Bond pricing zero-coupon bonds formula

All bonds except zero-coupon bonds accrue interest on a daily basis that is then paid out on the coupon date. As mentioned earlier, the formulas discussed so far calculate bonds prices as of a coupon payment date, so that no accrued interest is incorporated in the price. In all major bond markets, the convention is to quote this so-called clean price. 

Expression (3.14) is the formula for pricing zero-coupon bonds when the spot rate is the nonconstant instantaneous risk-free rate r(s) described above. This is the rate used in formulas (3.12), for valuing a money market account, and (3.15), for pricing a risk-free zero-coupon... 

The price process under the new measure Tq, either is used to derive the formula for the zero-coupon bond option (see section (5.2.1)), the characteristic function in (5.2.2), or finally to compute the moments of the underlying random variable (section (5.3.3) and (5.3.4)). 

Thus, we end up with the well known Black and Scholes -like formula for the price of a European call option on a zero-coupon bond... 

As noted above, the bond market includes securities, known as zero-coupon bonds, or strips, that do not pay coupons. These are priced by setting C to 0 in the pricing equation. The only cash flow is the maturity payment, resulting in formula (1.18)... 

The relationship between the yield r t, T) of the zero-coupon bond and the short rate r t) can be expressed by equating the right-hand sides of equations (3.16) and (3-3) (the formula for deriving the zero-coupon bond price, repeated here as (3-17)). The result is (3.18). 

Market practitioners armed with a term-structure model next need to determine how this relates to the fluctuation of security prices that are related to interest rates. Most commonly, this means determining how the price T of a zero-coupon bond moves as the short rate r varies over time. The formula used for this determination is known as Itos lemma. It transforms the equation describing the dynamics of the bond price P into the stochastic process (4.5). 

In this model, the formula for deriving the price of a zero-coupon bond is (4.16). 

Thus far our coverage of valuation has been on fixed-rate coupon bonds. In this section we look at how to value credit-risky floaters. We begin our valuation discussion with the simplest possible case—a default risk-free floater with no embedded options. Suppose the floater pays cash flows quarterly and the coupon formula is 3-month LIBOR flat (i.e., the quoted margin is zero). The coupon reset and payment dates are assumed to coincide. Under these idealized circumstances, the floater s price will always equal par on the coupon reset dates. This result holds because the floater s new coupon rate is always reset to reflect the current market rate (e.g., 3-month LIBOR). Accordingly, on each coupon reset date, any change in interest rates (via the reference rate) is also reflected in the size of the floater s coupon payment. 

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