Divergencia sobre - CUADRO COMPARATIVO Modificaciones en el Reglamento de la Ley de Relaciones

¯¯

¯ F^t

, (3.1.2)

where eK = 1 + δK.

Proceeding along the same lines, we can derive the price at time t of the T_N–maturity interest rate floor

FF(t, TN) := EP

δ XN

j=1

B_T_j [K − L(Tj−1, Tj−1)]⁺

¯¯

¯Ft

= δ XN

j=1

B(t, Tj)E_PTj

£(K − L(Tj−1, Tj−1))⁺¯

¯ F_t¤ .

In the case, when the forward process 1 + δL(·, Tj−1) is taken as underlying, we obtain

FF(t, TN) = XN

j=1

B(t, Tj)E_PTj

·³K − 1 − δL(Te j−1, Tj−1)

´₊¯¯

¯¯ F^t

¸ .

In the remaining part of this section we present two pricing approaches based on for-mulas (3.1.1) and (3.1.2) respectively. Without loss of generality we present all results in the coming sections only for the purely discontinuous time-inhomogeneous L´evy processes.

This is done with regard of applications, using the class of generalized hyperbolic L´evy processes as driving ones.

3.1.1 The Libor rate approach

Let the price of the domestic cap be given by equation (3.1.1). In this case the underlying is a domestic forward Libor rate, which has with respect to equation (2.2.82) the following dynamics under the domestic forward measure P^T^j

dL(t, T_j−1) L(t−, T_j−1) =

R^d

¡exp¡

λ(t, Tj−1)^>x¢

− 1¢ ³

µ^L− ν_T^L_j

(dt, dx),

where the compensator ν_T^L_j of the jump measure µ^L is given by equation (2.2.89). Equa-tion (2.2.83) yields

L(T_j−1, T_j−1) = L(0, T_j−1) exp





TZj−1

λ(s, T_j−1)^>dL^T_s^j





= L(0, T_j−1) exp

Recall that under Assumption 2.2.13, X_T_j−1 is a special semimartingale with respect to the domestic forward measure P^T^j. One can easily check that its characteristics are given by

. With respect to equation (2.2.90), the P^T^j–compensator of the jump measure µ^L satisfies

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

Observe that due to approximation we have made, the characteristics of the driving process L^T^j and, consequently, of XTj−1 are deterministic. Hence, we remain in the class of PIIAC processes. Since X_T_j−1 is an exponentially special semimartingale, applying Lemma 1.4.12 (see also Remark 1.4.13), we obtain the following representation of its characteristic function:

In the purely discontinuous case the drift term condition given by equation (2.2.86) sim-plifies to

Thus, we obtain the following representation of the characteristic function of XTj−1

χ^T^j−1(u) = exp

According to equation (3.1.1) the price of the j-th caplet at time t = 0 is given by Cpl (0, T_j, K) = δB(0, T_j)E_PTj

where ρ(dx) is a Lebesgue density of the distribution of X_T_j−1. We make the following technical assumption (see Sections 3.1 and 3.2 in Raible (2000)).

Assumption 3.1.1 To guarantee that the distribution of X_T_j−1 is absolutely continuous with respect to the Lebesgue measure and has a bounded continuous density ρ(x), we assume that

Z _∞

−∞

¯¯χ^T^j−1(u)¯

¯ du < ∞.

To calculate the price of the caplet we use the method developed in Raible (2000) , which uses bilateral Laplace transforms. More specifically, we define w(x, K) := (x − K)⁺, so that the payoff of the j-th caplet can be written as δw(L(Tj−1, Tj−1), K). Additionally, we put v(x) := w(e^−x, 1) = (e^−x− 1)⁺.

The following theorem provides an analytical expression for the price of the interest rate caplet.

Theorem 3.1.2 Set ξ_j := ln (K) − ln (L(0, T_j−1)), and let R < −1 be chosen such that χ^T^j−1(iR) < ∞. Then the price of the j-th caplet Cpl (0, T_j, K) at time t = 0 is given by

Cpl (0, T_j, K) = δB(0, T_j)Kexp (ξ_jR) 2π

Z∞

−∞

exp (iuξ_j) L[v](R + iu)χ^T^j−1(iR − u) du,

where L[v] denotes the bilateral Laplace transform of v(x), i.e.

L[v](z) = Z∞

−∞

e^−zxv(x) dx, z = R + iu ∈ C, u ∈ R.

