Swaptions . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Payer and Receiver Swaptions . . . . . . . . . . . . . . . . .
3.3.2 Forward Swaptions . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Valuation in the Lognormal Model of Forward Libor Rates
3.3.4 Market Valuation Formula for Swaptions . . . . . . . . . .
3.3.5 Valuation in the Lognormal Model of Forward Swap Rates
3.3.6 Hedging of Swaptions . . . . . . . . . . . . . . . . . . . . .
3.4 Choice of Numeraire Portfolio . . . . . . . . . . . . . . . . . . . . .
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25
25
27
30
30
32
33
35
35
36
37
4 Markov-Functional Models
4.1 Terminal Swap Rate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Calibration of Markov-Functional Models . . . . . . . . . . . . . . . . . . . . . . . .
38
39
41
1
2
1
M.Rutkowski
Introduction
The last decade was marked by a rapidly growing interest in the arbitrage-free modelling of bond
market. Undoubtedly, one of the major achievements in this area was a new approach to the
term structure modelling proposed by Heath, Jarrow and Morton in their work published in 1992,
commonly known as the HJM methodology. One of its main features is that it covers a large variety
of previously proposed models and provides a unified approach to the modelling of instantaneous
interest rates and to the valuation of interest-rate sensitive derivatives. Let us give a very concise
description of the HJM approach (for a detailed account we refer, for instance, to Chapter 13 in
Musiela and Rutkowski (1997a)).
The HJM methodology is based on an exogenous specification of the dynamics of instantaneous,
continuously compounded forward rates f (t, T ). For any fixed maturity T ≤ T ∗ , the dynamics of
the forward rate f (t, T ) are
df (t, T ) = α(t, T ) dt + σ(t, T ) · dWt ,
where α and σ are adapted stochastic processes with values in R and Rd , respectively, and W is
a d-dimensional standard Brownian motion with respect to the underlying probability measure P
which plays the role of the real-world probability. More formally, for every fixed T ≤ T ∗ , where
T ∗ > 0 is the horizon date, we have
Z
Z
t
t
α(u, T ) du +
f (t, T ) = f (0, T ) +
0
0
σ(u, T ) · dWu
for some Borel-measurable function f (0, ·) : [0, T ∗] → R and stochastic processes applications α(·, T )
and σ(·, T ). Let us notice that, for any fixed maturity date T ≤ T ∗ , the initial condition f (0, T ) is
determined by the current value of the continuously compounded forward rate for the future date T
which prevails at time 0. In practical terms, the function f (0, T ) is determined by the current yield
curve, which can be estimated on the basis of observed market prices of bonds (and other relevant
instruments).
Let us denote by B(t, T ) the price at time t ≤ T of a unit zero-coupon bond which matures at
the date T ≤ T ∗ . In the present setup, the price B(t, T ) can be recovered from the formula
Z
B(t, T ) = exp −
T
f (t, u) du .
t
The problem of the absence of arbitrage opportunities in the bond market can be formulated in
terms of the existence of a suitably defined martingale measure. It appears that in an arbitrage-free
setting – that is, under the martingale measure – the drift coefficient α in the dynamics of the
intstantaneous forward rate is uniquely determined by the volatility coefficient σ, and a stochastic
process which can be interpreted as the market price of the interest-rate risk.
If we denote by P∗ the martingale measure for the bond market, and by W ∗ the associated
standard Brownian motion, then
dB(t, T ) = B(t, T ) rt dt + b(t, T ) · dWt∗ ,
where rt = f (t, t) is the short-term interest rate, and the bond price volatility b(t, T ) satisfies
Z
b(t, T ) = −
T
σ(t, u) du.
t
(1)
3
Modelling of Forward Libor and Swap Rates
Furthermore, it appears that in the special case when the coefficient σ follows a deterministic function, the valuation formulae for interest rate-sensitive derivatives are independent of the choice of
the risk premium. In this sense, the choice of a particular model from the broad class of HJM models
hinges uniquely on the specification of the volatility coefficient σ.
The HJM methodology appeared to be very successful both from the theoretical and practical
viewpoints. Since the HJM approach to the term structure modelling is based on an arbitrage-free
dynamics of the instantaneous continuously compounded forward rates, it requires a certain degree
of smoothness with respect to the tenor of the bond prices and their volatilities. For this reason,
working with such models is not always convenient.
An alternative construction of an arbitrage-free family of bond prices, making no reference to
the instantaneous rates, is in some circumstances more suitable. The first step in this direction was
done by Sandmann and Sondermann (1993), who focused on the effective annual interest rate. This
approach was further developed in ground-breaking papers by Miltersen et al. (1997) and Brace et
al. (1997), who proposed to model instead the family of forward Libor rates. The main goal was to
produce an arbitrage-free term structure model which would support the common practice of pricing
such interest-rate derivatives as caps and swaptions through a suitable version of Black’s formula.
This practical requirement enforces the lognormality of the forward Libor (or swap) rate under the
corresponding forward martingale measure.
Let us recall that, by market convention, the forward Libor rate over the future accrual pariod
[T, T + δ], as seen at time t, is set to satisfy
1 + δL(t, T ) =
or equivalently,
B(t, T )
,
B(t, T + δ)
B(t, T ) − B(t, T + δ)
.
B(t, T + δ)
The last formula makes it obvious that the volatility of the forward Libor rate is not deterministic
if the bond price volatility follows a determinisitc function. For this reason the Black’s formula
for caps is manifestly incompatible with the Gaussian HJM model – that is, the HJM model in
which the bond price volatility b(t, T ) is deterministic. Consequently, the “market formula” for caps
cannot be derived in this setup (though the value of a cap is given by a closed-form expression in
the Gaussian HJM framework). On the other hand, it is interesting to notice that Brace et al.
(1997) parametrize their version of the lognormal forward Libor model introduced by Miltersen et
al. (1997) with a piecewise constant volatility function. They need to consider smooth volatility
functions in order to analyse the model in the HJM framework, however. The backward induction
approach to the modelling of forward Libor and swap rate developed in Musiela and Rutkowski
(1997) and Jamshidian (1997) overcomes this technical difficulty. In addition, in contrast to the
previous papers, it allows also for the modelling of forward Libor (and swap) rates associated with
accrual periods of differing lengths.
It should be stressed that a similar (but not identical) approach to the modelling of market rate
was developed in a series of papers by Hunt et al. (1996, 1997) and Hunt and Kennedy (1997,
1998). Since special emphasis is put here on the existence of the underlying low-dimensional Markov
process that governs directly the dynamics of interest rates, this alternative approach is termed
the Markov-functional approach. This property leads to a considerable simplification in numerical
procedures associated with the model’s implementation. Another important feature of this approach
is its ability of providing a perfect fit to market prices of a given family of interest-rate options (e.g.,
a family of caps with a fixed maturity and varying strike level).
Another tractable term structure model – which is beyond the scope of the present text, however
– is the rational lognormal model proposed by Flesaker and Hughston (1996a, 1996b) (see also
Rutkowski (1997) and Jin and Glasserman (1997) in this regard).
Let us finally mention that we use throughout the notation adopted in Musiela and Rutkowski
(1997a). The interested reader is referred to this monograph for more details on term structure
modelling as well as for the general background.
L(t, T ) =
4
M.Rutkowski
2
Modelling of Forward Libor Rates
In this section, we present various approaches to the modelling of forward Libor rates. Due to
the limited space, we focus on model’s construction and its basic properties and the valuation of
the most typical derivatives. For further details, the interested reader is referred to the original
papers: Musiela and Sondermann (1993), Sandmann and Sondermann (1993), Goldys et al. (1994),
Sandmann et al. (1995), Brace et al. (1997), Jamshidian (1997, 1999), Miltersen et al. (1997), Musiela
and Rutkowski (1997b), Rady (1997), Sandmann and Sondermann (1997), Rutkowski (1998, 1999),
Yasuoka (1998), and Glasserman and Kou (1999). The issues related to the model’s implementation1
are extensively treated in Brace (1996), Andersen and Andreasen (1997), Sidenius (1997), Brace et
al. (1998), Musiela and Sawa (1998), Hull and White (1999), Schlögl (1999), Yasuoka (1999), Lotz
and Schlögl (1999), and Glasserman and Zhao (2000), Brace and Womersley (2000), and Dun et
al. (2000).
2.1
Forward and Futures Libor Rates
Our first task is to examine these properties of forward and futures contracts related to the notion
of the Libor rate which are universal; that is, which do not rely on specific assumptions imposed on
a particular model of the term structure of interest rates. To this end, we fix an index j, and we
consider various interest-rate sensitive derivatives related to the period [Tj , Tj+1 ]. To be more specific,
we shall focus in this section on single-period forward swaps – that is, forward rate agreements.
We need to introduce some notation. We assume that we are given a prespecified collection of
reset/settlement dates 0 < T0 < T1 < · · · < Tn = T ∗ , referred to as the tenor structure. Also,
we denote δj = Tj − Tj−1 for j = 1, . . . , n. We write B(t, Tj ) to denote the price at time t of a
Tj -maturity zero-coupon bond. P∗ is the spot martingale measure, while for any j = 0, . . . , n we
write PTj to denote the forward martingale measure associated with the date Tj . The corresponding d-dimensional Brownian motions are denoted by W ∗ and W Tj , respectively. Also, we write
FB (t, T, U ) = B(t, T )/B(t, U ) so that
FB (t, Tj+1 , Tj ) =
B(t, Tj+1 )
,
B(t, Tj )
∀ t ∈ [0, Tj ],
is the forward price at time t of the Tj+1 -maturity zero-coupon bond for the settlement date Tj .
We use the symbol πt (X) to denote the value (i.e., the arbitrage price) at time t of a European
contingent claim X. Finally, we shall use the letter E for the Doléans exponential, for instance,
Z
Z ·
Z t
1 t
γu · dWu∗ = exp
γu · dWu∗ −
|γu |2 du ,
Et
2 0
0
0
where the dot ‘ · ’ and | · | stand for the inner product and Euclidean norm in Rd , respectively.
2.1.1
Single-period Swaps Settled in Arrears
Let us first consider a single-period swap agreement settled in arrears; i.e., with the reset date Tj
and the settlement date Tj+1 (multi-period interest rate swaps are examined in Section 3). By
the contractual features, the long party pays δj+1 κ and receives B −1 (Tj , Tj+1 ) − 1 at time Tj+1 .
Equivalently, he pays an amount Y1 = 1 + δj+1 κ and receives Y2 = B −1 (Tj , Tj+1 ) at this date. The
values at time t ≤ Tj of these payoffs are
πt (Y1 ) = B(t, Tj+1 ) 1 + δj+1 κ , πt (Y2 ) = B(t, Tj ).
The second equality above is trivial, since the payoff Y2 is equivalent to the unit payoff at time Tj .
Consequently, for any fixed t ≤ Tj , the value of the forward swap rate, which makes the contract
1 In
particular, an arbitrage-free discretization of the lognormal model of forward Libor rates.
5
Modelling of Forward Libor and Swap Rates
worthless at time t, can be found by solving for κ = κ(t, Tj , Tj+1 ) the following equation
πt (Y2 ) − πt (Y1 ) = B(t, Tj ) − B(t, Tj+1 ) 1 + δj+1 κ = 0.
It is thus apparent that
κ(t, Tj , Tj+1 ) =
B(t, Tj ) − B(t, Tj+1 )
,
δj+1 B(t, Tj+1 )
∀ t ∈ [0, Tj ].
Note that the forward swap rate κ(t, Tj , Tj+1 ) coincides with the forward Libor rate L(t, Tj ) which,
by the market convention, is set to satisfy
1 + δj+1 L(t, Tj ) =
B(t, Tj )
= E PTj+1 (B −1 (Tj , Tj+1 ) | Ft )
B(t, Tj+1 )
(2)
for every t ∈ [0, Tj ]. Let us notice that the last equality is a consequence of the definition of the
forward measure PTj+1 . We conclude that in order to determine the forward Libor rate L(·, Tj ), it is
enough to find the forward price FX (t, Tj+1 ) at time t of the contingent claim X = B −1 (Tj , Tj+1 )
in the forward contact that settles at time Tj+1 . Indeed, it is well known (see, for instance, Musiela
and Rutkowski (1997a)) that
FX (t, Tj+1 ) = B(t, Tj+1 ) E PTj+1 (B −1 (Tj , Tj+1 ) | Ft ).
Furthermore, it is evident that the process L(·, Tj ) follows necessarily a martingale under the forward
probability measure PTj+1 . Recall that in the Heath-Jarrow-Morton framework, we have, under
PTj+1 ,
T
(3)
dFB (t, Tj , Tj+1 ) = FB (t, Tj , Tj+1 ) b(t, Tj ) − b(t, Tj+1 ) · dWt j+1 ,
where, for each maturity date T, the process b(·, T ) represents the price volatility of the T -maturity
zero-coupon bond. On the other hand, if the process L(·, Tj ) is strictly positive, it can be shown to
admit the following representation2
Tj+1
dL(t, Tj ) = L(t, Tj )λ(t, Tj ) · dWt
,
where λ(·, Tj ) is an adapted stochastic process which satifies mild integrability conditions. Combining the last two formulae with (2), we arrive at the following fundamental relationship, which plays
an essential role in the construction of the lognormal model of forward Libor rates,
δj+1 L(t, Tj )
λ(t, Tj ) = b(t, Tj ) − b(t, Tj+1 ),
1 + δj+1 L(t, Tj )
∀ t ∈ [0, Tj ].
(4)
For instance, in the construction which is based on the backward induction, relationship (4) will allow
us to determine the forward measure for the date Tj , provided that PTj+1 , W Tj+1 and the volatility
λ(t, Tj ) of the forward Libor rate L(·, Tj−1 ) are known. One may assume, for instance, that λ(·, Tj )
is a prespecified deterministic function). Recall that in the Heath-Jarrow-Morton framework3 the
Radon-Nikodým density of PTj with respect to PTj+1 is known to satisfy
Z ·
dPTj
T
= ETj
(5)
b(t, Tj ) − b(t, Tj+1 ) · dWt j+1 .
dPTj+1
0
In view of (4), we thus have
dPTj
= ETj
dPTj+1
2 This
3 See
Z
0
·
δj+1 L(t, Tj )
T
λ(t, Tj ) · dWt j+1 .
1 + δj+1 L(t, Tj )
representation is a consequence of the martingale representation property of the standard Brownian motion.
Heath et al. (1992).
6
M.Rutkowski
For our further purposes, it is also useful to observe that this density admits the following representation
dPTj
= cFB (Tj , Tj , Tj+1 ) = c 1 + δj+1 L(Tj , Tj ) , PTj+1 -a.s.,
(6)
dPTj+1
where c > 0 is the normalizing constant, and thus
dPTj
dPTj+1
= cFB (t, Tj , Tj+1 ) = c 1 + δj+1 L(t, Tj ) ,
PTj+1 -a.s.
|Ft
Finally, the dynamics of the process L(·, Tj ) under the probability measure PTj are given by a
somewhat involved stochastic differential equation
δ L(t, T )|λ(t, T )|2
T
j+1
j
j
dt + λ(t, Tj ) · dWt j .
dL(t, Tj ) = L(t, Tj )
1 + δj+1 L(t, Tj )
As we shall see in what follows, it is nevertheless not hard to determine the probability law of L(·, Tj )
under the forward measure PTj – at least in the case of the deterministic volatility λ(·, Tj ) of the
forward Libor rate.
2.1.2
Single-period Swaps Settled in Advance
Consider now a similar swap which is, however, settled in advance – that is, at time Tj . Our
first goal is to determine the forward swap rate implied by such a contract. Note that under
the present assumptions, the long party (formally) pays an amount Y1 = 1 + δj+1 κ and receives
Y2 = B −1 (Tj , Tj+1 ) at the settlement date Tj (which coincides here with the reset date). The values
at time t ≤ Tj of these payoffs admit the following representations
πt (Y1 ) = B(t, Tj ) 1 + δj+1 κ , πt (Y2 ) = B(t, Tj )E PTj (B −1 (Tj , Tj+1 ) | Ft ).
The value κ = κ̂(t, Tj , Tj+1 ) of the modified forward swap rate, which makes the swap agreement
settled in advance worthless at time t, can be found from the equality
πt (Y2 ) − πt (Y1 ) = B(t, Tj ) E PTj (B −1 (Tj , Tj+1 ) | Ft ) − (1 + δj+1 κ) = 0.
It is clear that
−1
E PTj (B −1 (Tj , Tj+1 ) | Ft ) − 1 .
κ̂(t, Tj , Tj+1 ) = δj+1
We are in a position to introduce the modified forward Libor rate L̃(t, Tj ) by setting, for every
t ∈ [0, Tj ],
−1
L̃(t, Tj ) := δj+1
E PTj (B −1 (Tj , Tj+1 ) | Ft ) − 1 .
Let us make two remarks. First, it is clear that finding of the modified forward Libor rate L̃(·, Tj ) is
formally equivalent to finding the forward price of the claim B −1 (Tj , Tj+1 ) for the settlement date
Tj .4 Second, it is useful to observe that
L̃(t, Tj ) = E PTj
1 − B(T , T ) j
j+1 Ft = E PTj (L(Tj , Tj ) | Ft ).
