Relationship Between the VIX and SP500 Revisited

A recent post on Bloomberg website entitled Rising VIX Paints Doubt on S&P 500 Rally pointed out an interesting observation:

While the S&P 500 Index rose to an all-time high for a second day, the advance was accompanied by a gain in an options-derived gauge of trader stress that usually moves in the opposite direction

The article refers to a well-known phenomenon that under normal market conditions, the VIX and SP500 indices are negatively correlated, i.e. they tend to move in the opposite direction. However, when the market is nervous or in a panic mode, the VIX/SP500 relationship can break down, and the indices start to move out of whack.

In this post we revisit the relationship between the SP500 and VIX indices and attempt to quantify their dislocation.  Knowing the SP500/VIX relationship and the frequency of dislocation will help options traders to better hedge their portfolios and ES/VX futures arbitrageurs to spot opportunities.

We first investigate the correlation between the SP500 daily returns and change in the VIX index [1]. The graph below depicts the daily changes in VIX as a function of SP500 daily returns from 1990 to 2016.

Volatility index and S&P500 index correaltion
Daily point change in VIX v.s. SP500 return

We observe that there is a high degree of correlation between the daily SP500 returns and daily changes in the VIX. We calculated the correlation and it is -0.79 [2].

We next attempt to quantify the SP500/VIX dislocation. To do so, we calculated the residuals. The graph below shows the residuals from January to December 2016.

residuals VIX/SP500 regression
Residuals of VIX/SP500

Under normal market conditions, the residuals are small, reflecting the fact that SPX and VIX are highly (and negatively) correlated, and they often move in lockstep. However, under a market stress or nervous condition, SPX and VIX can get out of line and the residuals become large.

We counted the percentage of occurrences where the absolute values of the residuals exceed 1% and 2% respectively. Table below summarizes the results

Threshold Percentage of Occurrences
1% 17.6%
2% 3.9%

We observe that the absolute values of SP500/VIX residuals exceed 1% about 17.6% of the time. This means that a delta-neutral options portfolio will experience a daily PnL fluctuation in the order of magnitude of 1 vega about 17% of the time, i.e. about 14 times per year.  The dislocation occurs not infrequently.

The Table also shows that divergence greater than 2% occurs less frequently, about 3.8% of the time. This year, 2% dislocation happened during the January selloff,  Brexit and the US presidential election.

Most of the time this kind of divergence is unpredictable. It can lead to a marked-to-market loss which can force the trader out of his position and realize the loss. So the key in managing an options portfolio is to construct positions such that if a divergence occurs, then the loss is limited and within the allowable limit.

 

Footnotes:

[1] We note that under different contexts, the percentage change in VIX can be used in a correlation study.  In this post, however, we choose to use the change in the VIX as measured by daily point difference. We do so because the change in VIX can be related directly to Vega PnL of an options portfolio.

[2]  The scope of this post is not to study the predictability of the linear regression model, but to estimate the frequency of SP500/VIX divergence.  Therefore, we applied linear regression to the whole data set from 1990 to 2016. For more accurate hedges, traders should use shorter time periods with frequent recalibration.

Volatility Trading Strategies, a Comparison of Volatility Risk Premium and Roll Yield Strategies

Volatility trading strategies

In previous posts, we presented 2 volatility trading strategies: one strategy is based on the volatility risk premium (VRP) and the other on the volatility term structure, or roll yield (RY).  In this post we present a detailed comparison of these 2 strategies and analyze their recent performance.

The first strategy (VRP) is based on the volatility risk premium.  The trading rules are as follows [1]:

Buy (or Cover) VXX  if VIX index <= 5D average of 10D HV of SP500

Sell (or Short) VXX  if VIX index > 5D average of 10D HV of SP500

The second strategy (RY) is based on the contango/backwardation state of the volatility term structure. The trading rules are as follows:

Buy VXX if 5-Day Moving Average of VIX/VXV >=1 (i.e. backwardation)

Sell  VXX if 5-Day Moving Average of  VIX/VXV  < 1 (i.e. contango)

Table below presents the backtested results from January 2009 to December 2016. The starting capital is $10000 and is fully invested in each trade (different position sizing scheme will yield different ending values for the portfolios. But the percentage return of each trade remains the same)

