Dr. Pawel LachowiczIn quantitative finance, the Volatility of Volatility (also referred to as Vol-of-Vol or VoV) is an important parameter for pricing various derivative products (e.g. Volatility Dispersion Swaps) and its correct estimation is frequently desired. VoV is usually a single number treated an an input parameter.
One can make a couple of assumptions to start working within VoV framework. Namely, (1) there is one single VoV for all assets belonging to specific kind (e.g. equities, FX, or cryptocurrencies); (2) this unknown VoV has a certain distribution due to variability of its measurement; (3) a high percentile of this distribution (e.g. 95%) would lead to a conservative estimate of this single VoV; and (4) volatilities are log-normally distributed, which mean the only variation (from the mean) would be purely due to uncertainty in the VoV parameter.
In this post, we develop three methodology variants that can be used in the VoV estimation based on raw time-series. One of them, by making the final decision at the end, can be selected for as the best one and the resulting number plugged into, inter alia, crypto-derivative pricing models.
Variant 1 $-$ Vol-of-Vol estimation based on the most severe stress period
Let a raw time-series denoted as:
$$
\{ x_i \} \ \mbox{for} \ i = 1, …, M
$$ where $M$ stands for a number of data points, is transformed into a time-series of daily log returns:
$$
r_i = \log\left( \frac{x_i}{x_{i-1}} \right) \ \mbox{for} \ i = 2, …, M .
$$ Next, the $K$ day long rolling window is being applied to it with a step of one day. For each window the volatility has been calculated as:
$$
\sigma_d = \sqrt{ \frac{1}{K-1} \sum^{K}_{k=1} \left( r_k – \ \bar{r} \right)^2 } .
$$ $L$ windows form a set of
$$
\{ \sigma_{d_j} \} \ \mbox{for} \ j = 1, …, L
$$ from which we pick up the highest volatility (most severe one) and translate it to the annualised volatility as follows:
$$
\sigma_{ann} = \max\left( \left\{ \sigma_{d_j} \right\} \right) \sqrt{365} .
$$ For a wide portfolio of $N$ cryptocurrencies, a set of annualised volatilities can be composed, denoted as:
$$
\{ \sigma_{ann_j} \} \ \mbox{for} \ n = 1,…, N
$$ based on which the Vol-of-Vol, $\eta$, has been derived as:
$$
\eta = N^{-1} \sum^{N}_{n=1} \sigma_{ann_n}
$$ with an uncertainty of:
$$
\Delta\eta = \frac{1}{\sqrt{N}} \left[ \frac{1}{N-1} \sum^{N}_{n=1} \left( \sigma_{ann_n} -\ \bar{\sigma_{ann}} \right)^2 \right]^{-1/2}
$$ or
$$
\eta \pm \Delta\eta .
$$ In this Variant, we aim to select such stress period among $L$ scenarios characterised by the highest level of volatility at the cryptocurrency level. The mean of the distribution of such volatilities, i.e. VoV, would be conservatively high however the location of the stress period per individual cryptocurrency would differ.
The VoV in this Variant becomes a function of three parameters: a number of crypto-assets $N$; a number of data points $M$; and a rolling window length $K$. In short,
$$
\eta(N, M, K) .
$$
Variant 2 $-$ VoV estimation based on entire time-series
In this Variant, instead of considering a rolling window over the log return time-series, the entire span of $M-1$ data points (per crypto-asset) is considered. Therefore, we note a difference conferring to Variant 1 formulae, i.e.
$$
\sigma_d = \sqrt{ \frac{1}{M-1} \sum^{M-1}_{m=1} \left( r_m – \ \bar{r} \right)^2 }
$$ $$
\sigma_{ann} = \max\left( \left\{ \sigma_{d} \right\} \right) \sqrt{365} .
$$ and for $N$ cryptocurrencies:
$$
\{ \sigma_{ann_j} \} \ \mbox{for} \ n = 1,…, N
$$
we determine Vol-of-Vol in the same was as above:
$$
\eta = N^{-1} \sum^{N}_{n=1} \sigma_{ann_n}
$$ with the corresponding uncertainty. The expected estimations of VoV in this Variant become a function of two parameters: a number of cryptoassets $N$, and a number of data points (M), or
$$
\eta(N,M) .
$$
Variant 3 $-$ VoV estimation based on cross-section among all cryptocurrencies
If we have a look at the daily log returns time-series:
$$
\{ r_i \} \ \mbox{for} \ i = 2, …, M
$$ then, in fact, the index $i$ is a function of time (timestamp) and allows us to rewrite it as:
$$
\{ r_i \} \equiv \{ r_i(t_i) \} .
$$ Given that, we calculate the daily volatility cross-section as follows:
$$
\sigma_d(t_i) = \sqrt{ \frac{1}{N-1} \sum^{N}_{n=1} \left( r_n(t_i) – \bar{r} \right)^2 }
$$ where $N$ denotes a number of crypto-assets we consider (portfolio), and
$$
\sigma_{ann}(t_i) = \sigma_d(t_i) \sqrt{365}
$$ followed by Vol-of-Vol expressed as:
$$
\eta = (M-1)^{-1} \sum^{M-1}_{m=1} \sigma_{ann_j}(t_i) .
$$ The VoV in this Variant remains a function of two parameters: a number of crypto-assets $N$ and a number of data points $M$, i.e.
$$
\eta(N,M).
$$ By taking a different vantage point here, we aim to determine VoV for a basket of $N$ crypto-assets by consideration of daily volatility as a function of time, therefore extending a number of measurements from $N$ (Variant 1 and 2) to $M-1$.
End Note
Above, we presented three different methods of calculation of Vol-of-Vol parameter. This post would not be complete if we allow ourselves to omit a mention about Heston pricing model for derivative products. Heston model, inter alia, depends on rate of mean reversion of variance under risk-neutral dynamics, long-term mean of variance under risk-neutral dynamics, initial variance under risk-neutral dynamics, correlation between returns and variances under risk-neutral dynamics, and… VoV. Therefore, an ability to plug into model the most optimal value of VoV becomes of paramount importance.
Explore Further
→ Quantitative Analysis of a Sample Drawn from the Unknown Continuous Population
→ In Search for Stablecoin’s Perfect Peg: Application of Negative Hypergeometric Distribution
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