MATH802 Advanced Financial Modelling &Analytics

Semester 1,2025

Written Project

Due Date:2nd June 2025,4:00pm

Instructions:

·The project should be submitted as a single PDF document through Canvas by the due date.You should also submit all SAS program files..

·Do not submit your files as a zip folder.

·Your project file should include this cover page.

·Whereprogrammingin SASis required,ascreenshot/image of the relevant output shouldbe provided in the PDF and a SAS program file should be submitted.SAS code will be assessed based on its accuracy and elegance.

·Failure to follow these instructions will result in the deduction of marks.

Late Submission: Late projects,without an approved extension,willbe subject to a deduction of 5% (one grade e.g.from C+to C)of the total mark available for each 24-hour period,or part thereof,up to a maximum of five calendar days.Projects over five days late willnot normally be accepted or marked and students will receive a DNC(Did Not Complete)for that assessment.You may find more information from the URL link:

https://Www.aut.ac.nz/being-a-student

Originality: This project is an individual piece of work. You are encouraged to discuss the project with your lecturers and classmates,however,the work you submit must be your own.Projects that show high similarities to work submitted by other students will be investigated for plagiarism and treated very seriously.Plagiarism software,such as TurnItIn,may be used to electronically compare submissions to those of other students and to documents on the internet.Talk to the lecturer if you have any questions about this requirement.

Introduction

In this project,you will design and evaluate hedging strategies for an options portfolio using real-world data from the U.S.financialmarkets.The primary emphasis will be on delta hedging,while also exploring more advanced techniques  such as gamma and vega hedging to address second-order and volatility- related risks.You will employ the Black-Scholes model to compute the Greeks(Delta,Gamma,Theta, Vega,and Rho)and integrate them into your hedging framework.

Project Steps

1.Data Collection&Preprocessing

2.Greeks Calculation using Black-Scholes Model

3.Designing Hedging Strategies

4.Report

Step 1:Data Collection and Preprocessing

You are expected to collect relevant and high-quality financial data from reliable sources.The dataset should include historical price data for both the underlying asset(e.g.,a NASDAQ-listed stock or index) and its corresponding options.Preprocessing should involve handling missing values,ensuring data consistency,aligning dates across different datasets,etc.

Suggested  Data Sources

·Stock,Option and Volatility:Yahoo Finance

·Risk-Free Rate:U.S.Treasury Yield(10-year)from FRED

Step 2:Greeks Calculation for Black-Scholes Model Implementation

Implement the Black-Scholes formula in SASto compute the option prices

Call Price:

C=S·N(d₁)-K·erT.N(d₂)

Put Price:

P=K·erT.N(-d2)-S·N(-d₁)

Where:

d1 = σ√T/ln(S/K)+(r +σ2/2)T, d₂=d1-σ√T

Compute the Greeks for each option:

Implementation:

· Use numerical differentiation if closed-form solutions are complex.

· Visualize the option Greeks across a range of strike prices and maturities,and perform a detailed analysis of their behavior and implications.

· Discuss how the sensitivities of option Greeks vary with different levels of volatility. Suggested Stocks for Analysis

· Stocks: Select an individual stock like AAPL etc.,orindex like NASDAQ-100.

· Option: Select options with maturities ranging from  1month to  12 months.

Step 3:Hedging Strategy

Delta-Hedging

Implement a delta-hedging strategy for a hypothetical options portfolio.Follow the steps below and document your process,including calculations,data sources,and  assumptions.

1.Portfolio Setup:

·Selecta real-world  U.S.-listed  stock(e.g.,AAPL,SPY)and an associated  optionposition,either long  or  short(e.g.,1  contract=100  shares).

·Collect market data:current stock price,optionstrike price,time to expiration,implied volatility, and risk-free rate.

2.Hedge Execution:

·Calculate  the  portfolio's  total  Delta(e.g.,for   1  contract,△×100).

·Neutralize the portfolio's Delta by taking an offsetting position in the underlying stock(e.g., buy or sell△×100 shares).

·Explain how this position achieves delta neutrality and mitigates price risk.

3.Rebalancing:

·Simulate price changes in the underlying stock(e.g.,±5%over one day or one week)based on the  historical  information.

·Recalculate the option's Delta after the price change.

·Adjust the stock position to restore delta neutrality.Quantify the number of shares bought or sold.

·Discuss the  frequency  of rebalancing  and potential  challenges.

4. Analysis:

·Compare the profit/loss(P/L)of  the  hedged  portfolio(option  +stock)versus an unhedged portfolio (option only)under the simulated price change.

·Provide a brief analysis of the hedge's effectiveness in reducing risk.

Advanced Strategies

Explore an advancedhedging technique—either,Gamma(T)orVega(v)hedging,to manage risks beyond those addressed by Delta hedging.

·Design and implement a hedging strategy that manages Gamma or Vega risk,using additional options or other appropriate financial instruments.

·Calculate and present the portfolio's Gammaor Vega exposures before and after implementing the hedging strategy.

·Analyze the portfolio's performance before and after implementing the hedging strategy,focusing on the reduction of volatility exposure and overall risk.

·Provide a brief explanation of howit complements delta-hedging.

·Discuss any limitations,trade-offs,or practical considerations involved in maintaining a Gamma- or Vega-neutral portfolio.

Step 4:Report

Deliverables

·A  comprehensive  report  that  includes  all  relevant  calculations,data  sources,and  step-by-step details of the hedging strategy.

·Clear and well-labeledvisualizations that supportyour analysis,accompaniedbythoughtful interpretation.

·A concise 1-2 page discussion summarizing your key findings,insights,and reflections.

Report  Structure

1.Introduction (Objectives &Background)

2.Data Description(Sources,Cleaning Process)

3. Methodology (Black-Scholes,Greeks,Hedging)

4. Implementation(Portfolio  construction,Calculations,etc.)

5.Analysis &Results(Greeks Analysis,Hedging Performance)

6. Discussion(Key insights and findings,practical implications of hedging.)

7. Conclusion(Summary,Limitations)

Grading Criteria