RY VRP
Initial capital 10000 10000
Ending capital 179297.46 323309.02
Net Profit 169297.46 313309.02
Net Profit % 1692.97% 3133.09%
Exposure % 99.47% 99.19%
Net Risk Adjusted Return % 1702.07% 3158.54%
Annual Return % 44.22% 55.43%
Risk Adjusted Return % 44.46% 55.88%
Max. system % drawdown -50.07% -79.47%
Number of trades trades 32 55
Winners 15 (46.88 %) 38 (69.09 %)

We observe that RY produced less trades, has a lower annualized return, but less drawdown than VRP. The graph below depicts the portfolio equities for the 2 strategies.

volatility risk premium and roll yield strategies
Portfolio equity for the VRP and RY strategies

It is seen from the graph that VRP suffered a big loss during the selloff of Aug 2015, while RY performed much better. In the next section we will investigate the reasons behind the drawdown.

Performance during August 2015

The graph below depicts the 10-day HV of SP500 (blue solid line), its 5-day moving average (blue dashed line), the VIX index (red solid line) and its 5-day moving average (red dashed line) during July and August 2015. As we can see, an entry signal to go short was generated on July 21 (red arrow). The trade stayed short until an exit signal was triggered on Aug 31 (blue arrow).  The system exited the trade with a large loss.

volatility risk premium relative value arbitrage
10-day Historical Volatility and VIX

The reason why the system stayed in the trade while SP500 was going down is that during that period, the VIX was always higher than 5D MA of 10D HV.  This means that 10D HV was not a good approximate for the actual volatility during this highly volatile period. Recall that the expectation value of the future realized volatility is not observable. This drawdown provides a clear example that estimating actual volatility is not a trivial task.

By contrast, the RY strategy was more responsive to the change in market condition. It went long during the Aug selloff (blue arrow in the graph below) and exited the trade with a gain. The responsiveness is due to the fact that both VIX and VXV used to generate trading signals are observable. The graph below shows VIX/VXV ratio (black line) and its 5D moving average (red line).

volatility term structure relative value arbitrage
VIX/VXV ratio

 

In summary, we prefer the RY strategy because of its responsiveness and lower drawdown. Both variables used in this strategy are observable. The VRP, despite being based on a good ground, suffers from a drawback that one of its variables is not observable. To improve it, one should come up with a better estimate for the expectation value of the future realized volatility.  This task is, however, not trivial.

References

[1] T Cooper, Easy Volatility Investing, SSRN, 2013

 

 

 

October was Favorable for Short Volatility Strategies

Last month was particularly favorable for short volatility strategies. In this post, we will investigate the reasons behind it.

First, the main PnL driver of a delta neutral, short gamma and short vega strategy is the spread between the implied volatility (IV) and the subsequently realized volatility (RV) of returns. Trading strategies such as long butterfly is profitable when, during the life of the position, RV is low compared to IV. The graph below shows the difference between IV and RV for SP500 during the last 5 months. (Note that RV is shifted by 1 month, so that IV-RV presents accurately the spread between the implied volatility and the volatility realized during the following month). As we can see from the graph, IV-RV was high, around 4%-7%, during October (the area around the “10/16” mark). Hence short volatility strategies were generally profitable during October.

implied minus realized volatilities
IV-RV spreads of SP500

The second reason for the profitability is more subtle. The graph below shows the IV-RV spreads in function of monthly returns. As we can see, there is a high degree of correlation between IV-RV and the monthly returns. In fact, we calculated the correlation for the last 10 years and it is 0.69

IV-RV spreads v.s. SP500 monthly returns

This means that when IV-RV is high, SP500 usually trends up. This was the case, for example, during the month after Brexit (see the area around the “07/16” mark on the first graph). However, when the market trends, the cost of hedging in order to keep the position delta neutral is high. By contrast, even though IV-RV was high in October, the market moved in a range, thus helping us to minimize our hedging costs. This factor therefore contributed to the profitability of short volatility strategies.

In summary, October was favorable for short volatility strategies due to the high IV-RV spread and the range bound nature of the market.

A Volatility Term Structure Based Trading Strategy

In previous 2 articles, we explored a volatility trading strategy based on the volatility risk premium (VRP).  The strategy performed well up until August 2015, and then it suffered a big loss during the August selloff.

In this article, we explore another volatility trading strategy, also discussed in Ref [1]. This strategy is based on the volatility term structure [2].

