Университеттің 85 жылдығына арналған Қазіргі заманғы математика
Университеттің 85 жылдығына арналған «Қазіргі заманғы математика
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| Университеттің 85 жылдығына арналған «Қазіргі заманғы математика:
проблемалары және қолданыстары» III халықаралық Тайманов оқуларының материалдар жинағы, 25 қараша, 2022 жыл 495 ζ is volatility,K is the strike price of the asset, T is the maturity date, t is time, r is the interest rate, S is the price of the underlying stock, P(S,t) is the value of the put option, C(S,t) is the call type parameter value. We have a hypothesis regarding assets and markets. Here is a list of hypotheses: 1. The interest rate is risk-free, the volatility is known, and they are constant over time. 2. No dividends until maturity. The definition of a dividend according to the Cambridge Dictionary is the profit of a company that must be paid to the owners of shares. 3. Efficient market. This means that the market movement is unpredictable. 4. No transaction costs, taxes and commissions when buying an option.According to the Cambridge Dictionary, transaction costs are defined as the amount of money paid for the sale or purchase of something in addition to the price of the thing. Whereas tax means the amount of money that must be paid to the government based on the company's income or payment for services or goods that a person has bought. The cost for someone who sells goods, which is directly related to the amount sold or the system that uses such payments, is called a commission. 5. Consider only the European type of option and no principle of arbitration. The principle of arbitrage describes the process of trading the same things with different values, or someone has the opportunity to buy and sell the same instruments with different values, thereby making a profit without risk. So, the principle of the absence of arbitration can be summarized as follows: you cannot get a risk-free profit in the market. 6. The normal distribution describes the interest rate of change, while the level of the asset price at the time of repayment is expressed logarithmically by the normal distribution. According to these assumptions considered, the value of the option will depend on the value of the underlying stock and time, as well as on the parameters that are taken as constant. The solution of the Equation is the formulas by Scholes and Black for the call and put options type. The following equationsare formulas of the model named after Scholes and Black. For the call: For the put: 𝐺(𝑆, 𝑡) = 𝐶(𝑆, 𝑡) = 𝑆𝑁(𝑑 1 ) − 𝐾𝑒 –rc 𝑁(𝑑 2 ) (1) 𝐺(𝑆, 𝑡) = 𝑃(𝑆, 𝑡) = −𝑆𝑁(−𝑑 1 ) + 𝐾𝑒 –rc 𝑁(−𝑑 2 ) (2) where 𝜏 = 𝑇 − 𝑡 , 𝑁(𝑑 1 ), 𝑁(𝑑 2 ) are a function of cumulative distribution for the normal distribution. 1 d – x 2 𝑁(𝑑) = ƒ 𝑒 2 𝑑𝑆 √2𝜋 –∞ 𝑆 𝜎 2 𝑑 1 = 𝑑 2 = ln (𝐾) + (𝑟 + 𝜎√𝑇 𝑆 ln (𝐾) + (𝑟 − 𝜎√𝑇 2 ) 𝜎 2 2 ) 𝑁(𝑑) means that the probability of the random variable which is obeys to the Gaussian distribution is less than the value of 𝑑 and it is between 0 ≤ 𝑁 ≤ 1 . Let us consider the solution of Equation (18) with a boundary condition 𝐺(𝑆, 0) = max (𝑆 − 𝐾, 0) . We remove the factors of 𝑆 at the derivatives. Let us move on to a new variable 𝑦 = ln (𝑆) . Using (𝐺(𝑆)) ' = 𝐺 ' (𝑆) · 𝑆 ' and (𝐺(𝑆)) '' = 𝐺 '' (𝑆) · 𝑆 ' · 𝑆 ' + 𝐺 ' (𝑆) · 𝑆 '' , we get |