Model of LIBOR Rates

9.4 Model of LIBOR Rates

The Heath-Jarrow-Morton methodology of term structure modelling presented in the previous sec-tion is based on the arbitrage-free dynamics of instantaneous, continuously compounded forward rates. The assumption that instantaneous rates exist is not always convenient, since it requires a certain degree of smoothness with respect to the tenor (i.e., maturity) of bond prices and their volatil-ities. An alternative construction of an arbitrage-free family of bond prices, making no reference to the instantaneous, continuously compounded rates, is in some circumstances more suitable.

By definition, the forward δ-LIBOR rate³ 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 ) = FB(t, T, T + δ), ∀ t ∈ [0, T ]. (9.41) 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 ]. (9.42) In particular, the initial term structure of forward LIBOR rates satisfies

L(0, T ) = f_s(0, T, T + δ) = δ⁻¹

 B(0, T ) B(0, T + δ)− 1

. (9.43)

Under the forward measure P_{T +δ}, we have

dL(t, T ) = δ⁻¹F_B(t, T, T + δ) γ(t, T, T + δ)· dWt^{T +δ},

where W_t^{T +δ} and P_{T +δ} are yet unspecified. This means that L(·, T ) solves the equation

dL(t, T ) = δ⁻¹(1 + δL(t, T )) γ(t, T, T + δ)· dWt^{T +δ}, (9.44) subject to the initial condition (9.43). Suppose that forward LIBOR rates L(t, T ) are strictly positive.

Then formula (9.44) can be rewritten as follows

dL(t, T ) = L(t, T ) λ(t, T )· dWt^{T +δ}, (9.45) where for any t∈ [0, T ]

λ(t, T ) =1 + δL(t, T )

δL(t, T ) γ(t, T, T + δ). (9.46)

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 R^d-valued, bounded, deterministic function⁴ 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 P (0, T ), T ∈ [0, T^∗], and thus an initial term structure L(0, T ) of forward LIBOR rates

L(0, T ) = δ⁻¹

 P (0, T ) P (0, T + δ)− 1

, ∀ T ∈ [0, T^∗− δ]. (9.47)

3In practice,several types of LIBOR rates occur,e.g.,3-month LIBOR and 6-month LIBOR. For ease of exposition, we consider a fixed maturity δ.

4Volatility λ could follow a stochastic process; we deliberately focus here on a lognormal model of forward LIBOR rates in which λ is deterministic.

9.4.1 Discrete-tenor Case

We start by studying a discrete-tenor version of a lognormal model of forward LIBOR rates. It should be stressed that a so-called discrete-tenor model still possesses certain continuous-time features; for instance, forward LIBOR rates follow continuous-time processes. For ease of notation, we shall assume that the horizon date T^∗is a multiple of δ, say T^∗= M δ for a natural M. We shall focus on a finite number of dates, T_m^∗ = T^∗−mδ for m = 1, . . . , M −1. The construction is based on backward induction, therefore we start by defining the forward LIBOR rate with the longest maturity, L(t, T₁^∗).

We postulate that the rate L(t, T₁^∗) is governed under the probability measure P by the following SDE

dL(t, T₁^∗) = L(t, T₁^∗) λ(t, T₁^∗)· dWt, (9.48) with the initial condition

L(0, T₁^∗) = δ⁻¹

P (0, T₁^∗) P (0, T^∗)− 1

. (9.49)

Put another way, we postulate that for every t∈ [0, T1^∗]

L(t, T₁^∗) = δ⁻¹

P (0, T₁^∗) P (0, T^∗)− 1

Et

 · 0

λ(u, T₁^∗)· dWu

. (9.50)

Since P (0, T₁^∗) > P (0, T^∗), it is clear that L(t, T₁^∗) follows a strictly positive continuous martingale under P. Also, for any fixed t≤ T₁^∗, the random variable L(t, T₁^∗) has a lognormal probability law under P. The next step is to define the forward LIBOR rate for the date T₂^∗,

γ(t, T₁^∗, T^∗) = δL(t, T₁^∗)

