Monte Carlo Simulation Techniques

A "sampled success metric" is a performance measure or evaluation criterion calculated from a sample or subset of data rather than the entire population. Its calculation often involves higher costs per sample, such as manual review, leading to a trade-off between sample size and metric accuracy/sensitivity. In this tech blog, written by the data science team from Shopify, the discussion revolves around how the team leverages Monte Carlo simulation to understand metric variability under various scenarios to help the team make the right trade-offs. Initially, the team defines simulation metrics to describe the variability of the sampled success metric. For instance, if the actual success metric is decreasing over time, the metric could indicate how many months of sampled success metric would show a decrease, termed as "1-month decreases observed". Then, the team defines the distribution to run the Monte Carlo simulation. Monte Carlo simulation, a computational technique using random sampling to estimate outcomes of complex systems or processes with uncertain inputs, draws samples from a dedicated distribution that matches business needs. Based on past observations, the team’s application follows a Poisson distribution. Next comes the massive simulation phase, where the team runs multiple simulations for one parameter and then changes various parameters to simulate different scenarios. The goal is to quantify how much the sample mean will differ from the underlying population mean given realistic assumptions. The final result provides a clear statistical distribution of how much extra sample size could lead to metrics variability decrease and increased accuracy. This case study demonstrates that Monte Carlo simulation could be a valuable toolkit to add to your decision-making and data science knowledge. #datascience #analytics #metrics #algorithms #simulation #montecarlo #decisionmaking – – –  Check out the "Snacks Weekly on Data Science" podcast and subscribe, where I explain in more detail the concepts discussed in this and future posts:    -- Spotify: https://lnkd.in/gKgaMvbh   -- Apple Podcast: https://lnkd.in/gj6aPBBY    -- Youtube: https://lnkd.in/gcwPeBmR https://lnkd.in/dKnrZzzV 

📊 Advancing Commodity Option Pricing: The Power of Monte Carlo Simulations In the unpredictable landscape of energy and commodity markets, conventional pricing models like Black-Scholes often struggle to keep pace with real-world dynamics. The complexity introduced by factors such as geopolitical tensions, supply chain disruptions, and economic fluctuations calls for more sophisticated tools. 📈 New Article Highlight: I’m excited to share my recent exploration into Monte Carlo simulations for pricing commodity options using Python. This article covers: • The implementation of Geometric Brownian Motion (GBM) and its role in generating realistic price paths. • Essential calculations such as annualized volatility and the present value of expected payoffs. • The implementation of these models in Python, providing a step-by-step guide for practical application. Additionally, I touch upon enhancements like stochastic volatility models that offer improved realism in modeling commodity prices, capturing unpredictable shifts and high-risk scenarios more effectively. #OptionPricing #MonteCarloSimulation #CommodityTrading #EnergyMarkets #PythonProgramming #FinancialRisk #AdvancedFinance

Monte Carlo simulation 🎲 is a powerful tool in quantitative finance, helping to price financial instruments and options with complex features, path dependencies, or stochastic processes. By simulating thousands of possible outcomes, it provides a reliable approximation of expected payoffs. 📊📊 Here are some key applications of Monte Carlo simulation in asset and option pricing:👇👇 1. Pricing Path-Dependent Options 🛤️ - Asian options: Payoffs depend on the average price of the underlying asset over a period. Monte Carlo helps calculate the average and resulting payoffs. - Barrier options: Simulates price paths to determine if knock-in or knock-out barriers are breached. - Cliquet options: Models periodic resets and their impact on payoffs. 2) Pricing Exotic Options 🧩 - Rainbow options: Simulates correlated price paths of multiple underlying assets. - Basket options: Evaluates payoffs based on the weighted sum of several assets. - Digital options: Estimates binary payoffs based on whether certain conditions are met. 3) American Option Pricing 📈 Monte Carlo is used with Least-Squares Monte Carlo (LSMC) to incorporate early exercise features by estimating the continuation value through regression. 4) Credit Derivative Pricing 💳 - Credit default swaps (CDS): Simulates default probabilities and loss distributions. - Collateralized debt obligations (CDOs): Models cash flows by simulating defaults in a pool of underlying assets. 5) Real Options Analysis 🚀 Used in corporate finance to value investment decisions such as expanding, delaying, or abandoning projects by simulating cash flows under different market scenarios. Monte Carlo simulations are indispensable in quantitative finance for their flexibility and ability to model real-world complexities 🌟. Where have you applied Monte Carlo in your work? Share your thoughts below! 👇 #QuantFinance #MonteCarloSimulation #FinancialModeling #QuantitativeFinance #OptionPricing #StochasticProcesses #AssetPricing #DerivativesPricing #RiskManagement #ExoticOptions #FinancialEngineering #FinancialAnalysis #VolatilityModeling #MonteCarloMethods #QuantResearch #FinanceTech #DataScienceForFinance #MathematicalFinance #PortfolioManagement #FinanceInnovation

