  We study the problem of finding probability densities that match given
European call option prices. To allow prior information about such a density to
be taken into account, we generalise the algorithm presented in Neri and
Schneider (2011) to find the maximum entropy density of an asset price to the
relative entropy case. This is applied to study the impact the choice of prior
density has in two market scenarios. In the first scenario, call option prices
are prescribed at only a small number of strikes, and we see that the choice of
prior, or indeed its omission, yields notably different densities. The second
scenario is given by CBOE option price data for S&P500 index options at a large
number of strikes. Prior information is now considered to be given by
calibrated Heston, Schoebel-Zhu or Variance Gamma models. We find that the
resulting digital option prices are essentially the same as those given by the
(non-relative) Buchen-Kelly density itself. In other words, in a sufficiently
liquid market the influence of the prior density seems to vanish almost
completely. Finally, we study variance swaps and derive a simple formula
relating the fair variance swap rate to entropy. Then we show, again, that the
prior loses its influence on the fair variance swap rate as the number of
strikes increases.
