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УДК 528.1
CRUSTAL MOTION DERIVED FROM GPS
MEASUREMENTS IN NORTH AND
NORTHEAST CHINA: IMPLICATIONS FOR
TECTONICS
Guojie Meng
First Crust Monitoring and Application Center
China Earthquake Administration, Tianjing, China
E-mail:
mengguojie66@gmail.com
Xinkang Hu, Xiaoning Su, Honglin Jin, Guangyu Fu
Institute of Earthquake Science
China Earthquake Administration, Beijing, China
E-mail:
mengguojie66@gmail.com
Keywords:
: GPS measurement, strain accumulation, coseismic and postseismic
deformation.
North China and Northeast China Northeast China are generally considered to
be with relatively stable continental interior, in contrast to the other regions of
China. Nonetheless, the two areas are characterized by intermediate to strong
seismic activity. More than 2 scores of M > 7.0 earthquakes took place in North
China in last several millenniums, and several deep-focus earthquakes have been
recorded in the southernmost part of northeast China, adjoining the Far East,
Russia. We have collected GPS observations from 1999 to 2011, and derived an
integrated velocity field using GAMIT/GLOBK software. We have re-calculated
velocities with respect to the assumed AMU microplate, and reconfirmed almost
absence of significant relative movements within the area under study. For the
north China, composed of 4 tectonic microblocks and 3 active belts, we employ
an integrating model of blocks and dislocations to fit the observed GPS velocities
by solving an optimal problem using the simulated annealing algorithm. The
following conclusions are drawn from the modeling: The four main blocks show
very close Euler poles with clockwise angular rotation velocities. The 3 active
tectonic belts are found to have different motions in terms of slip rate and locking
depth. All the 3 tectonic belts display relatively high locking ratio, implying
that they are prone to accumulate strain. We also document the far-field co-
seismic and post-seismic displacements of the 2011 Great Tohoku earthquake
detected by GPS stations in the study area. About 2300 km away from the Great
Tohoku earthquake, the co-seismic offsets recorded by GPS sites are around
10mm. Post-seismic deformation displays a heterogeneous pattern, implying a
complex causative mechanism.
Сборник материалов XXXVII Дальневосточной Математической Школы-Семинара
имени академика Е.В. Золотова, Владивосток, 8 – 14 сентября 2013 г.
УДК 519.248:62-192+519.176
BOUNDS FOR PRICES OF FINANCIAL
ASIAN-TYPE OPTIONS
Alexander Novikov
University of Technology, Sydney, Australia and Steklov Mathematical Institute of RAS, Russia
PO Box 123, Broadway, Department of Mathematical Sciences, University of Technology,
Sydney, NSW 2007
E-mail:
Alex.Novikov@uts.edu.au
Nino Kordzakhia
Macquarie University, Sydney, Australia
Balaclava Rd, North Ryde NSW 2109, Australia
E-mail:
nkordzak@hotmail.com
Keywords:
Asian options; Lower and upper bounds; VWAP.
In the context of dealing with financial risk management problems it is desirable
to have accurate bounds for option prices in situations when pricing formulae do
not exist in the closed form. A unified approach for obtaining upper and lower
bounds for Asian-type options is proposed in this paper. The bounds obtained
are applicable to the continuous and discrete-time frameworks for the case of
time-dependent interest rates. A numerical example is provided to illustrate the
accuracy of the bounds.
Introduction
We aim to obtain accurate bounds for option prices
C
T
=
Ee
−
R
T
F
T
(
S
)
,
where
R
t
=
Z
t
0
r
s
ds
,
r
s
is an interest rate,
F
T
(
S
)
is an Asian-type payoff of the option
written on the stock price
S
= (
S
t
,
0
6
t
6
T
)
, T
is the maturity time. (We assume that
all random processes are defined on the same
filtered probability space (
Ω
,
{
F
t
}
t
>
0
, P
)
).
The typical payoff for Asian-type options is
F
T
(
S
) = (
T
Z
0
(
S
u
−
K
)
dµ
(
u
))
+
,
(1)
where
x
+
= max[
x,
0] = (
−
x
)
−
for any
x, K
is a fixed strike,
µ
(
u
)
is a distribution
function on the interval
[0
, T
]
.
Using the notation
h
=
T
Z
0
h
u
µ
(
du
)
, h
∈
H,
Сборник материалов XXXVII Дальневосточной Математической Школы-Семинара
имени академика Е.В. Золотова, Владивосток, 8 – 14 сентября 2013 г.
257
where
H
is the class of adapted random processes
h
= (
h
s
,
0
6
s
6
T
)
such that
Z
T
0
|
h
u
|
µ
(
du
) =
|
h
|
<
∞
a.s.,
we can rewrite (1) as follows
F
T
(
S
) = (
S
−
K
)
+
= (
S
−
K
)
+
.
