Price scales in financial markets
Scales refers to the currencies in finance, the fluctuating in time and
transactions occurs at random time with random intensities for this
reason great care must be taken in the selection of the most appropriate
variable to be studied, taking into a account to implicit assumption
associated with each possible choice.
The price unit of financial goods is usually the currency of the country
in which the particular financial markt is located. The value of the
currency is not constant in time. A currency can change its value
because of : inflation, economic growth and fluctuations in global
currency market. The economic growth of country is itself a random
variable.
Econophysics, 4th Year Levy stochastic process and limit theorem
Inflation
Inflation is the rate at which general level of prices for goods and
services is rising and consequently, the purchasing power of currency is
falling. So inflation is a sustained increase in the general price level of
goods and sevices in economy over a period of time. As inflation
increases, the percentage of goods and services that can be purchased
with the same amount of oney decreases.
Hyperinflation
In Hyperinflation, change in money and price level are so large indeed
hyperinflation is generally defined as inflation that exceeds 50% per
month. This means that the price level increases more than a hundred
fold over the cource of a year.
Econophysics, 4th Year Levy stochastic process and limit theorem
Economic Growth or Economic Recession
In economy a recession is a business cycle contracting. it is generally
slow down in economic activity. recession generally occurs when there is
a widespread drop in spending (an adverse demand shock). This may be
triggered by various events, such as financial crisis, an external trade
shock, an adverse supply etc.
A recession is a general down turn in any types of trades. A recession is
associated with high unemployement, slowly gross domestic product and
high inflation, high interest rate.
High inflation occurs due to limited liquidity.
Econophysics, 4th Year Levy stochastic process and limit theorem
Fluctuations of money (Fluctuations in the Global Currency)
Currency refers to money. The fluctuations of currency means different
of monitary values in different country. Fluctuation of currency depends
upon own country capacity How this country is develope, there
industriallized system, their trade system and economic trade system.
Basically, curency rate is self determined process but this is highly
influence by natural resources of particular country. GDP of this
country, monthly income of people, life style of people etc.
Econophysics, 4th Year Levy stochastic process and limit theorem
Investigate Price Change
Let us define 𝑦(𝑡) as the price of financial assest at time t. Then after
time Δ𝑡, the price value becomes 𝑦(𝑡 + Δ𝑡). Net change in price level is
given by
𝑧(𝑡) = 𝑦(𝑡 + Δ𝑡) − 𝑦(𝑡)
= 𝑦(𝑡) + 𝑦
𝜕𝑦
𝜕𝑡
+
𝑦
2
2!
𝜕
2
𝑦
𝜕𝑡
2
+ ....
− 𝑦(𝑡) using Taylor expansion
= 𝑦
𝜕𝑦
𝜕𝑡
(Price level change in equiibrium condition)
Merit of this approach:
1. Non-linear or stochastic transformation are not needed
2. The problem is that this definition is seriously affected by change in
scale.
Econophysics, 4th Year Levy stochastic process and limit theorem
Deflated or discounted price changes
𝑍
𝐷
(𝑡) = [𝑦(𝑡 + Δ𝑡) − 𝑦(𝑡)]𝐷(𝑡)
where D(t) is the deflection factor or a discounting factor.
Deflaction
When the overall price level decrease so that inflaction rate becomes
negative, it is called deflaction. It is negative of inflation. The merit of
this approach are that,
1. non-linear tranformations are not needed.
2. The gains possible with riskless investiments are accounted by D(t).
The problem is that deflactions and discounting factors are
unpredictable over the long term and three is no unique choice of D(t).
Econophysics, 4th Year Levy stochastic process and limit theorem
Analyze Returns
𝑅(𝑡) =
𝑦(𝑡 + Δ𝑡) − 𝑦(𝑡)
𝑦(𝑡)
=
𝑧(𝑡)
𝑦(𝑡)
The merit of this approach is that returns 𝑅(𝑡) provided a direct
percentage of gain or loss in a given time period. One can study the
successive difference of the natural logarithm of price.
