Calculation of Pi Using the Monte Carlo Method
by Eve Andersson

Home : Pi : One Calculation
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The "Monte Carlo Method" is a method of solving problems using statistics. Given the probability, P, that an event will occur in certain conditions, a computer can be used to generate those conditions repeatedly. The number of times the event occurs divided by the number of times the conditions are generated should be approximately equal to P.

How this program works:
If a circle of radius R is inscribed inside a square with side length 2R, then the area of the circle will be pi*R^2 and the area of the square will be (2R)^2. So the ratio of the area of the circle to the area of the square will be pi/4. This means that, if you pick N points at random inside the square, approximately N*pi/4 of those points should fall inside the circle. This program picks points at random inside the square. It then checks to see if the point is inside the circle (it knows it's inside the circle if x^2 + y^2 < R^2, where x and y are the coordinates of the point and R is the radius of the circle). The program keeps track of how many points it's picked so far (N) and how many of those points fell inside the circle (M). Pi is then approximated as follows:

4*M
pi = ---
N

Although the Monte Carlo Method is often useful for solving problems in physics and mathematics which cannot be solved by analytical means, it is a rather slow method of calculating pi. To calculate each significant digit there will have to be about 10 times as many trials as to calculate the preceding significant digit.Great Holiday Gifts!

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Eve Andersson (eve@eveandersson.com)
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Comments
Regarin the program
Hi eveander This is suresh here, doin my master in computers in ASU .As i'm a graduate student, actully we had a topic on monte carlo (pi...

...MonteCarloSimulation
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. MonteCarlosimulation (also known as the MonteCarlo Method) lets you see all the possible outcomes of your decisions and assess the impact of risk, allowing for better decision making under uncertainty
What is MonteCarlosimulation?
MonteCarlosimulation 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.
MonteCarlosimulation 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 for broke and for the most conservative decision—along with all possible consequences for middle-of-the-road decisions.
The technique was first used by scientists working on the atom bomb; it...

...April 2010
‘The problems of MonteCarloSimulation’ by David Nawrocki
This article describes the problems associated with using the MonteCarloSimulation Model
as a tool for determining future investment outcomes for investors. The tool is widely used
by Financial Advisors as a means of showing investors future returns on investments. The
article discusses why the use of MonteCarloSimulation in financial planning is difficult and
can lead to incorrect decisions which can have a detrimental impact on investors’
expectations of expected returns. The article tells us that MonteCarloSimulation uses
assumptions based on normal distributions and correlation coefficients of zero, neither of
which are real in the financial markets.
The article discusses why MonteCarloSimulation should only be used when there is no data available or it is too expensive to implement and why other methods may provide the same or better answers without being assumptive.
The author uses evidence from previous authors highlighting the problems with MonteCarloSimulation and the use of alternatives as a more accurate way of forecasting future returns for an investor. There are four alternatives shown and discussed, however, the...

...Application of MonteCarloSimulation in Capital Budgeting
| |
|by Prit, Aug 2, 2008 |
|The usefulness of MontecarloSimulation in Capital Budgeting and the processes involved in MonteCarloSimulation. It also |
|highlights the advantages in some situation compared to other deterministic models where uncertainty is the norm. |
|[pic] |
|Capital budgeting is an important area in Financial Management. Capital Budgeting means the investment in capital projects and|
|identify the projects, which has the highest value adding to the company at the cost of capital. It uses net present value of |
|future cash flows discounted at the appropriate cost of capital and compares it with initial investment and to see whether it |
|is a positive net present value. If the present value is less than the initial investment then the project is rejected. That |
|is the net present value is dependent on future cash flows. |
|In a deterministic model the cash flows are forecasted as a single figure and scenarios are considered one by one and |...

...Preface
This is a book about MonteCarlo methods from the perspective of ﬁnancial engineering. MonteCarlosimulation has become an essential tool in the pricing of derivative securities and in risk management; these applications have, in turn, stimulated research into new MonteCarlo techniques and renewed interest in some old techniques. This is also a book about ﬁnancial engineering from the perspective of MonteCarlo methods. One of the best ways to develop an understanding of a model of, say, the term structure of interest rates is to implement a simulation of the model; and ﬁnding ways to improve the eﬃciency of a simulation motivates a deeper investigation into properties of a model. My intended audience is a mix of graduate students in ﬁnancial engineering, researchers interested in the application of MonteCarlo methods in ﬁnance, and practitioners implementing models in industry. This book has grown out of lecture notes I have used over several years at Columbia, for a semester at Princeton, and for a short course at Aarhus University. These classes have been attended by masters and doctoral students in engineering, the mathematical and physical sciences, and ﬁnance. The selection of topics has also been inﬂuenced by my experiences in developing and delivering professional...

