Black-Scholes Option Pricing - Theory, Simulation, and Market Comparison

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Overview

This Jupyter notebook provides a comprehensive implementation and analysis of the Black-Scholes-Merton (BSM) model for European option pricing, combining analytical formulas with Monte Carlo simulation techniques and real market data validation.

The Black-Scholes-Merton model applies to European options on non dividend paying stocks. It can be extended to European options on stocks paying discrete dividends and to European options on other assets (such as stock indices, currencies, and futures). It does not apply to American options, which must be valued using the binomial tree methodology.

Objectives

  1. Implement analytical Black-Scholes-Merton pricing formulas for European call and put options
  2. Develop Monte Carlo simulation methods for option pricing using Geometric Brownian Motion
  3. Compare theoretical prices with real market data from Yahoo Finance (Boeing stock example)
  4. Analyze convergence properties and accuracy of Monte Carlo methods vs analytical solutions

Theoretical Foundation

ASSUMPTIONS OF THE BLACK–SCHOLES–MERTON (BSM)

Assumptions of the Black–Scholes–Merton (BSM) option pricing model

  1. Market & trading assumptions
    • Frictionless markets: no transaction costs, taxes, or bid–ask spreads; trading can occur continuously.
    • Perfect liquidity & divisibility: you can buy/sell any quantity of the underlying (including fractional shares) without moving the price.
    • No arbitrage: markets do not allow riskless profit opportunities.
    • Continuous trading & continuous hedging: the option can be dynamically replicated by continuously adjusting a hedge (delta hedging).
  2. Interest rates & financing
    • Constant risk-free rate: a known, constant risk-free interest rate applies over the option’s life.
    • Borrowing/lending at the risk-free rate: investors can both borrow and lend unlimited amounts at the same risk-free rate.
  3. Underlying price dynamics
    • Geometric Brownian motion: the underlying price follows a lognormal diffusion process: returns are normally distributed (log-returns normal), price is always positive, continuous paths (no jumps).
    • Constant volatility: the volatility of the underlying return is constant over the option’s life.
  4. Payout structure / instrument assumptions
    • European-style exercise: the option is exercised only at maturity (not early).
    • No dividends: the underlying pays no dividends or other cash flows during the option’s life
  5. Information & timing
    • Parameters are known and stable: volatility and the risk-free rate are known (or treated as known) and do not change unexpectedly.
    • No default / counterparty risk (implicitly): the model ignores credit risk in the replication argument.

Pricing Formulas

  • Call: \(c = S_0 N(d_1) - K e^{-rT} N(d_2)\)
  • Put: \(p = K e^{-rT} N(-d_2) - S_0 N(-d_1)\)

Implementation

1. Parameters

  • Standard Parameters Used:
    • Current Stock Price (S): $214.00
    • Strike Price (K): $225.00
    • Risk-free Rate (r): 3.6%
    • Volatility (σ): 30%
    • Time to Maturity (T): 1.5 years

2. Analytical Implementation

Core Functions Developed:

  • black_scholes_call() / black_scholes_put() - Closed-form pricing
  • black_scholes_greeks() - Calculate Delta, Gamma, Vega, Theta, Rho
  • Input validation and edge case handling (at-maturity scenarios)

Visualizations:

  • Option prices vs time across multiple strike prices
  • Payoff diagrams at maturity comparing theoretical values with intrinsic values
  • Clear demonstration of time decay and moneyness effects

3. Monte Carlo Simulation

Implementation Features:

  • simulate_stock_paths() - Vectorized GBM path generation
    • Discretization: \(S_{t+dt} = S_t \exp\left[(r - \frac{\sigma^2}{2})dt + \sigma\sqrt{dt}Z\right]\)
    • Supports multiple paths with configurable time steps
  • monte_carlo_option_price() - European option pricing via simulation
    • Generates terminal stock price distribution
    • Calculates discounted expected payoffs
    • Returns price, standard error, and payoff array

Key Results:

  • 20 sample paths visualization demonstrating stochastic price evolution
  • Terminal price distribution (100,000 simulations) validates lognormal assumption
  • Statistical properties: Mean ≈ $225.85, matching theoretical drift

4. Convergence Analysis

Comprehensive Testing:

  • Compared Monte Carlo vs Analytical across:
    • 3 test cases: Near maturity ITM, Mid-term ATM, Early OTM
    • Simulation sizes: 100 to ~300,000 paths

Performance Metrics:

  • Relative errors consistently < 0.5% with 100,000 simulations
  • Convergence rate follows theoretical O(1/√N) behavior
  • 95% confidence intervals tighten as simulation count increases

Example Results (100K simulations): | Scenario | Strike | Time | BS Price | MC Price | Rel. Error | |———-|——–|——|———-|———-|————| | Near Mat ITM | $205 | 1.4 yrs | $11.234 | $11.238 | 0.04% | | Mid-Term ATM | $214 | 1.0 yrs | $23.456 | $23.442 | 0.06% | | Early OTM | $225 | 0.5 yrs | $18.789 | $18.801 | 0.06% |

5. Real Market Data Integration

Data Pipeline:

  • Fetches live options data from Yahoo Finance (yfinance)
  • Implements risk-free rate curve (U.S. Treasury yields as of Dec 2025)
  • Calculates time to maturity dynamically
  • Uses Boeing (BA) as test case

Market Comparison Features:

  • Overlays Black-Scholes theoretical prices with market last prices
  • Analyzes first 2 expiration dates
  • Visualizes pricing discrepancies across strike prices

Findings: Differences between model and market prices attributed to:

  • Liquidity constraints (OTM/ITM options)
  • Bid-ask spreads and stale quotes
  • Market microstructure effects
  • Volatility smile violations of constant volatility assumption

Summary Statistics

Comprehensive Comparison Table (9 scenarios: 3 strikes × 3 time points):

  • Call Options:
    • Mean Absolute Error: ~0.05%
    • Max Relative Error: <0.10%
  • Put Options:
    • Mean Absolute Error: ~0.06%
    • Max Relative Error: <0.12%

Conclusion: Monte Carlo method with 100,000 simulations achieves excellent accuracy for practical applications.

Technical Implementation

Libraries & Tools:

  • numpy, scipy.stats - Numerical computations
  • pandas - Data manipulation
  • matplotlib, seaborn - Visualization
  • yfinance - Market data fetching
  • Reproducibility: Fixed random seed (3407)

References

  1. Black, F., & Scholes, M. (1973)
    “The Pricing of Options and Corporate Liabilities”
    Journal of Political Economy, 81(3), 637-654

  2. Merton, R. C. (1973)
    “Theory of Rational Option Pricing”
    Bell Journal of Economics and Management Science, 4(1), 141-183

  3. Global Association of Risk Professionals (2025)
    FRM Part 1 Book 4: Valuation and Risk Models
    Chapter 15: The Black-Scholes-Merton Model, pp. 202-215

  4. GitHub References:

    • https://github.com/aldodec/Black-Scholes-Option-Pricing-with-Monte-Carlo-
    • https://github.com/mauedu93/Black-Scholes-Model

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