What is Monte Carlo Simulation?
Monte Carlo Simulation is a method of solving probabilistic problems that involves numerically ‘seeing’ many alternative scenarios or games in order to quantify statistical attributes such as expectancies, variances, and probability of specific outcomes. In finance, simulations are used to predict future behaviour of shares, currency rates, and interest rates. This is done to analyse portfolio performance or price derivatives.
A Short Example
We have a complex investment portfolio and want to know the likelihood of losing money in the coming year, as our bonus depends on profitability. To assess this probability, we can simulate the evolution of our portfolio’s various components over the next year. To achieve this, we need a model that accounts for the random behaviour of each asset, as well as any potential correlation.
Numerical simulations can solve some deterministic issues, such as determining the value of 
A Deep Dive into Monte Carlo Simulation
Simulations can effectively address probabilistic challenges. To determine the likelihood of tossing heads, simply toss the coin repeatedly. More information on this and its significance to finance will follow shortly. Many deterministic issues can be handled using this method, as long as a probabilistic equivalent exists.
Buffon’s needle is a well-known example of this, with its problem and solution dating back to 1777. Create parallel lines on a table, one inch apart. Drop a one-inch-long needle onto this table. Using simple trigonometry, the likelihood of the needle reaching one of the lines is 
Conduct multiple experiments to approximate
Certain differential equations may correlate with probabilistic approaches. Stanislaw Ulam, inspired by a card game, developed this strategy while working on the Manhattan Project to create nuclear weapons. Nicholas Metropolis, a colleague, named this idea ‘Monte Carlo’.
Monte Carlo simulations are used in ļ¬nancial problems for solving two types of problems:
- Analysing a portfolio’s statistical features can reveal predicted returns, risk, potential drawbacks, profit or loss probability, and more.
- To determine the value of derivatives, use the theoretical link between option values and expected payoff in a risk-neutral random walk.
Exploring Portfolio Statistics
The most successful quantitative models treat investments as random walks.These models are based on a complex mathematical theory, yet understanding three basic notions is all that is required to comprehend their role in portfolio analysis.
To begin, develop an algorithm for the random evolution of basic investments. In equities, this is frequently the lognormal random walk. (If you are familiar with the actual/risk-neutral distinction, please note that the real random walk will be used here.)
This can be represented on a spreadsheet or in code by adding a random return to show how a stock price fluctuates over time. In the fixed-income world, the BGM model can be used to predict the evolution of interest rates over different maturities. Credit models may simulate a company’s insolvency at random.
When modelling multiple investments, it’s important to include their interrelationships. Correlations are commonly used to achieve this.
After simulating basic investments, it’s important to create models for more complex contracts, such as options, derivatives, and contingent claims. This requires knowledge of derivative theory. Understand thisĀ second notion.
You may analyse your portfolio’s data by simulating hundreds of potential possibilities. This method can be used to estimate classical Value at Risk and other metrics.
Conclusion
The results of risk-neutral pricing show that in popular derivatives theories, the value of an option can be determined as the present value of the expected reward under a risk-neutral random walk. Calculating expectations for a single contract is similar to portfolio analysis, but with a risk-neutral approach instead of the genuine random walk. Pricing models, while commonly represented as deterministic partial differential equations, can be solved probabilistically, as highlighted by Stanislaw Ulam for non-financial situations. Phelim Boyle, an actuary, first introduced this derivative pricing method in 1977.
Monte Carlo is widely used for both probabilistic and deterministic situations due to its ease of implementation.
Related Readings
- Modern Portfolio Theory in Finance
- Arbitrage in Quantitative Finance: All You Need To Know
- Modelling Approaches in Quantitative Finance: All You Need To Know
- Put-Call Parity: All You Need To Know