Volatility
Although the general public tends to have a misconception about one of the most common uses of volatility, which is to assume its predictive effect on future volatility or asset price movements, it is important to note that realized volatility is a fundamental tool in the field of quantitative finance.
There are numerous applications of realized volatility, but I will highlight the ones we consider most important.
Uses of Volatility: Analysis of Relative Implied Volatility
We use realized historical volatility over different periods to determine the price of implied volatility. For example, if we want to calculate the price of Implied Volatility (IV) for the SPX index for a given month and we only have the historical time series of the SPX and the event calendar for the next month, we can estimate the average volatility observed over 24-hour periods, one week, one month, among others.
$$IV_{\text{est}} = \frac{1}{N}\sum_{i=1}^{N}\sigma_{\text{hist},i} \times \text{event adjustment}_i$$
To evaluate implied volatility, we use realized historical volatility as a starting point. However, we must also consider information about economic events that will occur in the next month (such as the FOMC Meeting, NFP, GDP, etc.), as these deterministic factors influence the valuation of implied volatility and can increase the variability of the underlying asset.
To obtain an approximate estimate in urgent situations, it is possible to use historical volatility during similar events. This involves taking the average between the mean volatility during ON events (active) and the mean volatility during OFF events (inactive). This way, an intermediate volatility is obtained that, while not precise, will be more useful than having no reference at all. It is important to note that this approximation has limitations and does not replace a thorough analysis, but it can provide a provisional estimate in situations where more precise information is not available.
Uses of Volatility: Estimation of Mean Reversion / Trend Direction (Variance Ratio)
Volatility is commonly used to detect trends or mean reversion using realized volatility estimators, especially in delta hedging strategies. A widely used indicator is the variance ratio, which is a simple relationship between high-frequency variance (GK) and C2C variance.
$$VR = \frac{\sigma^2_{\text{intraday}}}{\sigma^2_{c2c}} \quad \begin{cases} VR > 1 & \Rightarrow \text{trending} \\ VR < 1 & \Rightarrow \text{mean-reverting} \end{cases}$$
The underlying idea is intuitive: assets that experience mean reversion tend to have higher intraday volatility, while C2C variance remains moderate. This suggests that the price reflects the long-term trend, and the variance ratio provides information about short-term changes in the asset.
Uses of Volatility: Analyzing Realized Volatility Across Market Regimes
Another important use of volatility is the analysis of realized volatility, which allows us to study its dynamics during changes in market regimes. If we work with derivatives, we have an additional advantage, as we can analyze the realized volatility of both the underlying asset and the derivative. Although it is not an exact prediction, using past volatility as an approximation allows us to make assumptions about future events.
For example, if we observe that negative data at FOMC meetings has caused volatility four times greater than positive data, we can use this information as an approximation to anticipate possible future scenarios. However, it is essential to note that each market dynamic is unique before and after events, so this information should not be considered a precise tool for predicting the future. It is important to remember that we do not have the ability to accurately foresee market movements.
Conclusions on the Use of Volatility
In conclusion, the uses of volatility and the microstructure of volatility are fundamental concepts for understanding financial markets. However, it is important to note that realized volatility is not a reliable indicator or a precise predictor of future volatility. There are numerous factors that influence market behavior and the price of financial assets.
Therefore, it is advisable to consider other factors such as C2C, Parkinson volatility, Garman volatility, and the analysis of relative implied volatility to obtain a better understanding of volatility. These additional indicators and tools can provide a more complete and precise view of volatility in financial markets.
VIX1D
Now let's focus on the indicator that shows us current volatility, at the next expiration. Since VIX is designed to estimate changes in volatility at 30 days, and with the rise of 0DTE products, CBOE has taken matters into its own hands, creating new calculation methodologies for its most popular indices.
On April 24, 2023, CBOE published a Whitepaper explaining a new index they have put into operation. Quoting the authors literally:
The VIX1D index has been designed to "estimate" the EXPECTED volatility at one day.
