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Risk Neutral Pricing and Financial Mathematics: A Primer by John L. Teall, Peter M. Knopf

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Chapter 7

Derivatives Pricing and Applications of Stochastic Calculus

Chapter 7 begins the presentation of the most important applications of the first six chapters of mathematical tools, including the Black–Scholes solution to option pricing. The self-financing replicating portfolio is defined and created as a preface to the Black–Scholes derivation. Plain vanilla options are first priced using the martingale approach and then through the development of the Black–Scholes partial differential equation. We discuss its solution for the case of a European call. From an applications perspective, we discuss implied volatility and other methods to obtain the return variance on underlying stock. We discuss the problem of smiles and smirks and derive ...

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