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

0.6

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?

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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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