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An Analysis of Dynamic Applications of Black-Scholes

Aileen Wang Period 5. An Analysis of Dynamic Applications of Black-Scholes. Purpose. Investigate Black-Scholes model Apply the B-S model to an American market Dynamic trading vs. fixed-time trading. What is option trading?. Option trading is a variation of market trading Calls and puts

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An Analysis of Dynamic Applications of Black-Scholes

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  1. Aileen Wang Period 5 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 • Fee charged for making a contract • Owner (buyer) of the contract has the option to exercise or to not exercise contract at time of maturity • More controlled • Investor has more clout • 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. Limitations • Disregarding extreme moves • B-S model does not cater to external influences • Influences are not predictable • Unforeseen risk = risk that cannot be avoided by investors

  17. Limitations • Instant, cost-free trading • Real life trading is not instant • Investor through system • System orders on trading floor • Order is put in • Not cost free either • Commission on stock trades • Fee for setting an option contract

  18. Limitations • Continuous time and continuous trading • Several gaps in time on the stock exchange • Big gaps: Friday afternoon – Monday morning • Daily gaps: Tuesday afternoon – Wednesday morning • Holidays: Stock exchange is not open for Christmas • Company/news changes during the gaps are not reflected in the market immediately

  19. Limitations • Standard trading • Standard trading means that currency exchange rates stay constant over time • FOREX • Market for currency trading • In the real world, currencies vary on a daily basis just like stocks

  20. 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

  21. Concluding thoughts • Models • Models are only models • Models cannot ever predict human behavior • Business and investing is not purely mathematical • Always use proper human judgment • Never rely on models to make all the decisions, financially or otherwise

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