7/15/2026
It’s been about one and a half years (ish??) since I posted anything on this blog. Last time it was quantum error correction. This time I want to use the same tools and concepts to apply them to a different research area – stock options
Insurance companies and trading desks price options millions of times a day. Every time you see a bid/ask on an option chain, something ran a pricing model to get there. This project is about whether a quantum computer could someday do that same job with far fewer calculations than the classical methods we use today.
TWO METHODS OF PRICING AN OPTION
Lets say you want to determine the fair value of a call option, a right to buy a share before some future moment at a fixed price. Of course, there is a closed form solution for a simple case (it is called the Black-Scholes formula). But the most interesting feature of this project is the two methods of pricing an option numerically, since in case of complicated options the latter may not exist.
Monte Carlo simulation. Simulate thousands of possible stock price paths, calculate payoff of the option in each path, average the values obtained and finally, discount back to the present day. It is the classical method and it works well. The drawback is the poor convergence rate – in order to reduce your error twice, you will need four times as many simulations.
Quantum Amplitude Estimation (QAE). Rather than sampling the path one-by-one, we can embed the probability distribution of future prices into a quantum computer via a quantum circuit, using an estimation algorithm which (at least theoretically) uses many fewer evaluations in order to achieve the same accuracy. Specifically, QAE promises a quadratic speed-up where the classical error decays as 1/√N while QAE decays as 1/N (Jong et al., 2023)
This project I will try to test these two methods to face each other – as the latter is quite new and still an emerging concept with the development of more quantum computing tech.
Interesting research
One study from earlier this year discovered the surprising thing that I wasn’t sure about going in: QAE could give an exact payoff of zero for an expected option payoff, even though the classical calculation would make it clear that it was positive. This occurs when the number of qubits being used is not sufficient for properly modeling the low probability ‘in the money’ tail of the distribution of prices (Innan et al., 2026)
The question that I’ll be investigating this summer is therefore not “Does QAE converge faster?” but rather “How many qubits does QAE need before it starts to converge faster?”
Some baseline testing for code
Without yet messing around with quantum code, I ran the classical Monte Carlo simulation for the vanilla case in textbooks ($100 stock, $100 strike, 20% volatility, 5% risk free rate, one year to maturity). The Black-Scholes solution of the same problem comes out to be $10.45.
Now, here’s how the error of the classical solution decreases as I keep increasing the number of simulations paths (each data point is an average of 8 trials):
| Samples | Average error vs. exact price |
|---|---|
| 100 | $1.03 |
| 500 | $0.34 |
| 1,000 | $0.42 |
| 5,000 | $0.15 |
| 10,000 | $0.12 |
| 50,000 | $0.04 |
| 100,000 | $0.04 |
Pretty much linear downward trend most of the way, although the data point for the 1,000 samples was slightly above the data point for 500 samples. Not a bug in the program, but that’s just how sampling noise behaves even when 8 trials are averaged together. It serves as an early lesson: one trial of any of these numbers can lie to you, something that became very important on the quantum side.
Next post
Next steps will be for me to actually get back on VSCODE and qiskit to start testing out some possible code and debugging. I will once again be using AI to help me code debug and understand many processes which I may not, as some of these concepts in quantum computing are very confusing and I’m simply trying to do research with the knowledge I have.
Sources used
Schneppat, J. O. (2025, November 27). Quantum Amplitude Estimation: Precise Quantum Computation for the Future of AI [Video]. YouTube. https://www.youtube.com/watch?v=V__BHOzbhXU
Investopedia. (2025, November 23). Understanding option prices: A guide to valuing derivatives. Investopedia. https://www.investopedia.com/articles/optioninvestor/09/buying-options.asp
Innan, N., Kashif, M., Marchisio, A., Usman, M.M., Shafique, M. (2026). Quantum Amplitude Estimation in Practice: A Case Study in Option Pricing. In: Barbaresco, F., Gerin, F. (eds) Quantum Engineering Sciences and Technologies for Industry and Services. QUEST-IS 2025. Communications in Computer and Information Science, vol 2743. Springer, Cham. https://doi.org/10.1007/978-3-032-13852-1_35
Jong, E., Sævarsson, B., Jóhannsson, H., & Chatzivasileiadis, S. (2023). Quantum Amplitude Estimation for Probabilistic Methods in Power Systems. arXiv preprint arXiv:2309.17299.
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