A Multi-Factor Volatility Model Adds Flexibility and Keeps Markovian Dynamics

If you care about how markets price future turbulence, the shape of volatility over time matters. This paper extends the Hobson and Rogers path-dependent volatility model, which links today’s variance to how far the log-price sits from its exponential moving average. The new quadratic variance function version, or QHR model, gives more flexibility in the moving average filter while keeping the model Markovian, meaning its future state depends on the present state rather than the full past. The paper also shows that this quadratic form lets the author write down weak-stationarity conditions for the variance process and explicit formulas for forward variance. Under stationarity, both the variance and the squared price increments follow an ARMA autocorrelation structure. For the simplest scalar case, the stationary distribution is a translated and rescaled Pearson type IV random variable. A numerical example then illustrates the model’s implied volatility surface and the term structures of forward variance, at-the-money volatility, and at-the-money skew.

A volatility model can act like a memory trick for markets. It watches how far today’s log price sits from a smoothed past average. That gap then sets the spot variance, which is the market’s instant wiggle room. The surprise here is simple. You can give that memory more shape without breaking the clean state update that makes the model usable. In this paper, the multi-factor Quadratic Hobson and Rogers, or QHR, model does that job. It aims at VIX-style pricing, where the future path of volatility matters as much as today’s level. The model keeps the story compact enough to follow, even though the memory filter becomes richer.

What the new QHR model changes

The original Hobson and Rogers model ties variance to an offset from an exponential moving average. An exponential moving average is a smoothed history that gives recent prices more weight. The QHR model keeps that basic idea. It adds a multi-factor setup and a quadratic variance function, which means the variance can depend on the offset in a curved way. That extra shape matters. It lets the model define the moving average filter with more freedom. It also keeps the dynamics Markovian, so the present state still carries all the needed memory. Under stationarity, the model gives explicit forward variance. It also shows that variance and squared price changes follow an ARMA pattern, which is a mix of short-run memory and longer-run carryover.

Why the quadratic form matters

The quadratic variance function does more than add curvature. It gives clear weak-stationarity conditions, meaning the variance process can settle into a stable long-run pattern. That is a useful test for any model that claims to describe market noise over time. In the simplest scalar case, the steady-state law has a named form. It is a translated and rescaled Pearson type IV random variable. That result gives a precise description of the long-run shape of the variance. The paper then uses a numerical exercise to show the model’s qualitative behavior. The exercise looks at the implied volatility surface, plus the term structures of forward variance, at-the-money volatility, and at-the-money skew.

What this means for pricing volatility

For traders and quants, the value lies in the balance. The model is not stuck with one rigid memory shape. It is also not forced into a tangled state that is hard to handle. That balance matters for instruments tied to future volatility, especially VIX derivatives. The paper’s setup keeps the process autonomous, which means the equations do not depend on time itself. It also keeps the offset non-dimensional, like a ratio rather than a raw level. That helps the model stay neat under scaling. The result is a framework that can say more about the future path of variance without giving up the mathematical grip that makes pricing possible.

The sharp test still ahead

The next hard test is whether the same flexibility still works in market data beyond the paper’s numerical exercise. The model already points to four targets: the implied volatility surface, forward variance, at-the-money volatility, and at-the-money skew. Those are the shapes a pricing model must match if it wants to live inside real markets. The most interesting open point is not whether the model can remember the past. It already can. The question is whether this richer quadratic memory can keep that same control when the market path gets rough. That is the real prize promised by the QHR setup.