Coupon-bearing bond

The owner of a swaption with strike price K maturing at time 7b, has the right to enter at time To the underlying forward swap settled in arreas. A swaption may also be seen as an option on a coupon bearing bond (see e.g. Musiela and Rutkowski [61]). 

Starting from the payoff function of a European option on a coupon bearing bond we can write the option price at the exercise date Tq as follows... 

On the contrary, there exists no closed-form solution for an option on a coupon bearing bond for multi-factor models. Furthermore, the characteristic function cannot be computed in closed-form and the Fourier inversion techniques are widely useless. Nevertheless, the moments of the underlying random variable can be computed and the lEE approach is applicable. 

For that reason, we derived the lEE techmque in chapter (4). Then the computation of the option priee on a coupon bearing bond is possible, even if there exists no semi-closed characteristic function. 

Eirst, we have to rewrite the date-t price of an option on a coupon bearing bond (2.4) as a function of the random variable V (7b, 7) ) leading to... 

As in section (5.3), we apply the IFF for the computation of an option on a coupon bearing bond, by deriving the solution of the transform... 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

Many Euro government bonds can be stripped, breaking them down into each of the single payments that they involve, that is, one flow for each remaining coupon payment and another one for the principal. With this procedure an n-year maturity coupon-bearing bond is transformed into n + 1 strips (zero coupon bonds), which can be traded separately in the market. Yet this market is much less liquid in the Eurozone than in the United States. 

Consider a bond paying a periodic cash payment p at times Ti,T2,...,T , and the principal at maturity T = T j. A coupon bond can be mapped into a portfolio of discount bonds with corresponding maturities (under one source of uncertainty, that is one factor model). The value of a coupon bearing bond at time t [Pg.594]

Hence, the value at time 0 of a European call option with maturity Tq and strike price K on the coupon bearing bond, under the one-factor HJM model described above, is given by... 

It is clear now that this is the same as a European call option on a coupon-bearing bond when the exercise price is equal to 1. 

The true yield measure derived in the previous section is not as straightforward as the one given earlier for the T-bill. Because a T-bill has only a single cash flow, its maturity value is known, so its return is easily calculated as its increase in value from start to maturity. Investors know that money put into a 90-day T-bill with a yield of 5 percent will have grown by 5 percent, compounded semiannually, at the end of three months. No such certainty is possible with coupon-bearing bonds. Consider although the investors in the 90-day T-bill are assured of a 5 percent yield after ninety days, they don t know what their investment will be worth after, say, sixty days or at what yield they will be able to reinvest their money when the bill matures. Such uncertainties don t effect the return of the short-term bill, but they have a critical impact on the return of coupon bonds. 

Treasury bond A negotiable, coupon-bearing debt obligation issued by the U.S. government and backed by its full faith and credit, having a maturity of more than seven years. Interest is paid semiannually. Exempt from state and local taxes. Also called U.S. Treasury bond. 

Although the introduction of the euro has helped to homogenise the characteristics and maturities of the bonds issued, there are still some differences between these countries supplied assets. Euro-denominated fixed coupon bonds bear the brunt of these countries issuance, but there are also some other types of bonds being issued by these countries. 

Interest rate risk comes three ways. First you will buy a bond bearing an 8 percent coupon rate, and interest rates will rise to 9 percent. This is an opportunity cost since investors who bought bonds yielding 8 percent are earning less than those who purchased bonds yielding 9 percent. 

---