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Abstract Monte Carlo simulation is a legitimate and widely used technique for dealing with uncertainty in many aspects of business operations. The purpose of this report is to explore the application of this technique to the stock volality and to test its accuracy by comparing the result computed by Monte Carlo Estimate with the result of Black-Schole model and the Variance Reduction by Antitheric Variattes. The mathematical computer softwear application that we use to compute and test the relationship between the sample size and the accuracy of Monte Carlo Simulation is itshapeMathematica. It also provides numerical and geometrical evidence for our conclusion.

0.1

Introduction to Monte Carlo Simulaion

Monte Carlo Option Price is a method often used in Mathematical ﬁnance to calculate the value of an option with multiple sources of uncertainties and random features, such as changing interest rates, stock prices or exchange rates, etc.. This method is called Monte Carlo simulation, naming after the city of Monte Carlo, which is noted for its casinos. In my project, I use Mathematica, a mathematics computer software, we can easily create a sequence of random number indicating the uncertainties that we might have for the stock prices for example.

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Pricing Financial Options by Flipping a Coin

A distcrete model for change in price of a stock over a time interval [0,T] is √ Sn+1 = Sn + µSn ∆t + σSn εn+1 ∆t, S0 = s (1) where Sn = Stn is the stock price at time tn = n∆t, n = 0, 1, ..., N − 1, ∆t = T /N , µ is the annual growth rate of the stock, and σ is a measure of the stocks annual price volatility or tendency to ﬂuctuate. Highly volatile stocks have large values of σ. Each term in sequence ε1 , ε2 ... takes on the value of 1 or -1 depending on the outcoming value of a coin tossing experiment, heads or tails respectively. In other words, for each n=1,2,... εn = 1 with probability = 1/2 −1 with probability = 1/2 (2)

By using Mathematica, it is very easy to create a sequence of random number. With this sequence, the equation (1) can then be used to simulate a sample path or trajectory of stock prices, {s, S1 , S2 , ..., SN }. For our purpose here, it has been shown as a relatively accurate method of pricing options and very useful for options that depend on paths.

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0.3.1

Proof of highly volatile stocks have large values for σ σ = 0.01

Let us simulate several sample trajectories of (1) for the following parameter values and plot the trajectoris: µ = 0.12, σ = 0.01, T = 1, s = $40, N = 254. The following ﬁgures are the graphs that we got for σ = 0.01 Since we have ε to be the value generated by ﬂipping a coin, it gives us arbitrary values and thus, we have diﬀerent graphs for parameter σ = 0.01, µ = 0.12, T = 1, s = $40, N = 254 This is another possibile graph. 1

Figure 1: Let M be the value of S254 of diﬀerent trajectories, k is the number of trajectories

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σ = 0.7

Then we repeated the experiment using the value of σ = 0.7 for the volality and other parameters remain the same. Similarly, it should have a number of diﬀerent graphs due to the arbitrary value of ε we generated by Mathematica. The following ﬁgures are the graphs that we got for σ = 0.7 Conclusion: From the two experiments above with the large diﬀerent σ and constant other parameters, we can tell that the larger the σ, the greater degree of variability in their behavior forthe ε ’s it is permissible to use random number generator that creates normally distributed random numbers with mean zero and variance one. Recall that the standard normal distribution has the bell-shape with a standard deviation of 1.0 and standard normal random variable has a mean of zero.

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0.4.1

Monte Carlo Method vs. Black-Scholes Model

Monte Carlo Method and its computing

Monte Carlo Method √ In the formular (1), the random terms Sn εn+1 ∆t on the right-hand side can be consider as shocks or distrubances that model functuations in the stock price. After repeatedly simulating stock price trajectories, as we did in the previous chapter, and computing appropriate averages, it is possible to obtain estimates of the price of a European call option, a type of

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Figure 2: S254 = 45.415, σ = 0.01 ﬁnancial derivative. A statistical simulation algorithm of this type is what we known as ”Monte Carlo method” A European call option is a contract between two parties, a holder and a writer, whereby, for a premium paid to the writer, the holder can purchase the stock at a future date T (the expiration date) at a price K (the strike price) agreed upon in the contract. If the buyer elect to exercise the option on the expiration date, the writer is obligated to sell the inderlying stock to the buyer at the price K, the strike price. Thus, the option has a payoﬀ function f (S) = max(S − K, 0) (3) where S = S(T ) is the price of the underlying stock at the time T when the option expires. This equation (3) produces one possible option value at expiration and after computing this thousands of times in order to obtain a feel for the possible error in estimating the price. Equation(3) is also known as the value of the option at time T since if S(T ) > K, the holder can purchase, at price K, stock with market value S(T) and thereby make a proﬁt equal toS(T ) − K not counting the option premium. However, on the other hand, if S(T ) < K, the holder will simply let the option expire since there would be no reason to purchase stock at a price that exceeds the market value. In other words, the option valuation problem is determine the correct and fair price of the option at the time that the holder and writer enter into the contract. In order to estimate the price call of a call option using a Monte Carlo method, an ensemble SN = S (k) (T ), k = 1, ...M 3

