I have a friend that traded options for 20 years and he trades now in a purely discretionary capacity.  He always says things like "Roman, AI is a crazy bubble, I've seen this before",  "Just wait until this AI bubble pops", "I wonder if the US will have a debt crisis before this AI bubble pops".  He, along with all of these folks on LinkedIn posting about an impending doomsday, always give me a chuckle. This letter is dedicated to him (don't worry he knows, hey Jay!).

There is a quantitative problem with shouting doomsday: you never have to answer the phone.  These folks aren't getting squeezed out of their shorts, and they're usually still triple long even with these strong opinions.  I dislike the act of rocking the boat for the sake of it, or to try and look like Michael Burry to those that haven't seen the distribution of WWII fighter plane damage.  Reversion in this context will always be an artifact of a negative catalyst which is never known a priori.

It's much like Cramer's Mad Money, he never actually has to answer for his picks that are wrong, but he can always cite his picks that were correct.  No shade to Cramer at all, I love what he does, but I would enjoy more transparency for the casual viewer that doesn't understand qualitative survivorship bias.  

That being said, this is your weekly reminder to master your quantitative skills so you can make your own informed decisions in the face of uncertainty to not be fooled by the casual (and typically baseless) qualitative statement!

With that I will leave you to the Weekly Guild Letter.  I hope you enjoy, and I hope you learn something!

- Roman

Some things in markets don’t evolve smoothly at all: they arrive. A trade hits the tape. A news headline drops. A quote updates. A block order sweeps the book. A jump fires off in a price series. All of these are “events,” and the markets are basically a never-ending stream of them.

Once you start looking at financial data through that lens, you realize how much of what we care about is really an arrival process. Trades arrive. Orders arrive. Information arrives. Volatility shocks arrive. Even liquidity itself, it shows up in bursts, disappears, then shows up again.  The simplest way to model this flow of events is the Poisson Arrival Process. 

Here’s what the model looks like:

You pick an intensity (lambda, perhaps estimated from historical data), the average rate at which events occur, and the process takes care of the randomness. Sometimes events cluster, sometimes there's a lull, but the idea is the same: the world doesn’t update every millisecond; it updates whenever something happens.

📈 Model applications

The Poisson arrival process is the cleanest way to model “things that show up whenever they feel like it.” Its waiting times follow an exponential distribution, which is memoryless, meaning the process doesn’t care how long it’s been since the last event, the probability of the next one arriving is exactly the same. Add independent increments, and you get a structure where arrivals in disjoint intervals never influence each other. It’s simple, almost too simple, but that’s why it works: information hits, trades print, jumps occur, orders queue and clear, each arrival starts fresh. The Poisson process is the backbone behind the intuition that markets have “their own clock,” and that risk often comes not from where you are on the calendar, but from the next arrival you don’t see coming.

You see Poisson processes all over quant finance once you know where to look. Merton’s jump-diffusion uses a Poisson clock to trigger discontinuities in price. Hawkes processes enrich that idea by letting arrivals spark more arrivals, capturing clustering in order flow and volatility. Credit risk models use Poisson-style intensities to represent default events. Market microstructure research uses them to describe trade arrivals, order submissions, and cancellations. Even limit-order-book models lean on Poisson arrivals to approximate how liquidity comes and goes. In every case, the Poisson component is doing the same job: giving randomness a schedule, giving shocks a mechanism, turning “stuff happens” into something you can simulate, price, and manage.

🎓 A little story

This isn’t a story but a remark from me (Roman, 👋 hi!) about the unreasonable effectiveness of the memoryless property. 

We see this in the real world all the time. Of course there’s dependency everywhere, but independence can be a surprisingly clean way to represent that dependence in some cases: you’re sitting outside and for ten minutes nothing happens and then BOOM three cars fly by. Markets behave the same way. No trading for ages, then suddenly size hits the tape and everything lights up. The exponential waiting time doesn’t capture the full richness of that behavior, but it captures the feeling of it: every arrival is a fresh start.

When I first learned about arrival processes, I expected something far more complicated. Markets are messy, order flow has structure, information is clustered… why start with something as naive as a memoryless process? But then the intuition landed: sometimes the simplicity reveals the structure. A Poisson process doesn’t try to model every microdependency; it says, “Look, events come when they come, and if that’s what matters for your model, start here.”

And the surprising part is how often that works. Modeling trade arrivals, jump times, quote updates, order cancellations, information releases, so many “events” in quant finance behave well enough under a Poisson clock that the mathematics becomes tractable without losing the spirit of the phenomenon.

Good models aren’t the ones that cram in every detail; they’re the ones that capture the right mechanics with the least machinery. The Poisson process is exactly that kind of model. It doesn’t predict when the next event happens, it gives you the right way to think about the uncertainty of when it happens.

💡 Takeaway

In practice, nothing in markets is perfectly Poisson, but plenty of things behave Poisson-ish when viewed through the right lens: order arrivals, trade prints, news bursts, jump times, liquidity shocks. Each model that uses a Poisson clock, whether it’s Merton’s jump diffusion, Hawkes-style contagion models, or market microstructure arrival models, is really asking the same question: how does risk enter the system, and how often?

Of course, no arrival model is complete, and we’ve only scratched the surface of point processes. But without the Poisson framework, we’d have no tractable way to talk about timing risk. With it, we get a clean, powerful structure for turning unpredictable events into something measurable, one arrival at a time.

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