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Monte Carlo Simulation
Monte Carlo Simulation is a computational technique used to model the impact of uncertainty and risk in decision-making processes. It involves running multiple simulations using random input variables to analyze the range of possible outcomes and their probabilities. In business, it is commonly used for financial forecasting, risk assessment, and optimization of complex systems.
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10 Key excerpts on "Monte Carlo Simulation"
eBook - PDF- Roberta S. Russell, Bernard W. Taylor(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
Some problems are difficult to solve analytically because they consist of random variables represented by probability distributions. Thus, a large proportion of the applications of simulations are for probabilistic models. The term Monte Carlo has become synonymous with probabilistic simulation in recent years. However, the Monte Carlo technique can be more narrowly defined as a technique for selecting numbers randomly from a probability distribution (i.e., sampling) for use in a trial (computer) run of a simulation. As such, the Monte Carlo technique is not a type of sim- ulation model but rather a mathematical process used within a simulation. The name Monte Carlo is appropriate, since the basic principle behind the process is the same as in the operation of a gambling casino in Monaco. In Monaco, devices like rou- lette wheels, dice, and playing cards produce numbered results at random from well-defined simulation An approach to operational problem solving in which a real-world problem situation is replicated within a mathematical model. SUPPLEMENT TO CHAPTER 13 Operational Decision-Making Tools: Simulation Monte Carlo technique A method for selecting numbers randomly from a probability distribution for use in a simulation model. 626 S U P P L E M E N T 1 3 Operational Decision-Making Tools: Simulation populations. For example, a 7 resulting from thrown dice is a random value from a population of 11 possible numbers (i.e., 2 through 12). This same process is employed, in principle, in the Monte Carlo process used in simulation models. The Monte Carlo process of selecting random numbers according to a probability distri- bution will be demonstrated using the following example. The manager of ComputerWorld, a store that sells computers and related equipment, is attempting to determine how many lap- tops the store should order each week.
eBook - PDF- Roberta S. Russell, Bernard W. Taylor(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Thus, a large proportion of the applications of simu- lations are for probabilistic models. The term Monte Carlo has become synonymous with probabilistic simulation in recent years. However, the Monte Carlo technique can be more narrowly defined as a technique for selecting numbers randomly from a probability distribution (i.e., sampling) for use in a trial (computer) run of a simulation. As such, the Monte Carlo technique is not a type of sim- ulation model but rather a mathematical process used within a simulation. The name Monte Carlo is appropriate, since the basic principle behind the process is the same as in the operation of a gambling casino in Monaco. In Monaco devices like roulette wheels, dice, and playing cards produce numbered results at random from well-defined pop- ulations. For example, a 7 resulting from thrown dice is a random value from a population of 11 possible numbers (i.e., 2 through 12). This same process is employed, in principle, in the Monte Carlo process used in simulation models. The Monte Carlo process of selecting random numbers according to a probability distri- bution will be demonstrated using the following example. The manager of ComputerWorld, a store that sells computers and related equipment, is attempting to determine how many lap- tops the store should order each week. A primary consideration in this decision is the average number of laptops that the store will sell each week and the average weekly revenue gener- ated from the sale of laptops. A laptop sells for $4300. The number of laptops demanded each Monte Carlo technique A method for selecting numbers randomly from a probability distribution for use in a simulation. SUPPLEMENT TO CHAPTER 13 Monte Carlo Simulation 591 week is a random variable (which we will define as x) that ranges from 0 to 4. From past sales records, the manager has determined the frequency of demand for laptops for the past 100 weeks.
eBook - PDF- Paul Goodwin, George Wright(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
In most practical problems there will be a large number of factors, and also the possible values which each of the factors can assume may be very large or infinite. Consider, for example, the possible levels of the costs of launch, manufactur- ing and distribution we might experience. All of this means that there will be a large or infinite number of combinations of circumstances which could affect the return on 188 APPLYING SIMULATION TO DECISION PROBLEMS the investment. In such situations it is clearly impractical to use an approach such as a probability tree to calculate the probability of each of these combinations of circum- stances occurring. One answer to our problem is to use a versatile and easily understood technique called Monte Carlo Simulation. This involves the use of a computer to generate a large number of possible combinations of circumstances which might occur if a particular course of action is chosen. When the simulation is performed the more likely combina- tion of circumstances will be generated most often, while very unlikely combinations will rarely be generated. For each combination the payoff which the decision maker would receive is calculated and, by counting the frequency with which a particular payoff occurred in the simulation, a decision maker is able to estimate the probability that the payoff will be received. Because this method also enables the risk associated with a course of action to be assessed, it is often referred to as risk analysis (although some authors use the term to include methods other than simulation such as mathematical assessment of risk). Monte Carlo Simulation is demonstrated in the next section. Monte Carlo Simulation As we stated earlier, a computer is normally used to carry out Monte Carlo simula- tion, but we will use the following simplified problem to illustrate how the technique works.
