Exchange traded products with the short exposure to the implied volatility of the S&P 500 index have been proliferating prior to “Volatility Black Monday” on the 5th of February 2018. To investigate the crash of short volatility products, I will analyse the intraday risk of these products to steep intraday declines in the S&P 500 index. As a result, I will demonstrate that these products have been poorly designed from the beginning having too strong sensitivity to a margin call on a short notice. In fact, I estimate that the empirical probability of such a margin call has been high. To understand the performance of product with the short exposure to the VIX, I will make an interesting connection between the short volatility strategy and leveraged strategies in the S&P 500 index and investment grade bonds. Finally, I will discuss some ways to reduce the drawdown risk of short volatility products.
- Exchange traded products (ETPs) for investing in volatility may not be appropriate for retail investors because, to deliver the lasting performance in the long-term, these products need risk controls and dynamic rebalancing to avoid steep drawdowns and to optimise the carry costs from the VIX futures curve.
- The convexity of VIX changes and the sensitivity of changes in the VIX futures to changes in the S&P 500 index is extremely high in regimes with low and moderate levels of the implied volatility. As a result, a margin call on short volatility ETPs is more likely to occur in periods with low to medium volatility rather than in periods with high volatility.
- Without proper risk-control on the notional exposure, ETPs with the short VIX exposure are too sensitive to the intraday margin calls on a very short notice. Empirically, in the regimes with medium volatility, an intraday decline of 7% in the S&P 500 index is expected lead to 80-100% spike in the VIX futures and, as a result, to margin calls for short volatility ETPs.
- Short volatility ETNs provide with a leveraged beta exposure to the performance of the S&P 500 index, there is no alpha in these strategies. This leveraged exposure can be replicated using either S&P 500 index with leverage of 4.2 to 1 or with investment grade bonds with leverage of 9.6 to 1. All these strategies perform similarly well in a bull market accompanied by a small realized volatility and significant roll yields, yet these leveraged strategies are subject to a margin call on daily basis.
Investing in Implied Volatility
There are two main ways to invest in products and strategies linked to the implied volatility of the S&P 500 index:
- S&P 500 index options with and without delta-hedging,
- VIX futures.
In my blog post I discussed the implementation of these strategies at the level of an institutional portfolio including delta-hedging, volatility-targeting, and risk attribution. For retail investors, exchange traded products (ETPs) provide a straightforward way to get exposure to the implied volatility of the flagship S&P 500 index using VIX futures. Available ETPs include both exchange traded funds (ETFs), which require a physical replication by their providers, and exchange traded notes (ETNs), which in essence are structured products with rule-based payoffs guaranteed by their issuers.
The distinct characteristics of all volatility ETPs is the notional exposure to the VIX including:
- Long volatility exposure obtained by buying VIX futures,
- Short volatility exposure obtained by selling VIX futures.
Long volatility ETPs benefit from fast and big declines in the S&P500 index because the implied volatility is negatively correlated to the performance of S&P500 index. Because of the strong negative correlation between the long volatility ETPs and the S&P500 index, investors have been attracted to include these ETPs for portfolio hedging. However, there are significant, yet implicit, costs of holding the long volatility ETPs. These costs arise from the contago effect observed in the VIX futures when the VIX futures trade at a premium to the spot value of the VIX. As a result, the long volatility ETPs have significant roll costs incurred by buying longer dates futures at a premium to the settled contracts. In fact, the contago effect is present almost 80% of the time which makes the long VIX ETPs to be prohibitive investments in the long run.
In opposite, short volatility ETPs are expected to benefit from the contago effect by selling VIX futures and realizing roll premiums in normal periods at the expense of suffering in stressed periods from large declines in the S&P500 index.
First long volatility ETPs were launched in 2010 when the risk-aversion was high in the aftermath of the global financial crisis and the demand for hedging products was high. Short volatility ETPs were also launched around the same time. Short volatility products have proliferated in the past two years prior to 2018 because the realized volatility of the S&P 500 index has been at low level while the roll premiums have been very high. The combined effect of high roll premiums along with low realized has resulted in large yet smooth returns on short volatility ETPs in 2016 and 2017. Without doubt, the smooth performance of past two years has resulted in crowding in these products prior to the crash in February 2018.
