Volume 03 Issue 10-2023
310
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
310-315
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
A
BSTRACT
This article outlines asset pricing models. In these models, the price of an asset changes randomly over
time. The initial models are very simple
–
price fluctuations are binomial. Based on these models, more
complex ones are shown, which already have practical significance and are used in real financial
calculations.
K
EYWORDS
Floating interest rate, random variable, bank account, equation for stock price dynamics, Brownian motion,
payoff function.
I
NTRODUCTION
The simplest binomial model
There is opinion among practitioners
–
financiers
the prices follow certain rhythms, cycles, trends.
Nowadays, with the development of computer
technology and computer networks that connect
the whole world into a single whole, price
behavior can be seen on a computer screen in real
Journal
Website:
http://sciencebring.co
m/index.php/ijasr
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Research Article
STOCHASTIC ASSET PRICING MODELS
Submission Date:
October 20, 2023,
Accepted Date:
October 25, 2023,
Published Date:
October 30, 2023
Crossref doi:
https://doi.org/10.37547/ijasr-03-10-48
Saipnazarov Shaylovbek Aktamovich
Associate Professor, Candidate Of Pedagogical Sciences, Tashkent State University Of Economics,
Uzbekistan
Sultanmuratova Dilrabo Shaylavbekovna
Senior Lecturer Of Tashkent International University Of Financial Management And Technology,
Uzbekistan
Fayziyev Javlon Abduvoxidovich
Senior Lecturer Of Tashkent State University Of Economics, Uzbekistan
Volume 03 Issue 10-2023
311
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
310-315
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
time. The so-called technical analyses claims that
certain parts of the price charts are repeated, and
from the initial section of such a characteristic
pattern, one can understand how the chart will go
further. This is the possibility of predicting price
behavior. In order to answer the question of
whether price movements are predictable, many
studies have been carried out. They brought an
unexpected and paradoxical result: most likely,
prices
change
completely
randomly,
approximately in the same way as the speeds of
gas molecules change in chaotic Brownian
motion. This question has not been finally
resolved and, apparently, mill never be resolved,
since again and again successful financiers will
appear, confident that they can predict the future
behavior of prices.
This article outlines 4 asset pricing
models. In these models, the price of an asset
changes randomly over time. The first two models
are very simple
–
price fluctuations have only two
values, which is why these models are called
binomial. On the basis of these models, more
complex ones are built, which already have
practical significance and are used in real
financial calculations. In this model, the s
–
price
of an asset without any special restrictions, such
as the price of a bond with redemption (at the
time of maturity, the price is equal to the face
value of the bond), for example, this is the price of
a share. Let the unit of time be a day. Then the
price of the asset by the end of the n-th day will be
n
x
x
S
S
+
+
+
=
1
0
where
−
0
S
is the price at the
beginning of the observation,
n
i
x
i
2
,
1
=
-
independent and equally distributed random
variables that take the values
1
,
1
+
−
with a
probability of 0,5.
Figure
–
1 shows the so-called binomial tree. Price behavior can be represented as a random
movement along this tree from left to right.
Let’s find the mathematica
l expectation and variance
of the random variable
n
S
. We have
=
=
+
=
n
i
i
n
S
x
M
S
M
S
M
1
0
0
S
S
0
1
2
3
Volume 03 Issue 10-2023
311
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
310-315
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
since the mathematical expectation of each random
i
x
is 0. Further, due to the
fig
–
1. Binomial tree. independence of the random variable and
i
x
, the variance of their sum is
equal to the sum of their variances. But the variance of each random variable
i
x
is 1, hence
n
S
D
n
=
.
Denote
n
x
x
+
+
1
by
n
X
. The probability that out of “
n
” random variable
i
x
k
took the value +1,
and the remaining
(
)
k
n
−
took the value
–
1, is equal to
( )
n
k
n
C
5
,
0
. The distribution series
3
2
1
,
,
X
X
X
are
shown in Fig
–
2.
1
X
-1
1
2
X
-2
0
2
3
X
-3
-1
1
3
P
0,5
0,5
P
0,25
0,5
0,25
P
0,125 0,375 0,378 0,125
Fig
–
2. Distribution series
3
2
1
,
,
X
X
X
For
10
n
, one can already use the central theorem, which states that the sun of a large number of
independent and identically distributed terms is approximately distributed according to the normal law.
(
)
(
) (
)
n
n
S
S
P
n
−
−
0
where
is the Laplace function. It follows that for
(
)
9973
,
0
3
10
0
=
−
n
S
S
P
n
n
.
In particular, with
16
=
n
we have
(
)
9973
,
0
12
0
=
−
S
S
P
n
, i.e. in 16 days the price will change by no more
than 12 units (it is assumed that
0
S
significantly exceeds 12).
