Week 5 lecture 1

8/7/2019 Week 5 lecture 1

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1

Monte Carlo Simulation



2

Monte Carlo Simulation and Options

When used to value European stock options, Monte

Carlo simulation involves the following steps:

1. Simulate 1 path for the stock price in a risk neutral

world

2. Calculate the payoff from the stock option

3. Repeat steps 1 and 2 many times to get many sample

payoff

4. Calculate mean payoff

5. Discount mean payoff at risk free rate to get an

estimate of the value of the option



3

Sampling Stock Price Movements� In a risk neutral world the process for a stock

price is

� We can simulate a path by choosing timesteps of length (t and using the discrete

version of this

where I is a random sample from J(0,1)

t S t S S (IW(Q!( Ö

d S S d t S dz ! Q W



4

A More Accurate Approach

t t et S t t S

t t t S t t S

d z dt S d

(I(Q

!(

(I(Q!(

Q!

or

is this of version discrete The

Use

2/Ö

2

2

2

)()(

2/Ö)(ln)(ln

2/Öln



5

Sampling from Normal

Distribution� One simple way to obtain a sample

from J(0,1) is to generate 12 random

numbers between 0.0 & 1.0, take thesum, and subtract 6.0

� In Excel =NORMSINV(RAND()) gives

a random sample from J(0,1)� In Matlab: µrandn¶ generates random

samples from J(0,1)



6

Standard Errors

� When N is sufficiently large, the priceestimate has a normal distribution with the

following parameters:

± Mean: the true price of the contract

± Standard deviation: with [ the standard

deviation of the discounted payoffs and N the

number of simulated paths

� is called the ³standard error´ of the

estimated priceN

[

N

[



7

Confidence Intervals

� The standard error of the estimate of the

option price is the standard deviation of

th

e discounted payoffs given by th

esimulation trials divided by the square root

of the number of observations.

� Remember: the estimate for the price is

the sample mean of the sample of generated prices



8

Extension

When a derivative depends on several

underlying variables we can simulate

paths for eac

hof t

hem in a risk-neutralworld to calculate the values for the

derivative



9

To Obtain 2 Correlated Normal

Samples� Obtain independent (uncorrelated) normal

samples x1 and x2

� We get two series I and I2

with correlation

as follows:

2

212

11

1 VV!I!I

xx

x



10

Cholesky Decomposition� Obtain n independent (uncorrelated) normal

samples x1«xn

� We want to obtain n series I and In with

correlation matrix :

� We first calculate the Cholesky decomposition A

of V:

¼¼¼¼

½

»

¬¬¬¬«

!

1

11

1,1

,1

21

112

nnn

nn

n

VV

V

VVV

V

.

1/

/

.

AA

aa

a

a

AT

nnn

!

¼¼¼¼

½

»

¬¬¬¬«

! such that0

00

1

22

11

..

1/

/1/

.



11

Cholesky Decomposition (cont¶d)

� The n series I and In with correlation matrix

are then obtained as:

� Remark: A only exists if V is indeed acorrelation matrix

¹¹¹

º

¸

©©©

ª

¨

!¹¹¹

º

¸

©©©

ª

¨

nn x

x

//

11

I

I



12

Cholesky Decomposition (cont¶d)

Special case

� If V is a 2x2 matrix:

� Then

¼½»¬«!

1121

12

V

VV

¼½»¬« ! 21

01VV

A



13

Application of Monte Carlo Simulation

� Monte Carlo simulation can deal with path dependent options, and optionswith complex payoffs.

� It can easily be used to price optionsdependent on several underlyingstate variables.

BUT

� It cannot easily deal with American-style options.



14

Determining Greek Letters

For (

1.Make a small change to asset price

2.Carry out the simulation again using the samerandom number streams

3.Estimate ( as the change in the option price

divided by the change in the asset price

Proceed in a similar manner for other Greek letters



15

1. Antithetic variable technique

2. Control variate technique

3. Importance sampling4. Stratified sampling

5. Moment matching

Variance Reduction Techniques



16

1. Antithetic variable technique

� Given the random series I1«In used togenerate one discounted payoff f 1, we use the

series - I1«-In to generate a second

discounted payoff f 2.

� The average of the two discounted payoffs isused a single estimate f:

� The number of independent estimates N is

given by the number of

� The standard error is given by

2

21 f f f

!

sf '

N

[



17

� Why does it work ?

� Each time we have drawn a random sample of I¶s that is unusually high, the series ±I is

unusually low, and vice versa.

� So the two series compensate each other.



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2. Control variate technique

� Suppose we want to obtain the price f A of aderivative A.

� Assume that there is another derivative B similar to the A, but of which we have an analytic

expression for the price.

� We generate price estimates f*A and f*B usingthe same I¶s.

� The price estimate f A for A is given by:

where f B is the known true price of B calculatedanalytically.

**

BBAA f f f f !



19

� Why does it work ?

� From the following expression for f A:

� One see that this technique adds the term to the

simulated price f*A for the derivative A.

� is the difference between the simulated price for

derivative B, and its known true price. It picks up, and

corrects for, any overestimation or underestimation.

*

BBf f

*

BB f f

**

BBAAf f f f !



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3. Moment matching

� For each path we store ale the Ii¶s.

� We calculate the mean m, and the standard

deviation W of the sample of Ii¶s.

� We define a new series of Ii¶s as:

� This way, the mean of series of Ii¶s used togenerate the path is exactly zero, and its

standard deviation is exactly one.

W

I I

mi

i

!*



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4. Stratified sampling

�W

e typically can generate a series of Ii¶s asfollows:

� With the ui¶s drawn from a uniform random

distribution on (0,1).� For any sample of M Ii¶s, the u

i¶s will never be

distributed perfectly uniform.

� We can achieve this by generating the Ii¶s as:

)(1

iiu

*!I

¹º

¸©ª

¨ *!

M

i

i

5.01I



2

3

5. Quasi-Random Sequences

� With stratified sampling, we need to determine the

number M of Ii¶s at the start, and stick to it.

� If we set M=100,000, but only use the first 90,000 ui¶s,

we will be missing all the 10% biggest values.

� Quasi-random sequences are series of ui¶s that are

spread uniformly over (0,1), just as with stratified

sampling.

� But extra values for the ui¶ are always set such that they

fill in the gaps left between the previous values.

� Quasi-random sequences are generated using

equations. There aren¶t random at all. They just appear

to be so.



2

4

Example of quasi-random numbers: The Sobel

sequence in two dimensions

Week 5 lecture 1

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