Categories

Archives

The January Barometer

The January Barometer, sometimes also referred to as January effect (although the January effect is a different thing), is the theory according to which market performances during the month of January can be used to predict the trend for the rest of the year. The January Barometer theory is often summarized by the saying: “As goes January, so goes the year”. The practical implications are rather straightforward: if the asset class under examination has a positive return in January, the theory suggests that it will be a bullish year while a negative performance should predict a bearish trend for the upcoming months. The January Barometer theory is based on the assumption that many fund managers and institutional investors, particularly those who are interested in medium–to–long term investments, at the beginning of each new year, will tend to place their positions already discounting their view on the next 6 to 12 months. Consequentially, the theory suggests that, should the view be negative, portfolio managers will position themselves on the short side of the market at the beginning of January. Otherwise, should they think the price action will go up in the next 6 to 12 months, they will go long. Hence, the theory is based on the fact that the buying or selling pressure generated by the amount of money allocated in the market by big players in January should move the price in the direction of their forecasts. This is the theory but as Albert Einstein stated: “In theory, theory and practice are the same. In practice, they are not”. The present research aims to investigate and study the reliability of the January Barometer theory in order to assess, under a probabilistic point of view, what are the chances to actually earn consistent profits if applied to financial markets. The research has been carried out on major equity indices in the world that have been, in turn, subdivided for geographical location: North America (S&P500, NASDAQ Composite), Europe (DAX30, FTSE100), AustralAsia (Nikkei225, ASX200) and Emerging Markets (Hang Seng, Bovespa, BSE Sensex). The dataset that has been used consists of index prices ranging from January 2004 until December 2014 implying 11 years worth of data. The back testing analyses that will be exposed and commented are simply aimed to understand if January’s market returns matched, if not in size at least in sign, with yearly ones. Hence, the combination negative return in January/negative yearly return as well as the combination positive return in January/positive yearly return will all be calculated as success cases. On the other hand, all non–matching returns will be treated as a failure of the theory. The results of the analyses will be presented and explained by geographical location, hence, the first equity indices that will be analyzed are the North American ones:

yearly returns: S&P500, NASDAQ Composite

The above reported table shows, in the first column, January’s lognormal returns calculated for the S&P500 while the second one lists the total return for each year. The right hand side of the table provides the same calculations but based on the NASDAQ Composite index. First of all, it is worth noting that, over the last 11 years, the average return for the S&P500 index has been +4.89% while the NASDAQ Composite yielded an average of +6.78%. Secondly, both equity indices have experienced similar fluctuations during the crises but the NASDAQ has consistently outperformed the S&P500 since the 2012 so far. The following chart plots the success/failure rates of the January Barometer theory applied to the aforementioned asset classes over the last 11 years:

success/failure rate: S&P500, NASDAQ

The success rate for both S&P500 and NASDAQ Composite is 54.5% while the failure rate is 45.5%. These numbers simply mean that the JB theory held true for 6 years but it unsuccessfully predicted yearly returns in 5 occasions. Nevertheless, the 54.5% success rate definitely does not fall within the category “reliable strategies” but different results could potentially come from different markets. The next table presents the results of our analyses on the German DAX30 and the British FTSE100:

yearly return: DAX, FTSE100

The tables evidently show that during the financial crises in 2008, the selected European markets performed as poorly as their American counterparties (although the British FTSE100 did not violate the –40% threshold). The last 3 years have gone quite well as far as returns are concerned although in the 2014 the British FTSE100 (–2.62%) underperformed the DAX30 (+3.06%). On average, the German index, over the last 11 years, has yielded a +7.26% return while long term investors in the British market managed to gain only +2.34%. The next graph plots the success/failure rates of the January Barometer theory applied on the aforementioned European asset classes:

success/failure rate: DAX, FTSE100

The results are very different from the ones observed for the overseas equity indices. First of all, the JB theory applied to the DAX30 has higher failures (54.5%) than successes (45.5%). In particular, the January Barometer strategy has successfully predicted future returns 5 times but it has failed 6 times implying that even in this market it led to poor results. On the other hand, the JB applied to the British FTSE100 shows a 63.6% winning chance while the losing probability is only 36.4%. In this case, the returns registered in the first month of the year since 2004 proved to be good forecasters for yearly returns. Numerically, the JB strategy would have been profitable in 7 cases while it would have failed only in 4 occasions. The next table lists the historical returns for the AustralAsian region:

yearly return: NIKKEI225, ASX200

The equity indices that have been used as a proxy for the AustralAsian geographic region are the Japanese Nikkei225 and the Australian ASX200. The 2008 was a very negative year also in AustrlAsia, in fact, both yearly returns are very close to the –50% threshold. However, also the European credit crisis in 2011 has dramatically influenced the aforementioned indices: –20.58% for the Nikkei225 and –13.16% for the ASX200. The data for the average returns, calculated over the last 11 years, show that the Nikkei225 yielded a +2.89% while the Australian equity index returned an average of +3.57% to potential long term investors. Did the January Barometer strategy provide any extra gains? The next chart will attempt to answer this question:

success/failure rate: NIKKEI225, ASX200

The Japanese index displays the usual 54.5% / 45.5% split between success and failure rates implying that the JB strategy proved profitable only 6 times and failed in 5 cases. Conversely, the success rate for the ASX200 is 63.6% (7 successes) while the failure rate is only 36.4% (4 failures) implying that the JB strategy was definitely more profitable when applied to the Australian index than to its Japanese equivalent. The last table of the present research focuses on the past performances of a geo–economic, rather than merely geographic, area: emerging markets.

yearly return: Hang Seng, Bovespa, BSE Sensex

The equity indices that have been selected for this section are the Hang Seng (Hong Kong), the Bovespa (Brazil) and the BSE Sensex (India). The first thing to notice is that during the crisis the losses incurred by these markets have been larger than the ones observed for the indices we have mentioned so far. In fact during the credit crunch, with the exclusion of the Brazilian Bovespa that in 2008 had a negative return of “only” 46.25%, the Hang Seng yielded a –55.43% while the Indian return for the same year was a remarkable –70.64%. However, the scenario considerably changes if the 2011–2014 time interval is taken into account. In fact, the Indian equity index, even including the drop in the 2011, has consistently outperformed the other two. As far as long term average returns are concerned, since the 2004 so far, the Hong Kong’s Hang Seng yielded a +5.43%, the Brazilian Bovespa returned a +5.81% while the Indian BSE Sensex averaged an impressive +13.68%. Emerging markets average returns are clearly very high but the BSE managed to outperform all other equity indices considered in the present research (the German DAX30 ranks second with a +7.26% and it is immediately followed by the NASDAQ Composite index that yielded an average return of +6.78%). The next chart plots the success/failure rates obtained by running the January Barometer strategy on the Hang Seng, Bovespa and BSE Sensex:

success/failure rate: Hang Seng, Bovespa, BSE Sensex

The success rates for the BSE Sensex (54.5%) and Bovespa (45.5%) do not really seem to provide a hedge (although the strategy proved more profitable on the Indian equity index than on the Brazilian one). The most significant information that can be extracted from this analysis is the high failure rate on the Hong Kong’s index (63.6%). In particular, the January Barometer strategy has consistently failed to predict yearly returns on the Hang Seng index for 7 years while it proved successful only in 4 cases.

All in all, the analyses just conducted on the 9 equity indices considered in the study seems to point out that the JB strategy does not to provide any particular hedge for investors. Nevertheless while wrangling data some interesting patterns have emerged. The patterns that have been detected can be grouped in two categories: the winning pattern and the losing pattern.

Winning Pattern

The January Barometer strategy can only yield two mutually exclusive outcomes: it is either profitable or not (there is a chance that the yearly return could eventually be 0% but the associated probability, over 252 trading days, is so low that we have voluntarily excluded it from the scenario analysis). The winning pattern has been extracted by simply running a frequency analysis on the times the JB strategy proved to be profitable. The JB strategy basically states that January’s return should match the yearly one, however, it makes no difference between negative or positive returns. Consequentially, as stated at the beginning of the research, a positive January’s return in a positive year would be counted as a success but a negative January’s return in a negative year would be also counted as a success. The most important thing for the strategy to work is a match between returns. The questions this section is trying to give an answer to are: is there a pattern among success cases? Do success cases have anything in common? Do bullish success cases outnumber bearish ones or vice versa? Before showing the results, it is worth reminding that 11 years worth of data on 9 different asset classes have been filtered implying a total of 99 observations. 52 out of 99 observations fall into the winning pattern category while the remaining 47 are axiomatically assigned into the losing pattern one. The following pie chart attempts to summarize the results obtained from data mining the data associated to the first pattern:

successes: bullish year vs bearish year

The success cases, for the combination January’s positive return in a positive yielding year (green area called BULL), are 34 while the success cases, for the combination January’s negative return in a bearish year (red area called BEAR), are only 18. Clearly, the outcome of this frequency analysis is a consequence of the overall trend in each year but it seems that the JB strategy, when profitable, works best in positive performing years rather than in negative ones (winning pattern). In order to understand why this is the cases we have to proceed and mine the data for failure cases.

Losing Pattern

In the previous section it has been observed that, numerically speaking, the majority of success cases happened during the combination January’s positive return in a positive yielding year. It has also been stated that the JB strategy “works best in positive performing years rather than in negative ones”. However, in order to understand why this is case, the present section will analyze all failure cases. If the JB strategy did not work, it axiomatically implies that January’s return did not match the sign of the yearly one. Consequentially, there are only two scenarios to consider when analyzing failure cases: the return in January was positive but the yearly one turned out to be negative or the year started by yielding a negative return in January but it ended with a positive performance. The next pie chart plots all the failure cases and groups them into two categories: BEAR TO BULL (negative return in January but positive on a yearly basis) and BULL TO BEAR (positive return in January but negative on a yearly basis):

failures: bear to bull years vs bull to bear years

The above reported chart is evidently displaying that, among failures, there is a very frequent pattern: the vast majority of failures in the JB strategy are due to years starting out with a negative return but that subsequently yield a positive one (losing pattern). Numerically speaking, the BEAR TO BULL cases counted 39 observations while the BULL TO BEAR ones are only 8.

Conclusion

  • The January Barometer strategy does not seem to provide consistent profits at least for the considered asset classes and the selected time frame
  • The January Barometer strategy successfully predicted yearly returns in 52 cases (52.5% of the total observations)
  • 34 of the 52 success cases come from the combination January’s positive return in a positive yielding year while only 18 come from the combination January’s negative return in a negative yielding year
  • The January Barometer strategy failed to predict yearly returns in 47 cases (47.5% of the total observations)
  • 39 of the 47 failure cases fall within the BEAR TO BULL category where January yielded a negative return but the year ended up with a positive performance
  • Only 8 failure cases fall within the BULL TO BEAR category where January yielded a positive return but the year ended up with a negative performance

 

The present research can be expanded in many ways. In fact, potential research developments could come from increasing the datasets in order to allow 20 or 30 years worth of observations, from including more equity indices or by expanding the analysis to different types of asset classes such as treasury bonds, commodities and currencies.

The Crack Spread

The crack spread is probably the most important financial strategy within the energy industry. The price of the crack spread is so crucial that it is closely monitored by commercials, hedge funds, banks, energy companies and governments. The value of the crack spread summarizes and combines, in 1 strategy, the price of crude oil and its two most important derivatives: diesel and gasoline. The entire oil industry, which still plays a very significant role as far as energy production is concerned, is strongly and inevitably linked to the performance of the crack spread. The present research investigates the structure of the aforementioned strategy and the quantitative relationships of and among its components.

