derivative pair trading

INDEX

Chapter

Contents Page No.

1 Introduction 7

1.1 Equity Market Neutral Strategy 7

2 Pair Trading 13

2.1 Key Characteristics 14

2.2 Instruments Used 16

2.3 Benefits of Pair Trading 17

3 Strategy 19

4 Pair Trading Model 20

4.2 Screening Pairs 20

4.3 Trading Rules 20

4.4 Trading Period 21

4.5 Excess Return Computation 22

4.6 Strategy Profits 22

4.7 Risk Control 23

5 Steps in Pair Trading 26

6 Cases of Pair Trading 27

6.1 Punjab National Bank – Bank of India 27

6.2 Aban offshore - Shiv-Vani oil & Gas Exploration 37

6.3 Punjab National Bank – Bank of Baroda 32

7 Conclusion 41

8 Bibliography 42

1 | P a g e

Executive summary

This project links uninformed demand shocks with the profits and risks of pairs trading. Usually

employed by sophisticated investors, pair trading is a relative value strategy that simultaneously

buys one stock while selling another. In a market with limited risk bearing capacity, uninformed

demand shocks cause temporary price pressure. A pair of stock prices that have historically

moved together diverge when subjected to differential shocks. Uninformed buying is shown to

be the dominant factor behind the divergence. A strategy that sells the higher priced stock and

buys the lower priced stock earns returns in excess of the market. The marked-to-market returns

of a pairs trading strategy are highly correlated with uninformed demand shocks in the

underlying shares. Measuring pairs trading profits represents a concise way to quantify the costs

of liquidity provision (i.e., the costs of keeping relative prices in line.)

It introduced to me strategies like risk management in convergence trading and understand the

importance and implications of such strategies as market neutral strategies. It provided me a rare

opportunity to understand importance of various statistical tools in the world of investments.

strategies like pair trading explore the temporary mispricing between assets and develops a

framework to take advantage of this temporary mispricing.

Market provides opportunities and it is individuals who should look at this opportunity in correct

perspective. Pair’ trading is useful in markets full of uncertainty.

2 | P a g e

INTRODUCTION

Pairs trading is a type of relative value strategy that buys an overpriced security and

simultaneously sells a similar, underpriced security. Traders typically track a pair of securities

whose prices move together. When prices diverge, they buy the down stock and simultaneously

sell the up stock. Traders profit if prices converge but lose money if prices diverge further. Pairs

trading has generated hundreds of millions of dollars in profits for companies such as Morgan

Stanley and D.E. Shaw.

Studying pairs trading broadens our understanding of financial markets. Because pairs trading

entails risk taking. Profits need not be thought of as coming from a narrow Wall St. strategy.

Rather, readers can think of these arbitrageurs as playing a vital role in the relative pricing of

securities. Profits are compensation for performing this service. Equivalently, readers can think

of profits as compensation for providing liquidity during times of differential market stress (e.g.,

stresses that affect some stocks but not others.)

Surprisingly, relative value strategies have received little attention in the academic literature. The

most notable paper is by Gatev, Goetzmann, and Rouwenhorst (2003) and offers a

comprehensive analysis. The authors use daily US data from 1962 to 2002. They show a simple

pairs trading rule produces excess returns of 11% per annum. Returns have high risk-adjusted

alphas, low exposure to known sources of systematic risk, cover reasonable transaction costs,

and do not come from short-term return reversals as documented in Lehmann (1990). It is

observed that uninformed trading shocks can explain the profitability of pairs trading.

Second, and much more importantly, we link uninformed trading shocks to the profitability of

pairs trading. We show that uninformed net buying is significantly correlated with a pair’s initial

price divergence. Additionally, uninformed trading is a significant “factor” in explaining the

strategy’s marked-to-market returns. These results suggest that pairs-trading strategies are

profitable because they identify situations with temporary price pressure. The strategy has low

risk because a position is effectively hedged by an offsetting position with similar factor

3 | P a g e

loadings. Execution is simplified and costs kept to a minimum because the offsetting position is

limited to a single stock.

Market neutral strategies are trading strategies that are widely used by hedge funds or proprietary

traders. A trader goes long certain instruments while shorting others in such a way that his

portfolio has no net exposure to broad market moves. The goal is to profit from relative miss

pricings between related instruments—going long those that are perceived to be under priced

while going short those that are perceived to be over priced—while avoiding systematic risk.

Market neutral strategies are sometimes called relative value strategies.

Pairs vs. spreads

Many traders think of a pair as a “spread” trade, but this comparison is not quite accurate. A

spread trade creates either net long or net short exposure, but a properly executed pairs trade is

dollar-neutral. By maintaining a market-neutral position, the effects of market direction can be

largely eliminated from the trade.

Consider the following comparison of a spread trade vs. a pairs trade:

Stock A: 20 per share

Stock B: 10 per share

Spread trade

Long 100 shares of stock A: 2,000

Short 100 shares of stock B: 1,000

Net long: 10 per share (1,000)

This is a hedged, bullish position.

Pairs trade

Long 100 shares of Stock A: 2,000

Short 200 shares of Stock B: 2,000

4 | P a g e

Net long/short: 0

This is a true market-neutral position.

Scenario 1

Both stocks rise 50 percent.

Stock A: 30

Stock B: 15

Scenario 2

Both stocks fall 50 percent.

Stock A: 10

Stock B: 5

A spread trade is a market bet with a built-in hedge, while a pairs trade is a market-neutral

position. In the first scenario’s bull market, the spread trade gains 500 (Stock A’s 1,000 profit -

Stock B’s 500 loss), and the pairs trade is flat (Stock A’s 1,000 profit - Stock B’s 1,000 loss).

In the second scenario’s bear market, however, the spread trade loses 500 (Stock A’s 1,000 loss -

Stock B’s 500 profit) as the pairs trade stays flat. Here, the spread trade loses money despite both

stocks dropping by an equal percentage. In both scenarios, the specifics of either stock had no

effect on price — the entire move is explained by the broader market fluctuations.