Ironically, the long-term performance of both long and short VIX ETPs is very poor. In the table below I report the total and annualized performance on VXX, which is a major ETP providing long exposure to constant maturity one-month VIX futures, and on XIV, which is a major ETP shorting VIX futures, from the inception of XIV on 12th March 2010 to 8th February 2018. The starting prices for XIV and VXX ETNs are adjusted for splits. In comparison, I also provide the % change in the VIX and one-month VIX future over the same period.
|Asset||Exposure to VIX||12-Mar-10||08-Feb-18||Total Return||Annual Return|
We see that, in fact, the long VIX ETN – VXX – fared the worst and wiped out all the equity despite increases in the VIX and one-month VIX future over the same period. The reason is simple – the roll costs in FIX futures are prohibitive in the long term to be constantly buying VIX futures. The short VIX ETN – XIV – fared better than its long counterpart notwithstanding 96% wipe-out on the 6th February. It is clear that, while shorting VIX futures is profitable almost 80% of the time because of sizable roll benefits, the strategy must dynamically adjust allocations to avoid steep drawdowns.
Constant maturity one-month VIX futures
The VIX is an aggregated measure of one-month implied volatilities on the S&P 500 index. The VIX cannot be traded directly with the trading possible only using futures and options on the VIX. At its settlement date, the VIX futures settles into the VIX spot value derived from the index options with maturity of one month. As a result, the VIX futures is a derivative contract with value linked to the S&P 500 index and its implied volatility. I published a technical paper back in 2008 on the subject of valuation of VIX futures and options.
The most common underlying for volatility ETPs is the constant maturity one-month VIX futures. The constant maturity futures is a basket of two futures with deterministic weights that change daily. At the start of the new settlement month on the third Wednesday of each month, the constant maturity futures is allocated 100% to the front month VIX futures. The exposure is then shifted linearly every day towards the second month futures proportional to the number of days to the next settlement Wednesday.
The key advantage of using the constant maturity VIX futures is that it provides a homogenous exposure to the VIX futures rolls and the sensitivity to the S&P500 index. Indeed, the VIX index is the most volatile and sensitive to the S&P500 index. The front month VIX futures becomes also more volatile as it moves closer to its settlement day. Using constant maturity VIX futures makes the performance of the investing strategy less sensitive to the VIX roll dates.
Throughout the note I will apply the notation “VIX1m futures” for the constant maturity one-month VIX futures.
What happened on Volatility Black Monday?
The key short-term risk of short volatility ETPs arises from the high sensitivity to steep negative performance on the S&P 500 index. In the next section I will develop a statistical model to analyse the intraday risk of short volatility ETPs. Now I will start by giving an account of events that happened on 5th February 2018.
The figure below illustrates the key market variables and their joint intraday dynamics (for the convenience of illustration I plot the negative performance of the S&P500 index and short volatility ETP XIV by reversing the sign of their negative performance):
- The blue dotted line is the intraday percentage performance of the S&P500 index futures from Friday’s close. At the time of NYSE close at 16:00 (NYC time), the negative performance stood at 4.02%.
- The red line is the relative performance of the constant maturity one-month VIX futures inferred from the intraday price of the February and March VIX futures contracts. At 16:00, the VIX1m futures increased by about 38%.
- The orange line is the prediction of the performance of the VIX1m futures using the regression model which applies the intraday performance of the S&P 500 index future to predict the performance of the VIX1m futures, as I describe in the next section. At 16:00, given the negative performance of the S&P 500 index of 4%, the model predicted change of about 31% in the VIX1m futures which is quite close to the actual change of 38%.
- The green dashed line is the intraday negative performance of the short volatility XIV ETN. At 16:00, the realized negative performance on XIV stood at 14.32%.
We see that up until 12:45, all variables moved in sync: the S&P index declined by 0.5% while XIV lost about 2.6% that was close to both the increase in the VIX1m futures of 2.2% and the predicted value using the S&P 500 index intraday performance. From about 13:00, the decline in the index started to accelerate, which was followed by the fast acceleration of the VIX1m futures. Meanwhile, the performance of the VIX1m futures started to exceed the predicted change using the intraday performance of the S&P 500 index. Surprisingly, from about 12:45 to 15:00, the negative performance on XIV ETN rather followed the predicted performance of the VIX1m futures but not the actual intraday performance on the VIX1m futures.
