Final Projects

Mathematical Finance Project Guide

Final Projects

This page gives project options and expectations for the final project in Mathematical Finance.

The GitHub repository for the book is:

https://github.com/wanghemath/Book-MathFinance

The final project should demonstrate that you can connect:

\[ \text{financial question} \quad \longrightarrow \quad \text{mathematical model} \quad \longrightarrow \quad \text{Python implementation} \quad \longrightarrow \quad \text{numerical experiment} \quad \longrightarrow \quad \text{financial interpretation}. \]

The project does not need to be long, but it should be clear, reproducible, and mathematically meaningful.

Project goals

A successful project should show that you can:

formulate a financial modeling question precisely;
identify the mathematical assumptions behind the model;
implement the model in Python;
test the model numerically;
visualize and interpret the results;
explain limitations, model risk, and possible extensions.

Recommended project format

Students may complete the project individually or in small teams, depending on instructor policy.

A typical final project has three deliverables.

Deliverable	Suggested length	Purpose
Proposal	1–2 pages	State the question, model, data/simulation plan, and expected output.
Notebook	One clean `.ipynb` file	Provide reproducible code, numerical experiments, and figures.
Report	6–10 pages	Explain the mathematics, computation, results, and interpretation.

A short presentation may also be added if the course includes a final presentation session.

Project expectations

The project should include the following components.

1. Financial question

Begin with a clear question, such as:

How accurately do different pricing methods agree for the same option?
How does hedging error change with rebalancing frequency?
How does an implied-volatility smile affect option prices?
How does estimation error affect portfolio optimization?
How well does a VaR model pass a backtest?
How do Vasicek and CIR models generate different yield curves?

2. Mathematical model

State the model clearly. For example:

binomial model;
geometric Brownian motion;
Black–Scholes model;
stochastic volatility model;
Vasicek or CIR short-rate model;
HJM forward-rate model;
mean-variance portfolio model;
CAPM or multi-factor model;
VaR or Expected Shortfall risk model.

Students should explicitly list the assumptions.

3. Computational implementation

The project should include Python code that is organized into reusable functions.

For example:

def black_scholes_call(S0, K, T, r, sigma):
    # return the Black--Scholes call price
    pass

Good code should be readable, reproducible, and commented when necessary.

4. Numerical experiment

The project should include at least one serious numerical experiment.

Examples:

convergence as the number of binomial steps increases;
Monte Carlo confidence intervals as the number of paths increases;
hedging error distribution under different rebalancing frequencies;
implied-volatility smile fitting across strikes;
portfolio weights under different covariance estimates;
VaR exceptions over a rolling backtest;
yield-curve shapes under different short-rate parameters.

5. Interpretation

The project should not stop at code output. Students should explain:

what the figures show;
whether the numerical result matches theory;
where the model is reliable;
where the model is fragile;
what limitations remain.

Project topic list

The following topics are designed to align with the 18 Colab labs.

ID	Project topic	Closely related labs
P1	Binomial, Black–Scholes, Monte Carlo, and PDE Price Comparison	Labs 2, 3, 4, 8
P2	Delta Hedging Error under Discrete Rebalancing and Model Misspecification	Labs 4, 5, 18
P3	Implied Volatility Smile Construction and Calibration	Labs 4, 5, 17, 18
P4	Monte Carlo Variance Reduction for Option Pricing	Labs 3, 6, 7
P5	Pricing Asian, Barrier, and Lookback Options	Labs 3, 6, 7
P6	Finite Difference Methods for European and American Options	Labs 4, 8
P7	Change of Measure and Likelihood-Ratio Pricing	Labs 9, 10
P8	Vasicek and CIR Yield-Curve Modeling	Labs 11, 12
P9	HJM Forward-Curve Simulation and Yield-Curve Risk	Labs 11, 12, 13
P10	Mean-Variance Portfolio Optimization with Estimation Risk	Labs 14, 15
P11	CAPM and Multi-Factor Risk Decomposition	Labs 14, 15
P12	Value at Risk, Expected Shortfall, and Backtesting	Labs 15, 16
P13	Heston Stochastic Volatility and Volatility Skew	Labs 5, 17, 18
P14	Integrated Option Portfolio: Calibration, Hedging, and Risk	Labs 4, 5, 16, 17, 18
P15	A Student-Designed Mathematical Finance Project	Any relevant labs

Project P1: Binomial, Black–Scholes, Monte Carlo, and PDE Price Comparison

Main question

How do different option-pricing methods compare when applied to the same European option?

Suggested tasks

Price a European call or put using a binomial tree.
Price the same option using Black–Scholes.
Price the same option using Monte Carlo simulation.
Price the same option using a finite-difference method.
Compare accuracy, speed, and sensitivity to numerical parameters.

