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Center for Nonlinear Studies |
Derivatives, the modern financial instruments, play an important role in the recent financial world crisis and getting a reliable estimation of their fair value has a central role in a lot of financial activities: Risk Management, Market Making Strategies, Arbitrage Opportunities, etc. On the other hand the need of managing simple financial models with few parameters with clear interpretation is of fundamental importance for a better understanding of the market situation and to avoid untenable time consuming simulations. To this end, the Black-Scholes (BS) model for option pricing and its simple generalizations is often used by practitioners because of its simplicity instead of other more complex models that require a bigger technical effort for their implementation, not always justified in terms of financial returns. In this framework we derive the distribution function of financial returns reversing the BS expression with volatility dependent on strike price (volatility smile). We show that this approach based on a volatility smile leads to relative minima for the distribution function (“bad” probabilities) never observed in real data and, in the worst cases, negative probabilities. We show that these undesirable effects can be eliminated by requiring “adiabatic” conditions on the volatility smile.
Host: Gennady Berman