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1、Lecture 6 Back testing statistical arbitrage strategies Marco Avellaneda G63 2936 001 Spring Semester 2009 Simulation of trading Profit Loss simplicityfor 0 and 0005 0bps 5 assume will weTypically period ofstart at account in theequity scommission clearing impact market stockshort on cash for receiv
2、edinterest stock longon cash for paidinterest cashfor rate referenceor rate Funds Fed periodover stock ofreturn adjusted dividend period ofstart at the stock in investment r nE rr rr r nR niQ n ni ni Basic P L equation Equity ValueMarket Short ValueMarket Long ratio leverage 1 1 1 1 1 1 1 n N i ni N
3、 i nini N i ni N i nini N i ninnn E Q QQQtr QtrRQtErEE Examples of Leverage Long only Long only Reg T margin acct Long short Neutral Reg T Long short Equal target position in each stock EQ N E Q ESL ESEL EL E L i iimax max 2 2 1 6 3 0 3 1 2 2 130 30 Investment funds Sharpe Ratio r S t E EE tN E EE t
4、N N n n nn N n n nn 1 1 periods periods 1 2 1 1 periods 2 1 1 1 periods Expected return over simulation period Variance over simulation period Sharpe Ratio The Sharpe ratio measures returns above the risk free rate It is independent of the leverage of the strategy dimensionless tdWdttXmtdX tdXdtt t
5、tP tdP tS tdS iiiiii iii i k k N k ik i i factors 1 Modeling the Evolution of Stock Residuals Daily sampling frequency Statistical Estimation Window 3 months 60 business days Stock returns a sum of the market return and a residual process Residual drift component expected excess return above mkt inc
6、rement of a stationary process Ornstein Uhlenbeck AR 1 process Defactoring using ETFs 0 jiji ij i ETF jiji RR If stock I is in sector j otherwise Regress returns on sector ETF returns In some cases we construct synthetic ETFs e g if the ETF did not exist in the past These are taken to be Capitalizat
7、ion Weighted Trading Signals Introduce the s score for each stock ieq ii i mtX ts 50 0 ifposition short Close 50 0 ifposition long Close 25 1 ifposition short Open 25 1 ifposition longOpen i i i i s s s s Example The trading signal seen graphically Open short Open long S i 1 25 1 25 0 50 0 50 Sell s
8、tock here hedge Buy stock here hedge t S score of JPM vs XLF Including the drift in signal generation ieq ii i ieqi i ii ieqi i iieqi iieqii iiii ii ii sss s s tXm dt tXtdXE mod ETFAbs Alpha BetaKappaReversion daysEquiVolAbs m HHH0 20 0 69 38 7 4 3 3 IYR0 11 0 90 39 6 2 1 8 IYT0 18 0 97 41 6 4 3 0 R
9、KH0 10 0 98 39 6 2 1 7 RTH0 17 1 02 39 6 3 2 7 SMH0 19 1 01 40 6 4 3 2 UTH0 09 0 81 42 6 2 1 4 XLF0 11 0 83 42 6 2 1 8 XLI0 15 1 15 42 6 3 2 4 XLK0 17 1 03 42 6 3 2 7 XLP0 12 1 01 42 6 2 2 0 XLV0 14 1 05 38 7 3 2 5 XLY0 16 1 03 39 6 3 2 5 Total0 15 0 96 40 6 3 2 4 Statistics on the Estimated OU Para
10、meters Average over 2006 2007 Correction to the s score due to alpha is of the order of 0 15 small Constructing market neutral portfolios from signals Large diversified trading universe of U S equities 1000 names Select within the trading universe those stocks that have a trading signal large magnit
11、ude of s score and open trades Monitor for closing trades through s score as well Keep all sectors beta neutral by using ETFs to balance the portfolio and maintain sector neutrality Leverage 2 2 i e 2 long 2 short for 1 of capital Expected Volatility for this Leverage 7 annualized 1 V averagemedaily
12、 volu Using the daily trading volume construct a residual process which measures the change in price per share Estimate AR 1 OU process for the new process Y t This makes deviations on unusually high volume more likely so the signal is weaker usual residual A Jan 2007 Oct 2008 Short covering after e
13、arnings surprise makes the stock run up ETF signals trading time vs actual time ETF signals in trading time Sharpe Ratios PCA trading time vs actual time 15 PCA signals in trading time Sharpe Ratios Focus on the crash of 2007 2007 Flat small losses year with cataclysmic dips in August and November L
14、everage 2 2 August 2007 comparison with Khandani Lo Comparing performance of MR strategies Jan 2007 March 2008 Sector view in Aug 2007 Tech Consumer vs Financials Real Estate Aug 2007 Consumer discretionary L S more volatile than Financials ETFs in trading time outperformed PCAs after 2005 Best strategy among those tested Comparing performance of MR strategies Jan 2007 March 2008
