Monte Carlo method in simulations - Revision history

Louis: /* Markov-Chain-Monte-Carlo Simulation */

2021-01-24T16:09:14Z

‎Markov-Chain-Monte-Carlo Simulation

Louis at 16:05, 24 January 2021

2021-01-24T16:05:17Z

Louis: /* [Markov-Chain-Monte-Carlo Simulation] */

2021-01-24T16:04:38Z

‎[Markov-Chain-Monte-Carlo Simulation]

Louis at 16:03, 24 January 2021

2021-01-24T16:03:44Z

Louis: /* Applications of the Monte Carlo Simulation */

2021-01-24T07:14:45Z

‎Applications of the Monte Carlo Simulation

Louis: /* Introduction */

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‎Introduction

Louis at 07:07, 24 January 2021

2021-01-24T07:07:30Z

Louis: /* Examples of Monte Carlo Simulations for Students */

2021-01-24T07:04:29Z

‎Examples of Monte Carlo Simulations for Students

Louis: /* Examples of Monte Carlo Simulations for Students */

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‎Examples of Monte Carlo Simulations for Students

Louis: /* Examples of Monte Carlo Simulations for Students */

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‎Examples of Monte Carlo Simulations for Students

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 == Markov-Chain-Monte-Carlo Simulation ==
-In reality, classical Monte Carlo simulation can sometimes reach its limits. This is especially the case when simulations are generated that have hundreds or thousands of unknown parameters. Therefore, Markov chain Monte Carlo simulations have been used since the 1950s. This was first done by Metropolis et at. (1953) to simulate a fluid in equilibrium with its gas state. They discovered that they did not need to simulate the exact dynamics, but only Markov chains that had the same equilibrium distribution. This is called the Metropolis algorithm. This was further developed in the 1970s into the Metropolis-Hastings algorithm, which is still highly relevant today. In a Markov chain, the variables follow a sequence of random elements X1,X2,...., where the conditional distribution of Xn+1 depends on Xn. Most Markov Chains in Monte Carlo simulations have a stationary transition probability if the conditional distribution of Xn +1 does not depend on Xn. In a simulation, for example, a sample of the generated variables is taken after a certain number of simulations and its properties are compared with the desired distribution. Depending on this, the next numbers are then simulated. With each further check, the distribution converges better and better with the desired one. Most Markov chains in simulation have an infinite state space, i.e. an infinitely large set of values of Xn. The theory of the Markov chain Monte Carlo simulation is the same as that of a conventional Monte Carlo simulation, but the statistical dependence of the random variables of the Markov chain changes the standard error.
+In reality, classical Monte Carlo simulation can sometimes reach its limits. This is especially the case when simulations are generated that have hundreds or thousands of unknown parameters. Therefore, Markov chain Monte Carlo simulations have been used since the 1950s. This was first done by Metropolis et at. (1953) to simulate a fluid in equilibrium with its gas state. They discovered that they did not need to simulate the exact dynamics, but only Markov chains that had the same equilibrium distribution. This is called the Metropolis algorithm. This was further developed in the 1970s into the Metropolis-Hastings algorithm, which is still highly relevant today. In a Markov chain, the variables follow a sequence of random elements X1,X2,...., where the conditional distribution of Xn+1 depends on Xn. Most Markov Chains in Monte Carlo simulations have a stationary transition probability if the conditional distribution of Xn +1 does not depend on Xn. In a simulation, for example, a sample of the generated variables is taken after a certain number of simulations and its properties are compared with the desired distribution. Depending on this, the next numbers are then simulated. With each further check, the distribution converges better and better with the desired one. Most Markov chains in simulation have an infinite state space, i.e. an infinitely large set of values of Xn. The theory of the Markov chain Monte Carlo simulation is the same as that of a conventional Monte Carlo simulation, but the statistical dependence of the random variables of the Markov chain changes the standard error.<ref name="Chapman">Brook, Steve; Gelman, Andrew; Jones, Galin L.; Meng, Xiao-Li. Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC. 2012. USA </ref>
 == Characteristics Of A High Quality Monte Carlo Simulation ==

