2 Wilmott magazine

Recipe for a … Hangover? formulate a number of volatility scenarios, value a derivative contract using each one of them, and average the values with certain weights. This more likely than not will introduce a volatility smile effect in valuation. It what follows we will call such models “mixture models”. They are different from what is known conventionally as “stochastic volatility models ” as the latter specify a stochastic process for volatility, whereas the former only specify a discrete distribution for it. It has long been realized (Hull and White 1987) that for European-style options, the actual volatility path has no bearing on the option value, as only the average volatility (what we call “term volatility ” to dis-tinguish it from instantaneous volatility) between now and the option’s expiry matters. Because of that, mixture models can legitimately be applied to European options. Non-dynamic nature of stochastic volatility is of no importance as only the total effect of randomness on the ter