...Technology Group, Silver Spring , MD, USA
Correspondence: Young Hoon Kwak, Associate Professor of Project Management, Department of
Decision Sciences, School of Business, The George Washington University, Washington, DC 20052,
USA. E-mail: kwak@gwu.edu
A b stra ct
MonteCarlosimulation is a useful technique for modeling and analyzing real-world
systems and situations. This paper is a conceptual paper that explores the applications
ofMonteCarlosimulation for managing project risks and uncertainties. The benefits
of MonteCarlosimulation are using quantified data, allowing project managers to
better justify and communicate their arguments when senior management is pushing
for unrealistic project expectations. Proper risk management education, training, and
advancements in computing technology combined with MonteCarlosimulation software allow project managers to implement the method easily. In the field of project
management, MonteCarlosimulation can quantify the effects of risk and uncertainty
in project schedules and budgets, giving the project manager a statistical indicator of
project performance such as target project completion date and budget.
Key wo rds
MonteCarlosimulation, project...

...| |
1. Research one of the MonteCarlo analysis Products listed in the Topic Notes
I reviewed the following products that developed MonteCarlo analysis package:
MonteCarloSimulation within Microsoft Excel Data Analysis and Business
Palisade's @RiskModeling
Oracle's Crystal Ball,
RiskDecision's Predict! Risk Controller
I really found two of the four solutions excellent.
1. MonteCarloSimulation within Mocrosoft Excel
I really was amazed by by MonteCarloSimulation that is available within the Excel Software. The cost of the book Microsoft Excel Data Analysis and Business Modeling by Wayne L Winston costs only $39.99 and includes a practice CD. This solution is outstanding and major companies utilize it such as General Motors, Proctor & Gamble and Eli Lilly. This Excel based solution is the most cost effective. However, the user has to learn and practice the various steps to get to the final results. The users have to learn the various formulas, such as =Rand(), Vlookup, COUNTIF, NORMINV(rand(),mu,sigma) for normal random variable, NORMINV(p, mu, sigma) for normal random varianve with a mean and a standard deviation. Finally, they have to learn...

...MonteCarloSimulation in Finance for Calculating European Options Value
1. Introduction
An option is a financial instrument whose value depends on a value of underlying security. Options trade started in 1973 at the Chicago Board Options Exchange (Hull, Fundamentals of futures and options markets 2008). Nowadays, options have become a crucial tool in finance; they have become valuable both for financial institutions and investors. Options are attractive to investors since they have great effect in reducing risk in investment. Throughout this independent study, we will be working within the European option. A European option is a vanilla option. A vanilla option is a standard type of option contract with no special features except the simple expiry date (the date in the contract) and the predetermined strike price (the agreed price at which a particular option price can be exercised). There are two types of European option; European call options and European put options. European call options give the owner the right, but not the obligation, to buy an agreed quantity of underlying securities on a certain time (the expiration date), for a specified price (the strike price). The expiration date is called the day until maturity. The seller of call options is obligated to sell the security should the buyer so decide. Vice versa, European put options give the owner the right, but not the obligation, to sell an agree quantity of an...

...SIMULATION
• WHAT is Simulation ?
• WHY is Simulation required ?
• HOW is Simulation applied ?
• WHERE is Simulation used ?
DEFINITION
• Simulation is a representation of reality through the use of model or
other device, which will react in the same manner as reality under a
given set of conditions.
• Simulation is the use of system model that has the designed
characteristic of reality in order to produce the essence of actual
operation.
• According to Donald G. Malcolm, simulation model may be defined
as one which depicts the working of a large scale system of men,
machines, materials and information operating over a period of time
in a simulated environment of the actual real world conditions.
NEED FOR SIMULATION
• Many real world problems which cannot be represented by a
mathematical model.
• Tool for tackling the complicated problem of managerial decisionmaking.
• Utilizes a computerized model.
• To represent actual decision-making under conditions of uncertainty
for evaluating alternative courses of action based upon facts and
assumptions.
MONTECARLO TECHNIQUE
STEPS:
1. Setting up a probability distribution for variables to be analyzed.
2. Building a cumulative probability distribution for each random
variable.
3. Generate random numbers .
4. Conduct the...