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Regression Analysis and QuantitativeTrading Strategies

χTrading

Butterfly Spread Strategy

Michael Beven

June 3, 2016

University of Chicago | Financial Mathematics

1 / 25χTrading – Butterfly Spread Strategy

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Overview

1 Strategy

2 Construction

3 Backtesting

4 Risk Management


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Strategy

Butterfly Strategy

Exploits fluctuations in the implied volatility of options

Buys (sells) a call and put at-the-money to the nearest strike (levels of 5)

Sells (buys) a call and put B units above and below the nearestat-the-money strike respectively

Butterfly is re-evaluated daily

Spread is hedge-able using delta


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Strategy

Butterfly Strategy

Outer options (wings) reduce capital requirements

Robust strategy with controllable results

Applicable to several options markets

Backtest suggests annualized returns of 14.7%

Backtest suggests Sharpe ratio of 0.09

Backtest suggests Sortino ratio of 0.18


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Strategy

Investment Universe and Securities

Potential underlying assets: all exchange traded securities with options

Requires liquid options with multiple strikes

Four options (two at-the-money and two wings), and

The underlying for an optional delta hedge

Can be long or short the butterfly


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Strategy

Investment Universe and SecuritiesA payoff illustration:

0 5 10 15 20 25 30Underlying

-15

-10

-5

0

5

10

15

Pay

off

Butterfly Strategy

ATM Call K=15Wing Call K=20ATM Put K=15Wing Put K=10Overall

Figure:


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Strategy

Competitive Edge

Employs exponentially weighted moving averages (EWMA) for analyzingimplied volatility

Flexibility – adjustable days used for EWMA

Ability to exploit multiple markets

Adds alpha to an already diversified portfolio

Low cost structureWing options trade on the opposite side to at-the-money options

Does not require view on market direction


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Construction

Model ConstructionThis strategy takes advantage of the changing implied volatility level:

1 Obtain daily implied volatilities of optionsEntire strip of strikesAccurate minimization techniques

2 Track implied volatility (at-the-money and near strikes) against EWMASearch for large deviations between EWMA and implied volatility

3 Buy or sell the butterfly spread appropriatelyCaptures returns from increasing and decreasing volatility


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Construction

Empirical ExplorationOptions can only exist at certain strikes when far from maturity. TheS&P 500 E-mini June 2016 contracts are assessed:

ESM6P_1800 ESM6P_1900 ESM6P_2000 ESM6P_2100 ESM6P_2200 ESM6P_2300Strike

0

50

100

150

200

250

300

350

Pri

ce

Call and Put Prices at 2015-06-25, S&P 500 E-mini June 2016

CallsPuts

Figure:9 / 25

χTrading – Butterfly Spread Strategy

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Construction

Empirical ExplorationMissing prices can be interpolated for backtesting purposes:


0

50

100

150

200

250

300

350

Pri

ce

Call and Put Prices (Interpolated) at 2015-06-25, S&P 500 E-mini June 2016

CallsPuts

Figure:10 / 25


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Construction

Empirical ExplorationMost important is the change in level of implied volatility (Sinclair,2008). Calls and puts have slightly different implied volatilities:


0.12

0.14

0.16

0.18

0.20

Vola

tilit

y

Call and Put Implied Volatilities at 2015-06-25, S&P 500 E-mini June 2016

CallsPuts

Figure:11 / 25


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Construction

Empirical ExplorationAverage index level: 2006, standard deviation of index level: 73.75

Jul 2015

Aug 2015

Sep 2015

Oct 2015

Nov 2015

Dec 2015

Jan 2016

Feb 2016

Mar 2016

Apr 2016

May 2016

Date

1800

1850

1900

1950

2000

2050

2100

2150

Index L

evel

S&P 500 E-mini Futures

Figure:12 / 25


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Construction

Empirical ExplorationA shorter history for EWMA increases agility; a longer historyincreases stability:

