All the explicit formulae for the valuation of lookback and barrier options available in the financial literature assume continuous monitoring of the underlying asset. In practice, however, monitoring is always discrete, and the gap between continuously and discretely monitored option values can be very large. In this paper, we provide explicit formulae for discretely monitored lookback and barrier options. They allow for non-constant volatility, interest rate, dividend rate and barrier parameters that vary as step functions of time. They can deal with any number and spacing of monitoring dates. They are not restricted to particular payoffs or strike price specifications. We also provide a simple rule for the numerical integration of these high-dimensional formulae, as well as an efficient interpolation method.
Discretely monitored lookback and barrier options : a semi-analytical approach