Bonds arbitrage-free value

Given the spot rates for each maturity, the arbitrage-free value of a government bond for which the spot rates apply is found as follows ... 

EXHIBIT 3.4 Determination of the Arbitrage-Free Value of an 8%, 10-Year Government Bond... 

For example, consider a 10-year government bond denominated in euros with an 8% coupon rate. Suppose that coupon payments are delivered semiannually and the annual spot rates are shown in the fourth column of Exhibit 3.4. The third column of the exhibit shows the cash flow every six months. The last column shows the present value of each cash flow discounted at the corresponding spot rate. The total in the last column is the arbitrage-free value of the bond, 115.2619. 

To value a nongovernment bond, the arbitrage-free value is found by adding a suitable spread to the government spot rates. A spot rate curve can be created using any benchmark such as LIBOR. 

The bond and the money market are both risk-free and have identical payouts at time T, and neither will generate any cash flow between now and time T. Since the interest rates involved are constant, the bond must have a value equal to the initial investment in the money market account g -r(r q other words, equation (3.13) must hold. This is a restriction placed on the zero-coupon bond price by the requirement for markets to be arbitrage-free. 

A short-rate model can be used to derive a complete term structure. We can illustrate this by showing how the model can be used to price discount bonds of any maturity. The derivation is not shown here. Let P t, T) be the price of a risk-free zero-coupon bond at time t maturing at time T that has a maturity value of 1. This price is a random process, although we know that the price at time T will be 1. Assume that an investor holds this bond, which has been financed by borrowing funds of value C,. Therefore, at any time t the value of the short cash position must be C,= —P(t, T) otherwise, there would be an arbitrage position. The value of the short cash position is growing at a rate dictated by the short-term risk-free rate r, and this rate is given by... 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

The fair price of a convertible bond is the one that provides no opportunity for arbitrage profit that is, it precludes a trading strategy of running simultaneous but opposite positions in the convertible and the underlying equity in order to realize a profit. Under this approach we consider now an application of the binomial model to value a convertible security. Following the usual conditions of an option pricing model such as Black-Scholes (1973) or Cox-Ross-Rubinstein (1979), we assume no dividend payments, no transaction costs, a risk-free interest rate, and no bid-offer spreads. 

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