{"version":"https://jsonfeed.org/version/1.1","title":"Zac Kienzle - Blog","home_page_url":"https://zackienzle.com/blog","feed_url":"https://zackienzle.com/blog/feed.json","description":"Personal website featuring experience, projects, thesis, and research notes.","language":"en-AU","authors":[{"name":"Zac Kienzle","url":"https://zackienzle.com"}],"items":[{"id":"https://zackienzle.com/blog/black-scholes-pde","url":"https://zackienzle.com/blog/black-scholes-pde","title":"The Black-Scholes Equation","summary":"How an option is priced by hedging away its risk. The Black-Scholes partial differential equation derived from Ito's formula and a delta hedge, its risk-neutral solution as a discounted expected payoff, and the closed-form Black-Scholes formula for a European call.","date_published":"2026-06-30T00:00:00.000Z","date_modified":"2026-06-30T00:00:00.000Z","tags":["quantitative-finance","stochastic-calculus","derivatives"]},{"id":"https://zackienzle.com/blog/mean-variance-portfolio","url":"https://zackienzle.com/blog/mean-variance-portfolio","title":"The Mean-Variance Portfolio","summary":"How to trade return against risk optimally. The Markowitz problem of minimising variance at a target return, its closed-form solution by Lagrange multipliers, the efficient frontier it traces, and the tangency portfolio maximising the Sharpe ratio.","date_published":"2026-06-29T00:00:00.000Z","date_modified":"2026-06-29T00:00:00.000Z","tags":["quantitative-finance","optimization","portfolio"]},{"id":"https://zackienzle.com/blog/residues-and-contour-integration","url":"https://zackienzle.com/blog/residues-and-contour-integration","title":"Residues and Contour Integration","summary":"Evaluating integrals by what a function leaves behind at its poles. The Laurent series and the residue, the residue theorem, and a contour computation of the Cauchy characteristic function.","date_published":"2026-06-28T00:00:00.000Z","date_modified":"2026-06-28T00:00:00.000Z","tags":["complex-analysis","integration"]},{"id":"https://zackienzle.com/blog/holomorphic-functions-and-cauchy","url":"https://zackienzle.com/blog/holomorphic-functions-and-cauchy","title":"Holomorphic Functions and Cauchy's Theorem","summary":"Why complex differentiability is so much stronger than real. The Cauchy-Riemann equations, Goursat's vanishing contour integral, the Cauchy integral formula recovering a function from its boundary values, and the analyticity and Liouville theorems that follow.","date_published":"2026-06-27T00:00:00.000Z","date_modified":"2026-06-27T00:00:00.000Z","tags":["complex-analysis"]},{"id":"https://zackienzle.com/blog/positive-definite-matrices","url":"https://zackienzle.com/blog/positive-definite-matrices","title":"Positive Definite Matrices","summary":"The matrices that act as squared lengths. Positive semidefinite matrices characterised by nonnegative eigenvalues, by a Gram factorisation, and by a square root, the Cholesky factorisation of a positive definite matrix, and the covariance matrix as the canonical example.","date_published":"2026-06-26T00:00:00.000Z","date_modified":"2026-06-26T00:00:00.000Z","tags":["linear-algebra","optimization","statistics"]},{"id":"https://zackienzle.com/blog/eigenvalues-and-the-spectral-theorem","url":"https://zackienzle.com/blog/eigenvalues-and-the-spectral-theorem","title":"Eigenvalues and the Spectral Theorem","summary":"How a symmetric matrix is diagonalised. Eigenvalues and eigenvectors, the real spectrum and orthogonal eigenvectors of a symmetric matrix, the spectral theorem diagonalising it by an orthonormal basis, and the variational characterisation behind principal component analysis.","date_published":"2026-06-25T00:00:00.000Z","date_modified":"2026-06-25T00:00:00.000Z","tags":["linear-algebra","spectral-theory"]},{"id":"https://zackienzle.com/blog/convex-optimization-and-kkt","url":"https://zackienzle.com/blog/convex-optimization-and-kkt","title":"Convex Duality and the KKT Conditions","summary":"How