Frontiers of Risk Management, Volume I by Dennis Cox

Author:Dennis Cox
Language: eng
Format: epub
Publisher: Business Expert Press
Published: 2018-07-04T10:19:57+00:00

CHAPTER 9

The Simple Art of Monte Carlo

Aaron Brown

Morgan Stanley

Introduction

Monte Carlo is a beautifully simple and counterintuitive idea: when a problem is too hard to solve rationally, do something random. It has application beyond applied mathematics. In the introduction to The Simple Art of Murder,1 Raymond Chandler explained the secret to keeping the plot moving in hard-boiled fiction, “[T]he demand was for constant action; if you stopped to think you were lost. When in doubt, have a man come through a door with a gun in his hand.” Thirty-three years later, the television show The A Team faced a similar problem. The show’s innovation was to have constant action sequences; there was no tolerance for thinking, investigating, or waiting. The screenwriters were forced to make such frequent use of the phrase “It’s so crazy, it just might work” to explain essentially random actions by the main characters, it became a tagline of the show. Of course, by that time the action stakes had been raised. A man with a gun was not enough—the A Team’s plan was usually something like build an armored assault vehicle out of a can of tuna fish and a used-up cigarette lighter, then drive down Main Street and hope the bad guys shoot and reveal their hideout.

Older readers will remember hand-soldered electronics with vacuum tubes. When these worked badly, people learned that a sharp slap on the side often corrected the problem. We used to tap meters and feather dials to get more accurate results. With modern solid-state components, randomness usually hurts rather than helps performance, but we still randomly jiggle papers to get them to line up and gently shake powders to get them smooth. Improvisational theatre teaches actors to respond to everything with, “yes, but . . . ,” and add some random element. “Yes” and “no” both stop the action. You need added randomness to keep the skit going and, of course, random mutations are the reason this chapter has been included in this book.

Pi and Canfield

A classic early application of Monte Carlo analysis is described in Hammersley and Handscomb’s Monte Carlo Methods.2 In 1864, an army officer named Fox was wounded in the American Civil War. He amused himself while recovering by tossing a needle onto a lined sheet of paper to determine the proportion of times the needle intersected a line. ­Georges-Louis Leclerc, Comte de Buffon, tried to calculate this proportion in 1777, so the experiment is known as “Buffon’s Needle.” Pierre-­Simon Laplace derived the correct mathematical expression in 1812. The probability is equal to two times the length of the needle divided by pi times the distance between the lines. Fox reversed the logic and used the experimental result to estimate pi (two times the length of the needle divided by the proportion of hits times the distance between the lines should equal pi in the long run). This is the key idea of Monte Carlo analysis, to use a random experiment to get an approximate answer to a mathematical problem.

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