## Simulation

A continuous-time price process can be simulated by taking a series of small time periods and then stepping the process forward period by period There are two natural ways to do this, and they are not exactly equivalent

First, consider the process in standard form deiined by (11.18). We take a basic period length At and set S(/o) = So, a given initial price at t = to. The corresponding simulation equation is

S(tk+]) - S(tk) = pS(tk)At + aS(tk)t(tk)VAt where the €(/*)'s are uncorrected normal random variables of mean 0 and standard deviation 1, This leads to

which is a multiplicative model, but the random coefficient is normal rather than log-normal, so this simulation method does not produce the lognormal price distributions that are characteristic of the underlying Ito process (in either of its forms).

A second approach is to use the log (or multiplicative) form (11 15) In discrete form this is

which is also a multiplicative model, but now the random coefficient is lognormal.

The two methods are different, but it can be shown that their differences tend to cancel in the long run. Hence in practice, either method is about as good as the other.

Example .11,3 (Simulation by two methods) Consider a stock with an initial price of \$10 and having v = 15% and a = 40% We take the basic time interval to be 1 week (At = 1 /52), and we simulate the stock behavior for 1 year. Both methods described in this subsection were applied using the same random e's, which were generated from a normal distribution of mean 0 and standard deviation 1. Table 111 gives the results. The first column shows the random variables dz = ¿J~A~t for that week. The second column lists the corresponding multiplicative factors The value Px is the simulated price using the standard method as represented by (1119) The fourth column shows the appropriate exponential factors for the second method, (11 20) The value P2 is the simulated price using that method. Note that even at the first step the results are not identical. However, overall the results are fairly close

TABLE 11.1

Simulation of Price Dynamics

TABLE 11.1

Simulation of Price Dynamics

 Week dz fi -f crdz Pi v -f- er dz Pi 0 10.0000 100000 1 .06476 00802 10 0802 00648 10.0650 2 - 19945 - 00664 10.0132 -00818 9 9830 3 - 83883 -.04211 9 5916 - 04365 9 5567 4 49609 .0.3194 9 8980 03040 9.8517 5 - 33892 -.01438 9.7557 - 01592 9.6961 6 1 ,39485 08180 10 5536 08026 10.5064 7 .61869 03B74 10,9625 .0.3720 10 9046 8 ,40201 02672 11.2554 02518 11,1827 9 - 71138 - .03503 10.8612 —.03656 10.7812 10 16937 01382 11.0113 01228 10.9144 11 1 19678 07081 11 7910 06927 11 6973 12 - 14408 - 00357 11 7489 - 0051i 11 6377 13 80590 0491.3 12.3261 .04759 12 2049 26 -1 23335 - 06,399 13.1428 - .06553 129157 39 68140 04222 17 6850 04068 17 3668 52 69955 .04.32,3 15.1230 .04169 14.7564

The price process is simulated by two methods Although they differ step by step, the overall results are similar

The price process is simulated by two methods Although they differ step by step, the overall results are similar 