Presentation Transcript

1. By: Joe Casey The Monte Carlo Method!!!
2. What is a Monte Carlo Method? The expression "Monte Carlo method" is actually very general. Monte Carlo methods are based on the use of random numbers and probability statistics to investigate problems. You can find MC methods used in everything from economics to nuclear physics to regulating the flow of traffic. A Monte Carlo method is a way of solving complex problems through approximation using many random numbers. They are very versatile, but are often slower and less accurate than other available methods.
3. Why Use the Monte Carlo Method? They tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithm. More broadly, Monte Carlo methods are useful for modeling events with significant uncertainty in inputs, such as the calculation of risk in business. The advantage of Monte Carlo methods over other techniques increases as the sources of uncertainty of the problem increase. Monte Carlo Methods are particularly useful in the valuation of options with multiple sources of uncertainty or with complicated features which would make them difficult to value through a straightforward Black-Scholes style computation. The technique is thus widely used in valuing Exotic options.
4. Overview! There is no single Monte Carlo method. Instead, the term describes a large and widely-used class of approaches. Essentially, the Monte Carlo method solves a problem by directly simulating the underlying (physical) process and then calculating the (average) result of the process. Because of their reliance on repeated computation of random or “pseudo-random” numbers, these methods are most suited to calculation by a computer However, these approaches tend to follow a particular pattern: 1. Define a Domain of Possible inputs 2. Generate Inputs randomly from the domain using a certain specified probability distribution 3. Perform a deterministic computation ( this means that given a particular input it will always produce the same output ) using inputs 4. Aggregate the results of the individual computations into a final result
5. In Most Basic Terms 1. Draw a random number 2. Process this random number in some way, for example plug it into an equation 3. Repeat steps 1 and 2 a large number of times 4. Analyze the cumulative results to find an estimation for a non random value
6. A little bit of History The term "Monte Carlo method" was coined in the 1940s by physicists working on nuclear weapon projects in the Los Alamos National Laboratory. The physicists were investigating radiation shielding and the distance that neutrons would likely travel through various materials. Despite having most of the necessary data, the problem could not be solved with analytical calculations. John von Neumann and Stanislaw Ulam suggested that the problem be solved by modeling the experiment on a computer using chance. The name is a reference to the Monte Carlo Casino in Monaco where Stanislaw Ulam's uncle would borrow money to gamble
7. A Simple Example of the Monte Carlo Method Monte Carlo Calculation of Pi We will use the unit circle circumscribed by a square However, it is easier to just use one quadrant of the circle. Sooooo.
8. Monte Carlo Calculation of Pi So lets pretend you are a horrible dart player. The worst. Every throw is completely random. Now, Imagine throwing darts at the unit circle Because your throws are completely random, The number of darts that land within the shaded unit circle is proportional to the area of the circle In other words, =
9. Continued Example If you remember your geometry, it is easy to show: If each dart thrown lands somewhere inside the square, the ratio of "hits" (in the shaded area) to "throws" will be one-fourth the value of pi.
10. Last one about pi, I swear! If you actually tried this experiment, you would soon realize that it takes a very large number of throws to get a decent value of pi...well over 1,000. To make things easy on ourselves, we can have computers generate random numbers. So, How? If we say our circle's radius is 1.0, for each throw we can generate two random numbers, an x and a y coordinate we can then use (x,y) to calculate the distance from the origin (0,0) using the Pythagorean theorem. If the distance from the origin is less than or equal to 1.0, it is within the shaded area and counts as a hit. Do this thousands (or millions) of times then average, and you will wind up with an estimate of the value of pi. How good it is depends on how many iterations (throws) are done.
11. Monte Carlo Methods for Pricing Options Mostly used to calculate the value of an option with multiple sources of uncertainty or with complicated features In terms of theory, Monte Carlo valuation relies on risk neutral valuation. This just means that the current value of all financial assets is equal to the expected future payoff of the asset discounted at the risk-free rate. Here is the pattern that is used: 1. Generate several thousand possible (but random) price paths for the underlying (or underlyings) via simulation 2. Then calculate the associated exercise value (aka the "payoff") of the option for each path. 3. These payoffs are then averaged 4. Discounted to today. This result is the value of the option
12. History of MC methods in Finance Monte Carlo methods were first introduced to finance in 1964 by David B. Hertz through his article in Harvard Business Review The Monte Carlo method was first suggested as a way to price options in 1977 by Phelim Boyle in his paper: “Options: A Monte Carlo Approach”
13. Pricing the Call Option with the Monte Carlo Method A European-style call option gives the right, but not the obligation, to "call" for a stock at a specified "strike" price K at a specified time T  If the stock price at time is such that  , the option holder "exercises" and collects   from the seller of the option. If  , the option expires worthless In either case the seller of the option collects a fee called the "premium" for entering into the agreement. We want to find out what that premium should be
14. Financial Derivatives We want to predict the price on a stock at time T. Black-scholes assumes a Brownian motion (random walk) to figure out what is most likely to happen to the price of a stock after time T To apply this to the Monte Carlo Method we will run many possible Brownian walks of a stock over some period of time T The method we will use is: 1. Perform many sample walks of the stock 2. Compute the value of the derivative for each sample walk 3. Average all the derivative values together to come up with an expected value for the derivative
15. Summary Monte Carlo methods can help solve problems that are too complicated to solve using equations, or problems for which no equations exist They are useful for problems which have lots of uncertainty in inputs They can also be used as an alternate way to solve problems that have equation solutions. Drawbacks: Monte Carlo methods are often slower and less accurate than solutions via equations.
16. Sources http://demonstrations.wolfram.com/MonteCarloValuationOfAnOption/ http://demonstrations.wolfram.com/MonteCarloEstimateForPi/ http://en.wikipedia.org/wiki/Monte_Carlo_method http://en.wikipedia.org/wiki/Monte_Carlo_methods_in_finance http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html http://en.wikipedia.org/wiki/Random_walk http://en.wikipedia.org/wiki/Monte_Carlo_methods_in_finance