Implied volatility is widely believed to be informationally superior to historical volatility, because it is the "markets" forecast of future volatility. But for S&P 100 index options, the most actively traded contract in the United States, we find implied volatility. In aggregate and across...

Exact closed-form valuation equations for traded derivative securities are rare. Numerical approximation, most commonly with Binomial and Trinomial lattice models, gives exact valuation in the limit, but convergence is non-monotonic and often slow, due to 'distribution error' and 'truncation...

There has been much discussion of risks tied to trading in derivatives, with some well-informed objective observers arguing that derivatives risks are not significantly greater or different from those associated with traditional financial instruments. Financial risks are often broken down into...