The divergence between XIV ETN and the VIX1m futures became most extreme around 15:10 when the S&P 500 futures dropped by 4.2%, the VIX1m futures increased by 45% with its predicted value rising to 34% while XIV ETN lost only 17% with the loss well above the fair estimate using the realized performance of VIX1m futures.
While the market somewhat recovered by 15:30, the S&P 500 index losses started to accelerate again up until the close at 16:00. In the period from 15:30 to 16:00, the loss on the index amounted to 4.23%, while the respective loss on the VIX1m futures and its statistical prediction amounted to 45% and 35%, respectively. Yet XIV ETN finished the trading session with about loss only of 15% well above the loss inferred by the VIX1m futures that XIV ETN is supposed to track.
The most interesting and yet mysterious incident happed right after cash market close at 16:00 while the futures market was still opened. From 16:00 to 16:15 the intraday gain in the VIX1m futures reached from 45% to nearly 100% indicating the default event for XIV ETN and other short volatility ETP. At the same time the loss on the S&P 500 index futures increased only by about 1.25% to 5.5%. As a result, the more than double spike in the VIX1m futures during this 15-minute period cannot be accounted by the decline in the S&P 500 index futures.
I can provide the following explanations for the erratic behaviour observed on the Volatility Black Monday.
- Since XIV ETN is an exchange traded note, there is no simple way to arbitrage the intraday discrepancies between the market price of the product and its fair value given market prices of product constituents. If XIV ETN were like a standard ETF that tracks an easily replicable and liquid basket of securities, the designated market participants could simultaneously sell short XIV and sell the VIX1m futures. This replicating portfolio would be then unwound at fair market prices using the redemption mechanism supported by the provider of this ETF product. The redemption mechanism is not supported by providers of ETNs so that there is no straghtforward way to correct for the intraday arbitrages between ETNs and their underlying baskets of securities.
- The provider of XIV ETN could be using other market participants to hedge the provider’s exposure to the volatility ETNs through swap agreements. Swap counterparties could have hedged their exposures through the VIX futures directly in line with their swap agreements linked to the actual performance of VIX1m futures so there was no one to force the convergence of XIV to its fair value.
- Retail investors were complacent in monitoring the futures market and reducing their exposure to XIV at favourable prices before the market close. The selling pressure from investors could have forced the market price XIV ETN downward closer to its fair value.
- Prior to and right after the market close at 16:00, all volatility ETPs need to rebalance their exposures to the VIX futures in line with their net asset value (NAV) outstanding at the end-of-day close prices. Given that the VIX 1m futures increased by 45%, the amount of the rebalancing was substantial because, on the one hand, short VIX ETPs needed to buy VIX futures to cover their short positions in line with their reduced NAVs and, on the other hand, long VIX ETPs needed to buy VIX futures to increase their exposure to the VIX futures in line with their increased NAVs. We could assume that product providers may have been delayed or underestimated their rebalancing. The double sided-demand from both the long and short sides could have been large indeed and the feedback effect could have lead to further spike in the VIX futures right after market close at 16:00.
In the retrospect, I can list the key reasons behind the crash of short volatility ETNs:
- The lack of the arbitrage mechanism enforcing the convergence to the true value (XIV diverged by almost 30% right prior the market close at 16:00).
- The rebalancing mechanism of both long and short VIX ETPs is poorly designed with marginal feedback effect (the double-sided demand after hours lead to a significant spike in the VIX1m futures leading to its intraday increase to 100%).
- The leverage and the volatility inherent in short volatility ETNs was underestimated in the face of potential margin calls on very short notice which could also happened (and it did happen in fact) outside of regular market hours with insufficient liquidity to absorb the demand.
Modelling the intraday risk of VIX futures
The this more technical section I will investigate is the following aspects.
- How it is likely that 4% decline in the S&P 500 index could lead to 45% and 100% spike in the VIX1m futures from the statistical point of view?
- What is the cut-off decline in the S&P 500 index that would lead to the marging call and the liquidation of short VIX ETPs?
- What is the historical likelihood of this type of liquidation event?
Sensitivity to S&P 500 index
The S&P 500 index is the key driver for short-term changes in the VIX and VIX futures while the volatility clustering and mean-revision being the drivers for the medium and long-term dynamics. Piotr Karasinksi and I published a technical paper describing the continuous time model, named as the beta stochastic volatility model, which links changes in the implied volatility to changes in the stock index. Here I will apply a discrete version of this model for daily observations. For my analysis I use the data from March 2004 until the 8th of February 2018, however, to estimate all regression models I use the data up to the 2nd of February to mitigate the impact of the large outlier observed on the 5th of February on the model predicative ability.
