Category: statistical arbitrage

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.

Brazil Beats Germany

The World Cup is over and Germany won the much coveted trophy. This World Cup will be remembered for its beautiful attacking-style games as well as one of the most crushing defeats in football history: the Host lost 1-7 to Germany in the semi final.

However, the Brazilians do not have to wait for another 4 years in order to have a chance to revenge against Germany. Their stock market has already beaten the German’s one. The chart below shows the ratio of the Brazil ETF (EWZ) v.s. Germany ETF (EWG) and we can see that the ratio is in an uptrend.

Statistically, the Brazilian market has underperformed over the last 3 years. It lost -29%, while the German market gained 45%. However, EWZ started bouncing at the beginning of this year from an oversold condition.

The bounce is also supported by fundamental factors: Brazil is an exporter of commodities and the uptrend in base metals lends support to its market recovery. Additionally, the upcoming Summer Olympics in Rio in 2016 will give a boost to the local economy.

Germany, on the other hand, is suffering from problems in Europe:   deflation threat, a weak euro, negative interest rates, geopolitical tension in Ukraine and Russia, and a possible bank failure in Portugal, just to name a few. This is a good opportunity for pair traders who want to take advantage of the divergence.

Brazil, Emerging Markets and the World Cup

There are now less than 2 weeks until the start of the 2014 FIFA World Cup of Soccer, which is the biggest sport event in the world. The event is being organized in Brazil. From an economic point of view, Brazil is one of the BRIC countries; it has underperformed the overall emerging market during the last 4-5 years.  The chart below shows the relative strength of Brazil ETF (EWZ) with respect to the emerging market ETF (EEM).

The ratio has been in a down trend for more than 4 years. We can observe, however, a rebound taking place in early 2014. Some analysts said that this rebound is supported in part by the preparation of the 2014 World Cup of Soccer and the 2016 summer Olympics.

Interestingly, Brazil is the favorite for winning the World Cup this year. It has the highest chance of winning the World Cup, followed by Argentina, Germany and Spain.

The implied probability of winning calculated from the various bookmakers odds is in the range of 20%-25%. A World Cup win can boost consumer confidence and hence the local economy in general. (We saw a similar situation before in 1998 when France won its first ever World Cup at home).

To play a potential recovery in Brazil, one can go long EWZ and hedge the downside with EMM.  If you worry about the negative impact of the host nation’s not winning the World Cup, you can hedge by laying against Brazil on a sport exchange.

Good luck and enjoy the Game!

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.

Another BRIC in the Wall

A good reward/risk trade is a one where fundamentals and technicals are aligned. We have seen two fundamentally similar countries (Canada/Australia) but they did not make a good pair for short-term trading. We have also seen two seemingly different economies (Australia/Indonesia) but that made a good pair.

There exist, however, some pairs that have good technical and fundamental relationships. India (INDL) and Russia (RUSL) are two emerging markets; they are part of the BRIC countries. The ratio chart (upper panel, below) exhibits a regular oscillating pattern, albeit somewhat volatile. The backtested equity line (lower panel) is, however, orderly upward.

Backtest results showed a winning percentage of 88% and a profit factor of 1.86, so this is a good pair to trade. As a bonus, these stocks are leveraged ETFs, hence the pair is also suitable for intra-day scalping.

 

Relative Pricing

Many popular trading strategies are based on some forms of fundamental or technical analysis. They attempt to value securities based on some fundamental multiples or technical indicators. These valuation techniques can be considered “absolute pricing”. Arbitrage trading strategies, on the other hand, are based on a so-called relative pricing. So what is relative pricing?

The theory and practice of relative pricing are derived from the principle of no arbitrage. Stephen A. Ross, a renowned professor of finance, is known for saying:

You can make even a parrot into a learned political economist—all he must learn are the two words “supply” and “demand”… To make the parrot into a learned financial economist, he only needs to learn the single word “arbitrage”.

What he was referring to is what financial economists call the principle of no risk-free arbitrage or the law of one price which states that: “Any two securities with identical future payouts, no matter how the future turns out, should have identical current prices.”

Relative pricing based on the principle of no risk-free arbitrage underlies most of the derivative pricing models in quantitative finance. That is, a security is valued based on the prices of other securities that are as similar to it as possible. For example an over-the-counter interest-rate swap is valued based on the prices of other traded swaps and not on, for example, some macro-economic factors. A bespoke basket option is valued based on the prices of its components’ vanilla options.

The principle of no risk-free arbitrage is employed in its original form in trading strategies such as convertible and volatility arbitrage. In statistical arbitrage  it is, however, relaxed; it normally involves stocks  which are similar but not 100% identical.

In summary, relative pricing based on the principle of no risk-free arbitrage is very different from absolute pricing. It is the foundation of many derivative pricing models and quantitative trading strategies.

Correlation Decreasing

Statistical arbitrage trading relies on, among other factors, the correlation between stocks. It is important to note, however, that correlation, like volatility, is not static, but time dependent and changing. Different market condition has a different level of correlation, and this has an important implication for stat-arb trading PnL.

We have been in a bull market lately, and it’s fairly common in bull markets for correlations to relax. The chart below depicts the CBOE Implied Correlation Index for SP500 stocks from November 2011 to March 2013. As we can see from the graph, the correlation is in a downtrend; it decreased from 80 % in Dec 2011 to about 55% in early March 2013.

The decrease in correlation explains in part why we have observed lots of dislocated pair relationships lately.  This dislocation increased the likelihood of pair divergence, hence one should exercise more caution when choosing pairs.

Models for Beating the Market

Edward Thorp is believed to be the first quantitative hedge fund manager. He first developed a winning blackjack strategy, and later started a successful hedge fund that exploited the pricing inefficiencies in the warrant and convertible markets. During the holidays I revisited one of his articles published in 2003 “A Perspective on Quantitative Finance, Models for Beating the Markets”. In this article Thorp recounted stories how he developed models for making money in blackjack and convertible bond hedging, respectively. According to him, developing a successful trading business  involves three steps:

  1. Idea,
  2. Development,
  3. Successful real world Implementation.

Most of the ideas (Step 1) in statistical arbitrage are more or less well known these days. To successfully build a quantitative trading business we need to complete Steps 2 and 3; we would need the following skills:

  1. Visionary,
  2. Quantitative,
  3. Entrepreneurial

Do you have the required relevant skills? If you’re missing one of these skills then learn it, improve it or team up with someone who already has it.

Happy Trading !!!

Arbitrage

Arbitrage is the process of buying assets in one market and selling them in another to profit from price differences. True arbitrage is both riskless and self-financing. In today’s modern financial markets with ultra-fast supercomputers riskless arbitrage rarely exists. Arbitrage strategies still work, but they’re often not risk-free. These strategies include (but not limited to):

  • Statistical arbitrage (pairs, basket trading): mostly involves equities and other instruments whose payoffs are linear.
  • Volatility arbitrage: involves different classes of options on a single or multiple underlyings. The payoffs of those options are not linear, i.e. they have convexities.
  • Convertible arbitrage: consists of a hybrid (equity + debt) instrument and a hedge.
  • Sport arbitrage: refers to inter-market arbitrage. It can also mean profiting from a bookmaker’s mispricing of sport matches.