Commodity traders and portfolio managers are no strangers to the concept of spread trading. In fact, many commodities are powerfully linked by common factors such as fundamentals, demand/supply dynamics, extraction/production procedures, import/export processes, shipping/transportation routes, geographical availability, geopolitical variables and so forth. There are many types of spreads within the commodity sector (natural gas vs power, copper vs aluminium, gold vs silver, platinum vs palladium) and all of them are based on some of the above mentioned common factors. The crack spread, the focus of the present research, is built using WTI Crude oil, RBOB gasoline and diesel prices. Nevertheless, it is worth mentioning that the term “crack spread” can also refer to another spread combination which involves WTI, RBOB and heating oil or to the less popular, but still frequently traded, European crack spread (Brent Crude vs European Gasoil).

The crack spread provides a fairly good approximation of the margin earned by refiners and this is precisely why this strategy is at the core of the oil industry. In fact, it simply expresses the relative value of the cost of crude oil with respect to refined products (gasoline and diesel). Refiners can enter a simple 1:1 crack spread (crude oil vs gasoline or crude oil vs diesel), however, in a typical refinery the gasoline output is usually two times larger than that of distillate fuel oils (diesel, heating oil, jet fuel oil, bunker oil, etc). Consequentially, it would be more appropriate to trade diversified crack spread constructions like a 3:2:1 spread or a 5:3:2 spread. The present analysis will focus on the 3:2:1 crack spread which is the most popular and useful construction because it meets the needs of many refiners. The formula for calculating the price of a crack spread is the following:

[(2 * RBOB $/bb + 1 * ULSD $/bb) – (3 * WTI $/bb)] / 3

The calculation is simple. However, RBOB and ULSD prices (or RBOB and heating oil prices) are expressed in $ per gallon and not in $ per barrel. Hence, they have to be multiplied by 42 (1 barrel contains 42 gallons) in order to express them in barrel terms. For example, on the 9th of June 2014, RBOB and NY Harbor Diesel closed at $3.049 and $2.886 per gallon respectively, hence, their “per–barrel prices” would automatically be ($3.049 *42) = $128.058 for RBOB and ($2.886*42) = $121.21 for ULSD.

Refiners are naturally long the crack because they have to purchase crude oil, refine it and then sell the products. Hence, the profit margin comes from the relation between crude prices and product prices while the biggest concerns for refiners come from any unwanted changes in such relation. Refiners predominantly fear increases in crude oil prices and drops in product prices because in this case their profit margin would shrink. The refiner, in order to lock in the price, will therefore cover its physical position on the financial market. Let’s assume a refiner decides to lock in the current margin because he fears an increase in the price of crude oil with respect to product prices. The best strategy he can implement involves selling a crack spread so that the long crude position will offset potential augments in oil prices while the short positions on products will counterbalance losses coming from potential plunges in diesel and/or gasoline prices. WTI is trading at $100.52, RBOB gasoline trades at $2.905 per gallon while NY Harbor diesel is at $2.927 per gallon. Consequently, RBOB gasoline and NY Harbor diesel in barrel terms are priced at ($2.905*42) =$122.01 and ($2.927*42) = $122.93. The 3:2:1 crack spread can now be constructed, so the refiner will purchase 3 WTI crude contracts at $100.52 and simultaneously sell 2 gasoline futures at $122.01 and 1 diesel futures at $122.93. The margin from the financial crack spread is equivalent to:

[(2 * $122.01 + 1 * $122.93) – (3 * $100.52)] / 3 = $21.79

The refiner has now locked in a $21.79 margin and it has secured its transaction. In fact, any potential rises in crude oil prices will be counterbalanced by a larger profit on the long WTI futures position while any potential plunges in gasoline or diesel prices will be offset by the gains on the financial short positions. Clearly, refiners tend to hedge looking forward so the contracts that will be used can expire in 1, 2 or 3 months from the implementation time depending on the delivery date. On the other hand, there are times where refiners are forced to sell crude oil and buy products. Consequentially, in order to hedge such exposure, they need to implement a reverse crack spread (also known as crack spread hedge). The reverse crack spread is just the opposite of a regular crack, in fact, it involves taking a short futures position on WTI crude oil and long positions on gasoline and diesel futures. Why would a refiner sell crude and buy products? Isn’t that counterproductive? Refineries work at full capacity to satisfy the demand for oil derivatives, however, forced shutdowns, due to machine breakdowns or unexpected technical problems, can happen. Contractual agreements must always be honoured, therefore, in the unfortunate event of a technical breakdown, refiners have to purchase products from someone else and deliver it to their clients as per contract. Furthermore, refiners tend to buy crude oil 2–3 months in advance so a breakdown would obligate them to sell the amount already delivered, or about to be delivered, because a technical “crash” would not allow them to refine it. These situations can easily happen, not that frequently, but they do happen. Hence, the best way to protect the business is to enter a reverse crack spread where crude oil is shorted and products are bought. The final margin is the difference between the operations on the physical market (selling WTI barrels and purchasing barrels of products) and the ones on the financial market (selling WTI futures while purchasing gasoline and diesel futures). Let’s assume that on the physical market the refiner faces a loss of $22.52 because he is forced to buy many thousands of barrels of products at a higher price due to the unexpected breakdown. On the other hand, thanks to the reverse crack strategy, he manages to lock in a $23.46 profit. The final margin would be $23.46 – $22.52 = $0.94.

Again, refiners do not usually buy products and sell crude unless obligated to do so, hence, the crack spread hedge is implemented only in particular occasions.

It is also worth mentioning that many refiners may want to enter different types of crack spread in order to cover the so–called energy basis risk. The basis risk is the difference in price between the same product delivered or traded in 2 different locations. The price difference between US Gulf Coast Ultra Low Sulphur Diesel and New York Harbor Ultra Low Sulphur Diesel is an example of basis risk. A refiner that uses only NY Harbour diesel will build a crack spread using NY Harbor ULSD futures while another one may want to take simultaneous positions on Gulf Coast and NY Harbor diesel because he refines them both.

The following section of the present research will quantitatively analyze the crack spread and it will separate each component of the strategy to study its behaviour and fluctuations. The first chart displays the price oscillations of the 3:2:1 crack spread that has been synthetically replicated using WTI crude, RBOB gasoline and NY Harbor diesel futures prices ranging from June 2006 to June 2014:

Crack Spread 3:2:1

It is evident that the margin earned by refiners has remarkably fluctuated over the past years and it is safe to say that geopolitical factors have strongly impacted its performance. The chart shows that the price of the crack spread has predominantly oscillated within $10 and $40 with occasional deviations from such channel. The price drop in 2008 and the violent spike in 2012 are clear examples of what happens to the margin when crude oil and products prices diverge in relative terms: crude oil goes up while products prices plunge or vice versa. The following graph summarizes the performance of the crack spread on a yearly basis:

Crack Spread average price

The figures reported in each bubble represent the average price for the crack spread in that year. The graph shows rather eloquently that, in the time interval 2006–2010, the price action moved between $13.3 and $17.3. However, the trend observed during the aforementioned 5 years was clearly bearish and the refiners’ margins decreased by $2–$3. The second part of the chart, instead, displays a diametrically opposite scenario. In fact, the data for the interval 2011–2014 show a violent explosion of the price action and a consequential widening of the margins. The highest average price was achieved in 2012 ($34.05) while 2013 and the first half of 2014 registered lower average prices ($25.55 and $23.77 respectively). It is interesting to notice that the price gap between 2010 and 2011 is as high as $15.94. Once again, the price jump was due to a divergence between the price of WTI, which at the beginning of 2011 dropped below $90 in several occasions, and the price of gasoline and diesel that, contrarily, remained constant. The initial imbalance between crude oil and products, however, persisted throughout the rest of the year even if the WTI recovered and moved back in the $100 area. It is important to point out that in November 2011 the Seaway pipeline project, which reversed the flow of crude oil, allowing its transportation from the Cushing to Houston’s vast refining area, got started. This project, which was completed on May 2012, has without a doubt contributed to increase the margin for refiners.

The next chart displays the fluctuations of the realized volatility for each component of the crack spread: WTI Crude Oil, RBOB gasoline and Ultra Low Sulphur diesel:

Volatility: WTI, Gasoline, Diesel

At a first glance, it is clear that the most volatile component is RBOB gasoline because the spikes in volatility are usually more violent in this market than in all the others. The second component, as far as volatility fluctuations are concerned, is WTI Crude Oil because its realized volatility is rather close to the RBOB one but slightly lower. The least volatile component of the entire crack spread is the diesel. In fact, it is evident that diesel realized volatility is well below the RBOB one and it is lower than the volatility curve observed for crude prices. The next graph, in order to provide a more accurate and quantitative approach, plots the distribution of the realized volatilities for each component and it ranks them:

Volatility Distribution: WTI, Gasoline, Diesel

The above reported chart eloquently confirms that RBOB is the asset class with the highest average volatility (37.73%), followed by WTI (27.97%) and ULSD (26.32%). It is important to point out that we are dealing with commodities and therefore it is not surprising to observe average volatilities well higher than the ones usually obtained by filtering equity indices data. The RBOB market is so volatile that its Low range values oscillate around 30.28% while WTI and diesel experience Low values fluctuating around 21.16% and 19.76% respectively. The same scenario can be easily noticed in the High segment of the distribution because, even in this case, the RBOB has the highest figure (46.61%) while WTI (35.76%) and ULSD (34.07%) rank lower. The examination of the extreme values, Minimum and Maximum, shows rather similar results. In particular, the Minimum segment has the identical ranking seen so far: RBOB is still the most volatile (16.68%), WTI is the second most volatile (11.15%) while diesel remains the third most volatile component (8.55%). The analysis of the Maximum segment, instead, provides some interesting evidence. Firstly, realized volatility spikes within energy markets can be rather violent and aggressive, therefore, it is not surprising to see values well beyond 100% for both RBOB gasoline (125.46%) and WTI Crude Oil (122.63%) while ULSD ranks again 3rd (81.26%).Secondly, it is very interesting to notice that in the Maximum segment, WTI values are very close to RBOB ones implying that extreme realized volatility explosions in the American crude oil market can be tremendously violent. Numerically speaking, a realized volatility explosion, that would cause the volatility to shift from the Medium segment to the High segment, would mean a 232.5% increase for RBOB gasoline but, in the case of WTI, it would imply an increment of 365.9%. It means that WTI extreme volatility explosions can be up to 57.3% more aggressive than the ones on RBOB which is, on average, the most volatile component of the crack.

