  The binomial tree method and the Monte Carlo (MC) method are popular methods
for solving option pricing problems. However in both methods there is a
trade-off between accuracy and speed of computation, both of which are
important in applications. We introduce a new method, the MC-Tree method, that
combines the MC method with the binomial tree method. It employs a mixing
distribution on the tree parameters, which are restricted to give prescribed
mean and variance. For the family of mixing densities proposed here, the
corresponding compound densities of the tree outcomes at final time are
obtained. Ideally the compound density would be (after a logarithmic
transformation of the asset prices) Gaussian. Using the fact that in general,
when mean and variance are prescribed, the maximum entropy distribution is
Gaussian, we look for mixing densities for which the corresponding compound
density has high entropy level. The compound densities that we obtain are not
exactly Gaussian, but have entropy values close to the maximum possible
Gaussian entropy. Furthermore we introduce techniques to correct for the
deviation from the ideal Gaussian pricing measure. One of these (distribution
correction technique) ensures that expectations calculated with the method are
taken with respect to the desired Gaussian measure. The other one
(bias-correction technique) ensures that the probability distributions used are
risk-neutral in each of the trees. Apart from option pricing, we apply our
techniques to develop an algorithm for calculation of the Credit Valuation
Adjustment (CVA) to the price of an American option. Numerical examples of the
workings of the MC-Tree approach are provided, which show good performance in
terms of accuracy and computational speed.
