Modeling Variance and Volatility Swap Strikes with Bergomi
Summary
The note outlines a model-based route to backtesting variance and volatility swaps when direct swap data is unavailable. It uses a one-factor Bergomi setup in which instantaneous variance follows a lognormal process correlated with the asset return. In this model, the variance-swap strike equals current variance at inception, while the volatility-swap strike is approximated with a vol-of-vol expansion. The difference gives a convexity adjustment that depends on current variance, vol-of-vol, and time to maturity.
The parameters can be estimated by calibrating to options on the underlying, and the answer points to calibrated option-model parameters as a possible data source. Once realized volatility has accumulated, the valuation expressions become more involved, though the note says an expansion remains tractable. The framework has material limits: the one-factor version implies a flat variance-swap term structure that shifts in parallel, which the answer considers unrealistic; a two-factor model may be more suitable. Model-free approximations are mentioned but not developed, and the response is not a full backtesting recipe.
Key ideas
- A calibrated options model can supply inputs for estimating variance- and volatility-swap strikes.
- In the stated one-factor model, the inception variance-swap strike equals instantaneous variance.
- The volatility-swap strike includes a convexity adjustment approximated through a vol-of-vol expansion.
- The one-factor model imposes a flat, parallel-moving variance-swap term structure, limiting realism.
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# Variance and vol swaps backtest
# Variance and vol swaps backtest
I am looking to build a backtesting tool for variance and vol swaps. Would anyone have an idea if there is available data for it or am I better off backtesting it using vanilla options and replicating the product?
## Answer by Frido (score 1)
https://quant.stackexchange.com/a/81414
Perhaps the easiest way to go about this is to choose a particular model for the options market in question.
For instance, suppose the market is described by the following model: \begin{gather} dS_t = \sqrt{v_t} S_t dW_t \\ dv_t = \omega v_t dZ_t \end{gather} with $E_t[dW_t dZ_t ] = \rho dt$. So this is a lognormal model for the instantaneous variance (a very fancy name for it is the "1-factor Bergomi model") where $\omega$ is the "vol-of-vol" (actually the vol of variance).
Under this model the varswap strike is simply $$ E_t \left[ \frac{1}{T-t} \int_t^T v_u du \right] = v_t $$ The volswap strike is not known in a closed-form / analytical solution. However, it is not too difficult to carry out a vol-of-vol expansion and to derive that the volswap strike is $$ E_t \left[ \sqrt{\frac{1}{T-t} \int_t^T v_u du} \; \right] = \sqrt{v_t} \left( 1 - \frac{ \omega^2 (T-t) }{24} \right) + O(\omega^3) $$ Hence under this model the convexity correction is $$ \sqrt{v_t} \,\frac{ \omega^2 (T-t) }{24} + O(\omega^3) $$ which I'm guessing is a metric or signal included in many a var vs. vol backtest.
The question is how to obtain $v_t$ and $\omega$: those can be obtained from the options market for $S_t$ by calibrating the model, and options data is readily available. If I am not mistaken, BBG provides calibrated parameters for SABR and Heston for some markets (eg FX). Not sure they support the Bergomi model yet though.
I have only written down the value of the swaps at inception. Once realized volatility has accumulated the expressions become slightly more complicated but still quite tractable (using a vol-of-vol expansion).
There are also so-called model-free approximations for the volswap, but that's another topic. Also, as you may notice the 1-factor Bergomi model implies a flat term-structure for varswap strikes and that the varswap strike term structure moves in parallel. That is not realistic, hence the 2-factor Bergomi model is preferable .
If my answer hasn't fully answered your question that's because entire papers/books can be written about your question as it is posed :)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.