Mathematical ideas

no-arbitrage pricing;
risk-neutral expectation;
binomial convergence;
Black–Scholes formula;
finite-difference discretization;
Monte Carlo confidence intervals.

Possible figures

price versus number of binomial steps;
Monte Carlo confidence interval versus number of paths;
finite-difference price surface;
error relative to Black–Scholes benchmark.

Project P2: Delta Hedging Error under Discrete Rebalancing and Model Misspecification

Main question

How does Delta hedging perform when rebalancing is discrete and the assumed volatility is wrong?

Suggested tasks

Simulate stock paths under GBM.
Sell a European call and Delta hedge it.
Rebalance daily, weekly, monthly, and continuously approximated by a fine grid.
Compare hedging error distributions.
Repeat when hedging volatility differs from realized volatility.

Mathematical ideas

Black–Scholes Delta;
self-financing portfolios;
discrete versus continuous hedging;
volatility misspecification;
hedging error.

Possible figures

histogram of hedging errors;
hedging error standard deviation versus rebalancing frequency;
hedging error mean versus assumed volatility.

Project P3: Implied Volatility Smile Construction and Calibration

Main question

How can option prices be converted into implied volatilities, and how can a volatility smile be fitted?

Suggested tasks

Generate or collect option prices across strikes.
Compute implied volatility by numerical root-finding.
Plot implied volatility against strike or log-moneyness.
Fit a simple quadratic smile model.
Use the fitted smile to price additional options.

Mathematical ideas

Black–Scholes monotonicity in volatility;
implied volatility inversion;
smile and skew;
calibration error;
interpolation and extrapolation risk.

Possible figures

market price versus strike;
implied volatility versus log-moneyness;
fitted smile curve;
residual calibration errors.

Project P4: Monte Carlo Variance Reduction for Option Pricing

Main question

How much can variance reduction improve Monte Carlo option pricing?

Suggested tasks

Price a European option by plain Monte Carlo.
Add antithetic variates.
Add control variates.
Add moment matching or stratified sampling.
Compare estimator variance and confidence intervals.

Mathematical ideas

standard error;
confidence intervals;
negative correlation;
control variate regression;
estimator efficiency.

Possible figures

error distribution across repeated simulations;
confidence interval width versus number of paths;
variance reduction factor by method.

Project P5: Pricing Asian, Barrier, and Lookback Options

Main question

How do path-dependent payoffs change option values?

Suggested tasks

Simulate risk-neutral GBM paths.
Price arithmetic Asian options.
Price barrier options.
Price lookback options.
Study sensitivity to volatility and monitoring frequency.

Mathematical ideas

path dependence;
running average;
running maximum and minimum;
barrier crossing;
Monte Carlo pricing.

Possible figures

example stock paths with payoff features marked;
option price versus volatility;
barrier option price versus barrier level;
monitoring-frequency effect.

Project P6: Finite Difference Methods for European and American Options

Main question

How can option-pricing PDEs be solved numerically?

Suggested tasks

Implement explicit, implicit, or Crank–Nicolson finite differences.
Compare numerical prices with Black–Scholes for European options.
Extend to an American put if appropriate.
Study stability and convergence.

Mathematical ideas

Black–Scholes PDE;
terminal condition;
boundary condition;
grid discretization;
stability;
early exercise.

Possible figures

option value surface;
error versus grid size;
early exercise boundary for an American put.

Project P7: Change of Measure and Likelihood-Ratio Pricing

Main question

How can risk-neutral pricing be computed by reweighting paths simulated under a different measure?

Suggested tasks

Simulate GBM under a physical drift.
Compute likelihood-ratio weights.
Reweight payoffs to estimate risk-neutral prices.
Compare with direct risk-neutral simulation.
Study the effective sample size of the weights.

Mathematical ideas

Radon–Nikodym derivative;
Girsanov theorem;
likelihood-ratio weighting;
risk-neutral pricing;
effective sample size.

Possible figures

distribution of likelihood-ratio weights;
weighted versus unweighted payoff estimates;
effective sample size versus drift difference.

Project P8: Vasicek and CIR Yield-Curve Modeling

Main question

How do Vasicek and CIR short-rate models generate different zero-coupon bond prices and yield curves?

Suggested tasks

Simulate Vasicek and CIR short rates.
Compute model-implied bond prices.
Plot model-implied yield curves.
Compare parameter sensitivities.
Calibrate simple parameters to a synthetic zero curve.

Mathematical ideas

short-rate discounting;
affine term structure;
mean reversion;
positive-rate constraint in CIR;
yield-curve calibration.

