1 Monte Carlo Method Ouldsaid Nour El-imane Mr Saadi Audit Diagnostic des systèmes d’information An Essay about: Monte-Carlo method Abstract Since the beginning of electronic computing, people have been interested in carrying out random experiments on a computer. Such Monte Carlo techniques are now an essential ingredient in many quantitative investigations.so what is monte carlo and why is it so important ? Introduction : Risk analysis is a part of every decision we make, we are Constantly faced with uncertainty, Ambiguity and variability. And even though we have unprecedented access to information we can’t accurately Predict the future . Monte Carlo simulation also known as the Monte Carlo the method Lets you see All the possible outcomes of your decision and assess the impact of risk allowing for a better decision making under uncertainty , Monte Carlo simulation uses random sampling and statistical modeling to estimate mathematical functions and mimic the operations of complex systems. Definition of Monte Carlo Simulation The Monte Carlo method was invented by scientists working on the atomic bomb in the 1940s, who named it for the city in Monaco famed for its casinos and games of chance. Its core idea is to use random samples of parameters or inputs to explore the behavior of a complex system or process. The scientists faced physics problems, such as models of neutron diffusion, that were too complex for an analytical solution -- so they had to be evaluated numerically. They had access to one of the earliest computers -- MANIAC -- but their models involved so many dimensions that exhaustive numerical evaluation was prohibitively slow. Monte Carlo simulation proved to be surprisingly effective at finding solutions to these problems. Monte Carlo simulation is a computerized mathematical technique that allows people to account for risk in quantitative analysis and decision making. The technique is used by professionals in such widely disparate fields as finance, project management, energy, manufacturing, engineering, research and development, insurance, oil & gas, transportation, and the environment. The simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action.. It shows the extreme possibilities—the outcomes of going 2 Monte Carlo Method for broke and for the most conservative decision—along with all possible consequences for middleof-the-road decisions. The Use of Monte Carlo Simulation: Whenever you need to make an estimate, forecast or decision where there is significant uncertainty, you'd be well advised to consider Monte Carlo simulation -- if you don't, your estimates or forecasts could be way off the mark, with adverse consequences for your decisions! Dr. Sam Savage, a noted authority on simulation and other quantitative methods, says "Many people, when faced with an uncertainty ... succumb to the temptation of replacing the uncertain number in question with a single average value. I call this the flaw of averages, and it is a fallacy as fundamental as the belief that the earth is flat." Most business activities, plans and processes are too complex for an analytical solution -- just like the physics problems of the 1940s. But you can build a spreadsheet model that lets you evaluate your plan numerically -- you can change numbers, ask 'what if' and see the results. This is straightforward if you have just one or two parameters to explore. But many business situations involve uncertainty in many dimensions -- for example, variable market demand, unknown plans of competitors, uncertainty in costs, and many others -- just like the physics problems in the 1940s. If your situation sounds like this, you may find that the Monte Carlo method is surprisingly effective for you as well. Knowledge Needed to Use monte carlo simulation To use Monte Carlo simulation, you must be able to build a quantitative model of your business activity, plan or process. One of the easiest and most popular ways to do this is to create a spreadsheet model using Microsoft Excel -- and use Frontline Systems' Risk Solver as a simulation tool. Other ways include writing code in a programming language such as Visual Basic, C++, C# or Java -- with Frontline's Solver Platform SDK -- or using a special-purpose simulation modeling language. You'll also need to learn (or review) the basics of probability and statistics. To deal with uncertainties in your model, you'll replace certain fixed numbers -- for example in spreadsheet cells - with functions that draw random samples from probability distributions. And to analyze the results of a simulation run, you'll use statistics such as the mean, standard deviation, and percentiles, as well as charts and graphs. Fortunately, there are great software tools (like ours!) to help you do this, backed by technical support and assistance. The importance of monte carlo method career wise : If your success depends on making good forecasts or managing activities that involve uncertainty, you can benefit in a big way from learning to use Monte Carlo simulation. By doing so, you can Avoid the Trap of the Flaw of Averages. As Dr. Sam Savage warns, "Plans based on average assumptions will be wrong on average." If you've ever found that projects came in later than you expected, losses were greater than you estimated as "worst case," or forecasts based on averages have gone awry -- you stand to benefit! 