(2)
In relation to discretely monitored options (
DMO
) or continuously monitored options
(
CMO
) the distribution function
µ
can be discrete or continuous respectively. This setup
also includes the case of call options on the volume-weighted average price (VWAP), that
is
A
T
:=
P
t
j
6
T
S
t
j
U
t
j
P
t
j
6
T
U
t
j
, F
T
(
S
) = (
A
T
−
K
)
+
,
where
U
t
j
is a traded volume at the moment
t
j
. By setting
µ
(
u
) =
P
t
j
6
u
U
t
j
P
t
j
6
T
U
t
j
,
0
6
u
6
T
we obtain the representations (1) and (2) for options on VWAP. The presentation of
classical Asian payoffs in the form (1) was mentioned by Rogers and Shi [9] and Veˇ
ceˇ
r
([13]) where they used the PDE approach for finding
C
T
for CMO under the geometric
Brownian motion (
gBm
) model and constant interest rates. The paper [9] generated a
flow of related results about lower and upper bounds under different settings. We would
like to mention here the pioneering paper by Curran [3] and the unpublished paper by
Thompson [11]; in fact, the latter contains some ideas which we are developing further
here. A similar approach was used by Albrecher et al ([1]). In addition, among other
recent papers on this topic we would like to mention the paper by Chen and Lyuu [2]
containing many numerical results for CMO under the gBm model, and the paper by
Lemmens et al [6] which discusses DMO based on bounds for geometric Levy processes. In
[6] comparisons to other approaches were presented; in particular, among other methods,
comparisons to the recursive integration method developed by Fusai and Meucci [4] and
the method utilising comonotonic bounds (e.g. [12]) were given. Note that all above cited
papers assume that the interest rate process is constant. The case of floating strikes, that
is options with the payoff
F
T
(
S
) = (
S
−
S
T
)
+
,
can be easily reduced to the case (1) and
is not discussed here. Below we develop a unified approach to obtaining lower and upper
bounds for Asian-type DMO and CMO including VWAP with general time-dependent
interest rates.
1.
Lower and Upper bounds
The main technical result, which we use for the derivation of lower and upper bounds
below, is given in the following
Theorem 1.
Let
z
be a real number. Then
C
T
= sup
z,h
∈
H
Ee
−
R
T
(
S
−
K
)
I
{
h > z
}
(3)
= inf
h
∈
H
Ee
−
R
T
(
S
−
K
(1 +
h
−
h
))
+
(4)
where both supremum and infimum are attained by taking
h
u
=
S
u
/K
(5)
Сборник материалов XXXVII Дальневосточной Математической Школы-Семинара
имени академика Е.В. Золотова, Владивосток, 8 – 14 сентября 2013 г.
258
and
z
= 1
.
Further we use the notation
X
t
:= log(
S
t
/S
0
)
and assume that the discounted process
e
−
R
t
S
t
=
S
0
e
X
t
−
R
t
is a martingale with respect
to the filtration
{
F
t
}
t
>
0
, as required by the non-arbitrage theory (see e.g. [5]). Theorem
1 implies that, for all
h
∈
H ,
the following lower and upper bounds hold
C
T
>
LB
:=
S
0
sup
z
Ee
−
R
T
(
e
X
−
K
S
0
)
I
{
h > z
}
,
(6)
C
T
6
U B
:=
S
0
Ee
−
R
T
(
e
X
−
K
S
0
(1 +
h
−
h
))
+
.
(7)
To find a process
h
producing accurate bounds we need to take into account a complexity
of calculations of the joint distribution of
(
X, h, h
)
.
Obviously, the problem can be made
computationally affordable when
h
u
is a linear function of
X
u
, that is under the choice
h
u
=
a
(
u
)
X
u
+
b
(
u
)
with some nonrandom functions
a
(
u
)
and
b
(
u
)
. Since both inequalities (6) and (7) are,
in fact, equalities when (5) holds, one may try to match the first moments of
h
u
and
S
u
/K
that is to set
Eh
u
=
E
(
S
u
/K
)
, V ar
(
h
u
) =
V ar
(
S
u
/K
)
.
Here we apply a simpler choice with
a
(
u
) =
a
=
const
and
b
(
u
) = 0
i.e.
h
u
=
aX
u
(8)
where the constant
a
needs to be chosen in the upper bound. For the latter case we have
C
T
>
LB
1 :=
S
0
sup
z
Ee
−
R
T
(
e
X
−
K
S
0
)
I
{
X > z
}
,
(9)
C
T
6
U B
1 :=
S
0
inf
a
Ee
−
R
T
(
e
X
−
K
S
0
(1 +
aX
−
aX
))
+
.
