Bergomi: forward variance as the modelling primitive
Heston models the instantaneous variance; Bergomi models the *forward* variance curve directly — the same way HJM models the forward rate curve. Variance swaps price exactly by construction.
Heston and other classical stochastic-vol models start from a state variable (instantaneous variance) and try to price variance swaps as outputs. Bergomi's insight is to flip that: take the forward-variance curve $\xi_0^u$ — the family of strikes for variance swaps at every future maturity — as the *modelling primitive*, and define dynamics on the whole curve. This is HJM for variance: by construction, today's model exactly reprices today's var-swap term structure, and the parameters left over govern the dynamics of vol-of-vol. This tutorial bootstraps the forward variance curve from var-swap quotes (using `variance_swaps`), writes the one-factor Bergomi SDE, and sets up the rough-vol extension.
β Intro Β· expand
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β Try first Β· expand
Worked example
Today's variance-swap term structure: $K_{\text{var}}(1\text{y}) = 0.04$, $K_{\text{var}}(2\text{y}) = 0.05$. Compute the implied forward variance $\xi_0^u$ on the segment $u \in [1, 2]$ (assuming a piecewise-flat forward-variance curve). Report the corresponding forward volatility.
β Worked example Β· expand
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β Practice Β· expand
Reflection
Bergomi makes the forward-variance curve the state. What advantage does this give over Heston's instantaneous-variance state when calibrating to a market with a complex term structure of var-swap quotes, and what does Bergomi still struggle with that pushes the field toward rough vol?