The paper proposes an approach to solving multidimensional systems of nonlinear equations based on the use of the OpenMP parallelization mechanism and the multicore architecture of modern computers. The software product, which performs the main function - the parallelization of the numerical solution of multidimensional SNE by the Newton method, is developed. The paper introduces the algorithm of parallelization of the Black-Scholes algorithm based on OpenMP technology for prediction of the option on the European market is proposed. The analysis of the speed and efficiency of calculations with different number of processor cores is carried out. As a result, appropriate estimates of the acceleration and efficiency coefficients were obtained. The proposed method is easily scaled to a different number of processor cores. A number of numerical experiments were conducted. The obtained results also indicate the possibility of further optimization of the computational process by developing the multi-core architecture of modern computers.
In a modern economic society it is impossible to do without forecasting. One of the approaches is to use the Black-Scholes method for prediction the behavior of the stock on the stock market for a certain time. The purpose of this work is to create a program that will allow you to quickly and effectively use the Black-Scholes method algorithm in practice. The paper introduces the algorithm of parallelization of the Black-Scholes algorithm based on OpenMP technology for prediction of the option on the European market is proposed. For implementation were used the Microsoft Visual Studio 2017 programming environment, the C ++ programming technology, and OpenMP standard (Open Multi-Processing).
Black-Scholes method requires fast execution of the program. One of the best ways to speed up the work of a software product is to use the directives of parallelization of modern OpenMP technology. It allows to compare the results obtained and to analyze the speed and efficiency of the program with different streams and the corresponding multi-core architecture of modern computers. The object of the study is a parallel algorithm for forecasting options in the European market. The object of the research is the use of OpenMP technology in the process of parallelizing the Black-Scholes method.
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Known [
The Black-Scholes method is one of the most effective stock market actions in
predicted behavior [
When using the Black-Scholes method, you should be prepared for the next: 1. The method requires a large amount of input data. 2. The method requires a powerful computer to quickly process and process these data. 3. The method has minimal error and risk.
Option requirements, the value of which can be determined by the Black-Scholes
method, limit its use for real options. So, using the method involves the assumption
that there is a fixed date of option execution or decision making. Therefore, as a rule,
this method is only suitable for options related to the scale of the project, in particular
the expansion options, and can not be used for options related to the duration of the
project and the reduction or refusal of the project. In fact, the limitation on the
application of the Black-Scholes method naturally arises because the method itself was
created to determine the value of financial options, for which a fixed date is
mandatory. For real options related to the term of execution, reduction or refusal of the
project, it is worth using the binomial method [
To successfully solve current tasks of current management of socio-economic
processes, it is necessary to carry out a thorough quantitative analysis of these processes
in order to forecast them. In our society a lot of things need to be predicted. Every day
we see or hear the weather forecast, the forecast of currency growth, the forecast of
changes in oil and gas prices, and others. For great precision, any method of
forecasting requires the processing of a large amount of data. The study of the theoretical
price for European options is also very important. You can do this by using the
BlackScholes Options pricing method [
This model is widely used in practice, among other things, can be used to evaluate all derivatives, including converting securities, and even to assess the equity of financially dependent firms.
According to the Black-Scholes model, the key element in determining the value of
an option is the expected volatility of the underlying asset. Depending on the
fluctuation of the asset, the price increases or decreases, which directly affects the cost of the
option. Thus, if the value of the option is known, then it is possible to determine the
level of volatility expected by the market [
The task is to develop a parallel algorithm for forecasting options in the European
market based on the Black-Scholes method and the modern OpenMP parallel
programming technology [
An option is a contract concluded between two investors, one of which writes out (sells) an option, and the other one buys it and acquires the right (but not obligation) in the term specified in the option term or to purchase a fixed asset at a fixed price ( in its composition may include, for example, futures contracts) of the person that the option, or sell him the asset.
In this regard, the following options are distinguished [
The owner (buyer) of the option pays the seller a kind of commission - the premium per unit of the underlying asset, which is called the price, or the full value of the option (like the price of the insurance policy).
The size of the bonus affects a whole set of factors: the ratio between the current (market and contract prices); term of the option; stability of the course of the underlying asset.
The buyer's risk in the options deal is within the premium paid to them; the seller's risk is not limited and the latter's income is based on premiums. The buyer's payment of the premium is obligatory and is made on the stock exchange through the payment.