a constrained minimum is certified. The Lagrangian and its dual function, weak duality, strong duality under Slater's condition from the supporting hyperplane theorem, and the Karush-Kuhn-Tucker conditions that characterise the optimum of a convex program.","date_published":"2026-06-24T00:00:00.000Z","date_modified":"2026-06-24T00:00:00.000Z","tags":["optimization","convex-analysis","duality"]},{"id":"https://zackienzle.com/blog/convex-sets-and-functions","url":"https://zackienzle.com/blog/convex-sets-and-functions","title":"Convex Sets and Functions","summary":"The geometry that makes minimisation tractable. Convex sets and functions and their first and second-order characterisations, the separating and supporting hyperplane theorems from projection onto a closed convex set, and the fact that a convex function's local minimum is global.","date_published":"2026-06-23T00:00:00.000Z","date_modified":"2026-06-23T00:00:00.000Z","tags":["optimization","convex-analysis","real-analysis"]},{"id":"https://zackienzle.com/blog/uniform-integrability","url":"https://zackienzle.com/blog/uniform-integrability","title":"Uniform Integrability and the Vitali Theorem","summary":"The exact condition for convergence in the mean. Uniform integrability, its characterisation by uniform absolute continuity, and the Vitali theorem that convergence in measure upgrades to L-one convergence exactly when the sequence is uniformly integrable.","date_published":"2026-06-22T00:00:00.000Z","date_modified":"2026-06-22T00:00:00.000Z","tags":["measure-theory","integration","probability"]},{"id":"https://zackienzle.com/blog/lp-spaces","url":"https://zackienzle.com/blog/lp-spaces","title":"The L-p Spaces","summary":"The Banach spaces of integrable powers. Young's inequality, the Holder and Minkowski inequalities that make the p-norm a norm, and the completeness theorem that promotes every L-p to a Banach space, the family of which L-squared is the one Hilbert member.","date_published":"2026-06-21T00:00:00.000Z","date_modified":"2026-06-21T00:00:00.000Z","tags":["measure-theory","functional-analysis","integration"]},{"id":"https://zackienzle.com/blog/stochastic-differential-equations","url":"https://zackienzle.com/blog/stochastic-differential-equations","title":"Stochastic Differential Equations","summary":"Equations driven by noise, and the theorem that they have a unique solution. The strong solution, Gronwall's inequality, uniqueness via the Lipschitz condition and the Ito isometry, and existence by Picard iteration with the factorial decay that makes the iterates converge.","date_published":"2026-06-20T00:00:00.000Z","date_modified":"2026-06-20T00:00:00.000Z","tags":["stochastic-processes","stochastic-calculus","sde"]},{"id":"https://zackienzle.com/blog/itos-formula","url":"https://zackienzle.com/blog/itos-formula","title":"Ito's Formula","summary":"The chain rule of stochastic calculus. The second-order Taylor expansion that the quadratic variation of Brownian motion forces, Ito's formula for a function of a Brownian motion and of an Ito process, the integration-by-parts rule, and the solution of geometric Brownian motion.","date_published":"2026-06-19T00:00:00.000Z","date_modified":"2026-06-19T00:00:00.000Z","tags":["stochastic-processes","stochastic-calculus","brownian-motion"]},{"id":"https://zackienzle.com/blog/quadratic-variation","url":"https://zackienzle.com/blog/quadratic-variation","title":"Quadratic Variation","summary":"Why Brownian motion needs its own calculus. The quadratic variation of a path, the theorem that Brownian motion accumulates quadratic variation equal to elapsed time, the resulting infinite total variation, and the covariation that gives the chain rule its extra term.","date_published":"2026-06-18T00:00:00.000Z","date_modified":"2026-06-18T00:00:00.000Z","tags":["stochastic-processes","brownian-motion","stochastic-calculus"]},{"id":"https://zackienzle.com/blog/series-and-power-series","url":"https://zackienzle.com/blog/series-and-power-series","title":"Series