Possible figures

simulated short-rate paths;
zero-coupon bond prices by maturity;
yield curves under different parameters;
calibration fit.

Project P9: HJM Forward-Curve Simulation and Yield-Curve Risk

Main question

How does the HJM framework model the evolution of the entire forward curve?

Suggested tasks

Specify an initial forward curve.
Choose one-factor or two-factor volatility functions.
Simulate forward curves under the HJM drift restriction.
Compute zero rates and discount factors.
Study principal components of yield-curve changes.

Mathematical ideas

instantaneous forward rates;
HJM no-arbitrage drift;
curve-valued stochastic processes;
factor structure;
yield-curve risk.

Possible figures

initial and simulated forward curves;
zero curves over time;
factor loading shapes;
principal components of curve changes.

Project P10: Mean-Variance Portfolio Optimization with Estimation Risk

Main question

How sensitive are optimized portfolios to estimated means and covariances?

Suggested tasks

Simulate or collect asset returns.
Estimate sample mean and covariance.
Compute global minimum-variance and tangency portfolios.
Add long-only or max-weight constraints.
Compare results under covariance shrinkage.

Mathematical ideas

portfolio expected return;
portfolio variance;
efficient frontier;
Sharpe ratio;
constrained optimization;
estimation risk.

Possible figures

efficient frontier;
random portfolios;
optimized portfolio weights;
out-of-sample performance comparison.

Project P11: CAPM and Multi-Factor Risk Decomposition

Main question

How can portfolio risk be decomposed into systematic and idiosyncratic components?

Suggested tasks

Simulate or collect factor returns and asset returns.
Estimate CAPM beta and alpha.
Estimate multi-factor exposures.
Decompose variance into factor and residual components.
Analyze portfolio factor exposure.

Mathematical ideas

linear regression;
beta;
alpha;
factor covariance;
idiosyncratic variance;
tracking error.

Possible figures

asset excess return versus market excess return;
estimated factor exposures;
systematic versus residual variance;
portfolio factor contribution chart.

Project P12: Value at Risk, Expected Shortfall, and Backtesting

Main question

How well do different risk models forecast portfolio losses?

Suggested tasks

Simulate or collect daily portfolio returns.
Compute historical VaR and ES.
Compute normal VaR and ES.
Implement rolling VaR forecasts.
Count exceptions and run coverage checks.
Compare performance in calm and volatile periods.

Mathematical ideas

loss distribution;
quantiles;
conditional tail expectation;
rolling estimation;
backtesting;
exception clustering.

Possible figures

returns and rolling VaR forecasts;
loss distribution with VaR and ES marked;
exception plot;
model comparison table.

Project P13: Heston Stochastic Volatility and Volatility Skew

Main question

How does stochastic volatility generate implied-volatility smiles and skews?

Suggested tasks

Simulate Heston stock and variance paths.
Price options under the Heston model.
Convert prices to Black–Scholes implied volatilities.
Study the effect of correlation, long-run variance, and vol-of-vol.
Compare with a constant-volatility Black–Scholes model.

Mathematical ideas

stochastic variance;
CIR-type variance process;
leverage effect;
option pricing under stochastic volatility;
implied-volatility skew.

Possible figures

stock and variance paths;
implied-volatility smile;
skew under different correlations;
option price comparison with Black–Scholes.

Project P14: Integrated Option Portfolio: Calibration, Hedging, and Risk

Main question

How can pricing, calibration, hedging, and risk measurement be combined in one workflow?

Suggested tasks

Construct a small option portfolio.
Generate or use option quotes.
Estimate implied volatilities.
Fit a smile or surface.
Compute Greeks.
Simulate hedging errors.
Compute portfolio VaR and ES.
Compare full revaluation with Delta-Gamma-Vega approximation.

Mathematical ideas

implied-volatility calibration;
Greeks;
hedging;
full revaluation;
risk approximation;
stress testing.

Possible figures

volatility smile or surface;
option portfolio value under scenarios;
hedging error histogram;
VaR and ES comparison.

Project P15: Student-Designed Mathematical Finance Project

Students may propose a custom project. A custom project should still include:

a clear mathematical finance question;
a model;
Python implementation;
numerical experiment;
financial interpretation;
discussion of limitations.

Possible custom directions include:

jump-diffusion option pricing;
local volatility;
rough volatility introduction;
credit risk modeling;
pairs trading and cointegration;
optimal execution;
reinforcement learning for hedging;
machine learning for volatility forecasting;
factor timing;
stress testing under macro scenarios.

Custom projects should be approved by the instructor before the main work begins.

Proposal template

Students may use the following proposal structure.

Project title:

Team members:

Main question:
What financial or mathematical question will the project answer?