3 Monte Carlo Method Go Beyond the Limits of 'What If' Analysis. A conventional spreadsheet model can take you only so far. If you've created models with best case, worst case and average case scenarios, only to find that the actual outcome was very different, you need Monte Carlo simulation! By exploring thousands of combinations for your 'what-if' factors and analyzing the full range of possible outcomes, you can get much more accurate results, with only a little extra work. Know What Factors Really Matter. Tools such as Frontline's Risk Solver enable you to quickly identify the high-impact factors in your model, using sensitivity analysis across thousands of Monte Carlo trials. It could take you hours to identify these factors using ordinary 'what if' analysis. Give Yourself a Competitive Advantage. If you're negotiating a deal, or simply competing in the marketplace, having a realistic idea of the probability of different outcomes -- when your opponent or competitor does not -- can enable you to strike a better bargain, choose the price that yields the most profit, or benefit in other ways. Be Better Prepared for Executive Decisions. The higher you go in an organization, the more you'll find yourself dealing with uncertainty. Simulation or risk analysis might not be essential for routine day-to-day, low-value decisions -- but you'll find it invaluable as you deal with higher-level, more strategic -- and higher-stakes -- decisions. Risk analysis is part of every decision we make. We are constantly faced with uncertainty, ambiguity, and variability. And even though we have unprecedented access to information, we can’t accurately predict the future. Monte Carlo simulation (also known as the Monte Carlo Method) lets you see all the possible outcomes of your decisions and assess the impact of risk, allowing for better decision making under uncertainty. How Monte Carlo Simulation Works : Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—a probability distribution—for any factor that has inherent uncertainty. It then calculates results over and over, each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete. Monte Carlo simulation produces distributions of possible outcome values. By using probability distributions, variables can have different probabilities of different outcomes occurring. Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis. 4 Monte Carlo Method Common probability distributions include: Normal Or “bell curve.” The user simply defines the mean or expected value and a standard deviation to describe the variation about the mean. Values in the middle near the mean are most likely to occur. It is symmetric and describes many natural phenomena such as people’s heights. Examples of variables described by normal distributions include inflation rates and energy prices. Lognormal Values are positively skewed, not symmetric like a normal distribution. It is used to represent values that don’t go below zero but have unlimited positive potential. Examples of variables described by lognormal distributions include real estate property values, stock prices, and oil reserves. Uniform All values have an equal chance of occurring, and the user simply defines the minimum and maximum. Examples of variables that could be uniformly distributed include manufacturing costs or future sales revenues for a new product. Triangular The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur. Variables that could be described by a triangular distribution include past sales history per unit of time and inventory levels. PERT The user defines the minimum, most likely, and maximum values, just like the triangular distribution. Values around the most likely are more likely to occur. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not 5 Monte Carlo Method as emphasized. An example of the use of a PERT distribution is to describe the duration of a task in a project management model. Discrete The user defines specific values that may occur and the likelihood of each. An example might be the results of a lawsuit: 20% chance of positive verdict, 30% change of negative verdict, 40% chance of settlement, and 10% chance of mistrial. During a Monte Carlo simulation, values are sampled at random from the input probability distributions. Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Monte Carlo simulation does this hundreds or thousands of times, and the result is a probability distribution of possible outcomes. In this way, Monte Carlo simulation provides a much more comprehensive view of what may happen. It tells you not only what could happen, but how likely it is to happen. Monte Carlo simulation provides a number of advantages over deterministic, or “singlepoint estimate” analysis: o Probabilistic Results. Results show not only what could happen, but how likely each outcome is. o Graphical Results. Because of the data a Monte Carlo simulation generates, it’s easy to create graphs of different outcomes and their chances of occurrence. This is important for communicating findings to other stakeholders. o Sensitivity Analysis. With just a few cases, deterministic analysis makes it difficult to see which variables impact the outcome the most. In Monte Carlo simulation, it’s easy to see which inputs had the biggest effect on bottom-line results. o Scenario Analysis: In deterministic models, it’s very difficult to model different combinations of values for different inputs to see the effects of truly different