Aileen WangPeriod 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

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

What is option trading?

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

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Scope of Study

Analysis of input variables

What are they? How will they be

obtained? What formulas are

necessary to calculate them?

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

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

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

Black-Scholes

Volatility

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%

AAPL – Sample Case

•Graphs comparing call and put values of expected versus actual.

AAPL – Sample Case

•Model is a good estimator for call, but put values tend to deviate as strike price increases

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?

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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)

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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

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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

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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

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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

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Limitations

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

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