Motivation:
Why is this question important?

Model:
What mathematical model will be used?
What assumptions does the model make?

Data or simulation plan:
Will the project use synthetic data, real data, or both?
How will the data be generated or obtained?

Computational plan:
What functions, algorithms, or numerical methods will be implemented?

Expected figures and tables:
What plots or tables will appear in the final report?

Risks and limitations:
What might make the project difficult?
What assumptions may be unrealistic?

Timeline:
What will be completed by each milestone?

Final report template

A recommended final report structure is:

1. Introduction
   - State the financial question.
   - Explain why it matters.

2. Mathematical model
   - Define variables and assumptions.
   - Present main equations.
   - Explain the pricing, hedging, optimization, or risk framework.

3. Computational method
   - Describe algorithms.
   - Explain simulation or numerical method.
   - Mention Python packages used.

4. Numerical experiments
   - Present figures and tables.
   - Explain parameter choices.
   - Compare results across methods or scenarios.

5. Interpretation
   - Explain what the results mean financially.
   - Discuss whether the results match theory.

6. Limitations and extensions
   - Discuss model assumptions.
   - Explain what could be improved.

7. Conclusion
   - Summarize the main finding.

8. References
   - Cite books, papers, data sources, and code resources.

Notebook expectations

The project notebook should be clean and reproducible.

A good notebook includes:

a title and short project summary;
imports in the first code cell;
fixed random seeds for simulation;
clearly named parameters;
reusable functions;
figures with titles and axis labels;
short markdown explanations between code blocks;
a conclusion cell summarizing findings.

Avoid notebooks that are only long blocks of code without explanation.

Presentation template

For a 7–10 minute presentation, use approximately 6–8 slides.

Slide	Content
1	Title, team, and main question
2	Financial motivation
3	Mathematical model
4	Computational method
5	Main results
6	Interpretation
7	Limitations
8	Conclusion and questions

Suggested milestones

Milestone	Suggested due date	Deliverable
Topic selection	Week 5	Choose topic and team
Proposal	Week 7	Submit 1–2 page proposal
Prototype notebook	Week 10	Basic code and first figures
Draft report	Week 12	Preliminary report
Final submission	Week 14	Final notebook, report, and optional slides

Grading rubric

A possible grading rubric is:

Category	Weight	Excellent work
Financial question and motivation	10%	The question is clear, meaningful, and well motivated.
Mathematical modeling	25%	Assumptions, variables, and equations are correct and clearly explained.
Computational implementation	25%	Code is correct, organized, reproducible, and appropriate for the model.
Numerical experiments	15%	Experiments are well designed and include useful comparisons.
Interpretation	15%	Results are explained in financial and mathematical terms.
Communication	10%	Report and notebook are clear, polished, and easy to follow.

Academic integrity and AI use

Students may use AI tools for help with coding, debugging, explanation, and brainstorming, subject to instructor policy. However, the final project must reflect the student’s own understanding.

A responsible AI-use statement could include:

AI tools were used for debugging and improving code readability. 
All mathematical modeling decisions, numerical experiments, interpretations, 
and final explanations were checked and written by the project team.

Students should cite external sources, including data sources, papers, textbooks, and substantial code resources.

Reproducibility checklist

Before submitting, check the following:

The notebook runs from top to bottom without errors.
Random seeds are set when simulations are used.
Figures have titles and axis labels.
Parameters are clearly stated.
The report explains all major equations.
Tables and figures are referenced in the text.
The conclusion answers the original question.
All sources are cited.
The submitted files have clear names.

Recommended file names

Use clear file names such as:

final-project-report.pdf
final-project-notebook.ipynb
final-project-slides.pdf

For teams, include a short project name:

heston-smile-project-report.pdf
heston-smile-project-notebook.ipynb
heston-smile-project-slides.pdf

Connection to the labs

The final project should extend the labs rather than simply repeat them.

A good project usually changes at least one of the following:

model parameters;
payoff structure;
numerical method;
market scenario;
risk measure;
calibration objective;
comparison benchmark;
data source;
interpretation question.

For example, a lab may price one option, while a project may compare pricing, hedging, and risk for an entire option portfolio.

Closing advice

Choose a project that is small enough to finish carefully.

A strong final project is not the one with the most complicated model. A strong project is one where the question is clear, the mathematics is correct, the code is reproducible, and the interpretation is thoughtful.

