Compiled by: Paul Thind – March 17, 2013 What is an Index? Any measure in the change in value of a property can be classified as an Index. One can, for example, start to measure the seasonal variation of temperature at a defined location. This measure over a specific number of days or minutes could be classified as an Index. Another example of an Index would be if one looked at average wind velocities at different locations over an area. The average of the measurements could be defined as the Wind Index for that area. In financial markets there are many indices which record the historical price (usually the Closing Price) of a given security or any number of them grouped together. There are innumerable economic indices, such as the development of a given countries inflation rate, unemployment rate, industrial production, currency value or the price of commodities. Indices are very useful in keeping track of the development in change. The Dow Jones Industrial Average is one of the oldest stock market indices. The price of Gold has been monitored for centuries. Stock Market Indices The development of a formal, regulated place or platform where people gather to exchange a common good has existed for a long time. The Amsterdam Stock Exchange is considered the oldest in the world. It was established in 1602 by the Dutch East India Company and it was the first company to issue stocks and bonds. It was later named the Amsterdam Bourse and was the first to begin trading securities. The Dow Jones Industrial Average (INDU Index) is a price-average of 30 blue-chip stocks that are “generally” the leaders in their industry is one of and most recognizable equity indices. INDU has been a widely followed indicator of the US stock market since 1928. It calculation predates to May 26, 1896. It didn’t always consist of 30 constituents. The INDU doesn’t cover transportation and utility companies for which there are the Dow Jones Transpiration and the Dow Jones Utilities Average.

The INDU doesn’t have strict rules for inclusion. According to the Dow Jones web site (http://www.djindexes.com/averages/) a stock typically is added to The Dow® only if the company has an excellent reputation, demonstrates sustained growth and is of interest to a large number of investors. Maintaining adequate sector representation within the indexes is also a consideration in the selection process.

When the Dow Jones Industrial Average was initially created, its values were calculated by simply adding up the component stock prices and dividing by the number of components. Today, the divisor is adjusted to smooth out the effects of stock splits and other composition changes.

The Standard and Poor’s 500 Index (SPX Index) is a market capitalization-weighted index of 500 stocks. This index is currently at a level of 1411.13 (August 24, 2012) and was developed with a base level of 10 for the 1941-43 base period. Although there are 500 companies in the Index, the top 45 companies represented 49.61% of the total weight as of close of 24 August 2012. Most countries now have their domestic stock exchanges and the corresponding stock market indices. Most use the market capitalization-weighted calculation method. Usually there are other criteria. Such as percentage of the issued shares which are listed on the exchange. In many developing countries, most big companies are owned by families which continue to have controlling stakes. The stock market listing is a mechanism to raise equity capital. Market Capitalization Stock Indices In many countries the largest companies are very much larger than the average. Market Capitalization indices have come to favor the largest companies. When these indices were conceived they were meant to be a simple reflection of the market structure, a rough measure of the relative importance of each company to economic value of the nations listed markets. In many countries, of course, the Private Sector is much bigger or very significant part of the economy. For many years it was not

possible to trade the indices as a whole. It was with the advent of the Capital Asset Pricing Model that the notion took hold that stock markets were the most efficient way to invest. It is no wonder that at least since the ideas of the stock markets being efficient and representing the value of the whole economy has been postulated investments have inadvertently flowed preferentially to the largest listed companies. The result is that the economic system favors large over small. This has been a major catalyst for the large getting ever bigger. The shares have been increasingly used as a currency for growing size through acquisitions. Thus if one invests in an Index such as the S&P 500 for the purpose of diversification, the objective is not achieved. At the time of writing, Apple Inc is the largest company in the United States, with a market capitalization of $621.71 billion followed by Exxon Mobile Corp with $406.43 billion market capitalization. While the stock market access favors the biggest, there are eventually limits to company size. These limits can be reached, either because businesses get too big to manage or competition erodes the financial advantages. Increasingly the regulatory framework aims to prevent monopolies in products and services by hindering mergers and acquisitions of companies or insisting on the breakup of monopolies. Market forces are still at work and in time limits to market size, pricing power or simple inability to find suitable companies to acquire can limit growth rates. Some companies can reach saturation in terms of their product offer. Automakers, detergent manufactures, telephone companies, airlines are some examples of companies which may have reached their saturations points for the time being for various reasons. The stock market indices can thus have very significant weightings in companies which are no longer growing. Despite this the growth of Index investing continues to advantage the big listed corporations. These recognitions on the short comings of the market indices have led to the growth of alternative market indices in recent years. Standard and Poor’s have come up with an equal weighted S&P 500 Index. The development of these alternative indices may itself improve the relative performance of the smaller companies. In time it may have the adverse effect of increasing the cost of capital for the larger companies and lead to reduced index performance. Custom Indices The custom index business is a relatively recent development in financial markets. Benchmark equity and bond indices have been around for a long time. The realization that the majority of active managers fail to match the performance of market indices such as S&P 500 or FTSE 100 led first to the development of index tracking mutual funds led by John Bogle, the founder of Vanguard Group, who started the First Index Investment Trust on December 31, 1975, which later became known as the Vanguard 500 Index Fund which aims to track the S&P 500 Index. This fund crossed $100 billion milestone in November 1999 and at the time of writing stands at $ 107.90 billion

