A New Procedure for Pricing Parisian Options

For many derivatives, the payoff at expiration depends on a function of one or more random variables. Even though each one may follow an easily-handled distribution such as the lognormal, applying the function, which may be as simple as just taking the average, leads to an intractable distribution. Sometimes, the problem can be solved with transform techniques, like Fourier or Laplace transforms, because the transform can change the function into a form that can be manipulated more easily. Applying an inverse transform gives the answer in terms of probabilities. But this is where the difficulty arises, because the inverse transform is often not amenable to easy solution. In this article Bernard et al. present a procedure for approximating a general Laplace transform with one that can be easily inverted. They demonstrate the use of the approach to price Parisian options, a class of barrier option that is activated only when the price of the underlying penetrates a given barrier and stays beyond it for a specified amount of time. Using their transform technique, accuracy is excellent and solution time becomes practically instantaneous.