It is well known that volatilities exhibit a term structure which is similar to the yield curve in the interest rate market. The picture below depicts the volatility term structure for SP500 as at August 31 2016 [3].

volatility term structure trading strategy
SP500 Volatility Term Structure at Aug 31 2016

Most of the time the term structure is in contango. This means that the back months have higher implied volatilities than the front months. However, during a market stress, the volatility term structure curve usually inverts. In this case we say that the volatility term structure curve is in backwardation (a similar phenomenon exists in the interest rate market which is called inversion of the yield curve).

The basic idea of the trading strategy is to use the state (contango/backwardation) of the volatility term structure as a timing mechanism. Specifically, we go long if the volatility term structure is in backwardation and go short otherwise.  To measure the slope of the term structure, we use the VIX and VXV volatility indices which represent the 1M and 3M implied volatilities of SP500 respectively.

The trading rules are as follows,

Buy VXX if 5-Day Moving Average of VIX/VXV >=1 (i.e. backwardation)

Sell  VXX if 5-Day Moving Average of  VIX/VXV  < 1 (i.e. contango)

The Table below presents the results

All trades Long trades Short trades
Initial capital 10000 10000 10000
Ending capital 177387.15 19232.01 168155.15
Net Profit 167387.15 9232.01 158155.15
Net Profit % 1673.87% 92.32% 1581.55%
Exposure % 99.44% 6.64% 92.80%
Net Risk Adjusted Return % 1683.22% 1390.38% 1704.17%
Annual Return % 46.07% 9.00% 45.05%
Risk Adjusted Return % 46.33% 135.54% 48.54%
All trades 30 15 (50.00 %) 15 (50.00 %)
 Avg. Profit/Loss 5579.57 615.47 10543.68
 Avg. Profit/Loss % 13.29% 2.29% 24.28%
 Avg. Bars Held 64.53 9.8 119.27
Winners 14 (46.67 %) 4 (13.33 %) 10 (33.33 %)
 Total Profit 208153.11 36602.85 171550.26
 Avg. Profit 14868.08 9150.71 17155.03
 Avg. Profit % 36.58% 25.17% 41.14%
 Avg. Bars Held 129.64 14.25 175.8
 Max. Consecutive 3 1 4
 Largest win 71040.59 18703.17 71040.59
 # bars in largest win 157 35 157
Losers 16 (53.33 %) 11 (36.67 %) 5 (16.67 %)
 Total Loss -40765.96 -27370.84 -13395.12
 Avg. Loss -2547.87 -2488.26 -2679.02
 Avg. Loss % -7.09% -6.02% -9.45%
 Avg. Bars Held 7.56 8.18 6.2
 Max. Consecutive 5 6 2
 Largest loss -9062.89 -8222.29 -9062.89
 # bars in largest loss 6 8 6
Max. trade drawdown -28211.89 -15668.21 -28211.89
Max. trade % drawdown -23.97% -18.20% -23.97%
Max. system drawdown -32794.28 -26555.13 -37915.18
Max. system % drawdown -50.07% -90.85% -34.31%
Recovery Factor 5.1 0.35 4.17
CAR/MaxDD 0.92 0.1 1.31
RAR/MaxDD 0.93 1.49 1.41
Profit Factor 5.11 1.34 12.81
Payoff Ratio 5.84 3.68 6.4
Standard Error 12109.91 6401.13 12526.9
Risk-Reward Ratio 1.49 0.15 1.36
Ulcer Index 11.25 42.48 8.24
Ulcer Performance Index 3.62 0.08 4.81
Sharpe Ratio of trades 0.8 0.53 0.97
K-Ratio 0.0745 0.0073 0.0683

The graph below shows the portfolio equity from 2009 up to August 2016.

volatility term structure trading strategy
Equity curve for trading strategy based on volatility term structure

The annual rerun is 46% and the drawdown is 50%. There are 2 interesting observations

  • This strategy did not suffer a large loss like the VRP strategy during the August selloff of last year
  • Long volatility trades are profitable

In the next installment we will compare the 2 strategies, volatility risk premium and roll yield, in details.

References

[1] T Cooper, Easy Volatility Investing, SSRN, 2013

[2]  Note that there is a so-called term structure risk premium in the options market that is not often discussed in the literature. The strategy discussed in this post, however, is not meant to exploit the term structure risk premium. It uses the term structure as a timing mechanism.

