Learning project exploring derivatives pricing models and machine learning in quantitative finance. Built from an astrophysics background with no prior finance training.
Three classical option pricing models (Black-Scholes, Monte Carlo, Binomial Tree) implemented from scratch, plus a neural network to predict implied volatility (IV). All tested on real SPY (S&P 500 ETF) options data from Yahoo Finance.
Each method approaches the same problem differently - analytical solution, stochastic simulation, and discrete approximation. Wanted to see how different mathematical frameworks converge to the same answer.
├── src/
│ ├── models/ # Three pricing implementations
│ ├── data/ # Download and clean market data
│ ├── ml/ # Neural network for IV prediction
│ └── analysis/ # Model comparisons
├── data/ # Raw SPY options chains
└── results/ # Plots and trained models
Coming from astrophysics, the equations looked familiar:
Black-Scholes partial differential equation (PDE) is the heat equation with drift. Stock price S evolves as:
∂V/∂t + ½σ²S²∂²V/∂S² + rS∂V/∂S - rV = 0
Change variables and this becomes the standard heat equation.
Monte Carlo simulation uses the same geometric Brownian motion I've seen in radiative transfer codes:
dS = μS dt + σS dW
Generate random price paths, calculate payoffs at expiration, average them. Exactly like Monte Carlo photon transport in radiative transfer calculations.
Binomial tree builds a lattice of possible future prices and works backwards from expiration. Similar to evolving hydrodynamic states in FLASH simulations, except in reverse time.
Downloaded 6,829 SPY option contracts, ended with 1,877 usable options after cleaning.
Removed:
- 313 expired contracts (time to expiration ≤ 0)
- 250 with bid-ask spreads >10% of price
- 1,652 with volume <10 contracts
- Options outside 0.85-1.15 moneyness range
- Implied volatility outliers (>200% or <1%)
Total filtered: 4,952 contracts (72.5%)
Can't compute reliable IV from illiquid options. Wide bid-ask spread means the market itself is uncertain about the price. Like trying to fit a spectrum with SNR < 2 - the data just isn't reliable enough.
Standard direction: volatility σ → option price V
Implied volatility: option price V → volatility σ
Used Newton-Raphson with vega (∂V/∂σ) as the derivative. Converges in 5-10 iterations for most options because vega is always positive - option value increases monotonically with volatility.
Key finding: IV isn't constant across strike prices. Options away from at-the-money (ATM) show higher IV than near-the-money options. This "volatility smile" means the Black-Scholes constant-volatility assumption is wrong.
Why the smile exists: The model assumes log-normal price distribution (Gaussian in log-space). Real markets have fatter tails - extreme moves happen more often than Gaussian predicts. Out-of-the-money (OTM) options price in this tail risk with higher IV.
All three methods solve the same problem under identical assumptions:
Black-Scholes: Analytical solution to the PDE. Instant computation.
Monte Carlo: Simulated 500,000 random price paths (parallelized with joblib). Averaged payoffs at expiration. Achieved ~900k simulations/second.
Binomial Tree: Discrete lattice with backward induction. Converges to Black-Scholes as number of steps increases.
Validated on textbook examples - all three agree within $0.01 for identical inputs.
Put-call parity verification: For European options, the relationship C - P = S - Ke^(-rT) must hold by arbitrage. Tested on all liquid options, difference <$0.000001. This proves the implementation is correct.
Built feedforward network in PyTorch:
- Architecture: 3 inputs → 64 neurons → 32 → 16 → 1 output
- Inputs: moneyness (S/K), time to expiration, option type (call/put)
- Target: implied volatility
- Data split: 70% training, 15% validation, 15% test
Started with standard decreasing layer pattern (64→32→16) commonly used for regression tasks. Kept it simple since only 3 input features - deeper networks would likely overfit on this small feature set.
Performance: 1.17% mean absolute error (MAE) on test set.
My Newton-Raphson solver achieved 2.53% MAE. The network does better because it's learning from Yahoo Finance's IV calculations, which likely use more sophisticated methods than basic Newton-Raphson. The network approximates whatever model Yahoo Finance uses internally.
