A popular model for stock prices is the following If p0 is t

A popular model for stock prices is the following. If p0 is the current stock price, then the price k periods from now, pk, (where the period could be a day, week, or any convenient unit of time, and k is any positive integer) is given by

Pk = p0. e^{( – 1/2 2).k+ . Z.Ök}

Where,

- e = the exponential function.

- = the mean percentage growth rate of per period.

- = the standard deviation of the growth rate per period.

- Z = is a normally distributed random variable with mean 0 and standard deviation 1.

Both and are typically estimated from actual stock price data, and they are typically expressed in decimal form, such as = 0.01 for a 1% mean growth rate. a. Suppose a period is defined as a month, the current price of the stock (as of the end of December 2010) is $75, = 0.006 and = 0.028 – Use simulation to obtain 500 possible stock price changes from the end 2010 to the end of December 2013. (Note: You can simulate a given change in one line and then copy it down) – Create a histogram of these changes to see whether the stock price change is at least approximately distributed. Also use the simulated data to estimate the mean price change and the standard deviation of the change. (5 marks)

b. Use simulation to generate the ending stock prices for each month in 2011. (Note: Use k = 1 again to get February’s price from January’s, and so on) – Then, use a data table to replicate the ending December 2011 stock price 500 times. Create a histogram of these 500 values. Analyse if these values appear to resemble a normal distribution.

(5 marks)

Solution

(a)