[3] The volatility term structure presented here is calculated based on VIX futures, which are the expectation values of 30-day forward implied volatility. Therefore, it is theoretically different from the term structure of spot volatilities which are calculated from SP500 options. Practically speaking, the 2 volatility term structures are highly correlated, and we use the futures curve in this article for illustration purposes.

Volatility Trading Strategy, a System Based on Volatility Risk Premium

Last year, we presented backtested results for a VXX trading strategy. The system’s logic is based upon the concept of volatility risk premium. In short, the trading rules are as follows:

Buy (or Cover) VXX  if VIX index <= 5D average of 10D HV of SP500

Sell (or Short) VXX  if VIX index > 5D average of 10D HV of SP500

The strategy performed well in backtest. In this follow-up post, we look at how it has performed since last year.  The Table below summarizes the results

All trades Long trades Short trades
Initial capital 10000 10000 10000
Ending capital 3870.55 9095.02 4775.53
Net Profit -6129.45 -904.98 -5224.47
Net Profit % -61.29% -9.05% -52.24%
Exposure % 100.00% 14.67% 85.33%
Net Risk Adjusted Return % -61.29% -61.68% -61.23%
Annual Return % -60.50% -8.86% -51.48%
Risk Adjusted Return % -60.50% -60.42% -60.33%
All trades 11 5 (45.45 %) 6 (54.55 %)
 Avg. Profit/Loss -557.22 -181 -870.74
 Avg. Profit/Loss % -3.35% -4.40% -2.47%
 Avg. Bars Held 24.55 8.6 37.83
Winners 6 (54.55 %) 2 (18.18 %) 4 (36.36 %)
 Total Profit 2366.97 365.38 2001.59
 Avg. Profit 394.49 182.69 500.4
 Avg. Profit % 13.75% 6.46% 17.40%
 Avg. Bars Held 32.5 7.5 45
 Max. Consecutive 2 2 2
 Largest win 1308.08 239.26 1308.08
 # bars in largest win 102 8 102
Losers 5 (45.45 %) 3 (27.27 %) 2 (18.18 %)
 Total Loss -8496.42 -1270.36 -7226.06
 Avg. Loss -1699.28 -423.45 -3613.03
 Avg. Loss % -23.87% -11.64% -42.21%
 Avg. Bars Held 15 9.33 23.5
 Max. Consecutive 2 2 1
 Largest loss -6656.33 -625.51 -6656.33
 # bars in largest loss 29 11 29

The strategy produced 11 trades with 6 trades (55%) being winners. However, it suffered a big loss during August. The graph below shows the portfolio equity since last August.

quantitative trading volatility
VRP volatility trading strategy

Large losses are typical of short volatility strategies. An interesting observation is that after the large drawdown, the strategy has recovered, as depicted by the upward trending equity line after August.  This is usually the case for short volatility strategies.

Despite the big loss, the overall return (not shown) is still positive. This means that the strategy has a positive expectancy. Drawdown can be minimized by using a correct position size, stop losses, and a good portfolio allocation scheme. Another solution is to construct limited-loss positions using VXX options.

 

Volatility Trading through VIX ETFs

It is well known that persistent biases exist in various markets. For example, in the tennis market, there exists a longshot bias. Similarly, financial markets exhibit a persistent bias called the risk premium. Formally, the risk premium is defined as

where EQ denotes the expectation value of X, a stochastic variable, in the risk-neutral world, and  EP denotes the expectation value of X in the real world. X can be, for example, commodity prices, FX rates, etc. See reference 1 for a thorough discussion of various risk premia in financial markets.

Of our particular interest is the case where X is the realized volatility (RV) of a stock or stock index. The risk premium in this case is often called volatility (or variance) risk premium (VRP).  If we substitute X in the above equation with the RV of SP500, then EQ (RV) becomes the VIX index, and the VRP becomes VIX- EP (RV).

Traders often try to exploit the VRP by trading listed options or OTC variance swaps. Another way to harvest the VRP is through trading VIX -based Exchange Trade Funds such as VXX.  In this post we explore the latter possibility.

The most difficult problem when designing a VRP-based trading strategy is that EP(RV), which is needed in order to calculate the VRP, is not observable. The best we can do is to use a quantitative method to estimate it.