So far, the concepts that have been discussed and expanded are:

1) The fundamental reasons behind the crack spread

2) How to construct a crack spread

3) Utilization of the crack spread for hedging purposes

4) Analysis of the crack spread price

5) Volatility analysis of the components

The last section of the present research will conclude the investigation on the components of the crack spread. The previous study showed that RBOB gasoline is the most volatile component of the strategy but, in order to quantitatively define which of the 3 asset classes influence crack spread prices the most, it is necessary to run a correlation analysis:

Crack Spread: Correlation Matrix

The correlation matrix is divided into 2 groups. The first group consists of 3 sets of bars containing the distribution of the correlation coefficients calculated by running the analysis between one component against the other: RBOB vs ULSD, WTI vs RBOB and WTI vs ULSD. The second group, instead, presents the distribution of the overall numerical relationships between each component and the crack spread itself: WTI vs Crack Spread, RBOB vs Crack Spread and ULSD vs Crack Spread. Medium correlation bars will be used as a proxy for long term correlation because they provide an assessment of the average connection among the variables under examination. The chart suggests that all components are well linked to each other, in fact, the RBOB/ULSD correlation is +0.70 while the WTI/RBOB rapport is +0.67. The strongest coefficient is observed for the WTI/ULSD pair and in this case the figure is as high as +0.86. The crack spread strategy, instead, displays a robust relationship with RBOB gasoline (+0.78), a very weak one with respect to diesel (+0.30) and an almost non–existent link to WTI (–0.09). The high correlation coefficient between gasoline and crack spread is due to the high volatile nature of the RBOB market. The high volatility in gasoline prices, in fact, is likely to produce sudden and more frequent changes to the price of the crack spread. It is important to mention that the strong linear relationship detected by the correlation analysis, between RBOB and crack spread , has been confirmed also by the regression analysis. In fact, the adjusted–R squared values obtained by regressing every single component against the crack spread concluded that RBOB gasoline is indeed the asset class that influences the price of the crack the most. Numerically speaking, the computed adjusted–R squared for the WTI/Crack Spread regression was 6.75% and it confirms the extremely low linear relationship between WTI and crack prices. The adjusted–R squared for the ULSD/Crack Spread regression was 24.65% and, even in this case, the weak connection is confirmed. The adjusted–R squared for the RBOB/Crack Spread regression was 45.5% and it robustly confirms the relevant linear relationship between gasoline prices and the price of the crack spread.

The Baltic Dry Index

The Baltic Dry Index measures shipping activity of raw materials around the world. In particular, the BDI provides a very efficient way to quantify and evaluate the strength of the global demand for commodities and raw materials. The Baltic Dry Index is compiled, on a daily basis, by the Baltic Exchange, and it is built thanks to the information gathered from the largest dry bulk shippers worldwide. Specifically, the Baltic Exchange collects the prices applied by dry bulk shippers for more than 20 shipping routes all over the world. The BDI is actually an average of 4 different components: The Baltic Capesize Index, The Baltic Panamax Index, the Baltic Supramax Index and the Baltic Handysize Index. What these indices are and what do they track? The fluctuations of the aforementioned indices are based upon the activity of 4 different types of ships: the Handysize, the Supramax, the Panamax and the Capemax. Let’s analyze them one at the time in order to highlight the difference among them:

1) Handysize Ships: they account for approximately 34% of the global fleet and can carry 15,000–35,000 dead weight tons of cargo

2) Supramax Ships: they account for approximately 37% of the global fleet and can carry 45,000–59,000 dead weight tons of cargo

3) Panamax Ships: they account for approximately 19% of the global fleet and can carry 60,000–80,000 dead weight tons of cargo. Panamax ships are the largest ships allowed through the Panama Canal

4) Capemax Ships: they account for approximately 10% of the global fleet and can carry more than 100,000 dead weight tons of cargo and they are too big to pass through the Panama Canal

As previously mentioned, every index is specifically created to keep track of the commercial activity connected to the 4 most important type of ship and this is precisely why the BDI is built upon them. Mathematically, the Baltic Dry Index is calculated using the following formula:

BDI = ((CapesizeTCavg + PanamaxTCavg + SupramaxTCavg + HandysizeTCavg) / 4 ) * 0.113473601)

The 0.113473601 is the multiplier introduced to standardize the calculation while the TCavg refers to the Time Charter average. Time Chartering is simply one of the ways to charter a tramp ship. Specifically, the charter will “book” the ship that best fits the size of the cargo for a specific period of time and for a pre–defined route. The hiring fees are expressed on a “per–ton” basis because they are structured on the amount of dead weight tons that are transported each month. It is important to point out that the transactional costs (bunker fuel and storage), on time chartering, are covered by the charterer itself. Consequentially, the TCavg is simply a quantification of the average cost for shipping raw materials on a established route on one the four dry bulk carriers.

The Baltic Dry Index is a very important leading indicator for worldwide business sentiment for several reasons:

1) It is difficult to modify because it is based on pure demand/supply changes which are, in turn, calculated on real orders

2) It is considered to be a leading indicator because orders are usually booked many months in advance (at least 2-3 months because of high intensity traffic in canals and potential port congestions. Think about the ship traffic concentration in the Panama and Suez canals)

3) It is a reliable index because business activities underlying the calculation are all legally certified, financed and paid upfront (none would hire a Panamax ship without having a paid order in place and none would place an order without actually needing raw materials and commodities)

4) It is a very good macroeconomic indicator, as far as shipping raw materials is concerned, because building a dry bulk carrier (whether it is a Handysize, a Supramax, a Panamax or a Capemax ship is irrelevant) takes many years. Therefore, their limited availability makes the tracking easier and more reliable because it means that the largest quantities of shipped raw materials have to necessarily be transported on one of those ships and, consequently, they are being accounted for in the calculation

The Baltic Dry Index is fairly straightforward to understand. In fact, higher or lower fluctuations simply imply a net increase or decrease in the demand for commodities and raw materials. Furthermore, the BDI is an efficient way to measure commodities’ demand because, given the fact that ships are limited and takes years to build, the amount of cargo that needs to be shipped will largely influence the oscillations of the index. The first chart of the present research displays the performance of the Baltic Dry Index since the 30th of June 2009 until the 27th of December 2013:

Baltic Dry Index

The above reported graph shows that the blue line (the actual BDI) has never fully recovered since the peak touched in November 2001 (4,661 points) while the lowest level ever touched was in February 2012 (647 points). It is evident that the 2011–2012 time interval has been rather flat in terms of shipping activity and demand for raw materials because both the quarterly and semi–annual trend lines oscillated laterally for many consecutive months. Nevertheless, the second half of the 2013 (from June onwards) showed an increased number of business activity but the recovery was far from being robust implying that the shipping of raw materials is still lower than it was before the credit crunch. The next chart will attempt to provide clue regarding the volatility of the index on different time perspectives:

Baltic Dry Index: Volatility Spectrum

First of all, it is important to mention that the Baltic Dry Index has a mean reverting volatility which tends to be confined within the 20%–40% thresholds. Also, the above reported graph suggests that the divergence between the volatility in the mid–term (red curve) and the volatility in the long–term (white curve) tends to be lower than the difference between short–term and medium–term volatilities. Volatility analysis is crucial in order to understand the fluctuations of the BDI which is why the next chart has been specifically created to show the volatility distribution of the volatility spectrum over the time period June 2009 – December 2013:

Baltic Dry Index: Volatility Distribution

The above reported chart displays the distribution of the volatility of the Baltic Dry Index in the short–term (ST VOL), medium–term (MT VOL) and long–term (LT VOL). The High, Mid–High, Medium, Mid–Low and Low sections correspond to the different volatility segments in the volatility spectrum. The sections will be now examined one at the time:

High: The ST volatility touched its highest point at 63.51% while MT and LT volatilities reached their respective tops at 62.41% and 54.20%

Mid–High: Mid–High volatilities are higher in the LT and MT than in the ST. In fact, mid–high volatility is 37.64% in the LT, equals 36.45% in the MT and it is approximately 35.05% in the ST

Medium: Medium volatility is higher in the long period than in the short one. Specifically, LT medium volatility is 32.25%, MT medium volatility is 30.36% and ST medium volatility is 27.12%

Mid–Low: Mid–Low volatilities are, even in this case, higher in the LT than in the ST. Specifically, LT mid–low volatility fluctuates around 25.51%, MT medium–low volatility is usually 24.31% while ST’s one is 19.19%

Low: LT volatility touched its lowest point at 13.83%, MT volatility minimum point was reached at 10.67% while in the ST the volatility has never got lower than 6.45%

The distribution analysis of the volatility spectrum has indeed provided very useful information. In fact, the calculation shows that mid–term and long–term volatilities, if we exclude the High segment, tend to systematically be more elevated than short–term volatility. Conversely, the highest spike in volatility was actually achieved in the short–term, although this volatility segment proved to be the least volatile of all. Why is that? What does this entail? The reason behind such phenomenon is the following: long and medium term volatilities are higher than the short term one as a consequence of the fact that the mean reverting pressure is lower in the LT and MT than in the ST. Specifically, such phenomenon implies that long–term and medium–term volatility explosions are more persistent and more ample, in terms of magnitude, and consequentially need more time to be “reabsorbed”. Besides, the fact that the highest volatility point ever reached by the Baltic Dry Index (63.51%) was actually achieved in the short–term implies that, in this segment, volatility explosions can be immediate and rather large although they tend to mean revert quickly. The strong persistency in medium and long term volatility explosions and the higher propensity to a quicker mean reverting process in the short–term volatility can be better understood by looking at the following serial correlation plot:

Baltic Dry Index: serial correlation

The chart shows 4 blocks. Each parallelepiped indicates the serial correlation amongst BDI data. The first on the left displays daily serial correlation, the second one from the left refers to weekly serial correlation, the third one from the left shows the monthly serial correlation while the last one refers to quarterly serial correlation. The interpretation of the above reported chart is fairly simple: the data from Baltic Dry Index tends to have a stronger bond in the very short term, a weak link on a weekly basis and they seem to have no relationship as far as monthly and quarterly data are concerned. The significant spread, between the first and the last two parallelepipeds, proves the point that short term volatility explosions tend to mean revert rather quickly and that actual data do not have any strong relationship in the long–term. The serial correlation plot also gives insights about the nature of the Baltic Dry Index itself: poor serial correlation in the long–term is a clear reflection of an ever–changing demand level for commodities and raw materials. The final chart highlights any inter–market relationship between the Baltic Dry Index and the most important asset classes in the world:

Baltic Dry Index: Correlation Matrix

The correlation matrix emphasizes the rapport with 3 markets in particular: Euro, American Treasury Bonds and German Bunds. The correlation between the Baltic Dry Index and Euro futures, over the 2009–2013 period, is without a doubt, the most robust (+0.47). The strong bond with the Single currency is predominantly due to the fact that approximately the 40% of the shipping transactional costs are due to bunker fuel. Bunker fuel (also known as fuel oil) is priced in US dollar but, since Euro is the largest currency component of the Dollar Index, every fluctuation in the European coinage will cause significant changes in the hiring fees charged to ship raw materials around the world. Consequently, the Baltic Dry Index, which accounts for the hiring fees charged for Handysize, Supramax, Panamax and Capemax ships, will be inevitably influenced by such variable. The remaining asset classes that display a good, although negative, correlation to the BDI are the 2 government debt world benchmarks: American T–Bonds and German Bunds. The reason American (-0.43) and German (-0.38) sovereign debt securities have an inverse link to the BDI is probably: hedging. In fact, the risk caused by taking positions on Baltic Dry Index futures and options contracts, traded through the Baltic Exchange, is usually counterbalanced via stable government bonds. The low correlation among the BDI and the so–called risky assets (DAX, WTI Crude Oil and Mini S&P500) is predominantly due to business cycles and commercial demand (commercials need to place their orders for raw materials 2–3 months ahead in order to account for production time, manage risk and ensure a continuous flow of commodity supply). On the other hand, the fact that Japanese Yen and Gold do not have a strong relationship with the BDI implies that fund managers, commercials and commodity traders prefer hedging their BDI market exposure using the aforementioned treasury markets.

