Monte Carlo Simulation - Missouri State University

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Monte Carlo Simulation
Brett Foster
Monte Carlo In A Nutshell
• Using a large number of simulated trials in
order to approximate a solution to a problem
• Generating random numbers
– Computer not required, though extremely helpful
A Brief History
• Earliest well documented use of Monte Carlo:
– Late 18th Century France
• Comte de Buffon performed an experiment involving
the repeated tossing of a needle onto a plane
• Wanted to determine probability of needle intersecting
a string
Modern Monte Carlo Method
• Named after Monte Carlo Casino in 1940’s by
a group of men working on the nuclear bomb
– John von Neumann, Stanislaw Ulam and Nicholas
Metropolis
• Ulam’s Uncle supposedly often frequented
this Casino
Onset of Computers
• Invention and popularization of computer
– Much more practical to implement Monte Carlo
– Popularity has taken off since 1950’s
Random Numbers
• Key aspect of computers: allow us to quickly
generate thousands of “random” numbers.
– Food for thought:
• Computers are ultimately deterministic devices
– How then can the numbers be “random?”
– We will leave this thought for the philosophers and assume
that we do actually have random numbers
A Simple Example
• Using Monte Carlo to approximate pi
– Will give us a better understanding of how to
implement and analyze a Monte Carlo Simulation
• Obviously, there are more efficient ways to figure out
digits of pi
• This example will help build a conceptual
understanding before looking at another example
• Consider a unit circle inscribed within a square
– The ratio of the area of the circle to the area of
the square is π/4
• We will use this key fact in order to estimate π by
randomly selecting points with the square, and
checking whether they are within the circle
• We will generate n points uniformly
distributed within the square
• Let X denote a binomial random variable
which takes the value 1 if a randomly
generated point falls within the circle and 0 if
the point falls outside of the circle
• X is a Bernoulli R.V. with p = π/4
• Notice: E(X) = π/4  E(4X) = π
• For convenience, let Y = 4X. Now, we estimate
π with
The Simulation:
• This simulation can easily be programmed into
a computer:
– I used VBA in Excel to write a short function to run
our simulation
• We input n, number of trials, and function will
repeatedly select points uniformly distributed within
the square and determine how many fall within the
circle
• Function computes sample mean and returns our
estimate for π :
VBA Code:
Function EstimatePi(n As Integer)
c=0
'loop through proceedure n times
For i = 1 To n
'generate point uniformly within square
x = -1 + 2 * Rnd()
y = -1 + 2 * Rnd()
'Check if within circle
If (x ^ 2 + y ^ 2 < 1) Then
c=c+1
End If
Next
'compute our estimate for pi
EstimatePi = 4 * c / n
End Function
How accurate is this method?
• Consider the variance of our estimate:
• Since X is a Bernoulli R.V. with p = π/4,
Var(X) = (π/4)(1- π/4), however we will
assume we don’t know π. Hence, we shall
compute the sample variance of X.
Sample Variance of X
• We can calculate this retrospectively, since X is
a Bernoulli R.V:
– Ex: When n = 1000, we estimate pi to be 3.188.
That is,
– Hence, 797 of our points fell within the circle,
making x = 1 for 797 of our trials, and zero for the
other 203
– Now,
can be easily calculated
Confidence Intervals
• Central limit theorem: tells us that
approximately normal (for large n)
• Then,
is
– the (1 – α) Confidence Interval is given by
where
is the inverse of the standard
normal cumulative distribution function
A note on variance
• The variance of our estimate decreases with n,
since
• As n changes, we have
• For large n, this gets very small
• What are the implications for Monte Carlo?
Results
n
c

