Bergomi Forward Variance Models

A weather app doesn't just tell you today's temperature and a drift rule — for planning a trip, you want the whole forecast curve: expected temperature for every day ahead. Classic stochastic volatility models like Heston work like the "temperature plus drift" approach: they model instantaneous variance and let a term structure emerge as a byproduct. Bergomi's forward variance models flip this around and start directly from the forecast curve itself.

The core object and how it moves

Define $\xi_t(u)$ as the forward variance seen from today $t$, for an instant $u$ in the future — the market's current price for "how much variance will be realized around date $u$," analogous to a forward interest rate. This whole curve is directly observable today from variance swap quotes (see forward variance curve). The model specifies how each point on this curve moves as new information arrives:

$$d\xi_t(u) = \omega\, \xi_t(u)\, dZ_t(u)$$

In words: the forward variance for horizon $u$, $\xi_t(u)$, moves proportionally to its own current level, scaled by $\omega$ (a volatility-of-volatility parameter — how jumpy the forecast itself is) and driven by a random shock $dZ_t(u)$ that can be correlated across different horizons $u$ and correlated with the underlying's own returns (this correlation is what generates the volatility skew). It's a lognormal-style dynamic for variance forecasts themselves, not for the stock price directly — the stock price dynamics come along separately, using the instantaneous variance $\xi_t(t)$ (the near-end of the curve) as the actual variance driving the underlying at each moment.

Worked example 1 — a shock at one horizon, felt everywhere correlated

Suppose today's forward variance curve gives $\xi_0(1\text{m}) = 0.04$ (20% vol, squared) and $\xi_0(6\text{m}) = 0.0625$ (25% vol). A single-factor Bergomi model with $\omega = 1.5$ and near-perfect correlation across horizons means a shock that moves the 1-month point up 10% tends to move the 6-month point similarly: new $\xi(1\text{m}) \approx 0.044$, new $\xi(6\text{m}) \approx 0.06875$. A two-factor version (a second, less correlated driver) lets the short and long ends move by different amounts — the front of the vol term structure is empirically more jumpy than the back, which a single flat-shock model can't reproduce.

Worked example 2 — where the skew comes from

The correlation $\rho$ between the underlying's return shocks and the forward variance shocks $dZ_t(u)$ generates the implied volatility skew, the same mechanism as in Heston applied point by point along the curve. With $\rho = -0.7$ (typical for an equity index: stock down, vol up), a 5% drop in the underlying tends to come with an increase across the forward variance curve, pushing $\xi_0(1\text{m})$ from 0.04 to roughly 0.046. Because the model is calibrated so this response matches quoted option skews at every maturity simultaneously, a Bergomi model fit to today's whole surface reprices the skew correctly after a spot move in a way a single-instantaneous-vol model often cannot.

What this means in practice

Bergomi-style models are the standard tool on exotic desks for pricing options whose payoff depends on the path of volatility (forward-starting options, cliquets, VIX options), because they're built to match the forward variance curve exactly by construction — a direct, observable market input, not something implied indirectly.

Practice

1. If a two-factor Bergomi model gives the short end of the forward variance curve a higher $\omega$ (vol-of-vol) than the long end, what pattern in the term structure's jumpiness does that reproduce?
2. Why is it a mistake to treat today's 6-month forward variance quote as a forecast that realized variance over the next six months will equal that number?