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Option Pricing and Estimation of Financial Models with R by Stefano M. Iacus

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Chapter 4

Numerical methods

4.1 Monte Carlo Method

Suppose we are given a random variable X and are interested in the evaluation of images/c04_I0001.gif where g( · ) is some known function. If we are able to draw n pseudo random numbers x1, … , xn from the distribution of X, then we can think about approximating images/c04_I0002.gif with the sample mean of the g(xi),

4.1 4.1

The expression (4.1) is not just symbolic but holds true in the sense of the law of large numbers whenever images/c04_I0004.gif. Moreover, the central limit theorem guarantees that

images/c04_I0005.gif

where N(m, s2) denotes the distribution of the Gaussian random variable with expected value m and variance s2. In the end, the number we estimate with simulations will have a deviation from the true expected value images/c04_I0006.gif of order images/c04_I0007.gif. Given that P(|Z| < 1.96)0.95, Z N(0, 1), one can construct an interval for the ...

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