Making Money with statistical Arbitrage: Generating Alpha in sideway Markets with this Option Strategy

Table of Contents

Part Ι
1.1 Abstract
1.2 Structure
1.4 List of Abbreviations
1.5 List of Figures and Tables

Part ΙΙ
2.1 Overview of the Hedge Fund Industry

Part ΙΙΙ
3.1 Hedge Fund Strategies Overview
3.2 Statistical Arbitrage in Detail
3.3 Performance Analysis

Part ΙV
4.1 State of the Art in Research
4.2 Principles of Garch
4.3 Introduction of a Semi -Variance Model
4.3.1 Methodology
4.3.2 Description of Market Data
4.3.3 Prediction Power
4.3.4 Risk Measurement
4.4 Backtest with Real Options
4.4.1 Out-of-Sample Market Data
4.4.2 Performance Comparison

Part V
5.1 Conclusion
5.2 Further Research

Part VI
6.1 List of Literature
6.2 Appendix

1.4 List of Abbreviations

illustration not visible in this excerpt

1.5 List of Figures and Tables

Figure 1: Hedge fund strategies overview

Figure 2: Average hedge fund strategy returns as of 2009 (Absolute-report (2010))

Figure 3: Chart pattern conversion to binary code

Figure 4: DAX-Chart from 1990-2010

Figure 5: JB-test on 30 days DAX returns

Figure 6: Negative and positive skew

Figure 7: Historical DAX upper and lower semi-variance

Figure 8: Model Accuracy on DAX

Figure 9: Garch and semi-variance-model in comparison.

Figure 10: Option spread strategy

Figure 11: Payoff-diagram of options spread strategy

Figure 12: Hedge fund annualized return comparison (CSFB (2209))

Figure 13: Hedge fund index performance (Absolute-report (2010))

Figure 14: Worldwide investment fund assets by (EFAMA (2010))

Figure 15: Composition of worldwide investment fund asset (EFAMA (2010))

Table 1: Monthly performance (CSFB (2010))

Table 2: Value at Risk (5%, 20day) for DAX, Garch and Semi-variance-model

Table 3: Shortfall of DAX, Garch and Semi-variance-model

Table 4: Overview of performance measures on backtest

Table 5: Options traded for the backtest

Table 6: Largest hedge funds (Market Folly(2010))

1.1 Abstract

In the following study I am going to present a short survey of the hedge fund industry, its regulation and the existent hedge fund strategies. Especially statistical arbitrage is explained in further detail and major performance measurement ratios are presented. In the second part, I am going to introduce a semi-variance model for statistical arbitrage. The model is compared to the standard Garch model, which is so often used in daily option trading, derivate pricing and risk management. Because investment returns are not equally distributed over time, sources for statistical arbitrage occur. The semi-variance model takes skewness into account and provides higher returns at lower volatility than the Garch model. The concept is aimed to be a synopsis of mean reversion and chart pattern detection. The computer model is generated with respect to Brownian motion and technical analysis and provides significant returns to the investment. As market efficiency hypothesis states the impossibility of arbitrage opportunities over the long run, on the other hand market anomalies significantly outstand. Connecting both elements creates a profitable trading system. The combination of both approaches delivers a sensible hedge fund concept. The out-of-sample backtest verifies out-performance and implies the need for further research in the area of higher moment CAPM and additional market timing strategies as sources of statistical arbitrage.

1.2 Structure

After giving a short survey of the hedge fund industry and the existent hedge fund strategies, statistical arbitrage is explained in further detail. In the second part a semi-variance model is compared to the Garch model. Its prediction power is stress tested and a backtest on the DAX30-Index performed. Therefore Value at Risk, Expected Shortfall and many other risk measurement ratios are taken into consideration. The appendix presents further ideas for research and possible expansions of the model.