Reference 2 tested various forms of EP(RV) estimate: GARCH, historical volatilities (HV). The author found that 10-day HV is the most effective. They then further smoothed out the HV by using a 5-day moving average in order to avoid whipsaws. The trading rules are as follows:

Buy (or Cover) VXX  if VIX index <= 5D average of 10D HV of SP500

Sell (or Short) VXX  if VIX index > 5D average of 10D HV of SP500

The Table below summarizes the trading strategy’s statistics. The starting capital is $10000. It is fully invested in each trade

All trades Long trades Short trades
Initial capital 10000 10000 10000
Ending capital 727774.91 -74688.89 812463.8
Net Profit 717774.91 -84688.89 802463.8
Net Profit % 7177.75% -846.89% 8024.64%
Exposure % 99.02% 6.58% 92.44%
Net Risk Adjusted Return % 7248.42% -12868.01% 8680.57%
Annual Return % 93.38% N/A 96.68%
Risk Adjusted Return % 94.29% N/A 104.58%
All trades 43 21 (48.84 %) 22 (51.16 %)
 Avg. Profit/Loss 16692.44 -4032.8 36475.63
 Avg. Profit/Loss % 12.04% 2.34% 21.29%
 Avg. Bars Held 38.79 6.14 69.95
Winners 32 (74.42 %) 12 (27.91 %) 20 (46.51 %)
 Total Profit 918340.82 94671.52 823669.31
 Avg. Profit 28698.15 7889.29 41183.47
 Avg. Profit % 18.13% 8.45% 23.93%
 Avg. Bars Held 47.47 5.92 72.4
 Max. Consecutive 9 5 8
 Largest win 198505.33 36704.45 198505.33
 # bars in largest win 78 7 78
Losers 11 (25.58 %) 9 (20.93 %) 2 (4.65 %)
 Total Loss -200565.92 -179360.41 -21205.5
 Avg. Loss -18233.27 -19928.93 -10602.75
 Avg. Loss % -5.69% -5.82% -5.10%
 Avg. Bars Held 13.55 6.44 45.5
 Max. Consecutive 3 4 1
 Largest loss -38701.76 -38701.76 -16516.3
 # bars in largest loss 4 4 41
Max. trade drawdown -300537.33 -38701.76 -300537.33
Max. trade % drawdown -46.12% -12.49% -46.12%
Max. system drawdown -331363.27 -164250.75 -300537.33
Max. system % drawdown -52.21% -95.16% -64.93%
Recovery Factor 2.17 -0.52 2.67
CAR/MaxDD 1.79 N/A 1.49
RAR/MaxDD 1.81 N/A 1.61
Profit Factor 4.58 0.53 38.84
Payoff Ratio 1.57 0.4 3.88
Standard Error 56932.62 32093.21 72795.73
Risk-Reward Ratio 1.67 -0.46 1.51
Ulcer Index 12.97 111.97 11.45
Ulcer Performance Index 6.78 N/A 7.97
Sharpe Ratio of trades 1.46 1.36 1.71
K-Ratio 0.0775 -0.0214 0.07

We observe that the short trades are profitable while the long ones lost money. The CARG is 93.4%, which is high, but so is the drawdown of -52%.  The graph below shows the portfolio equity

Volatility portfolio equity
VRP Volatility Trading Strategy

In summary, the VRP can be harvested through VIX ETF. However the drawdown is high. This strategy is viable if it is part of an asset allocation scheme. It would enhance the portfolio risk-adjusted return if we allocate, for example, 10% of our portfolio to this strategy, and the rest  to equity and fixed income investments.

References

[1 ]  A. Ilmanen, Expected Returns: An Investor’s Guide to Harvesting Market Rewards, John Wiley & Sons , 2011

[2] T Cooper, Easy Volatility Investing, SSRN, 2013

Volatility of Gold Pairs

It is commonly believed that commodity pairs are relatively easy to trade because their underlying stocks are pegged to a certain commodity market. Sometimes, however, this is not the case.

Gold stock pairs have been difficult to trade lately. One of the economic reasons is that as the gold spot declines, it approaches the production cost of around $1200 per ounce, and a small change in the spot would induce a big change (in percentage terms) in the profit margin of the producer. In other words, a small change in the spot would make a larger impact on the company’s profit and loss, thus causing a bigger fluctuation in the stock price. The big fluctuation magnifies the fundamental discrepancies inherent in the stocks of the pair. Consequently, deviations from the norm are likely due to more fundamental than statistical reasons. For quantitative traders who rely solely on statistics to make decision, it has been more difficult trading gold pairs profitably.

