Option pricing with transaction costs

Portfolio optimization is an important field of research within financial engineering. The aim of the optimization is to fins what is the best strategy for an investor when choosing how to allocate their money between a bank account and a constant number of risky assets. In our problem, the investor must pay transaction costs, meaning that every time he transfers money, he loses a certain percentage of the money transferred. Thus, we have made the assumption of proportional transaction costs. In a frictionless market, Merton has proven that the optimal policy consists of a constant proportion of wealth in the risky asset. This means that one must constantly rehedge the portfolio to keep this ratio constant regardless of the evolution of the risky asset´s value. When transaction costs are imposed, repeated rehedging becomes too expensive and the optimal policy of investment is different. The so-called transaction cost region will appear; the investor should buy, sell or stay idle depending on whether his position at current time is above, below or within this region. One can show that we can transform the portfolio optimization problem into a double obstacle problems. Using this latter form of the problem, we have created and algorithm unveiling the different transaction cost regions. The algorithm and results of this algorithm will be presented