Implementation And Numerical Issues

The two state variables, K and L, in terms of which the model is defined, allow, via Equation (16.1), to write for the rolled-up money-market account, B(t, T),

If the money-market account is chosen as numeraire, then preventing arbitrage is tantamount to stating that there must exist a (risk-neutral) measure Q under which the price at time f, V(f ), of a security of time-T payoff V(T) can be expressed as

The numerical procedure described in the following implicitly constructs a discrete-time counterpart of Equations (16.9) and (16.10), and is based on the same type of averaging/discounting procedure familiar from the BDT model.

The time-0 market price of a consol bond, in conjunction with the value of a suitable proxy for the short rate, uniquely determine the value at time 0 of the two state variables:

For numerical convenience, it is useful, as a preliminary step, to effect a transformation of variables from L, K to y = ln(L), u = In (A"). Figure 16.4 shows the geometry of the (y, u) lattice. From the initial node (root), the following procedure can then be employed: from the (0,0,0) node (where the first two indices denote the y and u variables, and the third the time slice) the (i j',1) nodes can be reached, with i, j = —1, 0, 1. (Notice that, as explained at greater length later y(2,2) u(-2,2)

y(1,1)u(-1,1) y(1,1)

u(0,1) y(1,1)u(1,1)

u(—2,2) y(0,1)

u(-1,1) y(o,o)

u(0,0) y(0,1)

u(1,1) y(0,2)

0 0

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