The trade must be market-neutral to ensure it won’t lose money unless there’s a change in

relative performance (i.e. one stock performs better than the other).

Similarly, if both stocks dropped by 5, the spread trade would remain flat even though Stock A

outperformed Stock B (Stock A loses 25 percent, while Stock B loses 50 percent). A trade can

only capture this relative performance if the trade is neutral.

Equity Market Neutral: An Overview

There are numerous strategies that generally fall under the market neutral umbrella. Market

neutral strategies are designed to benefit investors in all market conditions. They are deemed

market neutral because the direction of a particular market, whether up or down, should bare

5 | P a g e

little or no impact on the ability to generate a return. Such strategies include convertible

arbitrage, fixed income arbitrage, merger arbitrage and equity market neutral. Equity market

neutral, also known as statistical arbitrage or pairs trading, involves the trading of securities that

are interdependent. There is generally a price correlation between stocks in the same sector. If

the sector rallies, most of the companies go up in price. Share prices of competent companies

have a tendency to increase faster than that of weaker companies. The same holds true during

declines.

A market neutral manager in this case will go long on the competent companies and short-sell

the weaker companies in the same sector. Risk is mitigated through a consummated relationship

between the long and short positions in the portfolio. Diversifying across the entire market

breadth while pairing equal long and short positions within the same sectors provides for a

statistical advantage. The manager is concerned with capturing a return through the spread of the

long and short positions, as opposed to calling a market direction. The objective is to generate a

return without taking significant directional bets. If constructed properly, the elements of an

underlying stock market should have no bearing on the portfolio returns. An equity market

neutral strategy warrants more efficient use of information. Long-only mangers look at

companies and purchase their stock with the anticipation that the share price will increase in

value. If an underlying company does not meet the manager’s stringent purchasing requirements,

the company is passed over and the information is discarded. A market neutral manager will

utilize such negative information by short-selling the stock and positioning it in the portfolio to

reduce volatility.There are potentially three sources of returns in an equity market neutral

strategy. The first comes from the long side of the portfolio. Through a sophisticated process, the

manager determines what companies are suitable for buying. Profits are generated when the

stocks in the long portfolio rise in price. The second way to profit in an equity market neutral

strategy is from the short portfolio. Once again, companies are selected through a very

sophisticated process but, unlike the long portfolio, the manager looks for statistics that would

suggest that the share price of a company is unattractive. At this point, those particular shares are

borrowed from a broker and sold to generate the short portfolio. The idea behind short-selling is

to buy these same shares back at a future date for a lesser price and replenish what was borrowed

from the broker. Profits are generated if the stocks in the short portfolio decrease in price.

6 | P a g e

The third source of returns in the equity market neutral strategy comes from the proceeds of the

short sale. When the manager sells the stocks that were borrowed from the broker, cash is raised

from the proceeds and is typically reinvested in T-bills. This is known as the short rebate. Keep

in mind that stocks in the long and short portfolio are not randomly selected, but are chosen to

form a codependent relationship.

There are tremendous advantages to having an equity market neutral style in an investment

portfolio. Notably, returns are independent and uncorrelated to market direction. Volatility is

usually low. Returns are often attractive and constant regardless of market or economic

downturns. Equity market neutral strategies often complement other investment strategies,

providing a balanced and diversified portfolio.

As with any investment strategy, equity market neutral is not infallible. There are several

concerns with which managers are faced. The strategy by nature is extremely complex.

Sophisticated and expensive computer models are used to analyze data and assist in determining

long and short positions for the portfolio. The stock selection criteria can vary from manager to

manger, producing very different returns and levels of volatility.

Trading within an equity market neutral strategy can be very costly. The portfolio is turned over

often, to rebalance the long and short positions. As a result of the short positions, trading is

usually double that of a long-only portfolio.

Another concern is the limited availability of stocks for the short-sell. Simply put, not all stocks

can be shorted. Managers will short stocks that have a higher degree of liquidity. The quantity of

stocks obtainable for shorting may be limited. This can present capacity issues within the

portfolio. Managers are occasionally affected by the way the markets will value stocks as a

whole. If investors run up the stock price and reward companies that would generally be deemed

inefficient, a majority of the short positions in the portfolio could be perversely affected.

Managers will try to identify such a trend and compensate by reversing the shorts and

establishing long positions on the inefficient companies.

Style drift can often creep into the construction and management of an equity market neutral

portfolio. The emotional propensity or bias toward a specific sector or stock can lead to increased

7 | P a g e

volatility. Managers struggle to maintain a relatively small net exposure, (the difference between

the long and short positions in the portfolio).Despite these obstacles, equity market neutral

strategies have performed exceptionally well over the last ten years. The benefits definitely

outweigh the pitfalls. With the ability to generate an absolute return coupled with low volatility,

equity market neutral strategies are well suited for today’s hostile investment environment.

8 | P a g e

PAIR TRADING

Pairs trading refers to opposite positions in two different stocks or indices, that is, a long

(bullish) position in one stock and another short (bearish) position in another stock. The

objective is to make money on the relative price movements between them. The two stocks

might both go up, but the stock you are long will go up more and faster than the stock you are

short. Or, the two stocks might both go down, but the stock you are short will drop more and

faster than the stock you are long. One half of the pairs trade may be profitable, and the other

half of the pairs trade may lose money, but the goal is for the profits to exceed the losses.

The investment strategy we aim at implementing is a market neutral long/short strategy.

This implies that we will try to find shares with similar betas, where we believe one stock will

outperform the other one in the short term. By simultaneously taking both a long and short

position the beta of the pair equals zero and the performance generated equals alpha.

Pair of stock prices that have historically moved together in a correlated manner, diverge when

subjected to differential demand shocks. position on such a pair when its components diverge

and unwinding the position when they next converge has existed since the early periods of stock

trading.

