Modelling of Forward Libor and Swap Rates

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and Swap Rates

Marek Rutkowski

Faculty of Mathematics and Information Science

Warsaw University of Technology, 00-661 Warszawa, Poland

1 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

f (t, T ) = f (0, T ) + Z _t

0 α(u, T ) du +

Z _t

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

B(t, T ) = exp − Z _T t f (t, u) du .

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 ) r_tdt + b(t, T )· dW_t∗,

where r_t= f (t, t) is the short-term interest rate, and the bond price volatility b(t, T ) satisfies

b(t, T ) =− Z T

t

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Furthermore, it appears that in the special case when the coefficient σ follows a deterministic func-tion, 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 ) = B(t, T ) B(t, T + δ), or equivalently,

L(t, T ) = 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.

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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), Schl¨ogl (1999), Yasuoka (1999), Lotz and Schl¨ogl (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 P_T_j to denote the forward martingale measure associated with the date T_j. The correspond-ing d-dimensional Brownian motions are denoted by W∗ and WTj_{, respectively. Also, we write}

F_B(t, T, U ) = B(t, T )/B(t, U ) so that

FB(t, Tj+1, Tj) =

B(t, Tj+1)

B(t, T_j) , ∀ t ∈ [0, Tj],

is the forward price at time t of the T_j+1-maturity zero-coupon bond for the settlement date T_j. We use the symbol π_t(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 Dol´eans exponential, for instance,

Et Z · 0 γu· dW ∗ u = exp Z t 0 γu· dW ∗ u − 1 2 Z t 0 |γu| 2_du_,

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+1κ and receives B−1(Tj, Tj+1)− 1 at time Tj+1.

Equivalently, he pays an amount Y1= 1 + δj+1κ and receives Y2= B−1(Tj, Tj+1) at this date. The

values at time t≤ T_j of these payoffs are

πt(Y1) = B(t, Tj+1) 1 + δj+1κ

, πt(Y2) = B(t, Tj).

The second equality above is trivial, since the payoff Y₂ is equivalent to the unit payoff at time T_j. Consequently, for any fixed t≤ T_j, the value of the forward swap rate, which makes the contract

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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+1κ

= 0. It is thus apparent that

κ(t, Tj, Tj+1) =

B(t, Tj)− B(t, Tj+1)

δj+1B(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+1L(t, Tj) =

B(t, Tj)

B(t, Tj+1)

=E_P_Tj+1(B−1(Tj, Tj+1)| Ft) (2)

for every t∈ [0, T_j]. Let us notice that the last equality is a consequence of the definition of the forward measurePTj+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)EPTj+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 P_T_j+1. Recall that in the Heath-Jarrow-Morton framework, we have, under PTj+1,

dF_B(t, T_j, T_j+1) = F_B(t, T_j, T_j+1) b(t, T_j)− b(t, T_j+1)· dW_tTj+1, (3) 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

dL(t, T_j) = L(t, T_j)λ(t, T_j)· dW_tTj+1,

where λ(·, T_j) is an adapted stochastic process which satifies mild integrability conditions. Combin-ing the last two formulae with (2), we arrive at the followCombin-ing fundamental relationship, which plays an essential role in the construction of the lognormal model of forward Libor rates,

δj+1L(t, Tj)

1 + δ_j+1L(t, T_j)λ(t, Tj) = b(t, Tj)− b(t, Tj+1), ∀ 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 T_j, provided thatP_T_j+1, WTj+1 _{and the volatility}

λ(t, T_j) of the forward Libor rate L(·, T_j−1) are known. One may assume, for instance, that λ(·, T_j) is a prespecified deterministic function). Recall that in the Heath-Jarrow-Morton framework3 the Radon-Nikod´ym density ofPTj with respect toPTj+1 is known to satisfy

dP_T_j dPTj+1 =E_T_j Z · 0 b(t, T_j)− b(t, T_j+1)· dW_tTj+1 . (5)

In view of (4), we thus have dPTj dPTj+1 =ETj Z · 0 δj+1L(t, Tj) 1 + δj+1L(t, Tj) λ(t, Tj)· dWtTj+1 .