In this simplest model, prices cannot rise systematically, as, for example, the price of a zero-coupon
bond rises as it nears redemption. It is also clear that the expected return on an asset is 0. Therefore, the
risk-free rate must be equal to 0 (many observations show that the expected return on any risky asset
cannot be less than the risk-free rate). All these considerations make this model suitable only for some
explanatory illustrative calculations.
Binomial Cox-Ross-Rubinstein model
Suppose
that we have two types of assets at our disposal. A bank account of value “
B
” with a constant
interest rate “
r
”, such that its value at the end of the
n
–
th time period is equal to
(
)
0
1
B
r
B
n
n
+
=
and an
asset of value
S
with a random rate of return
i
f
. Here the rates
i
f
- are independent and identically
distributed random variables, taking two values -
b
a
,
, and
b
r
a
with probabilities
q
and
p
, i.e. the
interest rate is floating. In this case, the price of the asset at time
n
is equal
(
)
i
n
i
f
S
t
+
=
1
1
0
0
.
Volume 03 Issue 10-2023
312
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
310-315
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
Let
1
1
,
1
−
=
−
=
b
, where
1
, we have
=
=
=
−
−
−
a
f
if
S
b
f
if
S
S
n
n
n
n
n
,
,
1
1
1
If we introduce
a
random variable
1
=
т
with probability
q
and
p
, them
n
S
S
n
+
+
=
1
0
Obviously, in this case, the price of the asset
S
wanders over the set
n
k
S
k
,
1
,
0
=
.
Let’s find the
mathematical expectation of the price at the
n
–
th moment of time:
(
)
=
+
=
n
i
i
n
f
S
S
1
0
1
Since the random the value
(
)
n
i
f
i
,
,
1
,
1
=
+
, independent, the mean their works is equal to the
product of their mathematical expectations, so
(
)
=
+
+
=
+
=
n
i
n
i
n
bp
aq
S
f
M
S
S
M
1
0
0
1
1
(1)
Note that the securities market is called risk-neutral if investing in a bank account and in stocks gives, on
average, the same result. In our case, this means that if
0
0
B
S
=
, then the equality
(
)
bp
aq
S
r
S
n
+
+
=
+
1
1
0
0
. From here we can find the probability
p
, corresponding to such
a
market:
(
)
r
bp
p
a
=
+
−
1
or
a
b
a
r
p
−
−
=
General exponential binomial model
In the course of research on the behavior of prices, it was found that it is not the prices themselves that
randomly wander, but their logarithms, i.e.
n
H
n
l
S
S
0
=
where
n
n
h
h
H
+
+
=
1
and random variables are
(
)
n
i
h
i
,
,
1
=
- independent and approximately the
same.
From this we can conclude from the central limit theorem that the values of
n
H
for
10
n
are
distributed approximately according to the normal law. The parameters of this law: the mathematical
expectation and variance are completely determined by the mathematical expectations of the random
variable
i
h
and their variances. Let us replace discrete time with continuous time. Then, in particular, it
Volume 03 Issue 10-2023
313
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
310-315
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
turns out that for any moment
t
and any
t
T
natural logarithm of the price ratio
(
) ( )
t
s
T
t
S
/
+
is
distributed according to the normal law.
When the natural logarithm of a random variable is distributed according to the normal law, then
the distribution of the random variable itself is called lognormal. It can be proved that if
( )
Y
ln
is normally
distributed with parameters
G
,
, then
2
/
2
G
e
Y
M
+
=
and
( )
(
)
1
2
2
2
−
=
+
G
G
a
e
e
Y
D
. So, in the general
binomial model, the ratio of prices over any time interval is distributed lognormally.
Asset pricing models with
continuous time
The dynamics of the bank account
t
B
with continuous accrual of interest at the rate has the form:
0
,
0
=
t
e
B
B
t
t
Calculating the differential from both parts, we get
t
t
e
B
dB
0
=
or
dt
B
dB
t
t
=
From here we get:
t
B
dB
t
t
=
/
, i.e. the relative capital gain is proportional to time and interest rate.
Samuelson in 1965 introduced a similar equation for the dynamics of stock prices
t
S
:
( )
t
W
G
t
d
S
dS
t
t
+
=
,
(2)
where
- is the growth rate or rate of return,
G
- is the coefficient of volatility or random variability.
Randomness in price fluctuations is described by value
( )
t
w
d
- stochastic differential from the process of
Brownian motion
( )
t
w
, which is determined by the following properties:
1)
Process increments
( )
t
w
on non
–
overlapping time intervals are independent of each other;
2)
The increment
( )
( )
s
w
t
w
−
at
s
t
has zero mathematical expectation and variance equal to
s
t
−
, i.e.
