Forward Variance for Hedging VIX Options in a Rough Volatility Model

Summary

The document asks how to construct forward variance data for replicating a VIX option hedging experiment under a rough stochastic volatility model. It describes a setup in which VIX is modeled from fractional Brownian motion and the hedge uses a model delta together with a forward variance quantity, defined as a conditional expectation of future squared VIX under the risk-neutral measure. The experiment referenced by the question uses synthetic forward variance curves derived from S&P index options.

The author asks whether realized variance swaps could supply that quantity, and seeks clarification on the initialization of an at-the-money VIX option and its hedge ratio. No answers are included, so the document does not establish a calculation procedure, equivalence between realized and forward variance, or the precise meaning of the model hedge ratio. Its value is primarily in identifying implementation questions that matter when reproducing a model-based backtest; conclusions would require the referenced paper’s conventions and market data details.

Key ideas

• The described hedge uses a VIX option delta and a forward variance instrument.
• Forward variance is framed as a risk-neutral conditional expectation of future squared VIX.
• The referenced experiment builds synthetic forward variance curves from S&P index options.
• The document asks whether realized variance swap data can substitute, but supplies no answer.
• The at-the-money strike convention and model hedge ratio also remain unresolved.

Tags

# Compute Forward Variance Swap


# Compute Forward Variance Swap












I'm trying to compute a forward variance like this paper https://arxiv.org/pdf/2105.04073.pdf.

The paper shows that under rough stochastic volatility model assumption, options can be hedged with the underlying and variance swap.

I'm particularly interested in etablished a baseline for my work by replicating the experiment hedging VIX option which is modeled by

$$VIX_t = C e^{X_t}, \quad X_t = \sigma W^H_t,$$ where $W^H$ is a fractional Brownian motion.

I can follow the paper to the part of getting delta hedge which is

- $dP_t = N(d_t^1) dVIX_t$.

- $dP_t = \frac{N(d^1_t)e^{-\frac{1}{2}}(c^T(T) - c^T(t))}{2\sqrt{F^T_t}}dF_t^T$.

Here, $F^t$ is the forward variance ($F^T_t = \mathbb{E}_Q[VIX^2_T|\mathcal{F}_t]$).

To perform back test, the paper mentioned that the back test uses synthetic forward variance curve data, computed from S&P index options.

I wonder how the forward variance is computed here? Can I compute it using realized variance swap.

Additional questions after editting:

> The initial value of the hedging portfolio is initialized with the ATM VIX option price with maturity 1.5 months computed within the model

Does this mean that the strike $K$ is set to $S_0$?

> the quantity of the hedging asset in the portfolio is initialized with the corresponding model-based hedge ratio.

Does "model-based hedge ration" mean the delta hedge in the above formula?

(I'm from ML background, so I may not know many basic concepts in finance)

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