The price of the underlying asset (its unit), agreed in the option contract, is called the exercise price (exercise) of the option, or the contract price.
Here it should be emphasized that there are three types of different options within the options for buying (calling) and selling (put) shares, each of which has its own peculiarities.
These include: Internal options - have a strike price below the current market price for the put option. This means that the buyer of such an option can immediately exercise his right and earn a net income. However, given that each participant in options trading is interested in profitable transactions, then, as a rule, the premium on internal options always covers the specified price difference. With respect to such options, there is a notion of intrinsic price. It is always equal to the difference between the market price and the execution price. Market options - have an exercise price equal to or very close to the price of the underlying stock at the time of the option sale. External Options - characterized by the fact that their exercise prices are well above the base stock price under call options and significantly lower for put options.
Thus, from these types of options we can see the relationship between the contract
price and the risk [
Our program will use the classic options, they are based on the fact that the options that were the subject of the agreement from the beginning, firmly fixed the future price of the basis (or the value of the calculated indicator). In conventional classic options, the principle of constant price applies throughout the life of the contract.
In 1973, F. Black and M. Scholes published an article, The Pricing of Options and
Corporate Liabilities, in the Journal of Political Economy. The model proposed in this
article has radically changed the approach to the analysis of options and other
securities. To determine the value of the option, the authors proposed a formula whose
starting elements are known except one, and even this single element can be estimated in a
reasonable approximation [
Black and Scholes made a number of baseline assumptions that many researchers are working to validate. Among these tenets are: 1. You can estimate the fluctuations (standard deviation) of the stock return. 2. There is a constant interest rate on risk-free investments over time. 3. There is no cost to conclude the agreement; when concluding agreements without coverage for a term (transactions with a short position), the seller receives money immediately. 4. Taxes do not matter. 5. No dividends. 6. The share price is a random value; the price for period t has a log-normal distribution. 7. Trading is carried out continuously.
Obviously, this assumption is an idealization of real market power. Therefore, they should not be considered as joining people designed to use these models.
The model's conclusion is based on the concept of risk-free hedging. By buying
stocks and at the same time selling call options on these stocks, an investor can
construct a risk-free position where earnings on the shares will accurately offset losses on
the options, and vice versa [
The risk-free hedged position should yield a profit at a rate equal to the risk-free interest rate, otherwise there would be an opportunity to remove the arbitrage income and investors, trying to take advantage of this opportunity, would bring the option price to the equilibrium level determined by the model.
The Black-Scholes model is based on the assumption that future stock returns have a log-normal distribution with constant root mean square deviation - and that's all that is said about common stock returns. The expected return on them in this model affects the price of the option only indirectly, because of the stock price. 3.2
The option cost is calculated using the Black-Scholes formula developed to evaluate
financial options [
Where
F SN (d ) S N (d ) ,
1 0 2 d 1 ln ( 0.5 2 ) T (2) T d T , 1 (3)
Here, N (x) the probability that the deviation will be less under the conditions of the standard normal distribution (thus, and limit the range of values for the function of the standard normal distribution);
S - the current market value of the shares; S - the price of execution;
0 T -the expiration time of the option in years; - standard annual deviation of the stock price.
From the analysis of this formula, it follows that the option price is then higher when: the current market price of the stock is high (S); more time before the expiration of the option (T); greater risk.
Therefore, to increase the investment attractiveness of the project, it is more appropriate for companies to focus on increasing revenues rather than reducing costs.
The main difficulties that may arise when applying this model are related to obtaining reliable source data required for the calculation (time to implementation of the projected capabilities, values of variance and others). 3.3
At each iteration of the Black-Scholes algorithm, the following operations are performed: 1.
A pricing vector is created based on market price, risk, volatility, and time step. 1.1. We create an array of normally distributed random variables. 1.2. We calculate the estimated price at a given step. 1.3. Write down the value in the array of possible stock values. 1.4. We find the average of the array of all possible values at this step. This is an average value and will be our predicted price at some point.