PART II

2.1 Overview of the Hedge Fund Industry

The hedge fund industry has grown from as few as 300 funds in 1990 to more than a $2 trillion industry with 10,000 active funds today. EFAMA (2009) stated that investment fund assets worldwide reached EUR 15.9 trillion at the end of 2009. The very first hedge fund was started by Alfred W. Jones in 1949. By using leverage and short selling, he effectively "hedged" risk in the marketplace. Hedge funds can take both long and short positions, make concentrated investments, use leverage or derivatives, and invest in many markets. This is in sharp contrast to mutual funds, which are highly regulated and cannot easily take advantage the same breadth of investment instruments. Many hedge fund strategies seek to reduce market risk specifically by shorting equities or derivatives. There are a multitude of different strategies used by hedge fund managers today, and the term "hedge" doesn't always apply, since many of these funds are not hedged at all. In fact, many hedge funds take positions that are often highly speculative. Hedge funds are thought to provide returns that are uncorrelated with traditional investments. Their returns over a sustained period of time have outperformed standard equity and bond indexes with less volatility and less risk of loss than equities. Investing in hedge funds tends to be favoured by more sophisticated investors, who understand the private consequences of major stock market corrections. Many endowments and pension funds allocate assets to hedge funds . Most hedge funds are highly specialized, relying on the specific expertise of the manager or management team. Hedge funds are primarily organized as private partnerships to provide maximum flexibility in constructing a portfolio. Most hedge fund managers commit a portion of their wealth to the funds further aligning their interest with that of other investors. Many hedge fund strategies, particularly arbitrage strategies, are limited as to how much capital they can successfully employ before returns diminish. As a result, many successful hedge fund managers limit the amount of capital they will accept. It is important to understand the differences between the various hedge fund strategies. Investment returns, volatility, and risk vary among the different hedge fund strategies. Some strategies are able to deliver consistent returns with extremely low risk of loss, while others may be as or more volatile than mutual funds. The main purpose of hedge funds is to achieve positive returns independently of capital market movements.

Hedge Fund Risks

Investing in hedge funds implies several forms of risk. The appetite for risk can involve large losses (For example by strategies involving out the money option writing, leverage, short selling or high yield bonds, distressed securities and collateralized debt obligations based on sub-prime mortgages). Not only the known factors of operational and market risks have to be taken into account, but also the risks of transparency and regulation. Additionally to fewer jurisdictions from regulators, there are also less public disclosure requirements.

Fees

a) The manager's performance fee is calculated as a percentage of the fund's profits, usually counting both realized and unrealized profits. Performance fees align the interests of manager and investor more closely than flat fees do. Typically, hedge funds charge 20% of returns as a performance fee.
b) A high water mark means that the manager receives performance fees only on increases in the net asset value of the fund in excess of the highest net asset value it has previously achieved.
c) Some managers specify a hurdle rate, signifying that they will not charge a performance fee until the fund's annualized performance exceeds a benchmark rate.

The legal structure of a specific hedge fund – in particular its domicile and the type of legal entity used – is usually determined by the tax environment of the fund’s expected investors. Regulatory considerations will also play a role. Many hedge funds are established in offshore financial centres so that the fund can avoid paying tax on the increase in the value of its portfolio. An investor will still pay tax on any profit it makes when it realizes its investment, and the investment manager, usually based in a major financial centre, will pay tax on the fees that it receives for managing the fund.

Around 60% of hedge funds were registered in offshore locations in 2009. The Cayman Islands was the most popular registration location and accounted for 39% of the number of global hedge funds. It was followed by Delaware (US) 27%, British Virgin Islands 7% and Bermuda 5%. Around 5% of global hedge funds are registered in the EU, primarily in Ireland and Luxembourg

To verify a non-fraud, offshore management companies often provide the latest audited annual and semi-annual report without being obliged to. Moreover the contractual terms and conditions and sales prospectuses or equivalent documents, information on organization, management, investment policy, risk management and custodian bank or a comparable institution and information on investment restrictions, liquidity, the extent of the leverage and the carrying out of short sales are published.

PART III

3.1 Hedge Fund Strategies Overview

Abbildung in dieser Leseprobe nicht enthalten

Figure 1

In current practice, hedge funds strategies are classified following CSFB (2010) and HFR (20110). For classification in literature please read the papers by Injic and Heen (2003) or Agarwal and Naik (2000). The following overview is based on Dieter Kaiser (2003).The distinction between the different hedge fund strategies is often based on the following criteria: style, markets, instruments, exposure, sectors, method and diversification.

