The Effect of Correlation and Transaction Costs on the Pricing of Basket Options

In this article, we examine the problem of evaluating the option price of a European call option written on <italic>N</italic> underlying assets when there are proportional transaction costs in the market. Since the portfolio under consideration consists of multiple risky assets, which makes numerical methods formidable, we use perturbation analyses. The article extends the model for option pricing on uncorrelated assets, which was proposed by Atkinson and Alexandropoulos (2006; Pricing a European basket option in the presence of proportional transaction cost, <italic>Applied Mathematical Finance</italic>, 13(3), pp. 191--214). We determine optimal hedging strategies as well as option prices on both correlated and uncorrelated assets. The option valuation problem is obtained by comparing the maximized utility of wealth with and without option liability. The two stochastic control problems, which arise from the transaction costs, are transformed to free boundary and partial differential equation problems. Once the problems have been formulated, we establish optimal trading strategies for each of the portfolios. In addition, the optimal hedging strategies can be found by comparing the trading strategies of the two portfolios. We provide a general procedure for solving <italic>N</italic> risky assets, which shows that for ‘small’ correlations the <italic>N</italic> asset problem can be replaced by <italic>N</italic> (<italic>N</italic> − 1)/2 two-dimensional problems and give numerical examples for the two risky assets portfolios.