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An Analysis of Dynamic Applications of Black-Scholes

This study investigates the application of the Black-Scholes model in an American market and compares the generated call and put values with actual historical values. The study also explores the limitations of the model and suggests further research for future application.

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An Analysis of Dynamic Applications of Black-Scholes

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  1. Aileen Wang Period 5 Computer Systems Lab 2010 TJSTAR June 3, 2010 An Analysis of Dynamic Applications of Black-Scholes

  2. Purpose Investigate Black-Scholes model Apply the B-S model to an American market Dynamic trading vs. fixed-time trading

  3. What is option trading? • Option trading is a variation of market trading • Calls and puts • More controlled • Not necessarily at market price

  4. Questions To what kind of stock options is the Black-Scholes model most applicable to? Validity: How does Black-Scholes generated call and put values compare with the actual historical values? Variable factors: Stocks of a different industry (finance sector stocks vs. agriculture vs. technology) Different volatilities, different price levels

  5. Scope of Study • Analysis of input variables • What are they? • How will they be obtained? • What formulas are necessary to calculate them?

  6. Related Studies • 1973: Black-Scholes created • 1977: Boyle’s Monte Carlo option model • Uses Monte Carlo applications of finance • 1979: Cox, Ross, Rubenstien’s bionomial options pricing model • Uses the binomial tree and a discrete time-frame • Roll, Geske, and Whaley formula • American call, analytic solution

  7. Background Information • Black-Scholes: Two parts • Black-Scholes Model • Black-Scholes equation: partial differential equation • Catered to the European market • Definite time to maturity • American Market • Buy and sell at any time • More dynamic and violatile

  8. Procedure and Method • Main language: Java • Outputs: • Series of calls and puts • Spreadsheet, time-series plot • Inputs • Price • Volatility • Interest rate • Test data and historical data

  9. Black-Scholes

  10. Volatility

  11. AAPL – Sample Case • At a given time t, the stock price for AAPL was 239.94. • APPL options used are ranged from 90.00 to 190.00 in increasing increments of 5.00. • Three days until maturity, volatility of 20%, and a risk free rate of 0.35%

  12. AAPL – Sample Case • Graphs comparing call and put values of expected versus actual.

  13. AAPL – Sample Case • Model is a good estimator for call, but put values tend to deviate as strike price increases

  14. Limitations • Why doesn’t B-S always work? • Out of the money • Strike price is above stock price, option has no value • Disregards risk such as • Stock market crashes • Unexpected outside influences (terrorist attacks, mergers and acquisitions) • Typos?

  15. Limitations • B-S has many assumptions that are far from valid in real life: • Disregard of extreme moves • Assumes instant, cost-free trading • Continuous time and continuous trading • Standard trading (volatility risk of currency adjustments)

  16. Results • Explore • Option pricing with mathematics • Validity of the model • Comparing stocks of different volatility, industry, and nature • Further research • Comparison with other mathematical models • Application into markets in other countries

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