3.1.1 Long/Short Equity – make money on alpha

Overview

Long/short equity hedge fund managers buy undervalued stocks and sell the overvalued. Stock-picking is the key to outperformance in this discipline. Long-term holding, patience, and strong discipline are often required until the ultimate intrinsic value or potential worth is recognized by the market. Many managers use pair trades, where one stock is sold against another. Both stocks are of the same sector. If there are more stocks bought than sold, it is called a long bias. Vice versa a net short position is called a short bias. Typically a long bias implies directional trading in sense of a bull market.

Equity Hedge

The paradigm of equity hedge is absolute return achieved independently from the actual market movement. The fund can be market neutral or net long or net short - conditionally on the market situation. The manager combines leverage, short positions and stock picking. Partly derivatives are used.

Equity-Non-Hedge

Mutual funds concentrate on stock picking and research, whereas equity-non-hedge also has the freedom to apply leverage. This is the only distinction.

Short Selling

Short selling strategies sell securities without coverage. Thereby anticipating to rebuy them at a future date at a lower price. The manager assesses the securities to be overvalued.

3.1.2 Relative Value – Make Money on Spreads

Overview

Anomalies in the prices of identical underlyings at different exchanges are exploited. The riskless profit is called an arbitrage. Market risk should be decreased to zero. Market neutrality is not to be mistaken for “risk less”. Sometimes leverage up to 100 times of equity value is used in order to sufficiently generate profits on spreads.

Fixed Income Arbitrage

Bonds and other fixed income derivatives are traded in fixed income arbitrage. The strategy is based on historical correlation of yields followed by a change. Fund managers supposed the spread to mean revert to historical levels and take a profit. Often similar underlyings are traded like government bonds versus non government bonds with AAA rating

Credit Spread Arbitrage

Credit risks are valuated by the fund manager and the underlying securities are bought long or short. The strategy is similar to equity long/short, but dependant on the evaluation of the issuer’s credit rating. It can be via derivatives, asset backed securities, collateralized debt obligations or the underlying itself.

Capital-Structure-Arbitrage

This form of arbitrage profits from excess demand and psychological factors. Those are reflected in the short term imbalance of price between junior and senior debt.

Yield Curve Arbitrage

Given the actual yield curve, government bonds and its difference to corporate bonds, this hedge fund strategy gains value by yield curve changes.

Mortgage-Backed Securities

Hedge fund managers buy mortgage-back securities with the right of early repayment thus the risk premium is higher. The manager has to control two risks. If interest rate decreases, the house owner will refinances with lower interest rate and the security is paid out. On the other hand, if interest rate rises too far, the security will be reduced in price due to the initiation of higher yield instruments. To cover for duration exposure similar securities or derivatives are sold.

Convertible Bond Arbitrage

This strategy seeks to profit from the pricing of the embedded option in a convertible bond. Often used is a long convertible position with a corresponding short position in the underlying stock. Varying degrees of leverage are employed with this strategy. The combination results in a positive net cash flow whether the stock price increases or decreases.

Stock-Index-Arbitrage

A stock basket is traded against its representative index. Relative value discrepancies are skimmed with options or futures. Long or short positions in both directions are common.

Split Strike Conversions

Additionally to the underlying, a call with higher strike is sold, whereas a put with lower strike is bought. In both market phases the position will be profitable. In a down market the call premium delivers value, while the put hedges the underlying. In an up market the performance is limited to the call strike minus put premium. Only in a side market the paid put option premium is not offset by stock return. This strategy can also be inversed.

Statistical Arbitrage

Statistical arbitrage usually engages in very short term trading and residual mean reversion. Statistical computer models trade on quantitative models with high frequency. The portfolio often is market neutral.

3.1.3 Event Driven – make money on events

Overview

This strategy aims to profit from price imbalances resulting from a specific event or transaction in a life cycle of a business - for example, merger, hostile takeover, or leveraged buyout.

Merger Arbitrage

This strategy involves taking position in companies that are either currently or likely to be engaged in corporate mergers and acquisitions. Shares of the target are purchased, whereas stocks of the acquirer are sold. Risks are regulatory, financial, company-specific or antitrust reasons. The profit is made on the narrowing spread.

