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4.2.2. Modelling of function of a bias volatilnosti on the basis of exchange options for the future of the Russian Open Society "United Power Systems" at the moment of an estimation vnebirzhevyh options



Bias function volatilnosti to be according to procedure described in item 2.8.

Quoted sizes of awards of exchange options koll and put on the future of the Russian Open Society "United Power Systems" with one date ekspiratsii and strikes corresponding to them as of 01.04.05 are that:
For options koll on the Russian Open Society "United Power Systems" future:
P = (872, 520, 310, 155, 85, 50),
SC = (7500, 8000, 8500, 9000, 9500, 10000),
For options put on the Russian Open Society "United Power Systems" future:
Q = (12, 26, 77, 168, 326, 578),
SP = (6000, 6500, 7000, 7500, 8000, 8500).
Basic data for a finding internal volatilnostej exchange options are presented as follows:
"With" – in case of a finding internal volatilnosti for exchange options koll on the Russian Open Society "United Power Systems" future;
«p» – In case of a finding internal volatilnosti for exchange options put on the Russian Open Society "United Power Systems" future;
\"\"M = 8204 roubles;
S a-strike from set of strikes of exchange options koll SC;
S = or
S - A strike from set of strikes of exchange options put SP;
T = 0,19 (69 days/365 of days in a year);
\"\"R = 0,347 %;
Pi - the quoted award of an exchange option koll;
СM =
Qi - The quoted award of an exchange option put;
On Newton's procedure – Rafsona by means of built in function Newton RaphsonCollectorVol ("with" or «p»; M; S; T; R; P) it is found internal volatilnosti for each exchange option:
For exchange options koll on the Russian Open Society "United Power Systems" future:
VC1=Newton RaphsonCollectorVol ("with"; 8204; 7500; 0,19; 0,00347; 872) = 31,42 %;
VC2=Newton RaphsonCollectorVol ("with"; 8204; 8000; 0,19; 0,00347; 520) = 28,98 %;
VC3=Newton RaphsonCollectorVol ("with"; 8204; 8500; 0,19; 0,00347; 310) = 30,41 %;
VC4=Newton RaphsonCollectorVol ("with"; 8204; 9000; 0,19; 0,00347; 155) = 26,69 %;
VC5=Newton RaphsonCollectorVol ("with"; 8204; 9500; 0,19; 0,00347; 85) = 31,09 %;
VC6=Newton RaphsonCollectorVol ("with"; 8204; 10000; 0,19; 0,00347; 50) = 32,96 %.
For exchange options put on the Russian Open Society "United Power Systems" future:
VP1=Newton RaphsonCollectorVol («p»; 8204; 6000; 0,19; 0,00347; 12) = 37,56 %;
VP2=Newton RaphsonCollectorVol («p»; 8204; 6500; 0,19; 0,00347; 26) = 33,92 %;
VP3=Newton RaphsonCollectorVol («p»; 8204; 7000; 0,19; 0,00347; 77) = 33,57 %;
VP4=Newton RaphsonCollectorVol («p»; 8204; 7500; 0,19; 0,00347; 168) = 31,87 %;
VP5=Newton RaphsonCollectorVol («p»; 8204; 8000; 0,19; 0,00347; 326) = 30,09 %;
VP6=Newton RaphsonCollectorVol («p»; 8204; 8500; 0,19; 0,00347; 578) = 28,8 %.
As a result we will write down set internal volatilnostej VC for exchange options koll on the Russian Open Society "United Power Systems" future:
VC = (31,42 %, 29,98 %, 30,41 %, 26,69 %, 31,09 %, 32,96 %),
And set internal volatilnostej VP for exchange options put on the Russian Open Society "United Power Systems" future:
VP = (37,56 %, 33,92 %, 33,57 %, 31,87 %, 30,09 %, 28,8 %).

