Forward Variance and Ornstein–Uhlenbeck Dynamics in the Volatility Smile

Summary

The note explains how to interpret the instantaneous forward variance quantity in a model of volatility smiles. It distinguishes that quantity from a forward-starting variance swap, whose fair strike is constructed from the variances of two maturities. The discussion motivates representing the forward variance curve across maturities, though it does not develop that modeling rationale in detail.

For the proposed lognormal dynamics, applying Itô’s formula to the logarithm produces a stochastic integral and a variance correction. Factoring the maturity-dependent exponential separates the expression into a term that scales with time to maturity and a process driven by an exponentially weighted Brownian integral. That integral follows zero-mean Ornstein–Uhlenbeck dynamics. The note gives an algebraic explanation rather than empirical evidence, and it leaves broader questions about model assumptions and practical calibration unanswered.

Key ideas

• A forward-starting variance swap strike can be expressed using variance swap rates at two maturities.
• The instantaneous forward variance quantity is the limiting forward variance at a particular maturity.
• Applying Itô’s formula to log forward variance introduces a drift correction.
• An exponentially weighted Brownian integral has Ornstein–Uhlenbeck dynamics.

Tags

# Smile Dynamics - forward variance


# Smile Dynamics - forward variance












I was reading Smile Dynamics II by Lorenzo Bergomi. It is clear to me that on page 2

$$ V_t^{T_1,T_2}=\frac{(T_2-t)V^{T_2}_{t}-(T_1-t)V^{T_1}_{t}}{T_2-T_1} $$ is the fair strike of a forward-starting variance swap that starts accumulating variance at $T_1>t$ and matures at $T_2>T_1$. However, I find it difficult to conceptualize quantity $\xi_t^T=V_t^{T,T}$. What does it really represent? And why is it a good idea to model a term structure of such quantities in $T$?

Another basic question I had is the following: if the dynamics of $\xi_t^T$ is postulated to be lognormal in the first formula of page 2, then how come we end up with a zero-mean Ornstein-Uhlenbeck process in eq. (2.1)?

## Answer by Frido (score 2, accepted)

https://quant.stackexchange.com/a/75747

Ok, so $$ d\xi_t^T = \omega e^{-k(T-t)} \xi_t^T dW_t $$ where $W$ is standard Brownian. Then, just by applying Ito I hope you can see that $$ \log \xi_t^T / \xi_0^T = \omega \int_0^t e^{-k(T-u)} dW_u - \frac12 \omega^2 \int_0^t e^{-2k(T-u)} du $$ Now just write $$ e^{-k(T-u)} = e^{-k(T-t)}e^{-k(t-u)} $$ Then $$ \log \xi_t^T / \xi_0^T = \omega e^{-k(T-t)} X_t - \frac12 \omega^2 e^{-2k(T-t)} E_0[X_t^2] $$ with $X_t = \int_0^t e^{-k(t-u)} dW_u$ which is an O-U process.

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