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Chapter 26: Simulating Stock Prices

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Stock prices vs Returns

Not normally distributed
Normally distributed

Stock returns are normal

= mean return
= standard deviation of return
Z = random number, normally distributed
= 0, = 1 Simulated with Excel function Norm.S.Inv(rand())

Norm.S.Inv(Rand()) produces random numbers that are standard normally distributed. For more details, see Chapter 24, page 618.

Checking the simulation

Use Excel’s Frequency to see the distribution

1 + stock returns are lognormal
1 + stock return:

= mean return
= standard deviation of return

Simulate 1 + return: the growth of the stock price

1+return shows familiar “zigzag” noise

Frequency distribution of 1 + return is lognormal

1,000 simulations with Dt = 1 means 1,000 years of simulated stock growth!

Lognormal harder to see when Dt is small

1,000 simulations with Dt = 1/250 means 4 years of simulated stock growth. Much harder to see the difference between normal and lognormal.

Lognormal distribution,
m = 0, s = {1, 0.5, 0.25}
Source: http://en.wikipedia.org/wiki/Log-normal_distribution#mediaviewer/File:Some_log-normal_distributions.svg

When the mean is large relative to the sigma, the lognormal looks like the normal.

Source: http://www.vosesoftware.com/vosesoftware/ModelRiskHelp/index.htm#Distributions/Approximating_one_distribution_with_another/Normal_approximation_to_the_Lognormal_distribution.htm

Source: Wikipedia
Theorem: Sum of normals is normal

Multiplying continuous returns means adding them inside the Exp[ ] function

N different standard normal deviates

1 + growth is additive

The first two moments of the terminal stock price when the return is normally distributed

What does this mean?
The expected stock price increases with time and variance

This is why in risk-neutral pricing, the mean is often written as
The variance of the terminal stock price ST increases with time and variance

Comparing ST to E(ST), 100 simulations

Simulation of terminal stock price

Annual mean, mu11%
Annual sigma15%
Annual variance0.0225<-- =sigma^2 Deltat0.083333<-- =1/12 0.025519543 0.077841895 0.063577959 0.059302039 0.06760982 -0.07614611 0.021912562 -0.082427427 0.075097905 =mu*deltat+sigma*SQRT(deltat)*NORM .S.INV(RAND()) Stock return simulation STOCK RETURNS ARE NORMAL 1,000 simulations Count1000<-- =COUNT(A:A) Count1000<-- =COUNT(A:A) Mean0.0084<-- =AVERAGE(A:A) Mean10.14%<-- =AVERAGE(A:A)/deltat Sigma0.0442<-- =STDEV.S(A:A) Sigma15.31%<-- =G5*SQRT(1/deltat) Variance0.0020<-- =VAR.S(A:A) Variance0.0234<-- =G6/deltat Min-0.1197<-- =MIN(A:A) Max0.1369<-- =MAX(A:A) Return statistics Annualizing return statistics: Multiply by 1/ Annual mean, mu11% Annual sigma15% Annual variance0.0225<-- =sigma^2 Deltat0.0833<-- =1/12 BinsFrequency -0.03631<-- {=FREQUENCY(A:A,F12:F32)} -0.0197215 -0.0164125 -0.0097959 -0.00649105 -0.0031899 0.000132139 0.00344146 0.006749100 0.010057113 0.01336563 0.01667449 Frequency distribution of returns 020406080100120140160-0.0400-0.0300-0.0200-0.01000.00000.01000.02000.03000.0400Frequency distribution of returnsAnnual (mean,sigma) = (11%,15%), deltat = 0.004 Annual mean, mu12% Annual sigma33% Annual variance0.1089<-- =sigma^2 Deltat 1<-- 1 <-- =EXP(mu*deltat+sigma*SQRT(deltat)*NORM.S.INV(RAND())) <-- =EXP(mu*deltat+sigma*SQRT(deltat)*NORM.S.INV(RAND())) <-- =EXP(mu*deltat+sigma*SQRT(deltat)*NORM.S.INV(RAND())) <-- =EXP(mu*deltat+sigma*SQRT(deltat)*NORM.S.INV(RAND())) 1+RETURN = EXP(MU*Deltat+S*SQRT(Deltat)*Z IS LOGNORMAL Deltat =1.00; 1,000 simulations 0.00000.50001.00001.50002.00002.50003.00003.5000137731091451812172532893253613974334695055415776136496857217577938298659019379731+return = exp(mu*deltat+sigma*sqrt(deltat)*Z over 1000.00 yearsAnnual (mean,sigma) = (12%,33%), deltat = 1.000 01020304050607080901000.000.501.001.502.002.503.003.50Frequency distribution of 1+return = exp(mu*deltat+sigma*sqrt(deltat)*ZAnnual (mean,sigma) = (12%,33%), deltat = 1.000 010203040506070800.92000.94000.96000.98001.00001.02001.04001.06001.0800Frequency distribution of 1+return = exp(mu*deltat+sigma*sqrt(deltat)*ZAnnual (mean,sigma) = (12%,33%), deltat = 0.004 T4.00<-- =COUNT(A11:A1010)*deltat Average40.7983 Check40.6044<-- =S0*EXP((mu+sigma^2/2)*G3) Sigma11.7458<-- =STDEV.S(G11:G111) Check12.2502<-- =S0*EXP((mu+sigma^2/2)*G3)*SQRT(G3*(EXP(sigma^2)-1)) Simulation Simulation 26.627540.60438 135.595340.60438 238.203940.60438 336.260940.60438 445.977940.60438 557.248840.60438 640.721640.60438 766.704240.60438 Statistics for 101 simulations 101 simulations of terminal stock 20220exp*2*exp**T*exp12TESSTST /docProps/thumbnail.jpeg 程序代写 CS代考 加微信: cscodehelp QQ: 2235208643 Email: kyit630461@163.com

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