Pricing guaranteed return rate products and discretely sampled Asian options

ABSTRACT The present paper demonstrates a fast and numerically stable pricing algorithm that can determine the price of a guaranteed rate product, as well as its sensitivity to changes in the market (the Greeks) both for lognormal and jumpdiffusion asset price processes, with almost machine precision in a fraction of a second. In fact, the pricing algorithm only needs the assumption that the returns per period of the asset price process are independent; this enables evaluation for Lévy processes. Since guaranteed return rate products can be regarded as generalized discretely sampled Asian options, we can compute the price and Greeks for these Asian options as well. Using a new Laplace inversion technique developed by Den Iseger (2006a,b), we compute recursively the crucial densities at the sample times (needed for the computation of the price and the Greeks). This inversion technique computes the coefficients of a piecewise Legendre polynomial expansion for the original function if specific Laplace transform function values are known, and, conversely, obtains Laplace transform function values if specific values of the original function are known. We also present a technique to compute Greeks for path-dependent options in a lognormal and jump-diffusion model. This technique is based on a Girsanovtype drift adjustment.