2_{This representation is a consequence of the martingale representation property of the standard Brownian motion.} 3_{See Heath et al. (1992).}

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For our further purposes, it is also useful to observe that this density admits the following represen-tation dPTj dP_T_j+1 = cFB(Tj, Tj, Tj+1) = c 1 + δj+1L(Tj, Tj) , PTj+1-a.s., (6)

where c > 0 is the normalizing constant, and thus dPTj

dPTj+1_|F_t

= cFB(t, Tj, Tj+1) = c 1 + δj+1L(t, Tj)

, PTj+1-a.s.

Finally, the dynamics of the process L(·, Tj) under the probability measure PTj are given by a

somewhat involved stochastic differential equation

dL(t, T_j) = L(t, T_j) _δ j+1L(t, Tj)|λ(t, Tj)|2 1 + δj+1L(t, Tj) dt + λ(t, T_j)· dW_tTj .

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 T_j. 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+1κ 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+1κ

, πt(Y2) = B(t, Tj)EPTj(B−1(Tj, Tj+1)| Ft).

The value κ = ˆκ(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) EP_Tj(B−1(Tj, Tj+1)| Ft)− (1 + δj+1κ) = 0. It is clear that ˆ κ(t, Tj, Tj+1) = δj+1−1 EPTj(B−1(Tj, Tj+1)| Ft)− 1 .

We are in a position to introduce the modified forward Libor rate ˜L(t, Tj) by setting, for every

t∈ [0, Tj],

˜

L(t, Tj) := δj+1−1 EPTj(B−1(Tj, Tj+1)| Ft)− 1

.

Let us make two remarks. First, it is clear that finding of the modified forward Libor rate ˜L(·, 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

˜ L(t, T_j) =E_P_Tj ₁_{− B(T} j, Tj+1) δj+1B(Tj, Tj+1) Ft =E_P_Tj(L(T_j, T_j)| F_t). (7)

In particular, it is evident that at the reset date Tj the two kinds of forward Libor rates introduced

above coincide, since manifestly ˜

L(Tj, Tj) =

1− B(Tj, Tj+1)

δj+1B(Tj, Tj+1)

= L(Tj, Tj).

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To summarize, the ‘standard’ forward Libor rate L(·, Tj) satisfies

L(t, Tj) =EPTj+1(L(Tj, Tj)| Ft), ∀ t ∈ [0, Tj],

with the initial condition

L(0, Tj) =

B(0, T_j)− B(0, T_j+1) δj+1B(0, Tj+1)

. On the other hand, for the modified Libor rate ˜L(·, Tj) we have

˜

L(t, Tj) =EP_Tj( ˜L(Tj, Tj)| Ft), ∀ t ∈ [0, Tj],

with the initial condition ˜

L(0, Tj) = δj+1−1 EPTj(B−1(Tj, Tj+1))− 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(T_j, T_j) := 1− δ_j+1L(T_j, T_j).

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

E(t, Tj) :=EP∗(E(Tj, Tj)) = 2− EP∗ B−1(Tj, Tj+1)| Ft

(8) (recall thatP∗ 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, T}

j) satisfies

E(t, Tj) = 1− δj+1Lf(t, Tj), ∀ t ∈ [0, Tj]. (9)

It follows immediately from (8)–(9) that the following equality is valid 1 + δj+1Lf(t, Tj) =EP∗ B−1(Tj, Tj+1)| Ft

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Equivalently, we have

Lf(t, T_j) =E_P∗(L(T_j, T_j)| F_t) =E_P∗( ˜L(T_j, T_j)| F_t).

Note that in any term structure model, the futures Libor rate necessarily follows a martingale under the spot martingale measureP∗ (provided, of course, thatP∗ 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 continuous-and discrete-tenor setups.

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2.2.1 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 )· dW_t∗,

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 )· dW_tT.