In a tabular kind strikes and corresponding to them volatilnosti exchange options koll and put are presented as follows:
Strikes of exchange options for the Russian Open Society "United Power Systems" future (SP, Sc) Internal volatilnosti exchange options put on the Russian Open Society "United Power Systems" future (VP) Internal volatilnosti exchange options koll on the Russian Open Society "United Power Systems" future (VC)
6000 (0,6) 37,56 %
6500 (0,65) 33,92 %
7000 (0,7) 33,57 %
7500 (0,75) 31,87 % 31,42 %
8000 (0,8) 30,09 % 28,98 %
8500 (0,85) 28,80 % 30,41 %
9000 (0,9) 29,69 %
9500 (0,95) 31,09 %
10000 (1) 32,96 %

Table 4.14. Internal volatilnosti exchange options koll and put on the future of the Russian Open Society "United Power Systems" depending on a strike as of 01.04.05
In graphic submission of set internal volatilnostej look as follows:
\"Internal
Fig. 4.6. Internal volatilnosti exchange options koll and put on the future of the Russian Open Society "United Power Systems" depending on strikes of exchange options as of 01.04.05
Using linear approximation through property «the Format of a line of a trend» in software product EXCEL it is possible to find and construct some function characterising the general set volatilnostej. We receive following variants of functions of a bias volatilnosti exchange options depending on strikes of options koll and put:
1. Linear function of a bias volatilnosti with degree n = 1 depending on the strikes of exchange options reduced in 104 time (fig. 4.7.):
V (S/104) =-0,125 • (S/104) + 0,417 c R2 = 0,3508,
\"Linear
Fig. 4.7. Linear function of a bias volatilnosti V (S) in dependence of the strikes of exchange options reduced in 104 time on the Russian Open Society "United Power Systems" future
2. Square-law function internal volatilnosti cо degree n=2 depending on the strikes of exchange options reduced in 104 time (fig. 4.8 see.):
V (S/104) = 1,2713 • (S/104) 2 – 2,1591 • (S/104) + 1,2142 with R2 = 0,9211,

\"Square-law Fig. 4.8. Square-law function of a bias volatilnosti V (S) cо degree n=2 in dependence of the strikes of exchange options reduced in 104 time on the Russian Open Society "United Power Systems" future
3. Cubic function internal volatilnosti cо degree n=3 depending on the strikes of exchange options reduced in 104 time (fig. 4.9.):
V (S/104) = 0,0018 • (S/104) 3 - 0,0311 • (S/104) 2 +0,1295 • (S/104) + 0,3208
With R2 = 0,9336.
\"Cubic
Fig. 4.9. Cubic function of a bias volatilnosti V (S) cо degree n=3
Having combined three possible functions of a bias volatilnosti on one schedule, but, having replaced reduced strikes on usual it is possible to receive more evident picture of possible functions of a bias volatilnosti V (S):
\"Possible
Fig. 4.10. Possible functions of a bias volatilnosti V (S) c various degrees
Significance «R2» will be criterion at a choice of function of a bias volatilnosti. In this case it is necessary to choose bias function volatilnosti V (S) with degree n=2, that is a parabola, as at square-law and cubic function almost identical «R2»: 0,9211 and 0,9336, that is much more «R2» linear function-0,3508. And as the increase in degree of a multinomial essentially does not improve «R2» it is necessary to choose square-law bias functions volatilnosti (cм. Item 2.8.).
Result: at an estimation manufactured vnebirzhevyh options for the Russian Open Society "United Power Systems" future on 01.04.05 we will use a following equation of function of a bias volatilnosti for a finding internal volatilnosti for various strikes on an interval of strikes S є [6000; 10000]:
V (S/104) = 1,2713 • (S/104) 2 – 2,1591 • (S/104) + 1,2142

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A source: Pitchugin Igor Sergeevich. STRUCTURIZATION of OPTIONAL PRODUCTS ON THE BASIS OF the METHOD of OPTIMIZATION of FINAL MONETARY PAYMENTS. The dissertation on competition of a scientific degree of a Cand.Econ.Sci. Moscow - 2007. 2007

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  14. Programming in language VBA for a finding internal volatilnosti and estimations vnebirzhevyh options
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