(Bloomberg Ticker:

VFINX US Equity, Market Close 24 August, 2012). Strictly speaking the first S&P 500 index fund was started by John McQuown and David G. Booth at Wells Fargo and Rex Sinquefield at American National Bank in Chicago in 1973, but these were not made available to retail investors. Wells Fargo sold its indexing operations to Barclay’s Bank, which developed the iShares ETF business, which it sold to BlackRock in 2009, which became the world’s largest money manager. The success of market index, or beta, tracking funds led to the proliferation of similar products on every conceivable equity index as well as commodities such as Gold, Oil and Treasuries. This was aided by liquid futures and index swaps. The fact that most active managers do not beat beta indices after fees and expenses and acceptance of theory that concluded that markets are efficient (http://en.wikipedia.org/wiki/Eugene_Fama) and that market price fully reflects all available information and further that the marginal benefits of acting on information do not exceed the marginal cost (Jensen (1978)) gave further impetus to growth in index tracking (http://en.wikipedia.org/wiki/Jensen%27s_alpha).

The growth in index linked products took off with the rapid developments in derivatives markets and the parallel development of new types of wrappers in the form of Notes, Certificates and ETFs that could be listed on exchanges at low cost. In these formats it was also more efficient to offer capital protection and to sell directly to retail clients. Risk management tools such as CPPI (Constant Proportion Portfolio Insurance) and option pricing on indices, which is more difficult on unlisted funds, fed this growth. Benchmark bond and commodity indices have long existed, with the CRB being the first calculated by Commodity Research Bureau in 1957. The GSCI came in 1992, followed by DJ-AIG and RICI in 1998-1999. However, listed products on commodity and bond indices are quite recent. In recent years beta indices have been extended to include FX, rates, volatility, dividends, private equity, CTAs ,hedge funds. Even art and wine indices have been developed. The latter serve more as benchmarks instead of liquid investment vehicles. The goal to beat the performance of the market indices drives active fund management. In the form of rules based investing this is not a difficult task as market indices give the largest companies the highest weights. “Biggest” simply cannot continue to grow at high rates and many big companies remain big even when not performing. Given this, an obvious way to improve returns is to use equal weights. Other simple changes such as a re-balancing mechanism which takes profits on the performing stocks and redistributes into the cheaper ones achieves similar results. These types of index strategies can be termed enhanced beta as exposure is to a large number or all of market constituents. In addition to strong evidence that actively managed funds rarely beat market indices nothing illustrates this better than the research based indices published by S&P itself. According to S&P, the S&P ALL STARS Baskets include the highest ranked stocks in the S&P 500 or the S&P Europe 350 Indices, based on Standard & Poor’s world renowned, independent research and proprietary stock ranking system, STARS (Stock Appreciation Ranking System). STARS is a qualitative evaluation based on an analyst's determination of future appreciation potential of a specific common stock relative to its relevant S&P benchmark index based on a 12-month time horizon. The overarching investment philosophy driving the methodology is 'Growth at a Reasonable Price.' Construction of the Baskets is rules based, incorporating such factors as market capitalization, liquidity, and sector diversification. Each basket is equally weighted and rebalanced semi-annually to take into account any changes in ranking. We see that since 01/31/2005 to 8/31/2012 the SPX Index returned 4.49% per annum verses the All STARS Basket (SPALSTUS Index) giving only 2.27% per annum. The Fidelity Magellan Fund (FMAGX US Equity) invests in domestic and foreign company stocks has not beaten the SPX Index since 1985. It did extremely well in the years from 1963 to 1995. That was unlikely due to manager skills and was probably due to inside information. Alpha generation, or returns in excess of beta has been the focus of active managers, which they usually try to accomplish through stock picking. Unlike active management, transparent, rules based can improve on the beta indices in many ways. Simple momentum strategies which invest in the best performing sectors shows outperformance compared with market beta. Companies with strong balance sheets or those consistently paying out high dividends are also better long term investments compared with the markets. The first thematic custom index was developed by the author of this article in 2005

(Bloomberg ticker:

