The Paolucci Inequality

I hear the alarm bell in everyone's head going off.  All models are wrong, what if your action(s) fuck it up, whatever it is.  Well, was that your original intent?

Clearly not.  You thought that action would yield the desired outcome.  In fact, the downstream consequences of inaction are so severe, my conjecture suggests the Paolucci inequality holds in every case.

No rational agent acts against their own self interest (more generally their utility function, which may encompass irrational action toward oneself for another, still rational!).  We must accept that we are operating in a game of incomplete information.  That our actions may yield the unintended consequences.

Nevertheless, we can step up to the plate again, and again...and again.  That doesn't make failure any more pleasant, but it is a necessary psychological crutch to avoid white knuckling the helm.

Recently, I've been white knuckling the helm.  

I have dreams of improvement and expansion.  Dreams of running my hedge fund, visiting more universities for seminars, creating and building more.

After the loss of my Discord server, I was so scared of losing more of what I had built from nothing. I dug my feet into quicksand.  I tried to protect my garden at the expense of the garden.  I became paralyzed by choice and inaction.  I fell into the classic trap after rebuilding my server, brick by brick.  Then what?

Inaction.  

It feels safe, it's known, action requires uncertain, potentially negative outcomes.

Ephemerally, so afraid of regression and necessary iteration to make progress I refused to take steps forward.  It impacted my work, business, and personal relationships.  Getting out of bed was hard, but nothing was happening.

All of the sudden, the schedule of creating a video on Tuesday and Friday became suffocating.  I could do a 12 hour work day like nothing.  Now roughly a few hours in a week to produce these lectures felt impossible?  Inaction.

During this time I had some incredibly kind folks reach out via email offering words and discussions to help me loosen the grip on the helm.  They reminded me exactly why I do what I do.  Why I created Quant Guild in the first place, and where we are going which inevitably demands Action + Iteration.

To the folks that reached out to me, thank you.  We're just getting started.

- Roman

Volatility is not just stochastic. It is rough, persistent, and nonlinear.

The Quadratic Rough Heston model extends rough volatility by allowing variance to depend quadratically on an underlying rough factor. This additional structure helps capture features of the implied volatility surface that simpler rough models can struggle to reproduce.

The result combines rough volatility, stochastic variance, and nonlinear dynamics in a flexible framework for modern option pricing and volatility modeling.=

📚 Model Definition

Let's observe the quadratic rough Heston...

The Quadratic Rough Heston model models volatility through a rough latent variable, Z.

Variance is a quadratic function of Z, giving the model more flexibility than a standard linear variance specification.

The process Z is mean-reverting and stochastic, while the power-law kernel introduces memory and roughness. The Hurst parameter controls that roughness, and the correlation between price and volatility shocks captures the leverage effect.

The model combines rough volatility, mean reversion, and nonlinear variance dynamics in a single framework.

📈 Model Applications

The quadratic rough Heston was proposed to address limitations of standard rough volatility models, particularly their ability to jointly reproduce the SPX volatility surface and VIX derivatives.

The quadratic specification gives volatility additional flexibility while preserving the rough dynamics observed empirically.

In practice, this makes the model useful for SPX and VIX option pricing, joint calibration, volatility surface modeling, and volatility risk management. Its main appeal is fitting multiple volatility markets within one consistent framework.

🎓 A Little Story

When 0DTE options became pretty sexy, I started seeing rough volatility models everywhere.

Suddenly everyone was trying to explain and fit the extreme short-dated skew, where classical volatility models often struggle. 

Models like Quadratic Rough Heston became particularly interesting because rough dynamics are naturally suited to capturing the behavior of volatility at very short maturities.

It was one of those moments where a new market phenomenon made a relatively academic area of research feel very practical, very quickly.

💡Takeaway

Quadratic Rough Heston shows how roughness and nonlinear volatility dynamics can better capture the extreme behavior of modern option markets, particularly at short maturities.