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Frank–Wolfe Market Making: Notes

19 October 2026

Tutorial notes on combinatorial market making, from trade costs and coherent prices to Bregman projections and Frank–Wolfe as column generation. The full PDF includes a worked triangle market example, a Python program, and a discussion of profit certificates from time-limited integer solves.

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Some Facts about Bootstrapping

28 January 2025

Understanding the distribution of the sample standard deviation can be done using the delta method. By applying the multivariate central limit theorem and a Taylor series expansion, an asymptotic approximation is derived. While this first-order approach works under most conditions, it fails for edge cases like Bernoulli variables with $p = 0.5$, where zero gradients necessitate a second-order expansion, yielding a negative chi-squared distribution instead of a normal one.

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Probability of Constraint Satisfaction and Robust Optimization

29 January 2025

In portfolio optimization, Mean-Variance Optimization (MVO) relies on estimated inputs for expected returns $\boldsymbol{\mu}$ and covariance $\boldsymbol{Q}$, introducing uncertainty. While portfolios generated using true parameters meet return constraints, those based on estimates may not. Robust MVO addresses this risk by incorporating uncertainty into the model, improving reliability under estimation errors.

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