eBook - PDFDiscrete and Continuous Simulation
Theory and Practice
- Susmita Bandyopadhyay, Ranjan Bhattacharya(Authors)
- 2014(Publication Date)
- CRC Press(Publisher)
17 2 Monte Carlo Simulation 2.1 INTRODUCTION A simulation model can be deterministic or stochastic in nature. In a deterministic simulation model, the variables (both input and output variables) are not ran-dom variables, whereas in a stochastic simulation model, one or more variables are stochastic in nature. Monte Carlo Simulation is basically a stochastic simulation method and a sampling technique in which random sampling is basically used. An early version of Monte Carlo Simulation was observed in Buffon’s needle experiment in which the value of π could be calculated by dropping a needle on a floor that was made by several strips of wood. The real Monte Carlo Simulation was introduced first by Enrico Fermi [1] in 1930 while studying neutron diffusion. The word “Monte Carlo” was first coined by Von Neumann during the 1940s, that is, during World War II. Monte Carlo Simulation is used to solve problems that are difficult to represent by an exact mathematical model or problems in which the solution is difficult to obtain by a simple analytical method. Monte Carlo Simulation uses a user-defined probability distribution for any uncertain variable in a problem. For this reason, a Monte Carlo model is said to be a static method rather than a dynamic method. Thus, Monte Carlo Simulation generates random numbers and uses these numbers for a variety of problems, such as forecasting, risk analysis, and other estimations. Another important application of the Monte Carlo technique is to evaluate difficult integrals. Section 2.4 shows such evaluation through the use of various Monte Carlo methods. Consider, for example, the two examples of Monte Carlo Simulation prob-lems in Section 2.1.1. 2.1.1 E XAMPLES In this section, we show two examples to explain the use of Monte Carlo Simulation in various real-world problems. Example 1 ABC bakery keeps stock of a popular cake. The daily demands gathered from the experience of the manager are given in Table 2.1.
eBook - ePubEssentials of Pricing Analytics
Tools and Implementation with Excel
- Erik Haugom(Author)
- 2020(Publication Date)
- Routledge(Publisher)
Chapter 13Monte Carlo Simulation for pricing decisions
In most cases, managers or other decision-makers do not have perfect information about what the actual consequences of various pricing decisions will be like. Additionally, the properties of the various variables going into a pricing model may be hard to describe with smooth functions, and the pricing problem itself could be so complex that it is impossible to find analytical solutions. In such situations, Monte Carlo Simulation is a valuable tool that can be used to get an overview of the potential outcome of various pricing decisions. Examples of applications include, but are not limited to:• Estimating the probability of breaking-even from a given price change.• Estimating the market response from a range of pricing decisions.• Estimating competitors’ response from a range of pricing decisions.• Setting prices under uncertainty in general.• Setting dynamic prices under uncertainty with a special focus on some variables:• Weather, cannibalization, competitor actions.• Setting mark-down levels under uncertainty with a special focus on:• Response to various mark-down levels, cannibalization fraction.We shall, in this chapter, focus specifically on what Monte Carlo Simulation is, how it can be used when making pricing decisions, and how this powerful technique can be implemented in Excel. Specifically, the chapter will cover the following topics:• What is Monte Carlo Simulation?• How can simulation be used for pricing decisions?• The basics of Monte Carlo Simulations in Excel.• Examples of simulation models for pricing problems.13.1 What is Monte Carlo Simulation?
Monte Carlo Simulation is all about using the computer to draw random numbers in order to replicate elements of the real world. The term Monte Carlo is used because of the similarities between many gambling devices and the process behind simulation using the computer. A device such as the roulette wheel, for example, produces random numbers from a defined range (from 0 to 36). In practice, we could use a roulette wheel to perform simulation of many interesting real-life applications. There is only one drawback, though: It takes a very long time to obtain the random numbers we need. Many simulation projects require thousands of trials, and if you spend on average 30 seconds on each trial, it will take many hours, at best, to run one simple simulation experiment. This is the reason why we use the computer (using Excel for example) to draw the random numbers we need.