The starting point of this strategy is that stocks that have historically had the

same trading patterns (i.e. constant price ratio) will have so in the future as

well. If there is a deviation from the historical mean, this creates a trading opportunity, which

can be exploited. Gains are earned when the price relationship is restored

Pair trading is a non-directional, relative value investment strategy that seeks to identify two

companies with similar characteristics whose equity securities are currently trading at a price

relationship that is out of their historical trading range. This investment strategy entails buying

the undervalued security, while short selling the overvalued security, thereby maintaining market

neutrality.

9 | P a g e

Key Characterisitics

This definition lays out three main areas of focus that play out as subtexts to the overall idea of

pairs trading and must be considered and understood before the unified strategy will make sense:

Market neutrality, relative value or statistical arbitrage and technical analysis.

Market neutrality is the first of the three major features of pairs trading selected for investigation.

The term “market-neutral” has come to be a quite appealing label in the last several years and

can refer to a wide variety of strategies. Many investors mistake the term to mean “risk free”.

This misconception has been narrowly focused on in the marketing of these types of products,

and, often, the label is applied to anything that could be considered, even loosely, something that

reduces market exposure or systematic risk.

A market-neutral strategy derives its returns from the relationship between the performance of its

long positions and its short positions, regardless of whether this relationship is done on the

security or portfolio level.

The pairs system is essentially an arbitrage system that allows the trader to capture profits from

the divergence of two correlated stocks. Pairs trading contain elements of both relative value and

statistical arbitrage in that it often uses a statistical model as the initial screen for creating a

relative value trade. A careful pairs trader will perform several layers of analysis on top of the

model output before any pairs are actually executed

While it is certainly possible to create fundamentally driven pairs trades, the methodology

suggested uses technical to perform the majority of the analysis required before trading;

fundamentals are used simply as an overlay to ensure that there is no glaringly obvious reason to

avoid a trade not captured in the technical indicators examined.

When pair trading involves trading two correlated stocks; sell short one stock while

simultaneously buying the other. The position has “hedged” itself to the market and therefore the

market is free to do what it wants. If the market goes down, the short position should make

money. If it goes up, long position should make money. Of course, while each side the trade is

making money, there’s the other side that is losing money.

10 | P a g e

Correlation is calculated by dividing the covariance of the percentage changes of each stock or

index divided by the product of the standard deviations for the two stocks. Covariance is a

measure of the tendency of the two stocks or indices to move together, and dividing the

covariance by the standard deviations sets the correlation between +1 and -1. The question when

measuring the correlation coefficient between two stocks is about how much data to use. The

correlation calculated using six months of daily data will almost certainly be different from the

correlation and beta calculated using three years of monthly data. A good starting point is to use

the correlation for approximately the same number of days the stock is expected be held for the

pairs trade.

A technique that doesn’t rely on more sophisticated statistical tests is to look at a range of dates

for the calculations, say 30 days, 60 days, 90 days, and 120 days and see how similar the

correlations are between them. The more similar they are, the more possibility that the two

stocks or indices will continue to have that relationship.

Beta is another tool used in pairs trading that predicts the behavior of one stock based on

information about another stock. It is a coefficient that measures the magnitude of the

relationship between two stocks or indices and is calculated with a linear regression model.

In the regression, the set of one stock’s percentage returns is set as the independent variable, and

the other stock’s percentage returns is set as the dependent variable. The beta indicates the

magnitude of the relationship of the independent variable relative to the dependent variable. In

trading terms, beta indicates how much a stock will move when another stock or index moves

1%. Beta is usually displayed as the percentage that a stock moves against a particular index.

Beta is used to determine how many share of each stock to execute for the pairs trade.

Because beta measures the magnitude of the relationship between two stocks or indices, you can

apply beta to the delta of the positions to determine the quantity for each stock in the pair.

Remember that delta is an estimate of how much an option will change in value for a 1.00

change in the stock price.

For example, if stock A has a beta of 2.00 relative to stock B, then if stock B moves up 1%, then

stock A is expected to move up 2%. That means 200 shares of stock B are needed to have the

same potential risk/reward profile as 100 shares of stock A.

11 | P a g e

That way, one is not significantly riskier than the other, and the deltas can be roughly equivalent

in the pairs positions.

Once it is determined that how many deltas are for each stock or index of the pairs for trade,

trades that will give you the correct relative exposure can be found.

Instrument Used for Pair Trading

Pair trading strategy can be used

a. Between the STOCKS

b. Between the OPTIONS

c. Between the STOCKS and OPTIONS

Stocks are relatively easy to execute in actively traded stocks, but have virtually unlimited risk if

you’re wrong. That is, if you expect that a spread between two stocks will revert to a mean, but if

it does not, you can lose a lot of money on both the long and short stock positions of the pairs

trade. I use stocks only when I am highly confident in the trade.

Options are a good vehicle for pairs trading, and can simply be used as stock substitutes: long

calls for long stock, long puts for short stock. Options have limited risk, but can be tougher to

execute quickly.

Options also usually have higher “slippage” in execution than stocks do. Also, buying options

has its own risks, such as time decay and exposure to drops in implied volatility (vega). Option

spreads have many advantages, such as limited risk and reduced exposure to gamma, theta, and

vega. They can also used to create situations where you can still profit if the spread between the

pairs trade does not move the way you expect it will.

When looking for strategies comprising the pair, one should try to have roughly the same dollar

amount of risk between the positions. That is, probable gain or loss of roughly the same amount

between the two verticals. The reason for doing this is to have the ability for one half of the pairs

trade to make or lose as much as the other in the event that the pair does not move in the way

expected. Adjust the trade quantities to make the risk/reward of the long and short verticals

12 | P a g e

equal. That is, buy two 2.50-point verticals and sell one 5.00-point vertical. Such a position could

be considered to have equal risk and reward between the two verticals. The limited risk

characteristics of vertical spreads provides a natural “stop” for the pairs trade. When the sides of

the pairs trade are of equal risk and reward, for example selling a 5.00-point vertical and buying

a 5.00-point vertical, a credit is preferable. The initial credit allows for extreme moves in the

spread and still provide the potential for profit. When paying for a pair trade, that is, incurring a

debit upon execution, it is better to have one side to be able to make more money than the other.

