1 / 53

Dual Beta Model

Dual Beta Model. Ho Ken Jom, Li Wenru, Zhang Jian Department of Mathematics, NUS, 14 March 2011. The Reference Paper. Does beta react to market conditions? Estimates ‘bull’ & ‘bear’ betas using a nonlinear market model with an endogenous threshold parameter

tanuja
Download Presentation

Dual Beta Model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dual Beta Model Ho Ken Jom, Li Wenru, Zhang Jian Department of Mathematics, NUS, 14 March 2011

  2. The Reference Paper Does beta react to market conditions? Estimates ‘bull’ & ‘bear’ betas using a nonlinear market model with an endogenous threshold parameter -by George Woodward; Heather M.Anderson, Quantitative Finance, 25 March 2009

  3. Outline of The Presentation • The Model • Strategy • Back Testing

  4. Definition of “Bull” and “Bear” market Compare market index to a critical threshold value How to differentiate market states Trend-based scheme

  5. Market indicator- transition variable R* • R*= 12 month moving average of logarithmic returns • smoother • noise not “useful” Figure 1: Return on the market index Figure 2, The transition variable (R*) Jump in and out of market phases rapidly Much smoother path

  6. Models Dual-beta market model: : Market state indicator. critical threshold value for each industry Bear state: Bull state：

  7. Models Logistic Smooth Transition Market Model F is the logistic smooth transition function, : smoothness parameter When is large and negative When is large and positive

  8. Data • 24 industry groupings within the Australian Stock Exchange • Observations are monthly • Return series are calculated as the difference of the logarithms of prices.

  9. Data

  10. Estimation—LSTM model LSTM models： • 15 industries: significant at 5% • 6 industries: significant at 10% • 11: negative • 10: positive • “Smooth” transition • estimate LSTM

  11. Estimation—DBM model • Parameter estimates are almost identical

  12. Estimation—DBM model LSTM model • DBM fits the data well(R2) • Stocks spend more time in ‘bull’-market • ESS is not affected

  13. Summary • ‘Bull’ and ‘bear’ betas are significantly different for most industries • Transition between states is abrupt, supporting a dual-beta market modeling framework • For many industries, stocks spend more time in ‘bull’-market than ‘bear’-market states. • The risk associated with ‘bull’ states is not always smaller than the risk in ‘bear’ market states.

  14. Strategies for Dual Beta Market Model • Terms used • Model Calibration • Theory behind the strategy • Calculate the “fair” return of index • Search mispriced spots and trade • The shortage of the strategy

  15. Terms • Rm, Ri: Market/Index return • βu,βd: Beta in “up”/”down” state • α: Alpha • c: Threshold (defined on Rm) • Tu,Td: Upper and lower thresholds (for trading) • RMA: Rolling moving average • ε: the deviation from fair rate

  16. Bull & Bear • Paper suggests dual beta model • Differing betas for different market states • Relatively sudden transition • Suggests a market inefficiency • Use a 12-month MA to determine state • Market state only changes a few times (2-5 years)

  17. Up and Down • Attempts with a 12-month MA yielded losses • Not suitable for forecasting, trading on • Long-term/”slow” changes not useful • Trials with daily market data (Rm) successful • State can (and does) change from day to day • Investor sentiment • Smaller, but still significant changes

  18. Model • For a given c (Threshold value) partition data into “up”/”down” sets • Fit a dual beta (single alpha model) • Choose the c, α, βu,βd that provide the best fit (R2) • Look for significant change in beta

  19. Model Calibration • Calibration for single beta model (SBM) • Use the α of SBM to initiate a search for parameters of dual beta single alpha model(DBSAM) • The DBSAM should have a higher R-square as expected • Trade on pairs of high R-square

  20. Data Process • For a given set of estimated model parameters • Calculate Rm, compare with c to obtain state • Calculate Ri, compare with (α + βRm) • Check where difference lies w.r.t thresholds

  21. Fair Rate (Model Predicted rate) • For a given set of estimated model parameters • Calculate Rm, compare with c to obtain state • Calculate the fair rate of index return if(Rm<C) α + βd Rm if(Rm>C) α + βu Rm • The deviation is given as ε = Ri-(α+ βu Rm)

  22. Thresholds • Above +Tu: over-performing or “cheap” • Buy • Below –Tu: under-performing or “expensive” • (Short) Sell • Between ±Td: No significant difference (noise) • Close out positions

  23. Thresholds • Higher thresholds are more conservative • Buy only during bigger differences • Close out positions more quickly • Thresholds estimated by backtesting • Conservative thresholds give less gains/losses • Important to determine the best threshold

  24. Strategy • Impose a position limit +1, -1 • For p=0: if(ε >Tu), p = 1; else p =-1 • For p=1: if(abs(ε) <Td), p = 0, update P&L; else if(ε <-Tu), p = -1, update P&L • For p=-1: if(abs(ε) <Td), p = 0, update P&L; else if(ε >Tu), p = 1, update P&L