- Tomasz Nowakowski, Marek Młyńczak, Anna Jodejko-Pietruczuk, Sylwia Werbińska-Wojciechowska, Tomasz Nowakowski, Marek Mlynczak, Anna Jodejko-Pietruczuk, Sylwia Werbinska-Wojciechowska(Authors)
- 2014(Publication Date)
- CRC Press(Publisher)
Instead of asking how to interpret uncertainty in a Monte Carlo Simulation, it is more relevant to ask 1 INTRODUCTION Monte Carlo Simulation is a technique to propa- gate uncertainty in quantitative models, but that is all. It does not in itself provide any principles to integrate information or to treat uncertainty. Also, it does not adhere to a certain perspective on risk. If used for sensitivity analysis, where the effects on outputs by varying inputs are in focus, the interpretation of the results of a Monte Carlo Simulation does not depend on the interpretation assigned to its inputs and outputs. Sensitivity anal- ysis is in this sense unproblematic. The situation of concern is when Monte Carlo Simulation is used to quantify uncertainty in quan- tities in order to inform decisions. It is important to keep in mind that a Monte Carlo Simulation should not be associated to a particular framework. Introductory text books and common computer programs for Monte Carlo Simulation describe methods to quantify uncertainty in input param- eters and provide tools to describe the uncertainty in output. Vose (2008) present different statistical principles and argue for uncertainty being subjec- tive, at the same time as the impression is that prin- ciples from classical statistical statistics are seen as the first choice. The tutorial for @RISK (Palisade Corporation, 2013) rightly present the quantifica- tion and interpretation of uncertainty in a neutral way e.g. by talking about Monte Carlo Simulation as a technique “to combine all the uncertainties you identify in your modeling situation” leading to “results presented in the form of probability distribu- tions”. However, statements like “you can explicitly 1590 whose interpretation to consider. Most would say it is the interpretation by the decision makers.
eBook - PDF- Steven P. Millard, Nagaraj K. Neerchal(Authors)
- 2000(Publication Date)
- CRC Press(Publisher)
Monte Carlo Simulation Monte Carlo Simulation is a method of investigating the distribution of a random variable by simulating random numbers (Gentle, 1985). Usually, the random variable of interest, say Y, is some function of one or more other random variables: 13: Monte Carlo Simulation and Risk Assessment 737 ( ) ( ) 1 2 , , , k Y h X h X X X = = K (13.1) For example, Y may be an estimate of the median of a population with a Cauchy distribution, in which case the vector of random variables X repre- sents k independent and identically distributed observations from some par- ticular Cauchy distribution. As another example, Y may be the incremental lifetime cancer risk due to ingestion of soil contaminated with benzene (Thompson et al., 1992; Hamed and Bedient, 1997). In this case the random vector X may represent observations from several kinds of distributions that characterize exposure and dose-response, such as benzene concentration in the soil, soil ingestion rate, average body weight, the cancer potency factor for benzene, etc. These distributions may or may not be assumed to be inde- pendent of one another (Smith et al., 1992; Bukowski et al., 1995). Sometimes the input variables X 1 , X 2 , …, X k are called input parameters. This terminology can be confusing, however, since the input variables are of- ten random variables and therefore have distribution parameters associated with their probability distributions (e.g., mean and standard deviation for a normally distributed input variable). Sometimes the distribution of Y in Equation (13.1) can be derived ana- lytically based on statistical theory (Springer, 1979; Slob, 1994). Often, however, the function h is complicated and/or the elements of the random vector X involve several kinds of probability distributions, making it difficult or impossible to derive the exact distribution of Y. In this case, Monte Carlo Simulation can be used to approximate the distribution of Y.
