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R^d

£βⁱ(t, x, T, T^∗) − 1¤ ¡

µ^L− ν_T^L∗

¢(dt, dx)

− Z

R^d

£βⁱ(t, x, T, T^∗) − 1¤ξˆⁱ(t, x, T^∗)ν_T^L∗(dt, dx)

(2.2.80)

for 0 ≤ t ≤ T ≤ T^∗.

2.2.7 Discrete-tenor L´ evy domestic Libor rate model

Following the approach of Eberlein and Özkan (see again Eberlein and Özkan (2002)), we will model forward Libor rates (see Definition 2.1.5) with time-inhomogeneous Lévy processes. Let δ > 0 and T ≤ T^∗ − δ be fixed, then we define the domestic Libor rate L(t, T ) by

L(t, T ) = L(0, T ) exp



 Zt

λ(s, T )^>dL^{T +δ}_s





where

L^{T +δ}_t = Zt

b^{T +δ}_s ds + Zt

c_sdW_s^{T +δ}+ Zt

R^d

x¡

µ^L− ν_{T +δ}^L ¢

(ds, dx)

is a d–dimensional time-inhomogeneous L´evy process such that W^{T +δ} is a P^{T +δ}–standard Brownian motion in R^d, C(t) is a symmetric positive semidefinite d × d matrix, and c_t is a measurable version of the square root of C_t.

The random measure of jumps of L is given by µ^L, and ν_{T +δ}^L is its P^{T +δ}–compensator satisfying

ν_{T +δ}^L ([0, t], G) = Zt

R^d

1{[0,t]×G}(s, x)λ^{T +δ}_s (dx) ds, G ∈ B¡

R^d\ {0}¢ ,

where λ^{T +δ}_t is a measure on R^d, which integrates (|x|²∧ 1). Furthermore, for all t ∈ [0, T +δ]

holds λt({0}) = 0.

We assume that for all T ≤ T^∗− δ the following integrability conditions hold:

¯¯b^{T +δ}_s ¯

¯ ds < ∞, ZT

kCsk ds < ∞, ZT

|x|>1

exp¡ u^>x¢

λ^{T +δ}_s (dx) ds < ∞,

where |u| ≤ (1 + ²)M^{T +δ} for some constants M^{T +δ} > 1 and ² > 0. Hence, L^{T +δ}_t is a special semimartingale with respect to the domestic forward measure P^{T +δ}.

Assumption 2.2.12 The volatility of the domestic forward Libor rate process λ(·, T ) : [0, T ] 7→ R^d₊ is a bounded deterministic function such that λ(t, T ) = (0, . . . , 0)^> for t > T . Furthermore, there exist positive constants M and M such that M ≤ M and

0 < M ≤ |λ(·, T )| ≤ M < M^{T +δ} for each t ∈ [0, T ].

Observe that by Propositions 2.2.4 and 2.2.5 the volatility function λ(·, T ) is integrable with respect to the time-inhomogeneous L´evy process L^{T +δ}_t , and the process R^t

λ(s, T )^>dL^{T +δ}_s is a special semimartingale, which is also exponentially special under P^{T +δ}.

Furthermore, we assume that the following initial condition is satisfied L(0, T ) = 1

µ B(0, T ) B(0, T + δ) − 1

(2.2.81) for all T ∈ (0, T^∗− δ], where the initial term structure B(0, T ), T ∈ (0, T^∗] is strictly positive and decreasing in T .

The domestic Libor rate process must be a local martingale under P^{T +δ}. Applying Theo-rem 2.19 in Kallsen and Shiryaev (2002), the martingality of (L(t, T ))_0≤t≤T can be assured by choosing the drift term b^{T +δ}_t such that R^t

λ(s, T )^>b^{T +δ}_s ds is equal to the exponential compensator of L(t, T ). Applying Proposition 2.2.5, we get

λ(s, T )^>b^{T +δ}_s ds = − eK_t^L^{T +δ}(λ(·, T ))

= −1 2

λ(s, T )^>C_sλ(s, T ) ds

− Zt

R^d

£exp¡

λ(s, T )^>x¢

− 1 − λ(s, T )^>x¤

ν_{T +δ}^L (ds, dx),

where eK_t^L^{T +δ}(λ(·, T )) denotes the Laplace cumulant process of L^{T +δ} in λ(·, T ).

Hence, we obtain the following dynamics of L(·, T ) under the forward measure P^{T +δ}

dL(t, T ) = L(t−, T )



λ(t, T )^>c_tdW_t^{T +δ}+

+ Z

R^d

¡exp¡

λ(t, T )^>x¢

− 1¢ ¡

µ^L− ν_{T +δ}^L ¢

(dt, dx)



 .

