Interest-Rate Model As explained in the previous chapter, an interest-rate model is a probabilistic description of how interest rates can change over the life of a financial instrument being evaluated. An interest-rate model does this by making an assumption about the relationship between (1) the level of short-term interest rates, and (2) interest-rate volatility. Standard deviation of interest rates is used as the measure of interest-rate volatility The interest-rate models commonly used are arbitrage-free models based on how short-term interest rates can evolve (ie., change) over time. As explained in Chapter 16, models based solely on movements in the short-term interest rate are referred to as one- factor models Interest-Rate Lattice Exhibit 18-7 shows an example of the most basic type of interest-rate lattice or tree, a binomial interes In this model, i is assumed that interest rates can realize one of two possible rates in the next period. In the valuation model we present in this chapter, we will use the binomial model. Valuation models that assume that interest rates can take on three possible rates in the next period are called trinomial models. More complex models exist that assume that more than three possible rates in the next period can be realized. t-rate tree. The corresponding model is referred to as the binomial model 384 Chapter 18 Analysis of Bonds with Embedded Options Exhibit 18-7 3-Year Binomial Interest-Rate Tree 384 Chapter 18Analysis of Bonds with Embedded Options Exhibit 18-7 3-Year Binomial Intorest-Rate Tree IL SHL Todary 1 Year Yeans 3 Years Returning to the binoial interest rate tree in Exhibit 187, each node (bo irl) rep resents a time period that is equal to one year from the node to its left. Each node is labeded with an N, representing node, and a subscript that indicates the path that 1-year forward rates took to get to that node. H represents the higher of the two forward rates and L the lower af the two forward rales from the preceding ycar. For example, node NIcnsthat to get to that node the following path for 1-year rates occurred: The 1-year rate realized is the higher of the two rates in the first year and then the higher of the 1-year rates in the second year. Look first at the point nothing more than the current 1-year rate, or equivalently, the 1-year forward rate, which we denote by e In the model, a one-Eactor interest rate model is assumed. More specifi ally, it is assumed that the 1-year forward rate can evolve over time based on a random process called a lognormal random walk with a certain volatility. denoted N in Exhibit 18-7. This is the root of the tree and is We will use the following notation to describe the tree in the first year. Let -asumed volatility of the 1-year forward rate -the lower 1-year rate one year from now the higher 1-year rale one year from now The relationship between Fu and is as follows: where e is the base of the natural logarithm 2.71828. For example, suppose that is L074% and is 10% per year; then h" 4.074%(MH)-4.976% Note UhatNiis egelvaknt loN in the second year, and hat in the third year, N Isegatvalent to N Nuand that Nequialt to Nus We have simply weacted one label lor a node raher than cutter up the ngure with ucsryzn Chapter 18 Analysis of Bonds with Embedded Options 385 In the second year, there are three possible values for the 1-year rate, which we will denote as folows -ye rate in second year assuming the lower rate in the first year and the lower rate in the second year ye rate in second year assuming the hipher rate in the first year and the higher rate in the second year -1-yar rate in second year assuming the higher rate in the first year and the lower rate in the second year or equivalently the lower rate in the first year and the higher rate in the second year The relationship between r and Sohr example, if rin is 4.53%, then assuming once again that is 10% and and the other two 1-year rates is as follows 4.53%ette-6.757% ral-4.53%etie) .. 5.532% Exhibit 18 7 shows the notation for the binomial interest-rate tree in the third year. We can simplify the notation by letting , be the kower 1-year forward rate t years from now because all the other forward rates t years from now depend on that rate. Exhibit 18-8 shows the interest-rate tree using this simplified notation. Before we go on to show how to use this binomial interest rate tree to value bonds, Its focus on two isues here. First, what does the volatility parameter nthe expresuner represent? Second, how do we find the value of the bond at each node? Exhibit 18-8 3-Year Binomial Interest-Rate Tree with I-Year Forward Rates 1 Year 2 Years 3Years Lowwer 1-Year Forwsrd Rate 386 Chapter 18 Analysis of Bonds with Embedded Options Volatility and the Standard Deviation It can be shown that the standard deviation of the 1-year forward rate is equal torr.The standard deviation is a statistical measure of vokatility. For now, it is important to see that the process that we assumed generates the binomial interest-rate tree (or equiva- lently, the forward rates) implies that volatility is measured relative to the current level of rates. For example, ifo is 10% and the 1-year rate (G) is 4%, the standard deviation of the 1-year forward rate is 496 x10%-0.4% or 40 basis points. However, if the current 1-year rate is 12%, the standard deviation ofthe 1-year forward rate would be 12% x 10% or 120 basis points Determining the Value at a Node The answer to the second question about how we find the value of the bond at a node is as follows. First, calculate the bond's value at the two nodes to the right of the node where we want to obtain the bond's value. For example, in Exhibit 18-8, suppose that we want to determine the bond's value at node Nir The bond's value at node Nand N must be determined. Hold aside for now how we get these two values because, as we will see, the process involves starting from the last year in the tree and working backward to get the final solution we want, so these two values will be known. Effectively what we are saying is that if we are at some node, the value at that node will depend on the future cash flows. In turn, the future cash flows depend on (1) the bond's value one year from now, and (2) the coupon payment one year from now. The latter is known. The former depends on whether the 1-year rate is the higher or lower rate. The bond's value depending on whether the rate is the higher or lower rate is reported at the two nodes to the right of the node that is the focus of our attention. So the cash flow at a node will be either (1) the bond's value if the short rate is the higher rate plus the coupon payment, or (2) the bond's value if the short rate is the lower rate plus the coupon pay- ment For example, suppose that we are interested in the bond's value at NI The cash flow will be either the bond's valuc at Nuai plus the coupon payment or the bond's value at Nn plus the coupon payment. To get the bond's value at a node we follow the fundamental rule for valuation: The bond's value is the present value of the expected cash flows. The appropriate discount rate to use is the 1-year forward rate at the node. Now there are two presen this case: the present value if the 1-year rate is the higher rate and the value if it is the lower rate. Because it is assumed that the probability of both outcomes is equal, an average of the two present values is computed. This is illustrated in Exhibit 18-9 for any node assuming that the 1-year forward rate is r at the node where the valuation is sought and letting t values in the bond's value for the higher I-year rate the bonds value for the lower 1-year rate coupon payment V This can be sem ty notng at-1+2s. Then the sanlard devtation of one period forward rales is