Pure discount bonds

In this chapter we consider only default-free securities. We use interchangeably the notation ( ) = E - Ft). We shall denote by p(t,T) the price at time t of a pure discount bond with maturity T and obviously p(t,t) = p T,T)=l. 

This is the same model as that proposed by Rendleman and Bartter. This model is the only log-normal, single-factor model that leads to closed formulae for pure discount bonds. Nonetheless there is no closed formula for a European option on a pure discount bond. 

This model is similar to Dothan model being a log-normal short rate model. This model, however, does not lead to explicit formulae for pure discount bonds or for options contingent on them. In addition this is an... 

Bonds are traded generally over the counter. Futures contracts on bonds may be more liquid and may remove some of the modelling difficulties generated by the known value at maturity of the bonds. Hedging may be more efficient in this context using the futures contracts on pure discount bonds (provided they are liquid) rather than the bonds themselves. Chen provides closed-form solutions for futures and European futures options on pure discount bonds, under the Vasicek model. 

Ren-Raw Chen, Exact Solutions for Futures and European Futures Options on Pure Discount Bonds, Journal of Financial and Quantitative Analysis 27, no. 1 (March 1992), pp. 97-107. 

Initially the first formulas on pricing options on pure discount bonds used the Vasicek model for the term structure of interest rates. Thus, given that r follows equation (18.6), the price of a European call option with maturity Tq with exercise price fC on a discount bond maturing at T(Tq < T) is... 

One notable exception from this general class is the CIR model. There is a closed-formula for this case too. Following Clewlow and Strickland the price at time 0 of a European pure discount bond option is... 

Taking the same example as that developed to demonstrate the Vasicek model earlier, we now price the 3-year European call option on a 10-year pure discount bond using the CIR model for the short interest rates. Recall that face value is 1 and exercise price K is equal to 0.5. As in the example with the Vasicek model, we consider that o = 2% and tq = 3.75%. The CIR model overcomes the problem of negative interest rates (acknowledged as a problem for the Vasicek model) as long as 2a > o. This is true, for example, if we take a = 0.0189 and P = 0.24. Feeding this information into the above formulae is relatively tedious. A spreadsheet application is provided by Jackson and Staunton, After some work we get that the price of the call is... 

As discussed in chapter 1, there are two types of fixed-income securities zero-coupon bonds, also known as discount bonds or strips, and coupon bonds. A zero-coupon bond makes a single payment on its maturity date, while a coupon bond makes interest payments at regular dates up to and including its maturity date. A coupon bond may be regarded as a set of strips, with the payment on each coupon date and at maturity being equivalent to a zeto-coupon bond maturing on that date. This equivalence is not purely academic. Before the advent of the formal market in U.S. Treasury strips, a number of investment banks traded the cash flows of Treasury securities as separate zero-coupon securities. 

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