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:20260728T194034Z UID:0252d70d-9d49-43f4-86cc-c67b8394bde8 SEQUENCE:1 CREATED:20251121T091009Z DESCRIPTION: The thesis starts by explaining a practical problem: classic option models assume constant volatility and can’t explain the "\;sm ile"\; 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 meth od works on real data . First\, the text reviews the basics of option pric ing: no-arbitrage ideas\, how Black–Scholes is derived\, and why real ma rkets 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 pri ces 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\, a nd volatility-of-volatility shape option prices. Then come the numerical m ethods. Two approaches are built and compared. One simulates the joint mov ement of price and volatility step by step (Euler/Milstein). The other use s a path-integral trick to integrate out the price path and only simulate the volatility path\, which saves memory while staying accurate. Prices ar e 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 "\;surrog ate"\; for the heavy calculations. It predicts well and makes calibrat ion much faster. Finally\, the same framework is extended to path-dependen t products. By adding simple "\;potentials\,"\; it prices barrier and Asian options and checks the results against Monte Carlo. The work end s with a summary of accuracy\, speed\, and what each parameter does\, plus ideas for future improvements. LAST-MODIFIED:20251121T091009Z LOCATION:Sala V1.34 Edifício de Civil URL:http://df.vps.tecnico.ulisboa.pt/en/events/quantum-finance-path-integr als-and-hamiltonians-for-option-pricing/ X-ALT-DESC;FMTTYPE=text/html:

The thesis starts by explaining a practical problem: classic option models assume constant v olatility and can’t explain the "\;smile"\; seen in real markets . The goal is to use tools from physics to price options when volatility m oves 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 B lack–Scholes is derived\, and why real markets need a model where volati lity itself is random (the Merton–Garman equation).

Next\, the thesis rewrites these pricing equations in a physics style. Instead of on ly 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-vola tility shape option prices. Then come the numerical methods.

Two approaches are built and compared. One simulates the joint movement of pr ice and volatility step by step (Euler/Milstein). The other uses a path-in tegral trick to integrate out the price path and only simulate the volatil ity path\, which saves memory while staying accurate. Prices are turned in to implied volatilities so results can be compared fairly. With market dat a (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 "\;su rrogate"\; for the heavy calculations. It predicts well and makes cali bration much faster. Finally\, the same framework is extended to path-depe ndent products. By adding simple "\;potentials\,"\; it prices barr ier 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.

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