Университеттің 85 жылдығына арналған Қазіргі заманғы математика
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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 Samuelson wrote a paper ―Brownian Motion in the Stock Market‖ which was unpublished.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. |