Proof. Basically, we use the same arguments as in Theorem 3.2 in Raible (2000). Given ξ_j = ln (K/L(0, T_j−1)), we can rewrite equation (3.1.4) as

Cpl (0, T_j, K) = δB(0, T_j) Z∞

−∞

(L(0, T_j−1)e^x− K)⁺ρ(x) dx =

= δB(0, T_j)K Z∞

−∞

µL(0, T_j−1) K e^x− 1

¶₊

ρ(x) dx

= δB(0, T_j)K Z∞

−∞

v (ξ_j− x) ρ(x) dx

= δB(0, T_j)K(v ∗ ρ)(ξ_j) := V (ξ_j).

Note that x → e^−Rxv(x) is bounded and Z∞

−∞

e^−Rx|v(x)| dx = Z0

−∞

e^−Rx(e^−x− 1) dx = 1

R²+ R < ∞.

Furthermore,

Z∞

−∞

e^−Rx|ρ(x)| dx = χ^T^j−1(iR) < ∞.

Hence, we can apply Theorem B.2 in Raible (2000) to the functions v(x) and ρ(x) and express the bilateral Laplace transform of their convolution as the product of Laplace transforms of the factors.

L[V ](z) = δB(0, T_j)K L[v](z) L[ρ](z), z = R + iu, u ∈ R. (3.1.5) Since the integral that defines the bilateral Laplace transform L[V ](z) converges ab-solutely and ξ_j → V (ξ_j) is a continuous function, we can determine V_j by inverting the Laplace transform (see Theorem B.3 in Raible (2000)). In particular,

V (ξ_j) = 1 2πi

R+i∞Z

R−i∞

exp (ξ_jz) L[V ](z) dz

= 1 2π

Z∞

−∞

exp (ξ_j(R + iu)) L[V ](R + iu) du

= 1 2π lim

Y →∞

−Y

exp (ξ_j(R + iu)) L[V ](R + iu) du

if the limit exists. Equation (3.1.5) yields

V (ξ_j) = δB(0, T_j)Kexp (ξ_jR)

2π lim

Y →∞

−Y

exp (iuξ_j) L[v](R + iu) L[ρ](R + iu) du . (3.1.6)

Note that

L[ρ](R + iu) = Z∞

−∞

exp (−(R + iu)x) ρ(x) dx = Z∞

−∞

exp (i(iR − u)x) ρ(x) dx

= E_PTj

£exp¡

i(iR − u)X_T_j−1¢¤

= χ^T^j−1(iR − u).

Furthermore, for z ∈ C such that Im (z) = R we have

¯¯χ^T^j−1(z)¯

¯ = ¯

¯E_PTj

£exp¡

izXTj−1

¢¤¯¯

= ¯

¯E_PTj

£exp¡

(i Re (z) − Im (z)) X_T_j−1¢¤¯¯

≤ E_PTj

£¯¯exp¡

(i Re (z) − Im (z)) X_T_j−1¢¯¯¤

≤ E_PTj

£exp¡

− Im (z) X_T_j−1¢¤

= E_PTj

£exp¡

−RX_T_j−1¢¤

= χ^T^j−1(iR).

According to Raible (2000), we have

L[v](R + iu) = 1

(R + iu)(R + 1 + iu). One can easily check that R^∞

−∞

|L[v](R + iu)| du < ∞. Since |exp (iuξj)| = 1 and χ^T^j−1(iR) does not depend on u, the integral in (3.1.6) converges absolutely and the proof is complete.

¤ Consider a time-inhomogeneous L´evy process eL^T_t^j given by

Le^T_t^j := L^T_t^j − Zt

b^T_s^jds = Zt

R^d

µ^L− ν_T^L_j

(ds, dx).

Note that it has the same jumps as the process L^T_t^j, i.e ∆eL^T_t^j = R

R^d

xµ^L({t}, dx), but in contrast to L^T_t^j, we have E_PTj

hLe^T_t^j i

= 0, since eL^T_t^j is a P^T^j–martingale. Obviously, the characteristics of eL^T_t^j are given by

B_t^Lê = 0, C_t^Lê = 0, ν^Lê(dt, dx) = ν_T^L_j(dt, dx).

From Lemma 2.2.2 and Assumption 2.2.13 we infer that E_PTj

h exp

³ u^>Le^T_t^j

´i

< ∞ for all u with |u| ≤ (1 + ²)M^T^j.

Let us denote by χ^L_T_j(t, u) the characteristic function of eL^T_t^j, which in the purely dis-continuous case is given by

χ^L_T_j(t, u) = exp



 Zt

R^d

£exp¡ iu^>x¢

− 1 − iu^>x¤

ν_T^L_j(ds, dx)





= exp



 Zt

R^d

£exp¡ iu^>x¢

− 1 − iu^>x¤

β (s, x, T_j, T^∗) ν_T^L^∗(ds, dx)



 .