For example, selling a 2.00-point wide vertical and buying a 2.50-point wide vertical. In a very

large move in both underlying stocks or indices, the profits on the long vertical are potentially

greater than the loss on the short. So, a small debit is acceptable, as long as the long vertical’s

profits exceed both the loss on the short vertical and that initial debit. If both the long 2.50-point

and short 2.00-point verticals reach their maximum value, the profit of the long should offset the

loss on the short 2.00-point vertical and the initial debit. If both the long and short verticals reach

their minimum values, the loss on the pairs trade is restricted to that minimum debit. Thus, in

both cases where the pairs of stocks or indices make extreme price moves, there is a potential

profit to balance the potential loss.

Benefits of pair trading

What can Pair Trading do for You?Are you tired of trying to guess the market’s direction?

Would you like to learn how our professional traders have been consistently taking money out of

the market? Are you looking for a new strategy that can make you money and help minimize the

risk? Then pair trading may be perfect for you! Our pair trading method opens the door to

multi-layered trading strategies, facilitates diversification as you can trade many pairs at a time,

and allows you to potentially trade larger. All this helps with a trader's confidence. Whether the

market is traveling up or down or sideways, or moving fast or slow, the trader can generate

profits from trading the differential of two correlated stock any and every day.

When you study the price action of a pair you get very powerful results. Spread trading is trading

instruments that are by design quite prone to range bound trading. The chop can be easily

recognized, orders enveloped around the bids and asks of the pair stocks to participate in great

prints. Also, with predictability increased, the risk is reduced, and the options available to the

13 | P a g e

trader increase. A pair trader actually gets to respond to the action that the market is providing,

recognizes patterns and participates in a market neutral manner, not exposed, as you would be by

a long position only.

The markets have changed radically in the last six months, becoming largely random with only

brief periods of order. Consequently, a trader who focuses on trying to predict the overall market

direction or the direction of a single stock is often disappointed. Frequently, the exact opposite

outcome of what you think will happen, occurs. Many of the books written during the bubble

phase of the market in the 90's focus on trading momentum during volatility and predictable

order flow, and offer little help in consistently extracting profits in the current market climate.

14 | P a g e

STRATEGY

The starting point of this strategy is that stocks that have historically had the same trading

patterns (i.e. constant price ratio) will have so in the future as well. If there is a deviation from

the historical mean, this creates a trading opportunity, which can be exploited. Gains are earned

when the price relationship is restored.

Summary:

. find two stocks prices of which have historically moved together,

. mean reversion in the ratio of the prices, correlation is not key

. Gains earned when the historical price relationship is restored

. Free resources invested in risk-free interest rate.

Testing for the mean reversion

The challenge in this strategy is identifying stocks that tend to move together and therefore make

potential pairs. Our aim is to identify pairs of stocks with mean-reverting relative prices. To find

out if two stocks are mean-reverting the test conducted is the Dickey-Fuller test of the log ratio

of the pair. In the

A Dickey-Fuller test for determining stationarity in the log-ratio

yt = logAt −logBt of share prices A and B

_yt = μ + yt−1 + "t (17)

In other words, we are regressing yt on lagged values of yt.

the null hypothesis is that = 0, which means that the process is not mean reverting. If the null

hypothesis can be rejected on the 99% confidence level the price ratio is following a weak

stationary process and is thereby mean-reverting. Research has shown that if the confidence level

is relaxed, the pairs do not mean-revert good enough to generate satisfactory returns. This

implies that a very large number of regressions will be run to identify the pairs. If you have 200

stocks, you will have to run 19 900 regressions, which makes this quite computer-power and

time consuming.

15 | P a g e

PAIRS TRADING MODEL

Screening Pairs

For screening the rules follow the general outline of first "find stocks that move together, “and

second “take a long-short position when they diverge." A test requires that both of these steps

must be parameterized in some way. How do you identify "stocks that move together?" Need

they be in the same industry? Should they only be liquid stocks? How far do they have to

diverge before a position is put on? When is a position unwound? Some straightforward

choices about each of these questions are made. We put positions on at a two-standard deviation

spread, which might not always cover transactions costs even when stock prices converge.

By conducting this procedure, a large number of pairs will be generated. The problem is that all

of them do not have the same or similar betas, which makes it difficult for us to stay market

neutral. Therefore a trading rule is introduced regarding the spread of betas within a pair. The

beta spread must be no larger than 0.2, in order for a trade to be executed. The betas are

measured on a two-year rolling window on daily data. This gives mean-reverting pairs with a

limited beta spread, but to further eliminate the risk we also want to stay sector neutral. This

implies that we only want to open a position in a pair that is within the same sector. Due to the

different volatility within different sectors, we expect sectors showing high volatility to produce

very few pairs, while sectors with low volatility to generate more pairs. Another factor

influencing the number of pairs generated is the homogeneity of the sector. A sector like

Commercial services is expected to generate very few pairs, but Financials on the other hand

should give many trading opportunities. The reason why, is that companies within the Financial

sector have more homogenous operations and earnings.

Trading rules

The screening process described gives a large set of pairs that are both market and sector

neutral, which can be used to take positions. This should not be done randomly, since timing is

an important issue.

16 | P a g e

Basic rule will be to open a position when the ratio of two share prices hits the 2 rolling standard

deviation and close it when the ratio returns to the mean. However, we do not want to open a

position in a pair with a spread that is wide and getting wider. This can partly be avoided by the

following procedure: We actually want to open a position when the price ratio deviates with

more than two standard deviations from the 250 days rolling mean. The position is not opened

when the ratio breaks the two-standard-deviations

limit for the first time, but rather when it crosses it to revert to the mean again. we have an open

position when the pair is on its way back again

In Short:

--Open position when the ratio hits the 2 standard deviation band for two consecutive times.

