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Lorenzo Giordani
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Today’s yield curves
y
(
m
)
=
L
−
S
1
−
e
−
λ
m
λ
m
+
C
(
1
−
e
−
λ
m
λ
m
−
e
−
λ
m
)
y(m) = L - S\,\frac{1-e^{-\lambda m}}{\lambda m} + C\left(\frac{1-e^{-\lambda m}}{\lambda m} - e^{-\lambda m}\right)
y
(
m
)
=
L
−
S
λm
1
−
e
−
λm
+
C
(
λm
1
−
e
−
λm
−
e
−
λm
)
λ
=
0.731
\lambda = 0.731
λ
=
0.731
z
(
t
)
=
β
0
+
β
1
1
−
e
−
t
/
τ
t
/
τ
+
β
2
(
1
−
e
−
t
/
τ
t
/
τ
−
e
−
t
/
τ
)
z(t) = \beta_0 + \beta_1\,\frac{1-e^{-t/\tau}}{t/\tau} + \beta_2\left(\frac{1-e^{-t/\tau}}{t/\tau} - e^{-t/\tau}\right)
z
(
t
)
=
β
0
+
β
1
t
/
τ
1
−
e
−
t
/
τ
+
β
2
(
t
/
τ
1
−
e
−
t
/
τ
−
e
−
t
/
τ
)
z
(
t
)
=
β
0
+
β
1
1
−
e
−
t
/
τ
1
t
/
τ
1
+
β
2
(
1
−
e
−
t
/
τ
1
t
/
τ
1
−
e
−
t
/
τ
1
)
+
β
3
(
1
−
e
−
t
/
τ
2
t
/
τ
2
−
e
−
t
/
τ
2
)
\begin{aligned} z(t) = \beta_0 &+ \beta_1\,\frac{1-e^{-t/\tau_1}}{t/\tau_1} + \beta_2\left(\frac{1-e^{-t/\tau_1}}{t/\tau_1} - e^{-t/\tau_1}\right) \\ &+ \beta_3\left(\frac{1-e^{-t/\tau_2}}{t/\tau_2} - e^{-t/\tau_2}\right) \end{aligned}
z
(
t
)
=
β
0
+
β
1
t
/
τ
1
1
−
e
−
t
/
τ
1
+
β
2
(
t
/
τ
1
1
−
e
−
t
/
τ
1
−
e
−
t
/
τ
1
)
+
β
3
(
t
/
τ
2
1
−
e
−
t
/
τ
2
−
e
−
t
/
τ
2
)
P
^
i
=
∑
j
c
i
j
e
−
z
(
t
i
j
)
t
i
j
\hat P_i = \sum_j c_{ij}\, e^{-z(t_{ij})\, t_{ij}}
P
^
i
=
j
∑
c
ij
e
−
z
(
t
ij
)
t
ij
min
β
,
τ
∑
i
(
P
i
−
P
^
i
D
i
)
2
\min_{\beta,\,\tau}\ \sum_i \left(\frac{P_i - \hat P_i}{D_i}\right)^{2}
β
,
τ
min
i
∑
(
D
i
P
i
−
P
^
i
)
2
z
(
t
)
z(t)
z
(
t
)
f
(
t
)
=
d
d
t
[
t
z
(
t
)
]
f(t) = \tfrac{d}{dt}\big[\,t\,z(t)\big]
f
(
t
)
=
d
t
d
[
t
z
(
t
)
]
c
(
T
)
=
2
1
−
d
(
T
)
∑
j
d
(
t
j
)
c(T) = 2\,\frac{1 - d(T)}{\sum_j d(t_j)}
c
(
T
)
=
2
∑
j
d
(
t
j
)
1
−
d
(
T
)
d
(
t
)
=
e
−
z
(
t
)
t
d(t) = e^{-z(t)\,t}
d
(
t
)
=
e
−
z
(
t
)
t
β
0
\beta_0
β
0
β
1
\beta_1
β
1
β
2
\beta_2
β
2
β
3
\beta_3
β
3
τ
\tau
τ
τ
1
\tau_1
τ
1
τ
2
\tau_2
τ
2
β
\beta
β
i
i
i
c
i
j
c_{ij}
c
ij
P
i
P_i
P
i
D
i
D_i
D
i
t
t
t
t
j
t_j
t
j
T
T
T
L
L
L
S
S
S
C
C
C
m
m
m