Calibrating L-A Model to Chinese Stocks
Chun ChenSharalyn ChenFei LinHechen Yu
Agenda
•Background & motivations•Model & calibration algorithm•Sensitivity of parameters•Calibration Results
▫Volkswagen▫Air China▫China Rail Construction
•Conclusions
Background
•Project based on Avellaneda and Lipkin’s paper “A Dynamic Model for Hard-to-Borrow Stocks”
•Definitions:▫Short selling▫Hard-to-borrow stocks (HTB): insufficient float available for
lending▫Buy–ins: forcibly repurchase to cover short positions
•Phenomena associated with HTBs▫Artificially high prices and sharp drops due to buy-ins▫Examples, Volkswagen October 2008
A classic HTB example: Volkswagen
Motivation•61 stocks listed on both Hong Kong (H shares) and
Shanghai (A shares) stock exchanges▫51 out of 61 can be shorted in HK▫None can be shorted in Shanghai
•Same stock, but different price movements.• Can the price differentials be explained by the varying
degree of hard-to-borrowness?
Model
• St: stock price at time t• λt: buy-in rate • dNλt: Poisson process with intensity λ over (t, t+dt)• σ and κ : respective volatilities• ϒ: price elasticity of demand due to buy-ins• α: speed of mean reversion• X bar: long term equilibrium of Xt• β: impact of stock price change on buy-in intensity
Calibration Algorithm•6 dimensional optimization problem
•Minimize objective function▫max|pdfdata(r) – pdffitted(r)|▫Sum of differences in mean + variance + skewness +
kurtosis•Grid Search to identify good initial values•Then use Matlab fmincon for local optimization
σ κ ϒ α β X bar
FittingSensitivity of parameters: σ
0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.51
2
3
4
5
Tested parameter: σ , range: +-25%
σ
FittingSensitivity of parameters : σ
•1. σ
FittingSensitivity of parameters: ϒ
0.01 0.012 0.014 0.016 0.018 0.021.6
1.65
1.7
1.75
1.8
1.85
1.9
1.95
2
Tested parameter: γ , range: +-25
γ
FittingSensitivity of parameters: ϒ
•2. γ
FittingSensitivity of parameters: Xbar, α, β, κ
2.25 2.45 2.65 2.85 3.05 3.25 3.45 3.651.5
1.6
1.7
1.8
1.9
2
Tested parameter: α , range: +-25%
0.700000000000001 1.21.5
1.6
1.7
1.8
1.9
2
Tested parameter: β , range: +-25%
0.600000000000001 0.800000000000001 11.5
1.6
1.7
1.8
1.9
2
Tested parameter: κ , range: +-25%
2.25 2.45 2.65 2.85 3.05 3.25 3.451.5
1.6
1.7
1.8
1.9
2
Tested parameter: Xbar , range: +-25%
Calibration Results - VOW
Actual Returns
Fitted Returns
Mean 0.0065 0.0001
Std 0.1171 0.0618
Skewness
8.6589 -9.0180
Kurtosis 105.7043 123.6133
σ : 0.2296
κ: 0.0169
ϒ: 1
α: 1
β: 0.1365
X bar: 0.5
• Sample period: 1/1/2008 – 12/31/2008
Fitting
Gradual drift Sudden drop
• Imperfect fit due to:▫Fat tail returns of the actual stocks vs. Gaussian assumptions▫Model nature as below
Calibrating Chinese Stocks• In general, Chinese stock exhibit bubble effects with
A share price exceeding H share price•But this bubble effects are likely due to systematic
factors •Calibration divide into two categories:
▫1: Sample period including bubbles (Air China)▫2: Sample period without bubbles (Air Railway Con)
Calibration Category 1Sample period with bubbles (Air China)
Calibration Results – Air China
SH σ : 0.54
κ: 0.5
ϒ: 0.7 α: 5.6
β: 0.8 X bar: 0
HK
σ : 0.4 κ: 0.5
ϒ: -0.005
α: 5 β: 10 X bar: 3
• Sample period: 08/18/2006 – 04/19/2010• Fitted parameters:
Volatility
Low High
SH 4% 2.66 31.41
HK 3.6% 1.58 11.86
• Calibration shows that σ, ϒ, α and X bar are significantly different between SH and HK▫ σ: due to differences in actual volatilities▫ ϒ: due to differences in price ranges during
sample period
▫ X bar: reveals that “buy-in” intensity is higher for SH▫ α: reveals that the range of “buy-in” intensity fluctuation is higher in SH
Calibration Results – Air China•Generally good fit except A share kurtosis
SH Actu
SH Fit HK Actu
HK Fit
Mean 0.0028 0.0001 0.0013 0.0000
Std 0.0400 0.0551 0.0362 0.0252
Skewness
-0.0846 -7.2254 0.6069 0.0002
Kurtosis 3.5424 93.3621
8.5105 2.9954
Fitted kurtosis without extreme left tail points = 2.8653
Calibration Category 2Sample period without bubbles (China Railway Construction)
Calibration Results – China Rail Con• Sample period: 3/13/2008 – 4/19/2010• Fitted parameters:
Volatility
Low High
SH 2.6% 8.72 14.00
HK 3.3% 7.33 13.75
• Calibration shows that σ, κ, α and β are significantly different between SH and HK▫ σ: due to differences in actual volatilities▫ ϒ: no difference as price ranges are similar▫ κ, α and β: hard to detect individual impacts,
but can be interpreted together
SH σ : 0.28
κ: 0.3
ϒ: 0.015
α: 100
β: 10
X bar: 3
HK
σ : 0.4 κ: 0.8
ϒ: 0.015
α: 3 β: 1 X bar: 3
Calibration Results – China Rail Con• The net effect of κ, α and β is on the movement of “buy-in” intensity λ• λ varies between 0 and 40 for both A share and H share
A share
H share
Calibration Results – China Rail Con• Both parameter sets fit data well• ϒ =0.015 very small for both stocks
SH Act SH Fit HK Act SH Fit
Mean -0.0003 0.0000 0.0001 0.0000
Std 0.0264 0.0181 0.0327 0.0254
Skewness
-0.0162 -0.0351 0.0205 -0.0087
Kurtosis 6.2201 3.0105 10.5326 3.0018
A Share H Share
Conclusions• Calibration including period of bubbles (Air China)
▫ Calibrated L-A model have significant ϒ value, suggesting its ability to capture bubble effects, although such bubble is most likely due to systematic factors rather than HTB dynamics
• Calibration excluding period of bubbles (China Rail Con)▫ Calibrated L-A model have very small ϒ value
• Calibration could have multiple optimal parameters. It is essential to use multiple objective functions and criteria
Thank you! Questions?