CP-Bruno Dupire.qxd

prices as a function of volatility. If an option price is given by the market we can invert this relationship to get the implied volatility. If the model were perfect, this implied value would be the same for all option market prices, but reality shows this is not the case. Implied Black–Scholes volatilities strongly depend on the maturity and the strike of the European option under scrutiny. If the implied volatilities of at-the-money (ATM) options on the Nikkei 225 index are 20% for a maturity of six months and 18% for a maturity of one year, we are in the uncomfortable position of assuming that the Nikkei oscillates with a constant volatility of 20% for six months but also oscillates with a constant volatility of 18% for one year. It is easy to solve this paradox by allowing volatility to be timedependent, as Merton did (see Merton, 1973). The Nikkei would first exhibit an instantaneous volatility of 20% and subsequently a lower one, computed by a forward relationship to accommodate the one-year volatility. We now have a single process, compatible with the two option prices. From the term structure of implied volatilities we can infer a time-dependent instantaneous volatility, because the former is the quadratic mean of the latter. The spot process S is then governed by the following stochastic differential equation: