Illustration of Second Stage Suppose that the continuously compounded zero rates in the example in Figure 23.8 are as shown in Table 23.1. The value of <^o f 1-0- The value of a 0 is chosen to give the right price for a zero- coupon bond maturing at time St. That is, a 0 is set equal to the initial 5t-period interest rate. Because St = 1 in this example, a 0 = 0.03824. This defines the position of the initial node on the R-tree in Figure 23.9. The next step is to calculate the values of Qi,i, <2i,o> and 2i,-i- There is a probability of 0.1667 that the (1,1) node is reached and the discount rate for the first time step is 3.82%. The value of Q x , is therefore 0.1667e" 00382 = 0.1604. Similarly, Q, „ =0.6417 and Qi,_, =0.1604. Once Q\\, (2i,o> and 2i,-i have been calculated, we are in a position to determine ct\. This is chosen to give the right price for a zero-coupon bond maturing at time 2 St. Because SR — 0.01732 and St — 1, the price of this bond as seen at node B is £-< a i+ 001732 ) Similarly, the price as seen at node C is e~ a> and the price as seen at node D is e -( a i-° mi i 2 \ Th e price as seen at the initial node A is therefore From the initial term structure, this bond price should be e - °04512x2 = 0.9137. Substituting for the 2's in equation (23.28), we obtain This means that the central node at time St in the tree for R corresponds to an interest rate of 5.205% (see Figure 23.9). The next step is to calculate Q 22 , Qi,\ A Qifii Qh-M an d 62,-2- The calculations can be shortened by using previously determined Q values. Consider 22,1 as m example. This is the value of a security that pays off SI if node F is reached and zero otherwise. Node F can be reached only from nodes B and C. The interest rates at these nodes are 6.937% and 5.205%, respectively. The probabilities associated with the B-F and C-F branches are 0.6566 and 0.1667. The value at node B of a security that pays $1 at node F is therefore 0.6566e~ 006937 . The value at node C is O.1667e~ 005205 . The variable Q2X is 0.6566e-° 06937 times the present value of SI received at node B plus 0.1667e~ 005205 times the present value of $1 received at node C, that is, Similarly, <2 22 = 0.0182, 2 2 , 0 = 0.4736, Q 2 ,- { = 0.2033, and Q2_ 2 = 0.0189. The next step in producing the R-tree in Figure 23.9 is to calculate ot 2 . After that, the 2 3 ,/s can then be computed. We can then calculate a 3 , and so on. Once a m has been determined, the Q;,; for i = m + 1 can be calculated using Qm+\j = J2 G».J#(*> 7)exp[-(o! m +kSR)St] k where q(k, j) is the probability of moving from node (m, k) to node (m + 1, /) and the summation is taken over all values of k for which this is nonzero. where n m is the number of nodes on each side of the central node at time m St. The solution to this equation is Formulas for a's and Q 's To express the approach more formally, we suppose that the 6,-,/s have been determined for / < m (m A 0). The next step is to determine ot m so that the tree correctly prices a zero-coupon bond maturing at (m + I) St. The interest rate at node (m, j) is a m + jSR, so that the price of a zero- coupon bond maturing at time (m + \)St is given by P m+ i= £ Q mJ exp[-(c< m +j8R)8t] stock option trading | forex trader | making money |