Volatility indices, based upon the methodology of the Cboe volatility index (VIX), serve as measures of near-term market uncertainty across asset classes. They are constructed from out-of-the-money put and call premia using variance swap pricing. Volatility indices for fixed income markets are of particular importance, as they allow inferring market expectations about discount factors and credit premia, which have repercussions on all assets and the broader economy. There is a step-by-step construction plan for building a bespoke index for any rates market with liquid futures and options. Such a volatility index supports asset management in two ways. First, it is a valid basis for portfolio risk management and volatility targeting. Second, it can be used for extracting forward-looking market information, including changing probability quantiles for prices and rates, probabilities of certain extreme events, and the skewness of expectations.

The below quotes are from the paper. Headings, cursive text, and text in brackets has been added.

This post ties in with this site’s summary on macro trends, particularly the section on using financial data.

### Basic background of implied volatility indices

“__Implied volatility indices are constructed from out-of-the-money call and put option premia__. One of the reasons is that the market activity in out-of-the-money options is greater than in in-the-money options, because they have lower delta, and are thus cheaper to hedge and offer higher leverage.”

“The __calculation of VIX-type volatility indices across asset classes is based on variance swap pricing.__ A variance swap exchanges realised (historical) variance against the agreed ‘variance swap’ rate, which reflects the market-implied variance. The idea behind is that the price of a variance swap is obtained by replicating its payoff with a portfolio of a discrete set of put and call options, and the underlying futures…Such __replication can be achieved by the square of a VIX-type volatility index formula; it is an approximation of the expectation of the annualised variance of returns__ of the underlying asset over 30 days.”

“Volatility indices, built upon the methodology of Cboe VIX, have become __popular measures of market uncertainty over the short term, across a range of underlying asset classes__. They are easily interpretable as they reflect market pricing of subsequently realised volatility, implied from option prices, usually over the next 30 days…The development of volatility indices has a [long] history with a number of existing indices across equities, credit, commodities or FX.”

“An interest-rate volatility index offers __easily interpretable market expectations of interest rate uncertainty__, which is often tied to the future outcome of important market drivers, such as upcoming central bank decisions, political events, economic data, etc.

Fixed income option-implied volatility indices and products linked to them are already available in the U.S. (for US Treasury futures and interest rate swaps – Cboe TYVIX/SRVIX indices, ICE BofAML MOVE/SMOVE index) and Japan (for JGB futures – S&P/JPX JGB VIX)…There exists a euro version of ICE BofAML SMOVE index, which calculates the weighted average of normalized swaption implied volatilities on 2-year, 5-year, 10-year and 30-year euro interest rate swaps.”

### How to construct an implied volatility index for bonds

“We __apply the theory behind the variance swap pricing to calculate implied volatility indices from the options on European government bond futures__ for monthly expiries traded on Eurex exchange (Germany 2-year, 5-year, 10-year, France 10-year, and Italy 10-year). We use options on Schatz futures, Bobl futures, Bund futures, OAT futures, [and] BTP futures…We fix the maturity to a constant one-month period, 30 calendar days, i.e. the index measures the annualised implied volatility of underlying futures contracts over one month, expressed in price and basis points…due to enhanced liquidity in the front contracts.

“The step-by-step calculation of a fixed income volatility index, based on the square root of annualised variance, is well explained in TYVIX whitepaper [view here].

- We [use] the
__discrete approximation of variance fair value__…[which] returns the weighted sum of daily settlement prices of out-of-the-money call and put options on bond futures, with weights inversely proportional to the square of strike. Such weights ensure that the sensitivity of an option portfolio is not affected by changes in options’ implied volatilities… - The calculation
__gives the variance of the returns of an underlying bond futures contract__until maturity of the option… __One-month implied price volatility is then calculated by interpolating__between two listed options with closest expiration dates, for which the data are available…- To
__calculate volatility in basis points__, we use bond price-yield sensitivity relationship. We multiply the price volatility of the near and the next term option (expected % change in the price of the underlying Bund futures) by 100 and the futures price and divide it by the DV01 of the underlying futures.”

### Empirical features of a euro area bond volatility index

“The graph [below] shows the implied volatility index in case of Bund futures: the Bund volatility index…in basis points annualised.”

“In the case of equities, empirical evidence suggests that market downturn is almost always associated with heightened volatility, however, __fixed income volatility appears to be less directional; implied volatility increases both during a Bund sell-off and a Bund rally__, given its flight to safety status.”

“Bund implied volatility suggests __regime-like behaviour__: average Bund implied volatility appears to have moved from higher values in 2011-2012 to lower values in 2013-2014, then the average volatility between 2015 and the beginning of 2017 before decreasing again.”

“We can also observe __mean reversion; every sudden spike eventually returned towards calmer periods__, occasionally establishing a new regime. Volatility spikes and drops often reflect pricing of the uncertainty of a particular event: the volatility increases as the event date approaches, and drops after the event passed.”

