Options Valuation

An educational Python implementation of option-pricing models, covering closed-form European option valuation and simulation-based American option valuation.

What this project covers

European options — Black–Scholes

• Closed-form call and put pricing
• Delta, Gamma, and Vega
• Implied volatility solved with Newton–Raphson iteration
• Expiry and numerical edge-case handling

American options — Longstaff–Schwartz Monte Carlo

• Geometric Brownian Motion path simulation
• Backward induction through possible exercise dates
• In-the-money path selection
• Least-squares estimation of continuation value with a quadratic basis
• Pathwise comparison of immediate exercise versus continuation

flowchart LR
    A[Simulate stock paths] --> B[Set terminal payoffs]
    B --> C[Move backward one step]
    C --> D[Select in-the-money paths]
    D --> E[Regress future cash flows on current prices]
    E --> F{Exercise value greater?}
    F -- Yes --> G[Exercise now]
    F -- No --> H[Keep future cash flow]
    G --> I[Continue backward]
    H --> I

Loading

Benchmark

The American put example uses:

Parameter	Value
Initial stock price	36
Strike price	40
Time to maturity	1 year
Risk-free rate	6%
Volatility	20%
Simulations	10,000
Exercise steps	20

The Longstaff–Schwartz estimate is approximately $4.47. Because the method uses Monte Carlo simulation, the output varies slightly between runs.

Repository contents

• Options Valuation.ipynb — executable pricing models and verification examples
• American Option Pricer.pptx — presentation explaining the American option pricer
• Options Intro_Kevin.pptx — introductory options material
• Futures_Pricing.pptx — supporting futures-pricing material
• Option Project 3_Kevin.pptx — project presentation

Run locally

python -m venv .venv

Activate the environment, then install the dependencies and launch Jupyter:

python -m pip install -r requirements.txt
jupyter lab "Options Valuation.ipynb"

Model assumptions and limitations

• Constant risk-free rate and volatility
• Geometric Brownian Motion for the underlying price
• No dividends or transaction costs
• Quadratic regression basis for the continuation value
• Intended for learning and portfolio demonstration, not production trading or investment advice