(2.2.82)

Recall that FB(t, T, T +δ) = 1+δL(t, T ), therefore we derive from (2.2.82) that the forward process F_B(·, T, T + δ) satisfies

dF_B(t, T, T + δ)

F_B(t−, T, T + δ) = δL(t−, T ) 1 + δL(t−, T )



λ(t, T )^>c_tdW_t^{T +δ}

+ Z

R^d

¡exp¡

λ(t, T )^>x¢

− 1¢ ¡

µ^L− ν_{T +δ}^L ¢

(dt, dx)



 .

Observe that the coefficients











δL(t−, T )

1 + δL(t−, T )λ(t, T )^>ct, and δL(t−, T )

1 + δL(t−, T )

£exp¡

λ(t, T )^>x¢

− 1¤

are not deterministic. Since we want to stay within the class of PIIAC processes, we have to approximate the last equations by replacing the random term

l(t−, T ) := δL(t−, T ) (1 + δL(t−, T ))

with a deterministic one l(0, T ) = δL(0, T )/(1 + δL(0, T )). A similar approximation has been already used in Brace and Womersley (2000) for the derivation of the approximate swaption formula in the lognormal forward Libor model, as well as in Schl¨ogl (2002).

Notice that the quotient

l(t−, T ) = 1 − 1

1 + δL(t−, T ) = 1 − FB(t−, T + δ, T ) ∈ [0; 1].

The suggested approximation can be easily justified, since, in absolute terms, not only is l(t−, T ) bounded in [0; 1], but its absolute volatility is a couple of orders of magnitude less than the volatilities of the forwards. Following the approach of Musiela and Rutkowski (see Musiela and Rutkowski (1997a)), we build an arbitrage-free domestic bond market, which is based on the Libor model presented above, using backward induction procedure.

More specifically, given a discrete-tenor structure T = {T₀ < T₁ < T₂ < T_N < T_{N +1}= T^∗} with δ = T_j+1− T_j, j = 0, . . . , N, one can construct the forward measure for the date T_j−1 with j = 2, . . . , N + 1, provided that the forward measure P^T^j, the standard Brownian motion W^T^j, the compensator ν_T^L_j of the jump measure µ^L, and the volatility λ(·, Tj−1) of the domestic Libor rate L(·, T_j−1), are known. In particular, under the forward measure P^T^j with j = 1, . . . , N + 1 the domestic Libor rate is defined by

L(t, T_j−1) = L(0, T_j−1) exp



 Zt

λ(s, T_j−1)^>dL^T_s^j



 , (2.2.83)

where

L^T_t^j = Zt

b^T_s^jds + Zt

c_sdW_s^T^j+ Zt

R^d

µ^L− ν_T^L_j

(ds, dx), (2.2.84)

and for all t ≤ T_j−1 holds Zt



¯

¯b^T_s^j¯

¯ + kC_sk + Z

R^d

¡|x|²∧ 1¢

λ^T_s^j(dx)



 ds < ∞

for each j = 1, . . . , N + 1. Furthermore, we make the following integrability assumption.

Assumption 2.2.13 One can find such constants M > 1 and ² > 0, that for each j = 1, . . . , (N + 1) holds

TZj−1

|x|>1

exp¡ u^>x¢

λ^T_s^j(dx) ds < ∞, (2.2.85)

for |u| ≤ (1 + ²)M^T^j with M^T^j := M − (N − j + 1) fM and M − (N + 1) fM > max{1, M}, where we have set fM := M/(1 + ²) and M := M^T^{N +1} = M^T^∗.

Note that under Assumption 2.2.13, we have XN

k=1

λ(t, T_k) ≤ XN k=1

|λ(t, T_k)| ≤ NM < M

for all t ∈ [0, T_N]. Moreover, with respect to Proposition 2.2.4 the volatility function λ(·, T_j−1) is integrable with respect to L^T_t^j and the process R^t

λ(s, T_j−1)^>dL^Ts^j is a special semimartingale, which is also exponentially special.

To guarantee that L(·, T_j−1) is a local martingale under domestic forward measure P^T^j, we choose the drift term b^T^j such that R^t

λ(s, T_j−1)^>b^Ts^jds is equal to the exponen-tial compensator of R^t

λ(s, T_j−1)^>dL^Ts^j, where t ∈ (0, T_j−1] and j = 1, . . . , N + 1. Using Proposition 2.2.5, we obtain

λ(s, T_j−1)^>b^T_s^jds = − eK_t^L^Tj(λ(·, T_j−1))

= −1 2

λ(s, T_j−1)^>C_sλ(s, T_j−1) ds

− Zt

R^d

£exp¡

λ(s, Tj−1)^>x¢

− 1 − λ(s, Tj−1)^>x¤

ν_T^L_j(ds, dx).

