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Deriving Instantaneous Variance Drift from the Forward Variance Curve

Article Quant Q&A · Author: fwd_T
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Summary

The discussion explains how the dynamics of instantaneous variance follow from a family of forward variance claims. Since instantaneous variance at time t equals the forward variance maturing at t, its change can be split into the change in a nearby forward claim and the change in that claim’s value as its maturity advances. The latter is approximated by the maturity slope of the forward variance curve at its short end, while the claim’s value change contributes the martingale component.

This gives an interpretation of the drift of instantaneous variance as the local maturity slope of forward variance. The post sketches the argument using an infinitesimal time step and points to a historical predecessor, but does not develop the regularity assumptions or provide a full rigorous derivation. Its usefulness is conceptual, especially for understanding stochastic volatility models and variance-linked claims.

Key ideas

  • Instantaneous variance is identified with forward variance evaluated at the current time and maturity.
  • Its increment separates into a martingale change in a forward variance claim and a change from moving along the maturity curve.
  • The local maturity slope at the short end determines the drift contribution in the infinitesimal argument.
  • The explanation is informal and does not specify the mathematical assumptions needed for rigor.

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VolatilityDerivatives pricingStatistics
Full text
# Drift of stochastic variance as slope of the short end of the forward variance curve


# Drift of stochastic variance as slope of the short end of the forward variance curve












I was re-reading Chapter 6 of Stochastic Volatility Modeling by Lorenzo Bergomi. On page 203, he considers a forward variance of the following form: $$ d\xi_t^T=\lambda_t^T dZ_t^T, $$ where $Z_t^T$ is a standard Brownian motion for each $T>t$. Now he wants to prove on page 204 that: $$ dV_t=\frac{d \xi_t^T}{dT}\Biggr\rvert_{T=t} dt+\lambda_{t}^{t} dZ_t^t, $$ which means that the drift of a stochastic variance process $V_t^t=\xi_t^t$ is the slope at time $t$ of the short end of the forward variance curve. I am not entirely sure how he arrives at this formula. He says it is enough to differentiate the identity $V_t=\xi_t^t$, but I just don't see how one leads to other. Can you help me figure out?

## Answer by Frido (score 1, accepted)

https://quant.stackexchange.com/a/75838

The instantaneous variance $V_t = \xi_t^t$. So $$ dV_t = \xi^{t + dt}_{t+dt} - \xi^t_t $$ But $$ \xi^{t + dt}_{t+dt} = \xi^{t + dt}_{t} + d \xi^{t + dt}_{t} $$ The second term to the right of the equal sign, $d \xi^{t + dt}_{t}$, is a martingale as it is simply the change in the value of a claim. So $$ dV_t = d \xi^{t + dt}_{t} + (\xi^{t + dt}_{t} - \xi^t_t) $$ But the second term to the right of the equality above is the term structure of the claims $\{\xi_t^T\}$ as observed at time $t$, for $T \in [t,\infty)$. In other words $$ (\xi^{t + dt}_{t} - \xi^t_t) = \left( \frac{ d\xi_t^T}{dT} dT \right)_{T=t} $$ Hope this helps.

EDIT As an historical aside, Bergomi was not the first to imply the dynamics of instantaneous variance from variance swaps. That, afaik, was Dupire in his unpublished paper "Arbitrage pricing with stochastic volatility, BNP, 1992."

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.