HyperVolatility Researches related to the present one:

The US Dollar Index

Oil Fundamentals: Upstream, Midstream, Downstream & Geopolitics

Oil Fundamentals: Reserves and Import/Export Dynamics

Oil Fundamentals: Crude Oil Grades and Refining Process

The Oil Arbitrage: Brent vs WTI

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial

Option Greeks and Hedging Strategies

The aims of the actual research are, firstly, to present some of the most efficient methods to hedge option positions and, secondly, to show how important option Greeks are in volatility trading. It is worth mentioning that the present study has been completely developed by Liying Zhao (Quantitative Analyst at HyperVolatility) and all the simulations have been performed via the HyperVolatility Option Tool–Box. If you are interested in learning about the fundamentals of the various option Greeks please read the following studies Options Greeks: Delta, Gamma, Vega, Theta, Rho and Options Greeks: Vanna, Charm, Vomma, DvegaDtime .

In this research, we will assume that the implied volatilityσis not stochastic, which means that volatility is neither a function of time nor a function of the underlying price. As a practical matter, this is not true, since volatility constantly change over time and can hardly be explicitly forecasted. However, doing researches under the static–volatility framework, namely, the Generalized Black–Scholes–Merton (GBSM) framework, we can easily grasp the basic theories and then naturally extend them to stochastic volatility models.

Recall that the Generalized Black–Scholes–Merton formula for pricing European options is:

     C = Se (b-r)T N(d1) – Xe -rT N(d2)

P = Xe -rT N(–d2) – Se (b-r)T N(–d1)

Where

d1

d2

And N ( ) is the cumulative distribution function of the univariate standard normal distribution. C = Call Price, P = Put Price, S = Underlying Price, X = Strike Price, T = Time to Maturity, r = Risk–free Interest Rate, b = Cost of Carry Rate, σ= Implied Volatility.

Accordingly, first order GBSM option Greeks can be defined as sensitivities of the option price to one unit change in the input variables. Consequentially, second or third–order Greeks are the sensitivities of first or second–order Greeks to unit movements in various inputs. They can also be treated as various dimensions of risk exposures in an option position.

1. Risk Exposures

Differently from other papers on volatility trading, we will initially look at the Vega exposure of an option position.

1.1 Vega Exposure

Some of the variables in the option pricing formula, including the underlying price S, risk–free interest rate r and cost of carry rate b, can be directly collected from market sources. Strike price X and time to maturity T are agreed with the counterparties. However, the implied volatility σ, which is the market expectation towards the magnitude of the future underlying price fluctuations, cannot be explicitly derived from any market source. Hence, a number of trading opportunities arise. Likewise directional trading, if a trader believes that the future volatility will rise she should buy it while, if she has a downward bias on future volatility, she should sell it. How can a trader buy or sell volatility?

We already know that Vega measures option’s sensitivity to small movements in the implied volatility and it is identical and positive for both call and put options, therefore, a rise in volatility will lead to an increase in the option value and vice versa. As a result, options on the same underlying asset with the same strike price and expiry date may be priced differently by each trader since everyone can input her own implied volatility into the BSM pricing formula. Therefore, trading volatility could be, for simplicity, achieved by simply buying under–priced or selling over–priced options. For finding out if your implied volatility is higher or lower than the market one, you can refer to this research  that we previously posted.

Let’s assume an option trader is holding a so–called ‘naked’ short option position, where she has sold 1,000 out–of–the–money (OTM) call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30%, which is currently valued at $434.3. Suppose the market agreed implied volatility decreases to 20%, other things being equal, the option position is now valued at $70.6. Clearly, there is a mark–to–market profit of $363.7 (434.3-70.6) for this trader. This is a typical example of Vega exposure.

Figure 1 shows the Vega exposure of above the option position. It can be easily observed that the Vega exposure may augment or erode the position value in a non–linear manner:

option vega exposure

(Figure 1. Source: HyperVolatility Option Tool – Box)

A trader can achieve a given Vega exposure by buying or selling options and can make a profit from a better volatility forecast. However, the value of an option is not affected solely by the implied volatility because when exposed to Vega risk, the trader will simultaneously be exposed to other types of risks.

1.2 Theta Exposure

Theta is the change in option price with respect to the passage of time. It is also called ‘time decay’ because Theta is considered to be ‘always’ negative for long option positions. Given that all other variables are constants, option value declines over time, so Theta can be generally referred to as the ‘price’ one has to pay when buying options or the ‘reward’ one receives from selling options. However, this is not always true. It is worth noting that some researchers have reported that Theta can be positive for deep ITM put options on non–dividend–paying stocks. Nevertheless, according to our research, which is displayed in Figure 2, where X=$100, T=30 days, r=0.5%, b=0, σ =30%, the condition for positive Theta is not that rigorous:

option theta

(Figure 2. Source: HyperVolatility Option Tool – Box)

For deep in–the–money (ITM) options (with no other restrictions) Theta can be slightly greater than 0. In this case, Theta cannot be called ‘time decay’ any longer because the time passage, instead, adds value to bought options. This could be thought of as the compensation for option buyers who decide to ‘give up’ the opportunity to invest the premiums in risk–free assets.

Regarding the aforementioned instance, if an option trader has sold 1,000 OTM call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30%, the Theta exposure of her option position will be shaped like in Figure 3:

option theta exposure

(Figure 3. Source: HyperVolatility Option Tool – Box)

In the real world, none can stop time from elapsing so Theta risk is foreseeable and can hardly be neutralized. We should take Theta exposure into account but do not need to hedge it.

1.3 Interest Rate/Cost of Carry Exposure (Rho/Cost of Carry Rho)

The cost of carry rate b is equal to 0 for options on commodity futures and equals r–q for options on other underlying assets (for currency options, r is the risk–free interest rate of the domestic currency while q is the foreign currency’s interest rate; for stock options, r is the risk–free interest rate and q is the proportional dividend rate). The existence of r, q and b does have an influence on the value of the option. However, these variables are relatively determinated in a given period of time and their change in value has rather insignificant effects on the option price. Consequently, we will not go too deep into these parameters.

1.4 Delta Exposure

Delta is the sensitivity of the option price with regard to changes in the underlying price. If we recall the aforementioned scenario (where a trader has sold 1,000 OTM call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ =30%, valued at $434.3) if the underlying price moves downwards, the position can still be able to make profits because the options will have no intrinsic value and the seller can keep the premiums. However, if the underlying asset is traded at, say, $105, all other things being equal, the option value becomes $6,563.7, which leads to a notable mark–to–market loss of $6,129.4 (6,563.7-434.3) for the option writer. This is a typical example of Delta risk one has to face when trading volatility. Figure 4 depicts the Delta exposure of the above mentioned option position, where we can see that the change in the underlying asset price has significant influences on the value of an option position:

option delta exposure

(Figure 4. Source: HyperVolatility Option Tool – Box)

Compared to Theta, Rho and cost of carry exposures, Delta risk is definitely much more dominating in volatility trading and it should be hedged in order to isolate volatility exposure. Consequentially, the rest of this paper will focus on the introduction to various approaches for hedging the risk with respect to the movements of the underlying price.

2. Hedging Methods

At the beginning of this section, we should clearly define two confusable terms: hedging costs and transaction costs. Generally, hedging costs could consist of transaction costs and the losses caused by ‘buy high’ and ‘sell low’ transactions. Transaction costs can be broken down into commissions (paid to brokers, etc.) and the bid/ask spread. These two terms are usually confounded because both of them have positive relationships with hedging frequency.  Mixing these two terms up may be acceptable, but we should keep them clear in mind.

2.1 ‘Covered’ Positions

A ‘covered’ position is a static hedging method. To illustrate it, let us assume that an option trader has sold 1,000 OTM call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30%, and has gained $434.3 premium. Compared to the ‘naked’ position, this time, these options are sold simultaneously with some purchased underlying assets, say, 1,000 stocks at $90. In this case, if the stock price increases to a value above strike price (e.g. $105) at maturity, the counterparty will have the motivation to exercise these options at $100. Since the option writer has enough amount of stocks on hand to meet the exercise demand, ignoring any commission, she can still get a net profit of 1,000*(100-90) + 434.3= $10,434.3. On the contrary, if the stock price stays under the strike price (e.g. $85) upon expiry date, the seller’s premium is safe but has to suffer a loss in stock position which in total makes the trader a negative ‘profit’ of 1,000*(85-90) + 434.4=-$4565.7. The ‘covered’ position can offer some degrees of protection but also induces extra risks in the meantime. Thus, it is not a desirable hedging method.

2.2 ‘Stop-Loss’ Strategy

To avoid the risks incurred by stock prices’ downward trends in the previous instance the option seller could defer the purchase of stocks and monitor the movements of the stock market. If the stock price is higher than the strike price, 1,000 stocks will be bought as soon as possible and the trader will keep this position until the stock price will fall below the strike. This strategy seems like a combination of a ‘covered’ position and a ‘naked’ position, where the trader is ‘naked’ when the position is safe and he is ‘covered’ when the position is risky.

The ‘stop-loss’ strategy provides some degrees of guarantee for the trader to make profits from option position, regardless of the movements of the stock price. However, in reality, since this strategy involves ‘buy high’ and ‘sell low’ types of transactions, it can induce considerable hedging costs if the stock price fluctuates around the strike.

2.3 Delta–Hedging

A smarter method to hedge the risks from the movements of the underlying price is to directly link the amount of bought (sold) underlying asset to the Delta value of the option position in order to form a Delta– neutral portfolio. This approach is referred to as Delta hedging. How to set up a Delta–neutral position?

Again, if a trader has sold 1,000 call options priced with S=$90, X=$100, T=30 days, r=0.5%, b=0, σ=30% the Delta of her position will be -$119 (-1,000*0.119), which means that if the underlying increases by $1, the value of this position will accordingly decrease by $119. In order to offset this loss, the trader can buy 119 units of underlying, say, stocks. This stock position will give the trader $119 profit if the underlying increases by $1. On the other hand, if the stock price decreases by $1, the loss on the stock position will then be covered by the gain in the option position. This combined position seems to make the trader immunized to the movements of the underlying price.

However, in the case the underlying trades at $91, we can estimate that the new position Delta will be -$146. Obviously, 119 units of stocks can no longer offer full protection to the option position. As a result, the trader should rebalance her position by buying 27 more stocks to make it Delta–neutral again.

By doing this continuously, the trader can have her option position well protected and will enjoy the profit deriving from an improved volatility forecasting. Nevertheless, it should be noted that Delta–hedging also involves ‘buy high’ and ‘sell low’ operations which could cause a loss for every transaction related to the stock position. If the price of the underlying is considerably volatile, the Delta of the option position would change frequently, meaning the option trader has to adjust her stock position accordingly with a very high frequency. As a result, the cumulative hedging costs can reach an unaffordable level within a short period of time.

The aforementioned instance show that increasing hedge frequency is effective for eliminating Delta exposure but counterproductive as long as hedging costs are concerned. To reach a compromise between hedge frequency and hedging costs, the following strategies can be taken into considerations.

2.4 Delta–Gamma Hedging

In the last section, we have found that Delta hedging needs to be rebalanced along with the movements of the underlying. In fact, if we can make our Delta immune to changes in the underlying price, we would not need to re–hedge. Gamma hedging techniques can help us accomplishing this goal (recall that Gamma is the speed at which the Delta changes with respect to movements in the underlying price).

The previously reported example, where a trader has sold 1,000 call options priced with S=$90,X=$100,T=30days,r=0.5%,b=0 ,

σ =30% had a position Delta equal to -$119 and a Gamma of -$26. In order to make this position Gamma–neutral, the trader needs to buy some options that can offer a Gamma of $26. This can be easily done by buying 1,000 call or put options priced with the same parameters as the sold options. However, buying 1,000 call options would erode all the premiums the trader has gained while buying 1,000 put options would cost the trader more, since put options would be much more expensive in this instance. A positive net premium can be achieved by finding some cheaper options.