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

∂v ∂t + 1 2|λ(t, T )| 2_x2₍₁_{− x)}2∂2v ∂x2 = 0 (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-G¸atarek-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

1 + δL(t, T ) = F_B(t, T, T + δ), ∀ t ∈ [0, T ]. (12) 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

1 + δL(t, T ) = B(t, T )

B(t, T + δ), ∀ t ∈ [0, T ]. (13) In particular, the initial term structure of forward Libor rates satisfies

L(0, T ) = δ−1

_{B(0, T )} B(0, T + δ) − 1

. (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−1FB(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}

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where PT +δ is the forward measure for the date T + δ, and the asociated Wiener process WT +δ

equals

W_tT +δ= W_t∗− Z _t

0 b(u, T + δ) du, ∀ t ∈ [0, T + δ].

Put another way, the process L(·, T ) solves the equation

dL(t, T ) = δ−1(1 + δL(t, T )) γ(t, T, T + δ)· dW_tT +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 )· dW_tT +δ, (16) where for any t∈ [0, T ]

λ(t, T ) =1 + δL(t, T )

δL(t, T ) γ(t, T, T + δ). (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],

L(0, T ) = B(0, T )− B(0, T + δ)

δB(0, T + δ) . (18)

To construct a model satisfying (LR.1)–(LR.2), Brace et al. (1997) place themselves in the Heath-Jarrow-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 measureP∗ are governed by the following SDE

dL(t, T ) = µ(t, T ) dt + L(t, T )λ(t, T )· dW_t∗,

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 )· dW_t∗,

where σ∗(t, T ) =R_tTσ(t, u) du =−b(t, T ). On the other hand, the relationship (cf. (13))

1 + δL(t, T ) = exp Z T +δ T f (t, u) du (19)

is valid. Applying Itˆo’s formula to both sides of (19), and comparing the diffusion terms, we find that σ∗(t, T + δ)− σ∗(t, T ) = Z T +δ T σ(t, u) du = δL(t, T ) 1 + δL(t, T )λ(t, T ).

6_Volatility_{λ could well follow an adapted stochastic process; we deliberately focus here on a lognormal model of}

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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

b(t, T ) =−σ∗(t, T ) =− [δ−1_{(T −t)]} X k=1 δL(t, T− kδ) 1 + δL(t, T− kδ)λ(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 ) · dW_t∗.

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

convention, T−1 = 0). Let us denote δj = Tj− Tj−1 for j = 0, . . . , n. Then obviously Tj =

Pj i=0δi

for every j = 0, . . . , n. We find it convenient to denote, for m = 0, . . . , n,

T_m∗ = T∗−

n

X

j=n−m+1

δj= Tn−m.

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+1B(t, Tj+1)

, ∀ t ∈ [0, Tj].

Definition 2.2 For any j = 0, . . . , n, a probability measureP_T_j on (Ω,F_T_j), equivalent toP, is said to be the forward Libor measure for the date T_j if, for every k = 0, . . . , n the relative bond price

Un−j+1(t, Tk) :=

B(t, T_k) B(t, Tj)

, ∀ t ∈ [0, Tk∧ Tj],

follows a local martingale underPTj.

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 deterministicRd-valued function of time, anRd-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

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Our aim is to construct a family L(·, Tj), j = 0, . . . , n− 1 of forward Libor rates, a collection

of mutually equivalent probability measuresPTj, j = 1, . . . , n, and a family WTj, j = 1, . . . , n of

processes in such a way that: (i) for any j = 1, . . . , n the process WTj _{follows a d-dimensional}

standard Brownian motion under the probability measurePTj, (ii) for any j = 0, . . . , n− 1, the

forward Libor rate L(·, Tj) satisfies the SDE

dL(t, T_j) = L(t, T_j) λ(t, T_j)· dW_tTj+1, ∀ t ∈ [0, T_j], (21) with the initial condition

L(0, Tj) =

B(0, Tj)− B(0, Tj+1)

δ_j+1B(0, T_j+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 measureP by the following SDE7 dL(t, T₁∗) = L(t, T₁∗) λ(t, T₁∗)· dWt

L(0, T₁∗) =B(0, T

∗

1)− B(0, T∗) δnB(0, T∗)

. Put another way, we have

L(t, T₁∗) = B(0, T ∗ 1)− B(0, T∗) δ_nB(0, T∗) Et Z · 0 λ(u, T ∗ 1)· dWu .