𝑠(𝑡) = ln[𝑦(𝑡 + Δ𝑡)] − ln[𝑦(𝑡)]
= ln[𝑍 (𝑡) + 𝑦(𝑡)] − ln[𝑦(𝑡)] [∵ 𝑍 (𝑡) = 𝑦(𝑡 + Δ𝑡) − 𝑦(𝑡)]
= ln
1 +
𝑍 (𝑡)
𝑦(𝑡)
[∵ ln
(
1 + 𝑥
)
= 1 +
𝑥
2
2!
+
𝑥
3
3!
]
=
𝑍 (𝑡)
𝑦(𝑡)
= 𝑅(𝑡) [for high frequency data]
since, 𝑧(𝑡) is fast variance change, 𝑦(𝑡) is slow variance
Econophysics, 4th Year Levy stochastic process and limit theorem
So,
𝑅(𝑡) ≈ 𝐶
1
𝑍 (𝑡) [∵ 𝐶
1
=
1
𝑦(𝑡)
] (negligible)
More over if the total investigated time period is not too long then
𝐷(𝑡) ≈ 1
so,
𝑍 (𝑡) = 𝑍
𝐷
(𝑡)
For low inflation all four commonly used indicators are approximately
equal.
𝑆(𝑡) ≈ 𝑅(𝑡) ≈ 𝐶
1
𝑍 (𝑡) ≈ 𝑍
𝐷
(𝑡)
Econophysics, 4th Year Levy stochastic process and limit theorem
Time scale in financial market
We consider the approximate time scale to use for analyzing the market
data. Possible candidates for the correct time scale include:
The physical scale
The trading or market time
The number of transitions
As in the case of price scale unit, all the definations have merits and all
have problems. When examining price changes that take place
transaction occur. It is worth noting that each transaction occur at
random time involves random variable, the volume of the traded
financial goods.
Econophysics, 4th Year Levy stochastic process and limit theorem
Physical Time
Stock exchange close at night, over weekend and during holidays. A
similar limitations is also present in a global market such as foreign
exchange market. Although this market is active 24 hrs per day. The
social organization of business and the presence of biological cycles force
the markets activity to have temporal constraints in each financial
region of world. With the choice of physical time we donot know how to
model the stochastic dynamics of prices and arrival ofinformation during
hours in which the market is closed.
Econophysics, 4th Year Levy stochastic process and limit theorem
Trading Time
Trading time is well defined in stock exchanges – it is the time that
elapses during open market hours. In the foreign exchange market, it
coincides with the physical time. Empirical studies have tried to
determine the variance of log price changes observed from closure to
closure in financial markets. These studies show that the variance
determined by considering closure values of successive days is only
approximately 20% lower than the variance determined by considering
closure values across weekends. This empirical evidence supports the
choice of using trading time in the modeling of price dynamics. Indeed,
the trading time is the most common choice in research studies and in
the studies performed for the determination of volatility in option
pricing. However, problems also arise with this definition. Specifically,
(i) information affecting the dynamics of the price of a financial asset
can be released while the market is closed (or its activity is negligible in
a given financial area),
(ii) in high-frequency analyses overnight price changes are treated as
short- time price changes, and
Econophysics, 4th Year Levy stochastic process and limit theorem
(iii) the market activity is implicitly assumed to be uniform during
market hours.
This last assumption is not verified by empirical analyses. Trading
activity is not uniform during trading hours, either in terms of volume
or in number of contracts. Rather, a daily cycle is observed in market
data: the volatility is higher at the opening and closing hours, and
usually the lowest value of the day occurs during the middle hours.
Econophysics, 4th Year Levy stochastic process and limit theorem
Numericals
1. Check the stability
ln[𝜙(𝑞)] =
(
𝑖𝜇𝑞 − 𝛾|𝑞|
𝛼
[1 −𝑖𝛽
𝑞
|𝑞 |
𝑡𝑎𝑛(
𝜋
2
𝛼)]
𝑖𝜇𝑞 − 𝛾|𝑞| [1 +𝑖𝛽
𝑞
|𝑞 |
2
𝜋
ln |𝑞|]
2. If 𝑝(𝑥) =
4𝛽
𝜋
1
𝛽
2
+(𝑥−𝑎)
2
, find its characteristics.
