What Are Multivariate Quadratic Hawkes Processes?
Markets don’t just react to where prices are — they react to the path that got them there.
That’s where Multivariate Quadratic Hawkes Processes (MQHPs) come in.
These models go beyond classical Hawkes processes by capturing:
How past trends shape volatility (both small negative moves and large swings of any sign).
How shocks in one asset propagate to others (cross-leverage and cross-Zumbach effects).
How past covariances between assets can influence future joint volatility.
In short, MQHPs provide a richer and more realistic framework for understanding how risk spreads across markets — something traditional volatility models miss.
A Hawkes process is a type of self-exciting point process used in statistics and quantitative finance. In simple terms, it models events (like trades, price jumps, or market shocks) where each event increases the likelihood of future events happening soon after. It’s widely used to study clustering in high-frequency financial data.
Now, let’s build on that:
1. Quadratic Hawkes Processes
A Quadratic Hawkes Process (QHP) extends the classical Hawkes framework. Instead of events just depending linearly on past events, the intensity (i.e., how likely new events are) also depends quadratically on past returns or trends.
- This quadratic feature captures the fact that volatility depends on both the magnitude and the sign of past market moves.
- Example: Small downward moves in the S&P 500 raise volatility (the “leverage effect”), while large moves — up or down — push volatility higher regardless of direction (the “Zumbach effect”).
2. Multivariate Extension
A Multivariate Quadratic Hawkes Process (MQHP) studies these dynamics across multiple assets at once. Instead of just looking at how the past of one asset influences its own volatility, MQHP explores:
- Cross-leverage effects: The return of one asset affecting the volatility of another.
- Cross-Zumbach effects: Strong trends in one market (e.g., E-mini futures) influencing volatility in other assets (e.g., Treasuries).
- Covariance feedback: Past co-movements between two assets affecting their future joint volatility.
3. Why It Matters
In finance, MQHP provides a richer and more realistic framework to model how shocks propagate across markets. It moves beyond the assumption that volatility is only a function of current prices and recognizes that path dependence and cross-asset interactions are fundamental to risk and price dynamics.
Researchers like Jean-Philippe Bouchaud and collaborators have been at the forefront of proposing and empirically validating these models, showing they capture market phenomena that simpler models miss.
