BEGIN:VCALENDAR VERSION:2.0 PRODID:-//linuxsoftware.nz//NONSGML Joyous v1.4//EN BEGIN:VEVENT SUMMARY:QUANTUM FINANCE: Path Integrals and Hamiltonians for Option Pricin g DTSTART:20251124T100000Z DTEND:20251124T120000Z DTSTAMP:20260706T214451Z UID:0252d70d-9d49-43f4-86cc-c67b8394bde8 SEQUENCE:2 CREATED:20251121T091017Z DESCRIPTION:The thesis starts by explaining a practical problem: classic o ption models assume constant volatility and can’t explain the "\;smi le"\; seen in real markets. The goal is to use tools from physics to p rice options when volatility moves over time\, and to check that the metho d works on real data . First\, the text reviews the basics of option prici ng: no-arbitrage ideas\, how Black–Scholes is derived\, and why real mar kets need a model where volatility itself is random (the Merton–Garman e quation). Next\, the thesis rewrites these pricing equations in a physics style. Instead of only solving a differential equation\, it thinks of pric es 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\, an d volatility-of-volatility shape option prices. Then come the numerical me thods. Two approaches are built and compared. One simulates the joint move ment 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 t he 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 p arameters so the model’s implied volatilities match the market’s. To s peed this up\, a neural network is trained to act as a fast "\;surroga te"\; for the heavy calculations. It predicts well and makes calibrati on much faster. Finally\, the same framework is extended to path-dependent products. By adding simple "\;potentials\,"\; it prices barrier a nd 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. LAST-MODIFIED:20251121T091026Z LOCATION:Sala V1.34 Edifício de Civil URL:http://df.vps.tecnico.ulisboa.pt/pt/eventos/quantum-finance-path-integ rals-and-hamiltonians-for-option-pricing/ X-ALT-DESC;FMTTYPE=text/html:

The thesis starts b y explaining a practical problem: classic option models assume constant vo latility and can’t explain the "\;smile"\; seen in real markets. The goal is to use tools from physics to price options when volatility mo ves 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 Bl ack–Scholes is derived\, and why real markets need a model where volatil ity itself is random (the Merton–Garman equation).

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

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


To speed this up\, a neural network is trained to act as a fast "\;sur rogate"\; for the heavy calculations. It predicts well and makes calib ration much faster. Finally\, the same framework is extended to path-depen dent products. By adding simple "\;potentials\,"\; it prices barri er and Asian options and checks the results against Monte Carlo. The work ends with a summary of accuracy\, speed\, and what each parameter does\, p lus ideas for future improvements.

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