scenarios. Using Monte Carlo simulation, analysts can see exactly which inputs had which values together when certain outcomes occurred. This is invaluable for pursuing further analysis. o Correlation of Inputs. In Monte Carlo simulation, it’s possible to model interdependent relationships between input variables. It’s important for accuracy to represent how, in reality, when some factors goes up, others go up or down accordingly. An enhancement to Monte Carlo simulation is the use of Latin Hypercube sampling, which samples more accurately from the entire range of distribution functions. Areas of application : Monte Carlo methods are especially useful for simulating phenomena with significant uncertainty in inputs and systems with many coupled degrees of freedom. Areas of application include: Physical sciences: 6 Monte Carlo Method Monte Carlo methods are very important in computational physics, physical chemistry, and related applied fields, and have diverse applications from complicated quantum chromodynamics calculations to designing heat shields and aerodynamic forms as well as in modeling radiation transport for radiation dosimetry calculations In statistical physics Monte Carlo molecular modeling is an alternative to computational molecular dynamics, and Monte Carlo methods are used to compute statistical field theories of simple particle and polymer systems Quantum Monte Carlo methods solve the many-body problem for quantum systems. In radiation materials science, the binary collision approximation for simulating ion implantation is usually based on a Monte Carlo approach to select the next colliding atom. In experimental particle physics, Monte Carlo methods are used for designing detectors, understanding their behavior and comparing experimental data to theory. In astrophysics, they are used in such diverse manners as to model both galaxy evolution and microwave radiation transmission through a rough planetary surface. Monte Carlo methods are also used in the ensemble models that form the basis of modern weather forecasting. Engineering Monte Carlo methods are widely used in engineering for sensitivity analysis and quantitative probabilistic analysis in process design. The need arises from the interactive, co-linear and non-linear behavior of typical process simulations. For example, In microelectronics engineering, Monte Carlo methods are applied to analyze correlated and uncorrelated variations in analog and digital integrated circuits. In geostatistics and geometallurgy, Monte Carlo methods underpin the design of mineral processing flowsheets and contribute to quantitative risk analysis. In wind energy yield analysis, the predicted energy output of a wind farm during its lifetime is calculated giving different levels of uncertainty (P90, P50, etc.) impacts of pollution are simulated and diesel compared with petrol. In fluid dynamics, in particular rarefied gas dynamics, where the Boltzmann equation is solved for finite Knudsen number fluid flows using the direct simulation Monte Carlo method in combination with highly efficient computational algorithms. In autonomous robotics, Monte Carlo localization can determine the position of a robot. It is often applied to stochastic filters such as the Kalman filter or particle filter that forms the heart of the SLAM (simultaneous localization and mapping) algorithm. In telecommunications, when planning a wireless network, design must be proved to work for a wide variety of scenarios that depend mainly on the number of users, their locations and the services they want to use. Monte Carlo methods are typically used to generate these users and their states. The network performance is then evaluated and, if results are not satisfactory, the network design goes through an optimization process. In reliability engineering, Monte Carlo simulation is used to compute system-level response given the component-level response. For example, for a transportation network subject to 7 Monte Carlo Method an earthquake event, Monte Carlo simulation can be used to assess the k-terminal reliability of the network given the failure probability of its components, e.g. bridges, roadways, etc. In signal processing and Bayesian inference, particle filters and sequential Monte Carlo techniques are a class of mean field particle methods for sampling and computing the posterior distribution of a signal process given some noisy and partial observations using interacting empirical measures. Climate change and radiative forcing The Intergovernmental Panel on Climate Change relies on Monte Carlo methods in probability density function analysis of radiative forcing. Probability density function (PDF) of ERF due to total GHG, aerosol forcing and total anthropogenic forcing. The GHG consists of WMGHG, ozone and stratospheric water vapour. The PDFs are generated based on uncertainties provided in Table 8.6. The combination of the individual RF agents to derive total forcing over the Industrial Era are done by Monte Carlo simulations and based on the method in Boucher and Haywood (2001). PDF of the ERF from surface albedo changes and combined contrails and contrail-induced cirrus are included in the total anthropogenic forcing, but not shown as a separate PDF. We currently do not have ERF estimates for some forcing mechanisms: ozone, land use, solar, etc. Computational biology Monte Carlo methods are used in various fields of computational