δj+1 B(Tj , Tj+1 )
(7)
In particular, it is evident that at the reset date Tj the two kinds of forward Libor rates introduced
above coincide, since manifestly
L̃(Tj , Tj ) =
4 Recall
1 − B(Tj , Tj+1 )
= L(Tj , Tj ).
δj+1 B(Tj , Tj+1 )
that in the case of a forward Libor rate, the settlement date was Tj+1 .
7
Modelling of Forward Libor and Swap Rates
To summarize, the ‘standard’ forward Libor rate L(·, Tj ) satisfies
L(t, Tj ) = E PTj+1 (L(Tj , Tj ) | Ft ),
with the initial condition
L(0, Tj ) =
∀ t ∈ [0, Tj ],
B(0, Tj ) − B(0, Tj+1 )
.
δj+1 B(0, Tj+1 )
On the other hand, for the modified Libor rate L̃(·, Tj ) we have
L̃(t, Tj ) = E PTj (L̃(Tj , Tj ) | Ft ),
∀ t ∈ [0, Tj ],
with the initial condition
−1
E PTj (B −1 (Tj , Tj+1 )) − 1 .
L̃(0, Tj ) = δj+1
The calculation of the right-hand side above involve not only on the initial term structure, but also
the volatilities of bond prices (for more details, we refer to Rutkowski (1998)).
2.1.3
Eurodollar Futures Contracts
The next object of our studies is the futures Libor rate. A Eurodollar futures contract is a futures
contract in which the Libor rate plays the role of an underlying asset. By convention, at the contract’s
maturity date Tj , the quoted Eurodollar futures price, denoted by E(Tj , Tj ), is set to satisfy
E(Tj , Tj ) := 1 − δj+1 L(Tj , Tj ).
Equivalently, in terms of the zero-coupon bond price we have E(Tj , Tj ) = 2 − B −1 (Tj , Tj+1 ). From
the general theory, it follows that the Eurodollar futures price at time t ≤ Tj equals
(8)
E(t, Tj ) := E P∗ (E(Tj , Tj )) = 2 − E P∗ B −1 (Tj , Tj+1 ) | Ft
(recall that P∗ represents the spot martingale measure in a given model of the term structure). It
is thus natural to introduce the concept of the futures Libor rate, associated with the Eurodollar
futures contract, through the following definition.
Definition 2.1 Let E(t, Tj ) be the Eurodollar futures price at time t for the settlement date Tj .
The implied futures Libor rate Lf (t, Tj ) satisfies
E(t, Tj ) = 1 − δj+1 Lf (t, Tj ),
∀ t ∈ [0, Tj ].
It follows immediately from (8)–(9) that the following equality is valid
1 + δj+1 Lf (t, Tj ) = E P∗ B −1 (Tj , Tj+1 ) | Ft .
(9)
(10)
Equivalently, we have
Lf (t, Tj ) = E P∗ (L(Tj , Tj ) | Ft ) = E P∗ (L̃(Tj , Tj ) | Ft ).
Note that in any term structure model, the futures Libor rate necessarily follows a martingale under
the spot martingale measure P∗ (provided, of course, that P∗ is well-defined in this model).
2.2
Lognormal Models of Forward Libor Rates
We shall now describe alternative approaches to the modelling of forward Libor rates in a continuousand discrete-tenor setups.
8
2.2.1
M.Rutkowski
Miltersen-Sandmann-Sondermann Approach
The first attempt to provide a rigorous construction a lognormal model of forward Libor rates was
done by Miltersen et al. (1997). The interested reader is referred also to Musiela and Sondermann
(1993), Goldys et al. (1994), and Sandmann et al. (1995) for related previous studies. As a starting
point in their approach, Miltersen et al. (1997) postulate that the forward Libor rates process L(·, T )
satisfies
dL(t, T ) = µ(t, T ) dt + L(t, T )λ(t, T ) · dWt∗ ,
with a deterministic volatility function λ(·, T ) : [0, T ] → Rd . It is not difficult to deduce from the
last formula that the forward price of a zero-coupon bond satisfies
dF (t, T + δ, T ) = −F (t, T + δ, T ) 1 − F (t, T + δ, T ) λ(t, T ) · dWtT .
Subsequently, they focus on the partial differential equation satisfied by the function v = v(t, x),
which expresses the forward price of the bond option in terms of the forward bond price. It is
interesting to note that the PDE (11) was previously solved by Rady and Sandmann (1994) who
worked within a different framework, however.5 The PDE for the option’s price is
∂2v
∂v 1
+ |λ(t, T )|2 x2 (1 − x)2 2 = 0
∂t
2
∂x
(11)
with the terminal condition v(T, x) = (K − x)+ . As a result, Miltersen et al. (1997) obtained not
only the closed-form solution for the price of a bond option (this was already achieved in Rady and
Sandmann (1994)), but also the “market formula” for the caplet’s price. The rigorous approach
to the problem of existence of such a model was presented by Brace et al. (1997), who also worked
within the continuous-time Heath-Jarrow-Morton framework.
2.2.2
Brace-Ga̧tarek-Musiela Approach
To formally introduce the notion of a forward Libor rate, we assume that we are given a family
B(t, T ) of bond prices, and thus also the collection FB (t, T, U ) of forward processes. In contrast
to the previous section, we shall now assume that a strictly positive real number δ < T ∗ , which
represents the length of the accrual period, is fixed throughout. By definition, the forward δ-Libor
rate L(t, T ) for the future date T ≤ T ∗ − δ prevailing at time t is given by the conventional market
formula
(12)
1 + δL(t, T ) = FB (t, T, T + δ), ∀ t ∈ [0, T ].
The forward Libor rate L(t, T ) represents the add-on rate prevailing at time t over the future time
interval [T, T + δ]. We can also re-express L(t, T ) directly in terms of bond prices, as for any
T ∈ [0, T ∗ − δ], we have
B(t, T )
, ∀ t ∈ [0, T ].
(13)
1 + δL(t, T ) =
B(t, T + δ)
In particular, the initial term structure of forward Libor rates satisfies
L(0, T ) = δ −1
B(0, T )
−1 .
B(0, T + δ)
(14)
Given a family FB (t, T, T ∗) of forward processes, it is not hard to derive the dynamics of the
associated family of forward Libor rates. For instance, one finds that under the forward measure
PT +δ , we have
dL(t, T ) = δ −1 FB (t, T, T + δ) γ(t, T, T + δ) · dWtT +δ ,
5 In fact, they were concerned with the valuation of options on zero-coupon bonds for the term structure model
put forward by Bühler and Käsler (1989).
9
Modelling of Forward Libor and Swap Rates
where PT +δ is the forward measure for the date T + δ, and the asociated Wiener process W T +δ
equals
Z t
b(u, T + δ) du, ∀ t ∈ [0, T + δ].
WtT +δ = Wt∗ −
0
Put another way, the process L(·, T ) solves the equation
dL(t, T ) = δ −1 (1 + δL(t, T )) γ(t, T, T + δ) · dWtT +δ ,
(15)
subject to the initial condition (14). Suppose that forward Libor rates L(t, T ) are strictly positive.
Then formula (15) can be rewritten as follows
dL(t, T ) = L(t, T ) λ(t, T ) · dWtT +δ ,
(16)
where for any t ∈ [0, T ]
λ(t, T ) =
1 + δL(t, T )
γ(t, T, T + δ).
δL(t, T )
(17)
This shows that the collection of forward processes uniquely specifies the family of forward Libor
rates. The construction of a model of forward Libor rates relies on the following assumptions.
(LR.1) For any maturity T ≤ T ∗ − δ, we are given a Rd -valued, bounded deterministic function6
λ(·, T ), which represents the volatility of the forward Libor rate process L(·, T ).
(LR.2) We assume a strictly decreasing and strictly positive initial term structure B(0, T ), T ∈
[0, T ∗ ]. The associated initial term structure L(0, T ) of forward Libor rates satisfies, for every T ∈
[0, T ∗ −δ],
B(0, T ) − B(0, T + δ)
.
(18)
L(0, T ) =
δB(0, T + δ)
To construct a model satisfying (LR.1)–(LR.2), Brace et al. (1997) place themselves in the HeathJarrow-Morton setup and they assume that for every T ∈ [0, T ∗], the volatility b(t, T ) vanishes for
every t ∈ [(T − δ) ∨ 0, T ]. In essence, the construction elaborated in Brace et al. (1997) is based on
the forward induction, as opposed to the backward induction which we shall use in the next section.
They start by postulating that the dynamics of L(t, T ) under the spot martingale measure P∗ are
governed by the following SDE
dL(t, T ) = µ(t, T ) dt + L(t, T )λ(t, T ) · dWt∗ ,
where λ is a deterministic function, and the drift coefficient µ is unspecified. Recall that the
arbitrage-free dynamics of the instantaneous forward rate f (t, T ) are
df (t, T ) = σ(t, T ) · σ ∗ (t, T ) dt + σ(t, T ) · dWt∗ ,
where σ ∗ (t, T ) =
RT
t
σ(t, u) du = −b(t, T ). On the other hand, the relationship (cf. (13))
Z
T +δ
f (t, u) du
1 + δL(t, T ) = exp
(19)
T
is valid. Applying Itô’s formula to both sides of (19), and comparing the diffusion terms, we find
that
Z T +δ
δL(t, T )
λ(t, T ).
σ(t, u) du =
σ ∗ (t, T + δ) − σ ∗ (t, T ) =
1 + δL(t, T )
T
6 Volatility λ could well follow an adapted stochastic process; we deliberately focus here on a lognormal model of
forward Libor rates in which λ is deterministic.
10
M.Rutkowski
To solve the last equation for σ ∗ in terms of L, it is necessary to impose some sort of initial condition
on σ ∗ . For instance, by setting σ(t, T ) = 0 for 0 ≤ t ≤ T ≤ t+δ, we obtain the following relationship
[δ −1 (T −t)]
∗
b(t, T ) = −σ (t, T ) = −
X
k=1
δL(t, T − kδ)
λ(t, T − kδ).
1 + δL(t, T − kδ)
(20)
The existence and uniqueness of solutions to SDEs which govern the instantaneous forward rate
f (t, T ) and the forward Libor rate L(t, T ) for σ ∗ given by (20) can be shown using forward induction.
Taking this result for granted, we conclude that L(t, T ) satisfies, under the spot martingale measure
P∗
dL(t, T ) = L(t, T )σ ∗ (t, T ∗ + δ) · λ(t, T ) dt + L(t, T )λ(t, T ) · dWt∗ .
In this way, Brace et al. (1997) are able to completely specify their model of forward Libor rates.
2.2.3
Musiela-Rutkowski Approach
In this section, we describe an alternative approach to the modelling of forward Libor rates; the
construction presented below is a slight modification of that given by Musiela and Rutkowski (1997b).
Let us start by introducing some notation. We assume that we are given a prespecified collection
of reset/settlement dates 0 < T0 < T1 < · · · < Tn = T ∗ , referred to as the tenor structure (by
Pj
convention, T−1 = 0). Let us denote δj = Tj − Tj−1 for j = 0, . . . , n. Then obviously Tj = i=0 δi
for every j = 0, . . . , n. We find it convenient to denote, for m = 0, . . . , n,
∗
= T∗ −
Tm
n
X
δj = Tn−m .
j=n−m+1
For any j = 0, . . . , n − 1, we define the forward Libor rate L(·, Tj ) by setting
L(t, Tj ) =
B(t, Tj ) − B(t, Tj+1 )
,
δj+1 B(t, Tj+1 )
∀ t ∈ [0, Tj ].
Definition 2.2 For any j = 0, . . . , n, a probability measure PTj on (Ω, FTj ), equivalent to P, is said
to be the forward Libor measure for the date Tj if, for every k = 0, . . . , n the relative bond price
Un−j+1 (t, Tk ) :=
B(t, Tk )
,
B(t, Tj )
∀ t ∈ [0, Tk ∧ Tj ],
follows a local martingale under PTj .
It is clear that the notion of forward Libor measure is in fact identical with that of a forward
probability measure for a given date. Also, it is trivial to observe that the forward Libor rate
L(·, Tj ) necessarily follows a local martingale under the forward Libor measure for the date Tj+1 . If,
in addition, it is a strictly positive process, the existence of the associated volatility process can be
justified by standard arguments.
In our further development, we shall go the other way around; that is, we will assume that for any
date Tj , the volatility λ(·, Tj ) of the forward Libor rate L(·, Tj ) is exogenously given. In principle, it
can be a deterministic Rd -valued function of time, an Rd -valued function of the underlying forward
Libor rates, or it can follow a d-dimensional adapted stochastic process. For simplicity, we assume
throughout that the volatilities of forward Libor rates are bounded processes (or functions). To be
more specific, we make the following standing assumptions.
Assumptions (LR). We are given a family of bounded adapted processes λ(·, Tj ), j = 0, . . . , n − 1,
which represent the volatilities of forward Libor rates L(·, Tj ). In addition, we are given an initial
term structure of interest rates, specified by a family B(0, Tj ), j = 0, . . . , n, of bond prices. We
assume here that B(0, Tj ) > B(0, Tj+1 ) for j = 0, . . . , n − 1.
11
Modelling of Forward Libor and Swap Rates
Our aim is to construct a family L(·, Tj ), j = 0, . . . , n − 1 of forward Libor rates, a collection
of mutually equivalent probability measures PTj , j = 1, . . . , n, and a family W Tj , j = 1, . . . , n of
processes in such a way that: (i) for any j = 1, . . . , n the process W Tj follows a d-dimensional
standard Brownian motion under the probability measure PTj , (ii) for any j = 0, . . . , n − 1, the
forward Libor rate L(·, Tj ) satisfies the SDE
Tj+1
dL(t, Tj ) = L(t, Tj ) λ(t, Tj ) · dWt
with the initial condition
L(0, Tj ) =
,
∀ t ∈ [0, Tj ],
(21)
B(0, Tj ) − B(0, Tj+1 )
.
δj+1 B(0, Tj+1 )
As already mentioned, the construction of the model is based on backward induction, therefore we
start by defining the forward Libor rate with the longest maturity, i.e., Tn−1 . We postulate that
L(·, Tn−1 ) = L(·, T1∗ ) is governed under the underlying probability measure P by the following SDE7
dL(t, T1∗ ) = L(t, T1∗) λ(t, T1∗ ) · dWt
with the initial condition
L(0, T1∗ ) =
B(0, T1∗ ) − B(0, T ∗ )
.
δn B(0, T ∗ )
Put another way, we have
L(t, T1∗ ) =
B(0, T1∗ )
∗
B(0, T1∗ ) − B(0, T ∗ ) Et
δn B(0, T ∗ )
Z
0
·
λ(u, T1∗ ) · dWu .
L(·, T1∗ )
> B(0, T ), it is clear that the
follows a strictly positive martingale under
Since
PT ∗ = P. The next step is to define the forward Libor rate for the date T2∗ . For this purpose, we need
to introduce first the forward probability measure for the date T1∗ . By definition, it is a probability
measure Q, which is equivalent to P, and such that processes
U2 (t, Tk∗ ) =
B(t, Tk∗ )
B(t, T1∗ )
are Q-local martingales. It is important to observe that the process U2 (·, Tk∗ ) admits the following
representation
U1 (t, Tk∗ )
.
U2 (t, Tk∗ ) =
1 + δn L(t, T1∗ )
Let us formulate an auxiliary result, which is a straightforward consequence of Itô’s rule.
Lemma 2.1 Let G and H be real-valued adapted processes, such that
dGt = αt · dWt ,
dHt = βt · dWt .
Assume, in addition, that Ht > −1 for every t and denote Yt = (1 + Ht )−1 . Then
d(Yt Gt ) = Yt αt − Yt Gt βt · dWt − Yt βt dt .
It follows immediately from Lemma 2.1 that
dU2 (t, Tk∗ ) = ηtk · dWt −
δn L(t, T1∗ )
∗
λ(t,
T
)
dt
1
1 + δn L(t, T1∗ )
7 Notice that, for simplicity, we have chosen the underlying probability measure P to play the role of the forward
Libor measure for the date T ∗ . This choice is not essential, however.
12
M.Rutkowski
for a certain process η k . Therefore it is enough to find a probability measure under which the process
T1∗
Wt
Z
:= Wt −
t
0
δn L(u, T1∗)
λ(u, T1∗ ) du = Wt −
1 + δn L(u, T1∗ )
Z
t
0
γ(u, T1∗ ) du,
t ∈ [0, T1∗ ], follows a standard Brownian motion (the definition of γ(·, T1∗ ) is clear from the context).
This can be easily achieved using Girsanov’s theorem, as we may put
Z ·
dPT1∗
= ET1∗
γ(u, T1∗ ) · dWu , P-a.s.
dP
0
We are in a position to specify the dynamics of the forward Libor rate for the date T2∗ under PT1∗ ,
namely we postulate that
T∗
dL(t, T2∗ ) = L(t, T2∗ ) λ(t, T2∗ ) · dWt 1
with the initial condition
B(0, T2∗ ) − B(0, T1∗ )
.