Since B(0, T₁∗) > B(0, T∗), it is clear that the L(·, T₁∗) follows a strictly positive martingale under 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 measureQ, which is equivalent to P, and such that processes

U₂(t, T_k∗) = B(t, T

∗ k)

B(t, T₁∗)

areQ-local martingales. It is important to observe that the process U2(·, T_k∗) admits the following representation

U2(t, T_k∗) = U1(t, T

∗ k)

1 + δnL(t, T1∗).

Let us formulate an auxiliary result, which is a straightforward consequence of Itˆo’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 H_t>−1 for every t and denote Y_t= (1 + H_t)−1. Then d(YtGt) = Yt αt− YtGtβt

· dWt− Ytβtdt

. It follows immediately from Lemma 2.1 that

dU₂(t, T_k∗) = η_tk· dWt−

δnL(t, T1∗)

1 + δnL(t, T₁∗)

λ(t, T₁∗) dt

7_{Notice that, for simplicity, we have chosen the underlying probability measure} _P_{to play the role of the forward}

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for a certain process ηk_{. Therefore it is enough to find a probability measure under which the process} WT1∗ t := Wt− Z t 0 δnL(u, T1∗) 1 + δnL(u, T₁∗) λ(u, T₁∗) du = Wt− Z t 0 γ(u, T ∗ 1) 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

dP_T∗ 1 dP =ET1∗ Z · 0 γ(u, T ∗ 1)· dWu , P-a.s.

We are in a position to specify the dynamics of the forward Libor rate for the date T₂∗ under PT₁∗,

namely we postulate that

dL(t, T₂∗) = L(t, T₂∗) λ(t, T₂∗)· dWT1∗

t

L(0, T₂∗) =B(0, T

∗

2)− B(0, T1∗) δ_n−1B(0, T₁∗) .

Let us now assume that we have found processes L(·, T₁∗), . . . , L(·, T_m∗). This means, in particular, that the forward Libor measure PT_m−1∗ and the associated Brownian motion WT

∗

m−1 _{are already}

specified. Our aim is to determine the forward Libor measurePTm∗. It is easy to check that

Um+1(t, Tk∗) :=

B(t, T_k∗) B(t, T_m∗) =

Um(t, Tk∗)

1 + δ_n−mL(t, T_m∗). Using Lemma 2.1, we obtain the following relationship

WTm∗ t = W Tm−1∗ t − Z _t 0 δn−mL(u, Tm∗) 1 + δ_n−mL(u, T_m∗)λ(u, T ∗ m) du

for t∈ [0, T_m∗]. The forward Libor measureP_T∗

m can thus be easily found using Girsanov’s theorem.

Finally, we define the process L(·, T_m+1∗ ) as the solution to the SDE dL(t, T_m+1∗ ) = L(t, T_m+1∗ ) λ(t, T_m+1∗ )· dWTm∗

t

L(0, T_m+1∗ ) = B(0, T

∗

m+1)− B(0, Tm∗)

δn−mB(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 measurePTm+1.

Let us now examine the existence and uniqueness of the implied savings account,8 in a discrete-time setup. Intuitively, the value B_t∗ 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) =

FB(t, Tj, T∗)

F_B(t, T_j+1, T∗) =

B(t, Tj)

B(t, T_j+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 D¨oberlein and Schweizer (1998) and D¨oberlein et al. (1999) for further developments and the general uniqueness result.