biology, for example for Bayesian inference in phylogeny, or for studying biological systems such as genomes, proteins, or membranes. The systems can be studied in the coarse-grained or ab initio frameworks depending on the desired accuracy. Computer simulations allow us to monitor the local environment of a particular molecule to see if some chemical reaction is happening for instance. In cases where it is not feasible to conduct a physical experiment, thought experiments can be conducted (for instance: breaking bonds, introducing impurities at specific sites, changing the local/global structure, or introducing external fields). Computer graphics Path tracing, occasionally referred to as Monte Carlo ray tracing, renders a 3D scene by randomly tracing samples of possible light paths. Repeated sampling of any given pixel will eventually cause the average of the samples to converge on the correct solution of the rendering equation, making it one of the most physically accurate 3D graphics rendering methods in existence. Applied statistics The standards for Monte Carlo experiments in statistics were set by Sawilowsky. In applied statistics, Monte Carlo methods may be used for at least four purposes: 1. To compare competing statistics for small samples under realistic data conditions. Although type I error and power properties of statistics can be calculated for data drawn from classical theoretical distributions (e.g., normal curve, Cauchy distribution) 8 Monte Carlo Method for asymptotic conditions (i. e, infinite sample size and infinitesimally small treatment effect), real data often do not have such distributions. 2. To provide implementations of hypothesis tests that are more efficient than exact tests such as permutation tests (which are often impossible to compute) while being more accurate than critical values for asymptotic distributions. 3. To provide a random sample from the posterior distribution in Bayesian inference. This sample then approximates and summarizes all the essential features of the posterior. 4. To provide efficient random estimates of the Hessian matrix of the negative log-likelihood function that may be averaged to form an estimate of the Fisher information matrix. Monte Carlo methods are also a compromise between approximate randomization and permutation tests. An approximate randomization test is based on a specified subset of all permutations (which entails potentially enormous housekeeping of which permutations have been considered). The Monte Carlo approach is based on a specified number of randomly drawn permutations (exchanging a minor loss in precision if a permutation is drawn twice—or more frequently—for the efficiency of not having to track which permutations have already been selected). Artificial intelligence for games Main article: Monte Carlo tree search Monte Carlo methods have been developed into a technique called Monte-Carlo tree search that is useful for searching for the best move in a game. Possible moves are organized in a search tree and many random simulations are used to estimate the long-term potential of each move. A black box simulator represents the opponent's moves. The Monte Carlo tree search (MCTS) method has four steps 1. Starting at root node of the tree, select optimal child nodes until a leaf node is reached. 2. Expand the leaf node and choose one of its children. 3. Play a simulated game starting with that node. 4. Use the results of that simulated game to update the node and its ancestors. The net effect, over the course of many simulated games, is that the value of a node representing a move will go up or down, hopefully corresponding to whether or not that node represents a good move. Monte Carlo Tree Search has been used successfully to play games such as Go, Tantrix, Battleship, Havannah, and Arimaa. Design and visuals Monte Carlo methods are also efficient in solving coupled integral differential equations of radiation fields and energy transport, and thus these methods have been used in global illumination computations that produce photo-realistic images of virtual 3D models, with 9 Monte Carlo Method applications in videogames, architecture, design, computer generated films, and cinematic special effects. Search and rescue The US Coast Guard utilizes Monte Carlo methods within its computer modeling software SAROPS in order to calculate the probable locations of vessels during search and rescue operations. Each simulation can generate as many as ten thousand data points that are randomly distributed based upon provided variables. Search patterns are then generated based upon extrapolations of these data in order to optimize the probability of containment (POC) and the probability of detection (POD), which together will equal an overall probability of success (POS). Ultimately this serves as a practical application of probability distribution in order to provide the swiftest and most expedient method of rescue, saving both lives and resources. Finance and business Monte Carlo simulation is commonly used to evaluate the risk and uncertainty that would affect the outcome of different decision options. Monte Carlo simulation allows the business risk analyst to incorporate the total effects of uncertainty in variables like sales volume, commodity and labour prices, interest and exchange rates, as well as the effect of distinct risk events like the cancellation of a contract or the change of