QUANTUM FINANCE: Path Integrals and Hamiltonians for Option Pricing

The thesis starts by explaining a practical problem: classic option models assume constant volatility and can’t explain the "smile" seen in real markets. The goal is to use tools from physics to price options when volatility moves over time, and to check that the method works on real data . First, the text reviews the basics of option pricing: no-arbitrage ideas, how Black–Scholes is derived, and why real markets need a model where volatility itself is random (the Merton–Garman equation).

Next, the thesis rewrites these pricing equations in a physics style. Instead of only solving a differential equation, it thinks of prices as coming from a "kernel" or "propagator" that tells how today’s price depends on all possible future paths. This view helps us see clearly how parameters like correlation, mean reversion, and volatility-of-volatility shape option prices. Then come the numerical methods.

Two approaches are built and compared. One simulates the joint movement of price and volatility step by step (Euler/Milstein). The other uses a path-integral trick to integrate out the price path and only simulate the volatility path, which saves memory while staying accurate. Prices are turned into implied volatilities so results can be compared fairly. With market data (SPY options), the thesis sets up a calibration: pick model parameters so the model’s implied volatilities match the market’s.


To speed this up, a neural network is trained to act as a fast "surrogate" for the heavy calculations. It predicts well and makes calibration much faster. Finally, the same framework is extended to path-dependent products. By adding simple "potentials," it prices barrier and Asian options and checks the results against Monte Carlo. The work ends with a summary of accuracy, speed, and what each parameter does, plus ideas for future improvements.