and Power Series","summary":"Infinite sums and the functions they define. Convergence of series, absolute convergence and the root test, the radius of convergence of a power series by Cauchy-Hadamard, term-by-term differentiation, and the exponential series with its defining identity.","date_published":"2026-06-17T00:00:00.000Z","date_modified":"2026-06-17T00:00:00.000Z","tags":["real-analysis","series","power-series"]},{"id":"https://zackienzle.com/blog/riemann-integral","url":"https://zackienzle.com/blog/riemann-integral","title":"The Riemann Integral","summary":"Area as a limit of sums squeezed between over and under estimates. Upper and lower sums, the Riemann criterion for integrability, the integrability of continuous functions through uniform continuity, and the fundamental theorem of calculus tying the integral to the derivative.","date_published":"2026-06-16T00:00:00.000Z","date_modified":"2026-06-16T00:00:00.000Z","tags":["real-analysis","integration","calculus"]},{"id":"https://zackienzle.com/blog/differentiation","url":"https://zackienzle.com/blog/differentiation","title":"Differentiation and Taylor's Theorem","summary":"The derivative and what the mean value theorem extracts from it. The derivative as a limit, differentiability implying continuity, Fermat's interior-extremum principle, Rolle's theorem and the mean value theorem, and Taylor's theorem with the Lagrange remainder.","date_published":"2026-06-15T00:00:00.000Z","date_modified":"2026-06-15T00:00:00.000Z","tags":["real-analysis","calculus"]},{"id":"https://zackienzle.com/blog/the-karhunen-loeve-expansion","url":"https://zackienzle.com/blog/the-karhunen-loeve-expansion","title":"The Karhunen-Loeve Expansion","summary":"The Karhunen-Loeve expansion writes a stochastic process in the eigenbasis of its covariance operator, coordinates that are uncorrelated and mean-square optimal. We prove the expansion and its optimality, then derive it for Brownian motion and the Brownian bridge.","date_published":"2026-06-14T00:00:00.000Z","date_modified":"2026-06-14T00:00:00.000Z","tags":["stochastic-processes","functional-analysis","probability","dimensionality-reduction"]},{"id":"https://zackienzle.com/blog/price-formation-in-the-order-book","url":"https://zackienzle.com/blog/price-formation-in-the-order-book","title":"Price Formation in the Order Book","summary":"The models built on the order book mechanism. The efficient price is a martingale, the bounce and adverse selection set spreads, queue position sets fills, imbalance sets the microprice, Kyle's lambda turns information into impact, and Almgren-Chriss solves execution.","date_published":"2026-06-13T00:00:00.000Z","date_modified":"2026-06-13T00:00:00.000Z","tags":["market-microstructure","quantitative-finance","probability","stochastic-finance"]},{"id":"https://zackienzle.com/blog/the-limit-order-book","url":"https://zackienzle.com/blog/the-limit-order-book","title":"The Limit Order Book","summary":"How a limit order book works as a mechanism. Bids and asks, the price grid, every order type and time-in-force, price-time and pro-rata priority, matching, crossed and locked books, the opening and closing auction, hidden liquidity, and the message stream, with worked examples.","date_published":"2026-06-12T00:00:00.000Z","date_modified":"2026-06-12T00:00:00.000Z","tags":["market-microstructure","limit-order-book","trading"]},{"id":"https://zackienzle.com/blog/ornstein-uhlenbeck","url":"https://zackienzle.com/blog/ornstein-uhlenbeck","title":"The Ornstein-Uhlenbeck Process","summary":"The canonical mean-reverting diffusion. We solve the Ornstein-Uhlenbeck SDE in closed form, derive its Gaussian transition and stationary laws, read off the half-life, and reduce it to an exact AR(1) recursion for estimation.","date_published":"2026-06-11T00:00:00.000Z","date_modified":"2026-06-11T00:00:00.000Z","tags":["stochastic-processes","mean-reversion","quantitative-finance"]}]}