Derivatives Pricing · LSMC · Binomial Tree · Crank-Nicholson · Greeks · Delta Hedging · Python
FE-620 Final Project — MS Financial Engineering, Stevens Institute of Technology
Author: Swara Dave
This project implements and compares three numerical methods for pricing American options on SPY (S&P 500 ETF) across 1-month and 3-month maturities, using real market data from Bloomberg Terminal and Yahoo Finance.
Three pricing methods implemented:
- Least Squares Monte Carlo (LSMC) — Longstaff-Schwartz simulation-based approach
- Binomial Tree — Cox-Ross-Rubinstein discrete lattice model
- Crank-Nicholson Finite Difference — PDE-based method with second-order accuracy
| Strike | Actual | LSMC | Binomial Tree | Crank-Nicholson |
|---|---|---|---|---|
| 590 | 10.86 | 10.57 | 10.84 | 10.58 |
| 600 | 5.41 | 6.14 | 6.31 | 6.25 |
| 610 | 2.21 | 3.21 | 3.34 | 3.40 |
| Strike | Actual | LSMC | Binomial Tree | Crank-Nicholson |
|---|---|---|---|---|
| 585 | 33.97 | 31.92 | 33.20 | 33.19 |
| 600 | 22.58 | 22.63 | 23.58 | 23.52 |
| 615 | 13.22 | 15.19 | 15.86 | 15.75 |
Binomial Tree and Crank-Nicholson most closely match market prices. LSMC introduces bias from regression approximation but handles high-dimensional problems better.
- SPY prices — Yahoo Finance (3-month historical closing prices)
- Options data — Bloomberg Terminal (bid/ask for strikes 570–640)
- Risk-free rate — 13-week T-bill rate (IRX): 4.23%
- Historical volatility — Estimated at ~13–14% across 1–6 month lookback windows
LSMC (Longstaff-Schwartz Monte Carlo)
- Simulates GBM price paths; uses least-squares regression to estimate continuation value at each exercise date
- Determines early exercise optimality by comparing continuation value to immediate payoff
Binomial Tree (Cox-Ross-Rubinstein)
- Discrete recombining lattice; option value computed backward from expiration
- Early exercise handled explicitly at each node
Crank-Nicholson Finite Difference
- Discretizes Black-Scholes PDE using averaged explicit/implicit scheme
- Second-order accuracy; stable and fast convergence for single-asset options
- Delta — finite difference approximation: (V(S+h) - V(S-h)) / 2h
- Gamma — second derivative: (V(S+h) - 2V(S) + V(S-h)) / h²
- Vega — sensitivity to volatility
- Theta — sensitivity to time decay
Dynamic delta-neutral hedging implemented for SPY call option (Dec 6–13, 2024):
- Initial delta: 0.7152 → short 71.52 SPY shares
- Daily rebalancing based on spot price changes
- Total hedging cost tracked across 5 trading days
AmericanOptions-SPY/
├── 1_LSMC_MonteCarlo.ipynb # QMC/LSMC pricing implementation
├── 2_Volatility_DeltaHedging.ipynb # Volatility estimation & delta hedging
└── 3_BinomialTree_CrankNicholson_Greeks.ipynb # Binomial Tree, CN method & Greeks
- Clone the repo
- Install dependencies:
pip install numpy scipy pandas matplotlib yfinance- Run notebooks in order:
1_LSMC_MonteCarlo.ipynb— Monte Carlo pricing2_Volatility_DeltaHedging.ipynb— volatility & hedging3_BinomialTree_CrankNicholson_Greeks.ipynb— Binomial Tree, Crank-Nicholson & Greeks
Note: SPY historical prices are fetched automatically via
yfinance. Options market data (bid/ask for strikes 570–640) was sourced from Bloomberg Terminal as of Nov 20, 2024 and Dec 13, 2024. If you don't have Bloomberg access, substitute with options data from Yahoo Finance (yfinanceoptions chain) or CBOE for similar strike ranges.
Swara Dave — MS Financial Engineering, Stevens Institute of Technology
