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Option Pricing ModelStephen Yan-leung Cheung

Professor of Finance (Chair)

Department of Economics and Finance

City University of Hong Kong

March 1, 2001

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Two Main Pricing Models

1. Black-Scholes Model

2. Binomial Model

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Option value is a function of: Stock price Strike price Risk-free rate Time to maturity of the option Volatility Dividends expected during the life

of the option

1. Black-Scholes Model

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1. Black-Scholes Model

Call Put

Variable

Stock Price

Strike Price

Time to Maturity

(in general)

Volatility

Risk-free Rate

Dividends

Assessibility

Yes

Yes

Yes

?

Yes

Yes

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1. Black-Scholes Model

What is meant by volatility?

It is the standard deviation of the stock return over a period of time.

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Take observations P1, P2,……Pn over a period, where Pi is the closing price of stock

1. Black-Scholes Model

Stock return (Ri) is the proportionate

change

Ri = ln (Pi/Pi-1)

The historical volatility is the standard deviation of Ri (annualized)

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1. Black-Scholes Model

Problems:What is the “right” period for

estimation?

Should be the most recent period that is generally commensurate with the expected option life.

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1. Black-Scholes Model

Problems:What is the “right” interval to

calculate the stock return?

For annualized volatility, weekly return should be enough.

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1. Black-Scholes Model

Problems:What is the “right” price to

calculate the stock return?

Weekly closing price or weekly highest price.

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1. Black-Scholes Model

Problems:New issues?

Historical volatility of similar company of the same industry.

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2. Binomial Model

Another useful & popular technique for pricing a stock option is binomial pricing model

Assumption has to be made on the possible changes of stock price over the life of the option

The two methods give similar approximation and the same result under some circumstances

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Biases in the Black-Scholes Model

Assumptions: Stock price distribution Volatility is constant Large jumps in stock price

Empirical result shows these problems seem to be less pronounced as an option’s life increases

Examples Options' value as at 29-Dec-00 based on volatility estimated by:

30-days 60-days 90-days

China Online (Bermuda) Ltd 0.026 0.033 0.032

Dao Heng Bank Group Ltd. 13.996 14.786 15.450

Bank Of East Asia 6.139 6.476 6.496

E-Kong Group Limited 0.373 0.405 0.400

Celestial Asia Securities 0.163 0.173 0.166

Iregent Group Ltd 0.054 0.044 0.038

H. K. Construction Hldgs 0.225 0.222 0.208

401.Com Ltd 0.003 0.003 0.003

China Star Entertainment Ltd 0.136 0.146 0.138

Kerry Properties Ltd 5.209 4.742 4.617

Quality Healthcare Asia Ltd 0.796 0.720 0.802

Li & Fung Ltd 5.919 6.597 6.089

Pacific Concord Hldg 0.389 0.379 0.379

E-Life International Limited 0.100 0.130 0.124

Easyknit International Hldg 0.148 0.169 0.158

Yugang International 0.028 0.035 0.033

Tak Sing Alliance Holdings 0.042 0.050 0.049

Softbank Investment Intl Str 0.875 0.929 0.918

Guoco Group Ltd 5.419 5.544 5.670

China Resources Enterprises 3.470 3.829 3.946

Examples Options' value as at 29-Dec-00 based on volatility estimated by:

1-years 2-years 3-years 4-years 5-years

China Online (Bermuda) Ltd 0.035 0.043 0.039 0.037 0.035

Dao Heng Bank Group Ltd. 16.538 17.005 19.917 19.610 18.650

Bank Of East Asia 7.168 7.432 8.041 7.838 7.467

E-Kong Group Limited 0.427 0.427 0.421 0.419 0.423

Celestial Asia Securities 0.174 0.194 0.198 0.189 0.179

Iregent Group Ltd 0.035 0.026 0.024 N.A. N.A.

H. K. Construction Hldgs 0.213 0.195 0.230 0.232 0.214

401.Com Ltd 0.003 0.004 0.004 0.004 0.004

China Star Entertainment Ltd 0.163 0.158 0.155 0.156 0.159

Kerry Properties Ltd 4.872 5.183 5.408 5.118 N.A.

Quality Healthcare Asia Ltd 0.849 0.817 0.798 0.837 0.812

Li & Fung Ltd 6.723 6.174 5.945 5.695 5.524

Pacific Concord Hldg 0.482 0.453 0.459 0.460 0.447

E-Life International Limited 0.138 0.133 0.130 0.127 N.A.

Easyknit International Hldg 0.236 0.225 0.224 0.218 0.209

Yugang International 0.039 0.039 0.039 0.039 0.038

Tak Sing Alliance Holdings 0.059 0.056 0.055 0.056 0.054

Softbank Investment Intl Str 0.940 0.939 0.936 0.931 0.924

Guoco Group Ltd 6.068 6.047 6.777 6.611 6.368

China Resources Enterprises 3.963 3.935 4.383 4.536 4.383

Examples Options' value as at 29-Dec-00 based on volatility estimated by:

1-years 2-years 3-years 4-years 5-years

China Online (Bermuda) Ltd 0.035 0.043 0.039 0.037 0.035

Dao Heng Bank Group Ltd. 16.538 17.005 19.917 19.610 18.650

Bank Of East Asia 7.168 7.432 8.041 7.838 7.467

E-Kong Group Limited 0.427 0.427 0.421 0.419 0.423

Celestial Asia Securities 0.174 0.194 0.198 0.189 0.179

Iregent Group Ltd 0.035 0.026 0.024 N.A. N.A.

H. K. Construction Hldgs 0.213 0.195 0.230 0.232 0.214

401.Com Ltd 0.003 0.004 0.004 0.004 0.004

China Star Entertainment Ltd 0.163 0.158 0.155 0.156 0.159

Kerry Properties Ltd 4.872 5.183 5.408 5.118 N.A.

Quality Healthcare Asia Ltd 0.849 0.817 0.798 0.837 0.812

Li & Fung Ltd 6.723 6.174 5.945 5.695 5.524

Pacific Concord Hldg 0.482 0.453 0.459 0.460 0.447

E-Life International Limited 0.138 0.133 0.130 0.127 N.A.

Easyknit International Hldg 0.236 0.225 0.224 0.218 0.209

Yugang International 0.039 0.039 0.039 0.039 0.038

Tak Sing Alliance Holdings 0.059 0.056 0.055 0.056 0.054

Softbank Investment Intl Str 0.940 0.939 0.936 0.931 0.924

Guoco Group Ltd 6.068 6.047 6.777 6.611 6.368

China Resources Enterprises 3.963 3.935 4.383 4.536 4.383

Diff.

22.86%

20.43%

12.18%

1.91%

13.79%

45.83%

18.97%

33.33%

5.16%

11.00%

6.39

21.71%

7.83%

8.66%

12.92%

2.63%

9.26%

1.73%

12.07%

15.27%

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~ End ~