The variance risk premium manifests as a long-term difference between option-implied and expected realized asset price volatility. It compensates investors for taking short volatility risk, which typically comes with a positive correlation with the equity market and occasional outsized drawdowns.

A recent paper investigates a range of options-related strategies for earning the variance risk premium in the long run, including at-the-money straddle shorts, strangle shorts, butterfly spread shorts, delta-hedged shorts in call or put options, and variance swaps. Evidence since the mid-1990s suggests that variance is an attractive factor for the long run, particularly when positions take steady equal convexity exposure. Unlike other factor strategies, variance exposure has earned premia fairly consistently and typically recovered well from its intermittent large drawdowns.

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

Also, the variance risk premium in the summary has been defined as the difference between implied and realized price volatility, not the other way round, as in the paper.

This post ties in with this site’s summary on implicit subsidies, particularly the section on volatility markets.

### Basics of the variance risk premium

“The __variance risk premium is the deviation between… the option implied (risk-neutral) variance…[and] the expected (physical) variance__ realized over the life of the option…More loosely speaking, the variance risk premium is the expected return on the ‘variance factor’…Approaches to earn [this] factor premium can differ greatly. What they have in common, however, is that they provide exposure to variance changes.”

“The variance risk premium is a well-known feature in options markets and exists across asset classes and countries. There is strong evidence, in particular, of a [positive] variance risk premium in options on equity indexes [meaning that implied price volatility tends to be higher than realized volatility].”

“The performance of variance strategies is closely linked to the market for at least two reasons. First…the index is the underlying of the derivatives positions…Second, __variances tend to increase when the market goes down__…The structure of variance risk involves __rare, high-impact events__.”

### Basic strategies for gaining variance exposure

“[We] systematically __compare different design elements of variance-based investment strategies in terms of their suitability for long-term investors__…(i) The payoff problem: Which payoff profiles are appropriate? Which instruments should be used to create them? (ii) The leverage problem: Which risk level should be chosen? How can the ex-ante variance risk of different strategies be measured and compared? (iii) The finite maturity problem: Which maturities of derivatives should be chosen? When and how often should positions be rolled over?…Answering these questions is highly relevant to investors, especially institutions such as pension funds, insurance companies and university endowments.”

“The strategies under study are different ways to earn variance risk premia. More broadly, they are examples of factor investments because ‘stock index variance’ is just one among many equity-based factors.”

“A short position in an **at-the-money (ATM) straddle** [is an] intuitive approach to selling protection against rising variances. The short straddle limits market exposure because it __combines a short call, with negative delta, with a short put, [which has] positive delta. It consists of only two ATM instruments__, thus limiting transaction costs. The short straddle’s stylized payoff profile is shown in part (a) of [the figure below]. This payoff is generally consistent with an instrument showing variance exposure. The risk here is that large negative or positive price movements in the underlying could result in large negative returns–even medium price movements already lead to losses.”

“A first idea __to reduce the risk of the straddle is the short strangle, which involves shifting option strike prices from ATM to out-of-the-money (OTM), thereby avoiding losses in case of medium price changes__ in the underlying. It also reduces the magnitude of losses in case of large price moves in the underlying, compared with the straddle. Such a payoff profile is depicted in part (b) of [the figure below]. The strangle’s disadvantages include, first, that OTM options have less variance exposure than ATM options, and second, that the maximum payoff is lower than for a straddle. Finally, large negative returns can still occur since the payoff function has no lower bound.”

“To counter the risk of extreme losses, one could add a floor to the payoff profile__. By adding a long OTM call and a long OTM put, a straddle becomes a butterfly spread__, as shown in part (c) of [the figure below]. Limiting the downside risk, however, also comes with drawbacks. Since two long option positions enter the portfolio together with short positions, the portfolio’s overall variance exposure is reduced. Moreover, the additional long positions in calls and puts incur costs—both transaction costs and costs of capital for the option premiums.”

“A further approach is to sell **delta-hedged call or put options**, i.e., __puts or calls hedged with positions in the market index__. Delta-hedged options have a similar payoff structure as a straddle. Therefore, they have similar advantages, i.e., limited correlation with the market factor and a portfolio made up of only two instruments, but also similar disadvantages, i.e., potentially large downside risk. However, the long-term performance of delta-hedged options may differ considerably from that of a straddle strategy, since they contain different instruments. Differences in performance may be due to a different potential for leverage or to different transaction costs and margin requirements.”

“Finally, an investor can gain variance exposure through a **variance swap **[whose __payout is linked to the difference between realised variance and a strike level__], which is the most direct way to earn the variance risk premium. Variance swaps may also limit correlation with the market factor, since they are initially delta-neutral by construction. However, a variance swap can be viewed as a fairly complex option portfolio with potentially high transaction costs and leverage constraints. Moreover, since variance is calculated as the squared difference from the mean, a variance swap could be prone to extreme losses if the realized variance reaches a peak.”

### Empirical evaluation of variance strategies for U.S. equity

#### General points

“In a study of S&P 500 index options, we assess…variance risk premium strategies empirically. Our __data sample ranges from January 1996 to June 2021__, therefore including the two most consequential stock market crashes in recent decades…We use realistic assumptions about transaction costs and CBOE margin requirements for option positions.”

“For transaction costs, we use the ask price for option purchases (long positions) and the bid price for option sales (short positions). However, since institutional investors can probably trade at terms better than the quoted spread…we use an effective spread of 25% of the quoted spread as our baseline case…We assume that transactions in the underlying, necessary for delta hedging, are possible at a quoted bid-ask spread of 3 basis points.”