Let us assume that the trader has decided to choose, as a hedging tool, the call option priced with S=$90, X=$110, T=30 days, r=0.5%, b=0, σ=30%, with 0.011 Delta and 0.00374 Gamma. To offset his sold Gamma, the trader needs to buy 26/0.00374 = 6,952 units of this option which cost him $197.3 which leads to an extra Delta of 6,952*0.011=$76. At this point, the trader has a Gamma–neutral position with a net premium of $237 (434.3-197.3) and a new Delta of -$43(-119+76). Therefore, buying 43 units of underlying will provide the trader with Delta neutrality. Now, let’s suppose that the underlying trades at $91, the Delta of this position would become -$32 but since the trader had already purchased 43 units of stocks, she only needs to sell 12 units to make this position Delta–neutral. This is definitely a better practice than buying 27 units of stocks as explained in section 2.3 where the trader had only Delta neutralized but was still running a non–zero Gamma position.

However, Delta–Gamma Hedging is not as good as we expected. In order to explain this, let us look at Figure-5 which shows the Gamma curve for an option with S=$90, X=$110, T=30 days, r=0.5%, b=0, σ=30%:

option gamma

(Figure 5. Source: HyperVolatility Option Tool – Box)

We can see that Gamma is also changing along with the underlying. As the underlying comes closer to $91, Gamma increases to 0.01531 (it was 0.00374 when the underlying was at $90) which means that, at this point, the trader would need $107 (6,954*0.01531) Gamma and not $26 to offset her Gamma risk. Hence, she would have to buy more options. In other words, Gamma–hedging needs to be rebalanced as much as delta – hedging.

Delta–Gamma Hedging cannot offer full protection to the option position, but it can be deemed as a correction of the Delta–hedging error because it can reduce the size of each re–hedge and thus minimize costs.

From section 2.3 and 2.4, we can conclude that if Gamma is very small, we can solely use Delta hedging, or else we could adopt Delta–Gamma hedging. However, we should bear in mind that Delta–Gamma hedging is good only when Speed is small. Speed is the curvature of Gamma in terms of underlying price, which is shown in Figure – 6:

option speed

(Figure 6. Source: HyperVolatility Option Tool – Box)

Using the basic knowledge of Calculus or Taylor’s Series Expansion, we can prove that:

 Δ Option Value Delta * ΔS + ½ Gamma * ΔS + 1/6 Speed * ΔS

We can see that Delta hedging is good if Gamma and Speed are negligible while Delta–Gamma hedging is better when Speed is small enough. If any of the last two terms is significant, we should seek to find other hedging methods.

2.5 Hedging Based on Underlying Price Changes / Regular Time Intervals

To avoid infinite hedging costs, a trader can rebalance her Delta after the underlying price has moved by a certain amount. This method is based on the knowledge that the Delta risk in an option position is due to the underlying movements.

Another alternative to avoid over–frequent Delta hedging is to hedge at regular time intervals, where hedging frequency is reduced to a fixed level. This approach is sometimes employed by large financial institutions that may have option positions in several hundred underlying assets.

However, both the suitable ‘underlying price changes’ and ‘regular time intervals’ are relatively arbitrary. We know that choosing good values for these two parameters is important but so far we have not found any good method to find them.

2.6 Hedging by a Delta Band

There exist more advanced strategies involving hedging strategies based on Delta bands. They are effective for finding the best trade–off between risks and costs. Among those strategies, the Zakamouline band is the most feasible one. The Zakamouline band hedging rule is quite simple: when the Delta of our position moves outside of the band, we need to re–hedge and just pull it back to the edge of the band. However, the theory behind it and the derivation of it are not simple. We are going to address these issues in the next research report.

Figure-7 provides an example of the Zakamouline bands for hedging a short position consisting of 1,000 European call options priced with S=$90, X=$110, T=30 days, r=0.5%, b=0, σ=30%:

zakamouline bands

(Figure 7. Source: HyperVolatility Option Tool – Box)

In the next report, we will see how the Zakamouline band is derived, how to implement it, and will also see the comparison of Zakamouline band to other Delta bands in a quantitative manner.

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial

Extracting Implied Volatility: Newton-Raphson, Secant and Bisection Approaches

The aim of the present research is to identify an efficient method to extract implied volatility from options prices. It is worth mentioning that the present study has been completely developed by Liying Zhao (Options Engineer at HyperVolatility) and all the simulations have been performed via the HyperVolatility Option Tool – Box. Options traders, and by “traders” we obviously mean market makers too, are more concerned with volatility than premium itself because they know perfectly well that the latter is a consequence of the former. Therefore, it is essential to find out which implied volatility level has been used to price options in order to determine which trading strategy should be adopted. It goes without saying that if an option trader “discovers” some options priced with a lower implied volatility level he would immediately consider buying strategies while the opposite scenario would apply when there is a case of overpricing. At a first glance, things seem quite simple but what would happen if the implied volatility that you have extracted from market prices does not match the one employed by your counterparty? Chances are that if you extract a ‘wrong’ implied volatility figure from the ‘right’ market price you could probably be misled by it and eventually lose money.

There are many ways to “recuperate” the implied volatility from market prices and among them the Newton–Raphson (NR) method is undoubtedly one of the most popular employed by option traders. The question is, can the Newton–Raphson approach be trusted? In order to answer this question we first need to examine the NR method. The formula is the following:

formula1a

Where  CodeCogsEqn(3) if (i=0) is the initial guess for the implied volatility, c-sigma I is the option price derived from the initial guess, cm is the market price of the option, deltac-deltasigma is obviously the Vega in terms of the initial guess, and finally sigma I+1 would be the updated implied volatility. While the absolute value of cm-csigma falls into a desired degree, sigma I+1 would be the desired estimation of the implied volatility used to price the option. It is worth noting that the initial guessed values are figures that can be obtained by using other formulas, past experience but they can even be chosen randomly.

The NR method is very fast and, if the initial guesses are good the efficiency of the model is enhanced which is why many option traders prefer to use it. However, in order to use this method the value of Vega is needed. Fortunately, Vega can be easily derived following an analytical approach at least for European style options. What happens if we are trading American style options? Well, Vega can be numerically computed for American options; nevertheless, in this case the advantage of the NR method (that is high efficiency) is, more or less, sacrificed.

Let’s omit the aforementioned issue and let us focus on the accuracy of the NR method. We will now present a simulation where we will price an entire options chain written on a hypothetical commodity using the BSM model and then we will extract the implied volatility via the NR approach. Let us assume that we are trading European style commodity options where the underlying price equals $100, the risk–free interest rate is 0.5%, the cost of carry is 0, the implied volatility is 20% and there are 20 days left before expiration. Given those parameters, we can price an option chain with strike prices ranging from $50 to $150 by using the Black–Scholes–Merton (BSM) formula. Bear in mind we need to assume that the theoretical prices of the option chain are perfectly matched by market prices. Now, we can finally use market prices inversely with the NR method and we will theoretically get an implied volatility equal to 20% for each instrument in the option chain. The following chart help clarifying the concept:

implied  volatility

(Source: HyperVolatility Option Tool – Box)

The chart displays only 1 curve (which shows the implied volatility for put options) because the volatility curve for call options moves in the exact same way so there is a problem with overlapping. Nevertheless, we have got a ‘volatility skew’. Do not worry. This is not a real volatility skew. Volatility skews exist because Out-of-The-Money (OTM) and In-The-Money (ITM) options are more heavily traded. Nonetheless, we are not dealing with real trading conditions; instead, we just want to get back to the implied volatility we used to price these options. Thus, we can infer that this ‘volatility skew’ originates from the inaccuracy of the NR method. According to some deeper researches the Newton–Raphson method becomes very inaccurate when the strike of the option is more than 20% Away-From-The-Money (AFTM).

If we use the biased implied volatility that we previously obtained and plug it into the BSM formula to price the same option chain we would get a range of new option prices. Luckily, the correlation coefficient between the prices of the option chain with biased and unbiased implied volatility is equal to 1, which means, the bias of implied volatility for AFTM options has little effect on the pricing. Why? To explain this, let us consider the following Vega chart:

options vega

(Source: HyperVolatility Option Tool – Box)

It is quite clear from the chart that the implied volatility is only influential for slightly In-The-Money (ITM), or slightly Out-of-The-Money (OTM) options. In other words, shifts in the implied volatility level would significantly influence options whose strike is close to the ATM area but would not impact that much Away–From–The–Money options. Therefore, when the strike price is far away from the underlying price the change in implied volatility becomes less important when pricing an option. On the other hand, Near-The-Money (NTM) options are very sensitive to changes in implied volatility. For instance, using 10% or 30% implied volatility while we are pricing an Away–From–The–Money option ($100 underlying, $60 strike) would not show much difference in terms of premiums. For the same reason, when we take the reversed BSM formula to extract implied volatility from market prices for AFTM options, it is normal that the resulting implied volatility does not match the volatility level we used to price the option chain with in the first place. From a deeper angle, it is worth saying that Vega approaches zero when the underlying price is far away from strike price. Furthermore, small errors in the estimation of Vega, being it the NR formula’s denominator, can lead to large errors in the implied volatility.

Another approach that can be used to extract implied volatility from options prices is the so-called Secant Method. Some ‘mathematicians’, 3,000 years before Newton, developed a root–finding algorithm called Secant Method (SM) that uses a succession of roots of secant lines to approximate a root of a function. This method works similarly to the NR method and consequently it is just as fast as the NR approach. Moreover, the Secant Method does not require the knowledge and computation of Vega. Can we assert that the Secant Method is the perfect approach? Before answering this question, let’s see what the Secant Method is about:

formula2a

where cm is the market price,  sigma n is the result for each iteration, c sigma n is the theoretical price for each result. With two initial guesses sigma 0 and sigma 1 we can get sigma 2, but we are going to repeat this process until we get sigma n and sigma n-1 (which are close enough) and then sigma n that would be the desired answer.

To evaluate the reliability of this method we are going to let the statistics speak. Suppose we are trading a European style commodity options with underlying price equal to $100, risk–free interest rate 0.5%, cost of carry set to 0, implied volatility 20% and 20 days to maturity. Given the aforementioned parameters, we now price an option chain with strike prices ranging from $50 to $150 by using the BSM formula. However, this time we use the theoretical prices inversely but with the Secant Method and try to extract the implied volatility which should equal 20% for each option.

implied volatility

(Source: HyperVolatility Option Tool – Box)

The above reported chart clearly shows that we can perfectly recover the implied volatility from market prices for Near–The–Money options. Nevertheless, Away–From–The–Money options present a more complex scenario because at the ‘tails’ of the strike distribution we have an increased deal of inaccuracy. The main reason for the existence of these irregular observations is that this algorithm may not converge if the first derivative of the function in terms of the argument, Vega in this instance, is close to 0 (remember the Vega chart for AFTM options: Vega approaches 0 and here the convergence of the Secant Method is not guaranteed). It is also worth noting that in order to execute the Secant Method we need two initial guesses but in this case both the guessed values must be very close to the target value. Thus, this approach requires a great deal of preparatory work before the start of iterations otherwise one would run the risk to get worse–than–expected results.

Alternatively, we can try other ways to extract the implied volatility from market prices. One of them is the Bisection Method (BM), where we also need two initial volatility estimates: a ‘low’ estimation of the implied volatility sigma L , which would generate the option value c L and a ‘high’ volatility estimation, sigma H,generating the option value c H. The option market price, cm , then should fluctuate between c L and c H. The Bisection computation of the implied volatility is given by:

formula3a

where c sigma MINOR we replace sigma L with sigma I+1 while if c sigma MAJOR we replace sigma H with sigma I+1. When sigma H-L falls into the desired interval, sigma I+1 would be the desired implied volatility.

