research

Quantitative Finance

My current research centers on rough volatility, especially the rough Heston model. I am interested in numerical methods for the fractional Riccati equation, calibration, and the asymptotic regimes that govern long- and short-maturity option pricing.

The immediate technical questions are numerical and probabilistic, but the surrounding research program is broader:

• Rough volatility: rough Heston, fractional Riccati equations, high-order schemes, convergence, stability, and asymptotics.
• Fourier methods for option pricing: Carr-Madan FFT, Lewis contour integrals, characteristic functions, and the role of transform methods in derivative pricing.
• High-frequency finance: statistical inference for rough volatility using high-frequency option data, with attention to implementation constraints.
• Computational systems: C++ and low-level implementation as part of the practical language of quantitative finance, not just an engineering afterthought.

Time Series

Time series analysis is the second major branch of the research map. I am especially interested in reading time series through an operator-theoretic lens rather than only as a collection of econometric procedures.

The objects I find most compelling here include autocovariance operators, spectral methods, filtering, state-space representations, and the interface between stochastic processes and functional analysis. This perspective also connects naturally back to high-frequency finance, where data structure, sampling, and microstructure noise matter as much as the model.

Mathematical Foundations

The mathematical core of my work is:

• Measure theory and probability: the language for stochastic processes and financial models.
• Functional analysis: Hilbert spaces, Banach spaces, operators, duality, and spectral thinking.
• Fourier and complex analysis: transform methods, contour integrals, characteristic functions, and analytic structure.
• PDEs and integral equations: Green’s functions, kernels, and the bridge between local dynamics and global representation.

Functional analysis is the unifying lens. It gives a common grammar for objects that first appear unrelated: option pricing operators, quantum observables, time-series covariance operators, PDE solution kernels, and integral transforms.

Cross-Domain Themes

Several recurring themes connect the technical and philosophical sides of the project.

Functional analysis as a common language.
Hilbert space methods make it possible to compare mathematical finance, quantum mechanics, time series, and PDEs without reducing one field to another. The analogy is not merely decorative: many questions in these areas can be phrased as questions about operators, expectation values, spectra, kernels, and approximation.

Green’s functions and kernels.
Green’s functions appear in PDE, physics, and pricing theory as a way of turning local equations into global representations. I am interested in how this idea travels across domains, and in when the same formal object carries different conceptual meanings.

Quantum mechanics and quantitative finance.
Both fields can be read, at a certain level of abstraction, as theories of expectation values of operators. I am interested in the technical analogy and also in the limits of the analogy: where it illuminates, where it misleads, and what it says about mathematical modeling.

Philosophy of mathematics.
The technical work raises philosophical questions about why certain abstractions work so well. I am especially interested in Platonism, structuralism or relationism, and more deflationary views of mathematical truth. These are not separate from the mathematics for me; they shape how I think about the choice of language.

Humanistic Orbit

Outside the formal research program, I keep notes on Chinese intellectual history, world history, political philosophy, metaphysics, film, and the beauty of mathematics. I see these as part of the same intellectual project: understanding how abstract structures, historical forces, and forms of representation shape the way we reason.

I plan to open a writing section once a small set of essays is polished enough to stand on its own.

Ongoing Projects

A list of papers I am preparing for submission can be found on the publications page.