RBSZH20 Index). Recognizing that the water sector would be a long term high growth area, the index selected the largest and most liquid stocks from companies engaged in the water business. The S&P Custom/ABN AMRO Water Index was calculated by Standards and Poor’s. Another version included highest analysts recommendations to pick constituents from the liquid water companies universe. Certificates were listed on the European exchanges and the product was a huge success world-wide. The Index Methodology for the Water Index was quite simple. The first step is to create a list of companies which represent the Water Universe of Companies. These companies were drawn from a global search limited by first only Share Companies listed on official stock exchanges of Australia, Canada, New Zealand, EU Members, Norway Hong Kong, Singapore, Iceland, Switzerland, Japan and the USA. In addition to this rule Share Companies listed on official stock exchanges from Argentina, Bahrain, Brazil, Bulgaria, Chile, China, Colombia, Croatia, Egy[t, India, Indonesia, Israel,

Jordon, Korea, Malaysia, Mexico, Morocco, Nigeria, Oman, Pakistan, Peru, Philippines, Romania, Russia, Saudi Arabia, Slovakia, South Africa, Sri Lanka, Taiwan, Thailand, Turkey, Venezula, Zimbabwe were also included, but only the companies’ ADRs listed in New York or GDRs listed in London were considered. The key filter in the construction of all Custom Indices is the liquidity. Custom Indices are meant to be tradable. Liquidity is measured by the Average Daily Traded Volumes. Based on the size of the Universe it is usually common sense in determining how many constituents should be chosen which meet the liquidity criteria and the second criteria of Minimum Market Capitalisation. At the date of selection:

a. the Share Company must have a minimum total market capitalisation of min. USD 500,000,000 or the equivalent amount in another currency calculated by applying the Exchange Rate as published on the respective Bloomberg page <Bloomberg Code Equity DES>; if the company is covered by an analyst/analysts, at least 50% of the ANR for the respective Share Company shall be “buy” and/or “hold”. If a company is not or not yet covered by Analysts it may however qualify for Water Stocks Index membership or for the Water Reserve Universe;

b. at least 45% of the Share Company’s portion of business must derive from Water or Water related businesses as indicated in the last available quarterly and/or annual report. This percentage will be monitored on every Trading Day using the database of the financial market information provider Thomson (extel full reports section); Except from this rule are RWE AG and Suez SA due to their importance of water business each conducts on a global level.

c. the Share Company must show a 3 Month ADT of minimum USD 7,500,000. In case of the qualifying water companies, we realised that some very large multi-national companies had very big water business. The other businesses were in other utilities and waste management. Exclusion of these companies from the Water Universe didn’t make sense. *RWE AG and Suez SA have been initially selected as index components due to their importance of water business each conducts on a global level. The Water Stocks Index Components will be initially equally weighted (each Share Company at 10%) in the Water Stocks Index based on the Price of the Water Stocks Index Components on the Index Launch Date. The Index is re-weighted every year on the 1

st of October.

This Index is tracked in the form of a Certificate and is listed on the Swiss Exchange (ISIN: CH0023013623). What is significant is that since November 5, 2005, the Certificate, after all fees has performed much better than the Pictet’s actively Managed Water Fund (PICWAPA LX Equity). The annual return of 8.19% verses 5.75% for the Pictet Fund. The recognition of rapid growth in emerging markets, particularly the BRIC countries with their huge appetite for natural resources, led to development of indices capturing metals and mining companies, energy and infrastructure as well as indices focused on commodities and emerging markets in general. Not only were these indices expected to outperform the broader markets but were also to be used as building blocks. Indices capturing themes such as Renewable Energy, Luxury Goods, Precious Metals, Bio-fuels, Generic Drugs, Nano Technology, Sharia Compliant Indices, etc. followed. The RBS Global Metals and Mining TR Index (RBSZMMG) improve on the MSCI World Metal & Mining Index (MIGUMMIN). The difference in performance for the same dates from November 5, 2005 to August 26, 2012 is almost 1.50% higher, while the volatility is the same. What appeared from the launch of these indices is the Index methodology appears to be making improvement over diversified broad indices or funds. Our approach looks for:

(a) The most liquid stocks, and (b) A re-balancing mechanism which takes profits from the performing shares and reinvests in

the laggards.

(c) Exclusion of those companies that don’t meet minimum market capitalization criteria. There is sporadic evidence that investing in companies with high Environmental, Social Governance standards (ESG) yields better results in the long term. The initial emphasis in avoiding companies which sell unhealthy, addictive or life destroying products (tobacco, alcohol and weapons) has been widened to include CO2 emissions, equal opportunity, worker safety as well as responsible investing in terms of not being over leveraged and managing operational risks. The catastrophic spill by BP and the Fukushima Earth quake suggests that versions of Black Swans are not rare. Another approach to alpha generation is through the powerful concept of market timing. The RBS Alpha Indices rely on the basic premise that societies are organized around days of the week, month end, holidays and seasons. This organization impacts both behavior and money flows and can be captured in index construction. Indeed these alpha indices outperform market indices over medium and long terms. Long dated results shows that investing only for 6-7 days per month, in many cases produces superior returns compared with remaining invested for the whole time.