(k)

(4)

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Figure 3: S254 =45.1316, σ = 0.01 of M stock orices at expiration is generated using the diﬀerence equation √ (k) (k) (k) (k) (k) (k) S0 = s (5) Sn+1 = Sn + rSn ∆t + σSn εn+1 ∆t, Equation (5) is identical to equation (1) for eachk = 1, ..., M , except the growth rate µ is replaces by the annual interest r that it costs the writer to borrow money. Option pricing theory requires that the average value of (k) the payoﬀs f (SN 0, k = 1, ..., M be equal to the compounded total return ˆ obtained by investing the option premium, C(s), at rate r over the life of option, M 1 (k) ˆ f (sN ) = (1 + r∆t)N C(s). (6) M k=1 ˆ Solving(6) for C(s) yields the Monte Carlo estimate ˆ C(s) = (1 + r∆t)−N 1 M

M

f (sN ) k=1 (k)

(7)

ˆ for the option price. So, the Monte Carlo estimateC(s) is the present value of the average of the payoﬀs computed using rules of compound interest.

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Computing Monte Carlo Estimate

We use equation (7) to compute a Monte Carlo estimate of the value of a ﬁve 5 month call option, in other word T = 12 years, for the following parameter values: r = 0.06, σ = 0.2, N = 254, andK = $50. N is the number of times of steps for each trajectories.

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Figure 4: S254 =45.1881, σ = 0.01

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Comparing to the Exact Black-Scholes Formular

Monte Carlo has been used to price standard European options, but as we known that Black-Scholes model is the correct method of pricing these options, so it is not necessary to use Monte Carlo simulation. Here is the formular for exact Black-Scholes model: √ s K −d1 C(s) = erf c( √ ) − e−rT erf c(f rac−d2 2) (8) 2 2 2 where √ s σ2 1 d1 = √ [ln( ) + (r + )T ], d2 = d1 − σ T k 2 σ T and erfc(x) is the complementary error function, 2 erf c(x) = √ π

∞ x

(9)

e−t dt

2

(10)

Now we insert all data we have to the Black-Schole formula to check the accuracy of our results by comparing the Monte Carlo approximation with the value computed from exact Black-Schole formula. We generated BlackScholes Model with parameterr = 0.06, σ = 0.2, K = $50, k = 1, ..., M (whereT = N ∆t), N = 200. And we got: C(40) = 1.01189, C(45) = 2.71716 and C(50) = 5.49477. The error of the Monte Carlo Estimate seems to be very large. Thus we repeat the previous procedure and increased our sample size (M) to 50,000 and 100,000. Then, I made a chart to check if the accuracy of Monte Carlo Simulation increases by the increasing of the sample size.

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Figure 5: S254 =45.5859, σ = 0.01 Comparison of the accuracy of the Monte Carlo Estimate to the BlackSchole Model

MCE(1,000) ˆ C(40) = 1.67263 ˆ C(45) = 4.6343 ˆ C(50) = 8.1409

MCE (10,000) ˆ C(40) = 1.61767 ˆ C(45) = 4.24609 ˆ C(50) = 7.92441

MCE (50,000) ˆ C(40) = 1.76289 ˆ C(45) = 4.3014 ˆ C(50) = 7.7889

MCE (100,000) ˆ C(40) = 1.7496 ˆ C(45) = 4.2097 ˆ C(50) = 7.6864

Black-Schole Model BS(40) = 1.71179 BS(45) = 4.11716 BS(50) = 7.49477

By the comparing Monte Carlo Estimate and the Black-Schole modle, we can tell quite easily that the error gets smaller as the sample size increases. Here we calculated the relative error by the equation M onteCarloEstimate − Black − Scholemodle Black − ScholeM odle The results are error(1,000) E(40) = 0.60593 E(45) = 0.32330 E(50) = 0.24272 error (10,000) E(40) = 0.30232 E(45) = 0.19466 E(50) = 0.07819 error(50,000) E(40) = 0.14037 E(45) = 0.13862 E(50) = 0.05254 error (100,000) E(40) = 0.14037 E(45) = 0.08833 E(50) = 0.02157 Black-Schole Model BS(40) = 1.71179 BS(45) = 4.11716 BS(50) = 7.49477 (11)

