A Workout in Computational Finance

4 Mean Reversion and Trinomial Trees

In the previous chapters, we have dealt with the basic Black-Scholes model (and its discrete tree versions) for describing the stock price movement of an equity. In this chapter, we introduce several interest rate instruments (also named fixed income instruments) and a number of basic models, the so-called one-factor short rate models. These describe the stochastic behavior of the short rate, the fictitious underlying which is, by construction, the interest rate paid for an incremental time.

Fixed income instruments are of huge importance in financial markets. According to the Bank of International Settlement, the outstanding notional volume of over-the-counter¹ (OTC) derivatives was 600 (American) trillion USD by the end of 2010. The portion of OTC interest rate derivatives among these was 465 trillion, more than three quarters in terms of notional amount. (Source: http://www.bis.org/statistics/otcder/dt1920a.pdf).

4.1 SOME FIXED INCOME TERMS

Governments (or banks or corporations) issue bonds when they need money to fulfill their obligations and the income from taxes (revenues) does not suffice or when it can be foreseen that there will be a liquidity bottleneck in the near future. The investor in the bond pays a sum that is to be invested, say, 10 000 EUR to the German Republic, and obtains the fixed income of the annual interest rate on her investment. At the maturity of the bond, the invested sum is paid back to the investor. This scenario ...