-- Close position when the ratio hits the mean

Trading Period

Once we have paired up all liquid stocks in the formation period, we study the top 5 and 20 pairs

with the smallest historical distance measure. This last set is valuable because most of the top

pairs share certain characteristics. On the day following the last day of the pairs formation

period, we begin to trade according to a pre-specified rule. We chose rules based on the

proposition that we open a long-short position of when the pair prices have diverged by a certain

amount, and close the position when the prices have reverted. Following practice, we base our

rules for opening and closing positions on a standard deviation metric. We open a position in a

pair when prices diverge by more than two historical standard deviations, as estimated during the

pairs formation period. We unwind the position at the next crossing of the prices. If prices do

not cross before the end of the trading interval, gains or losses are calculated at the end of the

last trading day of the trading interval. Since the positions are effectively self-financing

portfolios, we report the payoffs by going one rupee short in the higher-priced stock and one

rupee long in the lower-priced stock.

17 | P a g e

Excess Return Computation

Because pairs may open and close at various points during the trading period, the calculation of

the excess return on a portfolio of pairs is a non-trivial issue. Pairs that open and converge

during the trading interval will have positive cash flows. Because pairs can re-open after initial

convergence, pairs can have multiple positive cash flows during the trading interval. Pairs that

open but do not converge will only have cash flows on the last day of the trading interval when

all positions are closed out. Therefore, the payoffs to pairs trading strategies are a set of positive

cash flows that are randomly distributed throughout the trading period, and a set of cash flows at

the end of the trading interval which can either be positive or negative. For each pair we can

have multiple cash flows during the trading interval, or we may have none in the case when

prices never diverge by more than two standard deviations during the trading interval. Because

the trading gains and losses are computed over long short positions of one rupee, the payoffs

have the interpretation of excess returns. The excess return on a pair during a trading interval is

computed as the sum of the payoffs during the trading interval.

We consider two measures of excess return on a portfolio of pairs: the return on committed

capital and the return on actual employed capital. The excess return on committed capital takes

the sum of the payoffs over all pairs during the trading period, and divides it by the number of

pairs in the portfolio. This measure of excess return is clearly conservative if a pair does not

trade for the whole of the trading period, we still include a dollar of committed capital in our

calculation of excess return. A hedge fund would presumably be more flexible in its sources and

uses of funds. In such case computing excess return relative to the actual capital employed

may give a more realistic measure of the trading profits. We calculate the excess return on

employed capital as the sum of the pair payoffs divided by the number of pairs that actually

open during the trading period. This is a conservative approach to computing the excess

return, because it implicitly assumes that all cash is received at the end of the trading period.

Because any cash flows during the trading interval is positive by construction, it ignores the fact

that these cash flows are received early, and understates the computed excess returns.

Strategy Profits

The excess returns for the pairs portfolios that are unrestricted in the sense that the matching

18 | P a g e

stocks do not necessarily belong to the same broad industry categories. There are diversification

benefits from combining multiple pairs in a portfolio. As the number of pairs in a portfolio

increases, the portfolio standard deviation falls, as does the range of the realized returns and the

frequency of negative portfolio excess return during a period. For example, during the full

sample period of 34 years, a portfolio of 20 pairs has only 6 six-month periods with negative

payoffs, while a portfolio of 5 pairs returns negative profits in 11 trading periods. The

distribution of pairs payoffs is skewed right and peaked relative to the normal

distribution.

Since pairs-trading is in essence a contrarian investment strategy, the returns may be biased

upward due to the bid-ask bounce. In particular, the strategy buys sells stocks that have done

well relative to their match and buys those that have done poorly. Part of any observed price

divergence is potentially due to price movements between bid and ask quotes: conditional on

divergence the winner’s price is more likely to be an ask quote and the loser’s price a bid quote.

Since we have used these same prices for the start of trading, our returns may be due to the fact

that we are implicitly buying at bid quotes (losers) and selling at ask quotes (winners). The

opposite is true at the second crossing (convergence): part of the drop in the winner’s price

can reflect a bid quote, and part of the rise of the loser’s price an ask quote . The excess returns

are still significantly positive in a statistical sense, the dramatic drop in the excess returns

suggests that a non-trivial portion of the profits in pair trading may be due to bid-ask bounce. It

is difficult to quantify which portion of the profit reduction is due to bid-ask bounce and which

portion stems from true mean reversion in prices due to rapid market adjustment. None-the-less,

this difference raises questions about the economic significance of our results when we include

transactions costs.

Risk control

Furthermore, there will be some additional rules to prevent us from loosing too much money on

one single trade. If the ratio develops in an unfavourable way, we will use a stop-loss and close

the position as we have lost 20% of the initial size of the position. Finally, we will never keep a

position for more that 50 days. On average, the mean reversion will occur in approximately 35

days , and there is no reason to wait for a pair to revert fully, if there is very little return to be

earned. The potential return to be earned must always be higher than the return earned on the

19 | P a g e

benchmark or in the fixed income market. The maximum holding period of a position is

therefore set to 50 days. This should be enough time for the pairs to revert, but also a short

enough time not to loose time value. The rules described are totally based on statistics and

predetermined numbers. In addition, there is a possibility for us to make our own decisions. If we

for example are aware of fundamentals that are not taken into account in the calculations and that

indicates that there will be no mean reversion for a specific pairs, we can of course avoid

investing in such pairs.

From the rules it can be concluded that we will open our last position no later than 50 days

before the trading game ends. The last 50 days we will spend trying to close the trades at the

most optimal points of time.