Conclusion: Monte Carlo Simulation gives the option price is a sample average, thus according to the most elementary principle of statistics, its standard deviation is the standard deviation of the sample divided by the square root of the sample size. So, the error reduces at the rate of 1 over the square root of the sample size. To sum up, the accuracy of Monte Carlo Simulation is increasing by increasing the size of the sample. 6

Figure 6: replaced previous value of sigma with 0.7

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Comparing to the Variance Reduction by Antitheric Variattes

Variance Reduction by Antithetic Variates is a simple and more widely used way to increase the accuracy of the Monte Carlo Simulation. It is the technique used in some certain situations with an additional increase in computational complexity is the method of antithetic variates. In order to achieve greater accuracy, one method of doing so is simple and automatically doubles the sample size with only a minimum increase in computational time. This is called the antithetic variate method. Because we are generating obervations of a standard normal random variable which is distributed with a mean of zero, a variance of 1.0 and symmetric, there is an equally likely chance of having drawn the observed value times−1. Thus, for each arbitrary ε we draw, there should be an artiﬁcially observed companion observation of −ε that can be legitimately created by us. This is the antithetic variate. For each k=1,...,M use the sequence ε1 , ..., εN −1

(k) (k)

(12)

(k) (k)

k+1 in equation (5) to simulate a payoﬀ f (SN )and also use the sequence −ε1 , ..., −εN −1 k− in equation (5) to simulate an associated payoﬀ f (SN ). Now the payoﬀs k+ k− are simulated inpairs f (SN ), f (SN ) This is the Mathematica Program that we ran to evaluate the Variance Reduction. After comparing Monte Carlo Simulation with Variance Reduction by Antithetic Variates, We made a table of data.

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Figure 7: S254 =47.084, σ = 0.7

k(1,000) V (40) = 1.74309 V (45) = 4.4789 V (50) = 7.94642

k(5,000) V (40) = 1.66685 V (45) = 4.305103 V (50) = 7.67707

k(50,000) V (40) = 1.69966 V (45) = 4.1778 V (50) = 7.62341

k (100,000) V (40) = 1.70674 V (45) = 4.099835 V (50) = 7.53716

BS Model BS(40) = 1.71179 BS(45) = 4.11716 BS(50) = 7.49477

While the table for the data of Monte Carlo Simulation we get under the same condition is: MCE(1,000) ˆ C(40) = 1.67263 ˆ C(45) = 4.6343 ˆ C(50) = 8.1409 MCE (5,000) ˆ C(40) = 1.72486 ˆ C(45) = 4.34609 ˆ C(50) = 7.92241 MCE (50,000) ˆ C(40) = 1.76289 ˆ C(45) = 4.3014 ˆ C(50) = 7.7889 MCE (100,000) ˆ C(40) = 1.7496 ˆ C(45) = 4.2097 ˆ C(50) = 7.6864

By the chart of Monte Carlo Estimate and the Variance Reduction by Antithetic Variates, we can tell quite easily that the error gets smaller as the sample size increases. Here we calculated the relative error by the equation M onteCarloEstimate − V arianceReductionbyAntitheticV ariates (13) V arianceReductionbyAntitheticV ariates The results are error(1,000) E(40) = 0.0402 E(45) = 0.0347 E(50) = 0.0245 error (50,000) E(40) = 0.0372 E(45) = 0.0296 E(50) = 0.0217 error (100,000) E(40) = 0.0251 E(45) = 0.0268 E(50) = 0.0198

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Figure 8: S254 =54.7495, σ = 0.7

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Random Walk ds = µSdt + σS β dw t (14)

Random Walk

this is the random walk, where ds is S(tk +∆t)−S(tk ), so this can be written as √ S(tk + ∆t) − S(tk ) = µS(tk )∆t + σS(tk )β εk ∆t (15) We randomly picked two diﬀerent positive numbers, one greater than 1 and one less than one. β as 0.5 and 2 and we made mathematica ran twice while keep every other variables constant. Here are some graphs that we plot using Mathematica. Figure 19 is a graph for β = 0.5 In addition, we plot the graphs when β = 2 as well. During Mathematica’s computation, we got some values that are overﬂowing. Within the values that are available, we plotted some corresponding graphs. Figure 20 is a graph for the random walk when β = 2. By using 1000 as a sample size, we used the same program to compute ˆM (10), CM (20), CM (40) and CM (160) when β = 0.5, where CM (10) is ˆ ˆ ˆ ˆ C ˆM (20) is the Monte Carlo the Monte Carlo estimate for sample size 1000, C ˆ simulation for sample size 2000 ans so forth. We get CM (10) = 3.80332, ˆM (20) = 4.75116, CM (40) = 6.09321 and CM (160) = 7.63962 . Now we ˆ ˆ C are using the equation log2 ( ˆ ˆ CM (10) − CM (20) ) ˆ ˆ CM (40) − CM (160) (16)