Let us now assess the accuracy of this method by using the same approach adopted for the Newton–Raphson formula. Again, suppose we are trading European style commodity options with underlying price equal to $100, risk-free interest rate 0.5%, cost of carry set to 0, implied volatility 20% and 20 days to maturity. We will now price an option chain with strike prices ranging from $50 to $150 by using the BSM formula. However, this time we employ the theoretical prices inversely and we will try to find the implied volatility via the Bisection Method:

chart4

(Source: HyperVolatility Option Tool – Box)

The chart highlights that the implied volatility become slightly inaccurate only for very deep ITM options and in this case the ‘confident interval’ is significantly enlarged. The advantages of the Bisection Method are that it can be implemented without the knowledge of Vega and it can be used to find the implied volatility when dealing with American options. On the other hand, it may still not be too attractive extracting implied volatility with this method because it is not as efficient as the Newton – Raphson formula. Furthermore, the Bisection Method is computationally expensive, hence, when dealing with a great amount of options the capacity of the computer adopted becomes crucial.

There is no perfect method to extract implied volatility from market prices without errors and with high speed. Nevertheless, the Newton–Raphson approach, despite its drawbacks, is the dominating method in this area. In fact, with good initial guesses the NR formula can be a very ‘smart’ method which provides accurate results particularly for Near–The–Money options with high efficiency. The Bisection Method is considered to be ‘dull’ but it provides larger ‘confident intervals’. Besides, the Secant Method is as fast as NR one but it is very hard to implement. Finally, it is worth mentioning that, in spite of what the majority of option traders believe, there actually are analytical expressions to extract the implied volatility from market prices. The problem with these closed formulas is that they are too complicated. In fact, each function can even have more than thirty parameters and each parameter, in turn, will have to be calibrated with a great deal of accuracy. Luckily, there is a lot of room for improvement in this field and perhaps future researches will propose an analytical solution to this problem that will manage to replace the Newton–Raphson approach.

In order to summarise the findings of the present research, please have a look at the following table where all the models and characteristics are not only listed but also ranked. The way to interpret the results is very simple: the higher the mark, the better. Even the section called “Difficulty” has been quantified in the same way, hence, an higher mark means that it is easier to implement:

chart5

It is clear enough that the Newton–Raphson method and the Bisection Method are the ones showing the highest marks. However, they are recommended not in all cases. In particular, the NR approach is more suitable when dealing with a large set of options while the Bisection Method has to be preferred when the data set available is rather small.

If you are interested in options trading or pricing you may want to read some HyperVolatility researches related to this topic:

“The Pricing of Commodity Options”

“Options Greeks: Delta, Gamma, Vega, Theta, Rho”

“Options Greeks: Vanna, Charm, Vomma, DvegaDtime”

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial

 

Commodities and Currencies: Inter-Market Analysis

Commodity markets are becoming increasingly important in today’s financial scenario. Many investors, traders and fund managers are now shifting their attention towards asset classes such as Crude Oil, Natural Gas, Gold, Silver, Wheat, Coffee or Sugar. Furthermore, the credit crunch brought a great deal of market instability and many equity indices, as well as single stocks, have been abandoned by or lost their appeal to many market players. There are several liquid commodities in the world but the present research will concentrate on, arguably, the heaviest in terms of volume: WTI Crude Oil, Gold and Henry Hub Natural Gas. It is important to point out that the European oil benchmark, the Brent Crude, has been deliberately omitted from the aforementioned list, although very liquid, but it will be the focus of another research. This research will present the results of an inter–market analysis where we compare and contrast the performances of above mentioned 3 commodities against a basket of currencies. The currency exchanges that have been selected are either very liquid (Euro, Japanese Yen, British Pound, Swiss Franc) or related to countries whose trade balances heavily rely on commodities and this is the case for South Africa, Canada, Australia and New Zealand (some market players can also refer to these exchanges as commodity currencies). The research has been conducted using a database consisting of daily data ranging from October 2011 until the 25th of June 2013. Consequently, our study aims to capture the medium–to–long term relationship that WTI Crude Oil, Gold and Henry Hub Natural Gas futures have with those exchanges. The first commodity that will be examined is the WTI Crude Oil:

WTI Crude

The American crude oil chart shows a very strong relationship with Euro, Swiss Franc and South African Rand. However, the correlation is strong and positive for Swiss Franc (+0.36) and Euro (+0.34) but it is negative for the South African currency (-0.34). The positive connection with the Euro is a consequence of the fact that the Single currency is responsible for the 56.7% of US Dollar Index fluctuations. The Swiss Franc, being a European currency, shows a robust positive correlation to the WTI (+0.36), however, the chart signals that the Swiss Franc is not a good hedging tool for the American oil market. Usually, the Swiss currency is adopted by many portfolio managers and market players as a hedge against volatile periods in risky assets but, when it comes to WTI oil, this is not the case. The negative correlation with the South African Rand (-0.34) is given by the fact that South Africa is the second largest oil refiner in Africa. Nevertheless, almost all the oil that is imported belongs to OPEC countries. Consequently, the Rand is more linked to the price of oils extracted from OPEC countries than to the American West Texas Intermediate. Finally, if we exclude a “mild correlation” with the Australian and Canadian Dollars, the remaining exchanges (Yens, British Pounds and New Zealand Dollars) do not show a great sensitivity to WTI price changes. The next market under examination will be Gold:

Gold

The gold market has very good positive connections with many asset classes. If we exclude New Zealand Dollars (+0.07), the Euro (+0.20) and the South African Rand (-0.65), gold futures are strongly bounded to many exchanges. The strong positive correlation to Australian Dollars (+0.62) is obvious because Australia is one of the largest gold exporters in the world while the +0.69 rapport with the Japanese Yen is a signal that many market players started to use again gold as a hedging tool for their portfolios. Furthermore, Canada (+0.57) has large gold mines while the UK (+0.63) is a strong importer and it primarily purchases gold for luxury goods or investment purposes. Once again the South African Rand is a standalone market because its strong negative correlation to gold futures (-0.65) is not just due to portfolio hedging but also to strong fundamental reasons. Specifically, the South African mining industry has been subject to many strikes in the last 3 – 4 years that greatly influenced its extraction rate. Consequently, the SA gold’s output precipitously fell and this could have caused a major loss of “sensitivity” between gold prices and the local currency. The next commodity market is the Henry Hub Natural Gas:

Henry Hub Natural Gas

The Henry Hub is the core distribution point in the natural gas pipeline scheme located in Louisiana, USA. Natural gas futures traded on the NYMEX have been named after it because of the great logistic importance of this area. The above reported chart shows an unstable negative correlation to Australian, Canadian Dollars and British Pounds (-0.22, -0.30 and -0.33 respectively), a rather weak but positive link to New Zealand Dollars and Euros and no correlation at all to Swiss Franc futures(-0.03). On the other hand, there are two currencies that are strongly correlated to natural gas prices: Japanese Yen and the South African Rand. The reasons Japanese Yens have a robust negative relationship (-0.57) to the Henry Hub is that the Asian country has recently doubled its nat–gas consumption but the local demand is almost entirely satisfied via imports (circa 4,500 billion cubic feet). Furthermore, the Japanese Yen is often considered to be a “safe haven” type of investment, hence, many market players will tend to buy the Asian currency when risky assets plummet. The negative relationship between Yens and nat–gas prices is an evident sign that many portfolio managers tend to hedge their natural gas positions with the Japanese Yen. Conversely, the great explosion in natural gas consumption in South Africa (over 160 billion cubic feet in 2011 against 58.2 billions in 2000) and the low import levels make the South African currency rather “susceptible” to oscillations in natural gas prices.

Let’s summarize the main findings in a few bullet points:

WTI CRUDE OIL

1) The correlation is strong and positive with Swiss Franc (+0.36) and Euro (+0.34) but it is negative for the South African currency (-0.34)

2) South Africa is the second largest oil refiner in Africa but almost all the in–flowing oil is imported from OPEC countries. This explains the negative correlation (-0.34) with the Rand

GOLD

1) Australia is one of the largest gold exporters in the world, hence, Australian Dollars are strongly connected to this commodity (+0.62)

2) The +0.69 rapport with the Japanese Yen implies that gold is still considered to be a hedging tool for portfolio risk

3) Canadian Dollars are positively correlated to gold (+0.57) because Canada has large gold mines

4) There is a positive and robust link between gold and British Pounds (+0.63) because UK is a strong importer and Great Britain buys gold for luxury goods or investment purposes

5) The South African gold’s output plunged in recent years due to several strikes in the mining industry. Hence, the drop in the SA gold extraction rate could have caused the negative connection between gold prices and the local currency (-0.65)

HENRY HUB NATURAL GAS

1) There is a robust negative relationship with Japanese Yens (-0.57) because Japan increased its natural gas consumption but it imports almost all the natural gas needed (circa 4,500 billion cubic feet)

2) Negative correlation with the Japanese Yen (-0.57) implies that natural gas portfolios tends to be hedged with the Asian currency

3) The large increase in natural gas consumption in South Africa (over 160 billion cubic feet in 2011 against 58.2 billions in 2000) and a strong local production strengthened the link between the South African currency and natural gas prices (+0.73)

All traders and investors interested in trading commodities and in particular Crude Oil and Gold are strongly advised to read the following HyperVolatility researches:

Commodity Volatility Indices: OVX and GVZ

“The Pricing of Commodity Options”

“Gold Market Performance in 2012”

“The Oil Arbitrage: Brent vs WTI”

“Trading Gold and Silver: A Realized Volatility Approach”

The HyperVolatility Forecast Service enables you to receive statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial.

The Pricing of Commodity Options

The present research will prove particularly useful to option traders. The analysis proposed by the HyperVolatility Team will explain, in a few bullet points, how the most popular commodity options pricing models behave and what the practical divergences in terms of prices are. The present study is very valuable to anyone interested in trading options because most trading platforms allow the trader to choose the model via which the theoretical value of the options will be calculated and consequently shown (the HyperVolatility Forecast Service provides market projections for many asset classes. Send an email to info@hypervolatility.com and get a free 14 days trial). The pricing models that will be analyzed are the Barone–Adesi –Whaley, the Bjerksund & Stensland (the 2002 version), the Black–76, the Binomial Tree and the classic Black–Scholes–Merton one. The models have been tested against each other and the following charts graphically show the divergence of 1 pricing model with respect to all others. The research has been performed assuming that the underlying asset (S) is a WTI crude oil futures contract, that the volatility (σ) is 20%, that the interest rate (r) is 0.5% and that the Cost of Carry is 0 (which is normal when dealing with commodity options).