( )
( )
(
)
( )
( )
(
)
s
t
s
w
t
w
D
s
w
t
w
M
−
=
−
=
−
,
0
3)
( )
0
0
=
w
It follows from the definition that
( )
(
)
(
) ( )
(
)
0
=
−
+
=
t
w
dt
t
w
M
t
w
d
M
and
( )
(
)
dt
t
w
d
D
=
The solution of stochastic equation (2) has the form
( )
( )
2
/
0
2
t
G
t
Gw
t
e
e
S
t
S
+
=
(3)
which is now commonly called geometric or economic Brownian motion.
It is easy to show that
( )
(
)
1
2
/
2
=
−
t
G
t
Gw
e
M
and therefore the market described by this model will be
risk neutral if
r
=
, because then
( ) ( )
(
)
t
B
t
S
M
=
with
0
0
B
S
=
. We also note that the first stochastic model
Volume 03 Issue 10-2023
314
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
310-315
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
of stock price dynamics was proposed by Bachelier in 1900, according to which
( )
( )
t
Gw
t
S
t
S
+
+
=
0
, i.e.
this model is close to the simplest binomial model.
Option Pricing in other stochastic models
Let
(
)
n
n
S
S
S
f
,
,
,
1
0
in the binomial cox
–
Ross
–
Rubinstein model be the payoff function describing the
obligations of the option seller. Then the fair price of option
n
С
(i.e., the premium that the buyer must pay
to the seller for the right to own this option) is found from the condition
(
)
(
)
n
n
n
n
S
S
S
f
M
r
С
,
,
,
1
1
0
*
=
+
,
(4)
where
*
M
is the mathematical expectation to the neutral market, i.e. when
a
b
a
r
P
P
−
−
=
=
*
Equality (4) means that the seller of the option, having received the amount
n
С
from the sale and deposited
it in
a
bank account, must, by time
n
, compensate on average his obligations to the buyer of the option,
i.e.
(
)
(
)
n
n
n
n
S
S
S
f
M
r
С
,
,
,
1
1
0
*
=
+
−
The right side of the last equality means the average costs of the option seller, discounted to the time of its
sale
(
)
0
=
n
. In the case of the European call option, the payoff function has the form
(
)
0
,
max
H
n
n
S
S
f
−
=
, where
H
S
is the contractual price, and then for
n
С
a more specific expression can be obtained
(
)
(
)
*
1
*
0
,
,
,
,
P
n
B
r
S
P
n
B
S
C
n
H
n
−
−
=
,
where
(
)
(
)
=
+
=
=
p
n
n
H
X
P
P
n
B
a
b
a
S
S
P
r
b
P
,
0
*
1
*
,
,
,
log
/
log
1
,
,
p
n
X
,
is a binomial distributed random variable taking values
n
k
,...,
1
,
0
=
with probabilities
.
k
n
k
k
n
q
p
C
−
In the Samuelsson model, the rational value of a call option
T
С
with payoff function
(
)
+
−
=
H
T
T
S
S
f
was obtained by Black and Scholes in 1973 and their famous formula is
( )
( )
n
T
E
t
d
e
S
d
S
С
2
1
0
−
=
−
,
(5)
where
,
,
/
2
log
1
2
2
0
1
T
G
d
d
T
G
T
G
r
S
S
d
E
−
=
+
+
=
( )
−
x
is the distribution function of the standard normal distribution.
Volume 03 Issue 10-2023
315
International Journal of Advance Scientific Research
(ISSN
–
2750-1396)
VOLUME
03
ISSUE
10
Pages:
310-315
SJIF
I
MPACT
FACTOR
(2021:
5.478
)
(2022:
5.636
)
(2023:
6.741
)
OCLC
–
1368736135
In the Black
–
Scholes formula, the value
( )
1
d
determines the sensitivity of the value of the option
C
to the value of the share
S
(when the stock price rises by 1 point (dollar, soum) the value of the option to
buy the right rises by
( )
1
d
points). In addition, a portfolio of one stock and
m
call options will be risk
–
free if
( )
1
/
1
d
m
=
. The value
( )
2
d
is interpreted as the probability of the option being realized, i.e. the
probability that the execution price
E
S
will exceed the share price at time
T.
Example.
Let the risk
–
free annual continuous interest rate
1
,
0
=
, the initial cast of the share
100
0
=
S
$, the maturity period 7 months (i.e.
365
/
210
=
T
). Find the national value of the option for the
right to buy at a price
70
=
E
S
$, if the volatility coefficient is
8
,
0
=
G
.
Solution.
Let’s use the Black –
Scholes formula. In our conditions
( )
( )
648
,
0
838
,
0
;
379
,
0
;
986
,
0
2
1
2
1
=
=
=
=
d
d
d
d
. From formula (5) we get
99
,
40
648
,
0
70
838
,
0
100
5753
,
0
1
,
0
=
−
=
−
e
C
$.
R
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Т
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.:
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, 2005.
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В
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