Solution
Characteristics function is,
𝜙(𝑞) =
∫
∞
−∞
𝑒
𝑖𝑞𝑥
𝑝(𝑥)𝑑𝑥
=
∫
4𝛽
𝜋
𝑒
𝑖𝑞𝑥
𝛽
2
+ (𝑥 − 𝑎)
2
𝑑𝑥
=
4𝛽
𝜋
∫
𝑒
𝑖𝑞𝑥
𝛽
2
+ (𝑥 − 𝑎)
2
𝑑𝑥
=
4𝛽
𝜋
𝑒
𝑎
∫
𝑒
𝑖𝑞 (𝑥−𝑎)
𝛽
2
+ (𝑥 − 𝑎)
2
𝑑𝑥
Econophysics, 4th Year Levy stochastic process and limit theorem
put 𝑥 − 𝑎 = 𝑡 and 𝑑𝑥 = 𝑑𝑡
=
4𝛽
𝜋
𝑒
𝑎
∫
𝑒
𝑖𝑞𝑡
𝛽
2
+ 𝑡
2
𝑑𝑥
=
4𝛽
𝜋
𝑒
𝑎
𝜋
𝑎
𝑒
−𝑡𝑎𝑞
=
4𝛽
𝜋
𝑒
𝑎(1−𝑡𝑞)
∴ 𝜙(𝑞) =
4𝛽
𝑎
𝑒
𝑎(1−𝑞𝑡)
3. If 𝑃
𝐿
(𝑥) =
Γ (1+𝛼) sin
(
𝜋 𝛼
2
)
𝜋 |𝑥 |
1+𝛼
, find characteristics function and also
calculate 𝜙
𝑛+1
(𝑞)
Solution
𝜙(𝑞) =
∫
∞
−∞
𝑒
𝑖𝑞𝑥
𝑝(𝑥)𝑑𝑥
=
∫
∞
−∞
𝑒
𝑖𝑞𝑥
Γ(1 + 𝛼) sin
𝜋 𝛼
2
𝜋|𝑥|
1+𝛼
𝑑𝑥
Econophysics, 4th Year Levy stochastic process and limit theorem
=
Γ(1 + 𝛼) sin
𝜋 𝛼
2
𝜋
∫
∞
−∞
𝑒
𝑖𝑞𝑥
|𝑥|
1+𝛼
𝑑𝑥
=
Γ(1 + 𝛼) sin
𝜋 𝛼
2
𝜋
𝜋
1 + (1 + 𝛼)
2
∴ 𝜙(𝑞) = 𝛼Γ(𝛼) sin
𝜋𝛼
2
1
1 + (1 + 𝛼)
2
and 𝜙
𝑛+1
(𝑞) = 𝜙
1
(𝑞).𝜙
2
(𝑞).....[𝜙
𝑛
(𝑞)]
[𝑛+1]
4. Characteristics function 𝜙
2
(𝑞) = 𝑒
−2𝛾𝑞
2
, find its corresponding
probability distribution function.
5. Characteristics function 𝜙(𝑞) = 𝑒𝑥 𝑝[𝑖𝜇𝑞 −
𝜎
2
2
𝑞
2
] find 𝜙
𝑛+1
(𝑞) . Check
stable process.
6. If 𝑃(𝑥) =
𝑒
−𝑥
𝑥
𝛾−1
Γ (𝛾)
find 𝜙
𝑛+1
(𝑞) and find its mean.
Ans: 𝜙
𝑛+1
= (1 − 𝑖𝑞)