“Overall, our results show that variance strategies can be attractive to the long-term investor if properly designed…__The variance factor translates into an attractive factor strategy for long-term investors,__ both as a stand-alone factor and as a complement to the market investment, despite being correlated with the market… __Variance strategies…consistently earn premiums over the entire study period__. The latter distinguishes variance strategies from other factor strategies, which have not generated premiums since the 2008 financial crisis…Correlation with the market is significantly positive, but varies across strategies due to different payoff profiles.”

#### Lessons from the maximum-exposure strategies

“We calculate the maximum possible exposure of trading strategies by levering the positions until the capital is fully used to buy long positions and provide margins on short positions.”

“Maximum exposure strategies differ greatly in terms of their return and risk profile. Some strategies exhibit risk levels that most investors would find excessive.”

“[The figure below shows] the accumulated wealth over time, as created by different strategies…[and] shows large differences among strategies. One group of extremely risky strategies consists of the straddle and the strangle. These strategies reach very high wealth levels by 2007 but collapse during the financial crisis in October 2008. While the straddle recovers quickly from this shock, the strangle barely survives and does not have enough capital to recover quickly. It is eventually hit by the market turmoil due to the Covid-19 pandemic and is forced into bankruptcy in March 2020. The straddle also takes a massive hit from the pandemic…The delta-hedged put and call strategies display different behavior. The delta-hedged put fluctuates more and shows a steeper trend. In contrast, the delta-hedged call has a very smooth path but does not seem to generate enough exposure for a significant upward movement, especially after the financial crisis. Finally, the variance swap combines a clear upward trend with a very smooth path, except during the most extreme months of the financial crisis and the Covid-19 pandemic.”

“[The table below] shows, by means of summary statistics, how the different strategies perform. Panel A reports the sample moments of monthly returns and Panel B describes downside risk. The first three measures of downside risk (VaR, CVaR, Max Loss) take a monthly perspective and refer to (potential) losses in the following month. This view is sufficient for a monthly investment horizon. A long-term investor, however, cares about the characteristics of the entire path. In particular, the ability of a given strategy to recover from an intermediate downturn is crucial. Therefore, Panel B offers four different drawdown statistics. The maximum drawdown (Max DD) is the maximum percentage loss of a strategy from its current maximum value to a trough. The average drawdown (Average DD) shows how far (on average over all months of the 25-year period) a strategy is from its previous maximum. Drawdown length indicates how many months it takes to reach a new wealth maximum at a given point in time. For this measure, we report the maximum number of months (Max DD Length) and the average (over all months of the 25-year period) number of months (Average DD Length).”

“[The table above] reveals several key differences between the strategies. First, compared to the strangle, the straddle’s returns have a higher mean, a lower standard deviation, are less skewed to the left, and have a lower kurtosis. The straddle is less risky than the strangle by all measures of downside risk. Therefore, under maximum exposure, replacing a straddle with a strangle does not reduce risk, especially for large downturns. Both straddle and strangle strategies are highly speculative, with high mean returns but massive downside risk. Second, for insight into the impact of floors we compare straddles to butterfly spreads and strangles to condor strangles. Floors result in less negative skewness and less kurtosis, but they also lower mean returns. Because of this cost, such floors may not be the best way to reduce risk. Indeed, the butterfly spread and the condor strangle display much more downside risk than the delta-hedged put, the delta-hedged call, and the variance swap. Third, the variance swap’s monthly returns show by far the most negative skewness and the highest kurtosis of all variance strategies. However, these characteristics do not necessarily imply high downside risk, as standard deviation and path dynamics also matter. Although the variance swap experiences some heavy losses (the maximum monthly loss is almost 37%), the relatively short drawdown length shows that the strategy can recover quite well from such losses.”

#### Lessons from the equal-exposure strategies

“Equalized exposure uses a model-free measure of convexity (gamma).”

“Equal exposure strategies also differ from each other, but much less than maximum exposure strategies.”

“As expected, __standardizing strategies by ex-ante gamma leads to a more homogeneous picture__, with the following key results. First, there is no clear evidence that a strangle helps reduce the risk of a straddle, as strangle returns are even more left-skewed and leptokurtic. In terms of downside risk, the different measures point in different directions. The straddle has a less negative VaR and CVaR, but a higher maximum loss. The maximum drawdown (DD) and maximum DD length are lower for the strangle, but the straddle offers lower average DD and average DD length. Second, while the butterfly’s and condor’s floors help with monthly skewness and kurtosis, they are not effective in reducing downside risk. Third, the delta-hedged put, delta-hedged call, and straddle have very similar paths and distributional properties. So if these behave differently under maximum exposure, it is due to their different potential for leverage. Indeed, the delta-hedged call requires going long in the index, which consumes considerable capital. In contrast, the delta-hedged put involves shorting the index, with only margin requirements. The delta-hedged put therefore has much greater leverage potential compared to the call.”

“A (delta-hedged) straddle requires a very small investment (long or short) in the index. In addition, the CBOE’s margin requirements for a short straddle are much lower than the sum of the margin requirements for a short call and a short put. Therefore, straddles can achieve much higher gammas than the other strategies. Fourth and last, the variance swap has the highest return of all the variance strategies while showing the lowest downside risk for the majority of the risk measures.”