## 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 defined 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

Sfo+i) - S(tk) = pS{tk)At + aS(tk)€{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

This leads to

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 = I /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 11.1 gives the results. The first column shows the random variables dz = ¿J~A~t for that week. I he 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 IT 1

Simulation of Price Dynamics

TABLE IT 1

Simulation of Price Dynamics

Week |
dz |
p -f erdz |
Pi |
v + 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 |
08380 |
10 5536 |
08026 |
10.5064 |

7 |
.61869 |
03B74 |
10,9625 |
.0.3720 |
10 9046 |

8 |
.40201 |
02672 |
11.2554 |
02518 |
11,1827 |

9 |
- 73138 |
- .0350.3 |
10.8612 |
—.03656 |
10.7812 |

10 |
16937 |
01382 |
11.0113 |
01228 |
10.9144 |

11 |
1 19678 |
07081 |
117910 |
06927 |
11 6973 |

12 |
- 14408 |
- 00357 |
11 7489 |
- 00511 |
11 6377 |

1.3 |
80590 |
0491.3 |
12.3261 |
.04759 |
12 2049 |

26 |
-I 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

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