--- title: "Final Projects" subtitle: "Mathematical Finance Project Guide" description: "Final project ideas, expectations, milestones, templates, and grading rubrics for the Mathematical Finance book." format: html: toc: true toc-depth: 3 number-sections: false jupyter: python3 --- # Final Projects This page gives project options and expectations for the final project in **Mathematical Finance**. The GitHub repository for the book is: [https://github.com/wanghemath/Book-MathFinance](https://github.com/wanghemath/Book-MathFinance) The final project should demonstrate that you can connect: $$ \text{financial question} \quad \longrightarrow \quad \text{mathematical model} \quad \longrightarrow \quad \text{Python implementation} \quad \longrightarrow \quad \text{numerical experiment} \quad \longrightarrow \quad \text{financial interpretation}. $$ The project does **not** need to be long, but it should be clear, reproducible, and mathematically meaningful. ## Project goals A successful project should show that you can: - formulate a financial modeling question precisely; - identify the mathematical assumptions behind the model; - implement the model in Python; - test the model numerically; - visualize and interpret the results; - explain limitations, model risk, and possible extensions. ## Recommended project format Students may complete the project individually or in small teams, depending on instructor policy. A typical final project has three deliverables. | Deliverable | Suggested length | Purpose | |---|---:|---| | Proposal | 1--2 pages | State the question, model, data/simulation plan, and expected output. | | Notebook | One clean `.ipynb` file | Provide reproducible code, numerical experiments, and figures. | | Report | 6--10 pages | Explain the mathematics, computation, results, and interpretation. | A short presentation may also be added if the course includes a final presentation session. ## Project expectations The project should include the following components. ### 1. Financial question Begin with a clear question, such as: - How accurately do different pricing methods agree for the same option? - How does hedging error change with rebalancing frequency? - How does an implied-volatility smile affect option prices? - How does estimation error affect portfolio optimization? - How well does a VaR model pass a backtest? - How do Vasicek and CIR models generate different yield curves? ### 2. Mathematical model State the model clearly. For example: - binomial model; - geometric Brownian motion; - Black--Scholes model; - stochastic volatility model; - Vasicek or CIR short-rate model; - HJM forward-rate model; - mean-variance portfolio model; - CAPM or multi-factor model; - VaR or Expected Shortfall risk model. Students should explicitly list the assumptions. ### 3. Computational implementation The project should include Python code that is organized into reusable functions. For example: ```python def black_scholes_call(S0, K, T, r, sigma): # return the Black--Scholes call price pass ``` Good code should be readable, reproducible, and commented when necessary. ### 4. Numerical experiment The project should include at least one serious numerical experiment. Examples: - convergence as the number of binomial steps increases; - Monte Carlo confidence intervals as the number of paths increases; - hedging error distribution under different rebalancing frequencies; - implied-volatility smile fitting across strikes; - portfolio weights under different covariance estimates; - VaR exceptions over a rolling backtest; - yield-curve shapes under different short-rate parameters. ### 5. Interpretation The project should not stop at code output. Students should explain: - what the figures show; - whether the numerical result matches theory; - where the model is reliable; - where the model is fragile; - what limitations remain. ## Project topic list The following topics are designed to align with the 18 Colab labs. | ID | Project topic | Closely related labs | |---|---|---| | P1 | Binomial, Black--Scholes, Monte Carlo, and PDE Price Comparison | Labs 2, 3, 4, 8 | | P2 | Delta Hedging Error under Discrete Rebalancing and Model Misspecification | Labs 4, 5, 18 | | P3 | Implied Volatility Smile Construction and Calibration | Labs 4, 5, 17, 18 | | P4 | Monte Carlo Variance Reduction for Option Pricing | Labs 3, 6, 7 | | P5 | Pricing Asian, Barrier, and Lookback Options | Labs 3, 6, 7 | | P6 | Finite Difference Methods for European and American Options | Labs 4, 8 | | P7 | Change of Measure and Likelihood-Ratio Pricing | Labs 9, 10 | | P8 | Vasicek and CIR Yield-Curve Modeling | Labs 11, 12 | | P9 | HJM Forward-Curve Simulation and Yield-Curve Risk | Labs 11, 12, 13 | | P10 | Mean-Variance Portfolio Optimization with Estimation Risk | Labs 14, 15 | | P11 | CAPM and Multi-Factor Risk Decomposition | Labs 14, 15 | | P12 | Value at Risk, Expected Shortfall, and Backtesting | Labs 15, 16 | | P13 | Heston Stochastic Volatility and Volatility Skew | Labs 5, 17, 18 | | P14 | Integrated Option Portfolio: Calibration, Hedging, and Risk | Labs 4, 5, 16, 17, 18 | | P15 | A Student-Designed Mathematical Finance Project | Any relevant labs | ## Project P1: Binomial, Black--Scholes, Monte Carlo, and PDE Price Comparison ### Main question How do different option-pricing methods compare when applied to the same European option? ### Suggested tasks - Price a European call or put using a binomial