δn−1 B(0, T1∗ )
L(0, T2∗ ) =
∗
Let us now assume that we have found processes L(·, T1∗ ), . . . , L(·, Tm
). This means, in particular,
∗
∗
and the associated Brownian motion W Tm−1 are already
that the forward Libor measure PTm−1
∗ . It is easy to check that
specified. Our aim is to determine the forward Libor measure PTm
Um+1 (t, Tk∗ ) :=
Um (t, Tk∗ )
B(t, Tk∗ )
=
.
∗)
∗)
B(t, Tm
1 + δn−m L(t, Tm
Using Lemma 2.1, we obtain the following relationship
T∗
Wt m
=
T∗
Wt m−1
Z
−
t
0
∗
δn−m L(u, Tm
)
∗
λ(u, Tm
) du
∗)
1 + δn−m L(u, Tm
∗
∗ can thus be easily found using Girsanov’s theorem.
]. The forward Libor measure PTm
for t ∈ [0, Tm
∗
) as the solution to the SDE
Finally, we define the process L(·, Tm+1
∗
Tm
∗
∗
∗
) = L(t, Tm+1
) λ(t, Tm+1
) · dWt
dL(t, Tm+1
with the initial condition
∗
)=
L(0, Tm+1
∗
∗
B(0, Tm+1
) − B(0, Tm
)
.
∗)
δn−m B(0, Tm
Remarks. (i) It is not difficult to check that equality (6) is satisfied within the present setup.
(ii) If the volatility coefficient λ(·, Tm ) : [0, Tn ] → Rd is a deterministic function, then for each
date t ∈ [0, Tm ] the random variable L(t, Tm ) has a lognormal probability law under the forward
probability measure PTm+1 .
Let us now examine the existence and uniqueness of the implied savings account,8 in a discretetime setup. Intuitively, the value Bt∗ of a savings account at time t can be interpreted as the cash
amount accumulated up to time t by rolling over a series of zero-coupon bonds with the shortest
maturities available. To find the process B ∗ in a discrete-tenor framework, we do not have to specify
explicitly all bond prices; the knowledge of forward bond prices is sufficient. Indeed, it is clear that
FB (t, Tj , Tj+1 ) =
B(t, Tj )
FB (t, Tj , T ∗ )
=
.
∗
FB (t, Tj+1 , T )
B(t, Tj+1 )
8 The interested reader is referred to Musiela and Rutkowski (1997b) for the definition of an implied savings
account in a continuous-time setup. See also Döberlein and Schweizer (1998) and Döberlein et al. (1999) for further
developments and the general uniqueness result.
13
Modelling of Forward Libor and Swap Rates
This in turn yields, upon setting t = Tj
FB (Tj , Tj , Tj+1 ) = 1/B(Tj , Tj+1 ),
(22)
so that the price B(Tj , Tj+1 ) of a single-period bond is uniquely specified for every j. Though the
bond that matures at time Tj does not physically exist after this date, it seems justifiable to consider
FB (Tj , Tj , Tj+1 ) as its forward value at time Tj for the next future date Tj+1 . In other words, the
spot value at time Tj+1 of one cash unit received at time Tj equals B −1 (Tj , Tj+1 ). The discrete-time
savings account B ∗ thus equals, for k = 0, . . . , n (recall that T−1 = 0),
BT∗k =
k
Y
k
Y
−1
FB Tj−1 , Tj−1 , Tj =
B Tj−1 , Tj
j=0
j=0
since, by convention, we set B0∗ = 1. Note that FB Tj−1 , Tj−1 , Tj = 1 + δL(Tj−1 , Tj ) > 1 for j =
0, . . . , n, and since BT∗j = FB (Tj−1 , Tj−1 , Tj ) BT∗j−1 , we find that BT∗j > BT∗j−1 for every j = 0, . . . , n.
We conclude that the implied savings account B ∗ follows a strictly increasing discrete-time process.
Let us define the probability measure P∗ , equivalent to P on (Ω, FT ∗ ), by the formula9
dP∗
= BT∗ ∗ B(0, T ∗ ),
dP
P-a.s.
(23)
The probability measure P∗ appears to be a plausible candidate for a spot martingale measure.
Indeed, if we set
(24)
B(Tl , Tk ) = E P∗ (BT∗l /BT∗k | FTl )
for every l ≤ k ≤ n, then in the case of l = k − 1, equality (24) coincides with (22). Let us observe
that it is not possible to uniquely determine the continuous-time dynamics of a bond price B(t, Tj )
within the framework of the discrete-tenor model of forward Libor rates (the specification of forward
Libor rates for all maturities is necessary for this purpose).
2.2.4
Jamshidian’s Approach
The backward induction approach to modelling of forward Libor rates presented in the preceding
section was re-examined and essentially generalized by Jamshidian (1997). In this section, we present
briefly his approach to the modelling of forward Libor rates. As made apparent in the preceding
section, in the direct modelling of Libor rates, no explicit reference is made to the bond price
processes, which are used to formally define a forward Libor rate through equality (13). Nevertheless,
to explain the idea that underpins Jamshidian’s approach, we shall temporarily assume that we are
given a family of bond prices B(t, Tj ) for the future dates Tj , j = 1, . . . , n. By definition, the spot
Libor measure is that probability measure equivalent to P, under which all relative bond prices
are local martingales, when the price process obtained by rolling over single-period bonds, is taken
as a numeraire. The existence of such a measure can be either postulated, or derived from other
conditions.10 Let us put, for t ∈ [0, T ∗ ] (as before T−1 = 0)
Y
m(t)
Gt = B(t, Tm(t) )
B −1 (Tj−1 , Tj ),
(25)
j=0
where
m(t) = inf {k = 0, 1, . . . |
k
X
δi ≥ t} = inf {k = 0, 1, . . . | Tk ≥ t}.
i=0
9 Recall that P plays the role of the forward Libor measure for the date T ∗ . Therefore, formula (23) is a consequence
of standard definition of a forward measure.
10 One may assume, e.g., that bond prices B(t, T ) satisfy the weak no-arbitrage condition, meaning that there exists
j
a probability measure P̃, equivalent to P, and such that all processes B(t, Tk )/B(t, T ∗ ) are P̃-local martingales.
14
M.Rutkowski
It is easily seen that Gt represents the wealth at time t of a portfolio which starts at time 0 with one
unit of cash invested in a zero-coupon bond of maturity T0 , and whose wealth is then reinvested at
each date Tj , j = 0, . . . , n − 1, in zero-coupon bonds which mature at the next date; that is, Tj+1 .
Definition 2.3 A spot Libor measure PL is a probability measure on (Ω, FT ∗ ) which is equivalent
to P, and such that for any j = 0, . . . , n the relative bond price B(t, Tj )/Gt follows a local martingale
under PL .
Note that
Y
m(t)
B(t, Tk+1 )/Gt =
−1
1 + δj L(Tj−1 , Tj−1 )
j=0
k
Y
1 + δj L(t, Tj−1 )
j=m(t)+1
so that all relative bond prices B(t, Tj )/Gt , j = 0, . . . , n are uniquely determined by a collection of
forward Libor rates. In this sense, G is the correct choice of the reference price process in the present
setting. We shall now concentrate on the derivation of the dynamics under PL of forward Libor rates
L(·, Tj ), j = 0, . . . , n − 1. Our aim is to show that these dynamics involve only the volatilities of
forward Libor rates (as opposed to volatilities of bond prices or other processes). Therefore, it
is possible to define the whole family of forward Libor rates simultaneously under one probability
measure (of course, this feature can also be deduced from the preceding construction). To facilitate
the derivation of the dynamics of L(·, Tj ), we postulate temporarily that bond prices B(t, Tj ) follow
Itô processes under the underlying probability measure P, more explicitly
(26)
dB(t, Tj ) = B(t, Tj ) a(t, Tj ) dt + b(t, Tj ) · dWt
for every j = 0, . . . , n, where, as before, W is a d-dimensional standard Brownian motion under an
underlying probability measure P (it should be stressed, however, that we do not assume here that
P is a forward (or spot) martingale measure). Combining (25) with (26), we obtain
(27)
dGt = Gt a(t, Tm(t) ) dt + b(t, Tm(t) ) · dWt .
Furthermore, by applying Itô’s rule to equality
1 + δj+1 L(t, Tj ) =
B(t, Tj )
,
B(t, Tj+1 )
(28)
we find that
dL(t, Tj ) = µ(t, Tj ) dt + ζ(t, Tj ) · dWt ,
where
µ(t, Tj ) =
B(t, Tj )
a(t, Tj ) − a(t, Tj+1 ) − ζ(t, Tj )b(t, Tj+1 )
δj+1 B(t, Tj+1 )
and
ζ(t, Tj ) =
B(t, Tj )
b(t, Tj ) − b(t, Tj+1 ) .
δj+1 B(t, Tj+1 )
(29)
Using (28) and the last formula, we arrive at the following relationship
b(t, Tm(t) ) − b(t, Tj+1 ) =
j
X
k=m(t)
δk+1 ζ(t, Tk )
.
1 + δk+1 L(t, Tk )
(30)
By definition of a spot Libor measure PL , each relative price B(t, Tj )/Gt follows a local martingale
under PL . Since, in addition, PL is assumed to be equivalent to P, it is clear that it is given by the
Doléans exponential, that is
Z ·
dPL
= ET ∗
hu · dWu , P-a.s.
dP
0
15
Modelling of Forward Libor and Swap Rates
for some adapted process h. It it not hard to check, using Itô’s rule, that h necessarily satisfies, for
t ∈ [0, Tj ],
a(t, Tj ) − a(t, Tm(t) ) = b(t, Tm(t) ) − ht · b(t, Tj ) − b(t, Tm(t) )
for every j = 0, . . . , n. Combining (29) with the last formula, we obtain
B(t, Tj )
a(t, Tj ) − a(t, Tj+1 ) = ζ(t, Tj ) · b(t, Tm(t) ) − ht ,
δj+1 B(t, Tj+1 )
and this in turn yields
dL(t, Tj ) = ζ(t, Tj ) ·
b(t, Tm(t) ) − b(t, Tj+1 ) − ht dt + dWt .
Using (30), we conclude that process L(·, Tj ) satisfies
j
X
dL(t, Tj ) =
k=m(t)
δk+1 ζ(t, Tk ) · ζ(t, Tj )
dt + ζ(t, Tj ) · dWtL ,
1 + δk+1 L(t, Tk )
Rt
where the process WtL = Wt − 0 hu du follows a d-dimensional standard Brownian motion under
the spot Libor measure PL . To further specify the model, we assume that processes ζ(t, Tj ), j =
0, . . . , n − 1, have the following form, for t ∈ [0, Tj ],
ζ(t, Tj ) = λj t, L(t, Tj ), L(t, Tj+1 ), . . . , L(t, Tn ) ,
where λj : [0, Tj ] × Rn−j+1 → Rd are given functions. In this way, we obtain a system of SDEs
dL(t, Tj ) =
j
X
k=m(t)
δk+1 λk (t, Lk (t)) · λj (t, Lj (t))
dt + λj (t, Lj (t)) · dWtL ,
1 + δk+1 L(t, Tk )
where we write Lj (t) = (L(t, Tj ), L(t, Tj+1 ), . . . , L(t, Tn )). Under mild regularity assumptions, this
system can be solved recursively, starting from L(·, Tn−1 ). The lognormal model of forward Libor
rates corresponds to the choice of ζ(t, Tj ) = λ(t, Tj )L(t, Tj ), where λ(·, Tj ) : [0, Tj ] → Rd is a
deterministic function for every j.
2.3
Dynamics of Libor Rates and Bond Prices
We assume that the volatilities of processes L(·, Tj ) follow deterministic functions. Put another
way, we place ourselves within the framework of the lognormal model of forward Libor rates. It is
interesting to note that in all approaches, there is a uniquely determined correspondence between
forward measures (and forward Brownian motions) associated with different dates T0 , . . . , Tn . On
the other hand, however, there is a considerable degree of ambiguity in the way in which the spot
martingale measure is specified (in some instances, it is not introduced at all). Consequently, the
futures Libor rate Lf (·, Tj ), which equals (cf. Section 2.1.3)
Lf (t, Tj ) = E P∗ (L(Tj , Tj ) | Ft ) = E P∗ (L̃(Tj , Tj ) | Ft ),
(31)
is not necessarily specified in the same way in various approaches to the lognormal model of forward
Libor rates. For this reason, we start by examining the distributional properties of forward Libor
rates, which are identicall in all abovementioned models.
For a given function g : R
→ R and a fixed date u ≤ Tj , we are interested in the following payoff
of the form X = g L(u, Tj ) which settles at time Tj . Particular cases of such payoffs are
X1 = g B −1 (Tj , Tj+1 ) , X2 = g B(Tj , Tj+1 ) , X3 = g FB (u, Tj+1 , Tj ) .
16
M.Rutkowski
Recall that
B −1 (Tj , Tj+1 ) = 1 + δj+1 L(Tj , Tj ) = 1 + δj+1 L̃(Tj , Tj ) = 1 + δj+1 Lf (Tj , Tj ).
The choice of the “pricing measure” is thus largely the matter of convenience. Similarly, we have
B(Tj , Tj+1 ) =
1
= FB (Tj , Tj+1 , Tj ).
1 + δj+1 L(Tj , Tj )
(32)
More generally, the forward price of a Tj+1 -maturity bond for the settlement date Tj equals
FB (u, Tj+1 , Tj ) =
1
B(u, Tj+1 )
=
.
B(u, Tj )
1 + δj+1 L(u, Tj )
(33)
Generally speaking, to value the claim X = g(L(u, Tj )) = g̃(FB (u, Tj+1 , Tj )) which settles at time
Tj we may use the formula
πt (X) = B(t, Tj )E PTj (X | Ft ),
∀ t ∈ [0, Tj ].
It is thus clear that to value a claim in the case u ≤ Tj , it is enough to know the dynamics of either
L(·, Tj ) or FB (·, Tj+1 , Tj ) under the forward probability measure PTj . If u = Tj , we may equally well
use the the dynamics, under PTj , of either L̃(·, Tj ) or Lf (·, Tj ). For instance,
πt (X1 )
= B(t, Tj )E PTj (B −1 (Tj , Tj+1 ) | Ft )
= B(t, Tj )E PTj (FB−1 (Tj , Tj+1 , Tj ) | Ft )
but also
πt (X1 ) = B(t, Tj ) 1 + δj+1 E PTj (Z(Tj ) | Ft ) ,
where Z(Tj ) = L(Tj , Tj ) = L̃(Tj , Tj ) = Lf (Tj , Tj ).
2.3.1
Dynamics of L(·, Tj ) under PTj
We shall now derive the transition probability density function (p.d.f.) of the process L(·, Tj ) under
the forward probability measure PTj . Let us first prove the following related result, due to Jamshidian
(1997).
Proposition 2.1 Let t ≤ u ≤ Tj . Then
E PTj L(u, Tj ) | Ft = L(t, Tj ) +
δj+1 Var PTj+1 L(u, Tj ) | Ft
1 + δj+1 L(t, Tj )
.
(34)
In the case of the lognormal model of Libor rates, we have
E PTj L(u, Tj ) | Ft
where
vj2 (t, u) = Var PTj+1
2
δj+1 L(t, Tj ) evj (t,u) − 1
= L(t, Tj ) 1 +
1 + δj+1 L(t, Tj )
Z
u
t
Z
λ(s, Tj ) · dWsTj+1 =
u
!
,
(35)
|λ(s, Tj )|2 ds.
(36)
t
In particular, the modified Libor rate L̃(t, Tj ) satisfies11
L̃(t, Tj ) = E PTj L(Tj , Tj ) | Ft
11 This
2
δj+1 L(t, Tj ) evj (t,Tj ) − 1
= L(t, Tj ) 1 +
1 + δj+1 L(t, Tj )
equality can be referred to as the convexity correction.
!
.
17
Modelling of Forward Libor and Swap Rates
Proof.
Combining (6) with the martingale property of the process L(·, Tj ) under PTj+1 , we obtain
E PTj L(u, Tj ) | Ft =
so that
E PTj+1 (1 + δj+1 L(u, Tj ))L(u, Tj ) | Ft
1 + δj+1 L(t, Tj )
δj+1 E PTj+1 (L(u, Tj ) − L(t, Tj ))2 | Ft
E PTj L(u, Tj ) | Ft = L(t, Tj ) +
1 + δj+1 L(t, Tj )
.
In the case of the lognormal model, we have
1
2
L(u, Tj ) = L(t, Tj ) eηj (t,u)− 2 vj (t,u) ,
Z
where
u
ηj (t, u) =
t
Consequently,
λ(s, Tj ) dWsTj+1 .
(37)
2
E PTj+1 (L(u, Tj ) − L(t, Tj ))2 | Ft = L2 (t, Tj ) evj (t,u) − 1 .
This gives the desired equality (35). The last asserted equality is a consequence of (7).
2
To derive the transition probability density function (p.d.f.) of the process L(·, Tj ), notice that
for any t ≤ u ≤ Tj , and any bounded Borel measurable function g : R → R we have
E PTj+1 g(L(u, Tj )) 1 + δj+1 L(u, Tj ) Ft
.