1 + δL(t, T₁^∗)λ(t, T₁^∗), ∀ t ∈ [0, T₁^∗]. (9.51) Given that the volatility γ(t, T₁^∗, T^∗) is determined by (9.51), the forward process F_B(t, T₁^∗, T^∗) is known to solve, under P

dFB(t, T₁^∗, T^∗) = FB(t, T₁^∗, T^∗) γ(t, T₁^∗, T^∗)· dWt (9.52) and the initial condition is FB(0, T₁^∗, T^∗) = P (0, T₁^∗)/P (0, T^∗). The forward process FB(t, T₁^∗, T^∗) is a continuous martingale under P, since the volatility γ(t, T₁^∗, T^∗) follows a bounded process. We introduce a d-dimensional process W^T1∗, which corresponds to the date T₁^∗, by setting

W_t^T1∗ = Wt−

 t 0

γ(u, T₁^∗, T^∗) du, ∀ t ∈ [0, T₁^∗]. (9.53)

Due to the boundedness of the process γ(t, T₁^∗, T^∗), the existence of the process W^T1∗and of the asso-ciated probability measure PT₁^∗, equivalent to P, under which the process W^T1∗ follows a Browinian motion, and which is given by the formula

dPT₁^∗

dP =ET₁^∗

 · 0

γ(u, T₁^∗, T^∗)· dWu

, P-a.s., (9.54)

is trivial. The process W^T1∗ may be interpreted as the forward Brownian motion for the date T₁^∗. We are in a position to specify the dynamics of the forward LIBOR rate for the date T₂^∗ under the forward probability measure P_T∗

1. Analogously to (9.48), we set

dL(t, T₂^∗) = L(t, T₂^∗) λ(t, T₂^∗)· dWt^T1∗, (9.55) with the initial condition

L(0, T₂^∗) = δ⁻¹

P (0, T₂^∗) P (0, T₁^∗)− 1

. (9.56)

9.4. MODEL OF LIBOR RATES 151

Solving equation (9.55) and comparing with (9.46) for T = T₂^∗, we obtain γ(t, T₂^∗, T₁^∗) = δL(t, T₂^∗)

1 + δL(t, T₂^∗)λ(t, T₂^∗), ∀ t ∈ [0, T₂^∗]. (9.57) To find γ(t, T₂^∗, T^∗), we make use of the relationship

γ(t, T₂^∗, T₁^∗) = γ(t, T₂^∗, T^∗)− γ(t, T1^∗, T^∗), ∀ t ∈ [0, T2^∗]. (9.58) Given the process γ(t, T₂^∗, T₁^∗), we can define the pair (W^T2∗, P_T2^∗) corresponding to the date T₂^∗ and so forth. By working backwards to the first relevant date T_M^∗₋₁ = δ, we construct a family of forward LIBOR rates L(t, T_m^∗), m = 1, . . . , M−1. Notice that the lognormal probability law of every process L(t, T_m^∗) under the corresponding forward probability measure PT_m^∗₋₁ is ensured. Indeed, for any m = 1, . . . , M− 1, we have

dL(t, T_m^∗) = L(t, T_m^∗) λ(t, T_m^∗)· dW_t^Tm−1∗ , (9.59) where W^Tm∗−1 is a standard Brownian motion under PT_m−1^∗ . This completes the derivation of the lognormal model of forward LIBOR rates in a discrete-tenor framework. Note that in fact we have simultaneously constructed a family of forward LIBOR rates and a family of associated forward processes. Let us now examine the existence and uniqueness of the implied savings account, in a discrete-time setting. The implied savings account is thus seen as a discrete-time process, B_t^∗, t = 0, δ, . . . , T^∗ = M δ. 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,

F_B(t, T_j, T_j+1) = F_B(t, T_j, T^∗)

F_B(t, T_j+1, T^∗)= B(t, T_j) B(t, T_j+1), where we write Tj= jδ. This in turn yields, upon setting t = Tj

FB(Tj, Tj, Tj+1) = 1/B(Tj, Tj+1), (9.60) so that the price B(Tj, Tj+1) of a one-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+1of one cash unit received at time Tj equals B⁻¹(Tj, Tj+1). The discrete-time savings account B^∗thus equals

B_T^∗_k= k j=1

FB

Tj−1, Tj−1, Tj

=

^k

j=1

B

Tj−1, Tj

 ⁻¹

for k = 0, . . . , M− 1, since by convention B₀^∗= 1. Note that FB

Tj, Tj, Tj+1

= 1 + δL(Tj, Tj+1) > 1

for j = 1, . . . , M− 1, and since

B_T^∗_j+1= F_B(T_j, T_j, T_j+1) B_T^∗_j,

we find that B_T^∗_j+1> B_T^∗_j for every j = 0, . . . , M− 1. We conclude that the implied savings account B^∗follows a strictly increasing discrete-time process. We define the probability measure P^∗∼ P on (Ω,FT^∗) by the formula

dP^∗

dP = B_T^∗∗P (0, T^∗), P-a.s. (9.61) The probability measure P^∗ appears to be a plausible candidate for a spot martingale measure.