𝑛+1
,
¯
𝑥 = 0
Econophysics, 4th Year Levy stochastic process and limit theorem
7. Find the characteristics fucntion of Cauchy distribution.
We have, Cauchy distribution has probability density function as
𝑝(𝑥) =
𝑣
𝜋(𝑣
2
+ 𝑥
2
)
− ∞ < 𝑥 < ∞
Now, characteristics fucntion of Cauchy distribution is
𝜙
𝑥
(𝑡) =
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
𝑝(𝑥)𝑑𝑥
=
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
𝑣
𝜋(𝑣
2
+ 𝑥
2
)
𝑑𝑥
=
𝑣
𝜋
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
(𝑣
2
+ 𝑥
2
)
𝑑𝑥
Now, using standard integral we have,
𝜙
𝑥
(𝑡) =
𝑣
𝜋
𝜋
𝑣
𝑒
−𝑣 |𝑡 |
𝑡 > 0
∴ 𝜙
𝑥
(𝑡) = 𝑒
−𝑣 |𝑡 |
; 𝑡 > 0
Econophysics, 4th Year Levy stochastic process and limit theorem
8. Characteristics fucntion of laplace distribution
We have probability density function of laplace distribution
𝑓 (𝑥) =
1
2
𝑒
−|𝑥 |
− ∞ < 𝑥 < ∞
Now the characteristics function is
𝜙
𝑥
(𝑡) =
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
𝑓 (𝑥)𝑑𝑥
=
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
1
2
𝑒
−|𝑥 |
𝑑𝑥
=
1
2
∫
∞
−∞
𝑒
𝑖𝑡 𝑥
𝑒
−|𝑥 |
𝑑𝑥
=
1
2
∫
0
−∞
𝑒
𝑖𝑡 𝑥
𝑒
𝑥
𝑑𝑥 +
∫
∞
0
𝑒
𝑖𝑡 𝑥
𝑒
−𝑥
𝑑𝑥
=
1
2
∫
0
−∞
𝑒
𝑥 (1+𝑖𝑡 )
𝑑𝑥 +
∫
∞
0
𝑒
−𝑥 (1−𝑖𝑡 )
𝑑𝑥
Econophysics, 4th Year Levy stochastic process and limit theorem
=
1
2
"
𝑒
𝑥 (1+𝑖𝑡 )
(1 + 𝑖𝑡)
0
∞
+
𝑒
−𝑥 (1−𝑖𝑡 )
−(1 − 𝑖𝑡)
∞
0
#
=
1
2(1 + 𝑖𝑡)
𝑒
0
− 𝑒
−∞
−
1
2(1 − 𝑖𝑡)
𝑒
−∞
− 𝑒
0
=
1
2
1
1 + 𝑖𝑡
+
1
1 − 𝑖𝑡
=
1
2
2
1 + 𝑡
2
∴ 𝜙
𝑥
(𝑡) =
1
1 + 𝑡
2
9. Characteristics function of Normal distribution
Probability density function of Gaussian distribution is normal
𝑝(𝑥) =
1
𝜎
√
2𝜋
𝑒
−
𝑥 −𝜇
2 𝜎
2
Econophysics, 4th Year Levy stochastic process and limit theorem
=
𝑒
𝑖𝑞𝜇
√
2𝜋
∫
∞
−∞
𝑒
𝑖𝑘 𝑧
𝑒
−𝛼𝑧
2
𝑑𝑧 [where, 𝑞𝜎 = 𝑘 and 𝛼 =
1
2
]
=
𝑒
𝑖𝑞𝜇
√
2𝜋
∫
∞
−∞
cos 𝑘𝑧𝑒
−𝛼𝑧
2
𝑑𝑧 +𝑖
∫
∞
−∞
sin 𝑘𝑧𝑒
−𝛼𝑧
2
𝑑𝑧
=
𝑒
𝑖𝑞𝜇
√
2𝜋
∫
∞
−∞
cos 𝑘𝑧𝑒
−𝛼𝑧
2
𝑑𝑧
=
𝑒
𝑖𝑞𝜇
√
2𝜋
𝜋
𝛼
𝑒
−𝑘
2
/4𝛼
Now, putting value of 𝛼 and 𝑘
𝜙(𝑞) =
𝑒
𝑖𝑞𝜇
√
2𝜋
𝜋
1/2
𝑒
−𝑞
2
𝜎
2
/(4×1/2)
= 𝑒
𝑖𝑞𝜇
𝑒
−𝑞
2
𝜎
2
/2
∴ 𝜙(𝑞) = 𝑒
𝑖𝑞𝜇−𝑞
2
𝜎
2
/2
Econophysics, 4th Year Levy stochastic process and limit theorem
