Barrier options are financial derivative contracts that are activated or deactivated according to the crossing of specified barriers by an underlying asset price. Exact models for pricing barrier options assume continuous monitoring of the underlying dynamics, usually a stock price. Barrier options in traded markets, however, nearly always assume less frequent observation, e.g. daily or weekly. These situations require approximate solutions to the pricing problem. We present a new approach to pricing such discretely monitored barrier options that may be applied in many realistic situations. In particular, we study daily monitored up-and-out call options of the European type with a single underlying stock. The approach is based on numerical approximation of the transition probability density associated with the stochastic differential equation describing the stock price dynamics, and provides accurate results in less than one second whenever a contract expires in a year or less. The flexibility of the method permits more complex underlying dynamics than the Black and Scholes paradigm, and its relative simplicity renders it quite easy to implement.

bstract: When we use stochastic differential equations as models of financial data that appear as time series, we have to estimate the equation parameters. For complex models this is not straightforward. Approximate maximum likelihood methods are useful tools for this purpose. We suggest the following approach: The likelihood function given by the time series and the parameters is estimated for fixed values of the parameter vector. We apply a standard optimization method that repeatedly calls the estimation procedure, with different parameter vectors as arguments, until the optimization converges. The likelihood function is the product of the transition probability densities given by the data. By solving the Fokker-Planck equation associated with the stochastic differential equation, one can obtain these probability densities. Exact, analytical solutions to the Fokker-Planck equation can rarely be found. We therefore apply a path integral method to find approximate solutions. This path integral method is based on the fact that solutions to stochastic differential equations are Markov processes. The time intervals between all the pairs of consecutive data are split in smaller partitions, so that the Euler-Maruyama method is fairly accurate. For all the pairs of data, the total probability law is then applied recursively with the delta distribution given by the first data point as the initial density. The propagating density is represented numerically on a finite, adaptive grid, and the Euler-Maruyama method provides an approximation to the conditional probability density that appears in the total probability law. The method is tested on artificially generated time series with known parameter vectors. It is seen that it yields satisfying parameter estimates for different models in quite reasonable CPU-time, even with thousands of data. It also compares favorably with other methods.