(10)
Note that the calculation of the lower bound (9) does not depend on a choice of the
constant
a
.
2.
The case of Gaussian returns
Here we suppose that the process
X
= (
X
u
,
0
6
u
6
T
)
is Gaussian. To simplify
the exposition we also suppose that the process
r
t
is nonrandom. The case of stochastic
interest rates which are independent of
S
t
,
can be treated in a similar way. The pair
(
X
u
, X
)
, obviously, has a Gaussian distribution with
Cov
(
X
u
, X
) =
T
Z
0
Cov
(
X
u
, X
s
)
dµ
(
s
)
,
(11)
V ar
(
X
) =
T
Z
0
T
Z
0
Cov
(
X
u
, X
s
)
dµ
(
u
)
dµ
(
s
)
.
(12)
Below we consider a numerical example which corresponds to the gBm model with
X
u
=
R
u
+
σW
u
−
σ
2
/
2
u,
Сборник материалов XXXVII Дальневосточной Математической Школы-Семинара
имени академика Е.В. Золотова, Владивосток, 8 – 14 сентября 2013 г.
259
where
W
u
is a standard Bm. For the case of DMO we assume that
µ
(
u
)
is an uniform
discrete distribution on (0,T] with jumps at points
u
i
=
i
N
T, i
= 1
, ..., N,
where
N
is the number of time units (e.g. trading days). From (11) we obtain
κ
(
u
i
) :=
cov
(
W
u
i
, W
) =
N
X
j
=1
min(
u
i
, s
j
)
T /N
=
u
i
(
T
−
u
i
2
+
T
2
N
)
,
V
N
:=
V ar
(
W
) =
T
3
(1 +
3
2
N
+
1
2
N
2
)
,
Note that letting
N
→ ∞
one can obtain the characteristics needed for the pricing
of CMO as well. For numerical illustrations and comparisons we consider the set of
parameters
S
0
=
K
= 100
, σ
= 0
.
3
,
the interest rate
r
s
= 0
.
09(1 +
c/
2 sin(2
πs
))
,
(13)
where the parameter
c
= 0
or
c
= 1
.
One can speed up calculations of the bounds using
the function
erf c
(
x
)
. For example, using the Girsanov transformation we have obtained
the following expression for the lower bound
LB
1 =
e
−
R
T
S
0
2
T N
max
z
[
X
i
e
R
ui
erf c
{
p
V
N
/
2(
z
−
σκ
(
u
i
))
} −
K
S
0
erf c
{
p
V
N
/
2
z
}
]
.
It takes less than a quarter of second with Mathematica for any
σ
to find this lower
bound. Computing the upper bounds UB2 is also relatively fast (up to 7 seconds using
Mathematica for fixed
a
) but essentially slower with use of the command
FindMinimum
in Mathematica. The optimal value of
a
for the upper bound (10) is usually found in the
interval
(0
.
7
,
1)
. In fact, we found that UB1 with the choice
a
= 1
produces a reasonable
accuracy (the errors are less than 0.5% for wide ranges of
T, σ
and
K
). In Table 1 the
numerical results for LB1 and UB1 obtained with Mathematica are reported with three
decimal digits. We provide the calculated bounds for two cases
c
= 0
and
c
= 1
in (13);
the results for
c
= 1
are formatted in bold and placed in brackets. As an estimate for
the price we consider a midpoint of the interval
(
LB
1
, U B
1)
:
ˆ
C
T
=
LB
1 +
U B
1
2
.
The following bound is valid for the relative error of
ˆ
C
T
:
|
ˆ
C
T
/C
T
−
1
|
100%
6
(
U B
1
/LB
1
−
1)50%
.
Table 1.
T
N
LB
1
U B
1
error
%
f or
ˆ
C
T
1
10
50
∞
12.162
(12.135)
11.782 (
11.785)
11.718 (
11.741
)
12.259 (
12.239
)
11.829 (
11.807
)
11.731
(11.769)
0.4 (
0.42
)
0.1
(0.11)
0.03
(0.11)
9
10
50
∞
56.344
(60.769)
56.073 (
60.066)
56.012 (
60.014)
57.233
(61.568)
56.419 (
60.506)
56.146 (
60.197)
0.78
(0.68)
0.3
(0.37)
0.1
7 (0.15)
As it might be anticipated, the prices for options with longer maturities (here
T
= 9
)
depend essentially on the term structure of interest rates.
Сборник материалов XXXVII Дальневосточной Математической Школы-Семинара
имени академика Е.В. Золотова, Владивосток, 8 – 14 сентября 2013 г.