Variance term premia are surcharges on traded volatility that compensate for bearing volatility risk in respect to underlying asset prices over different forward horizons. The premia tend to increase in financial market distress and decrease in market expansions. Variance term premia have historically helped predicting returns on various equity volatility derivatives. The premia themselves can be estimated based on variance swap forward rates and their decomposition into expected underlying price variance and risk premia. In particular, the variance term premia are obtained as the difference between forward swap rates and realized volatility forecasts, whereby the latter are related to a “volatility state vector”.

The post ties in with SRSV’s summary lecture on implicit subsidies, particularly the sections on popular subsidy-based strategies in the equity and volatility space.

*The below are excerpts from the paper. Emphasis and cursive text have been added. *

### What is a variance term premium?

“[*The variance term premium is*] the __price of risk for bearing realized variance shocks over [ various] horizons__… Variance term premia represent the term-structure of expected holding period returns from receiving fixed in variance swaps.”

“The estimated __variance term premia tend to increase during periods of financial distress and decrease during expansions__… Variance term premia can be as high as 5% to 10% during periods of financial distress such as the Asian financial crisis, LTCM crisis, financial crisis, and European sovereign debt crisis… This business cycle variation drives the return predictability of the model.”

“Across the variance swap curve, the paper finds that __long-end variance swap rates are primarily driven by risk premia whereas short-end variance swap rates are driven by both the quantity and price of volatility risk__… After a negative shock, long dated variance term premia tend to remain elevated while short dated variance term premia mean revert more quickly.”

### Why is the variance term premium important?

“Since the financial crisis, __rapid growth in the trading of S&P 500 index options and VIX futures__ has led to the development of separate derivatives markets where investors and firms can manage their volatility and stock market risk. As of 2016, the average open interest in VIX futures was over 414 thousand contracts per day, a more than 10-fold increase over the past decade.”

“Variance risk premia…predict the returns from selling volatility for different horizons, maturities, and products, including variance swaps, straddles, and VIX futures… __Beyond forecasting returns, the variance term premia also reflects how investors’ pricing of risk has changed over time__… Prior to the financial crisis, the term-structure of risk premia was relatively flat on average and the slope sometimes switched signs. After the crisis, long-dated risk premia increased relative to short-dated risk premia and have remained persistently high.”

“Volatility markets are integrated through the time-varying term structure of variance risk premia…There is significant evidence of market efficiency and integration across volatility markets. Synthetic variance swap rates constructed from index option prices closely track over-the-counter variance swap quotes.”

### How can variance term premia be calculated?

“__Variance swaps are over-the-counter derivatives that allow investors to hedge and speculate on volatility over different horizons__… The floating leg of a variance swap pays the realized variance of the underlying asset from the trade date until the maturity of the swap…The only cashflow occurs at maturity and is equal to the difference between the fixed variance swap rate and the floating amount of realized variance that the underlying asset exhibits over the life of the swap. The fixed rate is priced to make the swap costless to enter at the time of trade. __Variance swaps can be interpreted as a form of volatility insurance, with the fixed rate and maturity representing the insurance premium and length of coverage__. By trading variance swaps, investors give rise to a term structure of market implied volatility that embeds information about volatility expectations and risk premia over different horizons.”

*On variance swap markets and risk premia view post here.*

“The __model decomposes variance swap rates into…[1] realized variance forecasts and [2] variance term premia__. The realized variance forecasts measure the expected quantity of stock market volatility over different horizons. The variance term premia measure the expected holding period return from receiving fixed in variance swaps over different horizons.”

“I assume [*tha*t] __the systematic risk in the economy can be summarized by a vector of state variables that follows a stationary vector autoregression__. The first element is the logarithm of realized variance. The [*other*] variables can be any financial or macroeconomic variables that help to price the cross section of variance swap rates or explain the time series variation of variance swap returns.”

“Variance swap forwards decompose the variance swap curve into one-month swap rates with forward starting dates, similar to the relationship between forward rates and yields in fixed income…__Variance swap forwards are exponential affine [ time-invariant linear] in the state vector__.”

“To compute realized variance forecasts and the variance term premia in the model, variance swap rates can be decomposed [*into forecasts and premia*]…I __compute variance term premia by subtracting the realized variance forecasts from variance swap rates__…The __realized variance forecasts are obtained by [ estimating the coefficients of the coefficients of the state vector that drives volatility]__…[

*For estimation it is assumed that*] the state vector follows a monthly vector autoregression…The state variables in the empirical implementation include log realized variance and the first principal components of log variance swap rates.”