Distressed Securities

This strategy invests in illiquid debt and/or equity of firms in or near bankruptcy in order to profit from a potential recovery. As well as long positions in the stock, short positions in the debt are applied. These profits are generated from the market’s lack of understanding of the true value of the deeply discounted securities. Results generally are not dependent on the direction of the markets.

High Yield

Low rated bonds with no historical earnings or complete revenue track record, but high yield is part of the portfolio in this strategy. The implied risk is offset by put options on the bond or call options on stocks which would profit on the default of the underlying.

Special Situations

This strategy invests in event-driven situations such as mergers, hostile takeovers, reorganizations or leveraged buy outs. It may involve simultaneous purchase of stock in companies being acquired and the sale of stock in its acquirer, hoping to profit from the spread between the current market price and the ultimate purchase price of the company. The manager may also utilize derivatives to leverage returns and to hedge out interest rate or market risk. Results are generally not dependent on direction of market.

Regulation D

Private equity transactions can be found in this section. In exchange for equity a convertible is issued which is not exchange traded. The warrant has no fixed strike price. The hedge fund manager sells additionally to the long convertible position a stock and an option. At maturity the conversion is discussed with the company. Until maturity a sale of the convertible is not possible on exchange, because there is no liquid market.

3.1.4 Global Macro – make money on trends

Overview

An opportunistic, "top-down" approach is implemented by managers using this macroeconomic strategy. Trades are based upon major changes in the global economy, including interest rates and currencies, as well as changes in the economic policies of specific countries. Macro strategies use leverage. Relative valuation of financial instruments is within the strategy as well as market dynamics and sentiment. The manager tries to exploit perceived divergences between and within various asset classes. Returns are made by a correct forecast of global market performances.

Currencies

The fund manager encumbers with debt in a currency he sees overvalued and invests in a currency which is relatively undervalued. While the manager focuses on relative value trends, three-point-arbitrage is mentioned here as a spread strategy. In three-point-arbitrage currency is changed back over two other currencies (for example USD/EUR to EUR/CHF to CHF/USD).

Emerging Markets

Investments in equity or debt of emerging markets which tend to have higher inflation and volatile growth. Some emerging market countries are Brazil, China, India, and Russia. The major emerging market areas are Latin America, Eastern Europe, Asia, and the Pacific Rim. Various asset classes with different strategies are used. Short selling is not permitted in many emerging markets and therefore effective hedging is often not available and buy-and-hold-strategies are most common.

Market Timing

Market timing strategies switch among various asset classes to time price movements in different markets. Stocks, bonds, mutual funds, and money market funds are some of the asset classes used. Assets among different asset classes are allocated depending on the manager’s view of the economic or market outlook. Market movements and the timing entry and exit from markets are predicted by trend following strategies.

Commodities

A commodity strategy invests in material goods or its related futures. Examples are gold, oil, coffee or wheat. At the end of the holding period a profit can be generated by selling the underlying or the good. Arbitrage strategies can be applied hereby as well. Transition to the managed futures strategy is smooth.

3.2 Statistical Arbitrage in Detail

Origins of Statistical Arbitrage

Statistical arbitrage originated in the 1980s from the hedging demand created by Morgan Stanley's equity block trading desk operations. If the firm purchased a large block of shares, it would short a closely correlated stock to hedge against any major downturns in the market. Traders soon began to think of these pairs not as a block to be executed and its hedge, but rather two sides of a trading strategy aimed at profit making rather than simply hedging.

Statistical arbitrage refers to highly technical short-term mean-reversion strategies involving large numbers of securities, very short holding periods and substantial computational, trading, and IT infrastructure. It involves data mining and statistical methods, as well as automated trading systems. Statistical arbitrage is actually any strategy that is bottom-up, beta-neutral in approach and uses statistical or econometric techniques in order to provide signals for execution. Signals are often generated through a contrarian mean-reversion.

The goal is to construct a tradable stationary process so that trades are entered when the process reaches an extreme value, and exited when the process reverts to some mean value. Since market inefficiencies are generally small in magnitude, so transaction costs are one reason why inefficiencies remain.

Statistical arbitrage is subject to model weakness as well as stock-specific risk.

