Quant Finance Foundations.
The bedrock — valuation, pricing, and the mathematics every model inherits.
4 articles · each one with a runnable notebook
Geometric Brownian Motion
The canonical continuous-time model for asset prices. Why returns compound rather than add, the SDE and its closed-form solution, an exact simulation scheme — then calibrated to seven years of SPY and run out 1,000 paths over five years, with the terminal distribution checked against the theoretical log-normal.
Black–Scholes & the Greeks
Five observable inputs in, a fair price and a full risk report out. We replicate the option, derive the Greeks in closed form, then price and risk-map a 3-month QQQ option from real data — and check it two ways: put-call parity and Monte Carlo.
Bond Pricing, Duration & Convexity: Via US Treasuries — and the ETFs That Lived Through 2022
Price, duration and convexity — the three numbers that summarise a bond's rate risk. We build them in 20 lines, walk the price-yield curve and the duration ladder, then check the theory against SHY/IEF/TLT's real drawdowns in the worst bond year in modern history.
Heston vs Black–Scholes: Fitting the Volatility Smile
Black-Scholes assumes a single, constant volatility — so it predicts the same implied vol at every strike. The market flatly disagrees: implied vol curves into a smile/skew. The Heston model, by letting volatility itself be random, can bend to fit that curve. This piece pits the two against a real SPX-style implied-vol surface and measures who fits, by how much, and at what cost.