Summary:

. Stop loss at 20% of position value

. Beta spread < 0.2

. Sector neutrality

. Maximum holding period < 50 trading days

. 10 equally weighted positions

Risks

1. Through this strategy we do almost totally avoid the systematic market risk. The reason

there is still some market risk exposure, is that a minor beta spread is allowed for. In

order to find a sufficient number of pairs, we have to accept this beta spread, but the

spread is so small that in practise the market risk we are exposed to is ignorable.

2. Also the industry risk is eliminated, since we are only investing in pairs belonging to

the same industry.

3. The main risk we are being exposed to be then the risk of stock specific events that is the

risk of fundamental changes implying that the prices may never mean revert again, or at

least not within 50 days. In order to control for this risk we use the rules of stop-loss and

maximum holding period.

4. This risk is further reduced through diversification, which is obtained by simultaneously

investing in several pairs. Initially we plan to open approximately 10 different positions.

20 | P a g e

Finally, we do face the risk that the trading game does not last long enough. It might be

the case that our strategy is successful in the long run, but that a few short run failures

will ruin our overall excess return possibilities.

21 | P a g e

STEPS IN PAIR TRADING

1. Selecting pairs of highly correlated stocks from the same industry. Suggested correlation is

greater than 85%.

2. Find the price ratio of the pairs for the desired period (1 year) ,mean and the 3 standard

deviations on the either side.

3. Choose the pairs which are now showing a mean reversion.

4. Observe the one year price ratio of the pair and identify the resistance in terms of the nearest

standard deviation.

5 .Determine buy/sell for individual pairs and determine quantities according to available lot size

such that money value of buy is equal to money value of sell.

6. Hold the position till the immediate target and reverse to book profit.

7. Keep track if trade is not going according to your desired direction…..

a. book loss if spread is going against desired direction by more than 5% or 20 %

of the margin money is at risk

b. if pair is not reverting back and is dull without movement for 35 days.

8. Close the position if there is any potential announcement in coming days.

22 | P a g e

CASES OF PAIR TRADING

The pair taken as an example is from the Banking Sector

1. Punjab National Bank & Bank of India.

It has been observed that the Correlation between the stocks for the last 1 year is 0.90. This is

quite evident from the below graph which shows the relative price movement of the two stocks

in past one year. Also, one can notice that the two stocks mostly move in tandem with one

another. It is in the recent past that they have diverged from each other.

PRICE MOVEMENT

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

4/7/2008

5/7/2008

6/7/2008

7/7/2008

8/7/2008

9/7/2008

10/7/2008

11/7/2008

12/7/2008

1/7/2009

2/7/2009

Period

Pri

ce

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

PNB

BOI

After determining the correlation amongst the stocks, the next thing is to find out the Ratio of the

Prices of the stocks.

Ratio = Stock Price of PNB / Stock Price of BOI

23 | P a g e

This Ratio forms the basis of our findings. The next step is to find the average and the standard

deviation of the ratios for the past 1 year. The result for the same are as below:

Mean 1.7102Stdev 0.0914

On the basis of the mean and the Standard deviation (S. D.), the sigma levels are determined. We find

upto plus and minus 3 sigma levels as shown below.

MEAN-3 S.D.

MEAN-2 S.D.

MEAN-S.D.

MEANMEAN+

S.D.MEAN+2 S.D.

MEAN+3 S.D.

1.4359 1.5273 1.6187 1.7102 1.8016 1.8930 1.9845

Looking at the recent data one can notice the trend in the movement of the Price Ratio of the two

stocks.

24 | P a g e

Table shows the data for the month of February 2009 and some days of march when the Strategy is

initiated.

DATEPunjab National

Bank Bank of India RATIO2/2/2009 391.10 239.00 1.63642/3/2009 397.75 238.55 1.66742/4/2009 400.95 241.30 1.66162/5/2009 396.65 238.80 1.66102/6/2009 399.35 244.70 1.63202/9/2009 404.75 250.65 1.61482/10/2009 408.90 254.35 1.60762/11/2009 406.50 249.75 1.62762/12/2009 402.45 250.20 1.60852/13/2009 408.55 251.90 1.62192/16/2009 391.25 237.15 1.64982/17/2009 369.10 220.55 1.67352/18/2009 364.30 220.95 1.64882/19/2009 374.80 226.70 1.65332/20/2009 362.65 221.90 1.63432/24/2009 346.55 221.00 1.56812/25/2009 341.40 223.70 1.52622/26/2009 324.20 220.35 1.47132/27/2009 337.60 225.55 1.49683/2/2009 325.20 219.25 1.48323/3/2009 312.35 210.25 1.48563/4/2009 306.05 203.80 1.50173/5/2009 292.05 197.40 1.47953/6/2009 310.90 196.45 1.5826

In the graph below we find out that ratio reverts back to mean (mid Jan)) after nearly touching +

2 Standard Deviation. Since then ratio moved towards the mean and after crossing the mean it

has further moved below to cross - 2 standard deviation. (end Feb) before reverting back to mean

Recently it has started its journey back to mean and one can sense an opportunity to initiate a

long on PNB-BOI Pair.

25 | P a g e

P AIR T R AD ING

1.0000

1.2000

1.4000

1.6000

1.8000

2.0000

2.2000

P eriod

Ratio

(PNB

/ BO

I)

R atio

ME A N-3 S .D.

ME A N-2 S .D.

ME A N-S .D.

ME A N

ME A N+ S .D.

ME A N+ 2 S .D.

ME A N+ 3 S .D.

We initiate as on 6th MARCH 2009.

LONG PNB Futures at 310.90 (120 shares)

SHORT BOI Futures at 196.45 (200 shares)

Position Script Price when InitiatedNo. of lot Total Amt

Margin Amt

6-Mar-09 Long PNB 310.9 120 37308 7461.6 Short BOI 196.45 200 39290 7858

Total Margin 15319.6

26 | P a g e

The table below shows the movement of the ratio of the Price of the two stocks

DATE Punjab National Bank Bank of India RATIO3/9/2009 304.20 180.60 1.68443/12/2009 316.35 189.80 1.66683/13/2009 336.35 194.15 1.73243/16/2009 342.15 199.25 1.71723/17/2009 328.20 192.85 1.70183/18/2009 340.10 196.25 1.73303/19/2009 342.65 203.55 1.68343/20/2009 332.60 197.85 1.68113/23/2009 361.55 206.90 1.74753/24/2009 366.90 204.00 1.7985

We will hold our position till ratio reverts back to mean (1.71).We find that with time ratio starts

reverting back and as on 24th March 2009 ratio is 1.79.

