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June 2011
Black Scholes for TechiesAraik Grigoryan
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Outline
✤ Random behavior of assets
✤ Stochastic Differential Equation (SDE) for geometric Brownian motion
✤ Taylor Series
✤ Ito’s Lemma
✤ Black Scholes special portfolio and derivation of the Partial Differential Equation (PDE)
✤ Black Scholes assumptions and implications2
Random behavior of assets
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Return
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Mean of returns
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Standard deviation of returns
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Scaled return
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Returns as a frequency distribution
0
0.1
0.2
0.3
0.4
0.5
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0.7
-4 -3 -2 -1 0 1 2 3 4
Empirical PDF Normal PDF
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Returns as a rough model
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How does the mean change with time?
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How does the standard deviation change with time?
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Finally, a discrete-time model
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And now for something completely continuous
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Questions?
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Taylor series of a function of one variable
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Taylor series of a function of two variables
?
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What is dS2?
?17
What is dX2?
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Ito’s lemma
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Questions?
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Special portfolio
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Change in portfolio value
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To eliminate risk, carefully choose...
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Money in the bank also grows risk free
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Equate option and money in the bank portfolios
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The Black-Scholes equation
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The Black-Scholes equation, in words
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Solution for a call option
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Black-Scholes assumptions
✤ There are no dividends on the underlying.
✤ The underlying follows lognormal random walk. In reality, it does not have to be true but other forms may not have closed-form solutions and will have to be solved numerically.
✤ Interest rate r is a known function of time. In reality, it is not known in advance and is stochastic.
✤ There are no transaction costs.
✤ There are no arbitrage opportunities.29
Why is Θ proportional to Γ?
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Questions?
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References
✤ Paul Wilmott. Paul Wilmott Introduces Quantitative Finance. John Wiley and Sons, West Sussex, England, 2007.
✤ Fischer Black and Myron Scholes. “The Pricing of Options and Corporate Liabilities”. Journal of Political Economy 81 (3): 637-654.
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