Indeed, if we set

B(T_l, T_k) = E_P∗(B_T^∗_l/B_T^∗_k| FTl) (9.62) for every l≤ k ≤ M, then in the case of l = k − 1, equality (9.62) coincides with (9.60).

9.4.2 Continuous-tenor Case

By a continuous-tenor model we mean a model in which all forward LIBOR rates L(t, T ) with T ∈ [0, T^∗] are specified. Given the discrete-tenor skeleton constructed in the previous section, it is sufficient to fill the gaps between the discrete dates to produce a continuous-tenor model. To construct a model in which each forward LIBOR rate L(t, T ) follows a lognormal process under the forward measure for the date T + δ, we shall proceed by backward induction.

First step. We construct a discrete-tenor model using the previously described method.

Second step. We focus on the forward rates and forward measures for maturities T ∈ (T1^∗, T^∗).

In this case we do not have to take into account the forward LIBOR rates L(t, T ) (such rates do not exist in the present model after the date T₁^∗). From the previous step, we are given the values B^∗_T∗ 1

and B^∗_T∗ of a savings account. It is important to observe that B_T^∗∗

1 and B_T^∗∗ are FT₁^∗-measurable random variables. We start by defining a spot martingale measure P^∗ associated with the discrete-tenor model, using formula (9.61). Since the model needs to match a given initial term structure, we search for an increasing function α : [T₁^∗, T^∗]→ [0, 1] such that α(T1^∗) = 0, α(T^∗) = 1, and the process

ln B^∗_t = (1− α(t)) ln BT^∗₁^∗+ α(t) ln B_T^∗∗, ∀ t ∈ [T1^∗, T^∗], satisfies P (0, t) = EP^∗(1/B_t^∗) for every t∈ [T₁^∗, T^∗]. Since 0 < B_T^∗∗

1 < B_T^∗∗, and P (0, t), t∈ [T₁^∗, T^∗], is assumed to be a strictly decreasing function, a function α with desired properties exists and is unique.

Third step. In the previous step, we have constructed the savings account B_T^∗ for every T ∈ [T₁^∗, T^∗]. Hence the forward measure for any date T ∈ (T1^∗, T^∗) can be defined by setting

dPT

dP^∗ = 1

B_T^∗P (0, T ), P^∗-a.s. (9.63)

Combining (9.63) with (9.61), we obtain dPT

dP =dPT

dP^∗ dP^∗

dP = B_T^∗∗P (0, T^∗)

B^∗_TP (0, T ) , P-a.s., for every T ∈ [T1^∗, T^∗], so that

dP_T dP | Ft

= E_P

B_T^∗∗P (0, T^∗) B_T^∗P (0, T )

 Ft

, ∀ t ∈ [0, T ].

Exponential representation of the above martingale – that is, the formula dPT

dP | Ft

= P (0, T^∗) P (0, T ) Et

 · 0

γ(u, T, T^∗)· dW_u^∗

, ∀ t ∈ [0, T ],

yields the forward volatility γ(t, T, T^∗) for any maturity T ∈ (T1^∗, T^∗). This in turn allows us to define also the associated P_T-Brownian motion W^T. Given the forward measure P_T and the associated Brownian motion W^T, we define the forward LIBOR rate process L(t, T − δ) for arbitrary T ∈ (T₁^∗, T^∗) by setting (cf. (9.48)–(9.49))

dL(t, Tδ) = L(t, Tδ) λ(t, Tδ)· dW_t^T, where Tδ = T− δ, with initial condition

L(0, T_δ) = δ⁻¹

P (0, T_δ) P (0, T ) − 1

. Finally, we set (cf. (9.51))

γ(t, T₁^∗, T^∗) = δL(t, T₁^∗)

1 + δL(t, T₁^∗)λ(t, T₁^∗), ∀ t ∈ [0, T1^∗],

9.5. MODEL OF FORWARD SWAP RATES 153

In document Introduction to Arbitrage Pricing (Page 150-154)