Black-Scholes Model Implementation

---

Overview

A mathematical and computational implementation of the Black–Scholes model for European option pricing, including derivation sketches, explicit solutions, simulation of underlying paths, and numerical delta hedging verification. This project bridges continuous-time financial theory and practical discrete-time implementation.

black-scholes-model-implementation/
│
├── docs/
│   ├── BlackScholes_Derivation.pdf
│   └── BlackScholes_Derivation.tex
│
├── figures/
│   └── ...
│
├── notebooks/
│   ├── delta_hedging_analysis.ipynb
│   └── option_pricing_and_greeks.ipynb
│
├── src/
│   ├── delta_hedge.py
│   ├── option.py
│   └── underlying.py
│
├── .gitattributes
├── .gitignore
└── README.md

---

Features

• A pdf showcasing theory:
  • Assumptions.
  • Black-Scholes equation.
  • Heat equation vs Feynman-Kac formula, and equivalence.
  • Black-Scholes formula.
  • Explicit pricing functions for vanilla options.
• Option classes, with methods for pricing and Greeks.
• Addition operator overriding to instantiate linear combinations of vanilla calls and puts. E.g. butterfly spread.
• Simulation of geometric Brownian motion sample paths.
• Delta hedging strategy implementation.
• Delta hedging analysis:
  • PnL of hedged portfolio vs unhedged control group.
  • Convergence analysis of discretisation error vs rebalancing frequency.
  • Confirmation of drift-independence of delta hedge.
• Figures:
  • Sample underlying paths.
  • Payoff functions.
  • Option price and Greek surfaces.
  • Discrete hedging approximations.
  • PnL histograms.
  • Hedging error convergence using log-log plots.
  • Drift-independence.

---

Key Visualisations

• Call delta surface showing sensitivity of option value to the underlying. The transition from 0 to 1 across moneyness is clearly visible at maturity as the sigmoid slope converges to the step function.

• Simulated geometric Brownian motion path with corresponding option price, delta, and gamma evolution. Demonstrates dynamic sensitivity as maturity approaches.

• Comparison of unhedged and delta-hedged PnL. Hedging significantly reduces variance and confirms convergence toward theoretical replication as rebalancing frequency increases.

---

Technologies

• LaTeX (MikTex, TexMaker).
• Visual Studio Code.
• Git.
• Python.
• Jupyter.
• NumPy.
• SciPy.
• Matplotlib.
• Pandas.

---

Outcome and Conclusion

This project delivers a complete theoretical and computational implementation of the Black–Scholes model introduced by Fischer Black and Myron Scholes. The analytical pricing formula, Greeks, and PDE derivation were translated into reusable, object-oriented code, with support for structured payoffs via linear combinations of options.

Simulations of geometric Brownian motion and discrete delta hedging experiments validate key theoretical results:

• PnL converges towards zero under a hedged portfolio.
• Hedging error decreases with higher rebalancing frequency.
• Pricing and replication are drift-independent under lognormal dynamics.

Overall, the project bridges continuous-time financial theory and practical numerical implementation, providing a solid foundation for extending beyond the classical Black–Scholes model.

---

Future Extensions

• Add a pdf to mathematically analyse the theoretical discretisation error.
• Distribution analysis of PnL (e.g. with KDE).
• Volatility study:
  • Implied volatility solver.
  • Empirical data, and the volatility plots (smile, skew, surface).
  • PnL skew under volatility misspecification.
• Implementation and comparison (e.g. accuracy and tractability vs computational speed) of alternate pricing methods:
  • Explicit formula (already implemented).
  • Directly applying numerical integration of Black-Scholes formula.
  • FFT pricing.
  • Monte Carlo pricing.
• Black-Scholes extended model:
  • American option pricing (smooth pasting).
  • Dividend-paying assets (both continuous and discrete cases).
• Testing framework (pytest or unittest).
• Advanced hedging implementation:
  • Accounting for and minimising transaction costs.
  • Minimising hedging frequency by only adjusting delta when gamma hits a threshold.
• Exotic models:
  • Lévy processes (fat tails and jumps).
  • Stochastic volatility (Heston and rough Heston).

---