**Chapter 3 ****The**** Monte – Carlo Option Pricing Formula**

**3.1 Introduction**

Monte – Carlo simulation is a numerical method that is useful in many situations when no closed form solution is available, named after a famous casino in Monaco, where Stanislaw Ulam’s uncle would borrow money to gamble. The Monte – Carlo method can be used to simulate a wide range of stochastic processes, essentially the method solves a problem by directly simulating the physical process and then calculating the average result of the process. The benefits of the Monte – Carlo method over other techniques increases as the sources of the uncertainty of the problem increase.

“For more than three or four state variables, formulae such as Black-Scholes do not exist, while other standard approaches such as the Binomial Option pricing model face several difficulties and are not practical. In these cases, Monte – Carlo methods converge to the solution more quickly than other numerical methods, require less memory and are easier to program.”

The Monte – Carlo method is a very general process and thus is a valid approach in areas such as physics, chemistry and computer science to name but a few. The method was actually established by Stanislaw Ulam and John von Neumann while working on the Manhattan Project, the project to develop the first nuclear weapon during World War II. The method was then introduced in derivative valuation in finance by Phelim Boyle in 1977; some 30 years after the original simulations were established.

Stanislaw Ulam was a Polish Mathematician born April 13th 1909 in Lwow, Galicia, then in Austria – Hungary. Ulam obtained his PhD from the Polytechnic Institute in Lvov in 1933, and made major contributions in many fields from pure mathematics to physics. Some of Ulam’s more prestigious discoveries include solving the problem of how to initiate fusion in the hydrogen bomb, participating in the Manhattan Project and proposing the Teller – Ulam design of thermonuclear weapons. Ulam also invented nuclear pulse propulsion and developed a number of mathematical tools in number theory, set theory, ergodic theory, and algebraic theory. Ulam was also an early advocate of using computers to conduct mathematical experiments.

John von Neumann was born December 28th 1903 in Budapest, Austria – Hungary, and he received his PhD in mathematics from the University of Budapest in 1925. Von Neumann is “generally regarded as one of the foremost mathematicians of the 20th Century.”

**“**Von Neumann was a child prodigy. When only six years old he could divide eight-digit numbers in his head.”

Von Neumann was one of the original members of the Institute for advanced study also known as the ‘demigods’, at Princeton University, along with Albert Einstein, J. Robert Oppenheimer, Kurt Godel and Erwin Panofsky. Von Neumann was most notably known for his work on game theory, von Neumann algebras, von Neumann architecture and cellular

automata.

“There’s an old joke about the Fermi Paradox, a problem which occurred to Enrico Fermi one day at Los Alamos: where are They? If there are intelligent aliens out there in the universe, why aren’t they here yet? A million years is nothing, as the universe reckons things, but, judging from our own track-record, a species that much older than us would have technology which would blow our minds, pretty close to limits set by physical laws. Leo Szilard is supposed to have answered Fermi: Maybe they’re already here, and you just call them Hungarians.”

Phelim Boyle was born on a farm in Lavey, County Londonderry, Northern Ireland in 1941. Boyle received his PhD in applied mathematics from Trinity College, Dublin in 1970. Boyle was the first person to apply the Monte – Carlo simulation to option pricing in 1977, and is also known for his contributions in the area of quantitative finance which includes the use of the Trinomial method to price options. Boyle also received the prestigious International Association of Financial Engineers Financial Engineer of the Year in 2005.

Source by Scott E McClelland