In order to parallel this algorithm, we propose to use OpenMP concurrent
programming technology, which can be considered as a high-level add-on over Pthreads
(or similar thread libraries). OpenMP implements multi-threaded computations, in
which the master thread creates a set of slave threads and the task is shared between
them. It is assumed that threads run in parallel on a machine with multiple processors
(the number of processors need not be greater than or equal to the number of threads)
[
The tasks that flow in parallel, as well as the data needed to perform these tasks,
are described using the specific preprocessor directives of the respective language
pragm. The number of streams created can be controlled by the program itself, by
calling library procedures and externally, using environment variables [
Advantages of this technology: with the idea of "incremental parallelization", OpenMP is ideal for developers seeking to quickly parallel their computing programs with large, parallel cycles. The developer does not create a new concurrent program, but simply sequentially adds OpenMP directives to the program text; OpenMP is a rather flexible mechanism, which gives the developer great control over the behavior of the parallel program; it is envisaged that the OpenMP program on a single-processor platform can be used as a serial program, ie there is no need to support serial and parallel versions. OpenMP directives are simply ignored by a serial compiler, and stubs can be invoked to call OpenMP procedures; support for the so-called "orphan" (detached) directives, that is, directives for synchronization and division of labor may not be directly within the lexical context of a parallel domain.
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The first step is to create the file ml_data.csv, which records the input data required for our method. Namely: 180 numbers, which in percentage show the behavior of the stock in the European market for 180 minutes. The data is written in one line, without spaces, but through commas. calcVolatility(calculates volatility); find2DMean(finds the average value in the stock data array); randGen(generates an array of random numbers based on the normal distribution); runBlackScholesModel(designs the Black-Scholes model).
The result of the program execution is written to the opt.scv file, which will have 180 data describing the behavior of the price at each time step as a percentage.
calcVolatility(float spotPrice, int timesteps): a function that calculates volatility based on stock input.
Stages of execution: Reading the file. The data in the file is recorded in a single tape. We read the tape, close the file. We turn a string into a stream. Extract values read from a file and write them to an array. We find the average value of the calculated prices in one minute. Calculation of market volatility as standard deviation. The function returns volatility as a percentage.
find2DMean(float **matrix, int numLoops, int timesteps):a function that finds the mean of the 2D array through the first index inLoops.
Stages of execution: A private copy of the 'sum' variable is created for each thread. At the end of the run, all private copies of the variable that are written to the global variable are available We calculate the average value by columns Return the mean
randGen(float mean, float stdDev):randGen function Creates a random number based on the standard distribution of the mean of 0.0 and standard deviation of 1.0. runBlackScholesModel (float spotPrice, int timesteps, float riskRate, float volatility): a function that designs the Black-Scholes model.
Stages of execution: 1. Create an array of normally distributed random variables, an array of stock prices at each time interval. 2.
We put a stock price at the beginning of data processing equal to the market price. 3. Fill the array with random normal values.
We force the Black-Scholes method to calculate the stock price for the next period of time.
С S exp r 2
rand , 2 (5) where: С - predicted value; S - market price; r - risk (0.001); - volatile; 1/ t ; rand – normally distributed random variable.
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In Fig. 4. Black-Scholes option forecasting schedule for the next 180 minutes is presented.
In the experimental study, Black-Shouse stock prediction was performed in parallel with one, two, four and eight threads on eight nuclear processors using OpenMP technology. In the Table 1 shows the results of numerical experiments.
Analyzing the results, we can conclude that with the increase in the number of flows, the time during which the algorithm of the program finds forecasting the option price in the European market is almost halved.
The acceleration coefficient according to the formula is calculated in the paper. In the Table 2 shows the value of the coefficient when the number of threads varies. The graph of acceleration change depending on the number of threads on the octa-core processor is presented in Fig. 5.
The results shown in Fig. 5, show that the developed parallel algorithm provides normal scalability, that is, the task execution time decreases proportionally as the number of threads increases. The highest acceleration is achieved by using 8 threads on an eight-core processor.
The coefficient of efficiency is calculated by the formula E p S p / p . In the Table 3 shows the value of the coefficient when the number of threads varies. A graph of performance changes depending on the number of threads on an octa-core processor is presented in Fig. 6. The subject area is analyzed in detail. The most universal method of finding effective investment solutions - the Black-Scholes method, has been studied and evaluated. Has been developed a program that enables the calculation of options forecasts in the European market over a given time. To speed up the forecast, the timing of the algorithm's performance has been significantly improved through the use of OpenMP concurrent programming technology and its many-stream properties. In doing so, we were able to compare the speed of program execution with varying the number of threads on the octa-core computer architecture.