E PTj g(L(u, Tj )) | Ft =
1 + δj+1 L(t, Tj )
The following simple lemma appears to be useful.
Lemma 2.2 Let ζ be a nonnegative random variable on a probability space (Ω, F , P) with the probability density function fP . Let Q be a probability measure equivalent to P. Suppose that for any
bounded Borel measurable function g : R → R we have
E P (g(ζ)) = E Q (1 + ζ)g(ζ) .
Then the p.d.f. fQ of ζ under Q satisfies fP (y) = (1 + y)fQ (y).
Proof.
The assertion is in fact trivial since, by assumption,
Z ∞
Z ∞
g(y)fP (y) dy =
g(y)(1 + y)fQ (y) dy
−∞
−∞
for any bounded Borel measurable function g : R → R.
2
Assume the lognormal model of Libor rates and fix x ∈ R. Recall that for any t ≥ u we have
L(u, Tj ) = L(t, Tj ) e
ηj (t,u)− 12 Var PT
j+1
(ηj (t,u))
,
where ηj (t, u) is given by (37) (so that it is independent of the σ-field Ft ). Markovian property of
L(·, Tj ) under the forward measure PTj+1 is thus apparent. Denote by pL (t, x; u, y) the transition
p.d.f. under PTj+1 of the process L(·, Tj ). Elementary calculations involving Gaussian densities yield
pL (t, x; u, y) =
=
PTj+1 {L(u, Tj ) = y | L(t, Tj ) = x}
(
2 )
ln(y/x) + 12 vj2 (t, u)
1
√
exp −
2vj2 (t, u)
2πvj (t, u)y
18
M.Rutkowski
for any x, y > 0 and t < u. Taking into account Lemma 2.2, we conclude that the transition p.d.f.
of the process12 L(·, Tj ), under the forward probability measure PTj , satisfies
p̃L (t, x; u, y) = PTj {L(u, Tj ) = y | L(t, Tj ) = x} =
1 + δj+1 y
pL (t, x; u, y).
1 + δj+1 x
We are in a position to state the following result, which can be used, for instance, to value a
contingent claim of the form X = h(L(Tj )) which settles at time Tj (cf. Schmidt (1996)).
Corollary 2.1 The transition p.d.f. under PTj of the forward Libor rate L(·, Tj ) equals, for any
t < u and x, y > 0,
(
2 )
ln(y/x) + 12 vj2 (t, u)
1 + δj+1 y
.
exp −
p̃L (t, x; u, y) = √
2vj2 (t, u)
2πvj (t, u) y(1 + δj+1 x)
2.3.2
Dynamics of FB (·, Tj+1 , Tj ) under PTj
Observe that the forward bond price FB (·, Tj+1 , Tj ) satisfies
FB (t, Tj+1 , Tj ) =
1
B(t, Tj+1 )
=
.
B(t, Tj )
1 + δj+1 L(t, Tj )
(38)
First, this implies that in the lognormal model of Libor rates, the dynamics of the forward bond
price FB (·, Tj+1 , Tj ) are governed by the following stochastic differential equation, under PTj ,
T
dFB (t) = −FB (t) 1 − FB (t) λ(t, Tj ) · dWt j ,
(39)
where we write FB (t) = FB (t, Tj+1 , Tj ). If the initial condition satisfies 0 < FB (0) < 1, this equation
can be shown to admit a unique strong solution (it satisfies 0 < FB (t) < 1 for every t > 0). This
makes clear that the process FB (·, Tj+1 , Tj ) – and thus also the process L(·, Tj ) – are Markovian
under PTj . Using Corollary 2.1 and relationship (38), one can find the transition p.d.f. of the Markov
process FB (·, Tj+1 , Tj ) under PTj ; that is,
pB (t, x; u, y) = PTj {FB (u, Tj+1 , Tj ) = y | FB (t, Tj+1 , Tj ) = x}.
We have the following result (see Rady and Sandmann (1994), Miltersen et al. (1997), and Jamshidian
(1997)).
Corollary 2.2 The transition p.d.f. under PTj of the forward bond price FB (·, Tj+1 , Tj ) equals, for
any t < u and arbitrary 0 < x, y < 1,
 2 
x(1−y)

1 2


ln y(1−x) + 2 vj (t, u) 
x
.
exp −
pB (t, x; u, y) = √


2vj2 (t, u)
2πvj (t, u)y 2 (1 − y)


Proof.
Let us fix x ∈ (0, 1). Using (38), it is easy to show that
1−x
1 − y
; u,
,
pB (t, x; u, y) = δ −1 y −2 p̃L t,
δx
δy
where δ = δj+1 . The formula now follows from Corollary 2.1.
2
Let us observe that the results of this section can be applied to value the so-called irregular cash
flows, such as caps or floors settled in advance (for more details on this issue we refer to Schmidt
(1996)).
12 The Markov property of L(·, T ) under P
j
Tj can be easily deduced from the Markovian features of the forward
price FB (·, Tj , Tj+1 ) under PTj (see formulae (38)–(39)).
19
Modelling of Forward Libor and Swap Rates
2.4
Caps and Floors
An interest rate cap (known also as a ceiling rate agreement) is a contractual arrangement where
the grantor (seller) has an obligation to pay cash to the holder (buyer) if a particular interest rate
exceeds a mutually agreed level at some future date or dates. Similarly, in an interest rate floor, the
grantor has an obligation to pay cash to the holder if the interest rate is below a preassigned level.
When cash is paid to the holder, the holder’s net position is equivalent to borrowing (or depositing)
at a rate fixed at that agreed level. This assumes that the holder of a cap (or floor) agreement also
holds an underlying asset (such as a deposit) or an underlying liability (such as a loan). Finally, the
holder is not affected by the agreement if the interest rate is ultimately more favorable to him than
the agreed level. This feature of a cap (or floor) agreement makes it similar to an option. Specifically,
a forward start cap (or a forward start floor) is a strip of caplets (floorlets), each of which is a call
(put) option on a forward rate, respectively. Let us denote by κ and by δj the cap strike rate and
the length of the accrual period, respectively. We shall check that an interest rate caplet (i.e., one
leg of a cap) may also be seen as a put option with strike price 1 (per dollar of notional principal)
which expires at the caplet start day on a discount bond with face value 1 + κδj which matures at
the caplet end date.
Similarly to swap agreements, interest rate caps and floors may be settled either in arrears or
in advance. In a forward cap or floor, which starts at time T0 , and is settled in arrears at dates
Tj , j = 1, . . . , n, the cash flows at times Tj are Np (L(Tj−1 ) − κ)+ δj and Np (κ − L(Tj−1 ))+ δj ,
respectively, where Np stands for the notional principal (recall that δj = Tj − Tj−1 ). As usual, the
rate L(Tj−1 ) = L(Tj−1 , Tj−1 ) is determined at the reset date Tj−1 , and it satisfies
B(Tj−1 , Tj )−1 = 1 + δj L(Tj−1 ).
(40)
The price at time t ≤ T0 of a forward cap, denoted by FCt , is (we set Np = 1)
FCt
=
=
n
X
j=1
n
X
E P∗
B
t
(L(Tj−1 ) − κ)+ δj Ft
BTj
B(t, Tj ) E PTj (L(Tj−1 ) − κ)+ δj Ft .
(41)
j=1
On the other hand, since the cash flow of the j th caplet at time Tj is manifestly a FTj−1 -measurable
random variable, we may directly express the value of the cap in terms of expectations under forward
measures PTj−1 , j = 1, . . . , n. Indeed, we have
FCt =
n
X
B(t, Tj−1 ) E PTj−1 B(Tj−1 , Tj )(L(Tj−1 ) − κ)+ δj Ft .
(42)
j=1
Consequently, using (40) we get equality
FCt =
n
X
B(t, Tj−1 ) E PTj−1
+ 1 − δ̃j B(Tj−1 , Tj ) Ft ,
(43)
j=1
which is valid for every t ∈ [0, T ]. It is apparent that a caplet is essentially equivalent to a put option
on a zero-coupon bond; it may also be seen as an option on a single-period swap.
The equivalence of a cap and a put option on a zero-coupon bond can be explained in an intuitive
way. For this purpose, it is enough to examine two basic features of both contracts: the exercise set
and the payoff value. Let us consider the j th caplet. A caplet is exercised at time Tj−1 if and only
if L(Tj−1 ) − κ > 0, or equivalently, if
B(Tj−1 , Tj )−1 = 1 + L(Tj−1 )(Tj − Tj−1 ) > 1 + κδj = δ̃j .
20
M.Rutkowski
The last inequality holds whenever δ̃j B(Tj−1 , Tj ) < 1. This shows that both of the considered
options are exercised in the same circumstances. If exercised, the caplet pays δj (L(Tj−1 ) − κ) at
time Tj , or equivalently
δj B(Tj−1 , Tj )(L(Tj−1 ) − κ) = 1 − δ̃j B(Tj−1 , Tj ) = δ̃j δ̃j−1 − B(Tj−1 , Tj )
at time Tj−1 . This shows once again that the j th caplet, with strike level κ and nominal value 1, is
essentially equivalent to a put option with strike price (1 + κδj )−1 and nominal value δ̃j = (1 + κδj )
written on the corresponding zero-coupon bond with maturity Tj .
The analysis of a floor contract can be done long the simlar lines. By definition, the j th floorlet
pays (κ − L(Tj−1 ))+ at time Tj . Therefore,
FFt =
n
X
E P∗
j=1
but also
FFt =
n
X
B
t
(κ − L(Tj−1 ))+ δj Ft ,
BTj
B(t, Tj−1 ) E PTj−1
δ̃j B(Tj−1 , Tj ) − 1
+ Ft .
(44)
(45)
j=1
Combining (41) with (44) (or (43) with (45)), we obtain the following cap-floor parity relationship
FCt − FFt =
n
X
B(t, Tj−1 ) − δ̃j B(t, Tj )
(46)
j=1
which is also an immediate consequence of the no-arbitrage property, so that it does not depend on
model’s choice.
2.4.1
Market Valuation Formula for Caps and Floors
The main motivation for the introduction of a lognormal model of Libor rates was the market practice
of pricing caps and swaptions by means of Black-Scholes-like formulae. For this reason, we shall
first describe how market practitioners value caps. The formulae commonly used by practitioners
assume that the underlying instrument follows a geometric Brownian motion under some probability
measure, Q say. Since the formal definition of this probability measure is not available, we shall
informally refer to Q as the market probability.
Let us consider an interest rate cap with expiry date T and fixed strike level κ. Market practice
is to price the option assuming that the underlying forward interest rate process is lognormally
distributed with zero drift. Let us first consider a caplet – that is, one leg of a cap. Assume that the
forward Libor rate L(t, T ), t ∈ [0, T ], for the accrual period of length δ follows a geometric Brownian
motion under the “market probability”, Q say. More specifically
dL(t, T ) = L(t, T )σ dWt ,
(47)
where W follows a one-dimensional standard Brownian motion under Q, and σ is a strictly positive
constant. The unique solution of (47) is
(48)
L(t, T ) = L(0, T ) exp σWt − 12 σ 2 t2 , ∀ t ∈ [0, T ],
where the initial condition is derived from the yield curve Y (0, T ), namely
B(0, T )
= exp (T + δ)Y (0, T + δ) − T Y (0, T ) .
1 + δL(0, T ) =
B(0, T + δ)
The “market price” at time t of a caplet with expiry date T and strike level κ is calculated by means
of the formula
FC t = δB(t, T + δ) E Q (L(T, T ) − κ)+ Ft .
21
Modelling of Forward Libor and Swap Rates
More explicitly, for any t ∈ [0, T ] we have
FC t = δB(t, T + δ) L(t, T )N ê1 (t, T ) − κN ê2 (t, T ) ,
(49)
where N is the standard Gaussian cumulative distribution function
Z x
2
1
e−z /2 dz, ∀ x ∈ R,
N (x) = √
2π −∞
and
ê1,2 (t, T ) =
ln(L(t, T )/κ) ± 12 v̂02 (t, T )
v̂0 (t, T )
with v̂02 (t, T ) = σ 2 (T − t). This means that market practitioners price caplets using Black’s formula,
with discount from the settlement date T + δ.
A cap settled in arrears at times Tj , j = 1, . . . , n, where Tj − Tj−1 = δj , T0 = T, is priced by
the formula
n
X
δj B(t, Tj ) L(t, Tj−1 )N êj1 (t) − κN êj2 (t) ,
(50)
FCt =
j=1
where for every j = 0, . . . , n − 1
êj1,2 (t) =
ln(L(t, Tj−1 )/κ) ±
v̂j (t)
1
2
v̂j2 (t)
(51)
and v̂j2 (t) = (Tj−1 − t)σj2 for some constants σj , j = 1, . . . , n. Apparently, the market assumes that
for any maturity Tj , the corresponding forward Libor rate has a lognormal probability law under the
“market probability”. The value of a floor can be easily derived by combining (50)–(51) with the
cap-floor parity relationship (46). As we shall see in what follows, the valuation formulae obtained
for caps and floors in the lognormal model of forward Libor rates agree with the market practice.
2.4.2
Valuation in the Lognormal Model of Forward Libor Rates
We shall now examine the valuation of caps within the lognormal model of forward Libor rates of
Section 2.2.3. The dynamics of the forward Libor rate L(t, Tj−1 ) under the forward probability
measure PTj are
T
(52)
dL(t, Tj−1 ) = L(t, Tj−1 ) λ(t, Tj−1 ) · dWt j ,
where W Tj follows a d-dimensional Brownian motion under the forward measure PTj , and λ(·, Tj−1 ) :
[0, Tj−1 ] → Rd is a deterministic function. Consequently, for every t ∈ [0, Tj−1 ] we have
Z ·
λ(u, Tj−1 ) · dWuTj .
L(t, Tj−1 ) = L(0, Tj−1 )Et
0
In the present setup, the cap valuation formula (53) was first established by Miltersen et al. (1997),
who focused on the dynamics of the forward Libor rate for a given date. Equality (53) was subsequently rederived through a probabilistic approach in Goldys (1997) and Rady (1997). Finally, the
same result was established by means of the forward measure approach in Brace et al. (1997). The
following proposition is a consequence of formula (42), combined with the dynamics (52). As before,
N is the standard Gaussian probability distribution function.
Proposition 2.2 Consider an interest rate cap with strike level κ, settled in arrears at times Tj , j =
1, . . . , n. Assuming the lognormal model of Libor rates, the price of a cap at time t ∈ [0, T ] equals
FCt =
n
X
j=1
n
X
j
j
δj B(t, Tj ) L(t, Tj−1 )N ẽ1 (t) − κN ẽ2 (t) =
FC jt ,
j=1
(53)
22
M.Rutkowski
where FC jt stands for the price at time t of the j th caplet for n j = 1, . . . , n
ẽj1,2 (t) =
ln(L(t, Tj−1 )/κ) ±
ṽj (t)
and
ṽj2 (t)
Z
Tj−1
=
1
2
ṽj2 (t)
|λ(u, Tj−1 )|2 du.
t
Proof. We fix j and we consider the j th caplet. It is clear that its payoff at time Tj admits the
representation
(54)
FC jTj = δj (L(Tj−1 ) − κ)+ = δj L(Tj−1 ) ID − δj κ ID ,
where D = {L(Tj−1 ) > K} is the exercise set. Since the caplet settles at time Tj , it is convenient
to use the forward measure PTj to find its arbitrage price. We have
FC jt = B(t, Tj )E PTj FC jTj | Ft ),
∀ t ∈ [0, Tj ].
Obviously, it is enough to find the value of a caplet for t ∈ [0, Tj−1 ]. In view of (54), it is clear that
we need to evaluate the following conditional expectations
FC jt = δj B(t, Tj ) E PTj L(Tj−1 ) ID Ft − κδj B(t, Tj ) PTj (D | Ft )
= δj B(t, Tj )(I1 − I2 ),
where the meaning of I1 and I2 is clear from the context. Recall that L(Tj−1 ) is given by the formula
Z
Tj−1
L(Tj−1 ) = L(t, Tj−1 ) exp
t
λ(u, Tj−1 ) · dWuTj −
1
2
Z
Tj−1
|λ(u, Tj−1 )|2 du .
t
Since λ(·, Tj−1 ) is a deterministic function, the probability law under PTj of the Itô integral
Z
Tj−1
λ(u, Tj−1 ) · dWuTj
ζ(t, Tj−1 ) =
t
is Gaussian, with zero mean and the variance
Z
Var PTj (ζ(t, Tj−1 )) =
Tj−1
|λ(u, Tj−1 )|2 du.
t
Therfore, it is straightforward to show that13
I2 = κ N
ln(L(t, Tj−1 ) − ln κ − 12 vj2 (t)
vj (t)
!
.
