Университеттің 85 жылдығына арналған Қазіргі заманғы математика



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EQUATION 
 
TULEGENOVA E.N.
1
, Candidate of Sciences in Economics 
etulegenova80@mail.ru 
DAUTBAEVA A.O.
1
, Candidate of Sciences in Technical 
 
ALIASKAR A.
1
Master’s degree 
1
the Republic of Kazakhstan, Korkyt Ata Kyzylorda University 
 
To evaluate the cost of the option, a mathematical formula has been studied since early 
1960s year. However, the clearly work to describe the option’scost was occurred in 1973 by 
Fisher Black and Myron Scholes when they published their work about the costing model of 
option type. The model of option costing named after Scholes and Black is the standard form to 
evaluate the option’scost and firstly used as the standard model on the Chicago Board Options 
Exchange (known as CBOE) and American Stock Exchange (or AMEX) [2]. In April 1973, the 
CBOE began trading and in 1975 year the costing model was starting used by traders. The 
formula of costingby Fisher and Myron is the formula that is frangible to jump and tail events. 
However, we should note that Merton, Black and Scholes never did not come up with a new 
formula, they are received already existing and known mathematical formulas to calculate the 
option’s cost. 
A risk – free evaluation was researched by De Finetti, Ramsey, Arrow Debre and Savage 
which is the main principle of the theory named after Scholesand Black. However, they did not 
use in research until the 1970s. The options were not traded in the stock exchange until the 
1973s. Only in 1973 call type option were traded while put type option in the 1977s. 
Samuelson wrote a paper ―Brownian Motion in the Stock Market‖ which was 
unpublished. In 1965, Paul Samuelson came out with work ―Rational Theory of Warrant 
Pricing‖ [13]. He was very close to obtain the partial differential form of the equation by Scholes 
and Black and the model of option costing. Because the stock’s cost dynamics in the model 
follows an Ito process with constant characteristics and this hypothesis dating on Samuelson’s 
work which is adapted from Louis Bachelier’s doctoral thesis in 1900. Bachelier’s model of 
option costing accepted that stock’scosts obey the normal distribution (see Figure – 4). However, 
this model well described for stock values with short time because for stock costs over a long 
time become negative. The financial asset can end up worth nothing, however it is cost cannot be 
negative. That is why the minimum possible cost of the financial asset is zero. By using 
Bachelier’s work,Samuelson presumed that the basic financial asset’s return followed a 
lognormal distribution (see Figure – 5). He cannot take the costing model because after taking 
the mathematical formula for the warrant’s cost, here was two unknown variables such as an 
expected return on the asset and on the warrants. It was difficult to estimate them. He could get 
the model of asset costing - cost movements if Samuelson had found a way to estimating the 
amount of these variables. The way to obtaining the values of these variables was the no – 
arbitrage principle. But he had used this principle already. Most of his work considered only call 
type. He reduced his assumptions in research by believe on consistency in series of time under 
stationary. However, he might have come up with the new costing model of option if he has not 
been using those hypotheses. 
Before deriving the PDE, we denote the main parameters that describe the Black–Scholes 
option model. 




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