The statistical relationship on which the model is based may be spurious, or may break down due to changes in the distribution of returns on the underlying assets. Factors which the model may not be aware of having exposure to, could become the significant drivers of price action in the markets, and the inverse applies also. On a stock-specific level, there is risk of M&A activity or even default for an individual name. Such an event would immediately end any historical relationship assumed from empirical statistical analysis.

Statistical arbitrage has also caused some major problems, however. The most readily apparent was the Long Term Capital Management collapse, which almost left the market in ruins.

In the academic literature statistical arbitrage is opposed to deterministic arbitrage. In deterministic arbitrage a sure profit can be obtained from being long some securities and short others.

a) For arbitrage of typ 1, there is a sure payoff in the future, without the merit to pay anything today or even getting some money already now.

[Abbildung in dieser Leseprobe nicht enthalten](1)

[Abbildung in dieser Leseprobe nicht enthalten](2)

where p’ is the price vector and N denotes the quantities of involved assets in a matrix. X is the payoff matrix.

b) Arbitrage of typ 2 delivers a payment today, but there is surely no payment in the future.

[Abbildung in dieser Leseprobe nicht enthalten](3)

[Abbildung in dieser Leseprobe nicht enthalten](4)

c) The combined contingent claim of both formulas must thus have a price of:

[Abbildung in dieser Leseprobe nicht enthalten](5)

If this price is violated, an arbitrage opportunity is possible by the Law of One Price. The portfolio can be replicated and sold with profit.

In statistical arbitrage there is a statistical mispricing of one or more assets based on the expected value of these assets. Statistical arbitrage conjectures statistical mispricing or price relationships that are true in expectation, in the long run when repeating a trading strategy.

[Abbildung in dieser Leseprobe nicht enthalten](6)

Statistical arbitrage opportunity is a zero-cost trading strategy for which the conditional expected payoff in each final state of the economy is nonnegative. Unlike a pure arbitrage opportunity, a statistical arbitrage opportunity can have negative payoffs provided that the average payoff in each final state is non- negative (Bondarenko (2003)).

[Abbildung in dieser Leseprobe nicht enthalten](7)

In the later analysis of statistical arbitrage option strategy the expected payoff will play a major role.

3.3 Performance Analysis

As many hedge fund manager aim to outperform the market, we have to find means how to compare the outcome of their results. As many strategies can provide very high returns, they often are also very volatile. The relation of profit and the amount of risk to achieve this result have to be in a reasonable proportion. The following chapter should help in getting a clearer view. Useful insights can be delivered by alpha, standard deviation, lower partial moments, drawdown and finally Value at Risk models.

Alpha Measures the value that an investment manager produces, by comparing the manager's performance to that of a risk-free investment (usually a Treasury bill). Alpha can also be used as a measure of residual risk, relative to the market in which a fund participates.

[Abbildung in dieser Leseprobe nicht enthalten](8)

where Abbildung in dieser Leseprobe nicht enthalten is the portfolio’s return, Abbildung in dieser Leseprobe nicht enthalten the return of the market, Abbildung in dieser Leseprobe nicht enthalten the risk-free rate and Abbildung in dieser Leseprobe nicht enthalten is the sensitivity of the expected asset returns to the expected market return. A fund manager thus only has positive alpha, if the funds performance is lager than the risk-free rate and moreover provides a risk adequate profit compared to the whole market as benchmark.

Performance Measures

A detailed overview of many performance measures is given by Martin Eling (2006).

There are many important ratios in literature. The maybe most important and most cited is the sharpe ratio by Modigliani(1997).

[Abbildung in dieser Leseprobe nicht enthalten](9)

where Abbildung in dieser Leseprobe nicht enthalten is the portfolio’s return, Abbildung in dieser Leseprobe nicht enthalten the risk- free rate which is sometimes also adjusted to a higher hurdle rate for more exact benchmarking and the standard deviation Abbildung in dieser Leseprobe nicht enthalten. This ratio compares the excess return to the volatility inhered.

Lower Partial Moments

Lower partial moments calculate the lower deviation from the given expected return. They are often measured by higher exponents to take investor’s risk aversion into account.