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This in turn yields, upon setting t = Tj

FB(Tj, Tj, Tj+1) = 1/B(Tj, Tj+1), (22)

so that the price B(T_j, T_j+1) of a single-period bond is uniquely specified for every j. Though the bond that matures at time T_j does not physically exist after this date, it seems justifiable to consider F_B(T_j, T_j, T_j+1) as its forward value at time T_j for the next future date T_j+1. In other words, the spot value at time T_j+1of one cash unit received at time T_jequals B−1(T_j, T_j+1). The discrete-time savings account B∗thus equals, for k = 0, . . . , n (recall that T₋₁= 0),

B_T∗_k= k Y j=0 FB Tj−1, Tj−1, Tj = _Yk j=0 B Tj−1, Tj −1

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 B∗_T_j = FB(Tj−1, Tj−1, Tj) BT∗j−1, we find that BT∗j > B∗Tj−1for every j = 0, . . . , n.

We conclude that the implied savings account B∗follows a strictly increasing discrete-time process. Let us define the probability measureP∗, equivalent to P on (Ω, F_T∗), by the formula9

dP∗ dP = B

∗

T∗B(0, T∗), P-a.s. (23)

The probability measure P∗ appears to be a plausible candidate for a spot martingale measure. Indeed, if we set

B(Tl, Tk) =EP∗(B_T∗_l/B_T∗_k| FTl) (24)

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)

Gt= B(t, Tm(t)) m(t)_Y j=0 B−1(Tj−1, Tj), (25) where m(t) = inf{k = 0, 1, . . . | k X i=0 δi≥ t} = inf {k = 0, 1, . . . | Tk≥ t}.

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}_j_{) satisfy the weak no-arbitrage condition, meaning that there exists}

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It is easily seen that Gtrepresents 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 measurePL is a probability measure on (Ω,FT∗) which is equivalent

toP, and such that for any j = 0, . . . , n the relative bond price B(t, Tj)/Gtfollows a local martingale

underPL. Note that B(t, Tk+1)/Gt= m(t)_Y j=0 1 + δjL(Tj−1, Tj−1) ₋₁ Yk j=m(t)+1 1 + δjL(t, Tj−1)

so that all relative bond prices B(t, T_j)/G_t, 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 underPLof 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

Itˆo processes under the underlying probability measureP, more explicitly dB(t, Tj) = B(t, Tj) a(t, Tj) dt + b(t, Tj)· dWt

(26) for every j = 0, . . . , n, where, as before, W is a d-dimensional standard Brownian motion under an underlying probability measureP (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

dG_t= G_t a(t, T_m(t)) dt + b(t, T_m(t))· dW_t. (27) Furthermore, by applying Itˆo’s rule to equality

1 + δj+1L(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) δj+1B(t, Tj+1) a(t, Tj)− a(t, Tj+1) − ζ(t, Tj)b(t, Tj+1) and ζ(t, Tj) = B(t, Tj) δ_j+1B(t, T_j+1) b(t, Tj)− b(t, Tj+1) . (29)

Using (28) and the last formula, we arrive at the following relationship

b(t, T_m(t))− b(t, T_j+1) = j X k=m(t) δ_k+1ζ(t, T_k) 1 + δk+1L(t, Tk) . (30)

By definition of a spot Libor measurePL, 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 Dol´eans exponential, that is

dPL dP =ET∗ Z · 0 hu· dWu , P-a.s.

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for some adapted process h. It it not hard to check, using Itˆo’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) δj+1B(t, Tj+1) a(t, Tj)− a(t, Tj+1) = ζ(t, Tj)· b(t, Tm(t))− ht , and this in turn yields

dL(t, Tj) = ζ(t, Tj)· b(t, T_m(t))− b(t, Tj+1)− ht dt + dWt . Using (30), we conclude that process L(·, Tj) satisfies

dL(t, T_j) = j X k=m(t) δk+1ζ(t, Tk)· ζ(t, Tj) 1 + δk+1L(t, Tk) dt + ζ(t, T_j)· dW_tL,

where the process W_tL = Wt−

R_t

0hudu follows a d-dimensional standard Brownian motion under

the spot Libor measurePL. To further specify the model, we assume that processes ζ(t, T_j), j = 0, . . . , n− 1, have the following form, for t ∈ [0, T_j],

