Monte Carlo for the Newbies

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The purpose of this paper is to be an introduction to Monte Carlo techniques used in nance, mainly to price complicated products, called exotics. This paper has been designed for a on-expert public and focuses on giving a broad overview of Monte Carlo techniques from a practical point of view and presents also some improvements that can be done to increase the eciency.

Contents
1 Origin 3
1.1 Geometrical application . . . . . . . . . . . . . . . . . . . . . 3
2 Financial modelling 5
2.1 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Classic Call case . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Results obtained . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Why does this work ? . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Asian options . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 General case for 1 asset . . . . . . . . . . . . . . . . . . . . . 8
2.7 General case for n assets . . . . . . . . . . . . . . . . . . . . . 8
3 How to build a Monte Carlo pricer 9
3.1 Approach used . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Generation of random numbers . . . . . . . . . . . . . . . . . 9
3.2.1 Introduction to the problem . . . . . . . . . . . . . . . 9
3.2.2 The dierent types of generators . . . . . . . . . . . . 9
3.2.3 From uniform to normal . . . . . . . . . . . . . . . . . 10
3.3 How to build correlated arrays . . . . . . . . . . . . . . . . . 11
4 Greek computation 12
4.1 What are these greeks ? . . . . . . . . . . . . . . . . . . . . . 12
4.2 Simple approach . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 More advanced methods . . . . . . . . . . . . . . . . . . . . . 13
4.3.1 Path dierenciation . . . . . . . . . . . . . . . . . . . 13
4.3.2 Other methods . . . . . . . . . . . . . . . . . . . . . . 14
5 Conclusion 14

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