Interest rates affect economic activities at all levels. Central banks, including the Federal Reserve (informally known as the Fed) target interest rates as a policy tool to influence economic activity. Interest rate derivatives are popular with investors who require customized cash flow needs or specific views on interest rate movements.
One of the key challenges that interest rate derivative traders face is to have a good and robust pricing procedure for these products. This involves understanding the complicated behavior of an individual interest rate movement. Several interest rate models have been proposed for financial studies. Some common models studied in finance are the Vasicek model, CIR model, and Hull-White model. These interest rate models involve modeling the short rate and rely on factors (or sources of uncertainty) with most of them using only one factor. Two-factor and multifactor interest rate models have been proposed.
In this chapter, we will cover the following topics:
- The yield curve in a normal environment and an inverted environment
- Valuing a zero-coupon bond using Python
- Bootstrapping a yield curve
- Calculating forward rates from the yield curve
- Calculating the yield to maturity and price of a bond
- Calculating the bond duration and convexity using Python
- Discussing short rate models as a function of the yield curve
- The Vasicek short rate model
- The Cox-Ingersoll-Ross short rate model
- The Rendleman and Bartter short rate model
- The Brennan and Schwartz short rate model
- Pricing a callable zero-coupon bond using finite differences
- Discussing methods of callable bond pricing
Corporations and governments issue fixed-income securities as a means of raising money. The owner of such debts lends money and expects to receive the principal when the debt matures. The issuer who wishes to borrow money may issue a fixed amount of interest payment during the lifetime of the debt at prespecified times.
The holder of debt securities, such as U.S. Treasury bills, notes, and bonds, faces the risk of default by the issuer. The federal government and municipal government are thought to face the least default risk since they can easily raise taxes and create more money to repay the outstanding debt dues.
Most bonds pay a fixed amount of interest semi-annually, while some pay quarterly, or annually. These interest payments are also referred to as coupons. They are quoted as a percentage of the face value or par amount of the bond on an annual basis. For example, a five-year $10,000 Treasury bond with a coupon rate of 5 percent pays coupons of $500 in each year, or coupons of $250 every six months, up till and including the maturity date. Should the interest rates drop and new T-bonds pay a 3 percent coupon rate, the buyer of the new bond will only receive coupons of $300 annually, while existing holders of the 5 percent bond will continue to receive $500 annually. As the characteristic of the bonds influences its prices so they're closely related to current levels of the interest rates in an inverse relationship manner, the value of the bond decreases as the interest rates increase. As interest rates decrease, bond prices increase.