Pages: 100

We begin our study of financial mathematics in this chapter by introducing the key concepts of rate-of-return and present value . With the stage thus set, we will be prepared to investigate a number of important ideas in the theory of interest: compound interest, annuities, loan repayment, and the valuation of bonds. Prerequisite mathematics for the first few sections of this chapter, in which time moves discretely, is simply standard precalculus material with an emphasis on geometric sequences and series (which will be reviewed). At the end of the chapter we will see how the ideas extend to continuous time models, for which both differential and integral calculus will be necessary. By the end of the chapter students should be prepared to make educated decisions in many financial situations in their personal lives. The groundwork for further study in financial mathematics is laid as well, including study of actuarial mathematics and preparation for the FM exam that is administered by the major actuarial societies. For this purpose, you may find the references Federer Vaaler [6] and Broverman [3] to be very useful. Both Chapters 1 and 2 of this book are heavily influenced by these works. Furthermore, the financial services industry is concerned with how to value firms and tradable assets, and students will acquire the background to continue the pursuit of problems related to this important area.

Pages: 54

One of the primary goals of financial mathematics is to give fair values to financial securities that trade in markets, such as futures, options, and, in this chapter, bonds.

Pages: 104

Very rarely in the preceding chapters have we allowed the possibility that financial quantities behave unpredictably. But variable interest rates on loans do happen, investors may miss payments into annuities at random times, bonds can default, and, most important, some of the most interesting investment objects, like common stocks, definitely do not have a predictable evolution in value over time. Clearly the assumption of determinism of all inputs to a financial problem is inadequate to model the real world.

Pages: 34

One of the main thrusts in the subject of financial mathematics is the study of the optimal distribution of wealth among collections of risky assets called portfolios . We will consider mostly portfolios of stocks in this chapter, and also limit our attention to single-time-step problems. In such problems, there is a current time “now” and a time “later” at which we observe the performance of our investment. We have a finite collection of stocks in which we are interested in investing. Our task is to decide how to allocate money among the stocks, in view of the tradeoffs between return and riskiness that are natural in a financial market.

Pages: 40

Much of the field of financial mathematics has been devoted in recent years to the theory and application of financial derivatives , that is, assets whose values depend on some other underlying asset. Derivatives provide opportunities for investment speculation that can produce great rewards (in percentage terms), but also incur great losses. Derivatives can also be used in tandem with other assets to hedge , that is, to provide protection against risk. In this chapter we will introduce derivatives in a discrete-time environment whose properties are fairly well known. In Section 5.1, the basic language of derivative pricing theory and the key idea of arbitrage are introduced. Anti-arbitrage assumptions will be used in Section 5.2 to find derivative values in single time period models. These results are generalized rather easily to multiple time period models for simple derivatives in Section 5.3.

Pages: 48

The field of Financial Mathematics contains many interesting topics beyond what has been covered so far, and we will close this book by introducing a few of them.