~Russian Bear’Z Blog~

This article may be useful for those people who don t know anything about the concept of volatility as well as for those who have been trading Vega for a long time. Volatility is a fixed magnitude that reflects the price alterations and depends on the way we measure them. In contrast to ATR (Average True Range) volatility is an absolute magnitude. It means that if ATR is 300 per share of Sberbank and 0.1 per share of RAO EES it doesn’t mean that concerning Sberbank the risk factor is higher or lower as the main role plays a capitalization ratio. Thus, knowing this magnitude we can draw a conclusion on what asset the price is more flexible.

Volatility can be of two types – historical and implied. Historical volatility is such a volatility the estimation of which is carried out on the grounds of historical data on this asset. The implied volatility is such a volatility that will reflect the price of the last option deal with the near paying off and strike if we place it in the formula of Black-Scholes. Consequently, the implied volatility makes sense only if we for this asset the options are traded.

Сontinuation:

Knowing the historical data we can estimate the volatility differently. All such methods have different aims, for example: the least fault of the 5-day tardiness. Once I found a site in Internet concerning the determination of the levels for the most liquid American shares on the ground of the statistical understanding of volatility and you are offered to trade them using the strategy of rebound. Running a few steps forward I would like to explain that such a strategy is based on the consequence of the Central Limit Theorem that reads as follows: if it is known that volatility V is 30% and price P of the share XXX is 50, the price on the next day will lie in the range of:

[P-PV/(100√	365	), P+PV/(100√	365	)]

with the probability 0.68. And if the volatility is double the probability amounts to 0.95. The point is that in this situation we deal with the statistical probability without any probabilistic model (space). That’s why it is difficult to draw a conclusion about the results. The forthcoming volatility is quite another matter. As at the least there are few people who are interested in the probability of this event. So, if the historical volatility is of your interest we would examine the question in details

The first and perhaps the most frequently occurring method of valuation of the historical volatility for some period on the basis of the standard deviation is:

E =	n Σ i=1	ln	ci-1 ci

D =	n Σ i=1	(ln	ci-1 ci	- E)2

Vh = 100√	365D

n – period of computing
c_i – price of closing in the descending sequence (0 – today, 1 – yesterday, etc.).
Е – average logarithms of price.
D – standard deviation of prices.
V_h– historical volatility.

Consequently historical volatility is the probable deviation of price which can happen in a year.
Often for the upper value of the forthcoming prices this change-over is used:

Pexp = Pe	V/(100√ 365D )	= P+PV/(100√	365D	)

It is obvious that under the small values of V such an exactness is convenient for everybody.

The main disadvantage of this method is that the price at the beginning of the period could have changed first greatly and than weakly and that would give us the same result in the case of the opposite situation. For this purpose the regressive models of the formation of distributions ARCH/GARCH are used. The model of volatility ARCH(p) for the non-zero mathematical expectations is calculated in this way

δi-1 = ln	ci-1 ci	, для i=1,…, p, where p

Va = √	α0D + p Σ i=1 αi δi2

p – order of ARCH-model.
δ_i – difference between the closing prices.

αi – fixed factors of the model,	p Σ i=1	αi = 1.

D – standard deviation for some period.
V_a – volatility ARCH(p).

And in the model GARCH(p,q) is present the regression from the result:

Vgi = √	β0Va2 + q Σ i=1 βi(Vgi)2

βi – fixed factors of the model,	q Σ i=0	βi = 1.

V_gⁱ – volatility GARCH(p, q) in the descending order (0 – the valuation for today, 1 – yesterday, … q days ago).

We have only to select the coefficients α, β and we can use them for their purpose. Some specialists recommend α₀ 0.95 for some instruments and 0.98 for the others, but I believe you d better select the coefficients for yourself or for some specific aim. As for me I usually use GARCH (10,10) with the coefficients α(0.5, 0.075, 0.075, 0.075, 0.075, 0.05, 0.05, 0.025, 0.025, 0.025, 0.025) and β(0.4, 0.09, 0.09, 0.09, 0.09, 0.06, 0.06, 0.03, 0.03, 0.03, 0.03). In this way on the daily charts of RAO EES (fig. 1)looks such a model in a comparison with volatility calculated on the basis of standard deviation:

Fig. 1.
(Volatility GARCH (10, 10) – blue line, volatility on the base
of the standard deviation – red line)

In both cases the 20-day window was used. The fact that the blue line is above the red line is good as the forthcoming volatility at the last day was even greater – 32 %. And also the blue line has the advance dynamics. That is the code of this indicator in TS Omega:


{|||||||||||||||||||||||||||||||||||||||||||||||||||||
~ GARCH Volatility Indicator
~ EasyLanguage code for Omega TradeStation
~ whiteline Group Copyright (c) 2004
|||||||||||||||||||||||||||||||||||||||||||||||||||||}
Input:
    period(numeric);
var:
    d1(0.3),
    d2(0.3),
    d3(0.2),
    d4(0.1),
    d5(0.1),
    r1(0.3),
    r2(0.3),
    r3(0.2),
    r4(0.1),
    r5(0.1),
    ii(0),
    Base(0),
    Diff(0),
    AvgDiff(0),
    ARCH(0),
    GARCH(0);
Diff = log(c/c[1]);
AvgDiff = Average(Diff, period);
Base = 0;
for ii = 0 to period-1
begin
    Base = Base + 365*Square(Diff[ii] - AvgDiff)/period;
end;
ARCH = 0.5*Base + 0.5*365*(
    (Square(Diff[0])+Square(Diff[1]))*d1 +
    (Square(Diff[2])+Square(Diff[3]))*d2 +
    (Square(Diff[4])+Square(Diff[5]))*d3 +
    (Square(Diff[6])+Square(Diff[7]))*d4 +
    (Square(Diff[8])+Square(Diff[9]))*d5
    )/2;
GARCH = 0.4*ARCH + 0.6*(
    (GARCH[1]+GARCH[2])*r1 +
    (GARCH[3]+GARCH[4])*r2 +
    (GARCH[5]+GARCH[6])*r3 +
    (GARCH[7]+GARCH[8])*r4 +
    (GARCH[9]+GARCH[10])*r5
    )/2;
plot1(SquareRoot(GARCH), “GARCH Vola”, blue);


Once one of my friends suggested examining the volatility «upwards» and «downward» and added that it is a well-known fact. After several searches in Internet and in literature I have not found anything of the kind and on the contrary found the negation of existence of such differentiations. But I decided that this idea is not so bad and frankly speaking exactly this model I use in the option trade. Thus, this idea is this: it is necessary to divide the row Vⁱ in 2 components Uⁱ, Dⁱ, in such a way that for each i the equality holds true Vⁱ = (Uⁱ+Dⁱ)/2 and for the row Uⁱ the connection with the growth of the prices (correlation in this way) takes place while for Dⁱ – with the decrease. Everything is made for GARCH-model(Fig. 2.):

Fig. 2. Differentiation of volatility
(U – green line, D – red, V – blue)

In the middle are represented these lines – the volatility upwards, the volatility downward and the correlation with the price below. As indicated in the picture the correlation of GARCH-model takes place between the correlations of its components and then the average value is always equal to “the prototypa”.

Статьи по теме:
Разбиение волатильности

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