As previously mentioned, the study will examine 1 pricing model at the time and, in order to avoid confusion and make things simpler, we decided to list the most important aspects below each graph:

Barone Adesi Whaley model

1) The Barone-Adesi-Whaley model overprices options when compared to other formulas. The pricing spread with respect to other models is on average between 0.06% and 0.08%

2) The Barone-Adesi-Whaley prices tend to get closer to other models as the expiration increases

3) The Barone-Adesi-Whaley model, on average, tends to overprice options with respect to the Binomial Tree (~ 0.16% higher) for short maturities. The trend is higher for out-of-the-money options and particularly for put options

4) The prices derived from the Bjerksund & Stensland model are always lower than Barone-Adesi-Whaley prices. The difference is bigger for 1 month options (~ 0.16%)

5) The Black-76 performs as well as the Black–Scholes–Merton model, however, their results overlap and that is why the Black-76 curve is not visible

6) The difference with the Black–Scholes–Merton model becomes larger as the expiration increases but it is not higher than 0.1%

 

Bjerksund & Stensland model

1) The Bjerksund & Stensland model under–prices options in respect to other models. On average the difference ranges between 0.05% – 0.06%

2) The under–pricing tends to reduce as the expiration increases

3) The Bjerksund & Stensland  model produces prices which are lower than the Barone–Adesi–Whaley one for any expiration

4) The Black–Scholes–Merton model approximates to the Bjerksund & Stensland one from the 8th month onwards

5) The Black–76 performed as well as the Black–Scholes–Merton model and that is why the overlapped curve cannot be seen  in the chart

6) The Binomial Tree approach shows the highest differential with respect to the Bjerksund & Stensland model. The divergence in pricing oscillates around 0.15%

 

Black-76 model

1) The Black–76 model over–prices options only with respect to the Bjerksund & Stensland one (almost 0.05%)

2) The divergence between Black–76 and Bjerksund & Stensland attenuates when longer expirations are approached

3) The Barone–Adesi–Whaley model prices are slightly higher than Black–76 ones and the discrepancy augments with the passage of time (between 0.08% and 0.1% for 10 months and 1 year expiring options respectively)

4) The Binomial Tree approach, if we exclude the short term, delivers higher prices than the Black–76 model but the divergence oscillates around the interval 0.03% – 0.04%

5) The Black–76 model performed as well as the Black–Scholes–Merton one and that is why the BSM curve is flat to 0. Needless to say that the Black–Scholes–Merton curve suggests that there is no difference in pricing

 

Binomial Tree model

1) The Binomial Tree under–prices options with respect to other models in the short term (around 2.5%) but the divergence is much lower for longer dated derivatives

2) The Barone–Adesi–Whaley model and the BSM model perform as well as the Black–76 one therefore their curves are hidden in the chart

3) The Bjerksund & Stensland model provided higher prices for short dated options but in the long term the Binomial Tree approach shows a slight over–pricing tendency with respect to the former. However, the spread is no higher than 0.04% – 0.05%

 

Black Scholes Merton model

1) The performances of the Black–Scholes–Merton formula with respect to other pricing models match perfectly well with the outcome generated by the Black–76 model

2) The above reported chart is identical to the graph extrapolated for the Black–76 model, in fact, the green curve does not move from the 0 axis

If you are interested in trading options you might want to read also the HyperVolatility researches entitled “Options Greeks: Delta, Gamma, Vega, Theta, Rho” and “Options Greeks: Vanna, Charm, Vomma, DvegaDtime”

The HyperVolatility Forecast Service enables you to receive the statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial.

The US Dollar Index

The Dollar Index is a very useful yet simple to understand tool for currency, commodity and equity traders. The index measures the fluctuations of the US Dollar against a basket of different currencies: Euro, Japanese Yen, Canadian Dollar, British Pound, Swedish Krona and Swiss Franc (the Forecast Service provides market projections for currencies and many other asset classes. Send an email to info@hypervolatility.com and get a free 14 days trial). In practical terms, the Dollar Index tries to quantify how much the US dollar is appreciating or depreciating with respect to some of the most important and traded currencies in the world. The reason the Dollar Index is a valuable instrument for many traders is primarily due to its correlation to other asset classes and particularly risky assets.

As previously mentioned, the Index tracks the performance of the American dollar against a basket of other currencies, however, each exchange rate has a different weight. Specifically, the Index is a weighted geometric mean whose components and respective weights are the following: Euro (56.7%), Japanese Yen (13.6%), British Pound (11.9%), Canadian Dollar (9.1%), Swedish Krona (4.2%) and Swiss Franc (3.6%). The Index has 100 as a benchmark value therefore every reading below this threshold would imply a depreciation of the American currency against the basket, primarily the Euro, while any value above it is obviously indicating an appreciation. It goes without saying that the reason the Dollar Index has been trading below the 100 level for so long is the axiomatic consequence of the strength of the Single currency. It is clear that the Euro, whose weight in the Dollar Index calculation is 56.7%, is the most important currency since its impact is by far the heaviest. Obviously, every appreciation of the Single currency will impact negatively on the performance of the dollar and the next chart evidently displays such relationship:

Dollar Index

The above reported chart evidently displays the natural inverse relationship between the 2 markets and the obvious contrary fluctuations can be used to hedge potential currency based portfolios. In particular, the Euro, which is considered to be a risky asset, is positively correlated to equity indices therefore a drop in the Single currency would automatically correspond to a rise in the Dollar index. However, the positive correlation amongst Euro futures and other equity indices (such as E-Mini S&P500, DAX or Nasdaq futures) would suggest that a down trend in the Single currency is likely to be accompanied by a retracement in many risky assets. Therefore, the Dollar Index, being a “contrarian market by default”, could be very useful to detect market nervousness and cover unstable positions in equities. In order to better understand how much the American currency is linked to its components we will analyze the volatility fluctuations of the most important exchanges: Euro, British Pound and Japanese Yen futures. The reason we chose these 3 currencies is related to their weights, in fact, the combination of the aforementioned asset classes is responsible for 82.2% of the Dollar Index’s oscillation (Euro 56.7% + Japanese Yen 13.6% + British Pound 11.9%).

Dollar Index

The calculation tracks the volatility performance of the Dollar Index, Euro futures, Japanese Yen futures and British Pound futures. It is evident that the oscillations are more or less correlated, in fact, the volatility decreased for all asset classes in January, April – May, August – September and November – December but drastically augmented in February, June – July and October. Undoubtedly, every market behaved differently in the short term but the macro movements are very correlated each other. This information is very important to forex traders because it proves that the Dollar Index, in terms of magnitude, moves as much as the other currencies implying that it could be used as a hedging tool when the Euro or the British Pound are in downtrend (the behaviour of the Japanese Yen is different and it will be treated separately later on). Let’s have a look at the relationships amongst the aforementioned exchanges from a quantitative point of view:

Dollar Index

The table presents the correlation and covariance of each asset class against the Dollar Index and the results stress what have been previously stated: the Index obviously has an extremely strong negative correlation to the price of the Euro (-0.93) and a strong negative correlation to the price of the British Pound (-0.66). However, Euro and British Pound volatilities display a strong positive connection to the Dollar Index, in fact, the correlation coefficients are +0.79 and +0.69 respectively. The positive connection in volatility is an obvious consequence of the fact that the rise in the Dollar Index will always be proportioned to the depreciation of the remaining currencies, hence, the movement will be fairly similar in terms of magnitude. The covariance of Euro and British Pound prices with respect to the value of the dollar (the covariance measures the mutual variability of 2 random variables) is negative, -0.16 and -5.04 respectively, which is another natural consequence of the appreciation of the American currency against European exchanges. On the other hand, the volatility covariance is positive, 1.95 and 1.73 respectively, implying that Euro and British Pound volatilities rise when the volatility of the Dollar Index rises and vice versa. The reason volatilities move in the same direction is because both markets observe a leverage effect process: the volatility tends to rise when the price action is in downtrend. Specifically, a drop in Euro and British Pound futures would probably cause an increase in market volatility but a rise in the Dollar Index would increase its volatility too. In other words, risky assets such as Euro or British Pound futures follow a leverage effect process while the Dollar Index is governed by an inverted leverage effect: the volatility increases with a larger buying pressure and decreases in case of a sell–off. Consequently, If Euro and British Pound futures are plunging they are effectively depreciating against the Dollar. Therefore, the volatilities of the 2 European asset classes, following a leverage effect process, will increase while the appreciation of the Dollar will push the Dollar Index up and the buying pressure would increase its variance too.

The Japanese Yen, instead, needs a different approach. First of all, it is necessary to remind that the Asian currency is often used as a hedging tool in portfolio management because many market participants rush to buy Yen when equities drop. Secondly, it is important to point out that the “safe haven role” played by the Japanese Yen has a clear implication: the volatility rises when the market heads north (if you are interested in hedging portfolio risk you may want to read the HyperVolatility research entitled “Portfolio Hedging: Risky Assets vs Safe Havens”). As we can see, Japanese Yen futures, as well as the Dollar Index, are popular assets when equity indices and single stocks are plummeting and their volatilities are both driven by an inverted leverage effect process. The positive price correlation between Dollar Index and Japanese Yen futures (+0.10) and the weak covariance (-0.26) evidently proves the point just made. The price correlation is clearly very low and the negative, although feeble, covariance indicates that the buying pressure was sometimes stronger for the Asian currency. The numbers suggest that in 2012 the Dollar Index and Japanese Yen futures have been both used as a hedging tool but in a diverse fashion because the buying pressure was evidently different.

Conclusions

The Dollar Index is a very simple to understand tool for traders and it is worth monitoring when investing. Here are some important points to bear in mind:

1) The benchmark value for the Dollar Index is 100. If the Index is below this level the dollar is depreciating against other currencies while a reading above this threshold implies an appreciation of the American currency

2) The Dollar Index is a weighted geometric mean whose components and respective weights are the following: Euro (56.7%), Japanese Yen (13.6%), British Pound (11.9%), Canadian Dollar (9.1%), Swedish Krona (4.2%) and Swiss Franc (3.6%)

3) The Euro is the currency that influences Dollar Index’s fluctuations the most

4) The volatility of the Dollar Index rises with an increasing buying pressure and decreases when the market goes down

5) Euro, Japanese Yen and British Pound are responsible for the 82.2% oscillations in the Dollar Index

6) The Dollar Index can be effectively used to hedge positions in Euro and British Pound futures

7) Japanese Yen futures do not show much correlation to the Dollar Index because of their “safe haven” characteristics

8) The Dollar Index, given its “contrarian nature” is a rather good asset to hold when risky assets (equities, stocks, etc) plunge

The HyperVolatility Forecast Service enables you to receive the statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial.

Options Greeks: Vanna, Charm, Vomma, DvegaDtime

The present article deals with second order Options Greeks and it constitutes the second part of a previously published article entitled “Options Greeks: Delta,Gamma,Vega,Theta,Rho”. Before getting started it is important to highlight the great contribution that Liying Zhao (Options Analyst at HyperVolatility) gave to this report. All the calculations and numerical simulations that will be shown and commented are entirely provided by Mr Zhao.

Second-order Greeks are sensitivities of first-order Greeks to small changes in different parameters. Mathematically, second-order Greeks are nothing else but the second-order partial derivatives of option prices with respect to different variables. In practical terms, they measure how fast first order options Greeks (Delta, Vega, Theta, Rho) are going to change with respect to underlying price fluctuations, volatility, interest rate changes and time decay. Specifically, we will go through Vanna, Charm (otherwise known as Delta Bleed), Vomma and DvegaDtime. It is important to point out that all charts have been produced by assuming that the underlying asset is a futures contract on WTI crude oil, the ATM strike (X) is 100, risk-free interest rate (r) is 0.5%, implied volatility is 10% while the cost of carry (b) is 0 (which is the case when dealing with commodity options).

