Forward price, calculation

From Figure 7.3, we see fliat to price a very Iraig-dated bond off the yield of the 30-year government bond would lead to errors. The unbiased expectations hypothesis suggests that 100-year bond yields are essentially identical to 30-year yields however, this is in fact incorrect. The theoretical 100-year yield in fact will be approximately 20-25 basis points lower. This reflects the convexity bias in longer dated yields. In our illustration, we used a hypothetical scenario where only three possible interest-rate states were permitted. Dybvig and Marshall showed that in a more realistic environment, with forward rates calculated using a Monte Carlo simulation, similar observations would result. Therefore, the observations have a practical relevance. 

The forward price is calculated only for the purpose of incorporating repo interest it should not he confused with a forward interest rate, which is the interest rate for a term starting in the future and which is calculated from a spot interest rate. Nor should it be taken to he an indication of what the market price of the hond might be at the time of trade termination, the price of which could differ greatly from the sell/buyback forward price. 

Some of the newer models refer to parameters that are difficult to observe or measure direcdy. In practice, this limits their application much as B-S is limited. Usually the problem has to do with calibratii the model properly, which is crucial to implementing it. Galibration entails inputtii actual market data to create the parameters for calculating prices. A model for calculating the prices of options in the U.S. market, for example, would use U.S. dollar money market, futures, and swap rates to build the zero-coupon yield curve. Multifactor models in the mold of Heath-Jarrow-Morton employ the correlation coefficients between forward rates and the term structure to calculate the volatility inputs for their price calculations. 

This is convenient because this means that the price at time t of a zero-coupon bond maturing at T is given by Equation (3.7), and forward rates can be calculated from the current term structure or vice versa. 

Forward rates may be calculated using the discount function or spot interest rates. If spot interest rates are known, then the bond price equation can be set as ... 

An important aspect of trading these bonds is using expectations of future monthly changes in linking indices, provided by economists, to calculate expected forward real yields and expected forward break-even inflation. Making assumptions about future price index levels allows these forward aggregates to be calculated in the same way that forward nominal bond prices and yields are worked out. 

In this example, one counterparty sells 10 million nominal of the UKT 5.75% 2009 at the spot price of 104.60, this being the market price of the bond at the time. The consideration for this trade is the market value of the stock, which is 10,505,560 as before. Repo interest is calculated on this amount at the rate of 5.75% for one week, and from this the termination proceeds are calculated. The termination proceeds are divided by the nominal amount of stock to obtain the forward dirty price of the bond on the termination date. For various reasons, the main one being that settlement systems deal in clean prices, we require the forward clean price, which is obtained by subtracting from the forward dirty price the accrued interest on the bond on the termination date. At the start of the trade the 5.75% 2009 had 29 days accrued interest, therefore on termination this figure will be 29 + 7 or 36 days. 

Let f t,s) be the forward rate at time s > 0 calculated at time t < s. The instantaneous forward rate at time t to borrow at time T can be calculated from the bond prices using... 

The reverse works as well if forward rates are known then bond prices can be calculated via... 

It is obvious that = 0 and therefore the volatility of the forward rate determines the drift as well. In other words all that is needed for the HJM methodology is the volatility of the bond prices. The short rates are easily calculated from the forward rates. Once a model for short rates is determined under the risk-neutral measure Q the bond prices are calculated from... 

To illustrate this, consider the 3-year swap used to demonstrate how to calculate the swap rate. Suppose that one year later, interest rates change as shown in Columns (4) and (6) in Exhibit 19.8. In Colnmn (4) shows the current 3-month EURIBOR. In Column (5) are the EURIBOR futures price for each period. These rates are used to compute the forward rates in Column (6). Note that the interest rates have increased one year later since the rates in Exhibit 19.8 are greater than those in Exhibit 19.3. As in Exhibit 19.3, the current 3-month EURIBOR and the forward rates are used to compute the floating-rate payments. These payments are shown in Column (8) of Exhibit 19.8. 

Calculating a forward rate is equivalent to estimating what interest rate will be applicable for a loan beginning at some point in the future. This process exploits the law of one price, or no arbitrage. Consider a loan that begins at time T and matures at T+ 1. The process of calculating the rate for that loan is similar to the one described above. Start with the simultaneous purchase at time t of one unit of a bond with a term of T+ 1 for price P t, 7+1) and sale of p amount of a bond with a term of T for price P t, T). The net cash position at t must be zero, sop must be... 

The next step is to use this tree to describe the bond s price evolution, ignoring its call feature. The tree is constructed from the final date backwards, using the bond s ex-coupon values. At each node, the ex-coupon bond price is equal to the sum of the expected value plus the coupon six months forward, discounted at the appropriate six-month yield. At year 3, the bond s price at all the nodes is 100.00, its ex-coupon par value. At year 2.5, the bond s price at the highest yield, 7-782 percent, is calculated by using this rate to discount the bond s expected price six months forward. The price in six months in both the up and the down state is 103.00—the ex-coupon value plus the final coupon payment. The bond s price at this node, therefore, is derived using the risk-neutral pricing formula as follows ... 

A more accurate approach m ht be the one used to price interest tate swaps to calculate the present values of future cash flows usit discount tates determined by the markets view on where interest rates will be at those points. These expected rates ate known as forward interest rates. Forward rates, however, are implied, and a YTM derived using them is as speculative as one calculated using the conventional formula. This is because the real market interest rate at any time is invariably different from the one implied earlier in the forward markets. So a YTM calculation made using forward rates would not equal the yield actually realized either. The zero-coupon rate, it will be demonstrated later, is the true interest tate for any term to maturity. Still, despite the limitations imposed by its underlying assumptions, the YTM is the main measure of return used in the markets. 

Thus OAS is a general stochastic model, with discount rates derived from the standard benchmark term structure of interest rates. This is an advantage over more traditional methods in which a single discount rate is used. The calculated spread is a spread over risk-free forward rates, accounting for both interest-rate uncertainty and the price of default risk. As with any methodol-ogy, OAS has both strengths and weaknesses however, it provides more realistic analysis than the traditional yield-to-maturity approach. Hence, it has been widely adopted by investots since its introduction in the late 1980s. 

Using the spot rate structure at Table 12.1, the price of this bond is calculated to be 98.21. This would be the bonds fair value if it were liquid and default free. Assume, however, that the bond is a corporate bond and carries an element of default risk, and is priced at 97.00. What spread over the risk-free price does this indicate We require the spread over the implied forward rate that would result in a discounted price of 97.00. Using iteration, this is found to be 67.6 basis points. The calculation is... 

Futures exchanges use the letters H, M, U, and Z to refer to the contract months March, June, September, and December. The June 2004 contract, for example, is denoted by M4. Forward rates can be calculated for any term, starting on any date. Figure 18.1 shows the futures prices on that day and the interest rate that each price implies. The stub is the interest rate from the current date to the expiry of the first, or front... 

---