10. For the given probability distribution 𝑝(𝑥) = 𝑎𝑒
−|𝑥 |
; −∞ < 𝑥 < ∞
where a is a normalization constant. Show that 𝑎 =
1
2
, mean
(𝜇
′
1
) = 0; (𝜎) =
√
2
Solution: Here,
𝑝(𝑥) = 𝑎𝑒
−|𝑥 |
using normalization condition
∫
∞
−∞
𝑎𝑒
−|𝑥 |
= 1
=⇒ 𝑎
∗
∫
0
−∞
𝑒
−𝑥
𝑑𝑥 + 𝑎
∫
∞
0
𝑒
−𝑥
𝑑𝑥 = 1
=⇒ 2𝑎
∫
∞
0
𝑒
−𝑥
𝑑𝑥 = 1
=⇒ 2𝑎[−𝑒
−𝑥
]
∞
0
= 1
=⇒ 𝑎 =
1
2
∴ 𝑝(𝑥) =
1
2
𝑒
−|𝑥 |
Econophysics, 4th Year Levy stochastic process and limit theorem
By applying convolution theorem
𝑓 [𝑓 (𝑥) ∗ 𝑔(𝑥)] = 𝑜 𝑓 [𝑓 (𝑥)] ∗ 𝑜 𝑓 [𝑔(𝑥)]
= 𝐹 (𝑞)𝐺 (𝑞)
Let us take
𝑠
2
= 𝑥
1
+ 𝑥
2
The probability density function of two independent and identically
distributed random variable is given by convolution of two probability
density function of each random variables.
𝑃
𝑠
(𝑠
2
) = 𝑝(𝑥
1
) ⊗ 𝑝(𝑥
2
)
So, characteristics function of 𝑃(𝑠
2
) is,
𝜙
2
(𝑞) = 𝜙(𝑞) ⊗ 𝜙(𝑞)
= [𝜙(𝑞)]
2
Econophysics, 4th Year Levy stochastic process and limit theorem
If we take 𝑠 = 𝑥
1
+ 𝑥
2
+ ... + 𝑥
𝑛
i.e., random variable then characteristics
function of 𝑠
𝑛
is,
𝜙
𝑛
(𝑞) = 𝜙(𝑞).𝜙(𝑞)....𝜙(𝑞)
= [𝜙(𝑞)]
𝑛
So, characteristics fucntion for given function upto 𝑛𝑡ℎ order is,
𝜙
𝑛
(𝑞) =
𝑒
𝑖𝑞𝑏
− 𝑒
𝑖𝑞𝑎
𝑖𝑞(𝑏 − 𝑎)
𝑛
To show that given distribution is stable one;
𝜙
2
(𝑞) =
𝑒
𝑖𝑞𝑏
− 𝑒
𝑖𝑞𝑎
𝑖𝑞(𝑏 − 𝑎)
2
=
(𝑒
𝑖𝑞𝑏
− 𝑒
𝑖𝑞𝑎
)
2
−𝑞
2
(𝑏 − 𝑎)
2
Econophysics, 4th Year Levy stochastic process and limit theorem
By applying inverse fourier transform we got corresponding probability
density function is 𝑃
2
(𝑥). If we get 𝑃
2
(𝑥) identical with probability
density function of uniform distribution. The this distribution is called
stable otherwise not stable;
𝑃
2
(𝑥) =
1
2𝜋
∫
𝑏
𝑎
(𝑒
𝑖𝑞𝑏
− 𝑒
𝑖𝑞𝑎
)
2
−𝑞
2
(𝑏 − 𝑎)
2
𝑒
−𝑖𝑞𝑥
𝑑𝑥
=
1
−2𝜋𝑞
2
∫
𝑏
𝑎
(𝑒
𝑖𝑞𝑏
− 𝑒
𝑖𝑞𝑎
)
2
(𝑏 − 𝑎)
2
(cos 𝑞𝑥 − 𝑖 sin 𝑞𝑥)𝑑𝑥
=
1
−2𝜋𝑞
2
∫
𝑏
𝑎
𝑒
𝑖𝑞𝑏
− 𝑒
𝑖𝑞𝑎
𝑏 − 𝑎
2
cos 𝑞𝑥𝑑𝑥
which is not same with the uniform distribution. So, we can say that
this distribution is not stable.
Econophysics, 4th Year Levy stochastic process and limit theorem
