ABSTRACT
In this chapter we introduce Monte Carlo methods, a set of mathematical technique based on the computation of a sample mean https://www.w3.org/1998/Math/MathML"> X - https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003266730/1992ab88-6f81-434e-a8c9-92a1be4ca054/content/math0_4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where the sample is obtained by simulation. https://www.w3.org/1998/Math/MathML"> X - https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003266730/1992ab88-6f81-434e-a8c9-92a1be4ca054/content/math0_5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is an unbiased estimator of the theoretical expectation https://www.w3.org/1998/Math/MathML"> E X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003266730/1992ab88-6f81-434e-a8c9-92a1be4ca054/content/math0_6.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , which is more efficient as the number of simulations increase. Then, a simulation of the payoff function becomes a generic tool to compute European option prices.
As examples, we implement a Monte Carlo method for the Black-Scholes-Merton model to price:
A European call and put option.
A path-dependent option. In particular, a Barrier option.
An option depending on more than one underlying assets.
