Methods for determining value at risk
View Patent ↗A preferred embodiment comprises a method for determining value-at-risk based on tick-by-tick financial data. Major steps of the method comprise the following: (1) financial market transaction data is electronically received by a computer; (2) the received financial market transaction data is electronically; (3) a time series z is constructed that models the received financial market transaction data; (4) an exponential moving average operator is constructed; (5) an operator is constructed that is based on the exponential moving average operator; (6) a causal operator Ω[z] is constructed that is based on the iterated exponential moving average operator; (7) values of predictive factors are calculated; (8) the values calculated by the computer are stored in a computer readable medium, and (9) value-at-risk is calculated from the values stored in step (8).
1. A method of determining value-at-risk, comprising the steps of:
electronically receiving financial market transaction data over an electronic network;
electronically storing in a computer-readable medium said received financial market transaction data;
constructing an inhomogeneous time series z that represents said received financial market transaction data;
constructing an exponential moving average operator;
constructing an iterated exponential moving average operator based on said exponential moving average operator;
constructing a time-translation-invariant, causal operator Ω[z] that is a convolution operator with kernel ω and that is based on said iterated exponential moving average operator;
electronically calculating values of one or more predictive factors relating to said time series z, wherein said one or more predictive factors are defined in terms of said operator Ω[z];
electronically storing in a computer readable medium said calculated values of one or more predictive factors; and
electronically calculating value-at-risk from said calculated values.
2. The method of claim 1 , wherein said operator Ω[z] has the form:
Ω
[
z
]
(
t
)
=
∫
-
∞
t
ⅆ
t
′
ω
(
t
-
t
′
)
z
(
t
′
)
=
∫
0
∞
ⅆ
t
′
ω
(
t
′
)
z
(
t
-
t
′
)
.
3. The method of claim 1 , wherein said exponential moving average operator EMA[τ; z] has the form:
EMA
[
τ
;
z
]
=
μ
EMA
[
τ
;
z
]
(
t
n
-
1
)
+
(
v
-
μ
)
z
n
-
1
+
(
1
-
v
)
z
n
,
where
α
=
t
n
-
t
n
-
1
τ
,
μ
=
ⅇ
-
α
,
and
ν depends on a chosen interpolation procedure.
4. The method of claim 1 , wherein said operator Ω[z] is a differential operator Δ[τ] that has the form:
Δ[τ]=γ(EMA [ατ, 1 ]+EMA[ατ 2]−2 EMA [αβτ, 4]), where γ is fixed so that the integral of the kernel of the differential operator from the origin to the first zero is 1; α is fixed by a normalization condition that requires Δ[τ; c]=0 for a constant c; and β is chosen in order to get a short tail for the kernel of the differential operator Δ[τ].
5. The method of claim 4 wherein said one or more predictive factors comprises a return of the form r[τ]=Δ[τ; x], where x represents a logarithmic price.
6. The method of claim 1 wherein said one or more predictive factors comprises a momentum of the form x−EMA[τ; x], where x represents a logarithmic price.
7. The method of claim 1 wherein said one or more predictive factors comprises a volatility.
8. The method of claim 7 wherein said volatility is of the form:
Volatility[τ,τ′, p;z]=MNorm
Volatility
[
τ
,
τ
′
,
p
;
z
]
=
MNorm
[
τ
2
,
p
;
Δ
[
τ
′
;
z
]
]
,
where
MNorm[τ,p;z]=MA[τ;|z| p ] 1/p , and
MA
[
τ
,
n
]
=
1
n
∑
k
=
1
n
EMA
[
τ
′
,
k
]
,
with
τ
′
=
2
τ
n
+
1
,
and where p satisfies 0<p≦2, and τ′ is a time horizon of a return r[τ]=Δ[τ; x], where x represents a logarithmic price.
9. The method of claim 1 , wherein said exponential moving average operator EMA[τ; z] has the form:
EMA
[
τ
;
z
]
=
μ
EMA
[
τ
;
z
]
(
t
n
-
1
)
+
(
v
-
μ
)
z
n
-
1
+
(
1
-
v
)
z
n
,
where
α
=
t
n
-
t
n
-
1
τ
,
μ
=
ⅇ
-
α
,
and
ν
=
1
-
μ
α
,
corresponding to a linear interpolation procedure.
10. The method of claim 1 , wherein said exponential moving average operator EMA[τ; z] has the form:
EMA
[
τ
;
z
]
=
μ
EMA
[
τ
;
z
]
(
t
n
-
1
)
+
(
v
-
μ
)
z
n
-
1
+
(
1
-
v
)
z
n
,
where
α
=
t
n
-
t
n
-
1
τ
,
μ
=
ⅇ
-
α
,
and
ν=1, corresponding to a previous point interpolation procedure.
11. The method of claim 1 , wherein said exponential moving average operator EMA[τ; z] has the form:
EMA
[
τ
;
z
]
=
μ
EMA
[
τ
;
z
]
(
t
n
-
1
)
+
(
v
-
μ
)
z
n
-
1
+
(
1
-
v
)
z
n
,
where
α
=
t
n
-
t
n
-
1
τ
,
μ
=
ⅇ
-
α
,
and
ν=μ, corresponding to a next point interpolation procedure.