DUPIRE’S EQUATION FOR BUBBLES

Abstract. We study Dupire’s equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset fol-lows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied. 1