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	In reality, classical Monte Carlo simulation can sometimes reach its limits. This is especially the case when simulations are generated that have hundreds or thousands of unknown parameters. Therefore, Markov chain Monte Carlo simulations have been used since the 1950s. This was first done by Metropolis et at. (1953) to simulate a fluid in equilibrium with its gas state. They discovered that they did not need to simulate the exact dynamics, but only Markov chains that had the same equilibrium distribution. This is called the Metropolis algorithm. This was further developed in the 1970s into the Metropolis-Hastings algorithm, which is still highly relevant today. In a Markov chain, the variables follow a sequence of random elements X1,X2,...., where the conditional distribution of Xn+1 depends on Xn. Most Markov Chains in Monte Carlo simulations have a stationary transition probability if the conditional distribution of Xn +1 does not depend on Xn. In a simulation, for example, a sample of the generated variables is taken after a certain number of simulations and its properties are compared with the desired distribution. Depending on this, the next numbers are then simulated. With each further check, the distribution converges better and better with the desired one. Most Markov chains in simulation have an infinite state space, i.e. an infinitely large set of values of Xn. The theory of the Markov chain Monte Carlo simulation is the same as that of a conventional Monte Carlo simulation, but the statistical dependence of the random variables of the Markov chain changes the standard error.		In reality, classical Monte Carlo simulation can sometimes reach its limits. This is especially the case when simulations are generated that have hundreds or thousands of unknown parameters. Therefore, Markov chain Monte Carlo simulations have been used since the 1950s. This was first done by Metropolis et at. (1953) to simulate a fluid in equilibrium with its gas state. They discovered that they did not need to simulate the exact dynamics, but only Markov chains that had the same equilibrium distribution. This is called the Metropolis algorithm. This was further developed in the 1970s into the Metropolis-Hastings algorithm, which is still highly relevant today. In a Markov chain, the variables follow a sequence of random elements X1,X2,...., where the conditional distribution of Xn+1 depends on Xn. Most Markov Chains in Monte Carlo simulations have a stationary transition probability if the conditional distribution of Xn +1 does not depend on Xn. In a simulation, for example, a sample of the generated variables is taken after a certain number of simulations and its properties are compared with the desired distribution. Depending on this, the next numbers are then simulated. With each further check, the distribution converges better and better with the desired one. Most Markov chains in simulation have an infinite state space, i.e. an infinitely large set of values of Xn. The theory of the Markov chain Monte Carlo simulation is the same as that of a conventional Monte Carlo simulation, but the statistical dependence of the random variables of the Markov chain changes the standard error.
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	== Characteristics Of A High Quality Monte Carlo Simulation ==		== Characteristics Of A High Quality Monte Carlo Simulation ==

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−	== [Markov-Chain-Monte-Carlo Simulation] ==	+	== Markov-Chain-Monte-Carlo Simulation ==

		+	In reality, classical Monte Carlo simulation can sometimes reach its limits. This is especially the case when simulations are generated that have hundreds or thousands of unknown parameters. Therefore, Markov chain Monte Carlo simulations have been used since the 1950s. This was first done by Metropolis et at. (1953) to simulate a fluid in equilibrium with its gas state. They discovered that they did not need to simulate the exact dynamics, but only Markov chains that had the same equilibrium distribution. This is called the Metropolis algorithm. This was further developed in the 1970s into the Metropolis-Hastings algorithm, which is still highly relevant today. In a Markov chain, the variables follow a sequence of random elements X1,X2,...., where the conditional distribution of Xn+1 depends on Xn. Most Markov Chains in Monte Carlo simulations have a stationary transition probability if the conditional distribution of Xn +1 does not depend on Xn. In a simulation, for example, a sample of the generated variables is taken after a certain number of simulations and its properties are compared with the desired distribution. Depending on this, the next numbers are then simulated. With each further check, the distribution converges better and better with the desired one. Most Markov chains in simulation have an infinite state space, i.e. an infinitely large set of values of Xn. The theory of the Markov chain Monte Carlo simulation is the same as that of a conventional Monte Carlo simulation, but the statistical dependence of the random variables of the Markov chain changes the standard error.

	== Characteristics Of A High Quality Monte Carlo Simulation ==		== Characteristics Of A High Quality Monte Carlo Simulation ==