To model the sensitivity of the VIX futures to the S&P 500 index, I estimate the following regression linking % daily returns on the VIX1m futures to % daily returns on the S&P 500 index and their square:
VIX1m EOD Return = Beta * (S&P500 EOD Return) + Convexity * (S&P500 EOD Return)^2
The estimate of the Beta parameter is -2.4 and the estimate of the Convexity parameter is 7. These estimates imply that for -10% daily decline in the index, the VIX1m futures is expected to increase by 31% which is attributed to 24% from the linear beta term and to 7% from the quadratic convexity term. The quadratic term is important because the volatility changes asymmetrically for large negative changes in the S&P 500 index.
In the figure below, I illustrate the actual data and the model prediction. Generally, the regression model fits the data adequately with high explanatory power of about 60%. However, the model does not account for the most important outlier – the spike of nearly 100% that occurred on the 5th of February 2018 when the index declined by 5%. In fact, the simple regression model implies that the S&P 500 index must experience a daily decline of nearly 26% for the VIX1m to increase by 100%. Are we missing something?
VIX futures sensitivity conditional on the volatility level
It turns out that an important variable in the sensitivity analysis is the level of the volatility. In practice we frequently observe large and rapid changes in both the index and its implied volatility with magnitudes well beyond ranges predicted using recent realized volatilities.
First, I will incorporate the conditioning on the volatility level to improve the simple regression model. Specifically, I assume that there are four regimes of the volatility with equal probabilities of 25%. Then using the daily data for VIX 1m futures, I find the 25%, 50%, 75% quantiles that serve as cut-off levels for these regimes. Each of the four regimes then has the equal 25% empirical probability. The figure below illustrates the labaling of regimes and their cut-off levels.
Second, I will focus on the intraday changes of the S&P 500 index and the VIX1m futures. The reason is that, as a rule, the intraday changes are larger in the magnitude than end-of-day changes and, as result, it is important to account for potential margin calls intraday and outside of market hours. Because both the S&P 500 index futures and VIX futures trade during the same extended market hours and there is strong negative correlation between the two, I expect the intraday highs in VIX1m futures to follow closely intraday lows in S&P 500 futures.
I apply the following regression with the intraday data:
VIX1m High Return = Beta * (S&P500 Low Return) + Convexity * (S&P500 Low Return)^2
Where the High and Low returns are defined using the intraday high and low Prices, respectively:
High Return=Day High Price / Previous Close -1
Low Return=Day Low Price / Previous Close-1
The high/low returns are allocated to the four samples, with the same size, according to the level of the VIX1m future close price on the previous trading day. Finally, I apply the above regression model for each of the four samples to estimate the model parameters conditional on the volatility regime.
In figure below, I show the estimated beta across different regimes. The regime of volatility does not appear to have a strong influence on the first order sensitivity of VIX futures to the S&P 500 index. Compared to the estimate of -2.4 for unconditional regression with daily returns, there is stronger sensitivity only in the third regime with moderate high volatility.
In the figure below, I show the estimated convexity of changes in the VIX1m futures to the squared returns on the S&P 500 index across the four regimes of volatility levels. In a stark contrast to the linear sensitivity measure, the convexity measure is strongly dependent on the regimes of volatility. The convexity is extremely high in the low or moderate low regimes implying that the VIX1m futures is expected to change very fast even for relatively small declines in the S&P 500 index. For an example, -10% intraday drop in the S&P 500 index implies that only due to the convexity effect the VIX1m futures is expected to change by 109% and 133% in the low and moderate low regimes, respectively. In opposite, the convexity becomes insignificant in high volatility regimes with the linear sensitivity to changes in the S&P 500 index driving the response in the VIX1m futures.
Predictions using conditional regression
In the figure below I illustrate the actual intraday high changes in the VIX1m futures vs intraday low changes in the S&P 500 index futures and the prediction by the estimated regression model conditional on the value of the volatility observed on the previous close day. We see that, because of the strong convexity effect in the low and moderate low volatility regimes, an intraday decline of -7% is expected to result in the intraday spike in the VIX1m futures of more than 80% which is the level of liquidation of short volatility ETPs.
