Option Pricing and CVA Calculations using the Monte Carlo-Tree (MC-Tree) Method

Abstract: 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.