Using Monte Carlo to Price Continuously Monitored Barrier Options on Discontinuously Traded Underlyings

Continuously monitored down-and-out calls and up-and-out puts have become very popular during the last decade, mainly because they can be interpreted as simple leveraged positions. Their deltas stay close to unity and their Greeks are small, particularly in low interest rate environments. However, due to discontinuous trading, barrier options are subject to gap risk which is not captured by incumbent pricing models. In this paper we present two Monte Carlo methods and a novel variance reduction technique to value knockout barrier options if the underlying has periodic trading halts: the adaptive step method which lets the sample step depend on the distance from the barrier, and the min-end simulation which samples minimum values during continuous intervals. To increase the Monte Carloïs power, we use the Modigliani Miller parity which says that the value of a knockout barrier option is equal to the difference between the underlying and a bond secured by the underlying.