Jul 2015

Aug 2015

Sep 2015

Oct 2015

Nov 2015

Dec 2015

Jan 2016

Feb 2016

Mar 2016

Apr 2016

May 2016

Date

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

Vola

tilit

y

EWMA Predicted vs. Actual Implied Volatility

Actual Implied VolatilityEWMA Predicted Implied Volatility

Figure:13 / 25


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Construction

Empirical ExplorationChoosing thresholds to buy/sell the butterfly is quantiles based:

0.0 0.2 0.4 0.6 0.8 1.0Quantile

0.025

0.020

0.015

0.010

0.005

0.000

0.005

0.010

0.015

Diffe

rence

Quantiles of EWMA Predicted Minus Actual Implied Volatility

Figure:14 / 25


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Construction

Empirical ExplorationEWMA predictions are higher on average compared to realized impliedvolatility. The difference is also negatively skewed:

0.025 0.020 0.015 0.010 0.005 0.000 0.005 0.010 0.015Difference

0

2

4

6

8

10

12

14

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Frequency

Histogram of EWMA Predicted Minus Actual Implied Volatility

Figure:15 / 25


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Construction

Empirical ExplorationButterfly wingspan: 20, EWMA history: 10 days, buy/sell thresholdsare ±0.3 standard deviations from the mean. 185 trades are made:

Jul 2015

Aug 2015

Sep 2015

Oct 2015

Nov 2015

Dec 2015

Jan 2016

Feb 2016

Mar 2016

Apr 2016

May 2016

Date

2

1

0

1

2

3

4

5

Daily

Pro

fit

Cumulative Profit

Figure:16 / 25


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Construction

Implications of Empirical Exploration

Strategy performs best when closer to maturity (within 6 months)

Call and put implied volatilities are averaged around the at-the-moneystrike for an overall level

Maximum cost of the spread is half the wingspan (arbitrage arguments)

Capital set at 3 times the above is cost


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Construction

Implications of Empirical ExplorationCapital set at $30 yields a profit over 13.3%:

Jul 2015

Aug 2015

Sep 2015

Oct 2015

Nov 2015

Dec 2015

Jan 2016

Feb 2016

Mar 2016

Apr 2016

May 2016

Date

10

5

0

5

10

15

Perc

enta

ge R

etu

rn

Cumulative Returns

Figure:18 / 25


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Construction

Implementation

Positions are reviewed at the beginning of each day

Position sizes are scaled to the desired level of investment

Refrain from delta hedging for symmetrical volatility trading

Set buying and selling thresholds based on the empirical differencebetween EWMA predicted and realized implied volatilities


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Construction

Investment Process

Mechanism of strategy diversifies well in a factor-based portfolio

Can apply the butterfly strategy across different asset classes

Candidate assets are selected through a rigorous process

Chosen assets are reassessed for each option expiration


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Backtesting

Results

June 2015 - May 2016

Fees and costs are to be incorporated

Strategy focuses on liquid assets

Long and short butterfly positions

Consistent performance over time


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Backtesting

Return and Risk

Cumulative return: 13.3%

Annualized cumulative return: 14.7%

Annualized standard deviation of returns: 11.7%

Sharpe ratio: 0.09

Sortino ratio: 0.18

Maximum drawdown: 7.5%


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Risk Management

Portfolio Structure

Possible diversified portfolio of butterfly spreads

Aim for diversification benefits in markets with different foundations ofvolatility

Diversification across different option expiration dates


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Risk Management

Process

Frequent revision of positions

Wings of butterfly are stops

Monitor returns versus capital outlay


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About the Manager

Michael Beven graduated as a National Merit Scholar from theAustralian National University in 2012 with a Bachelor of ActuarialStudies and a Bachelor of Finance (Majors in Quantitative andCorporate Finance). He is currently completing a Master of Science inFinancial Mathematics at the University of Chicago. Prior to startingat the University of Chicago, Michael was an Actuarial Analyst inSydney at Quantium, a data analytics focused actuarial consultancy. Hehas also interned at Macquarie Group and Westpac Institutional Bankin quantitative risk analytics. Michael is deeply interested in electronictrading and sees Chicago as a great place to progress his career.


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