eBook - PDFHandbook in Monte Carlo Simulation
Applications in Financial Engineering, Risk Management, and Economics
- Paolo Brandimarte(Author)
- 2014(Publication Date)
- Wiley(Publisher)
Part One Overview and Motivation Chapter One Introduction to Monte Carlo Methods The term Monte Carlo is typically associated with the process of modeling and simulating a system affected by randomness: Several random scenarios are gen-erated, and relevant statistics are gathered in order to assess, e.g., the perfor-mance of a decision policy or the value of an asset. Stated as such, it sounds like a fairly easy task from a conceptual point of view, even though some pro-gramming craft might be needed. Although it is certainly true that Monte Carlo methods are extremely flexible and valuable tools, quite often the last resort ap-proach for overly complicated problems impervious to a more mathematically elegant treatment, it is also true that running a bad Monte Carlo Simulation is very easy as well. There are several reasons why this may happen: • We are using a wrong model of uncertainty: - Because we are using an unrealistic probability distribution -Or because we are missing some link among the underlying risk factors - Or because some unknown parameters have been poorly estimated - Or because the very nature of uncertainty in our problem does not lend itself to a stochastic representation • The output estimates are not reliable enough, i.e., the estimator variance is so large that a much larger sample size is required. • There is a systematic error in the estimates, which could be low or high biased. • The way we generate sample paths, possibly discretizing a continuous-time model, induces a non-negligible error. • We are using poor random variate generators. • There is some possibly subtle bug in the computer program implementing the method. Some of these issues are technical and can be addressed by the techniques that we will explore in this book, but others are more conceptual in nature and point out a few intrinsic and inescapable limitations of Monte Carlo methods; it is wise not to forget them, while exploring the richness and power of the approach. 3
- Hui Wang(Author)
- 2012(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 4 Monte Carlo Simulation Monte Carlo Simulation is a very flexible and powerful tool for estimating integrals and expected values. Since most of the quantitative analysis in finance or risk management involves computing quantities that are indeed expected values, Monte Carlo Simulation is widely used in the financial industry. This chapter aims to give a quick introduction to Monte Carlo Simulation, as well as its pros and cons. 4.1 Basics of Monte Carlo Simulation Consider the generic question of estimating the expected value of a func-tion of some random variable X : μ = E [ h ( X )] . A plain Monte Carlo Simulation scheme can be roughly divided into two steps: 1. Generate samples, or independent identically distributed (iid) ran-dom variables X 1 , X 2 , . . ., X n , that have the same distribution as X . 2. The estimate of the expected value μ is defined to be the sample av-erage ˆ μ = 1 n [ h ( X 1 ) + h ( X 2 ) + · · · + h ( X n )] . Sometimes we simply refer to the samples X 1 , X 2 , . . ., X n as iid copies of X . The number of samples n is the sample size , which is usually chosen to be a large number. It should be noted that μ , the quantity we wish to estimate, is 68 CHAPTER 4. Monte Carlo Simulation an unknown fixed number, whereas the Monte Carlo estimate ˆ μ is a random variable . The value of ˆ μ will vary depending on the samples. It is possible to design many different Monte Carlo Simulation algo-rithms for estimating the same expected value μ . We briefly mention a cou-ple of alternatives. (a) Importance Sampling: Assuming that X admits a density f , we can write μ = integraldisplay R h ( x ) f ( x ) dx . Consider an arbitrary nonzero density function g ( x ) . It follows that μ = integraldisplay R h ( x ) f ( x ) g ( x ) · g ( x ) dx = E bracketleftbigg h ( Y ) f ( Y ) g ( Y ) bracketrightbigg , where Y is a random variable with density g .
eBook - PDFBusiness Analytics
The Art of Modeling With Spreadsheets
- Stephen G. Powell, Kenneth R. Baker(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
Select- ing uncertain parameters is a natural outgrowth of sensitivity analysis. Although many probability models are available, only a few are used routinely by business analysts. We have described these models (the Bernoulli, binomial, uniform, triangular, nor- mal, and custom distributions), and we have indicated the most appropriate applications for each. Ensuring that simulation results are sufficiently precise is primarily a matter of choosing a suitable run length. However, we have stressed the importance of determining how much precision is really needed in a given analysis before making this decision. Finally, simulations produce probability distributions as outcomes, which are by their nature complex. We have emphasized the importance of summarizing and simplifying these results, in order to make them useful to managers. Although simulation is more sophisticated than simple spreadsheet modeling, it is one of the most widely used of the advanced management science tools. It is relatively easy for business analysts to learn, and it provides a means for express- ing and understanding the uncertainty that is so prevalent in business. It does require some familiarity with probability, the language of uncertainty, but many analysts have this familiar- ity, and most can acquire what they need. Often, the bigger challenge with simulation is translating the results into a form that managers can understand and act upon. SUGGESTED READINGS Evans, J. R., and D.L. Olson. 2002. Introduction to Simulation and Risk Analysis. 2nd ed. Upper Saddle River, NJ: Prentice Hall. This full-length text is devoted to simulation using Crystal Ball, a software alternative to Analytic Solver Platform. The coverage of Crystal Ball is somewhat less detailed than the coverage of Analytic Solver Platform in this chapter, but there are many examples and applications. Discrete event simulation using ProModel is also covered.
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