[Abbildung in dieser Leseprobe nicht enthalten](10)

where Abbildung in dieser Leseprobe nicht enthalten is the average return and T is the number of regarded periods. First order partial moments can be interpreted as failure probability while lower partial moment of second can also be interpreted as the semi-variance.

Related performance measures are Omega Abbildung in dieser Leseprobe nicht enthalten by Shadwick and Keating (2002), Sortino-RatioAbbildung in dieser Leseprobe nicht enthaltenby Sortino and van der Meer (1991) and Kappa Abbildung in dieser Leseprobe nicht enthalten by Kaplan and Knowless (2004). All three measures set relative outperformance in relation to the adjusted risk.

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[Abbildung in dieser Leseprobe nicht enthalten](12)

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Upside-Potential-Ratio

Opposite to the average return is the upside-potential-ratio by Sortino, van der Meer and Plantinga ( 1999).

[Abbildung in dieser Leseprobe nicht enthalten](14)

Here upside movements are compared to downward movements relative to the average return or benchmark.

Drawdown

Young (1991), Kestern (1996) and Burke (1994) use a maximum drawdown method.

This method is often used in commodity trading since spot prices do not fall down to zero. Goods tend to retain a minimum price as opposed to defaulted equities.

Young (1991) refers to the maximum drawdown Abbildung in dieser Leseprobe nicht enthalten in his Calmar-ratio.

[Abbildung in dieser Leseprobe nicht enthalten] (15)

Whereas Kestern (1996) considers the mean drawdown to be a better performance indicator in his Sterling-ratio.

[Abbildung in dieser Leseprobe nicht enthalten] (16)

Burke (1994) expands this thought by adjusting the average drawdown to a ratio similar to the variance calculation procedure. Bigger drawdowns thus affect the ratio more than small drawdowns. The Burke-ratio is denoted as

[Abbildung in dieser Leseprobe nicht enthalten](17)

Value At Risk

Under Value at Risk the approximate loss with a specific probability is calculated under normal distribution assumption.

[Abbildung in dieser Leseprobe nicht enthalten](18)

where Abbildung in dieser Leseprobe nicht enthaltenis the assigned value in the standard normal distribution.

Adjusting the Value at Risk to CVaR, we get the maximum shortfall. This ratio explains the expected loss in the case of a significance level violation.

[Abbildung in dieser Leseprobe nicht enthalten](19)

The Value at Risk can also be modified by a Cornish-Fischer-method and adjusted to skew and kurtosis.

Abbildung in dieser Leseprobe nicht enthalten

[Abbildung in dieser Leseprobe nicht enthalten] (20)

where Abbildung in dieser Leseprobe nicht enthalten is the skew and Abbildung in dieser Leseprobe nicht enthalten the excess of kurtosis. Both are defined as:

[Abbildung in dieser Leseprobe nicht enthalten] (21)

[Abbildung in dieser Leseprobe nicht enthalten] (22)

In application we get the Excess Return on Value at Risk by Dowd (2000)

[Abbildung in dieser Leseprobe nicht enthalten] (23)

The Conditional Sharpe ratio by Artzner, Delbaen, Eber and Heath (1999)

[Abbildung in dieser Leseprobe nicht enthalten] (24)

And the Modified Sharpe Ratio by Gregoriou and Gueyie (2003)

[Abbildung in dieser Leseprobe nicht enthalten] (25)

Please read Martin Eling (2006) for a complete analysis of all performance measurement ratios introduced above. The performance measurement ratios are necessary and will be applied later to the model.

Reality Check

Hedge funds returns have differed strongly in the year 2009. Convertible Arbitrage and long/short equity have been the most successful strategies in 2009. One of the weakest performers has been equity market neutral. The following figure displays the wide spread of returns between fund performance within one strategy.

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Figure 2

CSFB (2010) provides a survey of compared sharpe ratios. The global macro strategy showed the highest sharpe ratio of 0,85. Dedicated short bias has been the worst strategy with -0,34.

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Table 1

This short overview does not focus on describing all the differences in historical data and discussing the advantages and disadvantages regarding portfolio selection, but wants to introduce a semi-variance-model for statistical arbitrage.

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