To evaluate I1 , we introduce an auxiliary probability measure P̂Tj , equivalent to PTj on (Ω, FTj−1 ),
by setting
Z ·
dP̂Tj
= ETj−1
λ(u, Tj−1 ) · dWuTj .
dPTj
0
Then the process Ŵ Tj given by the formula
Z t
T
T
λ(u, Tj−1 ) du,
Ŵt j = Wt j −
0
13 See,
∀ t ∈ [0, Tj−1 ],
for instance, the proof of the Black-Scholes formula in Musiela and Rutkowski (1997a).
23
Modelling of Forward Libor and Swap Rates
follows the d-dimensional standard Brownian motion under P̂Tj . Furthermore, the forward price
L(Tj−1 ) admits the representation under P̂Tj , for t ∈ [0, Tj−1 ]
Z
Tj−1
L(Tj−1 ) = L(t, Tj−1 ) exp
t
λj−1 (u) · dŴuTj +
1
2
Z
Tj−1
|λj−1 (u)|2 du
t
where we set λj−1 (u) = λ(u, Tj−1 ). Since
Z
I1 = L(t, Tj−1 )E PTj ID exp
Tj−1
t
λj−1 (u) · dWuTj −
1
2
Z
Tj−1
|λj−1 (u)|2 du Ft
t
from the abstract Bayes rule we get I1 = L(t, Tj−1 ) P̂Tj (D | Ft ). Arguing in much the same way as
for I2 , we thus obtain
!
ln L(t, Tj−1 ) − ln κ + 12 vj2 (t)
.
I1 = L(t, Tj−1 ) N
vj (t)
2
This completes the proof of the proposition.
Once again, to derive the floors valuation formula, it is enough to make use of the cap-floor parity
(46).
2.4.3
Hedging of Caps and Floors
It is clear the replicating strategy for a cap is a simple sum of replicating strategies for caplets.
Therefore, it is enough to focus on a particular caplet. Let us denote by FC (t, Tj ) the forward price
of the j th caplet for the settlement date Tj . From (53), it is clear that
FC (t, Tj ) = δj L(t, Tj−1 )N ẽj1 (t) − κ N ẽj2 (t) ,
so that an application of Itô’s formula yields14
dFC (t, Tj ) = δj N ẽj1 (t) dL(t, Tj−1 ).
(55)
Let us consider the following self-financing trading strategy in the Tj -forward market. We start our
trade at time 0 with FC (0, Tj ) units of zero-coupon bonds.15 At any time t ≤ Tj−1 we assume
ψtj = N ẽj1 (t) positions in forward rate agreements (that is, single-period forward swaps) over the
period [Tj−1 , Tj ]. The associated gains/losses process V, in the Tj forward market,16 satisfies17
dVt = δj ψtj dL(t, Tj−1 ) = δj N ẽj1 (t) dL(t, Tj−1 ) = dFC (t, T )
with V0 = 0. Consequently,
Z
FC (Tj−1 , Tj ) = FC (0, Tj ) +
0
Tj−1
δj ψtj dL(t, Tj−1 ) = FC (0, Tj ) + VTj−1 .
It should be stressed that dynamic trading takes place on the interval [0, Tj−1 ] only, the gains/losses
(involving the initial investment) are incurred at time Tj , however. All quantities in the last formula
are expressed in units of Tj -maturity zero-coupon bonds. Also, the caplet’s payoff is known already at
time Tj−1 , so that it is completely specified by its forward price FC (Tj−1 , Tj ) = FC jTj−1 /B(Tj−1 , Tj ).
Therefore the last equality makes it clear that the strategy ψ introduced above does indeed replicate
the j th caplet.
14 The
calculations here are essentially the same as in the classic Black-Scholes model.
need thus to invest FC j0 = FC (0, Tj )B(0, Tj ) of cash at time 0.
16 That is, with the value expressed in units of T -maturity zero-coupon bonds.
j
17 To get a more intuitive insight in this formula, it is advisable to consider first a discretized version of ψ.
15 We
24
M.Rutkowski
It should be observed that formally the replicating strategy has also the second component, ηtj
say, which represents the number of forward contracts on Tj -maturity bond, with the settlement
date Tj . Since obviously FB (t, Tj , Tj ) = 1 for every t ≤ Tj , so that dFB (t, Tj , Tj ) = 0, for the
Tj -forward value of our strategy, we get Ṽt (ψ j , η j ) = ηtj = FC (t, Tj ) and
dṼt (ψ j , η j ) = ψtj δj dL(t, Tj−1 ) + ηtj dFB (t, Tj , Tj ) = δj N ẽj1 (t) dL(t, Tj−1 ).
It should be stressed however, with the exception for the initial investment at time 0 in Tj -maturity
bonds, no bonds trading is required for the caplet’s replication. In practical terms, the hedging of a
cap within the framework of the lognormal model of forward Libor rates in done exclusively through
dynamic trading in the underlying single-period swaps. Of course, the same remarks (and similar
calculations) apply also to floors. In this interpretation, the component η j simply represents the
future (i.e., as of time Tj−1 ) effects of a continuous trading in forward contracts.
Alternatively, the hedging of a cap can be done in the spot (i.e., cash) market, using two simple
portfolios of bonds. Indeed, it is easily seen that for the process
Vt (ψ j , η j ) = B(t, Tj−1 )Ṽt (ψ j , η j ) = FC jt
we have
Vt (ψ j , η j ) = ψtj B(t, Tj−1 ) − B(t, Tj ) + ηtj dFB (t, Tj , Tj )
and
dVt (ψ j , η j ) =
=
ψtj d B(t, Tj−1 ) − B(t, Tj ) + ηtj dB(t, Tj )
N ẽj1 (t) d B(t, Tj−1 ) − B(t, Tj ) + ηtj dB(t, Tj ).
This means that the components ψ j and η j now represent the number of units of portfolios B(t, Tj−1 )−
B(t, Tj ) and B(t, Tj ) held at time t.
2.4.4
Bond Options
We shall now give the bond option valuation formula within the framework of the lognormal model of
forward Libor rates. This result was first obtained by Rady and Sandmann (1994), who adopted the
PDE approach and who worked in a different setup (see also Goldys (1997), Miltersen et al. (1997),
and Rady (1997)). In the present framework, it is an immediate consequence of (53) combined with
(43).
Proposition 2.3 The price Ct at time t ≤ Tj−1 of a European call option, with expiration date
Tj−1 and strike price 0 < K < 1, written on a zero-coupon bond maturing at Tj = Tj−1 + δj , equals
(56)
Ct = (1 − K)B(t, Tj )N l1j (t) − K(B(t, Tj−1 ) − B(t, Tj ))N l2j (t) ,
where
j
(t)
l1,2
ln ((1 − K)B(t, Tj )) − ln K B(t, Tj−1 ) − B(t, Tj ) ± 12 ṽj (t)
=
ṽj (t)
and
ṽj2 (t) =
Z
Tj−1
|λ(u, Tj−1 )|2 du.
t
In view of (56), it is apparent that the replication of the bond option using the underlying bonds
of maturity Tj−1 and Tj is rather involved. This should be contrasted with the case of the Gaussian
Heath-Jarrow-Morton model18 in which hedging of bond options with the use of the underlying
bonds is straightforward. This illustrates the general feature that each particular way of modelling
the term structure is tailored to the specific class of derivatives and hedging instruments.
18 In
such a model the forward prices prices of bonds follow lognormal processes.
25
Modelling of Forward Libor and Swap Rates
3
Modelling of Forward Swap Rates
We shall first describe the most typical swap contracts and related options (the so-called swaptions).
Subsequently, we shall present a model of forward swap rates put forward by Jamshidian (1996,
1997). For the sake of expositional convenience, we shall follow the backward induction approach
due to Rutkowski (1999), however.
3.1
Interest Rate Swaps
Let us consider a forward (start) payer swap (that is, fixed-for-floating interest rate swap) settled in
arrears, with notional principal Np . As before, we consider a finite collection of dates 0 < T0 < T1 <
· · · < Tn so that δj = Tj − Tj−1 > 0 for every j = 1, . . . , n. The floating rate L(Tj−1 ) received at
time Tj is set at time Tj−1 by reference to the price of a zero-coupon bond over the period [Tj−1 , Tj ].
More specifically, L(Tj−1 ) is the spot Libor rate prevailing at time Tj−1 , so that it satisfies
B(Tj−1 , Tj )−1 = 1 + (Tj − Tj−1 )L(Tj−1 ) = 1 + δj L(Tj−1 ).
(57)
Recall that in general, the forward Libor rate L(t, Tj−1 ) for the future time period [Tj−1 , Tj ] of
length δj satisfies
B(t, Tj−1 )
= FB (t, Tj−1 , Tj ),
(58)
1 + δj L(t, Tj−1 ) =
B(t, Tj )
so that L(Tj−1 ) coincides with L(Tj−1 , Tj−1 ). At any date Tj , j = 1, . . . , n, the cash flows of a
forward payer swap are Np L(Tj−1 )δj and −Np κδj , where κ is a preassigned fixed rate of interest
(the cash flows of a forward receiver swap have the same size, but opposite signs). The number n,
which coincides with the number of payments, is referred to as the length of a swap, (for instance, the
length of a 3-year swap with quarterly settlement equals n = 12). The dates T0 , . . . , Tn−1 are known
as reset dates, and the dates T1 , . . . , Tn as settlement dates. We shall refer to the first reset date
T0 as the start date of a swap. Finally, the time interval [Tj−1 , Tj ] is referred to as the j th accrual
period. We may and do assume, without loss of generality, that the notional principal Np = 1.
The value at time t of a forward payer swap, which is denoted by FS t or FS t (κ), equals
FS t (κ) = E P∗
n
o
nX
Bt
(L(Tj−1 ) − κ)δj Ft .
B
T
j
j=1
Since
L(t, Tj−1 ) =
(59)
B(t, Tj−1 ) − B(t, Tj )
,
δj B(t, Tj )
it is clear that the process L(·, Tj−1 ) follows a martingale under the forward martingale measure
PTj . Therefore
FS t (κ)
=
=
n
X
j=1
n
X
B(t, Tj )E PTj (L(Tj−1 ) − κ)δj Ft
B(t, Tj ) (L(t, Tj−1 ) − κ)δj
j=1
=
n
X
B(t, Tj−1 ) − B(t, Tj ) − κδj B(t, Tj ) .
j=1
After rearranging, this yields
FS t (κ) = B(t, T0 ) −
n
X
j=1
cj B(t, Tj )
(60)
26
M.Rutkowski
for every t ∈ [0, T ], where cj = κδj for j = 1, . . . , n − 1, and cn = δ̃n = 1 + κδn . The last equality
makes clear that a forward payer swap settled in arrears is, essentially, a contract to deliver a specific
coupon-bearing bond and to receive in the same time a zero-coupon bond. Relationship (60) may
also be established through a straightforward comparison of the future cash flows from these bonds.
Note that (60) provides a simple method for the replication of a swap contract, independent of the
term structure model.
In the forward payer swap settled in advance – that is, in which each reset date is also a settlement date – the discounting method varies from country to country. In the U.S. and in many
European markets, the cash flows of a swap settled in advance at reset dates Tj , j = 0, . . . , n − 1,
are L(Tj )δj+1 (1 + L(Tj )δj+1 )−1 and −κδj+1 (1 + L(Tj )δj+1 )−1 . Therefore the value FS ∗∗
t (κ) at time
t of this swap is
FS ∗∗
t (κ) =
E P∗
n n−1
X Bt δj+1 (L(Tj ) − κ) o
Ft
BTj 1 + δj+1 L(Tj )
j=0
=
E P∗
o
n n−1
X Bt
(L(Tj ) − κ)δj+1 B(Tj , Tj+1 ) Ft
B
Tj
j=0
=
E P∗
n n−1
X
j=0
o
Bt
(L(Tj ) − κ)δj+1 Ft ,
BTj+1
which coincides with the value of the swap settled in arrears. Once again, this is by no means
surprising, since the payoffs L(Tj )δj+1 (1 + L(Tj )δj+1 )−1 and −κδj+1 (1 + L(Tj )δj+1 )−1 at time Tj
are easily seen to be equivalent to payoffs L(Tj )δj+1 and −κδj+1 respectively at time Tj+1 (recall
that 1 + L(Tj )δj+1 = B −1 (Tj , Tj+1 )).
In what follows, we shall restrict our attention to interest rate swaps settled in arrears. As
mentioned, a swap agreement is worthless at initiation. This important feature of a swap leads to
the following definition, which refers in fact to the more general concept of a forward swap. Basically,
a forward swap rate is that fixed rate of interest which makes a forward swap worthless.
Definition 3.1 The forward swap rate κ(t, T0 , n) at time t for the date T0 is that value of the fixed
rate κ which makes the value of the forward swap zero, i.e., that value of κ for which FS t (κ) = 0.
Using (60), we obtain
κ(t, T0 , n) = (B(t, T0 ) − B(t, Tn ))
n
X
−1
δj B(t, Tj )
.
(61)
j=1
A swap (swap rate, respectively) is the forward swap (forward swap rate, respectively) with t = T .
The swap rate, κ(T0 , T0 , n), equals
κ(T0 , T0 , n) = (1 − B(T0 , Tn ))
n
X
−1
δj B(T0 , Tj )
.
(62)
j=1
Note that the definition of a forward swap rate implicitly refers to a swap contract of length
n which starts at time T0 . It would thus be more correct to refer to κ(t, T0 , n) as the n-period
forward swap rate prevailing at time t, for the future date T0 . A forward swap rate is a rather
theoretical concept, as opposed to swap rates, which are quoted daily (subject to an appropriate
bid-ask spread) by financial institutions who offer interest rate swap contracts to their institutional
clients. In practice, swap agreements of various lengths are offered. Also, typically, the length of the
reference period varies over time; for instance, a 5-year swap may be settled quarterly during the
first three years, and semi-annually during the last two. Swap rates also play an important role as
a basis for several derivative instruments. For instance, an appropriate swap rate is commonly used
as a strike level for an option written on the value of a swap; that is, a swaption.
27
Modelling of Forward Libor and Swap Rates
Finally, it will be useful to express that value at time t of a given forward swap with fixed rate κ
in terms of the current value of the forward swap rate. Since obviously FS t (κ(t, T0 , n)) = 0, using
(60), we get
FS t (κ) = FS t (κ) − FS t (κ(t, T0 , n)) =
n
X
(κ(t, T0 , n) − κ)δj B(t, Tj ).
(63)
j=1
3.2
Lognormal Model of Forward Swap Rates
The lognormal model of forward swap rates was developed by Jamshidian (1996, 1997). In this
section, we follow Rutkowski (1999). We assume, as before, that the tenor structure 0 < T0 < T1 <
Pj
· · · < Tn = T ∗ is given. Recall that δj = Tj − Tj−1 for j = 1, . . . , n, and thus Tj = i=0 δi for every
j = 0, . . . , n. For any fixed j, we consider a fixed-for-floating forward (payer) swap which starts at
time Tj and has n − j accrual periods, whose consecutive lengths are δj+1 , . . . , δn . The fixed interest
rate paid at each of reset dates Tl for l = j + 1, . . . , n equals κ, and the corresponding floating rate,
L(Tl ), is found using the formula
B(Tl , Tl+1 )−1 = 1 + (Tl+1 − Tl )L(Tl ) = 1 + δl+1 L(Tl ),
i.e., it coincides with the Libor rate L(Tl , Tl ). It is not difficult to check, using no-arbitrage arguments,
that the value of such a swap equals, for t ∈ [0, Tj ] (by convention, the notional principal equals 1)
n
X
FS t (κ) = B(t, Tj ) −
cl B(t, Tl ),
l=j+1
where cl = κδl for l = j + 1, . . . , n − 1, and cn = 1 + κδn. Consequently, the associated forward swap
rate, κ(t, Tj , n − j), that is, that value of a fixed rate κ for which such a swap is worthless at time
t, is given by the formula
κ(t, Tj , n − j) =
B(t, Tj ) − B(t, Tn )
δj+1 B(t, Tj+1 ) + · · · + δn B(t, Tn )
(64)
for every t ∈ [0, Tj ], j = 0, . . . , n − 1. In this section, we consider the family of forward swap rates
κ̃(t, Tj ) = κ(t, Tj , n − j) for j = 0, . . . , n − 1. Let us stress that the underlying swap agreements differ
in length, however, they all have a common expiration date, T ∗ = Tn .
Suppose momentarily that we are given a family of bond prices B(t, Tm ), m = 1, . . . , n, on a
filtered probability space (Ω, F, P) equipped with a Brownian motion W. As in Section 2.1, we find
it convenient to postulate that P = PT ∗ is the forward measure for the date T ∗ , and the process
∗
W = W T is the corresponding Brownian motion. For any m = 1, . . . , n − 1, we introduce the
fixed-maturity coupon process G(m) by setting (recall that Tl∗ = Tn−l , in particular, T0∗ = Tn )
Gt (m) =
n
X
δl B(t, Tl ) =
l=n−m+1
m−1
X
δn−k B(t, Tk∗ )
(65)
k=0
for t ∈ [0, Tn−m+1 ]. A forward swap measure is that probability measure equivalent to P, which
corresponds to the choice of the fixed-maturity coupon process as a numeraire asset. We have the
following definition.