Real insight: IV has predictable structure based on just moneyness, time, and option type. Moneyness is likely the dominant feature - it directly drives the volatility smile pattern. Time to expiration matters for term structure (near-term vs long-term IV differences). Option type (call/put) has minimal impact since put-call parity means they share the same IV for identical strikes.
Single underlying: Only tested on SPY. Model probably won't generalize to individual stocks with different volatility characteristics.
Static snapshot: One day's data. Haven't tested how IV surface evolves over time or across different market regimes (calm vs crisis periods).
Basic features: Only used moneyness, time, and option type. Could add historical realized volatility, VIX (market volatility index), or term structure patterns.
European assumption: Real SPY options are American (can exercise early), but priced them as European. Doesn't matter much for SPY since no dividend, but technically incorrect.
No trading validation: Didn't backtest whether model predictions would generate profit. That requires transaction costs, slippage, bid-ask impact - whole other complexity level.
Validation:
- Tested against Hull's derivatives textbook examples
- Verified put-call parity (difference <$0.000001)
- Confirmed deep in-the-money options ≈ intrinsic value
- Checked binomial tree → Black-Scholes convergence
Parallelization: Monte Carlo distributed price path simulations across CPU cores using joblib, similar to parallelizing hydrodynamic calculations.
Neural network:
- PyTorch implementation
- ReLU activation, dropout regularization
- StandardScaler preprocessing (zero mean, unit variance)
- MSE loss, Adam optimizer
Model agreement:
- All three pricing methods converge within $0.01
- Put-call parity verified on all liquid options
- Newton-Raphson IV solver: 2.53% MAE vs market IV
- Neural network IV prediction: 1.17% MAE
Final dataset:
- 1,877 liquid options from 6,829 raw contracts
- Moneyness range: 0.85 - 1.15
- Expiration range: 2 to 839 days
- Spot price: $631.92 (October 3, 2024)
Volatility surface patterns:
- Clear smile: OTM options show higher IV than ATM
- Term structure: short-dated ~14% IV, long-dated ~18% IV
- Longer time = more uncertainty = higher option value
Market Price vs Black-Scholes:
Black-Scholes prices using market IV match actual market prices within $0.50 MAE. The model works well as a pricing framework when given correct volatility input.
Volatility Smile:
Clear smile pattern visible - options away from at-the-money show higher implied volatility. Calls and puts follow similar patterns. This confirms the Black-Scholes constant volatility assumption breaks down in real markets.
IV Prediction Error Distribution:
Newton-Raphson IV errors centered near zero with 2.53% MAE. Most errors within ±5%, showing the numerical method works reliably for liquid options.
Multi-asset training: Test on different stocks and ETFs to check generalization.
Time series analysis: Collect daily data over months to study IV dynamics and regime changes.
Better baseline: Compare to industry models (SABR, SVI) instead of basic numerical solver.
Feature engineering: Add Greeks, historical volatility, market regime indicators.
Trading validation: Backtest with realistic transaction costs to see if predictions add value. Real application would require: (1) identifying when model IV differs from market IV, (2) executing spreads to capture mispricing, (3) accounting for transaction costs and bid-ask slippage. This model isn't ready for that - it learned from Yahoo Finance's own calculations, so it won't find true mispricings. Would need independent IV calculation or comparison to alternative models (SABR, local vol) to generate actual trading signals.
# Install dependencies
pip install numpy pandas scipy matplotlib yfinance scikit-learn torch joblib
# Pipeline
python src/data/fetch_market_data.py # Download SPY options
python src/data/clean_data.py # Filter and clean
python src/data/compute_implied_vol.py # Newton-Raphson IV calculation
python src/ml/vol_predictor.py # Train neural network
python src/analysis/compare_methods.py # Generate comparison plots- Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities
- Hull, J. (2018). Options, Futures, and Other Derivatives
- Cox, Ross, & Rubinstein (1979). Option Pricing: A Simplified Approach
Note: Learning project by someone with zero finance background. Implements standard textbook methods but hasn't been validated for real trading.