This document provides an outline for a presentation on the Black-Scholes model for pricing options. It begins with an overview of the random behavior of asset prices and geometric Brownian motion. It then covers Ito's lemma and how it is used to derive the Black-Scholes partial differential equation. The document concludes by listing the key assumptions of the Black-Scholes model, including that the underlying asset price follows a lognormal random walk and that there are no arbitrage opportunities or transaction costs.
2. 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 implications
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29. 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.
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32. 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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