RB

SXA

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RB

SXA

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X In

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Annualised Return 0.78% -7.59% 3.45% 4.22%

YTD Return -6.18% 7.28% 3.85% 18.19%

1 Month Return 1.95% 6.86% 2.86% 9.09%

3 Month Return 3.80% 5.92% 6.96% 10.37%

6 Month Return -7.99% -5.98% -4.15% 1.55%

1 Year Return -6.38% 4.99% -2.95% 22.71%

3 Year Return 8.58% -14.27% 4.09% 26.29%

5 Year Return 0.50% -44.18% 7.85% -7.14%

Minimum Daily Return -17.21% -11.41% -11.20% -9.88%

Average Daily Return 0.01% -0.02% 0.02% 0.03%

Min. Monthly (Calendar) Return -18.64% -23.83% -10.66% -19.19%

Avg. Monthly (Calendar) Return 0.18% -0.46% 0.34% 0.52%

Maximum Drawdown -43.35% -61.37% -25.35% -54.77%

Annualised Volatility 20.01% 26.14% 14.10% 24.74%

Daily Standard Deviation 1.26% 1.65% 0.89% 1.56%

Monthly (Calendar) Standard Deviation 4.77% 6.10% 3.44% 5.99%

Return / Risk Ratio 0.04 -0.29 0.24 0.17

Return / Risk Negative Ratio 0.03 -0.36 0.16 0.22

Return / Maximum Drawdown Ratio 0.018 -0.124 0.136 0.077 The prevailing, economically damaging assumption is that stock market indices as represented by the S&P 500, Nikkei 225, FTSE100 and so on are efficient and in the long term the most optimal way to invest. Long term can turn out to be a very long time indeed. The active mangers use market benchmarks as references for performance measures as if that was the “risk free” benchmark. Past and recent stock market performance suggests that remaining invested through thick and thin does not achieve acceptable returns. There have been times when markets have moved sideways or down

for very long periods. Sometimes returns achieved over 10, 15 and even 20 years were lost in a matter of months. The Dow, for example, first crossed the 200 level in 1927 and then only decisively moved above it at the end of 1949. The Nikkei first cross the 10,000 level at the end of 1983, it is still below that level, even though the high point reached was in 1989. Equity Market Neutral Indices Market Neutral as the name implies are portfolio constructions that have no net exposure to an underlying asset class. Thus a portfolio of equities consisting of long and short positions in equal measure can be considered to be market neutral. Market neutrality can be expressed on many measures. The most common is some form of analysis of company fundamental value measures. Usually companies are chosen from each sector. Other simple methods look at being long of high momentum shares and short of low momentum shares. An anti-momentum approach (mean reversion) can also be used. The rationale here is that stocks that have oversold, bounce back to their means and stocks that have risen also correct back to their means. The general purpose of Market Neutral strategies is to try and isolate alpha and lowering risks compared with being long or short the whole time. In general Market Neutral Indices are long of one portfolio of stocks and short and another portfolio or short of an index such as the SPX Index. These are also considered to be market neutral because one can use the money from the short position to buy the long positions. RBS Alpha Centurion uses short term mean reversion for selecting long and short positions which are constantly managed in order to take advantage of movements of prices away from their mean prices and to keep market neutrality. There are a number of indices for different market regions. One can also be market neutral with just a pair of stocks. Companies in the same sector may end up experiencing quite different prospects. A recent example is the decline of Nokia and the rise of Apple Inc. When using a pair trade it is best to not just buy and sell an equal position in dollar terms. It is always good to pay attention to volatility of each stock and try and frequently match the volatility. Even when the fundamental story for a given company is much worse than the other, it can happen that the share get over sold and can bounce back and visa versa. It is better to use a signal for the direction of the trade. The trend of a fixed ratio of the two stocks should indicate the general direction of the long/short position. A better measure is the simple 20 day moving average of the risk adjusted returns of the trade. With any pair trading the changes in volatility on the position always need to be combined with the knowledge of which of the two pairs is contributing the volatility. One can use this to reverse of come out of the trade. In this example if Apple Inc is becoming more volatile, it is a signal to reverse the trade. Dynamic Strategies Our research shows quite conclusively that stock markets and investments in other universally tradable assets such as commodities, Gold, Oil, Credit is extremely risky if one remains invested for the long term. It came as a surprise to many that in high risk environments everything becomes correlated. The best performing assets were those that had embedded capital protection, which product providers achieve through de-risking. That one should reduce risk when volatility is high is obvious. At RBS a large number of dynamic indices which react to prevailing market conditions have been created. Many use volatility and market trend measures. Capping or targeting volatility at acceptable pre-set levels permits pricing of option strategies previously not possible. Astute investors have embraced these developments. Other RBS indices use an array of measures such as trend, volatility control, sentiment indicators, yield curves, mean reversion and so on to anticipate price development in order to de-risk, re-balance or switch into safer assets. Our market neutral strategies which use long and short positions also reduce risk and aim to achieve absolute returns.