(2.2.86) With this specification of a drift term, the Libor rate L(t, Tj−1) can be represented as a stochastic exponential of the form

L(t, T_j−1) = L(0, T_j−1) E



 Z·

λ(s, T_j−1)^>c_sdW_s^T^j

+ Z·

R^d

¡exp¡

λ(s, Tj−1)^>x¢

− 1¢ ³

µ^L− ν_T^L_j

(ds, dx)





Hence, domestic forward process admits the following representation:

F_B(t, T_j−1, T_j) = F_B(0, T_j−1, T_j) E¡

H^T^j−1¢

t, where

H_t^T^j−1 = Zt

l(s−, T_j−1)λ(s, T_j−1)^>c_sdW_s^T^j

+ Zt

R^d

l(s−, T_j−1)¡ exp¡

λ(s, T_j−1)^>x¢

− 1¢ ³

µ^L− ν_T^L_j´

(ds, dx).

To simplify the notation, we put

α (t, T_j−1, T_j)^> := l(t−, T_j−1)λ(t, T_j−1)^>c_t

≈ l(0, T_j−1)λ(t, T_j−1)^>c_t, β (t, x, T_j−1, T_j) := l(t−, T_j−1)¡

exp¡

λ(t, T_j−1)^>x¢

− 1¢ + 1

≈ l(0, T_j−1)¡ exp¡

λ(t, T_j−1)^>x¢

− 1¢ + 1.

(2.2.87)

The domestic forward martingale measure associated with the date T_j−1can be defined by setting its Radon–Nikod´ym derivative

dP^T^j−1

dP^T^j = F_B(T_j−1, T_j−1, T_j)

FB(0, Tj−1, Tj) = 1 + δL(T_j−1, T_j−1)

1 + δL(0, Tj−1) P^T^j–a.s.

for j = 2, . . . , N + 1. Then a standard Brownian motion and the P^T^j−1–compensator of the jump measure µ^L are defined by

W_t^T^j−1 = W_t^T^j− Zt

α (s, T_j−1, T_j) ds,

ν_T^L_j−1(dt, dx) = β (t, x, T_j−1, T_j) ν_T^L_j(dt, dx).

Observe that there is a following relationship between the driving time-inhomogeneous L´evy processes with respect to the forward measures P^T^j and P^T^j−1

L^T_t^j = L^T_t^j−1 + Dt(Tj−1, Tj) , where the process D_t(T_j−1, T_j) is given by

D_t(T_j−1, T_j) = Zt

¡b^T_s^j − b^T_s^j−1¢ ds +

c_sα(s, T_j−1, T_j) ds

+ Zt

R^d

x (β (s, x, Tj−1, Tj) − 1) ν_T^L_j(ds, dx).

(2.2.88)

Note also that for j = 1, ..., N we can establish the following connection between the domestic forward measures P^T^j and P^T^∗

W_t^T^j = W_t^T^∗− Zt

α (s, T_j, T^∗) ds,

ν_T^L_j(dt, dx) = β (t, x, T_j, T^∗) ν_T^L∗(dt, dx),

(2.2.89)

where We can avoid approximations in equations (2.2.87) and (2.2.90) by modelling the do-mestic forward rate F_B(t, T_j−1, T_j) = 1 + δL(t, T_j−1) (instead of Libor rate L(t, T_j−1)) for j = 1, . . . , N + 1 as an exponential of a the time-inhomogeneous L´evy process given by equation (2.2.84). This approach, however, leads to negative Libor rates (which is not necessarily inconceivable). On the other hand, it simplifies the form of the compensator of the jump measure µ^L under each forward measure (see equation (2.2.94)), allowing us to use the theory of Esscher transformation. This fact, in particular, makes the calibration procedure, which will be described in the next chapters, much easier allowing us to avoid a few tedious numerical calculations. Alternatively, we postulate the following dynamics of the domestic forward process under the domestic forward measure P^T^j

dL(t, T_j−1) with the initial condition

F_B(0, T_j−1, T_j) = B(0, T_j−1)

B(0, T_j) , j = 1, . . . , N + 1.

Consequently, the Libor rate L(t, Tj−1) satisfies dL(t, T_j−1)

L(t−, T_j−1) = 1 + δL(t−, T_j−1) δL(t−, T_j−1)



λ(t, Tj−1)^>c_tdW_t^T^j

Exactly as in the previous case dP^T^j−1

Thus, for j = 1, . . . , N we can establish the following connection between the domestic forward measures P^T^j and P^T^∗

Hence, the density process dP^T^j

dP^T^∗ admits the following representation:

dP^T^j

where C_t^T^∗ is a deterministic function given by

One can easily check that

C_t^T^∗ = exp

In document CUADRO COMPARATIVO Modificaciones en el Reglamento de la Ley de Relaciones Colectivas de Trabajo, realizadas por el Decreto Supremo Nº TR (página 27-32)