z=rnorm(500,0,1)
> mu=0.006
> s=0.028
> p0=75
> k=2.19
> pk=c()
> for(i in 1:500){
+ pk[i]=p0*exp(mu-0.5*(s^2))*k +s*z[i]*k
+ }
> pk
[1] 165.2057 165.1449 165.0822 165.2435 165.1780 165.1718 165.1946 165.2040
[9] 165.1988 165.2346 165.1639 165.1900 165.1292 165.0920 165.1454 165.2623
[17] 165.0291 165.2402 165.2515 165.1954 165.1600 165.0947 165.2279 165.2377
[25] 165.1542 165.2245 165.0437 165.1821 165.1493 165.1384 165.1898 165.2116
[33] 165.2826 165.1463 165.1421 165.0368 165.1419 165.2513 165.1535 165.0740
[41] 165.1421 165.1310 165.3182 165.0816 165.1701 165.1369 165.1831 165.1360
[49] 165.1011 165.1895 165.1033 165.1943 165.2009 165.3071 165.0960 165.1733
[57] 165.1945 165.2633 165.1754 165.1665 165.1685 165.1810 165.2076 165.1706
[65] 165.1605 165.1448 165.2989 165.2446 165.1836 165.2343 165.1015 165.1086
[73] 165.1203 165.1046 165.2279 165.1224 165.1458 165.2586 165.2274 165.1222
[81] 165.1535 165.1980 165.2010 165.2518 165.1766 165.1298 165.1485 165.1676
[89] 165.2135 165.0718 165.1491 165.2760 165.0768 165.1831 165.0985 165.1925
[97] 165.2574 165.0991 165.1087 165.2691 165.1548 165.0978 165.1800 165.2110
[105] 165.1601 165.1282 165.3058 165.1770 165.1873 165.2072 165.3228 165.1232
[113] 165.1617 165.1208 165.1467 165.2096 165.2166 165.1238 165.3875 165.2163
[121] 165.1980 165.2051 165.2569 165.2305 165.1846 165.0902 165.1419 165.1530
[129] 165.1579 165.0914 165.1491 165.1561 165.1001 165.1427 165.1871 165.2347
[137] 165.1922 165.1018 165.0111 165.2347 165.1172 165.1758 165.1857 165.0911
[145] 165.1182 165.1056 165.1182 165.1919 165.3362 165.1315 165.1915 165.2231
[153] 165.1715 165.1876 165.1731 165.1912 165.1914 165.2596 165.2073 165.1401
[161] 165.1328 165.1132 165.1943 165.1533 165.1693 165.1099 165.0605 165.1846
[169] 165.1964 165.1873 165.1609 165.1720 165.2550 164.9997 165.2069 165.1789
[177] 165.1433 165.0643 165.0975 165.1619 165.1628 165.0787 165.2363 165.1239
[185] 165.1150 165.3058 165.2390 165.0503 165.1855 165.1784 165.2462 165.1241
[193] 165.1508 165.1952 165.2257 165.1603 165.1630 165.1106 165.3259 165.1828
[201] 165.2432 165.3032 165.2151 165.2244 165.2415 165.1871 165.2700 165.1751
[209] 165.2435 165.0748 165.1573 165.1471 165.2211 165.0944 165.1431 165.2681
[217] 165.1830 165.1172 165.1408 165.2213 165.0872 165.2213 165.1546 165.0639
[225] 165.1817 165.1550 165.2200 165.3009 165.1159 165.1366 165.1204 165.2362
[233] 165.2859 165.1343 165.2028 165.2315 165.1618 165.2177 165.2018 165.2673
[241] 165.1843 165.1869 165.2421 165.2068 165.1692 165.1449 165.2223 165.0962
[249] 165.2356 165.2640 165.2485 165.2175 165.2169 165.1925 165.2507 165.2139
[257] 165.2940 165.1053 165.3150 165.2765 165.1669 165.1189 165.1540 165.2355
[265] 165.1812 165.2280 165.2485 165.2445 165.1591 165.1278 165.1278 165.1732
[273] 165.2153 165.1960 165.2275 165.1156 165.0904 165.1713 165.0502 165.2312
[281] 165.1845 165.2244 165.1385 165.2478 165.1830 165.1295 165.2020 165.1665
[289] 165.2417 165.1604 165.0898 165.1785 165.1321 165.0550 165.3503 165.1550
[297] 165.1344 165.2138 165.1269 165.2506 165.1140 165.1731 165.2238 165.0907
[305] 165.1886 165.1543 165.3017 165.1873 165.2055 165.1369 165.1800 165.2286
[313] 165.1689 165.1430 165.1349 165.1445 165.1281 165.2180 165.2456 165.1438
[321] 165.1853 165.1722 165.0532 165.1579 165.3220 165.2954 165.1941 165.2494
[329] 165.1471 165.1649 165.2489 165.1591 165.1127 165.1469 165.1471 165.1441
[337] 165.1820 165.0599 165.0901 165.2912 165.1674 165.1597 165.0993 165.1693
[345] 165.1375 165.1833 165.1637 165.1304 165.1300 165.2175 165.2468 165.2017
[353] 165.1419 165.0913 165.2205 165.1313 165.1245 165.1720 165.2319 165.1820
[361] 165.2918 165.2158 165.0801 165.2149 165.2478 165.1480 165.1068 165.1472
[369] 165.1426 165.1972 165.2287 165.0841 165.1841 165.1998 165.1938 165.1313
[377] 165.1883 165.1743 165.2353 165.0869 165.1219 165.1648 165.1025 165.2097
[385] 165.1679 165.3163 165.2009 165.1587 165.2414 165.1579 165.1322 165.1664
[393] 165.1401 165.2606 165.1979 165.2406 165.2336 165.1726 165.0771 165.1928
[401] 165.3004 165.1273 165.1885 165.1881 165.1250 165.2362 165.2021 165.1362
[409] 165.1983 165.1520 165.1875 165.3143 165.2013 165.2289 165.1846 165.1995
[417] 165.3260 165.1590 165.1414 165.2213 165.1504 165.1594 165.2333 165.1805
[425] 165.1606 165.2612 165.2120 165.2615 165.0235 165.2047 165.2054 165.0851
[433] 165.0485 165.1627 165.2109 165.2361 165.1577 165.1350 164.9430 165.2336
[441] 165.2235 165.2479 165.2383 165.2048 165.1190 165.2853 165.1466 165.1547
[449] 165.1056 165.1944 165.1018 165.1512 165.1216 165.1089 165.0671 165.1820
[457] 165.0466 165.1583 165.1192 165.2354 165.2184 165.1454 165.1630 165.2414
[465] 165.1214 165.2277 165.2889 165.1511 165.1778 165.1118 165.1681 165.1806
[473] 165.1722 165.0689 165.1030 165.2053 165.2401 165.2040 165.1235 165.1251
[481] 165.2507 165.2930 165.1410 165.1999 165.2000 165.2809 165.1264 165.1495
[489] 165.1957 165.1842 165.0698 165.1742 165.0637 165.1359 165.1809 165.2026
[497] 165.1772 165.2230 165.1097 165.2129
> hist(pk)
> mean(pk)
[1] 165.1763
> sd(pk)
[1] 0.06226998

(b) z=rnorm(500,0,1)
mu=0.006
s=0.028
p0=75
k=1
ck=c()
pk=c()
for(i in 1:500){
ck[i]=p0*exp(mu-0.5*(s^2))*k +s*z[i]*k
pk[i]=ck[i]*exp(mu-0.5*(s^2))*k +s*z[i]*k
}
hist(pk)