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Aileen WangPeriod 5
Computer Systems Lab 2010TJSTAR June 3, 2010
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
• More controlled
• 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
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