Convergence Error Monte Carlo
Contents |
help Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us
Monte Carlo Simulation Convergence
Learn more about Stack Overflow the company Business Learn more about hiring developers or monte carlo integration convergence posting ads with us Quantitative Finance beta Questions Tags Users Badges Unanswered Ask Question _ Quantitative Finance Stack Exchange is
Standard Error Monte Carlo
a question and answer site for finance professionals and academics. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers monte carlo error propagation are voted up and rise to the top What does “convergence” in Monte Carlo simulation mean? up vote 2 down vote favorite I have read about convergence in terms of MC simulation for derivative pricing, but I am not clear on what it exactly means. Let us suppose I price an option 100,000 paths twice and both result in the same option price. Does that mean 100,000 monte carlo error analysis paths has resulted in convergence? Also, in determining the number of paths to use for pricing, is getting the same option price with 2 different runs a factor? (Assumption is I am not reseeding the Random Number. So the sequence of Random Numbers between the two runs is different). option-pricing monte-carlo numerical-methods share|improve this question asked Apr 1 '15 at 20:01 Karthik Balasubramaniam 835 add a comment| 4 Answers 4 active oldest votes up vote 2 down vote accepted To keep things simple let's assume you have a perfect random number generator (i.e. I will discuss only the statistics not the numerics of the problem). I will also focus on the practical matter and gloss over some mathematical details. From a practical perspective "convergence" means that you will never get an exact answer from Monte-Carlo but increasingly good approximations. Try out your 100'000 paths example. The two values for the price of your option will be slightly different everytime you use a fresh, i.e. independent, sample. Two mathematical theorems are relevant to describe convergence: First, the law of large numbers, which says that the average of independent samples converges to the expected value (i.e. price) and the central limit theorem,
(red=1,..,10, blue=11,..,100, green=101,..,256). Points from Sobol sequence are more evenly distributed. In numerical analysis, quasi-Monte Carlo method is a method for numerical integration
Standard Deviation Monte Carlo
and solving some other problems using low-discrepancy sequences (also called confidence interval monte carlo quasi-random sequences or sub-random sequences). This is in contrast to the regular Monte Carlo method or
Variance Monte Carlo
Monte Carlo integration, which are based on sequences of pseudorandom numbers. Monte Carlo and quasi-Monte Carlo methods are stated in a similar way. The problem is http://quant.stackexchange.com/questions/17204/what-does-convergence-in-monte-carlo-simulation-mean to approximate the integral of a function f as the average of the function evaluated at a set of points x1, ..., xN: ∫ [ 0 , 1 ] s f ( u ) d u ≈ 1 N ∑ i = 1 N f ( x i ) . {\displaystyle \int https://en.wikipedia.org/wiki/Quasi-Monte_Carlo_method _{[0,1]^{s}}f(u)\,{\rm {d}}u\approx {\frac {1}{N}}\,\sum _{i=1}^{N}f(x_{i}).} Since we are integrating over the s-dimensional unit cube, each xi is a vector of s elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the xi are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Monte Carlo uses a pseudorandom sequence. The advantage of using low-discrepancy sequences is a faster rate of convergence. Quasi-Monte Carlo has a rate of convergence close to O(1/N), whereas the rate for the Monte Carlo method is O(N−0.5).[1] The Quasi-Monte Carlo method recently became popular in the area of mathematical finance or computational finance.[1] In these areas, high-dimensional numerical integrals, where the integral should be evaluated within a threshold ε, occur frequently. Hence, the Monte Carlo method and the quasi-Monte Carlo method are beneficial in these situations. Contents 1 Approximation error bounds of quasi-Monte Carlo 2 Monte Carlo and
be down. Please try the request again. Your cache administrator is webmaster. Generated Tue, 04 Oct 2016 23:06:42 GMT by s_hv1002 (squid/3.5.20)
be down. Please try the request again. Your cache administrator is webmaster. Generated Tue, 04 Oct 2016 23:06:42 GMT by s_hv1002 (squid/3.5.20)