Definition 3.2 For j = 0, . . . , n, a probability measure P̃Tj on (Ω, FTj ), equivalent to P, is said to
be the fixed-maturity forward swap measure for the date Tj if, for every k = 0, . . . , n, the relative
bond price
B(t, Tk )
B(t, Tk )
=
,
Zn−j+1 (t, Tk ) :=
Gt (n − j + 1)
δj B(t, Tj ) + · · · + δn B(t, Tn )
t ∈ [0, Tk ∧ Tj ], follows a local martingale under P̃Tj .
28
M.Rutkowski
Put another way, for any fixed m = 1, . . . , n + 1, the relative bond prices
Zm (t, Tk∗ ) =
B(t, Tk∗ )
B(t, Tk∗ )
=
,
∗
Gt (m)
δn−m+1 B(t, Tm−1
) + · · · + δn B(t, T ∗ )
∗
∗
], are bound to follow local martingales under the forward swap measure P̃Tm−1
. It
t ∈ [0, Tk∗ ∧ Tm−1
∗
∗
equals, for t ∈ [0, Tm
],
follows immediately from (64) that the forward swap rate for the date Tm
∗
)=
κ̃(t, Tm
or equivalently,
∗
B(t, Tm
) − B(t, T ∗ )
,
∗
δn−m+1 B(t, Tm−1
) + · · · + δn B(t, T ∗ )
∗
∗
) = Zm (t, Tm
) − Zm (t, T ∗ ).
κ̃(t, Tm
∗
∗
) also follows a local martingale under the forward swap measure P̃Tm−1
. Moreover,
Therefore κ̃(·, Tm
∗
∗
−1
∗
∗
since obviously Gt (1) = δn B(t, T ), it is evident that Z1 (t, Tk ) = δn FB (t, Tk , T ), and thus the
probability measure P̃T ∗ can be chosen to coincide with the forward martingale measure PT ∗ . Our
aim is to construct a model of forward swap rates through backward induction. As one might expect,
the underlying bond price processes will not be explicitly specified. We make the folowing standing
assumptions.
Assumptions (SR). We assume that we are given a family of bounded adapted processes ν(·, Tj ), j =
0, . . . , n − 1, which represent the volatilities of forward swap rates κ̃(·, Tj ). In addition, we are given
an initial term structure of interest rates, specified by a family B(0, Tj ), j = 0, . . . , n, of bond prices.
We assume that B(0, Tj ) > B(0, Tj+1 ) for j = 0, . . . , n − 1.
We wish to construct a family of forward swap rates in such a way that
Tj+1
dκ̃(t, Tj ) = κ̃(t, Tj )ν(t, Tj ) · dW̃t
(66)
for any j = 0, . . . , n − 1, where each process W̃ Tj+1 follows a standard Brownian motion under the
corresponding forward swap measure P̃Tj+1 . The model should also be consistent with the initial
term structure of interest rates, meaning that
κ̃(0, Tj ) =
B(0, Tj ) − B(0, T ∗ )
.
δj+1 B(0, Tj+1 ) + · · · + δn B(0, Tn )
(67)
We proceed by backward induction. The first step is to introduce the forward swap rate for the date
T1∗ by postulating that the forward swap rate κ̃(·, T1∗ ) solves the SDE
∗
dκ̃(t, T1∗ ) = κ̃(t, T1∗ )ν(t, T1∗ ) · dW̃tT ,
∗
∀ t ∈ [0, T1∗ ],
(68)
∗
where W̃ T = W T = W, with the initial condition
κ̃(0, T1∗ ) =
B(0, T1∗ ) − B(0, T ∗ )
.
δn B(0, T ∗ )
To specify the process κ̃(·, T2∗ ), we need first to introduce a forward swap measure P̃T1∗ and an asso∗
ciated Brownian motion W̃ T1 . To this end, notice that each process Z1 (·, Tk∗ ) = B(·, Tk∗ )/δn B(·, T ∗ )
follows a strictly positive local martingale under P̃T ∗ = PT ∗ . More specifically, we have
dZ1 (t, Tk∗ ) = Z1 (t, Tk∗ )γ1 (t, Tk∗ ) · dW̃tT
∗
(69)
for some adapted process γ1 (·, Tk∗ ). According to the definition of a fixed-maturity forward swap
measure, we postulate that for every k the process
Z2 (t, Tk∗ ) =
Z1 (t, Tk∗ )
B(t, Tk∗ )
=
δn−1 B(t, T1∗ ) + δn B(t, T ∗ )
1 + δn−1 Z1 (t, T1∗ )
29
Modelling of Forward Libor and Swap Rates
follows a local martingale under P̃T1∗ . Applying Lemma 2.1 to processes G = Z1 (·, Tk∗ ) and H =
δn−1 Z1 (·, T1∗ ), it is easy to see that for this property to hold, it suffices to assume that the process
∗
W̃ T1 , which is given by the formula
T1∗
W̃t
Z
∗
= W̃tT −
t
0
δn−1 Z1 (u, T1∗)
γ1 (u, T1∗ ) du,
1 + δn−1 Z1 (u, T1∗ )
t ∈ [0, T1∗], follows a Brownian motion under P̃T1∗ (the probability measure P̃T1∗ is yet unspecified,
but will be soon found through Girsanov’s theorem). Note that
B(t, T1∗ )
= κ̃(t, T1∗ ) + Z1 (t, T ∗ ) = κ̃(t, T1∗ ) + δn−1 .
δn B(t, T ∗ )
Z1 (t, T1∗ ) =
Differentiating both sides of the last equality, we get (cf. (68) and (69))
Z1 (t, T1∗ )γ1 (t, T1∗ ) = κ̃(t, T1∗ )ν(t, T1∗ ).
∗
Consequently, W̃ T1 is explicitly given by the formula
T1∗
W̃t
Z
∗
= W̃tT −
t
0
δn−1 κ̃(u, T1∗ )
ν(u, T1∗ ) du
1 + δn−1 δn−1 + δn−1 κ̃(u, T1∗ )
for t ∈ [0, T1∗ ]. We are in a position to define, using Girsanov’s theorem, the associated forward swap
measure P̃T1∗ . Subsequently, we introduce the process κ̃(·, T2∗ ), by postulating that it solves the SDE
T1∗
dκ̃(t, T2∗ ) = κ̃(t, T2∗ )ν(t, T2∗ ) · dW̃t
with the initial condition
κ̃(0, T2∗ ) =
B(0, T2∗ ) − B(0, T ∗ )
.
δn−1 B(0, T1∗ ) + δn B(0, T ∗ )
For the reader’s convenience, let us consider one more inductive step, in which we are looking for
κ̃(t, T3∗ ). We now consider processes
Z3 (t, Tk∗ ) =
Z2 (t, Tk∗ )
B(t, Tk∗ )
=
,
δn−2 B(t, T2∗ ) + δn−1 B(t, T1∗ ) + δn B(t, T ∗ )
1 + δn−2 Z2 (t, T2∗ )
so that
T2∗
W̃t
for t ∈
[0, T2∗].
T1∗
= W̃t
Z
−
0
t
δn−2 Z2 (u, T2∗ )
γ2 (u, T2∗ ) du
1 + δn−2 Z2 (u, T2∗)
It is useful to note that
Z2 (t, T2∗ ) =
B(t, T2∗ )
= κ̃(t, T2∗ ) + Z2 (t, T ∗ ),
δn−1 B(t, T1∗ ) + δn B(t, T ∗ )
where in turn
Z2 (t, T ∗ ) =
Z1 (t, T ∗ )
1 + δn−1 Z1 (t, T ∗ ) + δn−1 κ̃(t, T1∗ )
and the process Z1 (·, T ∗ ) is already known from the previous step (clearly, Z1 (·, T ∗ ) = 1/dn ). Differentiating the last equality, we may thus find the volatility of the process Z2 (·, T ∗ ), and consequently,
to define P̃T2∗ .
We now examine the general case. We proceed by induction with respect to m. Suppose that
∗
∗
), the forward swap measure P̃Tm−1
and the
we have found forward swap rates κ̃(·, T1∗ ), . . . , κ̃(·, Tm
∗
∗ , the
associated Brownian motion W̃ Tm−1 . Our aim is to determine the forward swap measure P̃Tm
30
M.Rutkowski
∗
∗
associated Brownian motion W̃ Tm , and the forward swap rate κ̃(·, Tm+1
). To this end, we postulate
that processes
B(t, Tk∗ )
B(t, Tk∗ )
=
∗ ) + · · · + δ B(t, T ∗ )
Gt (m + 1)
δn−m B(t, Tm
n
∗
Zm (t, Tk )
∗)
1 + δn−m Zm (t, Tm
Zm+1 (t, Tk∗ ) =
=
∗
∗ . In view of Lemma 2.1, applied to processes G = Zm (·, T ) and
follow local martingales under P̃Tm
k
∗
H = Zm (·, Tm ), it is clear that we may set
Z t
∗
∗
δn−m Zm (u, Tm
)
T∗
∗
γm (u, Tm
) du,
(70)
W̃t mδ = W̃tT −
∗
0 1 + δn−m Zm (u, Tm )
∗
]. Therefore it is sufficient to analyse the process
for t ∈ [0, Tm
∗
)=
Zm (t, Tm
∗
B(t, Tm
)
∗
= κ̃(t, Tm
) + Zm (t, T ∗ ).
∗
δn−m+1 B(t, Tm−1 ) + · · · + δn B(t, T ∗ )
To conclude, it is enough to notice that
Zm (t, T ∗ ) =
Zm−1 (t, T ∗ )
.
∗
1 + δn−m+1 Zm−1 (t, T ∗ ) + δn−m+1 κ̃(t, Tm−1
)
Indeed, from the preceding step, we know that the process Zm−1 (·, T ∗ ) is a (rational) function
∗
). Consequently, the process under the integral sign
of forward swap rates κ̃(·, T1∗ ), . . . , κ̃(·, Tm−1
∗
) and their
on the right-hand side of (70) can be expressed using the terms κ̃(·, T1∗ ), . . . , κ̃(·, Tm−1
volatilities (since the explicit formula is rather lengthy, it is not reported here). Having found the
∗
∗
∗ , we introduce the forward swap rate κ̃(·, T
process W̃ Tm and probability measure P̃Tm
m+1 ) through
(66)–(67), and so forth. If all volatilities are deterministic, the model is termed the lognormal model
of fixed-maturity forward swap rates.
3.3
Valuation of Swaptions
For a long time, Black’s swaptions formula was merely a (widely used) practical tool to value
swaptions. Indeed, the use of this formula was not supported by the existence of a reliable term
structure model. Valuation and hedging of swaptions based on the suitable version of Black’s formula
was analysed, for instance, in Neuberger (1990). The formal derivation of this heuristic results within
the framework of a well established term structure model was first achieved in Jamshidian (1997).
3.3.1
Payer and Receiver Swaptions
The owner of a payer (receiver, respectively) swaption with strike rate κ, maturing at time T = T0 ,
has the right to enter at time T the underlying forward payer (receiver, respectively) swap settled
in arrears.19 Because FS T (κ) is the value at time T of the payer swap with the fixed interest rate
κ, it is clear that the price of the payer swaption at time t equals
+ o
nB t
FS T (κ) Ft .
PS t = E P∗
BT
Using (59), we obtain
PS t = E
P∗
nB t
BT
E
P∗
n
X
+ o
BT
(L(Tj−1 ) − κ)δj FT
Ft .
B
Tj
j=1
(71)
19 By convention, the notional principal of the underlying swap (and thus also the notional principal of the swaption)
equals Np = 1.
31
Modelling of Forward Libor and Swap Rates
On the other hand, in view of (63) we also have
PS t = E P∗
nB t
BT
E P∗
n
X
+ o
BT
(κ(T, T, n) − κ)δj FT
Ft
B
Tj
j=1
(72)
The last equality yields
PS t
= E P∗
= E
P∗
= E P∗
= E P∗
n
nB X
+ o
BT
t
(κ(T, T, n) − κ)δj FT
E P∗
Ft
BT
B
T
j
j=1
nB
t
BT
E
P∗
n
nB X
t
BT
j=1
n
nB X
t
BT
n
X
o
BT
(κ(T, T, n) − κ)+ δj FT Ft
B
Tj
j=1
o
δj B(T, Tj )E PTj (κ(T, T, n) − κ)+ FT Ft
o
δj B(T, Tj )(κ(T, T, n) − κ)+ Ft
j=1
n
+ o
nB X
t
cj B(T, Tj ) Ft .
1−
= E P∗
BT
j=1
Similarly, for the receiver swaption, we have
RS t = E P∗
that is
RS t = E P∗
nB t
BT
+ o
− FS T (κ) Ft ,
n
nB X
+ o
BT
t
(κ − L(Tj−1 ))δj FT
E P∗
Ft ,
BT
BTj
j=1
(73)
where we write RS t to denote the price at time t of a receiver swaption. Consequently, reasoning
in much the same way as in the case of a payer swaption, we get
RS t
=
E P∗
=
E P∗
=
E P∗
n
nB X
+ o
BT
t
(κ − κ(T, T, n))δj FT
E P∗
Ft
BT
B
T
j
j=1
nB
t
BT
E P∗
n
X
o
BT
(κ − κ(T, T, n))+ δj FT Ft
B
Tj
j=1
n
n B X
t
BT
cj B(T, Tj ) − 1
+ o
Ft .
j=1
We shall first focus on a payer swaption. In view of (71), it is apparent that a payer swaption
is exercised at time T if and only if the value of the underlying swap is positive at this date. It
should be made clear that a swaption may be exercised by its owner only at its maturity date T. If
exercised, a swaption gives rise to a sequence of cash flows at prescribed future dates. By considering
the future cash flows from a swaption and from the corresponding market swap20 available at time
T, it is easily seen that the owner of a swaption is protected against the adverse movements of the
swap rate that may occur before time T. Suppose, for instance, that the swap rate at time T is
greater than κ. Then by combining the swaption with a market swap, the owner of a swaption with
exercise rate κ is entitled to enter at time T, at no additional cost, a swap contract in which the
20 At any time t, a market swap is that swap whose current value equals zero. Put more explicitly, it is the swap in
which the fixed rate κ equals the current swap rate.
32
M.Rutkowski
fixed rate is κ. If, on the contrary, the swap rate at time T is less than κ, the swaption is worthless,
but its owner is, of course, able to enter a market swap contract based on the current swap rate
κ(T, T, n) ≤ κ. Concluding, the fixed rate paid by the owner of a swaption who intends to initiate
a swap contract at time T will never be above the preassigned level κ.
Notice that we that we have shown, in particular, that
n
nB
X
o
BT
t
E P∗
(κ(T, T, n) − κ)+ δj FT Ft
(74)
PS t = E P∗
BT
B
Tj
j=1
This shows that a payer swaption is essentially equivalent a sequence of fixed payments dpj =
δj (κ(T, T, n) − κ)+ which are received at settlement dates T1 , . . . , Tn , but whose value is known
already at the expiry date T. In words, a payer swaption can be seen as a specific call option on
a forward swap rate, with fixed strike level κ. The exercise date of the option is T, but the payoff
takes place at each date T1 , . . . , Tn . This equivalence may also be derived by directly verifying that
the future cash flows from the following portfolios established at time T are identical: portfolio A –
a swaption and a market swap; and portfolio B – a just described call option on a swap rate and a
market swap. Indeed, both portfolios correspond to a payer swap with the fixed rate equal to κ.
Finally, equality
n
+ o
nB X
t
cj B(T, Tj ) Ft
1−
(75)
PS t = E P∗
BT
j=1
shows that the payer swaption may also be seen as a standard put option on a coupon-bearing bond
with the coupon rate κ, with exercise date T and strike price 1.
Similar remarks are valid for the receiver swaption. In particular, a receiver swaption can also be
viewed as a sequence of put options on a swap rate which are not allowed to be exercised separately.
At time T the long party receives the value of a sequence of cash flows, discounted from time
Tj , j = 1, . . . , n, to the date T, defined by δj (κ − κ(T, T, n))+ . On the other hand, a receiver
swaption may be seen as a call option, with strike price 1 and expiry date T, written on a coupon
bond with coupon rate equal to the strike rate κ of the underlying forward swap.
Let us finally mention the put-call parity relationship for swaptions. It follows easily from (71)–
(73) that PS t − RS t = FS t , i.e.,
Payer Swaption (t) − Receiver Swaption (t) = Forward Swap (t)
provided that both swaptions expire at the same date T (and have the same contractual features).
3.3.2
Forward Swaptions
Let us now consider a forward swaption. In this case, we assume that the expiry date T̂ of the
swaption precedes the initiation date T of the underlying payer swap – that is, T̂ ≤ T. Recall that
FS t (κ) =
n
X
κ(t, T, n) − κ δj B(t, Tj )
j=1
for t ∈ [0, T ]. It is thus clear that the payoff PS T̂ at expiry T̂ of the forward swaption (with strike
0) is either 0, if κ ≥ κ(T̂ , T, n), or
PS T̂ =
n
X
κ(T̂ , T, n) − κ δj B(T̂ , Tj )
j=1
if, on the contrary, inequality κ(T̂ , T, n) > κ holds. We conclude that the payoff PS T̂ of the forward
swaption can be represented in the following way
PS T̂ =
n
X
j=1
κ(T̂ , T, n) − κ
+
δj B(T̂ , Tj ).