Vanna: Vanna measures the movements of the delta with respect to small changes in implied volatility (1% change in implied volatility to be precise). Alternatively, it can also be interpreted as the fluctuations of vega with respect to small changes in the underlying price. The following chart shows how vanna oscillates with respect to changes in the underlying asset S:

Options VannaThe above reported chart clearly shows that vanna has positive values when the underlying price is higher than strike (in our case S>$100) and it has negative values when the underlying moves just below it (S<$100). What does that imply? The graph highlights the fact that vega moves much more when the underlying asset approaches the ATM strike ($100 in our case) but it tends to approximate 0 for OTM options. Consequently, the delta is very sensitive to changes in implied volatility when the ATM area is approached. However, it is important to point out that delta will not always increase if the underlying moves from, say, $80 to $100 because in many risky assets (stocks, equity indices, some currencies and commodities) the implied volatility is inversely correlated to the price action. As a result, if WTI futures go from $80 to $100 the implied volatility will probably head south and such a phenomenon would decrease vanna which, in turn, would diminish the value of delta.

Charm (or Delta Bleed): Charm measures delta’s sensitivity to a small movement in time to maturity (T). In practical terms, it shows how the delta is going to change with the passage of time. The next chart displays graphically the relationship between the aforementioned variables:

Options Charm

The chart suggests that, like in the case of vanna, the charm achieves its highest absolute values when the options are around the ATM area. Therefore, slightly in-the-money or out-of-the-money options will have the highest charm values. This makes sense because the greatest impact of time decay is precisely on options “floating” around the ATM zone. In fact, deep ITM options will behave almost like the underlying asset while OTM options with the passage of time will approach 0. Consequently, the deltas of slightly ITM or OTM options will be the most eroded by time. Charm is very important to options traders because if today the delta of your position or portfolio is 0.2 and charm is, for instance, 0.05 tomorrow your position will have a delta equal to 0.25. As we can clearly see, knowing the value of charm is crucial when hedging a position in order to keep it delta – neutral or minimize portfolio risk.

Vomma: Vomma measures how Vega is going to change with respect to implied volatility and it is normally expressed in order to quantify the influence on vega should the volatility oscillate by 1 point. The fluctuations of vomma with respect to S are shown in the next chart:

Options VommaAs displayed in the above reported chart out-of-the-money options have the highest vomma, while at-the-money options have a low vomma which means that vega remains almost constant with respect to volatility. The shape of vomma is something that every options trader should bear in mind while trading because it clearly confirms that the vega that will be influenced the most by a change in volatility will be the one of OTM options while the relationship with ATM options will be almost constant. This makes sense because a change in implied volatility would increase the probability of an OTM options to expire in-the-money and this is precisely why vomma is the highest around the OTM area.

DvegaDtime: DvegaDtime is the negative value of the partial derivative of vega in terms of time to maturity and it measures how fast vega is going to change with respect to the time decay. The next chart is a visual representation of its fluctuations with respect to the underlying asset S:

Options DvegaDtime

The above reported graph clearly displays that the influence of time decay on volatility exposure measured by vega is mostly felt in the ATM area especially for options with short time to maturity. The fact that DvegaDtime is mathematically expressed as negative derivatives makes sense because time decay is clearly a price that every options holder has to pay. In order to make things easier have a look at the plots of vega and theta because you will immediately realize that both volatility and time decay have their highest and lowest values in the ATM area. It goes without saying that ATM options have the highest volatility potential and therefore vega will be effected the most by the passage of time when the strike of our hypothetical options and the underlying price gets very close.

The HyperVolatility Forecast Service enables you to receive the statistical analysis and projections for 3 asset classes of your choice on a weekly basis. Every member can select up to 3 markets from the following list: E-Mini S&P500 futures, WTI Crude Oil futures, Euro futures, VIX Index, Gold futures, DAX futures, Treasury Bond futures, German Bund futures, Japanese Yen futures and FTSE/MIB futures.

Send us an email at info@hypervolatility.com with the list of the 3 asset classes you would like to receive the projections for and we will guarantee you a 14 day trial.

Options Greeks: Delta,Gamma,Vega,Theta,Rho

First of all I would like to give credit to Liying Zhao (Options Analyst at HyperVolatility) for helping me to conceptualize this article and provide the quantitative analysis necessary to develop it. The present report will be followed by a second one dealing with second order Greeks and how they work.

Options are way older than one might imagine. Aristotele mentioned options for the first time in the “Thales of Miletus” (624 to 527 B.C.), Dutch tulip traders began trading options at the beginning of 1600 while in 1968 stock options have been traded for the first time at the Chicago Board Options Exchange (CBOE). The pricing of options has always attracted academics and mathematicians but the first breakthrough in this field was pioneered at the beginning of the 1900 by Bachelier. He literally discovered a new way to look at option valuation, however, the real shift between academia and business occurred in 1973 when Black, Scholes and Merton developed the most popular and used option pricing model. Such a discovery opened an entire new era for both academics and market players. Being one of the most crucial financial derivatives in the global market, options are now widely adopted as an effective tool to leverage assets or control portfolio risk. Nowadays, it is easy to find articles, researches and studies on option pricing models but this article will instead focus on options Greeks and in particular first-order Greeks (derived in the BSM world). Options Greeks are important indicators for assessing the degree of risk coming from exogenous variables, in fact, they measure option premium’s sensitivities to small changes in different parameters. Mathematically, Greeks are the partial derivatives of the option price with respect to different factors such as volatility, interest rate and time decay.

The purpose of this article is to explain, as clearly as possible, how Options Greeks work but we will concentrate only on the most popular ones: Delta, Gamma, Vega (or Kappa), Theta and Rho. It is worth mentioning that all the charts that will be presented have been extrapolated by assuming that the underlying is a WTI futures contract, that the options have a strike price (X) of 100, that the risk-free interest rate (r) is 0.5%, that the cost of carry (b) is 0 while implied volatility is 10%.

Delta: Delta measures the sensitiveness of the option’s price to a $1 fluctuation in the underlying asset price. The chart displays how the Delta moves in respect to the underlying price S and time to maturity T:

options delta

The chart clearly shows that in-the-money call options have much higher Delta values than out-of-the-money options while ATM options have a Delta which oscillates around 0.5. Call options have a Delta which ranges between 0 and 1 and it gets higher as the underlying approaches the strike price of the option which means that out-of-the-money call options will have a Delta close to 0 while ITM options will have a Delta fluctuating around 1. Many traders think of Delta as the probability of an option expiring in-the-money but this interpretation is not correct because the N(d) term in its formula expresses the probability of the option expiring ITM but only in a risk-neutral world.  In real trading conditions higher Delta calls do have a higher probability to expire ITM than lower Delta ones, however, the number itself does not provide a reliable source of information because everything depends on the underlying. The Delta simply expresses the exposure of the options premium to the underlying: a positive Delta tells you that the premium will rise if the underlying asset will trend higher and it will decrease in the opposite scenario. Put options, instead, have a negative Delta which ranges between -1 and 0 and the below reported chart displays its fluctuation in respect to the underlying asset.

put option delta

It is easy to notice that as the underlying asset moves below the $100 threshold (the strike price of our hypothetical put option) the Delta approaches -1, which implies that ITM put options have a negative Delta close to -1, while OTM options have a Delta oscillating around 0. In practical trading the value of the Delta is very important because it tells you how the options premium is going to change in the case the underlying moves by $1. Let’s assume you purchase a 100 call options on crude oil with a Delta of +0.5 and the premium was $1,000.  If the option is at-the-money the WTI (the underlying asset) will be at $100 but if oil futures go up by $1 dollar to $101 the premium of your long call will move to $1,500. The same applies to put options but in this case the ATM Delta will be -0.5 and your long put option position will generate a profit if WTI futures move from $100 to $99.

Gamma: Gamma measures Delta’s sensitivity to a $1 movement in the underlying asset price and it is identical for both call and put options. Gamma reaches its maximum when the underlying price is a little bit smaller, not exactly equal, to the strike of the option and the chart shows quite evidently that for ATM option Gamma is significantly higher than for OTM and ITM options.

option gamma

The fact that Gamma is higher for ATM options makes sense because it is nothing but the quantification of how fast the Delta is going to change and an ATM option will have a very sensitive Delta because every single oscillation in the underlying asset will alter it.

How Gamma can help us in trading? How do we interpret it?

Again, the value of Gamma is simply telling you how fast the Delta will move in the case the underlying asset experience a $1 oscillation. Let’s assume we have an ATM call option on WTI with a Delta of +0.5 while futures prices are moving around $100 and Gamma is 0.08, what does that imply? The interpretation is rather simple: a 0.08 gamma is telling us that our ATM call, in the case the underlying moves by $1 to $101, will see its Delta increasing to +0.58 from +0.5

Vega (or Kappa): Vega is the option’s sensitivity to a 1% movement in implied volatility and it is identical for both call and put options. The below reported 3-D chart displays Vega as a function of the asset price and time to maturity for a WTI options with strike at 100, interest rate at 0.5% and implied volatility at 10% (the cost of carry is set to 0 because we are dealing with commodity options).

option vega

The chart clearly highlights the fact that Vega is much higher for ATM options than for ITM and OTM options. The shape of Vega as a function of the underlying asset price makes sense because ATM options have by far the highest volatility potential but what does Vega really tell us in real trading conditions? Again, Vega (or Kappa) measures the dollar change in case of a 1% shift in implied volatility, therefore, an at-the-money WTI options whose value is $1,000 with a Vega of , say, 100 will be worth $1,100 if the implied volatility moves from 20% to 21%. Vega is a very important risk measure for options traders because it estimates how your P/L is going to change as a function of implied volatility. Implied volatility is the key factor in options pricing because the price of a single options will vary according to this number and this is precisely why implied volatility and Vega are essential to options trading (the HyperVolatility Forecast service provides analytical, easy to understand projections and analysis on volatility and price action for traders and investors).

Theta: Theta measures option’s sensitivity to a small change in time to maturity (T). As time to maturity is always decreasing it is normal to express Theta as negative partial derivatives of the option price with respect to T.  Theta represents the time decay of option prices in terms of a 1 year move in time to maturity and to view the value of Theta for a 1 day move we should divide it by 365 or 252 (the number of trading days in one year). The below reported chart shows how Theta moves:

option theta

Theta is evidently negative for at-the-money options and the reason behind this phenomenon is that ATM options have the highest volatility potential, therefore, the impact of time decay is higher. Think of an option like an air balloon which loses a bit of air every day. The at-the-money options are right in the middle because they could become ITM or they could get back into the OTM “limbo” and therefore they contain a lot of air, consequently, if they have got more air than all other balloons they will lose more than others when the time passes. Let’s look at a practical example. Let’s assume we are long an ATM call option whose value is $1,000 and has a Theta equals to -25, if the day after both the underlying price and volatility are still where they were 1 day before our long call position will lose $25.

Rho: Rho is the option’s sensitivity to a change in the risk-free interest rate and the next chart summarises how it fluctuates with respect to the underlying asset:

option rho

ITM options are more influenced by changes in interest rates (negative Rho) because the premium of these options is higher and therefore a fluctuation in the cost of money (interest rate) would inevitably cause a higher impact on high-premium instruments. Furthermore, it is rather clear that long dated options are much more affected by changes in interest rates than short-dated derivatives. The below reported chart displays how Rho oscillates when dealing with put options:

put option rho

The Rho graph for put options mirrors what it has been stated for calls: ITM have a larger exposure than ATM and OTM put options to interest rate changes and long term derivatives are much more affected by Rho than in the short term (even in this case the 3-D graph displays negative values). As previously mentioned Rho measures how much the option’s premium is going to change when interest rates move by 1%. Hence, an increase in interest rates will augment the value of a hypothetical call option and the rise will be equal to Rho. In other words, the value of the call option will increase by $50 if interest rates move from 5% to 6% and our WTI call option has a premium of $1,000 but Rho equals 50.

As stated at the beginning of the present report this is only the first part and a second article dealing with second order Greeks will be posted soon.

Go back to top