tree. - Price the same option using Black--Scholes. - Price the same option using Monte Carlo simulation. - Price the same option using a finite-difference method. - Compare accuracy, speed, and sensitivity to numerical parameters. ### Mathematical ideas - no-arbitrage pricing; - risk-neutral expectation; - binomial convergence; - Black--Scholes formula; - finite-difference discretization; - Monte Carlo confidence intervals. ### Possible figures - price versus number of binomial steps; - Monte Carlo confidence interval versus number of paths; - finite-difference price surface; - error relative to Black--Scholes benchmark. ## Project P2: Delta Hedging Error under Discrete Rebalancing and Model Misspecification ### Main question How does Delta hedging perform when rebalancing is discrete and the assumed volatility is wrong? ### Suggested tasks - Simulate stock paths under GBM. - Sell a European call and Delta hedge it. - Rebalance daily, weekly, monthly, and continuously approximated by a fine grid. - Compare hedging error distributions. - Repeat when hedging volatility differs from realized volatility. ### Mathematical ideas - Black--Scholes Delta; - self-financing portfolios; - discrete versus continuous hedging; - volatility misspecification; - hedging error. ### Possible figures - histogram of hedging errors; - hedging error standard deviation versus rebalancing frequency; - hedging error mean versus assumed volatility. ## Project P3: Implied Volatility Smile Construction and Calibration ### Main question How can option prices be converted into implied volatilities, and how can a volatility smile be fitted? ### Suggested tasks - Generate or collect option prices across strikes. - Compute implied volatility by numerical root-finding. - Plot implied volatility against strike or log-moneyness. - Fit a simple quadratic smile model. - Use the fitted smile to price additional options. ### Mathematical ideas - Black--Scholes monotonicity in volatility; - implied volatility inversion; - smile and skew; - calibration error; - interpolation and extrapolation risk. ### Possible figures - market price versus strike; - implied volatility versus log-moneyness; - fitted smile curve; - residual calibration errors. ## Project P4: Monte Carlo Variance Reduction for Option Pricing ### Main question How much can variance reduction improve Monte Carlo option pricing? ### Suggested tasks - Price a European option by plain Monte Carlo. - Add antithetic variates. - Add control variates. - Add moment matching or stratified sampling. - Compare estimator variance and confidence intervals. ### Mathematical ideas - standard error; - confidence intervals; - negative correlation; - control variate regression; - estimator efficiency. ### Possible figures - error distribution across repeated simulations; - confidence interval width versus number of paths; - variance reduction factor by method. ## Project P5: Pricing Asian, Barrier, and Lookback Options ### Main question How do path-dependent payoffs change option values? ### Suggested tasks - Simulate risk-neutral GBM paths. - Price arithmetic Asian options. - Price barrier options. - Price lookback options. - Study sensitivity to volatility and monitoring frequency. ### Mathematical ideas - path dependence; - running average; - running maximum and minimum; - barrier crossing; - Monte Carlo pricing. ### Possible figures - example stock paths with payoff features marked; - option price versus volatility; - barrier option price versus barrier level; - monitoring-frequency effect. ## Project P6: Finite Difference Methods for European and American Options ### Main question How can option-pricing PDEs be solved numerically? ### Suggested tasks - Implement explicit, implicit, or Crank--Nicolson finite differences. - Compare numerical prices with Black--Scholes for European options. - Extend to an American put if appropriate. - Study stability and convergence. ### Mathematical ideas - Black--Scholes PDE; - terminal condition; - boundary condition; - grid discretization; - stability; - early exercise. ### Possible figures - option value surface; - error versus grid size; - early exercise boundary for an American put. ## Project P7: Change of Measure and Likelihood-Ratio Pricing ### Main question How can risk-neutral pricing be computed by reweighting paths simulated under a different measure? ### Suggested tasks - Simulate GBM under a physical drift. - Compute likelihood-ratio weights. - Reweight payoffs to estimate risk-neutral prices. - Compare with direct risk-neutral simulation. - Study the effective sample size of the weights. ### Mathematical ideas - Radon--Nikodym derivative; - Girsanov theorem; - likelihood-ratio weighting; - risk-neutral pricing; - effective sample size. ### Possible figures - distribution of likelihood-ratio weights; - weighted versus unweighted payoff estimates; - effective sample size versus drift difference. ## Project P8: Vasicek and CIR Yield-Curve Modeling ### Main question How do Vasicek and CIR short-rate models generate different zero-coupon bond prices and yield curves? ### Suggested tasks - Simulate Vasicek and CIR short rates. - Compute model-implied bond prices. - Plot model-implied yield curves. - Compare parameter sensitivities. - Calibrate simple parameters to a synthetic zero curve. ### Mathematical ideas - short-rate discounting; - affine term structure; - mean reversion; - positive-rate constraint in