When the Black/Scholes formula is inverted to calculate the volatilities implied by reported option prices, the volatility estimates differ across exercise prices and times to expiration. Such is typically the case for options written on an index of stock market prices. Options that are deep in-the-money or out-of-the-money have higher implied volatilities than at-the-money options. The failure of the Black/Scholes model to describe the structure of reported option prices is thought to arise from its constant variance assumption. In response, recent literature has developed variations of the deterministic volatility function (DVF) option valuation model. Rather than positing a structural form for the volatility function, these authors search for a binomial or trinomial lattice that achieves an exact cross-sectional fit of reported option prices.

In Discussion Paper No. 1369, Research Fellow Bernard Dumas, Jeff Fleming and Robert Whaley assess the stability of the implied deterministic volatility function for the S&P 500 index. Since valuation and risk management are measured in dollars and cents, they evaluate the stability of the estimated volatility function by examining how well it predicts future option prices. They estimate the volatility function based on the cross-section of reported option prices one week, and then examine the price deviations from theoretical values a week later. They find two ways in which the deterministic volatility function (DVF) approach does not work. First, as they move one week into the future, the ability of the volatility function, estimated seven days earlier, to price options deteriorates markedly. It deteriorates so much that an ad hoc approach, based on an internally inconsistent use of the Black/Scholes formula with time-varying implied volatilities, produces smaller pricing errors than does the DVF approach. Second, the approach also cannot serve as a reliable guide for the purpose of hedging options. Here again, the hedging errors generated by the DVF model are larger than those of an ad hoc, internally inconsistent model. The most likely interpretation of their findings is that volatility itself must be considered stochastic. That is, a separate stochastic process must be postulated for volatility, which then requires estimates of the volatility of volatility to be set.

Implied Volatility Functions: Empirical Tests
Bernard Dumas, Jeff Fleming and Robert E Whaley