100
84
3.36
1,000
797
3.188
10,000
7,797
3.1188
100,000
78,757
3.15028
1,000,000
785,327 3.141308
10,000,000 7,854,333 3.1417332
S2(X)
0.13576
0.16195
0.17179
0.16731
0.16859
0.16853
S2()
0.02172121
0.00259125
0.00027486
0.00002677
0.00000270
0.00000027
S()
0.14738
0.05090
0.01658
0.00517
0.00164
0.00052
95% CI LB
3.07113
3.08823
3.08631
3.14014
3.13809
3.14072
95% CI UB
3.64887
3.28777
3.15129
3.16042
3.14453
3.14275
• For n = 10,000,000 our estimate is accurate to three
decimal places when compared with π
– Not exactly efficient
Hypothetically…
• With this method for estimating π, how many
trials are necessary for us to be at least 95%
sure that our error is less than 1x10-10?
• About 1.036x1021
• Clearly, Monte Carlo is not best method
for figuring out digits of pi
– So what is a practical application???
A Real World Example: Valuing a
Financial Option
• An option is a financial derivative
– Its value is derived from some other object, such
as a stock
• Payoff is dependent upon the value of the stock at the
expiration date of the option
– For a call option, the owner has the right to buy or sell the
underlying asset at a specified price, known as the “Strike
Price,” at the “expiration date” specified in the option contract
Example
• Consider a call option written on Apple stock
with a strike price of $450 and an expiration
date in 6 months.
– In 6 months, if the stock price is above $450 (say
$475 perhaps), the owner will exercise the option
and purchase the stock for $450 (getting a payoff
of $25)
• If the stock price is below $450, the owner will simply
let the option expire, making a payoff of $0
• More succinctly, the payoff at expiration is
given by
Payoff = Max{0, S – 450}
Where S is the stock price and 450 is the strike price
– Note that the payoff is never negative!
• How can we value this sort of derivative?
– Requires some assumptions about the behavior of
stock prices:
• Rate of return on a stock is normally distributed
– Implies that stock prices are Lognormal
» If X is a normal random variable, then exp(X) is a
lognormal random variable
Specifically, we say
• Where we define:
is the current stock price
is the stock price at time t > 0
is the expected rate of return on the stock
is the variance of the annual rate of return on
the stock and its square root is called volatility
is a standard normal random variable
• Using historical data, one can easily obtain
estimates for the expected return and
volatility of a stock
• Under the lognormal stock price model,
predicting a future stock price boils down to
generating a standard normal random variable
The Black-Scholes Equation
• Most options (such as the one which we
examined earlier) can easily be valued using
the Black-Scholes equation
• So why would we want to use Monte-Carlo?
– Not all options can be valued with Black-Scholes
• Why not???
A rough description
• Recall, the payoff function for a call option
discussed previously was given by
Payoff = Max{0, S-450}
If S is lognormal, the payoff of the option has to do
with the probability that S > 450 (and by how
much)
– What if instead of a strike price of 450, we have a
strike price that is also a random variable? What if
that random variable is not lognormal?
• Consider an option which gives the owner the
right (but not the obligation) to purchase a
share of stock A for the average of the prices
of stock B and stock C
• Then, our payoff function is given by
• This is called a “basket option” and no formula
exists to calculate its value
Enter…. Monte Carlo!
• Scenario:
– Suppose that Apple would like to encourage
innovation and dedication by issuing its
employees a stock option which pays off if the
price of Apple stock exceeds the average of
Google and Amazon stock 1 year from now
– Why value this option?
• Accounting purposes: what is this worth?
• Employee interest: what is the value of this option?
• The payoff is that of a basket option:
Payoff = Max{0, Sapple - .5(Sgoogle + SAmazon)}
• How can we value this?
– Monte Carlo Simulation:
• Run many trials where we calculate the average future
value of the option for these trials. Then discount this to
the present value using the risk free rate
Parameters
• In order to run the simulation, we need to
estimate the expected return and annual
volatility for each of the three stocks
– For expected return, we can use the CAPM model
– Volatility is calculated using historical data
– Need starting price (end of day November 4th)
– Need risk free rate of return: LIBOR = 0.95%
–t=1
Stock
Apple
Google
Amazon
Expected Return ()
.087
.081
.074
Estimated Annual Volatility ()
0.2532
0.2917
0.3688
Starting price
395.28
600.95
211.22
Standard Normal
• In our first example, we only needed to
generate uniform random variables on (0,1)
– We now need to generate standard normal
random variables to simulate future stock prices
• Generate uniform random variables on (0,1)
• Treat as a percentile from standard normal distribution,
and take the inverse
– This gives you a random draw from the standard normal
distribution
Another Issue
• Stock prices do not move independently of
one another
– We will need to calculate the correlation
coefficient for the returns on Apple, Google and
Amazon stock. Again, we can use historical data:
Corr(Apple,Google) Corr(Google,Amazon)
,
,
0.6357
0.5258
Corr(Apple,Amazon)
,
0.5193
How do we generate correlated R.V.’s?
• Let Z(ap), Z(g), and Z(az) be the standard normal
random variables used to simuate the future stock
prices of Apple, Google, and Amazon respectively.
Let ‘s be independent standard normal random
variables Now, define:
• One can easily verify that Z(ap), Z(g) and Z(az)
are all standard normal random variables,
with
Corr(Z(ap), Z(g)) =
Corr(Z(g), Z(az)) =
Corr(Z(ap), Z(az)) =
The Simulation
• We now have all the information necessary to run
our simulation. The procedure can roughly be
described as follows:
1. Simulate the future stock price for each of the
companies
2. Compute the option’s payoff for the simulated future
stock prices
3. Repeat steps 1 and 2 until you have reached the
desired number of trials
4. Compute the sample mean of the option payoff
5. Discount at the risk free rate to determine our
estimate for the value of the option
• As before, this can be done using VBA in Excel:
Function ValueBasketCall(s1 As Double, s2 As Double, s3 As Double, C12 As Double, C13 As Double, C23 As Double, v1 As Double,
v2 As Double, v3 As Double, a1 As Double, a2 As Double, a3 As Double, r As Double, t As Double, n As Integer)
p=0
m1 = s1 * Exp((a1 - 0.5 * (v1) ^ 2) * t)
m2 = s2 * Exp((a2 - 0.5 * (v2) ^ 2) * t)
m3 = s3 * Exp((a3 - 0.5 * (v3) ^ 2) * t)
For i = 1 To n
u1 = Rnd()
Z1 = WorksheetFunction.NormSInv(u1)
u2 = Rnd()
X2 = WorksheetFunction.NormSInv(u2)
Z2 = C12 * Z1 + X2 * (1 - C12 ^ 2) ^ 0.5
u3 = Rnd()
X3 = WorksheetFunction.NormSInv(u3)
Z3 = C13 * Z1 + X1 * (C23 - C13 * C12) / ((1 - (C12) ^ 2) ^ 0.5) + X3 * ((1 - (C13) ^ 2 - ((C23 - C13 * C12) ^ 2 / (1 - C12 ^ 2))) ^ 0.5)
q1 = m1 * Exp(Z1 * v1 * (t) ^ 0.5)
q2 = m2 * Exp(Z2 * v2 * (t) ^ 0.5)
q3 = m3 * Exp(Z3 * v3 * (t) ^ 0.5)
payoff = WorksheetFunction.Max(0, q1 - 0.5 * (q2 + q3))
p = p + payoff
Next
ValueBasketCall = p * Exp(-r * t) / n
End Function
Analyzing the results:
• As in our pi example, the results are much
more meaningful if we have some idea of
their accuracy
• Define V as the value of the option for a single
trial. Then, our estimate is given by
• CLT 
is approximately normal
• Also,
• How do we calculate Var(V)?
• Recall how we previously retrospectively
determined the sample variance
– Since X was Bernoulli, we could recover the values
of X and estimate S2(X)
– That approach does not work here, since V is
certainly not Bernoulli. Need to store each V as
we go, and calculate S2(V) last
• My solution: Estimate S2(V) using 1,000,000
trials with n = 1, and use to compute sample
variance for simulations with n>1
• Estimated S2(V) = 2293.63
Results
 