  25. The Shortage of The Strategy • Vulnerable to Systematical Risk (suppose p=1 but the market drops during holding period, or p=-1, but the market rise) • Does not consider transaction cost • Solution: impose further restrictions -impose positive condition for each cycle of trading -when return is very high, don’t short, and vice versa; do not hold for a long time for a long position, etc…

  26. Back Testing • Setup • Stationary Trading • Dynamical Trading • Stress Testing • Future Works • Conclusion

  27. Setup • Data Selection • Using Spot rate of return instead of RMA • Calibrating dual beta single alpha model (DBSAM) • Parameterization of the strategy • Which return to use? • The state transition probability

  28. Data Selection • Market variable: S&P 500(^GSPC) • Target indices scanned: NYSE Composite (^NYA) NASDAQ Composite ( ^IXIC) Vanguard Index Trust 500 Index (VFINX) PHLX OIL SERVICE SECTOR INDEX ( ^OSX) Bank of America (BAC) • Period: 01/01/2009~04/03/2011 (546 days), by YahooEOD Add some plots to show the betas here

  29. NYA and S&P500, 1/1/2009~6/03/2011

  30. R-Square for the First 294 Days *Sector index and single stock has low R-square thus trade S&P500 & NYA

  31. Using Spot Rate of return RMA data yields a bad fitting, and even fails itself, so chose spot rate return of S&P500 for trading

  32. Calibration of DBSAM • First conduct an SBM OLS on the whole 294 days data • Then partition the data to lower wing and upper wing with different candidate threshold values • Then for each partition, generate α-grid (see below), search for α and the associate βd & βu such that the composite R-square is maximized • In my program, s = 0.01, i = -25, …,25, but is still expensive in computing

  33. OSX ~ S&P500 SBM Fitting R2 = 0.69, Beta = 1.46, ALPHA = 0.175

  34. OSX~NYA DBSAM Fitting R2 = 0.76, Beta Lower = 1.56, Beta Upper = 1.44, ALPHA = 0.415

  35. Parameterization of The Strategy nTrade number of trading days in US, set as 252 ND number of days in data, in our case this is 546 NRD number of days for regression NRD = 546-NBT NBT number of days for back-testing TH the threshold weight to partition the market E the average modeling error Tu the threshold to trigger a position in scales of E Td threshold to close a position in scales of E Limit position limit, set as +1, -1 S the resolution for search for alpha, fixed as 0.01 FEE fees, set as zero for the time-being

  36. Which Return to use? • Program uses daily holding period return • Test shows log return performs better • However, for the purpose of consistency, I keep using daily holding return

  37. State Transition Probability(1) Define modeling error at t, t=0,1,..,N as Et. Assume : For all positive E’s, sum over both sides Similarly, we have Once we know r, we can estimate the volatility using

  38. State Transition Probability(2) Now, suppose the initial error is Et which is less than Td. After 1 time unit, the error distribution is given as Then the probability that a long position will be triggered is Eventually this will enable us to estimate the holding period.

  39. Stationary Trading The model is calibrated using the first NRD days data, and back-tested against the left NBT days data, assuming the model is stationary • Search optimal Tu(trigger threshold) • Search optimal Td (closing threshold) • Search optimal TH (weighting threshold) • Different NBT (number of back-testing days) • High Frequency Trading

  40. Optimal Tu *Setting: NBT = 126, Td = 0.1, TH = 0.2

  41. Optimal Tu

  42. Optimal Td *Setting: NBT = 126, Tu = 5, TH = 0.2

  43. Optimal TH *Setting: NBT = 126, Td = 0.5, Tu = 5

  44. Different NBT

  45. High Frequency Trading with Different NBT *It can be observed that (i) DBSAM is better (ii) return increases with NBT (Setting: Td = 1, Tu = 2, TH = 0.3)

  46. Dynamic Trading • The issue of computing time -It takes 50s for Java to search the grid, for 252 days, weekly update, need 42m on Dell OptiFlex755 • Using C & CUDA -Reduce to 7m (on Nvidia GTX580) • Some results

  47. Results: Dynamical SBM vs Stationary SBM

  48. Results: Dynamic DBSAM vs Stationary DBSAM

  49. Possible Reason for Under-Performance My own OLS algorithm for regression passing through the origin (consistent with matlab) yield slightly different parameters from AlgoQuant.

  50. Stress Testing (stationary): against 2007~2009 Crisis • Case I: Regression before crisis and back testing in crisis(1 Jan 2007~6 Mar 2009) • Case II: Both in crisis (3 Jan 2008~6 March 2010) • Case III: Regression in crisis and back testing out crisis? But no such case!

More Related