Markets are complex, ever changing systems where prices move in seemingly chaotic ways – yet beneath the surface, patterns emerge. Traditional models often simplify market dynamics into discrete time steps, but in reality, price action evolves continuously, flowing from one state to another without clear-cut boundaries. What if we could capture this fluidity more accurately?

Seeing Markets Through Geometry and Probability

Algebraic geometry gives us tools to analyse curves and surfaces, helping identify subtle price action patterns that might otherwise go unnoticed. Imagine plotting market movements not just as candlesticks or lines, but as evolving geometric shapes – where support and resistance levels form natural boundaries, and breakouts represent shifts in the underlying structure.

Meanwhile, Continuous time Markov chains (CTMCs) offer a way to model market “regimes” (bullish, bearish, ranging) without forcing them into rigid time intervals. Unlike discrete models, which assume fixed transition points (e.g., end-of-day close), CTMCs acknowledge that markets can switch states at any moment—a more realistic reflection of how traders actually experience price movements.

Why would this matter?

Most trading systems rely on discrete-time models, treating price changes as a sequence of independent snapshots. Markets don’t operate in neat, evenly spaced intervals – they evolve in real-time, with trends accelerating, reversing or fading unpredictably. By combining:

• Algebraic geometry (to map price structures),
• Continuous-time Markov chains (to detect regime shifts dynamically),

we create a framework that respects the continuity of market behaviour rather than forcing it into artificial time buckets.

Potential Applications

• Regime Detection: Recognizing when the market shifts from trending to mean-reverting in real-time.
• Pattern Recognition: Using geometric properties to identify high-probability breakout or reversal zones.
• Adaptive Risk Management: Adjusting position sizing based on the current “state” of the market.

Embracing the Fluid Nature of Markets

The true power of this approach lies in its acknowledgment that markets are not just a series of disconnected price points—they are a continuous, evolving process. By integrating geometric pattern analysis with probabilistic state transitions, we move closer to modelling markets as they truly are: dynamic, interconnected, and ever-changing.

Perhaps the future of quantitative trading isn’t just about faster computations or more data—but about better frameworks that align with the organic nature of price movements.

This was more of a reflective post. What do you think? Can blending geometry and continuous-time models improve trading strategies? Let me know your thoughts!

Links of interest:
https://www.youtube.com/@theprobabilitychannel-prof8089
https://www.youtube.com/@NormalizedNerd
and if you REALLY want to 🙂 Homogeneous Poisson Process