Part I: Computational Finance
The nine chapters that constitute Part I of Projects in Scientific Computing survey the literature of computational finance from a perspective that is, by design, unusual. The conventional approach to financial engineering begins with the Black–Scholes equation and its extensions, proceeds through a catalog of models and calibration techniques, and arrives at implementation as a largely separate concern. The approach taken here inverts that order. It begins with the mathematical and computational infrastructure—probability theory, scientific computing, stochastic calculus—and builds outward toward financial applications, treating the financial problems as instances of a broader class of computational problems in applied mathematics and theoretical physics. The result is a collection of bibliographic essays that traces the intellectual ancestry of modern quantitative finance back to its roots in measure theory, information theory, statistical mechanics, and differential geometry.
The organization of Part I reflects this philosophy. The first five chapters develop the mathematical and computational substrate. Chapter 2 (Probability, Information, and Risk) establishes the foundational arc: from Kolmogorov’s measure-theoretic axioms and Jaynes’s reinterpretation of probability as logic, through Shannon’s information theory and its extensions into algorithmic complexity and information geometry, to the Kelly criterion and the mathematics of growth-optimal sequential decision-making. The thread that binds these three domains is entropy—Shannon entropy $H(X) = -\sum p(x_i) \log p(x_i)$—which appears first as a measure of uncertainty, then as the fundamental quantity in coding theory and channel capacity, and finally as the objective function for optimal bet-sizing. The Kelly–Shannon correspondence, as the chapter makes clear, is not an analogy but a precise mathematical identity: Kelly’s criterion maximizes the rate of information transmission through a gambling channel.
Chapter 3 (Scientific Computing) provides the engineering and algorithmic infrastructure that supports everything that follows. The treatment spans linear algebra (from Roman’s abstract algebraic perspective to Strang’s computational approach), numerical analysis (classical foundations through the German school and modern perspectives), Monte Carlo methods, and the systems-level concerns—programming languages, computer architecture, operating systems—that are typically assumed rather than taught. The emphasis on hardware-aware computing and the interplay between algorithmic design and architectural constraints distinguishes these notes from purely mathematical treatments of numerical methods.
Chapter 4 (Stochastic Calculus) is the mathematical backbone of derivative pricing, and the treatment here is substantially broader than what appears in standard financial engineering curricula. The chapter develops five successive generalizations of Brownian motion: flat-space stochastic differential equations (with careful attention to the equivalence between Langevin, Itô/Stratonovich, and Fokker–Planck descriptions), diffusion on curved Riemannian manifolds, Nelson’s stochastic mechanics (a program that derives quantum dynamics from an underlying diffusion process), relativistic extensions, and stochastic partial differential equations treated by field-theoretic methods. Two themes unify the presentation. The first is that discretization is a physical modeling choice, not a mathematical convention—the Itô–Stratonovich dilemma for multiplicative noise being the simplest instance of a problem that recurs in progressively more sophisticated settings. The second is the interplay between geometry and probability: connections, curvature, and the frame bundle enter naturally when diffusion is generalized to manifolds, and the same geometric apparatus reappears in the gauge-theoretic formulation of arbitrage developed in Chapter 8.
Chapter 5 (Fractional Calculus) extends the mathematical framework in a different direction. Where Chapter 4 generalizes the spatial setting of diffusion (from flat space to manifolds to spacetime), Chapter 5 generalizes the temporal and spatial regularity of the diffusion process itself. The point of departure is the empirical observation that many physical and financial systems do not behave like Brownian motion: asset returns exhibit heavy tails and occasional extreme moves, particles in disordered media undergo anomalous diffusion with mean-squared displacement scaling as $\langle x^2 \rangle \sim t^{2/\alpha}$ for $\alpha \neq 2$. The natural mathematical framework for these phenomena is the theory of Lévy flights, alpha-stable distributions, and fractional operators—principally the fractional Laplacian $(-\Delta)^{\alpha/2}$, which is the infinitesimal generator of a symmetric $\alpha$-stable Lévy process. The chapter traces a conceptual pipeline from stochastic processes through the mathematical apparatus of fractional operators into applications in classical mechanics, gravity, and quantum theory. The connection to finance is direct: jump-diffusion and Lévy models for asset prices are the natural generalization of geometric Brownian motion when the Gaussian assumption is relaxed.
Chapter 6 (Machine Learning) occupies a pivotal position in the collection. Written from within the tradition that views machine learning through the lens of statistical physics—learning as inference, architectures as priors, training as dynamics—the chapter is anchored by five canonical textbooks (Bishop 2006 and 2024, Goodfellow 2016, Murphy 2022 and 2023) and supplemented by sixty-four research papers that constitute the volatile frontier. The theoretical centerpiece is the neural network–quantum field theory correspondence: the observation that wide neural networks admit a field-theoretic description, with the partition function, free energy, and renormalization group flow of statistical mechanics providing the natural language for understanding generalization and feature learning. The applications are drawn from physics—gravitational wave detection, surrogate modeling of the Einstein field equations, neural PDE solvers—but the methods transfer directly to the financial domain, where the same problems of inference in high-dimensional probability distributions arise in portfolio optimization, risk estimation, and derivative pricing.