CIR; - yield-curve calibration. ### Possible figures - simulated short-rate paths; - zero-coupon bond prices by maturity; - yield curves under different parameters; - calibration fit. ## Project P9: HJM Forward-Curve Simulation and Yield-Curve Risk ### Main question How does the HJM framework model the evolution of the entire forward curve? ### Suggested tasks - Specify an initial forward curve. - Choose one-factor or two-factor volatility functions. - Simulate forward curves under the HJM drift restriction. - Compute zero rates and discount factors. - Study principal components of yield-curve changes. ### Mathematical ideas - instantaneous forward rates; - HJM no-arbitrage drift; - curve-valued stochastic processes; - factor structure; - yield-curve risk. ### Possible figures - initial and simulated forward curves; - zero curves over time; - factor loading shapes; - principal components of curve changes. ## Project P10: Mean-Variance Portfolio Optimization with Estimation Risk ### Main question How sensitive are optimized portfolios to estimated means and covariances? ### Suggested tasks - Simulate or collect asset returns. - Estimate sample mean and covariance. - Compute global minimum-variance and tangency portfolios. - Add long-only or max-weight constraints. - Compare results under covariance shrinkage. ### Mathematical ideas - portfolio expected return; - portfolio variance; - efficient frontier; - Sharpe ratio; - constrained optimization; - estimation risk. ### Possible figures - efficient frontier; - random portfolios; - optimized portfolio weights; - out-of-sample performance comparison. ## Project P11: CAPM and Multi-Factor Risk Decomposition ### Main question How can portfolio risk be decomposed into systematic and idiosyncratic components? ### Suggested tasks - Simulate or collect factor returns and asset returns. - Estimate CAPM beta and alpha. - Estimate multi-factor exposures. - Decompose variance into factor and residual components. - Analyze portfolio factor exposure. ### Mathematical ideas - linear regression; - beta; - alpha; - factor covariance; - idiosyncratic variance; - tracking error. ### Possible figures - asset excess return versus market excess return; - estimated factor exposures; - systematic versus residual variance; - portfolio factor contribution chart. ## Project P12: Value at Risk, Expected Shortfall, and Backtesting ### Main question How well do different risk models forecast portfolio losses? ### Suggested tasks - Simulate or collect daily portfolio returns. - Compute historical VaR and ES. - Compute normal VaR and ES. - Implement rolling VaR forecasts. - Count exceptions and run coverage checks. - Compare performance in calm and volatile periods. ### Mathematical ideas - loss distribution; - quantiles; - conditional tail expectation; - rolling estimation; - backtesting; - exception clustering. ### Possible figures - returns and rolling VaR forecasts; - loss distribution with VaR and ES marked; - exception plot; - model comparison table. ## Project P13: Heston Stochastic Volatility and Volatility Skew ### Main question How does stochastic volatility generate implied-volatility smiles and skews? ### Suggested tasks - Simulate Heston stock and variance paths. - Price options under the Heston model. - Convert prices to Black--Scholes implied volatilities. - Study the effect of correlation, long-run variance, and vol-of-vol. - Compare with a constant-volatility Black--Scholes model. ### Mathematical ideas - stochastic variance; - CIR-type variance process; - leverage effect; - option pricing under stochastic volatility; - implied-volatility skew. ### Possible figures - stock and variance paths; - implied-volatility smile; - skew under different correlations; - option price comparison with Black--Scholes. ## Project P14: Integrated Option Portfolio: Calibration, Hedging, and Risk ### Main question How can pricing, calibration, hedging, and risk measurement be combined in one workflow? ### Suggested tasks - Construct a small option portfolio. - Generate or use option quotes. - Estimate implied volatilities. - Fit a smile or surface. - Compute Greeks. - Simulate hedging errors. - Compute portfolio VaR and ES. - Compare full revaluation with Delta-Gamma-Vega approximation. ### Mathematical ideas - implied-volatility calibration; - Greeks; - hedging; - full revaluation; - risk approximation; - stress testing. ### Possible figures - volatility smile or surface; - option portfolio value under scenarios; - hedging error histogram; - VaR and ES comparison. ## Project P15: Student-Designed Mathematical Finance Project Students may propose a custom project. A custom project should still include: - a clear mathematical finance question; - a model; - Python implementation; - numerical experiment; - financial interpretation; - discussion of limitations. Possible custom directions include: - jump-diffusion option pricing; - local volatility; - rough volatility introduction; - credit risk modeling; - pairs trading and cointegration; - optimal execution; - reinforcement learning for hedging; - machine learning for volatility forecasting; - factor timing; - stress testing under macro scenarios. Custom projects should be approved by the instructor before the main work begins. ## Proposal template Students may use the following proposal structure. ```text Project title: Team members: Main question: What financial or mathematical question will the project answer? Motivation: Why is this question important? Model: What mathematical model will be used? What assumptions does the model make? Data or simulation plan: Will the project use synthetic data, real data, or both? How will the data be generated or obtained? Computational plan: What functions, algorithms, or numerical methods will be implemented? Expected figures and tables: What plots or tables will appear in the final report? Risks and limitations: What might make the project difficult? What assumptions may be unrealistic? Timeline: What will be completed by each milestone? ``` ## Final report template A recommended final report structure is: ```text 1. Introduction - State the financial question. - Explain why it matters. 