More recently we have been working to use volatility as an asset class as well as using this measure as a risk management tool. Future work is increasingly focused on identifying, monitoring and optimizing variables that explain or influence market prices and to then use these to structure products. Dynamic strategies add to our understanding of how markets work. RBS is a market leader in using security specific and exogenous variables in the construction of dynamically managed, transparent, rules based indices. Much still needs to be accomplished. Volatility Ether is one of the most volatile substances. Without a lid ether evaporates very quickly. In financial markets the value of financial assets can declines substantially without volatility control.

According to Wikipedia (http://en.wikipedia.org/wiki/Volatility_(finance), in finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices. An implied volatility is derived from the market price of a market traded derivative (in particular an option).

Current volatility can be calculated over any time period. It simply looks at variation of prices over a given number of days during a specified time period. In order to make comparisons the annualized volatility is used. Again the Wikipedia description is good enough to define volatility.

Mathematical definition (http://en.wikipedia.org/wiki/Volatility_(finance)#Mathematical_definition)

The annualized volatility σ is the standard deviation of the instrument's yearly logarithmic returns.

The generalized volatility σT for time horizon T in years is expressed as:

Therefore, if the daily logarithmic returns of a stock have a standard deviation of σSD and the time period of returns is P, the annualized volatility is

A common assumption is that P = 1/252 (there are 252 trading days in any given year). Then, if σSD = 0.01 the annualized volatility is

The monthly volatility (i.e., T = 1/12 of a year) would be

The formula used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a random walk, or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time

periods is more complicated. Some use the Lévy stability exponent α to extrapolate natural processes:

If α = 2 you get the Wiener process scaling relation, but some people believe α < 2 for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot, who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with α = 1.7. (See New Scientist, 19 April 1997.)

Why is volatility important for investors?

“He was drunk. He didn’t know what he was doing. One minute he appeared to be calm, the next he was excited. I didn’t know what to do. So I kept my distance. Suddenly he swung the bat in his hand and shattered everything and from then on nothing was the same again between us. We lost our possessions and trust in the relationship. It was all so complicated. Everything is now in ruins.”

Volatility is important because people don’t know how to react in conditions of euphoria or distress. In general people (money managers) panic or are forced to cut losses when there are wide swings in prices. The flight to safety is a natural instinct in living organisms. High risk conditions get extrapolated into the future and the resulting uncertainty leads to risk aversion.

Generally future estimates of volatility (risk) end up being too high compared with what is actually realized in due course as the future transpires. This is particularly so in time of high or rising volatility. Conversely when risk starts to dissipate, future or implied volatility can fall much more rapidly and the historical measure can lag. The observed relative measures of short and long term volatility (the volatility term structure) can thus be important in taking positions in volatility or the underlying asset on which these measures are being observed.

Because the causes of volatility (price swings) can be so varied and many, predicting volatility is extremely complicated. We shall address the issues relating to complexity in the sections below.

Volatility is important for investors because our financial markets have evolved in a way that investors can make bets on the possible future outcomes for asset prices. Instead of buying a risky asset by paying 100% of the value of the asset, investors can take a bet that prices will rise or fall. The cost of these bets (options) is very small in comparison. For example, at the time of writing the level of S&P 500 Index is 1402.80 (9

th Aug, 2012 closing price). This means that in order to have exposure to one

unit of S&P 500 an investor would have to come up with $1402.80. If the investor had a view that the S&P 500 was going to rise between now and December 22, 2012 the investor can pay $53.20 or 3.792% of the current price and have the right to buy the S&P 500 at $1400.00.

The Black-Scholes option pricing model makes certain assumptions about how asset prices evolve in the market place, but at its core lies the observed historical distribution of prices and the volatility of returns of the given asset.

Black–Scholes formula

The value of a call option for a non-dividend paying underlying stock in terms of the Black–Scholes parameters is:

The price of a corresponding put option based on put-call parity is:

For both:

is the cumulative distribution function of the standard normal distribution

is the time to maturity

is the spot price of the underlying asset

is the strike price

is the risk free rate (annual rate, expressed in terms of continuous compounding)

is the volatility of returns of the underlying asset

The terms are the probabilities of the option expiring in-the-money under the equivalent exponential martingale probability measure for the stock and the equivalent martingale probability measure for the risk free asset, respectively. The risk neutral probability density for the

stock price is

where is defined as above.

Specifically, is the probability that the call will be exercised provided one assumes that the

asset drift is the risk-free rate. , however, does not lend itself to a simple probability

interpretation. is correctly interpreted as the present value, using the risk-free interest rate, of the expected asset price at expiration, given that the asset price at expiration is above the exercise price.

Setting aside the complicated variables, the Black-Scholes formula essentially is a relationship between the option price and the stock (asset) prices which says that for in the money options the price of the option moves is equivalent to the movement of the price of the stock (asset). Any deviation from this relationship can be brought into line by increasing or decreasing the exposure to the stock (asset).