Hence we book profit and close both the positions by buying BOI Futures and selling PNB

Futures. The detailed Summary is as shown below.

PositionScript

Price when Initiated

No. of lot

Total Amt

Margin Amt

Price when closed

Profit/Loss

6-Mar-09 24-Mar-09 Long PNB 310.9 120 37308 7461.6 366.9 6720 Short BOI 196.45 200 39290 7858 204 -1510

Total Margin 15319.6

Net Profit/Loss 5210

Ratio when initiated 1.582591Expected Ratio 1.710159 R O I 34.01%

27 | P a g e

2. Punjab National Bank & Bank of Baroda.



in past one year. Also, one can notice that the two stocks mostly move in tandem with one

another.



Ratio = Stock Price of PNB / Stock Price of BOB

This Ratio forms the basis of our findings. The next step is to find the average and the standard deviation

of the ratios for the past 1 year. The result for the same are as below:

Mean 1.7578Stdev 0.1171

28 | P a g e



MEAN-3 S.D.

MEAN-2 S.D.

MEAN-S.D.

MEANMEAN+

S.D.MEAN+2

S.D.MEAN+

3 S.D.1.4066 1.5236 1.6407 1.7578 1.8748 1.9919 2.1089


stocks.

Table shows the data for the month of February 2009 and some days of march when the Strategy is

initiated.

DATE Punjab National Bank Bank of Baroda RATIO2/2/2009 391.10 246.60 1.58602/3/2009 397.75 244.15 1.62912/4/2009 400.95 246.05 1.62952/5/2009 396.65 245.70 1.61442/6/2009 399.35 247.15 1.61582/9/2009 404.75 250.55 1.61542/10/2009 408.90 249.75 1.63722/11/2009 406.50 244.85 1.66022/12/2009 402.45 243.00 1.65622/13/2009 408.55 247.25 1.65242/16/2009 391.25 236.20 1.65642/17/2009 369.10 225.20 1.63902/18/2009 364.30 228.35 1.59542/19/2009 374.80 231.70 1.61762/20/2009 362.65 224.50 1.61542/24/2009 346.55 215.80 1.60592/25/2009 341.40 216.90 1.57402/26/2009 324.20 212.35 1.52672/27/2009 337.60 220.10 1.53383/2/2009 325.20 217.00 1.49863/3/2009 312.35 207.70 1.50393/4/2009 306.05 204.15 1.49913/5/2009 292.05 195.55 1.49353/6/2009 310.90 192.40 1.6159

29 | P a g e

In the graph below we find out that ratio reverts back to mean (22nd Jan)) after touching + 1

Standard Deviation. Since then ratio moved towards the mean and after crossing the mean it has

further moved below to cross - 2 standard deviation. (start March) before reverting back to mean

Recently it has started its journey back to mean and one can sense an opportunity to initiate a

long on PNB-BOB Pair.


LONG PNB Futures at 310.90 (100 shares)

SHORT BOB Futures at 192.4 (150 shares)

30 | P a g e

Position Script Price when Initiated No. of lot Total AmtMargin Amt

6-Mar-09 Long PNB 310.9 100 31090 6218

Short BOB 192.4 150 28860 5772

Total Margin 11990

The table below shows the movement of the ratio of the Price of the two stocks after the pair is

initiated.

Date Punjab National Bank Bank of Baroda RATIO

3/9/2009 304.2 183.55 1.657314

3/12/2009 316.35 191.65 1.650665

3/13/2009 336.35 203.75 1.650798

3/16/2009 342.15 206.4 1.657703

3/17/2009 328.2 198.7 1.651736

3/18/2009 340.1 207.8 1.63667

3/19/2009 342.65 208.55 1.643011

3/20/2009 332.6 203.9 1.631192

3/23/2009 361.55 223.9 1.614783

3/24/2009 366.9 227.5 1.612747

3/25/2009 398.1 225.45 1.765802

3/26/2009 414.35 235.3 1.760943

3/27/2009 438.85 247.65 1.772057

We will hold our position till ratio reverts back to mean (1.75).We find that with time ratio starts

reverting back and as on 27th March 2009 ratio is 1.77.

Hence we book profit and close both the positions by buying BOB Futures and selling PNB

Futures. The detailed Summary is as shown below.

31 | P a g e

Position ScriptPrice when Initiated No. of lot

Total Amt

Margin Amt

Price when closed Profit/Loss

6-Mar-09 27-Mar-09 Long PNB 310.9 100 31090 6218 438.85 12795 Short BOB 192.4 150 28860 5772 247.65 -8287.5

Total Margin 11990Net Profit/Loss 4507.5

Current Ratio 1.6159Expected Ratio 1.7577 R O I 37.59%

32 | P a g e

3. Shiv-Vani Oil Exploration & Aban Offshore



in past one year.



Ratio = Stock Price of PNB / Stock Price of BOB

This Ratio forms the basis of our findings. The next step is to find the average and the standard deviation

of the ratios for the past 1 year. The result for the same are as below:

Mean 0.2256Stdev 0.0657

33 | P a g e



MEAN-3 S.D.

MEAN-2 S.D.

MEAN-S.D.

MEANMEAN+

S.D.MEAN+2

S.D.MEAN+3

S.D.0.0285 0.0942 0.1599 0.2256 0.2913 0.3570 0.4227


stocks. Table shows the data for the month of February 2009 and some days of march when the

Strategy is initiated.

