Assignment 3: Develop and Interpret Statistical Models for Prediction and Inference in AI – Index Fund Price Movements

University of Illinois Urbana-Champaign | FIN 580: Statistical and AI Models in Finance | Assignment 3 (Individual Project Report, Week 9, Fall 2026 term)

Students in FIN 580 at the University of Illinois Urbana-Champaign complete this Assignment 3 by designing, coding, and interpreting statistical models that forecast Index Fund price paths and quantify investment risk. The project centers on stochastic processes implemented in Python with real market data, directly supporting the course goal of building inference skills for AI-enhanced financial decision systems. Participants select at least one major US Index Fund or ETF, estimate parameters from historical series, run Monte Carlo simulations, and translate outputs into clear risk metrics and strategy recommendations. The finished report follows a fixed professional structure that mirrors standards used by quantitative analysts and risk teams.

FAQ

Which stochastic model should students use for Index Fund price simulation in this FIN 580 assignment?

Geometric Brownian Motion remains the core model because it captures the log-normal diffusion of asset prices under constant drift and volatility; students may extend it with regime-switching or fat-tailed adjustments when diagnostics show poor fit to the chosen fund’s returns.

How many simulation paths and what time horizon produce reliable risk statistics?

Most high-scoring reports run 1,000 to 10,000 paths over a 6- to 12-month horizon (126–252 trading days); convergence checks on Value-at-Risk and expected shortfall quantiles guide the final choice.

Where do students obtain clean adjusted price data and how do they annualize parameters?

The yfinance library pulls daily adjusted closes from Yahoo Finance; students compute log returns, then scale mean and standard deviation by 252 for annualized drift $μ$ and volatility $σ$ .

What AI integration opportunities should appear in the recommendations section?

Reports that earn top marks link simulation outputs to automated monitoring rules, dynamic position sizing, or hybrid signals that feed downstream machine-learning classifiers for regime detection.

Why This Matters in Practice

Portfolio managers and robo-advisors run similar Monte Carlo engines daily to size hedges, set stop-loss thresholds, and satisfy regulatory stress tests. Graduates who master parameter calibration and path interpretation move directly into quant risk or fintech roles where these exact outputs inform live capital allocation.

Learning Outcomes

By completing this assignment students will:

Select and justify stochastic processes appropriate for financial time series.
Implement simulation algorithms in Python using NumPy and SciPy.
Extract investment-risk metrics from simulated path distributions.
Communicate technical findings in a format ready for professional or regulatory audiences.
Evaluate opportunities to embed statistical models inside larger AI-driven monitoring systems.

Task Description and Requirements

Objective Design, implement, and interpret statistical models that predict and infer Index Fund price movements using Python. Simulate potential future price paths and assess associated investment risks to support data-driven financial decision-making.

Use Case A financial analyst models the movement of Index Fund prices with statistical models to predict future trends and assess investment risks. Using Python, you will implement and analyze different models.

Required Steps

Data Collection: Download historical daily adjusted close prices for one or more US Index Funds or ETFs (examples: SPY, VTI, QQQ) covering at least five full years.
Model Selection: Choose and justify appropriate stochastic processes; Geometric Brownian Motion is the baseline, yet extensions are welcome when justified by diagnostics.
Implementation: Code the selected models in Python with NumPy and SciPy; document all equations and assumptions clearly.
Simulation: Generate ensembles of future price paths (minimum 1,000 paths) over a forward horizon you select and defend.
Analysis: Compute statistical summaries, risk metrics (VaR, expected shortfall, probability of loss), and compare simulated distributions against historical behavior.
Reporting: Produce a single professional PDF report containing every section listed below; embed figures, summary tables, and concise code snippets where they add immediate value.

Report Structure (use these exact headings)

Abstract (150–250 words): Overview of the project, models used, key numerical findings, and main recommendations.
Methodology: Data source and preprocessing steps; full mathematical specification of each model including the stochastic differential equation and its solution; parameter estimation procedure; simulation algorithm; any diagnostic tests performed.
Results: Presentation of simulation outputs through figures (path fan charts, terminal-price histograms, quantile tables); statistical summaries; direct comparison of risk metrics across scenarios or funds.
Recommendations: Actionable investment insights or AI integration opportunities such as model-based trading signals, portfolio optimization rules, or automated risk-monitoring dashboards.
Conclusion: Concise summary of what the simulations reveal about future price behavior and risk; reflection on modeling limitations and learning gains.
References: APA 7th edition; minimum three sources published 2018–2026 plus data and library citations.

Technical Notes

Use vectorized NumPy operations for speed; avoid slow Python loops where possible.
Seed the random number generator for reproducibility and state the seed in the report.
All equations must appear in proper mathematical notation.
Figures require clear titles, axis labels, and legends; color-blind friendly palettes are expected.

Submission Upload one PDF report (file name: FIN580_Assignment3_YourNetID.pdf) plus a zipped folder containing the Jupyter notebook or .py scripts that generated every figure and table. Due date announced on the course LMS; late penalties follow department policy.

Marking Rubric

Data Collection & Preprocessing (15 %) Excellent: Complete five-year series, rigorous cleaning, transparent parameter annualization, and clear justification of chosen fund(s). Good: Minor gaps in documentation or slight inconsistencies in return calculation. Satisfactory: Data present but preprocessing steps incomplete or poorly explained. Needs Improvement: Missing data, incorrect adjustments, or no justification.

Model Selection & Implementation (25 %) Excellent: Correct GBM (or justified extension) equations, clean vectorized code, explicit assumptions stated, and parameter estimation method fully documented. Good: Minor coding inefficiencies or incomplete assumption discussion. Satisfactory: Model implemented but equations or code contain errors that affect results. Needs Improvement: Fundamental misunderstanding of the stochastic process.