(76)
33
Modelling of Forward Libor and Swap Rates
This means that, if exercised, the forward swaption gives rise to a payment δj (κ(T̂ , T, n) − κ) at
each settlement date Tj , j = 1, . . . , n. By substituting T̂ = T we recover, in a more intuitive way
and in a more general setting, the previously observed dual nature of the swaption: it may be seen
either as an option on the value of a particular (forward) swap or, equivalently, as an option on the
corresponding (forward) swap rate. It is also clear that the owner of a forward swaption is able to
enter at time T̂ (at no additional cost) into a forward payer swap with preassigned fixed interest
rate κ.
3.3.3
Valuation in the Lognormal Model of Forward Libor Rates
Recall that within the general framework, the price at time t ∈ [0, T0 ] of a payer swaption21 with
expiry date T = T0 and strike level κ equals
PS t = E P∗
n
n B X
+ o
BT
t
(L(Tj−1 ) − κ)δj FT
E P∗
Ft .
BT
B
Tj
j=1
Let D ∈ FT be the exercise set of a swaption; that is
D = {ω ∈ Ω | (κ(T, T, n) − κ)+ > 0} = {ω ∈ Ω |
n
X
cj B(T, Tj ) < 1}.
j=1
Lemma 3.1 The following equality holds for every t ∈ [0, T ]
PS t =
n
X
δj B(t, Tj ) E PTj (L(T, Tj−1 ) − κ) ID Ft .
(77)
j=1
Proof.
Since
PS t = E P∗
n
n B
X
o
BT
t
ID E P ∗
(L(Tj−1 ) − κ)δj FT Ft ,
BT
BTj
j=1
we have
PS t
n
n
X
o
Bt
= E P∗ E P∗
(L(Tj−1 ) − κ)δj ID FT Ft
B
T
j
j=1
=
n
X
B(t, Tj ) E PTj (L(Tj−1 ) − κ)δj ID Ft ,
j=1
where L(Tj−1 ) = L(Tj−1 , Tj−1 ). For any j = 1, . . . , n, we have
= E PTj E PTj L(Tj−1 ) − κ FT ID Ft
E PTj (L(Tj−1 ) − κ) ID Ft
= E PTj (L(T, Tj−1 ) − κ) ID Ft ,
since Ft ⊂ FT and the process L(t, Tj−1 ) is a PTj -martingale.
2
For any k = 1, . . . , n, we define the random variable ζk (t) by setting
Z
ζk (t) =
t
T
λ(u, Tk−1 ) · dWuTk ,
∀ t ∈ [0, T ],
(78)
21 Since the relationship PS − RS = FS is always valid, and the value of a forward swap is given by (60), it is
t
t
t
enough to examine the case of a payer swaption.
34
M.Rutkowski
and we write
λ2k (t) =
Z
T
|λ(u, Tk−1 )|2 du,
∀ t ∈ [0, T ].
(79)
t
Note that for every k = 1, . . . , n and t ∈ [0, T ], we have
2
L(T, Tk−1 ) = L(t, Tk−1 ) eζk (t)−λk (t)/2 .
Recall also that the processes W Tk satisfy the following relationship
Z t
δk+1 L(u, Tk )
Tk+1
Tk
λ(u, Tk ) du
= Wt +
Wt
1
+
δk+1 L(u, Tk )
0
for t ∈ [0, Tk ] and k = 0, . . . , n − 1. For ease of notation, we formulate the next result for t = 0
only; a general case can be treated along the same lines. For any fixed j, we denote by Gj the joint
probability distribution function of the n-dimensional random variable (ζ1 (0), . . . , ζn (0)) under the
forward measure PTj .
Proposition 3.1 Assume the lognormal model of Libor rates. The price at time 0 of a payer
swaption with expiry date T = T0 and strike level κ equals
Z n
X
2
δj B(0, Tj )
L(0, Tj−1 )eyj −λj (0)/2 − κ ID̃ dGj (y1 , . . . , yn ),
PS 0 =
Rn
j=1
where ID̃ = ID̃ (y1 , . . . , yn ), and D̃ stands for the set
j n
X
o
−1
n
Y
2
cj
<1 .
1 + δk L(0, Tk−1 ) eyk −λk (0)/2
D̃ = (y1 , . . . , yn ) ∈ Rn j=1
Proof.
k=1
Let us start by considering arbitrary t ∈ [0, T ]. Notice that
j
j
Y
Y
B(t, Tk )
B(t, Tj )
=
=
(FB (t, Tk−1 , Tk ))−1
B(t, T )
B(t, Tk−1 )
k=1
k=1
and thus, in view of (13), we have
B(T, Tj ) =
j Y
−1
.
1 + δk L(T, Tk−1 )
k=1
Consequently, the exercise set D can be re-expressed in terms of forward Libor rates. Indeed, we
have
j n
X
−1
o
n
Y
cj
<1 ,
1 + δk L(T, Tk−1 )
D = ω ∈ Ω
j=1
k=1
or more explicitly
j
n
X
o
−1
n
Y
2
cj
< 1 .
1 + δk L(t, Tk−1 ) eζk (t)−λk (t)/2
D = ω ∈ Ω
j=1
k=1
Let us put t = 0. In view of Lemma 3.1, to find the arbitrage price of a swaption at time 0,
it is sufficient to determine the joint law under the forward measure PTj of the random variable
(ζ1 (0), . . . , ζn (0)), where ζ1 (0), . . . , ζn (0) are given by (78). Note also that
j n
X
o
−1
n
Y
2
cj
< 1 .
1 + δk L(0, Tk−1 ) eζk (0)−λk (0)/2
D = ω ∈ Ω
j=1
k=1
This shows the validity of the valuation formula for t = 0. It is clear that it admits a rather
2
straightforward generalization to arbitrary 0 < t ≤ T.
35
Modelling of Forward Libor and Swap Rates
3.3.4
Market Valuation Formula for Swaptions
The commonly used formula for pricing swaptions, based on the assumption that the underlying
swap rate follows a geometric Brownian motion under the intuitively perceived “market probability”
Q, is given by Black’s swaption formula (cf. Neuberger (1990))
PS t =
n
X
B(t, Tj )δj κ(t, T, n)N h1 (t, T ) − κN h2 (t, T ) ,
(80)
j=1
where T = T0 is the swaption’s expiry date, and
h1,2 (t, T ) =
ln(κ(t, T, n)/κ) ± 12 σ 2 (T − t)
√
σ T −t
for some constant σ > 0. To examine formula (80) in an intuitive way, let us assume, for simplicity,
that t = 0. In this case, using general valuation results, we obtain the following equality
PS 0 =
n
X
δj B(0, Tj ) E PTj (κ(T, T, n) − κ)+ .
j=1
Apparently, market practitioners assume lognormal probability law for the swap rate κ(T, T, n)
under PTj . The swaption valuation formula obtained in the framework of the lognormal model of
Libor rates appears to be more involved. It reduces to the “market formula” (80) only in very
special circumstances. On the other hand, the swaption price derived within the lognormal model
of forward swap rates (see Section 3.2 below) agrees with the (80). More precisely, this holds for
a specific family of swaptions. This is by no means surprising, as the model was exactly tailored
to handle a particular family of swaptions, or rather, to analyze certain path-dependent swaptions
(such as Bermudan swaptions). The price of a cap in the lognormal model of swap rates is not given
by a closed-form expression, however.
3.3.5
Valuation in the Lognormal Model of Forward Swap Rates
For a fixed, but otherwise arbitrary, date Tj , j = 0, . . . , n − 1, we consider a swaption with expiry
date Tj , written on a forward payer swap settled in arrears. The underlying forward payer swap
starts at date Tj , has the fixed rate κ and n − j accrual periods. Such a swaption is referred to as
the j th swaption in what follows. Notice that the j th swaption can be seen as a contract which pays
to its owner the amount δk (κ(Tj , Tj , n − j) − κ)+ at each settlement date Tk , where k = j + 1, . . . , n
(recall that we assume that the notional principal Np = 1). Equivalently, the j th swaption pays an
amount
n
X
+
δk B(Tj , Tk ) κ̃(Tj , Tj ) − κ
Ỹ =
k=j+1
at maturity date Tj . It is useful to observe that Ỹ admits the following representation in terms of
the numeraire process G(n − j) introduced in Section 3.2 (cf. formula (65))
Ỹ = GTj (n − j) κ̃(Tj , Tj ) − κ
+
.
Recall that the model of fixed-maturity forward swap rates presented in Section 3.2 specifies the
dynamics of the process κ̃(·, Tj ) through the following SDE
Tj+1
dκ̃(t, Tj ) = κ̃(t, Tj )ν(t, Tj ) · dW̃t
,
where W̃ Tj+1 follows a standard d-dimensional Brownian motion under the corresponding forward
swap measure P̃Tj+1 . Recall that the definition of P̃Tj+1 implies that any process of the form
36
M.Rutkowski
B(t, Tk )/Gt (n − j), k = 0, . . . , n, is a local martingale under P̃Tj+1 . Furthermore, from the general considerations concerning the choice of a numeraire (see, e.g. Geman et al. (1995) or Musiela
and Rutkowski (1997a)) it is easy to see that the arbitrage price πt (X) of an attainable contingent
claim X = g(B(Tj , Tj+1 ), . . . , B(Tj , Tn )) equals, for t ∈ [0, Tj ],
G−1
πt (X) = Gt (n − j) E P̃T
Tj (n − j)X | Ft ,
j+1
provided that X settles at time Tj . Applying the last formula to the swaption’s payoff Ỹ , we obtain
the following representation for the arbitrage price PS jt at time t ∈ [0, Tj ] of the j th swaption
PS jt = πt (Ỹ ) = Gt (n − j) E P̃T
j+1
(κ̃(Tj , Tj ) − κ)+ | Ft .
We assume from now on that ν(·, Tj ) : [0, Tj ] → Rd is a bounded deterministic function. In other
words, we place ourselves within the framework of the lognormal model of fixed-maturity forward
swap rates. The proof of following result, due to Jamshidian (1996, 1997), is straightforward.
Proposition 3.2 For any j = 1, . . . , n − 1, the arbitrage price at time t ∈ [0, Tj ] of the j th swaption
equals
n
X
j
δk B(t, Tk ) κ̃(t, Tj )N h̃1 (t, Tj ) − κN h̃2 (t, Tj ) ,
PS t =
k=j+1
where N denotes the standard Gaussian cumulative distribution function, and
h̃1,2 (t, Tj ) =
with v 2 (t, Tj ) =
R Tj
t
ln(κ̃(t, Tj )/κ) ± 12 v 2 (t, Tj )
,
v(t, Tj )
|ν(u, Tj )|2 du.
Proof.
2
The proof of the proposition is quite similar to that of Proposition 2.2 and thus it is omitted.
3.3.6
Hedging of Swaptions
The replicating strategy for a swaption within the present framework has similar features as the
replicating strategy for a cap in the lognormal model of forward Libor rates. Therefore, we shall
focus mainly on differences between these two cases. Let us fix j, and let us denote by FS j (t, T ) the
relative price at time t ≤ Tj of the j th swaption, when the value process
Gt (n − j) =
n
X
δk B(t, Tk )
k=j+1
is chosen as a numeraire asset. From Proposition 3.2, we find easily that for every t ≤ Tj
FS j (t, Tj ) = κ̃(t, Tj )N h̃1 (t, Tj ) − κN h̃2 (t, Tj ) .
Applying Itô’s formula to the last expression, we obtain
dFS j (t, Tj ) = N h̃1 (t, Tj ) dκ̃(t, Tj ).
(81)
Let us consider the following self-financing trading strategy. We start our trade at time 0 with the
invested in the portfolio G(n − j).22 At any time
amount PS j0 of cash, which is then immediately
j
t ≤ Tj we assume ψt = N h1 (t, Tj ) positions in market forward swaps (of course, these swaps
Pn
22
One unit of portfolio G(n − j) costs
δ B(0, Tk )
k=j+1 k
at time 0.
37
Modelling of Forward Libor and Swap Rates
have the same starting date and tenor structure as the underlying forward swap). The associated
gains/losses process V, expressed in units of the numeraire asset G(n − j), satisfies
dVt = ψtj dκ̃(t, Tj ) = N h̃1 (t, Tj ) dκ̃(t, Tj ) = dFS j (t, Tj )
with V0 = 0. Consequently,
Z
FS j (Tj , Tj ) = FS j (0, Tj ) +
Tj
0
ψtj dκ̃(t, Tj ) = FS j (0, Tj ) + VTj .
Here the dynamic trading in market forward swaps takes place at any date t ∈ [0, Tj ], and all
gains/losses from trading (involving the initial investment) are expressed in units of G(n − j). The
last equality makes it clear that the strategy ψ j introduced above does indeed replicate the j th
swaption.
3.4
Choice of Numeraire Portfolio
Let us sumarize briefly the theoretic results which underpin the recent approaches to term structure
modelling. For the reader’s convenience, we shall restrict here our attention to the case of bond
portfolios.
Let us consider two particular portfolio of zero-coupon bonds, with value processes Vt1 and Vt2 .
Typically, we are interested in options to exchange one of this portfolios for another, at a given date
T. Let us write
(82)
CT = (VT1 − KVT2 ) = VT1 ID − KVT2 ID ,
where K > 0 is a constant, and D = {VT1 > KVT2 } is the exercise set. It is easy to check using the
abstract Bayes rule that the equality
V02 VT1
dP1
2 = V 1 V 2,
dP
0
T
P2 -a.s.,
(83)
links the martingale measures P1 and P2 associated with the choice of value processes V 1 and
V 2 as discount factors, respectively (both probability measures are considered here on (Ω, FT )).
Furthermore, the arbitrage price of the option admits the following representation
Ct = Vt1 P1 (D | Ft ) − KVt2 P2 (D | Ft ),
∀ t ∈ [0, T ],
(84)
where D = {VT1 > KVT2 }. To obtain the Black-Scholes like formula for the option’s price Ct , it is
enough to assume that the the relative price V 1 /V 2 follows a lognormal martingale under P2 , so
that
(85)
d (Vt1 /Vt2 ) = (Vt1 /Vt2 )γt1,2 · dWt1,2
for a deterministic function γ 1,2 : [0, T ] → Rd (for simplicity, we also assume that the function γ 1,2
is bounded). In view of (83), the Radon-Nikodým density of P1 with respect to P2 equals
Z ·
dP1
γu1,2 · dWu1,2 , P2 -a.s.,
(86)
2 = ET
dP
0
and thus the process
Wt2,1
=
Wt1,2
Z
−
t
0
γu1,2 du,
∀ t ∈ [0, T ],
is a standard Brownian motion under P2 . Reasoning in the much the same way as in the proof
of the classic Black-Scholes formula (see, for instance, the proof of Theorem 5.1.1 in Musiela and
Rutkowski (1997a)), we obtain
(87)
Ct = Vt1 N d1 (t, T ) − KVt2 N d2 (t, T ) ,
38
M.Rutkowski
where
d1,2 (t, T ) =
ln (Vt1 /Vt2 ) − ln K ±
v1,2 (t, T )
and
2
(t, T ) =
v1,2
Z
T
t
|γu1,2 |2 du,
1
2
2
v1,2
(t, T )
∀ t ∈ [0, T ].
Of course, the caps and swaptions valuation formulae in lognormal models described above can be
seen as special cases of (87). For the j th caplet, we take
Vt1 = B(t, Tj ) − B(t, Tj+1 ),
Vt2 = δj+1 B(t, Tj+1 ).
In the case of the j th swaption, we have
Vt1 = B(t, Tj ) − B(t, Tn ),
Vt2 =
n
X
δk B(t, Tk ).
k=j+1
The idea can be, of course, applied to other interest rate derivatives.
It is worthwhile to notice that in order to get the valuation result (87) for t = 0, it is enough
to assume that the random variable VT1 /VT2 has a lognormal probability law under the martingale
measure P2 . This simple observation underpins the construction of the so-called Markov-functional
interest rate models – this alternative approach to term structure modelling is briefly reviewed in
the next section.
A more straightforward generalization of lognormal models of the term structure was developed
by Andersen and Andreasen (1997). In this case, the assumption that the volatility is deterministic
is replaced by a suitable functional form of the volatility. The resulting models are capable to handle
the so-called volatility skew in observed option prices (empirical studies have shown that the implied
volatilities of observed caps and swaptions prices tend to be decreasing functions of the strike level).
The main focus in Andersen and Andreasen (1997) is on the use of the CEV process23 as a model
of the forward Libor rate. Put more explicitly, they generalize equality (21) by postulating that
Tj+1
dL(t, Tj ) = Lα (t, Tj ) λ(t, Tj ) · dWt
,
∀ t ∈ [0, Tj ],
where α > 0 is a strictly positive constant. They derive closed-form solutions for caplet prices
under the above specification of the dynamics of Libor rates with α 6= 1, in terms of the cumulative
distribution function of a non-central χ2 probability law. It appears that depending on the choice
of the parameter α, the implied Black’s volatilities of caplet prices, considered as a function of the
strike level κ > 0, exhibit downward- or upward-sloping skew.