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, T_j) = j X k=m(t) δ_k+1λ_k(t, L_k(t))· λ_j(t, L_j(t)) 1 + δk+1L(t, Tk) dt + λ_j(t, L_j(t))· dW_tL,

where we write L_j(t) = (L(t, T_j), L(t, T_j+1), . . . , L(t, T_n)). Under mild regularity assumptions, this system can be solved recursively, starting from L(·, T_n−1). The lognormal model of forward Libor rates corresponds to the choice of ζ(t, T_j) = λ(t, T_j)L(t, T_j), where λ(·, T_j) : [0, T_j] → 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, . . . , T_n. 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₍_{·, T}

j), which equals (cf. Section 2.1.3)

Lf(t, T_j) =E_P∗(L(T_j, T_j)| F_t) =E_P∗( ˜L(T_j, T_j)| F_t), (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

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Recall that

B−1(Tj, Tj+1) = 1 + δj+1L(Tj, Tj) = 1 + δj+1L(T˜ j, Tj) = 1 + δj+1Lf(Tj, Tj).

The choice of the “pricing measure” is thus largely the matter of convenience. Similarly, we have B(Tj, Tj+1) =

1

1 + δ_j+1L(T_j, T_j) = FB(Tj, Tj+1, Tj). (32) More generally, the forward price of a Tj+1-maturity bond for the settlement date Tj equals

F_B(u, T_j+1, T_j) = B(u, Tj+1) B(u, T_j) =

1

1 + δ_j+1L(u, T_j). (33) Generally speaking, to value the claim X = g(L(u, Tj)) = ˜g(FB(u, Tj+1, Tj)) which settles at time

Tj we may use the formula

πt(X) = B(t, Tj)EPTj(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 measurePTj. If u = Tj, we may equally well

use the the dynamics, underPTj, of either ˜L(·, Tj) or Lf(·, Tj). For instance,

πt(X1) = B(t, Tj)EPTj(B−1(Tj, Tj+1)| Ft) = B(t, Tj)EP_Tj(FB−1(Tj, Tj+1, Tj)| Ft) but also πt(X1) = B(t, Tj) 1 + δj+1EP_Tj(Z(Tj)| Ft) , where Z(T_j) = L(T_j, T_j) = ˜L(T_j, T_j) = Lf_(T j, Tj). 2.3.1 Dynamics of L(·, Tj) underPTj

We shall now derive the transition probability density function (p.d.f.) of the process L(·, Tj) under

the forward probability measurePTj. Let us first prove the following related result, due to Jamshidian

(1997).

Proposition 2.1 Let t≤ u ≤ Tj. Then

EPTj L(u, Tj)| Ft = L(t, Tj) + δj+1VarP_Tj+1 L(u, Tj)| Ft 1 + δ_j+1L(t, T_j) . (34) In the case of the lognormal model of Libor rates, we have

EPTj L(u, Tj)| Ft = L(t, Tj) 1 + δj+1L(t, Tj) ev 2 j(t,u)− 1 1 + δj+1L(t, Tj) ! , (35) where v_j2(t, u) = VarP_Tj+1 Z u t λ(s, Tj)· dWsTj+1 = Z _u t |λ(s, Tj)|2ds. (36)

In particular, the modified Libor rate ˜L(t, T_j) satisfies11

˜ L(t, Tj) =EPTj L(Tj, Tj)| Ft = L(t, Tj) 1 + δj+1L(t, Tj) ev 2 j(t,Tj)− 1 1 + δj+1L(t, Tj) ! .