It is clear from the data that the large declines in the S&P 500 index in high volatility regimes lead to muted changes in the VIX1m futures because the convexity effect is insignificant in these regimes. I will discuss the three outliers in the data:
- 5th of February 2018: Volatility Black Monday is the most outstanding realization which I already discussed in depth in the last section. This realization occurred in the regime with moderate low volatility.
- 24th of June 2016: BREXIT related overnight spike in the VIX1m futures also occurred in the regime with moderate low volatility. The model prediction, given the intraday low in the S&P 500 index, is actually quite accurate to the actual realization.
- 6th of May 2010: the flash crash occurred in the regime with high volatility and it is outside of model prediction in that regime.
Historical estimate of likelihood of Volatility Black Monday
I established that only steep intraday changes in the S&P 500 index in low and moderate low volatility regimes can lead to extreme changes in the VIX1m futures. In the figure below I illustrate again the regression conditional on volatility levels focusing on the tail events only. I mark the realizations in four different colours according to the volatility regime in which these realizations occurred.
The model implies that the decline in the S&P 500 below the threshold of about -7% is expected to lead to the spike in the VIX1m futures by more than 80% which is then would lead to liquidations of short volatility ETPs. The probability of S&P 500 intraday below the threshold is about 0.37%. This empirical estimate is obtained from 13 realizations out of 3490 observations using the sample starting from 2004. I also used S&P 500 data with longer time series and have not established significant deviation of this tail estimate.
The estimate of 0.37% would indicate that a default event is expected to occur every year. This is the upper bound because, on the one hand, the default event is highly conditional on low and moderate low volatility regimes. If we assume that the regime occurrence and tail events in the S&P 500 intraday returns are independent, then the fair estimate of the default event reduces in half to 0.18% with expected occurrence of once every second year. On the other hand, the clustering of the realized volatility strongly implies that tail negative returns on the S&P 500 index are only likely to occur in high volatility regime.
Comparison to leveraged strategies in S&P 500 index and IG credit
Clearly short volatility ETPs provide a leveraged exposure to the performance of the S&P 500 index. How does the performance of short VIX ETP compare to the performance of other bullish leveraged strategies with similar risk profile?
I apply the volatility targeting to make a comparison between the following strategies:
- Short VIX1m Futures: To obtain the performance on this strategy from 2006, I use Bloomberg replication index with the ticker SPVXSPI that tracks the short constant maturity VIX1m futures in the same way as XIV ETN.
- The leveraged strategy to the S&P 500 index: I apply SPY ETF to get the time series of this strategy.
- The leveraged strategy to investment grade (IG) credit bonds: I apply LQD ETF with to estimate the performance on this strategy.
In table below, I provide the estimated in-sample volatility of the underlying assets. In practice, the volatility targeting is implemented on a regular basis (daily, weekly, etc) using trailing estimate of the recent realized volatility. However, to keep the analysis simple, I just match the in-sample volatility of the three strategies. Also, I assume that leverage can be financed at zero rate (on the level of an institutional portfolio, it can be thought as a funded leveraged overlay strategy).
For the leveraged strategies, the rebalancing of is daily following the standard practice of leveraged ETFs. It is well known that the performance on the leveraged ETFs is reduced by volatility drag estimated as follows:
Volatility drag=0.5*(Leverage- Leverage^2)* Volatility^2
In table I also report the estimated volatility drag using the in-sample volatility. The volatility drag is detrimental to the leveraged strategies with daily rebalancing
|Strategy||Underlying Asset||Leverage||Volatility||Volatility drag p.a.|
|Short VIX||VIX1m Future||-1.00||67.00%||-45%|
|Leveraged IG credit||LQD||9.57||7.00%||-20%|
In figure below I show the performances of leveraged strategies from the 3rd of January 2006 to the 8th of February 2018. It is remarkable how the short VIX strategy was able to withstand the volatility drag because of the high roll yields from the VIX futures. The performance on the leveraged equity and credit strategies has been pale in comparison to the short VIX strategy up until the volatility crash on the 5th of February 2018.
In figure below, I show the running drawdowns on the three strategies. It is remarkable that in 2008 the short VIX strategy had a relatively smaller drawdown than other leveraged strategies and it recovered faster than the other two. This fast recovery of short volatility strategies is a typical one because the volatility premium is very high right after the crisis. The key point here is to be able to withstand the drawdown and high market volatility. Once again, I emphasize the need for a robust risk-controlled implementation of short volatility strategies.