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Figure 9: S254 =7.95396, σ = 0.7 to see the error. After plugging the numbers, we get this value of the equation to be -0.67621. This value is very close to 1 as desired. 2 By the same procedure, we used Mathematica to generate the value for ˆ ˆ ˆ ˆ CM (10), CM (20), CM (40) and CM (160) when β = 2. There are some overﬂow in the result which gives us trouble for going further and also tells us that when β > 1, there will be overﬂow.

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Figure 10: S254 =22.7805, σ = 0.7

Array of S for monte carlo estimate1.png

Figure 11: We ﬁrst array for S in order to make space of memory for the ˆ data we will get later. Here, P(k) is theC(k)

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Mathematica program for Monte Carlo Estimate1.png

Figure 12: This is the Matehamatica program we made and data we got for the diﬀerent S(254), after 10 times of computing, named as M[k]=S[255, k] ˆ in mathematica. Again, here P(k) is the C(k). So, according to the data we got, there are 10 diﬀerent P[k+1], which in other word, 10 diﬀerent Monte ˆ Carlo estimate C(k + 1) after 10 times of computing

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S(0)=40.png

Figure 13: This result is the Monte Carlo estimate corresponding to our current stock prices of S(0) = s = 40 after using 254 steps (the number of N) and M ∼ 10, 000 for each trajectories for each Monte Carlo estimate. =

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S(0)=45.png

Figure 14: This result is the Monte Carlo estimate corresponding to our current stock prices of S(0) = s = 45 after using 254 steps (the number of N) and M ∼ 10, 000 for each trajectories for each Monte Carlo estimate. =

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S(0)=50.png

Figure 15: This result is the Monte Carlo estimate corresponding to our current stock prices of S(0) = s = 50 after using 254 steps (the number of N) and M ∼ 10, 000 for each trajectories for each Monte Carlo estimate. =

Figure 16: This the program we wrote with Mathematica for the Variance Reduction by Antithetic Variattes. We have S for the positive ε and Y for the negative ε and we get the average of these value by adding them up and divide by 2.

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Figure 17: This the program we wrote with Mathematica for diﬀerent values of β.

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Figure 18: This is the graph when β = 0.5

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Figure 19: This is how the graph looks like when β = 0.5

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Figure 20: This is the graph when β = 2

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Figure 21: This is how the graph looks when β = 2

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...Monte Carlo Simulation The Monte Carlo Simulation encompasses “any technique of statistical sampling employed to approximate solutions to quantitative problems” (Monte Carlo Method, 2005). The Monte Carlo method simulates the full system many times, each randomly choosing a value for each variable from its probability distribution. The outcome is a probability distribution of the overall value of the system calculated through the iterations of the model. A standard approach to risk management of projects is outline by Project Management Institute (2004) that includes six processes: Risk Management Planning, Risk Identification, Risk Qualification, Risk Quantification, Risk Response Planning, and Risk Monitoring and Control. Monte Carlo is usually listed as a method to use during the Risk Quantification process to better quantify the risks to the project manager is able to justify a schedule reserve, budget reserve, or both to deal with issues that could adversely affect the project. Monte Carlo simulation, while not widely used in project management, does get some exposure through certain project management practices. This is primarily in the areas of cost and time management to quantify the risk level of a projects budget or planned completion date. In time management, Monte Carlo simulation may be applied to project schedules to quantify the confidence the project manager should have in the target competition date or total project duration. In cost management, the......

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...Carlos Slim Helu In 1902, Julián Slim Haddad, father of Carlos Slim Helú, arrived to Mexico from Lebanon. He was escaping from the Ottoman Empire, which at the time conscripted young men into its army. One of the markers of Carlos Slim Helu’s success has been his ability to buy and sell at the correct times. His investments in the downturn of the 1980s were the foundation of his wealth, and throughout the 1990s he continued to sell businesses which were successful then use the return to invest in others which were up and coming. Carlos Slim was born on january 28, 1940. He completes his professional studies in civil engineering at the UNAM. By 1965 Carlos Slim aquired companies like inmobiliaria Carso, casa de bolsa inversora Bursátil, embotelladora Jarritos del sur, and some others. Carso was incorporated in January 1966. Carso comes from the first syllable of the names Carlos and Soumaya, Mr. Slim's wife. The Mexican economic crash of 1982 was what delivered the opportunity to consolidate his wealth. In 1990, Grupo Carso and other Mexican investors acquired 10.4% of the companys stock, in partnership with SBC - 5% with an option for an additional 5% - and France Telecom 5%. Since 1990, Telmex has embraced a work culture where training, modernization, quality and customer service is a priority. Ten principles of grupo Carso. 1. Have always simple organizational structures, minimal hierarchical levels; provide human and in-house development of the executives;......