The variables

The most important variable is Delta, or the rate of change of the option price change with changes in the Spot price of the asset. Vega measures sensitivity of option value to volatility. Theta is the sensitivity to time. Rho measures sensitivity of option price to interest rates.

Gamma measures the rate of change of delta to spot prices. Other second and third order sensitivity measures are beyond the scope of this discussion.

Key points

The key point to make is that the Black-Scholes formula is not simple and as complex as it is, it is not a reflection of how real markets function. The Black–Scholes model of the market for a particular stock makes the following explicit assumptions:

There is no arbitrage opportunity (i.e., there is no way to make a riskless profit).

It is possible to borrow and lend cash at a known constant risk-free interest rate.

It is possible to buy and sell any amount, even fractional, of stock (this includes short selling).

The above transactions do not incur any fees or costs (i.e., frictionless market).

The stock price follows a geometric Brownian motion with constant drift and volatility.

The underlying security does not pay a dividend.

Black and Scholes showed that under these assumptions “it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock.

Real markets are far more complex, where prices do not smoothly transition to new levels driven by supply and demands. Event risks, such as the Earthquake in Fukushima, the crash in stock markets in 1987, disruptions which can arise due to outbreak of a virus, collapse of a company (Enron), discovery of false accounting (Madof), political tensions (Arab Spring) and financial crisis (Europe) can all have unforeseen consequences. Whilst it may be still possible to hedge a given exposure, the costs associated with such hedging when one is too late can be very significant.

Unlike the tossing of a coin or a game of roulette, in finance worked out probabilities are usually not realized probabilities.

The nature of volatility

Just as with human emotions, the nature of volatility is complex and hard to predict. One observes that the more emotional a person is, the harder it is to predict his or her actions. Actions themselves can have consequences, which can lead to more unpredictable behavior. In financial markets the anticipation of an event risk can cause volatility to rise. The event itself may cause further spikes not because the event poses further risks but because the reaction to the event triggers reactions in the market which take on a life of their own. It is the human reactions to a changing environment which causes reaction and thus impacts the movement in asset prices.

Since the 1960’s when the option pricing models were built there has been vast changes in the way markets function. Since those times global trade and investments have been liberalized, globalization has gained momentum as a direct result and through technology development, markets are less regulated, machine trading has come to the fore, dark pools have become prominent, hedge funds have gathered huge sums, the derivatives markets have grown, there is far more information and noise in the system and leverage in the system has increased. Increased leverage can translates to essentially more risks in the global financial markets. Quite often price movements result from disruptions rather than orderly discovery of prices. In modern finance, automated trading front runs the slower human reactions to change. The financial system is tilted to advantage the few. It is not a level playing field.

Most investors cannot be expected to remain abreast of the vast amount of information generated and be able to make decisions and react. The result is that most individual investors (even most professional investors, mutual fund managers and hedge fund managers) who participate in the financial markets lose money, compared with a simple long only market index, because markets have become a lot more complex. Algorithm driven volumes exceed order based turnover on stock exchanges.

Extreme volatility in recent years and lack of performance by the major stock markets for a number of years has shattered the theory (which was always only a theory) that there is a tenable relationship between risk and returns. One has to conclude that one cannot simply remain invested in the long term in order to have a rosy future. A lifetime of savings can get wiped out in a matter of months.

Waiting for asset prices to recover may take another lifetime and when adjusted for purchasing power parity from peak to the new peak, it may not even happen.

Volatility itself is very volatile. This means that when volatility increases, the extent or the duration of the volatility continuing to rise or how long it might stay elevated cannot be easily predicted. Volatility can spike and is particularly susceptible to contagion effects. In the financial system the transmission of risk can be both rapid or remain hidden ad become inflated until it bursts like a dam. For example, a bank or fund manager may be holding a risky asset and the value starts to erode. When values erode, liquidity usually declines, which effectively mean that their ability to sell the position becomes diminished. There comes a time that the market price may not reflect the true risk and the Mark to Market (MtM) of valuation can no longer be used. Market prices are usually inaccurate in these conditions as demand and supply for very small trades can end up being used for valuation. The realized prices for liquidating a portfolio may be considerably lower. The position may cause other funding constraints under such conditions and the holder of the risky asset may have to resort to selling their more liquid holdings. While this increases liquidity the risk on the books has become more concentrated.

What is available as tradable financial instruments on volatility?

A number of products which use volatility as an asset class or as a way to protect portfolios exist. The VIX Index is the most popular measure of volatility. It is a measure of the 30-day expected or implied volatility on option prices on the S&P 500 Index. CBOE SPX Volatility Index (VIX Index) reflects the market estimate of future volatility based on the average of the implied volatilities for a wide range of strikes. 1

st and 2

nd month expirations are used until 8 days from expiration, then the 2

nd and 3

rd are

used. The VIX index was launched in 1993 and revised in 2003. Exchange traded VIX Options were launched in 2004 and 2006.