There are differences in the calculations compared to the traditional VIX, which are:
- Time Decay of Near-Term (Current Day Expiry) Options Used to Calculate the VIX1D Index
- Calculation of the VIX1D Index After Near-Term Expired
- Use of Business Year and Minutes vs. Calendar Year and Minutes
- The Volatility of the VIX1D Index
Mathematical Definition
σ² = (2/T) ∑ [(ΔKi / Ki²) * e^(RT) * Q(Ki)] - (1/T) * [(F/K0) - 1]²
Selection Method (Near & Next Term)
The universe of options defined for calculating the VIX1D components is given by PM-Settled SPXW Options, or the so-called weeklies. The so-called t0 is defined as the options in the universe whose DTE is assigned to today. The one considered as Next term, or T+1, is given by the options in the universe whose DTE is assigned as the second closest, the first being the Near Term or T0.
Risk Free Rate Selection Method
To select the risk-free interest rates needed for calculating the VIX1D index, US Treasury curves, called CMT or Constant Maturity Treasury, are used for the maturities of the previously selected options. To assign their weight and composition in more detail, CBOE refers to "Interest Rate Calculation – Bounded Cubic Spline Interpolation of the Cboe Volatility Index Mathematics Methodology."
Variance Calculation
For the calculation of variances, they are based on the options defined in series 1. Including the bid and ask, and the option price for each series. Additionally, CBOE refers to this methodology for its calculation "3(a) Volatility Index Calculation – Single Term of the Cboe Volatility Index Mathematics Methodology."
Given its time horizon, the VIX1D index is calculated in time units according to its expiration. Converting ratios to Minutes.
$$
T=\left(M_{\text {Time to Expiry }}\right) / M_{\text {year }}
$$
Where:
- M Time To Expiry -> The number of RTH session days from calculation to expiration.
- M Year -> Number of days during the RTH session in a year.
Calculation and Publication
A simplified calculation would be:
$$
\text { VIX1D }=100 \times \sqrt{{T_{1} \sigma_{1}^{2}[\frac{M_{T_{2}}-M_{\mathrm{CM}}}{M_{T_{2}}-M_{T_{1}}}]+T_{2} \sigma_{2}^{2}[\frac{M_{\mathrm{CM}}-M_{T_{1}}}{M_{T_{2}}-M_{T_{1}}}] } \times \frac{M_{y e a r}}{M_{\mathrm{CM}}}}
$$
Where:
- $M_{T_{1}}$ -> Minutes until today's session close.
- $M_{\text {CM}}$ -> Minutes until tomorrow's session close.
You can find all the details in the official CBOE article here.
Mathematical Definition
$$
\sigma^{2} = \frac{2}{T} \sum_{i} \left( \frac{\Delta K_{i}}{K_{i}^{2}} e^{R T} Q\left(K_{i}\right) \right) - \frac{1}{T} \left[\frac{F}{K_{0}} - 1\right]^{2}
$$
$$
\Delta K_{i} = \frac{K_{i+1} - K_{i-1}}{2}
$$
Calculation and Publication
A simplified calculation would be:
$$
\text{VIX1D} = 100 \times \sqrt{ T_1 \sigma_1^2 \left( \frac{M_{T_2} - M_{\text{CM}}}{M_{T_2} - M_{T_1}} \right) + T_2 \sigma_2^2 \left( \frac{M_{\text{CM}} - M_{T_1}}{M_{T_2} - M_{T_1}} \right) \times \frac{M_{\text{year}}}{M_{\text{CM}}} }
$$
Where:
- $M_{T_1}$: Number of business minutes until the expiration of near-term options.
- $M_{\text{CM}}$: Number of business minutes during the RTH session $(6.75 \times 60 = 405)$.
- $M_{\text{year}}$: Number of business minutes in a year $(252 \times 6.75 \times 60 = 102,060)$.
- $T_i$: $\frac{M_{T_i}}{M_{\text{year}}}$.
- $\sigma_i^2$: Variance of term $i$ calculated in step 3.
