Arbitrage and derivative securities under fixed and proportional transaction costs.

The theory of stock price models under fixed transaction costs is a relatively little explored area of modern mathematical finance. This thesis introduces a fixed transaction cost process and uses it to not only explore the world of fixed transaction costs, but also combine fixed transaction costs with proportional transaction costs (so-called combined transaction costs). We prove a no-arbitrage equivalent condition for a model under combined transaction costs by re-visiting the existing no-arbitrage conditions for proportional and fixed costs respectively, and proving each of them, by taking a first principles approach. This result can be seen as the fundamental theorem for a model under combined transaction costs. This research on combined transaction costs also presents an extensive contribution to the analysis of European derivative securities. A distinction is made between the situation when the number of derivatives that can be traded on demand is limited compared to when it is unlimited. It will be shown that the ask and bid prices of a derivative security are unchanged by the presence of a fixed transaction cost when the derivative can be purchased in unlimited quantities. One of the main achievements here is a risk-neutral representation for the ask and bid prices of European derivative securities under combined transaction costs when the quantity of the derivatives that can be purchased on demand is restricted, as it overcomes hurdles connected to the lack of convexity which is involved in the combined cost ask price calculation algorithm