The next three chapters develop the financial applications proper. Chapter 7 (Quantitative Finance) surveys the literature from the perspective of statistical physics, complexity science, and applied mathematics. The emphasis is on empirical stylized facts—fat-tailed return distributions, volatility clustering, long-memory in order flow, power-law scaling—that emerge from statistical analysis and that constrain the space of admissible theoretical models. The chapter traces the intellectual migration of methods from physics to finance, beginning with the chaos-theoretic time series analysis of the late 1980s and proceeding through econophysics, market microstructure, asset pricing, portfolio theory, risk management, and complexity economics. The orthodox neoclassical treatment is included alongside the physics-informed alternatives, and the tension between these paradigms—equilibrium versus empirical measurement, Gaussian versus heavy-tailed, representative agent versus heterogeneous interacting agents—is treated as a productive intellectual feature of the field rather than a problem to be resolved.
Chapter 8 (Option Trading) narrows the focus to derivatives, organized along the natural pipeline of quantitative finance practice: mathematical foundations of arbitrage pricing, volatility surface modeling and calibration, interest rate term structure, numerical implementation, and trading practice. What distinguishes the treatment is a substantial section on geometric and field-theoretic approaches to derivative pricing. The heat-kernel methods used for short-maturity implied-volatility expansions are the same heat-kernel methods used in spectral geometry and quantum field theory on curved spacetimes. The gauge-theoretic reformulation of arbitrage employs the fiber-bundle mathematics of the Standard Model. The path-integral formalism for derivative pricing is, literally, the Feynman path integral transported from physics to finance. These are not decorative analogies. They reflect the universality of the mathematical structures of diffusion, symmetry, and fluctuation, and they provide both alternative computational tools and conceptual perspectives that clarify what standard models assume and where they break down.
Chapter 9 (Algorithmic Trading) serves as the capstone, weaving together nearly every thread developed in the preceding chapters. The treatment follows the pipeline from raw market data to deployed strategy: market microstructure foundations (the limit order book as a physical system), research hygiene and evaluation discipline (the problem of overfitting, nonstationarity, and capacity constraints), signal generation and forecasting, optimal execution and market making, portfolio construction and position sizing (connecting back to the Kelly criterion of Chapter 2), derivatives and convexity overlays (connecting to Chapter 8), infrastructure and operations, and an optional track on reinforcement learning. The chapter is deliberately bibliographic rather than prescriptive: it surveys the relevant literature, distinguishes established results from active research and speculation, and identifies connections to material developed elsewhere. The emphasis throughout is on mathematical structure rather than institutional detail, on evaluation discipline rather than performance claims, and on the recognition that the gap between an elegant model and a functioning trading system is wide—much of the difficulty lying in evaluation, not computation.
Chapter 10 (Quantum Computing) may appear, at first glance, tangential to computational finance. It is not. The chapter traces the field from Feynman’s 1982 observation that classical simulation of quantum systems encounters an exponential resource barrier, through the mathematical foundations of quantum information theory and computational complexity, to three application domains: quantum simulation of physical systems, quantum machine learning, and quantum finance. The last of these—quantum algorithms for portfolio optimization, derivative pricing, and annealing-based trading—connects directly to the financial applications developed in later chapters, while the first two provide the theoretical context that determines which quantum computational advantages are genuine and which are artifacts of incomplete classical comparison. The chapter’s treatment of the “dequantization caveat” is particularly relevant: several claimed quantum speedups for linear algebraic problems have been matched by improved classical algorithms, and this dynamic constrains expectations for quantum finance.
A few structural observations about Part I as a whole are worth making explicit. First, the source count is substantial: the nine chapters together survey roughly 400 primary sources spanning more than a century of literature, from Kolmogorov’s 1933 axioms to 2024 ArXiv preprints. The bibliography is selective and personal—it does not aim to be exhaustive—but it is coherent in its arc. Second, the cross-referencing between chapters is intentional and substantive. The MSRJD (Martin–Siggia–Rose–Janssen–de Dominicis) formalism appears in the stochastic calculus chapter, the fractional calculus chapter, and implicitly in the machine learning chapter’s treatment of Langevin dynamics and path integrals. The Kelly criterion connects probability and information theory (Chapter 2) to portfolio construction (Chapter 9) through the same entropy-maximization principle. The heat-kernel and gauge-theoretic methods of Chapter 8 draw directly on the differential geometry developed in Chapter 4. These are not thematic echoes; they are instances of the same mathematics appearing in different applied contexts.
Third, the collection exhibits a consistent methodological stance that might be called physics-informed computational finance. This means several things in practice: a preference for mathematical foundations over application-driven heuristics; attention to the physical content of discretization conventions; a willingness to deploy the full apparatus of modern mathematical physics (fiber bundles, path integrals, renormalization group, field theory) when it illuminates financial problems; and a persistent skepticism about models that assume away the empirical features—heavy tails, long-range dependence, non-stationarity—that the data actually exhibit. This stance does not dismiss conventional approaches but situates them within a broader mathematical landscape, making visible both their strengths and their structural limitations.
The notes are intended for readers with graduate-level training in physics, mathematics, or a cognate quantitative discipline who wish to understand computational finance not merely as a professional toolkit but as a subject with deep structural connections to the rest of mathematical science. Each chapter is designed to be readable independently, but the cumulative effect of reading them in sequence sketches a whole greater than the sum of its parts: a map of a literature that, at its best, reveals quantitative finance and mathematical physics to be two dialects of the same mathematical language.