DATESHIV-VANI OIL EXPLORATION ABAN OFFSHORE RATIO

2/2/2009 105.85 469.90 0.22532/3/2009 103.30 466.65 0.22142/4/2009 106.80 430.45 0.24812/5/2009 103.05 417.00 0.24712/6/2009 103.25 420.80 0.24542/9/2009 105.35 435.40 0.24202/10/2009 106.55 440.00 0.24222/11/2009 106.95 438.00 0.24422/12/2009 111.20 443.75 0.25062/13/2009 111.50 455.90 0.24462/16/2009 107.80 431.40 0.24992/17/2009 105.00 413.10 0.25422/18/2009 104.30 396.55 0.26302/19/2009 105.00 381.80 0.27502/20/2009 103.55 360.05 0.28762/24/2009 102.15 353.95 0.28862/25/2009 101.70 355.10 0.28642/26/2009 104.50 348.75 0.29962/27/2009 104.10 317.25 0.32813/2/2009 104.50 293.90 0.35563/3/2009 102.60 287.40 0.35703/4/2009 102.05 285.70 0.35723/5/2009 101.10 256.90 0.39353/6/2009 97.50 231.65 0.42093/9/2009 95.40 232.00 0.41123/12/2009 91.75 229.85 0.39923/13/2009 93.10 254.70 0.36553/16/2009 92.75 270.45 0.34293/17/2009 94.95 286.65 0.3312

34 | P a g e

In the graph below we find out that ratio moves towards + 2 standard deviation ( In Feb) and

crosses + 2 Standard Deviation (Early March). It further moves on to touch + 3 standard

deviation (2nd March). After that it has started its journey back to mean and has crossed the +2

standard deviation mark on 12th March. At this point one can sense an opportunity to initiate a

short on Shiv-Vani - Aban Pair.


SHORT SHIV-VANI Futures at 94.95 (250 shares)

LONG ABAN Futures at 286.65 (100 shares)

Position ScriptPrice when Initiated No. of lot Total Amt Margin Amt

17-Mar-09 Long Aban 286.65 100 28665 5733

Short Shiv-vani 94.95 250 23737.5 4747.5

35 | P a g e

The table below shows the movement of the ratio of the Price of the two stocks after the pair is

initiated.

DATESHIV-VANI OIL EXPLORATION ABAN OFFSHORE RATIO

3/18/2009 97 306.2 0.3167864

3/19/2009 95.5 318.95 0.29942

3/20/2009 101.4 326.25 0.3108046

3/23/2009 104.7 329.95 0.3173208

3/24/2009 96.05 329.6 0.2914138

3/25/2009 97.6 332.2 0.2937989

3/26/2009 98.8 367.1 0.2691365

3/27/2009 96.95 420.85 0.2303671

We will hold our position till ratio reverts back to mean (0.2256).We find that with time ratio

starts reverting back and as on 27th March 2009 ratio is 0.2303.

Hence we book profit and close both the positions by buying Shiv-Vani Futures and selling Aban

offshore Futures. The detailed Summary is as shown below.

Position ScriptPrice when Initiated No. of lot

Total Amt

Margin Amt

Price when closed Profit/Loss

17-Mar-09 27-Mar-09 Long Aban 286.65 100 28665 5733 420.85 13420

ShortShiv-vani 94.95 250 23737.5 4747.5 96.95 -500

Total Margin 10480.5Net Profit/Loss 12920

Current Ratio 0.3312Expected Ratio 0.2255 R O I 123.28%

36 | P a g e

Conclusion

Pair Trading is a market neutral strategy that allows you to take positions which usually result in

generating profits irrespective of overall market direction. Though it is not risk neutral strategy

but when combined with proper risk management mechanism it provides opportunity of

generating considerable returns. It is one of the strategies employed by hedge funds seeking

greater alpha returns.

We examine contrarian strategies based on the notion of co integrated prices in a reasonably

efficient market, known as Pairs Trading. We form pairs of stocks, which are close substitutes

according to correlation and minimum distance criterion using a metric in price space. We find

that trading suitably formed pairs of stocks exhibits profits, with are robust to conservative

estimates of transaction costs. These profits are uncorrelated to the sensex, however they do

exhibit some sensitivity to the spreads between small and large stocks and between value and

growth. Because the strategies are trading intensive, the profitability of the strategy clearly

depends upon the price and the impact of execution.

Larger players such as institutions are likely to have a relative advantage in their ability to

command leverage to take positions and there ability to execute trades cheaply. On the other

hand competition in the industry and the price impact of large trades may be important factors

limiting the scale of pairs trading.

Pairs trading performed well over difficult times for U.S. stocks. When the U.S. stock market

suffered a dramatic real decline from 1969 through 1980, the pairs strategy had some of its best

performance. Perhaps after its discovery in the early 1980’s by Tartaglia and others, competition

has decreased opportunity. On the other hand, pair trading might simply be more profitable in

times when the stock market performs poorly.

37 | P a g e

Bibliography

The Handbook of Pair TradingBy – Douglas S. Ehrman

Pairs Trading - Quantitative Methods and AnalysisBy – Ganapathy Vidyamurthy

Pairs Trading, Convergence Trading, Cointegration.www.pairtrader.com/mentoring.html

Pairs Trading .www.traders-mag.com.

Pair trading www.investopedia.com

Selection of right Pairswww.rightpairs.com

Pairs Trading Strategy and Statistical Arbitrage – by Dan Pipitonehttp://ezinearticles.com/?Pairs-Trading-Strategy-and-Statistical-Arbitrage&id=1996014

Simulated Trading An Analysis Of Pair Trading. By- Prof Andrei Simonov.

38 | P a g e

http://ezinearticles.com/?Pairs-Trading-Strategy-and-Statistical-Arbitrage&id=1996014

http://www.rightpairs.com/

http://www.investopedia.com/

http://www.pairtrader.com/mentoring.html

derivative pair trading

Documents