The key characteristic of the VIX Index is that it is negatively correlated to the stock markets. This is same as saying that when risks rise stock markets fall. It should be stressed and we have pointed out before that when stock markets fall, volatility can jump a lot more and that at low levels of stable volatility, stock markets can continue to rise. The implications are that volatility can be used as a Portfolio Insurance concept, meaning that a small allocation to the volatility index can keep the whole portfolio stable during times of high volatility. The graph below illustrates this point.

The VSTOXX is the equivalent for the DJ Euro STOXX 50 Index. There is also VNAA on the Nasdaq-100; VDAX Index on the DAX-30; VFTSE on the FTSE-100, VSMI on the SMI and VHSI on the Hang Seng Index, VNKY on the Nikkei None of these indices are investable. However there are futures contracts on these traded on various exchanges. Chicago Board of Options Exchange (CBOE) has futures on the VIX Index as well as on Brazil and Russian Indices. Deutsche Borse trades the futures on the DAX and so on. Some are more liquid than others. Some of the tickers for the futures contracts are:

UXA Index

VNAA Index

VDAX Index

VGA Index

As forward views can be expressed over different periods, the forwards markets are usually designed with fixed rolls for 1-Month, 2-Month, 3-Month etc. forwards.

There are volatility products offered as Exchange Traded Notes (ETNs) or Exchange Traded Funds (ETFs). VXX US Equity is an ETN offered by Barclay’s Bank. VIXY US Equity is an ETF offered by ProShares.

How does the VIX Futures differ from other Futures Markets?

Most futures markets trade on the assumption that a futures contracts held can either be rolled into a new futures position or the position has to be settled by taking delivery in the case of a long position or making delivery of the eligible underlying asset in case one is holding a short position in the futures. For example, when one buys a futures position the 10-Year German Government Bond there are eligible cash bonds which can be delivered. The same applies with Crude Oil, or Gold and all other asset classes such as currencies.

As it is difficult to deliver 500 shares just in the right proportions to match the final settlement price, in the case of equity futures such as the SPX Index, a settlement value is determined by the Futures Exchange on the day before the Final Settlement Date. The same applies to the VIX Futures. CBOE VIX Futures are cash settled at the open, always thirty days before a final settlement of S&P 500 options.

The liquidity of the VIX Futures is a major concern for an instrument that is designed to encourage hedging or even speculation on risk. The VIX market is not efficient in the sense that there is a balanced supply and demand for the instrument. It is more a case of a herd mentality where demand for risk protection is sought by everyone simultaneously and the number of providers of liquidity is limited.

Fair Value for the VIX Futures

Generally the Futures prices should not diverge very far from the Fair Value of the reference. In the case of commodities the Fair Value could be considered as the price of the spot and the cost of carry to buy the spot (and the cost of delivery). In Bond futures, it is simply the cost of buying and holding a deliverable bond, the earned accrued interest to the contract expiry date, minus the cost of funding to the expiry date. Calculation of Fair Value is thus quite easy. There is often arbitrage between future prices and the deliverable instruments. In large liquid markets such arbitrage should not persist.

In the case of the VIX Futures, the fair value calculation is more complex and unreliable. The fair value for the VIX cannot be computed using the cost of carry approach because the cost of the VIX position can become very substantially different from the VIX Futures because the VIX Futures is designed to reflect a consensus view of future (30-day) expected stock market volatility. Fair Value estimates for the VIX Futures are extremely difficult to calculate. The only observation one can make is that on the expiry date of the VIX futures contract the spot price and the Fair Value should be the same and this tends to happen.

VIX Futures, possibly because it is a relatively new market, tend to trade away from calculated Fair Value. This tendency may be due to the structure of the VIX Index itself as it impacts supply and demand. By design the option prices on which the VIX Index is calculated have ‘profit’ elements from the traders contributing prices and the number of contributors is limited to the active members who

can provide option quotes. This may be causing distortions between Fair Value of the VIX Futures and the corresponding estimates derived from S&P 500 calendar options. For further reading on VIX Fair Value calculation, please refer to: http://cfe.cboe.com/education/vixprimer/Features.aspx . The author believes that this methodology suggested by the CBOE is not correct for estimating Fair Value.

Short comings of the VIX Futures Market

The VIX Futures is an estimate of implied volatilities of 30-day options starting at some future date as we have explained. The S&P 500 Index options market is quite liquid. The VIX Futures market is a forward start date estimate on these option values. One can imagine that in time of high risk no one is willing to price instruments that estimate what might happen in the future to the one month volatility. The options that are traded on the exchanges only reflect the implied volatility of options with different expiry dates. These are not forward starting options. The VIX Futures values are trying to reflect the implied volatilities in hypothetical forward starting options.