2. Mathematical model - Define variables and assumptions. - Present main equations. - Explain the pricing, hedging, optimization, or risk framework. 3. Computational method - Describe algorithms. - Explain simulation or numerical method. - Mention Python packages used. 4. Numerical experiments - Present figures and tables. - Explain parameter choices. - Compare results across methods or scenarios. 5. Interpretation - Explain what the results mean financially. - Discuss whether the results match theory. 6. Limitations and extensions - Discuss model assumptions. - Explain what could be improved. 7. Conclusion - Summarize the main finding. 8. References - Cite books, papers, data sources, and code resources. ``` ## Notebook expectations The project notebook should be clean and reproducible. A good notebook includes: - a title and short project summary; - imports in the first code cell; - fixed random seeds for simulation; - clearly named parameters; - reusable functions; - figures with titles and axis labels; - short markdown explanations between code blocks; - a conclusion cell summarizing findings. Avoid notebooks that are only long blocks of code without explanation. ## Presentation template For a 7--10 minute presentation, use approximately 6--8 slides. | Slide | Content | |---:|---| | 1 | Title, team, and main question | | 2 | Financial motivation | | 3 | Mathematical model | | 4 | Computational method | | 5 | Main results | | 6 | Interpretation | | 7 | Limitations | | 8 | Conclusion and questions | ## Suggested milestones | Milestone | Suggested due date | Deliverable | |---|---|---| | Topic selection | Week 5 | Choose topic and team | | Proposal | Week 7 | Submit 1--2 page proposal | | Prototype notebook | Week 10 | Basic code and first figures | | Draft report | Week 12 | Preliminary report | | Final submission | Week 14 | Final notebook, report, and optional slides | ## Grading rubric A possible grading rubric is: | Category | Weight | Excellent work | |---|---:|---| | Financial question and motivation | 10% | The question is clear, meaningful, and well motivated. | | Mathematical modeling | 25% | Assumptions, variables, and equations are correct and clearly explained. | | Computational implementation | 25% | Code is correct, organized, reproducible, and appropriate for the model. | | Numerical experiments | 15% | Experiments are well designed and include useful comparisons. | | Interpretation | 15% | Results are explained in financial and mathematical terms. | | Communication | 10% | Report and notebook are clear, polished, and easy to follow. | ## Academic integrity and AI use Students may use AI tools for help with coding, debugging, explanation, and brainstorming, subject to instructor policy. However, the final project must reflect the student's own understanding. A responsible AI-use statement could include: ```text AI tools were used for debugging and improving code readability. All mathematical modeling decisions, numerical experiments, interpretations, and final explanations were checked and written by the project team. ``` Students should cite external sources, including data sources, papers, textbooks, and substantial code resources. ## Reproducibility checklist Before submitting, check the following: - The notebook runs from top to bottom without errors. - Random seeds are set when simulations are used. - Figures have titles and axis labels. - Parameters are clearly stated. - The report explains all major equations. - Tables and figures are referenced in the text. - The conclusion answers the original question. - All sources are cited. - The submitted files have clear names. ## Recommended file names Use clear file names such as: ```text final-project-report.pdf final-project-notebook.ipynb final-project-slides.pdf ``` For teams, include a short project name: ```text heston-smile-project-report.pdf heston-smile-project-notebook.ipynb heston-smile-project-slides.pdf ``` ## Connection to the labs The final project should extend the labs rather than simply repeat them. A good project usually changes at least one of the following: - model parameters; - payoff structure; - numerical method; - market scenario; - risk measure; - calibration objective; - comparison benchmark; - data source; - interpretation question. For example, a lab may price one option, while a project may compare pricing, hedging, and risk for an entire option portfolio. ## Closing advice Choose a project that is small enough to finish carefully. A strong final project is not the one with the most complicated model. A strong project is one where the question is clear, the mathematics is correct, the code is reproducible, and the interpretation is thoughtful.