I am not looking for your winning strategies. Just the basic stuff from where to start. Can anyone share their opinions about what should I read to hit the ground running?
Many of the strategies are motivated by objective functions (contour integrals) in the complex plane and the elements of complex linear spaces, so I'd recommend at least for an applied understanding:
- Saff, E. B., and Snider, A. D. Fundamentals of Complex Analysis with Applications to Engineering, Science and Mathematics.
In addition to Saff and Snider, I recommend you take a look at Stony Brook's AMS 523, Mathematics of High Frequency Finance course gives a good outline which includes Banach spaces, Cauchy theory, residue calculus, the Cauchy-Riemann equations, all of which are extremely important in the groundwork for a HFT strategy in the equities space. Note however that these classes of strategies are probably most perfected at SocGen/Latour (Tower Research) though, who have refined them thoroughly over the years.
Also central to HFT is the construction of abstract measures, Hilbert space and positive functionals, and the Lebesgue-Radon-Nikodym theorem is an important milestone. I like Terry Tao's excursion into measure theory, followed by Rudin's attack on the LRN theorem. Rudin's approach is particularly elegant if you pick up from the Riesz representation theorem from Riesz and Nagy's exposition directly:
- Tao, T. An Introduction to Measure Theory.
- Riesz, F.,and Sz.-Nagy, B. Functional Analysis.
- Rudin, W. Real and Complex Analysis.
There's important analogs between statistics and algebraic topology, and it should also come as no surprise that the team that wrote Reflex at GETCO (before they merged with KCG) included several PhDs on algebraic topology, and several of them later left to expand that work at Citadel Securities:
- May, J. P. Differential Forms in Algebraic Topology
- Bott, R. A Concise Course in Algebraic Topology
Complex manifold theory, schemes, algebraic affine and projective curves and varieties, and in general the solutions of discrete-time polynomial equations are a common issue if you're modeling high-frequency tick data. Renaissance Technologies and Two Sigma have locked in several of the leading researchers in this space at Harvard's physics department plus several of Zariski/Mumford's former students and have been refining this space of strategies for several years already. If that doesn't deter you from trying, these are the classical references in algebraic geometry:
- Hartshorne, R. Algebraic Geometry.
- Griffiths, P., and Harris, J. Principles of Algebraic Geometry
Gathmann's course at the University of Kaiserlautern had very well-written course notes that I found particularly useful when I had to learn algebraic geometry from scratch in my first 3-6 months at a HFT shop.
Lastly, I cannot emphasize enough the importance of renormalization, conformal field theory and gauge theory. This is the primary reason why the HFT teams (DRW, SIG equities etc.) are staffed mostly by physics PhDs rather than applied math PhDs, because it's a natural development before coursework in quantum field theory. Deligne's book was my favorite:
- Deligne, P. Quantum Fields and Strings
Of particular importance is the range of topological field theories that relate to Chern-Simons theory (half-named after Jim Simons from Rentec) and the Dijkgraaf-Witten theory. Sati and Schreiber also have a wonderful collection of the latest developments and literature in this area.