Chapter 17 - Arbitrage Theorem in a New Setting

Abstract

The motivation for the main tools in derivatives pricing was introduced in the simple model of Chapter 2. There we discussed a simple construction of synthetic (martingale) probabilities that played an essential role in the first part of this book. Because the setting was very simple, it was well-suited for motivating complex notions such as risk-neutral probabilities and the crucial role played by martingale tools. Chapter 2 considered a model where lending and borrowing at a constant risk-free rate was one of the three possible ways of investing, the other two being stocks and options written on these stocks. Throughout Chapter 2 interest rates were assumed to be constant and a discussion of interest-sensitive financial derivatives was deliberately omitted. Yet, in financial markets a large majority of the instruments that trade are interest-sensitive products. These are used to hedge, to arbitrage the interest rate risk, and to speculate on it. Relaxing the assumption of constant interest rates is, thus, essential. As mentioned in Chapter 16, relaxing the assumption of constant interest rates and then introducing complex interest rate derivatives creates a need for new mathematical tools, most of which were discovered only lately. This chapter attempts to motivate these notions and introduces the new tools by using a simple discrete-state approach similar to the one utilized in Chapter 2. However, the model is extended in new directions so that these new tools and concepts can also be easily understood. By expanding the simplified framework of Chapter 2, one can discuss at least three major additional results.