Monte Carlo simulations are a way of solving probabilistic problems by numerically 'imagining' many possible scenarios or games so as to calculate statistical properties such as expectations, variances or probabilities of certain outcomes. In finance we use such simulations to represent the future behaviour of equities, exchange rates, interest rates, etc., so as to either study the possible future performance of a portfolio or to price derivatives.

Example

We hold a complex portfolio of investments, we would like to know the probability of losing money over the next year since our bonus depends on our making a profit. We can estimate this probability by simulating how the individual components in our portfolio might evolve over the next year. This requires us to have a model for the random behaviour of each of the assets, including the relationship or correlation between them, if any.

Some problems which are completely deterministic can also be solved numerically by running simulations, most famously finding a value for n.

Long answer

It is clear enough that probabilistic problems can be solved by simulations. What is the probability of tossing heads with a coin, just toss the coin often enough and you will find the answer. More on this and its relevance to finance shortly. But many deterministic problems can also be solved this way, provided you can find a probabilistic equivalent of the deterministic problem. A famous example of this is Buffon's needle, a problem and solution dating back to 1777. Draw parallel lines on a table one inch apart. Drop a needle, also one inch long, onto this table. Simple trigonometry will show you that

the probability of the needle touching one of the lines is 2/n. So conduct many such experiments to get an approximation to n. Unfortunately because of the probabilistic nature of this method you will have to drop the needle many billions of times to find n accurate to half a dozen decimal places.

There can also be a relationship between certain types of differential equation and probabilistic methods. Stanislaw Ulam, inspired by a card game, invented this technique while working on the Manhattan Project towards the development of nuclear weapons. The name 'Monte Carlo' was given to this idea by his colleague Nicholas Metropolis.

Monte Carlo simulations are used in financial problems for solving two types of problems:

• Exploring the statistical properties of a portfolio of investments or cashflows to determine quantities such as expected returns, risk, possible downsides, probabilities of making certain profits or losses, etc.

• Finding the value of derivatives by exploiting the theoretical relationship between option values and expected payoff under a risk-neutral random walk.

Exploring portfolio statistics

The most successful quantitative models represent investments as random walks. There is a whole mathematical theory behind these models, but to appreciate the role they play in portfolio analysis you just need to understand three simple concepts.

First, you need an algorithm for how the most basic investments evolve randomly. In equities this is often the lognormal random walk. (If you know about the real/risk-neutral distinction then you should know that you will be using the real random walk here.) This can be represented on a spreadsheet or in code as how a stock price changes from one period to the next by adding on a random return. In the fixed-income world you may be using the BGM model to model how interest rates of various maturities evolve. In credit you may have a model that models the random bankruptcy of a company. If you have more than one such investment that you must model then you will also need to represent any interrelationships between them. This is often achieved by using correlations.

Once you can perform such simulations of the basic investments then you need to have models for more complicated contracts that depend on them, these are the options/derivatives/contingent claims. For this you need some theory, derivatives theory. This the second concept you must understand.

Finally, you will be able to simulate many thousands, or more, future scenarios for your portfolio and use the results to examine the statistics of this portfolio. This is, for example, how classical Value at Risk can be estimated, among other things.

Pricing derivatives

We know from the results of risk-neutral pricing that in the popular derivatives theories the value of an option can be calculated as the present value of the expected payoff under a risk-neutral random walk. And calculating expectations for a single contract is just a simple example of the above-mentioned portfolio analysis, but just for a single option and using the risk-neutral instead of the real random walk. Even though the pricing models can often be written as deterministic partial differential equations they can be solved in a probabilistic way, just as Stanislaw Ulam noted for other, non-financial, problems. This pricing methodology for derivatives was first proposed by the actuarially trained Phelim Boyle in 1977.

Whether you use Monte Carlo for probabilistic or deterministic problems the method is usually quite simple to implement in basic form and so is extremely popular in practice.

References and Further Reading

Boyle, P 1977 Options: a Monte Carlo approach. Journal of Financial Economics 4 323-338

Glasserman, P 2003 Monte Carlo Methods in Financial Engineering. Springer Verlag

Jackel, P 2002 Monte Carlo Methods in Finance. John Wiley & Sons Ltd

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