Simulation, Visualization & Statistical Summaries (20 %) Excellent: Large number of paths, informative figures, accurate risk metrics, and clear convergence diagnostics. Good: Adequate paths and figures; minor issues in quantile calculation or labeling. Satisfactory: Simulation runs but visualizations lack clarity or risk metrics contain calculation errors. Needs Improvement: Too few paths or missing key risk statistics.

Analysis, Recommendations & AI Integration (25 %) Excellent: Deep interpretation that links numerical outputs to concrete investment or monitoring actions; creative yet realistic AI augmentation ideas. Good: Solid interpretation with at least two practical recommendations. Satisfactory: Descriptive rather than analytical; recommendations remain generic. Needs Improvement: Little interpretation or recommendations absent.

Report Quality, Structure & Referencing (15 %) Excellent: Professional tone, flawless APA citations, tight abstract, seamless flow between sections, and polished figures. Good: Minor formatting or citation inconsistencies. Satisfactory: Some sections underdeveloped or referencing incomplete. Needs Improvement: Poor organization or missing required sections.

Learning Resources and Suggested References (APA 7th)

Students are expected to locate additional sources; the following recent works provide strong foundations for model justification and extensions:

Mensah, E. T. et al. (2023). Simulating stock prices using geometric Brownian motion: A modified approach under normal and convoluted distributional assumptions. Scientific African, 20, e01657. https://doi.org/10.1016/j.sciaf.2023.e01657

Chen, S. (2022). A fast Monte-Carlo simulation algorithm based on geometric Brownian motion. Proceedings of SPIE, 12163, 1216319. https://doi.org/10.1117/12.3102075

Kim, G., Choi, S.-Y., & Kim, Y. (2025). A diffusion-based generative model for financial time series via geometric Brownian motion. arXiv preprint arXiv:2507.19003. https://arxiv.org/abs/2507.19003

Additional reading on parameter estimation and risk metrics appears in the course readings folder and standard quantitative finance references from 2018 onward.

Sample Response Excerpt – Results and Early Recommendations

Students frequently choose the SPDR S&P 500 ETF Trust (SPY) because its long history and tight tracking of the broad market simplify parameter interpretation. Five years of daily adjusted closes downloaded via yfinance produced an annualized drift near 0.11 and volatility of 0.17 after log-return calculations and 252-day scaling. The Monte Carlo engine then generated 5,000 paths over the next 252 trading days using the exact solution of the geometric Brownian motion SDE. Terminal prices centered around an 11 percent gain relative to the final observed level, yet the left tail showed a 5 percent Value-at-Risk of roughly 21 percent, indicating material downside exposure even under the baseline model. Path visualizations revealed clusters of high dispersion that matched the realized volatility spikes of 2022, confirming the simulation preserved key statistical features of the original series. These quantiles translate directly into position-sizing rules that shrink exposure when simulated drawdown probability exceeds internal thresholds. The overall pattern supports the use of simulation-derived alerts inside automated risk systems rather than static allocation bands. This outcome aligns closely with the modified distributional framework presented in recent extensions of geometric Brownian motion (Mensah et al., 2023).

Refining Drift Estimation Amid Market Regimes

Historical drift estimates remain noisy because equity markets shift between expansion and contraction regimes on timescales shorter than a typical five-year calibration window. Chen (2022) demonstrated that fast Monte Carlo schemes still deliver stable quantile estimates even when drift is allowed to vary across two or three latent states; students who add a simple two-state Markov switching layer on top of baseline GBM often obtain tighter confidence bands around the 5 percent and 1 percent VaR numbers. In practice this means re-estimating $μ$ on rolling three-year windows or conditioning it on macroeconomic indicators before each simulation batch. The payoff appears in the recommendations section where dynamic drift produces earlier warning signals ahead of the 2022-style drawdowns. Regime-aware calibration also reduces the common student error of treating a single long-run $μ$ as fixed when the fund in question exhibits clear cyclical behavior.

Turning Path Distributions into Live Monitoring Rules

Once the report quantiles exist, the next operational step is to embed them in lightweight Python dashboards that poll fresh prices daily and recompute breach probabilities. Teams that treat the simulated 5 percent VaR as a hard risk budget can automatically scale position size or trigger option overlays the moment realized volatility pushes the forward distribution outside tolerance. A frequent implementation gap is the omission of transaction costs and slippage; adding even modest round-trip cost assumptions (5–10 basis points) materially changes the breakeven holding period for any signal derived from the paths. Another useful extension compares pure GBM quantiles against those produced by a lightweight LSTM that updates drift and volatility each week; the hybrid version typically narrows the inter-quartile range of terminal prices and therefore supplies more decisive position-size recommendations. Students who document both the statistical and the engineering choices in the recommendations section demonstrate the precise skill set FIN 580 targets for AI-augmented risk roles.

Assignment (Week 10 / Assignment 4)

Building directly on the stochastic foundation established in Assignment 3, students will replace or augment the constant drift and volatility of geometric Brownian motion with outputs from supervised or sequence models (LSTM, gradient boosting, or regime classifiers) trained on macroeconomic and sentiment features. The task requires a side-by-side backtest of pure GBM versus at least one hybrid specification over a true out-of-sample window, followed by a 1200- to 1800-word comparative report that quantifies improvement in VaR accuracy, turnover, and risk-adjusted returns. Submission includes updated fan charts, performance tables, and a short reflection on when the added model complexity justifies the extra computational cost in live trading or monitoring pipelines.

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