Volumes in the VIX Futures can vary widely during the day. Banks are not willing to, at the time of writing to give fills based on Volume Weighted Averages Prices (VWAP). Trying to execute trades in size at closing prices is almost impossible in any size.

Volatility Term Structure

If one extracts the volatility implied in the pricing of options, one can look at option prices for the SPX Index for different periods and see what the term structure for volatility is at any one time. Clearly if one assumes that volatility is a measure of risk or uncertainty in the markets, one can imagine that the term structure of volatility should contain important information of how volatility and hence asset prices are expected to develop in the future. At the time of writing (August 22

nd, 2012) the VIX Index is

at 15.02. The three month volatility index (VXV Index) is 18.80%.

One can see that if the term structure is positive sloping, (which is usually the case in normal market conditions) rolling a long position in the shorter term 1-Month VIX contract will result in a negative carry. Generally, the shorter dated volatility is more volatile as it reacts to current events and current events get discounted in the future. Another way to view this is that the further one goes to the future, the more likely that implied volatility will tend to the long term average.

Term structure can clearly shift from upward sloping to flat to downward sloping. There is thus possibility of term structure arbitrage or using the term structure in an optimal way for hedging risk. The shift in term structure from upward to downward sloping usually happens in times of extreme stress. Under normal conditions it is better to take long volatility positions in the longer dated VIX contracts as the negative carry on these is less than the front month contracts.

Portfolio Protection

The tradable instruments which aim to protect portfolios are usually long of volatility. Dynamic exposure to long volatility for protecting portfolio is of growing interest. The idea is both to lower ‘insurance’ costs and to improve risk/return by not having the volatility protection as a drag on performance of the portfolio. Clearly the time to be long of volatility in general is when volatility is rising. Buying protection is cheap when volatility is low. Volatility on equity indices usually does not fall below certain levels. For example, it would be very unusual to have a volatility of 10% p.a. for long periods on the SPX. Similarly it would be unusual to have a very high level of volatility, say above 70-80% p.a. for sustained periods. Another characteristic of the VIX Index is that it can spike or decline quite fast compared with the S&P 500. This is called ‘convexity’. This can have the advantage that the hedge ratio, just as in insurance, can be small. The aim should be to monitor and try and match the beta of the portfolio and the hedge and not to use the hedge the whole time.

An alternative approach to estimating the hedging is to use prices of call options on the VIX Index and a put option on the portfolio to be hedged (S&P 500 for example). The VIX futures contract that

closely matches the option premiums paid and the option premium received on the S&P 500 Index for a fixed maturity call and put should be used.

There is no direct relationship between volatility and price. Volatility only tells us to what extent prices can be expected to move from their mean over a given time period. It is a measure of the amplitude of the movement. Volatility measures in themselves do not say anything about the direction of prices. It could happen that in conditions of relative low volatility asset prices can continue to decline. The economic conditions may be on a slow declining path in an otherwise stable market, interest rates could be at lows. Demand and supply could be in close balance. On the other hand, absolute high levels of volatility are not necessarily bad for performance. In general the direction of volatility or the sign and degree of the slope of the volatility is important driver for direction of prices. Having said this, high volatility can be considered as a higher level of uncertainty in prices. Generally in times of uncertainty demand for risk declines and asset prices fall. In general this negative correlation between asset prices and volatility is quite persistent.

Volatility as an asset class

The use of Volatility as an asset class is increasing. There are several ways to use volatility in asset allocation:

1. Use volatility based products in asset allocation 2. Use measures of volatility to determine weights 3. Use volatility as an indicator for increasing or decreasing risk

Examples of tradable volatility instruments:

The S&P 500 VIX Futures Series are traded on the CBOE with different expiries. Ticker, UXA Index is the Active Contract. There is also a Mini Futures Contract on the VIX, ticker: MVI1 Index.

VXX US Equity – iPath S&P 500 VIX Short-Term Futures ETN aims to track the Short-Term VIX Futures TR Index after expenses (expense ratio is 0.89%).

This product has lost 94.93% of its value since its launch. It is not meant to be used as an investment product. It is a good short term hedging instrument or is used in arbitrage strategies by hedge funds, again on a short term basis. One can obviously see that shorting this instrument would have produced very high returns. The large loss has resulted mainly from the upwardly sloping term structure.

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There are several ETN and ETFs on the S&P 500 VIX Futures. VIX IM Equity tracks the performance of the S&P 500 VIX Futures Enhanced Roll Index (SPVIXETR Index)

JPUSSTBL Index – J.P. Morgan Strategic Volatility Beta Index is a strategy that uses the term structure of the VIX by being long of the longer dated VIX futures (2-Months) and short of the shorter dated VIX futures (1-Month). This shorting of the short dated futures mitigates the losses which result from the positively sloping term structure we mentioned above (contago). This index gained 124.41% over the last three years and since its launch on June 20, 2008 USD 100 invested would have grown to 472.25 as of today’s date (August 22, 2012).

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