Another interesting observation that the drawdown of -96% on the short VIX ETN that occurred on 5th February 2018 is similar in magnitude to the drawdown of -92% that occurred during the financial crisis in 2008. What is astonishing is the speed of this drawdown.
Finally, while the two leveraged strategies had a drawdown of nearly -99% in 2008, they did not default in the sense the outstanding equity was enough to cover the intraday lows in their leveraged portfolios.
As I mentioned, the volatility drag is detrimental for leveraged ETFs especially in periods with high realized volatility. In the below figure I show the performances from the 9th of March 2009 when the current bull market started. In this respect, the performance of the strategy shorting VIX futures tracks very closely both the leveraged equity and credit strategies, yet loosing years of gains in the crash on the 5th of February.
In the hindsight it is clear short volatility ETNs provide only a leveraged beta exposure to the S&P 500 index, there is no alpha in these strategies! These strategies perform the best in a bull market accompanied by a small realized volatility and significant roll yields.
Improving volatility strategies using risk controls
As we saw from empirical data, robust risk controls are absolutely necessary to mitigate the risk of short volatility strategies and products. I will discuss the two relatively simple ways to implement the risk controls: the volatility targeting and the signal filtering.
The reasoning behind the volatility targeting is to limit the risk of the strategy to the pre-specified target level. Because the realized volatility of the VIX1m futures is about 60% in average and the volatility of, say, the S&P 500 index is about 15%, the volatility targeting provides an adequate way to rescaling the strategy performance to improve the decision making and allocations in the portfolio context.
To implement the volatility targeting, I track the recent realized volatility of the VIX1m futures and match the exposure to the volatility-target. The model for volatility measurements and forecast can be implemented using either a very simple close-to-close estimator or more reliable estimators that apply both recent intraday and long-term data. For the below illustrations I use volatility targeting with weekly rebalancing and GARCH-type model for weekly forecast of the volatility.
The reasoning behind the signal filtering is to control the ratio of the expected reward to risk and to aim at reducing allocations when the reward-risk ratio is poor.
Was the crash of short volatility strategies on the 5th of February predictable beforehand? Well, let us look at the term structure of VIX futures in January/February 2018 illustrated in the below figure.
We see that at the beginning of January 2018, the VIX futures were in a steep contago, which is the most frequent state empirically. When the futures curve is in a steep contago, the short VIX strategy is highly profitable if the realized volatility remains low. However, from the mid of January right up until the week of the crash, the realized volatility started to increase and the VIX spot also increased relative to the VIX futures. During the second part of February and the first two days of February, the VIX term structure flattened while the expected volatility of the VIX1m futures increased. As a result, for almost two weeks prior to the crash the expected reward to risk ratio was poor so that a short strategy with filtering should have reduced its exposure.
As an example, I use my own signal filtering which I developed and applied well before February 2018 to dynamically rebalance volatility strategies.
Performance of risk-controlled strategies
In the two figures below, I show the performances and running drawdowns of strategies sorting VIX1m futures with volatility-targets of 10%, 20%, and 30% and the strategy with filter and vol-target of 10%, all in comparison to the unconstrained short strategy from January 2007 to the 8th of February 2018.
We see that the volatility targeting amounts to controlling the drawdown. All strategies with volatility targeting performed considerably better than the naïve strategy shorting the VIX1m futures.
Below I show the realized Sharpe on the strategies versus their realized maximum drawdown.
Below I show the 2018 YTD performance on all the strategies with volatility targeting. The volatility targeting helped to reduce the drawdown in half and in third for the strategies with vol-target of 30% and 20%, respectively. The strategy with the filtering avoided the loss because it deleveraged in the mid of January when the VIX futures curve became flat.
- Artur Sepp (2017) “Allocation to systematic volatility strategies using VIX futures, S&P 500 index puts, and delta-hedged long-short strategies”, https://artursepp.com/2017/09/20/allocation-to-systematic-volatility-strategies-using-vix-futures-sp-500-index-puts-and-delta-hedged-long-short-strategies/
- Piotr Karasinski, Artur Sepp (2012) “Beta stochastic volatility model”, Risk, October, 67-73, http://ssrn.com/abstract=2150614
- Artur Sepp (2008) “VIX option pricing in a jump-diffusion model”, Risk, April, 84-89, http://ssrn.com/abstract=1412339