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...Using Monte Carlo Tool for Risk Assessment CMGT/442 March 21, 2016 Craig McCormick Using Monte Carlo Tool for Risk Assessment A project can have many variables that can prevent it from completion. No one variable can effect a project more than a project’s risks. The key to having a successful completion of a project is through a risk assessment. It is the project manager’s job to identify potential risks within a project. Identifying these uncertainties will help to minimize the risks and their impact on the project. The Monte Carlo tool can be an asset for a project manager and risk assessment process. What is Monte Carlo? Monte Carlo is a simulation of mathematical technique that analyzes risk for decision making. The simulation uses different choices of action and gives a variety of probabilities and possible outcomes based on those actions. These possible outcomes are given in the most extreme situations, as well as with minor decisions. Additionally, the Monte Carlo simulation provides possible consequences of conservative decisions. Overall, “Monte Carlo analysis involves determining the impact of the identified risks by running simulations to identify the range of possible outcomes for a number of scenarios.” (Marom, 2010) How Monte Carlo Works Project risks are one of the big unknowns when it comes to risk assessment. Monte Carlo attempts to change these unknowns by using probability distributions. Probability distributions show that each risk variable...

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...Modeling Order Book Fluctuation by Monte Carlo Technique CONTENTS Page no. 1) Certificate 2 2) Acknowledgement 3 3) Abstract 5 4) Introduction 6 5) Simulation code 8 ➢ Order Book 8 ➢ Diffusion 9 ➢ Price and Annihilation 11 ➢ One Trade return 14 ➢ Waiting time between consecutive trades 16 ➢ Conditional return 19 ➢ Hurst curve 20 6) Results and Discussion 22 7) Summary 28 8) Future Prospects 29 9) References ...

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...Master in Management Organizational Behavior & Leadership Case #2: The Personality of Leaders The Personality of Carlos Ghosn: The $10 Billion Man Read the following dossier of article extracts and answer the final questions. 1.- “The $10 billion man” Feb 24th 2005, The Economist Having turned round Nissan, Carlos Ghosn is about to run Renault as well It is said that he could add $10 billion to the market value of Ford or General Motors with a stroke of his pen. But Carlos Ghosn is not about to sign up as chief executive of either firm. Instead, in May, he will become the boss of Renault, France 's second-largest carmaker, while continuing to head Nissan, Japan's number two car firm. To ease the transition, this week he named Toshiyuki Shiga as Nissan's chief operating officer. Although Renault and Nissan have cross-shareholdings and a deep alliance, their relationship deliberately stops well short of outright merger. Perhaps that is why it has been so successful, avoiding the integration pain that has marred, for instance, DaimlerBenz's takeover of Chrysler. In his book, “Shift: Inside Nissan's Historic Revival”, published in English last month, Mr Ghosn says that the strength of the alliance “can be found, on the one hand, in its respect for the identities of the two companies, and on the other, in the necessity of developing synergies.” Certainly the benefit has flowed both ways since the Franco -Japanese deal was done in 1999. First, Renault......

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...provide well - behaved experimental systems are increasingly providing a bridge between theory and experiment, for instance; the Monte Carlo method (MC) and the molecular-dynamics method (MD). In Monte Carlo method the exact dynamical behavior of a system is replaced by a stochastic process, whereas the MD methods are based on a simpler principle and consists of solving a system of Newton's equations for an N-body system. Stochastic simulation is some times called MC simulation (simulation is a numerical technique for conducting experiment on a digital computer, which involves certain types of mathematical and logical models that describe the behavior of the system over extended period of real time). The generally accepted birth date of the MC method is 1949, when an article entitled "The Monte Carlo Method" appeared, the American mathematicians J.Neyman and S.Ulam are considered to be its originator. The first successful application of this method to a problem of statistical thermodynamics dates back only to 1953, when Metropolis and co-workers studied "fluid" consisting of hard disks. In the nineteenth and early twentieth centuries, statistical problems were sometimes solved with the help of random selections, that is, in fact, by the MC method. Prior to the appearance of electronic computers, this method was not widely applicable since the simulation of random quantities by hand is a very laborious process. Thus, the beginning of the MC method as a highly universal......

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