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In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. [1] Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results.
Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin.
The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path {, …,} to also take {, …,}.The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the precision.
Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes.
The 5% Value at Risk of a hypothetical profit-and-loss probability density function. Value at risk (VaR) is a measure of the risk of loss of investment/capital.It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day.
using Monte Carlo integration. This integral is the expected value of (), where = + and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as ,,. Then the estimate is given by
Monte Carlo Methods allow for a compounding in the uncertainty. [7] For example, where the underlying is denominated in a foreign currency, an additional source of uncertainty will be the exchange rate : the underlying price and the exchange rate must be separately simulated and then combined to determine the value of the underlying in the ...
The goal of a multilevel Monte Carlo method is to approximate the expected value [] of the random variable that is the output of a stochastic simulation.Suppose this random variable cannot be simulated exactly, but there is a sequence of approximations ,, …, with increasing accuracy, but also increasing cost, that converges to as .