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?