We can look at this problem from the options pricing theoretic point of view.  The average production cost of $1200 per ounce can be considered a put option strike. If gold spot deeps below $1200, then the stock is considered in the money. Since late 2012 the “options” are near at the money, and gold stocks behave more or less like ATM options that have greater gamma risks.

In the chart below, the solid black line shows the ratio of two gold stocks, AngloGold Ashanti (AU) v.s. Harmony Gold Mining (HMY). As it can be seen, starting August 2012 (marked with the blue vertical line), the declining in the stock prices started accelerating, and the oscillation in the pair ratio started increasing accordingly. Put it differently, the gamma has caused a greater oscillation in the pair ratio.

In the future, if gold spot trends up, the “options” will get out of the money, and the gamma risk will decrease. In this case gold stock pairs are expected to behave more regularly, thus providing better trade opportunities for statistical arbitrage traders.

Why Do Arbitrage Opportunities Still Exist?

A question arbitrageurs are frequently asked is “why aren’t the pricing inefficiencies arbitraged away?”  This is a very legitimate question.

I believe that in some areas of trading and investment, the number of arbitrage opportunity is diminishing. Take, for example, statistical arbitrage; its profitability is decreasing due to the increasing popularity of the method, competition among traders and advancement in information technology. In other areas of trading, opportunities still exist and persist. For example, in option trading, the volatility risk premium seems to persist despite the fact that it has become widely known.  Here are some possible explanations for the persistence of the volatility risk premium:

  • Due to regulatory pressures, banks have to meet Value-at-Risk requirements and prevent shortfalls. Therefore, they buy out-of-money puts, or OTC variance swaps to hedge the tail risks.
  • Asset management firms that want to guarantee a minimum performance and maintain a good Sharpe ratio must buy protective puts.
  • The favorite long-shot bias plays a role in inflating the prices of the puts.
  • There might be some utility effects that the traditional option pricing models are not capable of taking into account.
  • There are difficulties in implementing and executing an investment strategy that exploits the volatility risk premium and that is at the same time within the limits of margin requirements and drawdown tolerance.

We believe, however, that with a good understanding of the sources of cheapness and expensiveness of volatility, a sensible trading plan can be worked out to exploit the volatility risk premium within reasonable risk limits. We love to hear your suggestion.

Current Implied Volatility is Cheap

The phenomenon of the favorite long-shot bias (or volatility risk premium) can be exploited in order to construct a profitable options trading strategy. Basically, such a strategy would consist of selling overpriced options and hedging the risks using cheaper (or more precisely less overpriced) options. As can be seen from the graph below, the implied volatility (IV, yellow line) is generally higher than the historical volatility (HV, blue line). However, as of late, IV deepened below HV, which is rather unusual. A premium selling strategy in this kind of volatility environment might still profitable. However, the risks will likely outweigh the rewards.

The cheapness of IV is also confirmed by the volatility cone. As can be seen below, at the money IV, (depicted in white diamonds) is below the current HV (depicted by the white line).

A long volatility trade will have a higher probability of success in this kind of market.

The Favorite Long-shot Bias in Option Markets

The favorite long-shot bias is a phenomenon that has been studied extensively in gambling markets. A question has arisen naturally: does this bias exist in financial markets?

In a paper entitled “The Favorite /Long-shot Bias in S&P 500 and FTSE 100 Index Futures Options: The Return to Bets and the Cost of Insurance”, Hodges, Tompkins and Ziemba examined whether the favorite/long-shot bias that has been found in gambling markets (particularly in horse racing markets) applies to options markets. The motivation came from the fact that puts and calls on stock index futures represent leveraged short or long positions on the index and their behavior might have similar features to racetrack bets.

The authors found that both call and put options on the S&P 500 (and FTSE 100 to some degree) with one and three months to expiration display a relationship between probabilities of finishing in the money and expected returns that is very similar to the favorite long-shot bias in horse racing markets. In another word, they found evidence that was consistent with the hypothesis that investors tend to overpay for put options as the expected cost of insurance and call options as low-cost, high-payoff gambles.

This finding is consistent with results published by other researchers under the general framework of the volatility risk premium which refers to the fact that implied volatility in equity index options often tends to overstate realized volatility. In fact, the favorite/long-shot bias provides an alternative explanation for the existence of volatility risk premium.

For practitioners, a question arose: how to structure a position that takes advantage of the favorite long-shot bias?