Proposition 3.1.3 Suppose that Assumption 2.2.13 holds. Then χ^T^j−1(iR) < ∞ holds for Proof. Making use of Lemma 2.2.2 combined with Assumption 2.2.13, we obtain that

χ^T^j−1(iR) = E_PTj

Hence, χ^T^j−1(iR) < ∞. Note that with respect to Assumption 2.2.13 the constants M, M, and fM = M/(1 + ²) are chosen so that for each j = 1, . . . , N + 1 holds

M − (N − j + 1) fM ≥ M − (N + 1) fM > M, therefore M −(N −j+1) fM −M

M > 0 for each j = 1, . . . , N + 1.

To show that equality (3.1.7) holds, we first have to check that the right-hand side of (3.1.7) is well defined, i.e. we have to prove that

χ^L_T_j(T_j−1, zλ (t, T_j−1)) < ∞ and χ^L_T_j(T_j−1, zλ (t, T_j−1)) < ∞.

With respect to Proposition 2.2.5 the process µ special. Hence, from Lemma 1.4.3 it follows that

is a complex valued exponentially special semimartingale. Applying Lemma 1.4.12, we get χ^L_T_j(T_j−1, zλ (t, T_j−1))

= exp





TZj−1

R^d

¡exp¡

izλ(s, Tj−1)^>x¢

− 1 − izλ(s, Tj−1)^>x¢

ν_T^L_j(ds, dx)



 . (3.1.8)

Let us now consider the second function on the right-hand side of (3.1.7). Due to the fact that |λ(t, T_j−1)| ≤ M < M^T^j for all t ∈ [0, T_j−1], the function χ^L_T_j(T_j−1, −iλ (t, T_j−1)) is well defined, and with Lemma 1.3.4, we obtain

χ^L_T_j(T_j−1, −iλ (t, T_j−1)) := E_PTj



exp





TZj−1

λ(s, T_j−1)^>deL^T_s^j









= exp





Tj−1

R^d

¡exp¡

λ(s, T_j−1)^>x¢

− 1 − λ(s, T_j−1)^>x¢

ν_T^L_j(ds, dx)



 .

(3.1.9)

Plugging equations (3.1.8) and (3.1.9) into the right-hand side of (3.1.7), one can easily check that equality (3.1.7) holds. This proves our assertion. ¤

In a completely analogous way, as it was done in Theorem 3.1.2, we can calculate the price of an interest rate floorlet.

Corollary 3.1.4 With ξ_j = ln (K) − ln (L(0, T_j−1)) and R ∈

0,M −(N −j+1) fM M

, the price of the j-th floorlet at time t = 0 is given by

Flr (0, T_j, K) := δB(0, T_j)Kexp (ξ_jR) 2π

Z∞

−∞

exp (iuξ_j) χ^T^j−1(iR − u)

(R + iu)(R + 1 + iu)du . At the first glance, the formulae for the price of the j-th caplet and corresponding floorlet are the same. But the difference lies in the values of parameter R. To price put options, we must choose R > 0 (see Table 3.1 in Raible (2000)). Of course, χ^T^j−1(iR) < ∞ should hold as well.

Remark 3.1.5 If the jumps magnitude |x| of the driving time-inhomogeneous L´evy process and the upper bound M of the domestic Libor rate volatility structure are sufficiently small, we can approximate the coefficient β(·, x, T_j, T^∗) given by equation (2.2.90) in the following way

β(t, x, T_j, T^∗) ≈ YN k=j

£l(0, T_k) exp¡

λ(t, T_k)^>x¢

+ 1 − l(0, T_k)¤

≈ YN k=j

£l(0, Tk)(1 + λ(t, Tk)^>x) + 1 − l(0, Tk)¤

≈ YN k=j

£1 + l(0, T_k)λ(t, T_k)^>x¤

≈ YN k=j

exp¡

l(0, T_k)λ(t, T_k)^>x¢

≈ exp¡

ψ_j(t)^>x¢ ,

where ψj(t) := P^N

k=j

l(0, Tk)λ(t, Tk). Then equation (3.1.3) simplifies to

ν_T^L_j(dt, dx) ≈ exp¡

ψ_j(t)^>x¢

ν_T^L^∗(dt, dx).

In document CUADRO COMPARATIVO Modificaciones en el Reglamento de la Ley de Relaciones Colectivas de Trabajo, realizadas por el Decreto Supremo Nº TR (página 42-46)