Optional events and jumps

Black & Scholes (1973) made several simplifying assumptionsto derive their option pricing formula, amongthem that the price of the underlying asset follows a continuouslognormal random walk with constant volatility.However, most assets have dynamics more complexthan those of such a one-factor lognormal model. Variationson the asset dynamics, including time- and spot-dependent volatility,stochastic volatility and jumps in spot, have been invoked to explain thevolatility smiles, skews and term structures observed in various markets.In this article, we focus on one way that asset dynamics can be distinctfrom a lognormal random walk: those cases in which event risk plays alarge role. The applications include stock prices with the potential to crash,bonds whose issuers may default, and a case we study here in detail, currencieswith the potential for devaluation.

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