“The Bund volatility index has been moving together with the US Treasury implied volatility index, TYVIX, with a few exceptions related to European events, such as developments in the euro area sovereign credit spreads or repricing of the ECB rate expectations.”

### Relevance of a bond volatility index for trading strategies

“Using the Bund volatility index on a sample of data from November 2010 to August 2019, we can observe that __the index correctly predicts the direction of subsequently realised volatility by 63%.__ Interestingly, __periods of lower realised volatility are predicted almost with certainty__ (if the volatility index value is lower than realised volatility over the last 30 days, then subsequently realised volatility will also be lower than the realised volatility over the last 30 days, with the likelihood of 93%). Implied volatility is likely to decrease if no important market-driving event is scheduled until the option’s expiration date, and realised volatility follows. __Periods of higher realised volatility are predicted less accurately__, at 56%. One possible explanation could be the existence of an implied volatility risk premium.”

*N.B.: This implies, among others, that a volatility index can support the volatility targeting of fixed income positions.*

“The graph [below] displays the Bund volatility index together with subsequently realised volatility (annualised standard deviation of logarithmic Bund futures returns) subsequently realised over the next 30 days).”

### Extracting implied probability distributions and useful metrics

“We use option prices to construct __forward-looking measures of uncertainty around the prices of underlying European government bond futures__…by recovering the probability distribution of possible outcomes of bond futures prices, implied from options.”

“Information content from option prices can complement a more established analysis of market forward prices and analysts’ forward price point estimates…__Option prices reflect the probability of being exercised through their time value and their strike price__. This probability evolves over time as prices and the realised volatility of the prices of the underlying asset change. Probability distributions from option prices are often recovered by assuming a specific form of the density function, or a specific stochastic process for the underlying asset, or by differentiating the option pricing function directly.”

“__Risk-neutral probability density functions can be recovered from the second derivative of the call price with respect to strike, discounted to its present value__…The idea is that the density function of the underlying futures equals the discounted second derivative of the call price with respect to the strike, which can be approximated by going long call options with strikes + and − , and going short two call options with strike K (what is known as a butterfly strategy).”

“To create a smooth shape of the density…__interpolation methods are used to increase the option data granularity within and outside the quoted range__. In this process, out-of-the-money option prices are first transformed into implied volatilities using the Black pricing formula…[Then] implied volatilities are interpolated and extrapolated…[Then] the sufficiently large number of interpolated volatilities is converted back into smoothed theoretical option prices…[Finally] second derivative of smoothed option prices with respect to strike is calculated numerically to obtain the probability density function.”

“[We decided on] __fitting the [quoted] volatility smile using the SABR volatility model__ [Stochastic Alpha, Beta, Rho model, view paper here], which has proven to capture the smile and extrapolate beyond the quoted range well, also on days of unexpected volatility spikes…we use the SABR model to fit implied volatility smile with sufficient granularity between strike prices. Once we convert a granular grid of smoothed implied volatilities into smoothed option prices, we numerically differentiate them with respect to their strike prices to obtain their implied densities.”

### How to use the probability information

“Tracking implied quantiles over time can be useful for monitoring changing probabilities of more or less extreme outcomes. __We can calculate quantiles of the distribution from the cumulative distribution function__…Since the quantile is the inverse CDF, we can recover strikes for specific probabilities…Quantile function returns the level of strike for a given option-implied probability; with this probability, the underlying bond futures will be at or below a certain strike.”

“To analyse uncertainty over time, we construct a constant 30-day maturity chart with different quantiles of the option-implied distribution, which show more and less probable outcomes. The difference between the quantiles __indicates the changing range of strikes that underlying futures will reach with given probabilities__.”

“From a portfolio management perspective, it may be useful to estimate option-__implied probabilities that futures prices (or yields) rise or fall by a certain amount__, or hit a specific level. We can retrieve such measures directly from the CDF in (12) by setting the strike at, for example, 1% or 5% lower than futures price F. The graph below shows the implied probability that Bund futures drop by 1% or 5%, in the next 30 days.”

“The probability that Bund futures fall by 1% in one month has been oscillating between 13% and 25% and increases with rising implied volatility. It is often more intuitive to express such loss probability in terms of change in yield in the underlying bond rather than change in the price of the futures contract. A 1% change in Bund futures price corresponds to about 11 basis point change in the yield of the underlying CTD Bund, on average. __We can use price-yield sensitivity relationship from (5) to estimate implied probability of gain or loss expressed in basis points change__.”

“Similar to risk reversal strategy, __we can calculate skewness to measure the asymmetry of a distribution__ and assess the balance of risks between expected large downward and upward moves in bond futures. For example, a positive skewness would indicate higher option-implied probability of a large downward move in yields, than a large upward move. We calculate the skewness from the mean, the standard deviation, and the third moment of the implied distribution.”

“__Another measure of the shape of the distribution and the uncertainty in the tail that complements high and low quantiles is kurtosis__, calculated from the fourth moment and the standard deviation…A kurtosis measure reveals if the variance of the distribution comes from the tails or the central part. These additional measures provide useful insights into the asymmetry and tails of the distribution over time.”