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Proof. Combining (6) with the martingale property of the process L(·, Tj) underPTj+1, we obtain EP_Tj L(u, Tj)| Ft = EPTj+1 (1 + δj+1L(u, Tj))L(u, Tj)| Ft 1 + δj+1L(t, Tj) so that EPTj L(u, Tj)| Ft = L(t, T_j) +δj+1EPTj+1 (L(u, Tj)− L(t, Tj)) 2_{| F} t 1 + δj+1L(t, Tj) . In the case of the lognormal model, we have

L(u, Tj) = L(t, Tj) eηj(t,u)− 1 2v2j(t,u)_, where ηj(t, u) = Z u t λ(s, Tj) dWsTj+1. (37) Consequently, EP_Tj+1 (L(u, Tj)− L(t, Tj))2| Ft = L2(t, Tj) ev 2 j(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

EPTj g(L(u, Tj))| Ft = EPTj+1 g(L(u, Tj)) 1 + δj+1L(u, Tj) Ft 1 + δ_j+1L(t, T_j) . 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 prob-ability density function f_P. Let Q be a probability measure equivalent to P. Suppose that for any bounded Borel measurable function g :R → R we have

EP(g(ζ)) =EQ (1 + ζ)g(ζ). Then the p.d.f. f_Q of ζ under Q satisfies f_P(y) = (1 + y)f_Q(y). Proof. The assertion is in fact trivial since, by assumption,

Z _∞

−∞g(y)fP(y) dy =

Z _∞

−∞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)−12VarP_Tj+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. underPTj+1of 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} = √ 1 2πvj(t, u)y exp ( − ln(y/x) +12vj2(t, u) 2 2v_j2(t, u) )

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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 measurePTj, satisfies

˜

pL(t, x; u, y) =PTj{L(u, Tj) = y| L(t, Tj) = x} =

1 + δj+1y

1 + δj+1x

pL(t, x; u, y).

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, ˜ pL(t, x; u, y) = 1 + δj+1y √ 2πvj(t, u) y(1 + δj+1x) exp ( − ln(y/x) +12vj2(t, u) ₂ 2v2_j(t, u) ) . 2.3.2 Dynamics of FB(·, Tj+1, Tj) under PTj

Observe that the forward bond price FB(·, Tj+1, Tj) satisfies

F_B(t, T_j+1, T_j) = B(t, Tj+1) B(t, Tj)

= 1

1 + δj+1L(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,

dFB(t) =−FB(t) 1− FB(t)

λ(t, Tj)· dWtTj, (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

underPTj. Using Corollary 2.1 and relationship (38), one can find the transition p.d.f. of the Markov

process FB(·, Tj+1, Tj) underPTj; that is,

p_B(t, x; u, y) =P_T_j{F_B(u, T_j+1, T_j) = y| F_B(t, T_j+1, T_j) = 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. underPTj of the forward bond price FB(·, Tj+1, Tj) equals, for

any t < u and arbitrary 0 < x, y < 1,

p_B(t, x; u, y) = √ x 2πv_j(t, u)y2(1− y) exp     − lnx(1−y)_y(1−x)+1₂v2_j(t, u) 2 2v_j2(t, u)     .

Proof. Let us fix x∈ (0, 1). Using (38), it is easy to show that p_B(t, x; u, y) = δ−1y−2p˜_L t,1− x δx ; u, 1− y 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}_j_{) under}_P

Tj can be easily deduced from the Markovian features of the forward priceF_B(·, T_j, T_j+1) underP_T_j (see formulae (38)–(39)).

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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 δ_j 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 + δjL(Tj−1). (40)

The price at time t≤ T₀ of a forward cap, denoted by FC_t, is (we set N_p= 1)

FC_t = n X j=1 EP∗ _B t BTj (L(T_j−1)− κ)+δ_j F_t = n X j=1 B(t, T_j)E_P_Tj (L(T_j−1)− κ)+δ_j F_t . (41)

On the other hand, since the cash flow of the jthcaplet at time Tj is manifestly aFTj−1-measurable

random variable, we may directly express the value of the cap in terms of expectations under forward measuresPTj−1, j = 1, . . . , n. Indeed, we have