a tax law. Monte Carlo methods in finance are often used to evaluate investments in projects at a business unit or corporate level, or to evaluate financial derivatives. They can be used to model project schedules, where simulations aggregate estimates for worst-case, best-case, and most likely durations for each task to determine outcomes for the overall project. Monte Carlo methods are also used in option pricing, default risk analysis. Additionally, they can be used to estimate the financial impact of medical interventions. Law A Monte Carlo approach was used for evaluating the potential value of a proposed program to help female petitioners in Wisconsin be successful in their applications for harassment and domestic abuse restraining orders. It was proposed to help women succeed in their petitions by providing them with greater advocacy thereby potentially reducing the risk of rape and physical assault. However, there were many variables in play that could not be estimated perfectly, including the effectiveness of restraining orders, the success rate of petitioners both with and without advocacy, and many others. The study ran trials that varied these variables to come up with an overall estimate of the success level of the proposed program as a whole. Example of Monte Carlo simulation : The Asset Price Modeling One way to employ a Monte Carlo simulation is to model possible movements of asset prices using Excel or a similar program. There are two components to an asset's price movements: drift, which is a constant directional movement, and a random input, which represents market volatility. By analyzing historical price data, you can determine the drift, standard deviation, variance, and average price movement for a security. These are the building blocks of a Monte Carlo simulation. 10 Monte Carlo Method To project one possible price trajectory, use the historical price data of the asset to generate a series of periodic daily returns using the natural logarithm (note that this equation differs from the usual percentage change formula): Periodic Daily Return=ln(Previous Day’s/ PriceDay’s Price) Next use the AVERAGE, STDEV.P, and VAR.P functions on the entire resulting series to obtain the average daily return, standard deviation, and variance inputs, respectively. The drift is equal to: Drift=Average Daily Return − Variance/2 where: Average Daily Return=Produced from Excel’s AVERAGE function from periodic daily returns series Variance=Produced from Excel’s VAR.P function from periodic daily returns series _ Alternatively, drift can be set to 0; this choice reflects a certain theoretical orientation, but the difference will not be huge, at least for shorter time frames. Next obtain a random input: Random Value=σ×NORMSINV(RAND()) where: σ=Standard deviation, produced from Excel’s STDEV.P function from periodic daily returns series NORMSINV and RAND=Excel functions The equation for the following day's price is: Next Day’s Price=Today’s Price×e (Drift+Random Value) _ To take e to a given power x in Excel, use the EXP function: EXP(x). Repeat this calculation the desired number of times (each repetition represents one day) to obtain a simulation of future price movement. By generating an arbitrary number of simulations, you can assess the probability that a security's price will follow given trajectory. Here is an example, showing around 30 projections for the Time Warner Inc's (TWX) stock for the remainder of November 2015: 11 Monte Carlo Method The frequencies of different outcomes generated by this simulation will form a normal distribution, that is, a bell curve. The most likely return is at the middle of the curve, meaning there is an equal chance that the actual return will be higher or lower than that value. The probability that the actual return will be within one standard deviation of the most probable ("expected") rate is 68%; that it will be within two standard deviations is 95%; and that it will be within three standard deviations is 99.7%. Still, there is no guarantee that the most expected outcome will occur, or that actual movements will not exceed the wildest projections. Crucially, Monte Carlo simulations ignore everything that is not built into the price movement (macro trends, company leadership, hype, cyclical factors); in other words, they assume perfectly efficient markets. For example, the fact that Time Warner lowered its guidance for the year on November 4 is not reflected here, except in the price movement for that day, the last value in the data; if that fact were accounted for, the bulk of simulations would probably not predict a modest rise in price. 12 Monte Carlo Method Conclusion Monte Carlo simulation provides one of the most useful generic approaches to statistical computing. During the last few years significant advances have taken place in the application and theory of Monte Carlo. Recent insights in adaptive estimation methods provide novel approaches to rareevent simulation and randomized optimization, keep in mind Monte Carlo methods are flexible and can take many sources of uncertainty, but they may not always be appropriate. In general, the method is preferable only if there are several sources of uncertainty . References : solver.com , monte carlo simulation and how it can help you . investopedia.com Monte Carlo Simulation Definition . palisade.com , Monte Carlo Simulation en.wikipedia.org , Monte Carlo method .