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	In the first example, "Monte Carlo Sampling", the final value of X can be determined directly by the density function. In fact, no simulation is necessary. The PTi function is generated several times and then its properties are evaluated on the basis of the random experiments. In the second example, "Monte Carlo Simulation", a dynamic process is present. The value of Xt depends on its precursor Xt-1. This generates different paths. The dynamic model in Monte Carlo is usually described by a discrete-time, a continuous-time or a discrete event model. These examples suggest the definition that the term "sampling" is only used when no dynamics are generated over a time course, whereas this is exactly what happens with "simulations". In reality, however, these differences are often hardly to be made, since in both variants high-dimensional integrals of a function are often examined in order to make statements about a probability or expectation. <ref name="BRANDIMARTE">Brandimarte, Paolo. Handbook in Monte Carlo Simulation - Applications in Financial Engineering, Risk Management, and Economics. 2014. Published by John Wiley & Sons, Inc., Hoboken, New Jersey.</ref>		In the first example, "Monte Carlo Sampling", the final value of X can be determined directly by the density function. In fact, no simulation is necessary. The PTi function is generated several times and then its properties are evaluated on the basis of the random experiments. In the second example, "Monte Carlo Simulation", a dynamic process is present. The value of Xt depends on its precursor Xt-1. This generates different paths. The dynamic model in Monte Carlo is usually described by a discrete-time, a continuous-time or a discrete event model. These examples suggest the definition that the term "sampling" is only used when no dynamics are generated over a time course, whereas this is exactly what happens with "simulations". In reality, however, these differences are often hardly to be made, since in both variants high-dimensional integrals of a function are often examined in order to make statements about a probability or expectation. <ref name="BRANDIMARTE">Brandimarte, Paolo. Handbook in Monte Carlo Simulation - Applications in Financial Engineering, Risk Management, and Economics. 2014. Published by John Wiley & Sons, Inc., Hoboken, New Jersey.</ref>
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		+	== [Markov-Chain-Monte-Carlo Simulation] ==
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	== Characteristics Of A High Quality Monte Carlo Simulation ==		== Characteristics Of A High Quality Monte Carlo Simulation ==

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 == Applications of the Monte Carlo Simulation ==
-A Monte Carlo simulation can be used practically everywhere where randomness plays a role. Thus, there are no limits to the applications of Monte Carlo simulation. Therefore, the application areas range from particle physics to engineering, finance and climate research. <ref name="MIT">Dizikies, Peter.
+A Monte Carlo simulation can be used practically everywhere where randomness plays a role. Thus, there are no limits to the applications of Monte Carlo simulation. Therefore, the application areas range from particle physics to engineering, finance and climate research to many more. <ref name="MIT">Dizikies, Peter.
 Explained: Monte Carlo simulations: Mathematical technique lets scientists make estimates in a probabilistic world. MIT News. [17-05-2010] Available at: https://news.mit.edu/2010/exp-monte-carlo-0517</ref>

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 == Introduction ==
-Monte Carlo simulation is a mathematical technique used to simulate the possible outcomes of an uncertain event. Instead of fixed inputs, a probability distribution is used as a parameter at at least one point. By generating random probabilities as an input, a set of different outcomes and the probability of this outcomes can be generated through repeated simulation. The generation of [[pseudo random numbers]] by computers plays an important role in this. <ref name="IBM"> IBM Cloud Education. What is Monte Carlo Simulation?[2020-01-24]. Available at: https://www.ibm.com/cloud/learn/monte-carlo-simulation#toc-what-is-mo-b7t4vgvP</ref>
+Monte Carlo simulation is a mathematical technique used to simulate the possible outcomes of an uncertain event. Instead of fixed inputs, a probability distribution is used as a parameter at at least one point. By generating random probabilities as an input, a set of different outcomes and the probability of this outcomes can be generated through repeated simulation. [[Pseudorandom number generators]] by computers plays an important role in this. <ref name="IBM"> IBM Cloud Education. What is Monte Carlo Simulation?[2020-01-24]. Available at: https://www.ibm.com/cloud/learn/monte-carlo-simulation#toc-what-is-mo-b7t4vgvP</ref>
 The first well-known experiment that used randomness to solve a problem was "Buffon's Needle". <ref name="Harrison">Harrison, Robert L. Introduction To Monte Carlo Simulation. AIP Conf Proc. 2010 January 5; 1204: 17–21. doi:10.1063/1.3295638</ref>  The French scientist Georges Louis LeClerc, Comte de Buffon asked in the 18th century the question with what probability a randomly thrown needle intersects a grid of parallel lines. Only a few years later he was able to prove that this probability is 2L/Dπ (D > L). Let L be the length of the needle and D the distance between the lines. <ref name="Richeson">Richeson, David. A π-less Buffon's Needle Problem. Mathematics Magazine , Dec., 2006, Vol. 79, No. 5 (Dec., 2006), pp. 385-389. Available at: hhttps://www.jstor.org/stable/27642977</ref> In his first experiments, LeClerc allegedly threw baguettes over his shoulder onto a tiled floor.<ref name="Harrison">Harrison, Robert L. Introduction To Monte Carlo Simulation. AIP Conf Proc. 2010 January 5; 1204: 17–21. doi:10.1063/1.3295638</ref> In 1812, the French scientist Pierre-Simon Laplace suggested that this experiment could be used to determine the number π.<ref name="Harrison">Harrison, Robert L. Introduction To Monte Carlo Simulation. AIP Conf Proc. 2010 January 5; 1204: 17–21. doi:10.1063/1.3295638</ref> In 1901, the scientist Mario Lazzarini was able to determine the number π to 6 decimal places by throwing needles on a grid of parallel lines, and that after only 3408 throws.<ref name="Harrison">Harrison, Robert L. Introduction To Monte Carlo Simulation. AIP Conf Proc. 2010 January 5; 1204: 17–21. doi:10.1063/1.3295638</ref> However, this result is also doubted by scientists or also described as a stroke of luck. This is because in order to determine the number π to 6 decimal places in a 95% confidence interval, one would theoretically have to throw at least 134 trillion needles.<ref name="Badger">Badger, Lee. Lazzarini's Lucky Approximation of π. Mathematics Magazine , Apr., 1994, Vol. 67, No. 2 (Apr., 1994), pp. 83-91. Available at: https://www.jstor.org/stable/2690682</ref>  Nowadays, these experiments can be simulated much faster and more efficiently with the help of computers. One possible way is included in the examples.