This standard Monte Carlo simulation says that there is an 83% probability of gold increasing in price over the next 12 months. The average of all simulated outcomes is for a 17.3% gain. That will put the gold price at over $2'900. The Monte Carlo simulation says there's a one in three the gold price will rise more than 25%. The full range of outcomes with probabilities is shown below. What is a Monte Carlo simulation? Monte Carlo simulations are used extensively in the finance and investment domain to model uncertainty and optimize decisions. In portfolio management, for example, Monte Carlo simulations can help investors analyze the potential performance of an asset or portfolio. By generating a large number of potential future price paths for the asset(s) based on historical volatility and drift, investors can estimate the probability of different outcomes. Monte Carlo simulations can be done on anything which has daily pricing, but it has the greatest predictive power when dealing with heavily traded assets, such as mega-cap stocks, indices, currencies, gold, heavily traded commodities, and Treasury Bonds. A Monte Carlo simulation is not that difficult to do. See the "tests" below for the methodology. You don't need to be a mathematician. Anybody with schoolboy maths could do it in an Excel Sheet. Overall, Monte Carlo simulations provide a computationally robust way to assess the probabilities of various outcomes across different asset classes. Here's what we know about gold: Data period: 2004-11-18 until 2024-08-30 Mean daily return: 0.04% Mean Annual Return: 10.18% Standard Deviation (1 day): 1.1% Annualized Standard Deviation: 17.53% The Monte Carlo Tests Each test comprises picking 365 random daily returns out of the 4979 daily returns available. By compounding the daily returns in a randomized way we can reach a possible 1 year return. The test is then repeated 100 times to give us 100 possible one year returns. That's 100 possible price targets. Here are the results: Starting Value of XAUUSD = $2500 Range of final values of at the end of August 2025: $1602 to $4957 90% of the final values are in the range $2197 to $3983 95% of the final values are in the range of $2013 to $4242 Average of final values: $2933 (Anything outside that range is probably caused by a black Swan event) Probability of final price exceeding starting price: 83% Probability of final price dropping more than 10%: 7% Probabilty of final price dropping more than 25%: 1% Probability of final price dropping more than 50%: ~~ Probability of final price rising more than 10%: 56% Probability of final price rising more than 25%: 32% Probability of final price rising more than 50%: 14% In comparison, running a Monte Carlo simulation on the S&P 500, indicates an 81% probability of the price rising and a 9% probability of the index dropping more than 10%. The average of all simulated outcomes is for a gain of 19.1%.

Day 3 of 7: Unlocking Quant Knowledge – Monte Carlo Method for VaR Welcome to Day 3! Today, we’re taking a closer look at the Monte Carlo Method for calculating Value at Risk (VaR)—with a real-world example to show how it works in risk management. Monte Carlo Recap (Please also refer to my post on Monte Carlo from last week) Unlike simpler VaR methods that assume normal returns or ignore how stocks move together, Monte Carlo simulations generate thousands of possible market scenarios. Each scenario reflects different daily (or monthly) returns for every stock, capturing correlations and offering a more realistic view of risk. A Real-World Example: Monthly VaR for a Tech Portfolio - Defining the Portfolio Imagine you have $1 million invested across three tech-oriented stocks: Tesla (40%), Apple (35%), and Microsoft (25%). Based on historical data, Tesla has an annual volatility of 30%, Apple 20%, and Microsoft 18%. You also gather correlation data to capture how these stocks move together during major tech events. - Adjusting for a 1-Month Horizon Since you want a 1-month VaR, you scale annual volatility accordingly to reflect realistic monthly fluctuations. - Running the Simulations Your risk system generates 10,000+ random future price paths for each stock over one month. Some scenarios show minor fluctuations, while others simulate market shocks where all three stocks decline sharply. - Identifying Potential Losses After ranking the simulated portfolio losses, you identify the 95th percentile loss. If the 95% VaR is $50,000, this means there is a 5% chance of losing more than $50,000 in one month. Why This Matters Captures Correlations: If Tesla and Apple both drop after weak earnings, your model reflects this relationship, unlike simpler methods that treat stocks as independent. Accounts for Tail Risk: Standard geometric Brownian motion assumes log-normal returns, which may underestimate extreme events, but Monte Carlo can incorporate fat-tailed distributions for a more accurate risk assessment. Supports Better Decisions: Understanding potential worst-case losses helps investors decide whether to hedge, rebalance, or adjust their exposure. Key Takeaways - Monte Carlo offers a more realistic risk assessment by factoring in changing correlations, dividends, and non-normal distributions. - Reliable inputs, particularly for volatility and correlations, are essential for accurate results. - While computationally demanding, Monte Carlo provides deeper insights than simpler VaR models. Fun Fact Monte Carlo simulations were named after the Monaco casino district, highlighting their reliance on probability and randomness to predict outcomes. Follow along, share your thoughts, and let’s master VaR together—how do you see Monte Carlo clarifying your investment risks? #QuantFinance #RiskManagement #ValueAtRisk #MonteCarlo #EquityPortfolio #FinancialModeling #Finance #Inestment #MarketRisk

>>Monte Carlo Simulation on Databricks — Scaling Risk Analytics the Smart Way<< In financial-risk modeling, accuracy is non-negotiable — but speed and scalability often get ignored. Monte Carlo simulations are powerful, but on a single Python kernel, they’re painfully slow. That’s where Databricks changes the game. The Pipeline: ADLS → PySpark → Delta → Power BI ADLS – Source market data, credit spreads, FX rates, and trade exposures. PySpark – Run 10K+ Monte Carlo iterations in parallel. Delta Lake – Persist exposure distributions with full lineage & versioning. Power BI – Visualize PFE, EPE, VaR, and stress outcomes in near real time. Each simulation becomes a reproducible, governed, and scalable process — not a one-off experiment. Why It Matters In credit-risk capital (SACCR), market-risk VaR, or stress testing, we’re not just crunching numbers — we’re quantifying uncertainty at scale. Runtime cut from hours to minutes Full audit trail via Delta + Purview Reproducible experiments tracked in MLflow Monte Carlo on Databricks = mathematical rigor meets distributed compute. #MonteCarloSimulation #DataEngineering #RiskAnalytics #Power_BI #AzureDatabricks #DeltaLake #DataQuality #CreditRisk #DataGovernance #FinancialModeling #FinTech #C2C #C2H #Opentowork