n
 
95 % CI LB
95% CI UB
100
24.20
22.9363
4.789
14.82
33.59
1000
32.38
2.2936
1.514
29.41
35.35
10000
29.72
0.2294
0.479
28.78
30.66
100000
29.81
0.0229
0.151
29.52
30.11
1000000
29.99
0.0023
0.048
29.90
30.09
• With n = 1,000,000 we are about 95% sure
that the actual value is within $0.10 of $29.99
– Probably accurate enough for most practical
purposes
Conclusion
• Monte Carlo provides a numerical method to
obtain an estimate for a solution that needs to
be close, but not necessarily exact
• We showed two specific examples, but the
applications of Monte Carlo are very diverse
• Questions???
Sources:
•
•
•
•
Bodie, Zvi, Alex Kane, and Alan Marcus. Essentials of Investments. Eighth Edition. New York: Mcgraw-Hill,
2010.
Kalos, Malvin H., and Paula A. Whitlock. Monte Carlo Methods Volume 1: Basics. New York: John Wiley &
Sons, 1986.
McDonald, Robert obert Lynch. Derivatives Markets. 2nd. Boston: Pearson Education Inc., 2006. Print.
Sobol, IIya. A Primer for the Monte Carlo Method. Boca Raton: CRC Press, 1994.

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