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 '''Evolution of Stock Prices'''
-The last two examples already showed how Monte Carlo works in general and how it can be worked with in Excel. However, one important point is still missing for a complete Monte Carlo simulation, namely the time series. In the previous examples, an event was simulated repeatedly, but there was no relationship between the individual simulations. A good example of a complete simulation is the development of share prices. It is assumed that the share prices develop in a standard normal distribution.
+The last two examples already showed how Monte Carlo works in general and how it can be done in Excel. However, one important point is still missing for a complete Monte Carlo simulation, namely the time series. In the previous examples, an event was simulated repeatedly, but there was no relationship between the individual simulations. A good example of a complete simulation is the evolution of stock prices. It is assumed that the stock prices develop in a standard normal distribution.
-For example, the Moneta Money Bank share can be considered. This is, for example, at €67.80. The aim is to find out how this share will develop over the next 30 trading days. To do this, we first have to look at the historical share prices. In the last 3 months, the share had an average daily return of 0.3783% with a standard deviation of 1.9824%.
+For example, the Moneta Money Bank share should be considered. This is, for example, at €67.80 at the moment. The aim is to find out how this share will develop over the next 30 trading days. To do this, we first have to look at the historical share prices. In the last 3 months, the share had an average daily return of 0.3783% with a standard deviation of 1.9824%.
 For the first day, we enter the share's starting price of €67.80. Then simulate the price change for that day. Since this is standard normal distributed by assumption, this can be done in Excel with the function =NORM.INV(RAND();0.3783;1.9824). This random price change is multiplied by the starting price of the share to calculate the price gains/losses. The sum of this and the starting price of the share gives the final value of the share. This is then used as the starting value for the second trading day. The same procedure is followed for the next 29 days.

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 The first step is to create three cubes. For this, the function =RANDBETWEEN(1;6) can be used in Excel. This function returns a random value between 1 and 6 with each update. The sum is then calculated in a fourth cell.
 [[File:3 dice 1.png]]
 In the second step, a data table is created in which the results of the individual simulations are collected. To do this, first the header of the output table is transferred. The cells below are then equated with the cells of the original table. Then the rows below are numbered consecutively up to the desired simulation number. This can be done automatically in Excel by setting the desired final value under Home->Fill-> Series.
 [[File:3 dice 2.png]]
 Excel can then run this simulation repeatedly in an automated manner. New random numbers are drawn each time. To do this, select the first row of the data table (with the transferred values) down to the last row in which results are to be saved. Then select Data -> What-if-analysis -> data table. For Column-input-cell select any free cell that Excel can use for calculation. Finally, you get the results of the individual simulations in the data table. For the number of points, for example, the following result is obtained after 10,000 simulations:
 [[File:3 dice 3.png]]
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 For the first day, we enter the share's starting price of €67.80. Then simulate the price change for that day. Since this is standard normal distributed by assumption, this can be done in Excel with the function =NORM.INV(RAND();0.3783;1.9824). This random price change is multiplied by the starting price of the share to calculate the price gains/losses. The sum of this and the starting price of the share gives the final value of the share. This is then used as the starting value for the second trading day. The same procedure is followed for the next 29 days.
 [[File:MMB-1.png]]
 This gives a possible price development over 30 days. In order to make a statement about the development, this simulation must be repeated again and again. The easiest way to do this is to use the Data Tables function in Excel. Statistical evaluations can then be carried out on the basis of these different simulated paths. A possible development after 1000 simulations is shown in the figure. On the basis of this, it can be said that the stock price is on average 75.62€ and with 95% probability does not fall below a value of 62.87€. It must be said, however, that the number of simulations is quite low at 1000.
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 [[File:MMB-2.png|600px]]
 == References ==
 <references />