The Black–Scholes equation to price an option on an asset can be derived by several different approaches. This chapter first discusses the concept of an equivalent martingale measure and then lays the foundation for risk-neutral valuation. Using these tools, one derivation of the Black–Scholes equation is presented based on a hedged portfolio.

A now well-known issue with the Black–Scholes model is the observation of a volatility smile. To account for the excess kurtosis in equity index or foreign exchange log-return data, deep in or out of the money options will trade at a value higher than that predicted by the Black–Scholes equation. Inverting the Black–Scholes equation shows a higher implied volatility based on the pricing of in and out of the money options compared to the volatility from at the money options. Furthermore, many assets do not exhibit large upward jumps but have a probability of sharp declines much higher than that predicted from a lognormal distribution. This actual negative skewness or possibly just investor fear of a large price decrease manifests as a volatility skew, that is, smirk, where out of the money puts are more valuable than predicted by the Black–Scholes equation.

Merton (1976) addressed the negative skewness and excess kurtosis in log-return data by augmenting the Black–Scholes framework with a jump component. It is discussed that Merton's assumption of a lognormal jump component allowed the derivation of a semianalytical ...

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