FC_t= n X j=1 B(t, Tj−1)EP_Tj−1 B(Tj−1, Tj)(L(Tj−1)− κ)+δj Ft . (42)

Consequently, using (40) we get equality

FC_t= n X j=1 B(t, Tj−1)EPTj−1 1− ˜δjB(Tj−1, Tj) ₊ Ft , (43)

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 jth caplet. A caplet is exercised at time T_j−1 if and only if L(T_j−1)− κ > 0, or equivalently, if

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The last inequality holds whenever ˜δjB(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

δjB(Tj−1, Tj)(L(Tj−1)− κ) = 1 − ˜δjB(Tj−1, Tj) = ˜δj δ˜−1j − B(Tj−1, Tj)

at time T_j−1. This shows once again that the jth 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 ˜δj= (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 jth floorlet pays (κ− L(Tj−1))+ at time Tj. Therefore,

FF_t= n X j=1 EP∗ _B t BTj (κ− L(Tj−1))+δj Ft , (44) but also FF_t= n X j=1 B(t, Tj−1)EPTj−1 ˜ δjB(Tj−1, Tj)− 1 ₊ Ft . (45)

Combining (41) with (44) (or (43) with (45)), we obtain the following cap-floor parity relationship

FC_t− FF_t= n X j=1 B(t, Tj−1)− ˜δjB(t, Tj) (46)

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 toQ 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 dW_t, (47)

where W follows a one-dimensional standard Brownian motion underQ, and σ is a strictly positive constant. The unique solution of (47) is

L(t, T ) = L(0, T ) exp σWt−1₂σ2t2

, ∀ t ∈ [0, T ], (48) where the initial condition is derived from the yield curve Y (0, T ), namely

1 + δL(0, T ) = B(0, T ) B(0, T + δ) = exp (T + δ)Y (0, T + δ)− T Y (0, T ) .

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)EQ (L(T, T )− κ)+ Ft .

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More explicitly, for any t∈ [0, T ] we have

FC_t= δB(t, T + δ)

L(t, T )N ˆe₁(t, T )− κN ˆe₂(t, T ), (49) where N is the standard Gaussian cumulative distribution function

N (x) = √1 2π Z _x −∞e −z2_/2 dz, ∀ x ∈ R, and ˆ e_1,2(t, T ) = ln(L(t, T )/κ)± 1 2vˆ20(t, T ) ˆ v₀(t, T )

with ˆv₀2(t, T ) = σ2(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 FC_t= n X j=1 δjB(t, Tj) L(t, Tj−1)N ˆej1(t) − κN ˆej 2(t) , (50)

where for every j = 0, . . . , n− 1

ˆ ej_1,2(t) = ln(L(t, Tj−1)/κ)± 1 2vˆj2(t) ˆ vj(t) (51)

and ˆv2_j(t) = (T_j−1− t)σ_j2for some constants σ_j, 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

measurePTj are

dL(t, Tj−1) = L(t, Tj−1) λ(t, Tj−1)· dWtTj, (52)

where WTj_{follows a d-dimensional Brownian motion under the forward measure}P

Tj, and λ(·, Tj−1) :

[0, Tj−1]→ Rd is a deterministic function. Consequently, for every t∈ [0, Tj−1] we have

L(t, Tj−1) = L(0, Tj−1)Et Z · 0 λ(u, Tj−1)· dW Tj u .

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 subse-quently 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

FC_t= n X j=1 δ_jB(t, T_j) L(t, T_j−1)N ˜ej₁(t)− κN ˜ej₂(t)= n X j=1 FCj_t, (53)

Modelling of Forward Libor and Swap Rates

and Swap Rates

Marek Rutkowski

Faculty of Mathematics and Information Science

Warsaw University of Technology, 00-661 Warszawa, Poland

Contents

1

Introduction

2

Modelling of Forward Libor Rates

2.1

Forward and Futures Libor Rates

2.2

Lognormal Models of Forward Libor Rates

2.3

Dynamics of Libor Rates and Bond Prices

2.4

Caps and Floors