Monte Carlo Simulation
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 outcome. In finance we use simulation to represent future behavior of rates, equities and stress testing, so to study future performance of portfolio or for pricing.
Best way to understand simulation is by simple example : we hold a portfolio of investments, we want to understand its probability of losing money over next year. We can estimate this probability by simulating how the individual components in our portfolio might evolve over next year. This requires us to understand and model random behavior of each of the assets, including the relationship or correlation between them. Problem which are deterministic can be solved numerically by running simulations.
It is clear enough that probabilistic problems can be solved by simulations. Best example of coin tossing heads, just toss the coin often enough and you will find answer. But this problem can be solved using simple simulation. In many deterministic problem can also e solved this way, provided you can find a probabilistic equivalent of deterministic problem. In our article we will focus on financial domain.
Monte Carlo simulation are used in financial problems for solving two type of problems:
(1) Exploring statistical properties of portfolio of investment or cash-flows to determine quantities such as expected returns, risk, possible downsides, probabilities of making profit or losses etc.
(2) Finding value of derivatives by exploiting the theoretical relationship between option values and expected payoff under a risk-neutral random walk.
Exploring Portfolio Statistic : The most successful quantitative models represent investments as random walks. There is whole mathematical theory behind these models, but to appreciate the role they play in portfolio analysis you just need to understand 3 simple concept.
(1) You need an algorithm for how the most basic investment evolve randomly. In equities this is often the lognormal random walk(If you know about real/risk-neutral distinction then you should know that you will be using the real random walk here). This can be mapped by changes in stock price from one period to next by adding random return. In credit you may model that models the random bankruptcy of company.
(2) Once you create simple model you simulate many thousands or more, future scenarios for your portfolio and use the result to examine the statistic of this portfolio. Best example in credit risk industry is to calculate VAR.