4
Markov-Functional Models
As shown in Section 2.2.4, the forward Libor or swap24 rates follow a multidimensional Markov
process under any of the associated forward measures. In principle, lognormal models can be easily
calibrated to market prices of caps (or swaptions), which is, of course, a nice feature of this class of
term structure models, as opposed to the classic models based on the specification of the dynamics of
(spot or forward) instantaneous rates. On the other hand, however, due to the high dimensionality
of the underlying Markov process, the efficient implementation of these models appears to be rather
difficult.
23 In the context of equity options, the CEV (constant elasticity of variance) process was first introduced in Cox
and Ross (1976).
24 The multidimensional SDE which governs the dynamics of the family of forward swap rates is more involved than
the SDE for the family of Libor rates, and thus it is not reported here. The interested reader is referred to Jamshidian
(1997).
Modelling of Forward Libor and Swap Rates
39
To circumvent this obstacle, an alternative approach was recently developed in a series of papers
by Hunt and Kennedy (1997, 1998) and Hunt et al. (1996, 1997).25 It is based on the introduction
of a low-dimensional Markov process which (by assumption) governs, through a simple functional
dependence, the dynamics of all other relevant stochastic processes. For this reason, these class
of term structure models is referred to as Markov-functional interest rate models. In economical
interpretation, the underlying Markov process is assumed to represent the state of the economy; it
is thus justified to refer to its component as “state variables.”
Formally, one starts by introducing a one- or multi-dimensional process M, which possesses the
Markov property under the terminal measure, where the generic term terminal measure is intended to
cover not only cases considered in previous sections, but also other suitable choices of the numeraire
portfolio. As already mentioned, the relevant processes, such as n particular the value process of the
numeraire portfolio and zero-coupon bond prices, are assumed to be functions of M. For instance,
if T ∗ > 0 is the horizon date, than for any t ≤ s ≤ T we have
B(s, T, M ) B(t, T, Mt )
s = E P̂
Ft
Vt (Mt )
Vs (Ms )
where Vt (Mt ), t ≤ T ∗ , is the value process of the numeraire portfolio, and P̂ is the associated
martingale measure. The notation B(t, T, Mt ) emphasizes the direct dependence of the bond price
on time variables, t and T, as well as on the state variable represented by the random variable
Mt . Note that the functional from B(t, T, Mt ) is not explicitly known, except for some very special
choices of dates t and T. In some instances, if may appear convenient to postulate that26
B(T, S, MT )
= A + B(S)MT
VT (MT )
and to derive further properties from the martingale feature of relative prices. In the next section,
we shall present a particular example of such an approach, in which we focus on the derivation of
a simple formula for the so-called convexity correction. Then, in Section 4.2, we shall discuss the
problem of calibration of the Markov-functional model.
4.1
Terminal Swap Rate Model
The terminal swap rate model – put forward by Hunt et al. (1996) – was primarily designed for
the purpose of the comparative pricing of non-standard swap contracts vis-a-vis plain vanilla swaps
(informally, this is referred to as convexity correction; cf. Schmidt (1996)). Let us consider, as usual,
a given collection of reset/settlements dates T0 , . . . , Tn . We assume that the market price at time
0 of the (plain vanilla) fixed-for-floating swaption is known. We postulate, in addition, that it is
given by Black’s formula for swaptions. Let us consider the family of bond prices B(T, S), where
the maturity date S ≥ T belongs to some set S of dates. We postulate that there exist constants A
and BS such that for any S ∈ S
where Gt (n) =
Pn
D(T, S) := B(T, S)G−1
T (n) = A + BS κ(T, T, n),
j=1 δj B(t, Tj ),
κ(t, T, n) =
(88)
and (cf. (64))
B(t, T ) − B(t, Tn )
B(t, T ) − B(t, Tn )
=
.
δ1 B(t, T1 ) + · · · + δn B(t, Tn )
Gt (n)
Using the martingale property of discounted bond price D(·, S) and forward swap rate κ(·, T, n)
under the corresponding forward swap measure associated with the choice of G(n) as a numeraire,
we get
D(t, S) = A + BS κ(t, T, n),
25 We present here only few examples of their approach. The interested reader is referred to the original papers for
a more detailed account.
26 See Hunt et al. (1996) for alternative kinds of the functional dependence, including exponential and geometric.
40
M.Rutkowski
or equivalently
B(t, S) = A(1 − B(t, Tn )) + BS Gt (n)
for every t ∈ [0, T ]. We thus see that condition (88) is rather stringent; it implies that the price of
any bond of maturity S from S can by represented as a linear combination of values of two particular
portfolios of bonds, with one coefficient independent of maturity date S. The problem whether such
an assumption can be supported by an arbitrage-free model of the term structure is not addressed
in Hunt et al. (1996).
Let us now focus on the derivation of values of constants A and BS . To this end, we assume that
equality (88) holds, in particular, for any S = Tj , j = 1, . . . , n. Then
A
n
X
δj +
j=1
n
X
δj BTj κ(T, T, n) = A(Tn − T0 ) +
j=1
n
X
δj BTj κ(T, T, n) = 1,
j=1
and thus
A = (Tn − T0 )−1 ,
n
X
δj BTj = 0.
(89)
j=1
Consequently, using the first equality above and the martingale property of D(·, S) and κ(·, T, n),
we obtain
−1
+ BS κ(0, T, n),
(90)
B(0, S)G−1
0 (n) = (Tn − T0 )
so that for each maturity in question the constant BS is also uniquely determined. Notice that the
second equality in (89) is also satisfied for this choice of BS .
Hunt and Kennedy (1997) argue that under (88) the problem of pricing irregular cashflows
becomes relatively easy to handle. To illustrate this point, assume that we wish to value the claim
X which settles at time T and admits the following representation
X=
m
X
ci B(T, Si )F,
i=1
where ci ’s are constants, and Si ∈ S for i = 1, . . . , m. We
assume that the FT -measurable random
variable F has the form F = F̃ B(T, S1 ), . . . , B(T, Sm ) for some function F̃ : Rm
+ → R. To be in
line with the notation introduced in Section 3.4, we denote
Vt1 = B(t, T ) − B(t, Tn ),
Vt2 =
n
X
δj B(t, Tj ) = Gt (n).
j=1
Using (88) and (89)–(90), we obtain
X=
m
X
ci A(1 − B(T, Tn )) + BSi GT (n) F = w1 VT1 F + w2 VT2 F,
i=1
where
w1 =
m
X
ci A,
w2 =
i=1
m
X
ci BSi .
i=1
In view of the discussion in Section 3.4, it is clear that
πt (X) = w1 Vt1 E P1 (F | Ft ) + w2 Vt2 E P2 (F | Ft ).
(91)
Under the assumption that the forward rate κ(·, T, n) follows a geometric Brownian motion under
the forward swap measure P2 , it follows also a lognormally distributed process under P1 (see the
discussion in Section 3.4). Consequently, under (88), the joint (conditional) probability law of
41
Modelling of Forward Libor and Swap Rates
random variables B(T, S1 ), . . . , B(T, Sm ) under probability measures P1 and P2 are explicitly known.
We conclude that the conditional expectations in (91) can be, in principle, evaluated.
Consider, for instance, a fixed-for-floating constant maturity swap.27 To value one leg of the
floating side of a constant maturity swap, consider a cashflow proportional to κ(T, T, n), which
takes place at some date M > T. Ignoring the constant, such a payoff is equivalent to the claim
X = B(T, M )κ(T, T, n) which settles at time T. Using (91), we obtain
πt (X) = BM Vt1 E P1 (κ(T, T, n) | Ft ) + AVt2 E P2 (κ(T, T, n) | Ft ).
Consequently, at time 0 we have
π0 (X) = BM (B(0, T ) − B(0, Tn ))κ(0, T, n)eσ
2
T
+ AG0 (n)κ(0, T, n),
where σ is the implied volatlity of the traded swaption with maturity date T. Using the formula for
BM , we get
2
π0 (X) = B(0, M ) − AG0 (n) κ(0, T, n)eσ T + AG0 (n)κ(0, T, n),
or finally
π0 (X) = B(0, M )κ(0, T, n) 1 + (1 − w)eσ
2
T
,
(92)
−1
where we write w = AG0 (n)B (0, M ). It should be stressed that the simple valuation result (92)
hinges on the strong assumption (88).
4.2
Calibration of Markov-Functional Models
The most important feature of Markov-functional models is the fact that their calibration to market
prices of plain vanilla derivatives is relatively easy to perform. For convenience, we shall focus here
on the calibration of the Markov-functional model of fixed-maturity forward swap rates. The case
of forward Libor rates can be dealt with in an analogous way. A more extensive discussion of this
issue can be found in Hunt et al. (1997).
First, we assume that the forward swap rate for the date Tn−1 follows a lognormal martingale under the corresponding forward measure PTn . More specifically, we postulate that the proces
κ̃(·, Tn−1 ) = κ(·, Tn−1 , 1) satisfies
dκ̃(t, Tn−1 ) = κ̃(t, Tn−1 )ν(t, Tn−1 ) dWt ,
(93)
where W is a Brownian motion under PTn and ν(·, Tn−1 ) is a strictly positive deterministic function.
If we take the process
Z t
ν(u, Tn−1 ) dWu
Mt =
0
as the driving Markov process for our model, then clearly
κ̃(Tn−1 , Tn−1 ) = κ̃(0, Tn−1 ) e
and
1
MTn−1 − 2
R Tn−1
0
ν 2 (u,Tn−1 ) du
−1
1 R Tn−1 2
M
−
ν (u,Tn−1 ) du
.
B(Tn−1 , Tn , MTn−1 ) = 1 + δn κ̃(0, Tn−1 ) e Tn−1 2 0
(94)
(95)
Suppose that we are given (digital) swaptions prices for all strikes κ > 0 and all expiration dates
T0 , . . . , Tn−1 . Our goal is to find the joint probability law of (κ̃(T0 , T0 ), . . . , κ̃(Tn−1 , Tn−1 )) under
PTn . This can be achieved by deriving the functional dependence of each rate κ̃(Tj , Tj ) on the
underlying Markov process; more specifically, we search for the function hj : R+ → R+ such that
27 Similarly as in the case of a plain-vanilla fixed-for-floating swap, in a constant maturity swap the fixed and floating
payments occur at regularly spaced dates. The amounts of floating payments are based not on a Libor rate, but on
some other swap rate, however.
42
M.Rutkowski
κ̃(Tj , Tj ) = hj (MTj ). To this end, we assume that for any j = 0, . . . , n − 1 there exists a strictly
increasing function hj such that this holds (in view of (94), this statement is valid for j = n − 1).
By the definition of the probability measure PTn , for i = j + 1, . . . , n
B(T , T ) B(T , T ) B(Tj , Ti )
i
i i
i = E PTn
FTi = E PTn
MTj
B(Tj , Tn )
B(Ti , Tn )
B(Ti , Tn )
since FTi = FTWi = FTMi . Therefore, if B(Ti , Tn ) = B(Ti , Tn , MTi ) we obtain
1
B(Tj , Ti )
= E PTn
MTj ,
B(Tj , Tn )
B(Ti , Tn , MTi )
so that the right-hand side in the formula above is a function of MTj . Consequently, for
GTj (n − j) =
n
X
δi B(Tj , Ti )
i=j+1
we get
n
X
GTj (n − j)
δi
=
E PTn
MTj = gj (MTj ),
B(Tj , Tn )
B(Ti , Tn , MTi )
i=j+1
(96)
where gj : R → R is a measurable function with strictly positive values. The right-hand side in (96)
can be evaluated using the transition p.d.f. pM (t, m; u, x) of the Markov process M, provided that
the functional form of B(Ti , Tn , MTi ) is known for every i = j + 1, . . . , n. To put it more explicitly
Z
n
X
δi pM (Tj , m; Ti , x)
dx.
gj (m) =
B(Ti , Tn , x)
i=j+1 R
(97)
We work back iteratively from the last relevant date Tn−1 . In the first step, i.e., when j = n − 2,
the functional form of B(Tn−1 , Tn , MTn−1 ) given by (95). Assume now that the functional forms of
B(Ti , Tn , MTi ) were already found for i = j + 1, . . . , n − 1. In order to determine B(Tj , Tn , MTj ), it
is enough to find the functional form of the swap rate κ̃(Tj , Tj ). Indeed, we have
κ̃(Tj , Tj ) =
and thus
B −1 (Tj , Tn ) = 1 + κ̃(Tj , Tj )
1 − B(Tj , Tn )
GTj (n − j)
GTj (n − j)
= 1 + hj (MTj )gj (MTj ).
B(Tj , Tn )
(98)
Our next goal is to show how to find the function hj , under the assumption that the functional
forms of bonds prices B(Ti , Tn , MTi ) are known for every i = j +1, . . . , n. To this end, we assume that
we are given all market prices of digital swaptions with expiration date Tj and any strictly positive
strike level κ. We find it convenient to represent the price at time 0 of the j th digital swaption, with
strike κ and expiration date Tj , in the following way28
DS j0 (κ) = B(0, Tn ) E PTn
G (n − j)
Tj
I {κ̃(Tj ,Tj )>κ}
B(Tj , Tn )
for j = 0, . . . , n − 2. Under the present assumptions, we obtain
DS j0 (κ) = B(0, Tn ) E PTn gj (MTj ) I {hj (MTj )>κ} ,
28 By definition, the j th digital swaption, with unit notional principal, pays the amount δ at time T for i =
i
i
j + 1, . . . , n whenever inequality κ̃(Tj , Tj ) > κ holds.
43
Modelling of Forward Libor and Swap Rates
or equivalently,
DS j0 (κ) = B(0, Tn ) E PTn gj (MTj ) I {MT
>h−1
(κ)}
j
j
.
Finally, if we denote by fM (x) = pM (0, 0; Tj , x) the p.d.f. of MTj under PTn then
Z
gj (x) I {x>h̃j (κ)} fM (x) dx,
DS j0 (κ) = B(0, Tn )
(99)
R
29
DS j0 : R+ → R+ is strictly
where we write ĥj = h−1
j . It is natural to assume that the function
decreasing as a function of the strike level κ, with
DS j0 (0) =
n
X
δi B(0, Ti ) = G0 (n − j)
i=j+1
and DS j0 (+∞) = 0. Since
E PTn gj (MTj ) = G0 (n − j)B −1 (0, Tn )
it can be deduced from (99) that ĥj (0) = −∞. On the other hand, condition DS j0 (+∞) = 0 implies
that ĥj (+∞) = +∞. Finally, the function ĥj implicitly defined through equality (99) is strictly
increasing, so that it admits an inverse function hj with desired properties. To wit, for hj = ĥ−1
j
we have: hj : R → R+ is strictly increasing, with hj (−∞) = 0 and hj (+∞) = +∞. This shows that
the procedure above leads to a reasonable specification of the functional form κ̃(Tj , Tj ) = hj (MTj ).
For the reader’s convenience, we shall recapitulate the main steps of the calibration procedure. In the first step, we numerically find the function hn−2 which expresses κ̃(Tn−2 , Tn−2 ) in
terms of MTn−2 . To this end, we need first to evaluate the function gn−2 using formula (97) with
B(Tn , Tn , x) = 1 and B(Tn−1 , Tn , x) given by (95)
In the second step, we first determine B(Tn−2 , Tn , x) using relationship (98), that is,
B −1 (Tn−2 , Tn , x) = 1 + hn−2 (x)gn−2 (x).
Then, we find gn−3 using (97), and subsequently we determine the rate κ̃(Tn−3 , Tn−3 ), or rather the
corresponding function hn−3 .
Continuing this procedure, we end up with the following representation of the finite family of
swap rates
(κ̃(T0 , T0 ), . . . , κ̃(Tn−1 , Tn−1 ) = g0 (MT0 ), . . . , gn−1 (MTn−1 ) .
This representation uniquely specifies the probability law of the considered family of swap rates
under the terminal forward measure PTn .
Remarks. In view of (93), the price at time t ≤ Tn−1 of the (n − 1)th digital swaption equals
(κ) = δn B(t, Tn ) PTn {κ̃(Tn−1 , Tn−1 ) > κ | Ft }
DS n−1
t
that is,
(κ) = δn B(t, Tn )N h̃2 (t, Tn−1 )
DS n−1
t
(100)
where N denotes the standard Gaussian cumulative distribution function, and the coefficient h̃2 is
given in the formulation of Proposition 3.2. Needless to say that formula (100) is not valid in the
present setup, even for t = 0, for any digital swaption with maturity T0 , . . . , Tn−2 . Moreover, it is
clear that assumption (93) is not necessary; we need only assume that the functional form of the
swap rate κ̃(Tn−1 , Tn−1 ) with respect to some underlying Markov process M is explicitly known
(and is a monotone function of MTn−1 ).
29 Recall that the function DS j represents the observed market prices of digital swaptions. Therefore, the foregoing
0
assumptions about the behaviour of this function are indeed quite natural.
44
M.Rutkowski
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