lecture 10 Monte carlo,ppt.ppt

7/27/2019 lecture 10 Monte carlo,ppt.ppt

http://slidepdf.com/reader/full/lecture-10-monte-carlopptppt 1/28

Monte Carlo



Monte Carlo studies are one of the earliest computer techniques,

dating back to the 1940’s, they allow us to analyse a range of

questions which are not directly tractable to an analytical solution.

Sometimes these problems may be simply so complex that we

can not work out the large sample (asymptotic) answer. In other

cases we may be interested in the answer for a small sample sizeor a limited number of states which are not easily calculated.

This technique has an enormous range of applications, pure

statistics, project appraisal, macroeconomic models, derivative

pricing – almost anywhere where there are either stochastic

processes or intractable mathematical formulas.



The essence of the technique is to assume a known world

(DGP),

to assume something about the statistical distribution of

certain events.

Then to numerically draw many realisations from this

distribution.

Calculate the outcome and analyse the resulting distribution

of solutions.



A simple example from Metropolis and Ulam

Suppose we wish to evaluate the relative volume of a region in 20

dimensional space where the region is defined by 20 non linearinequalities as follows,

0),...,,(

0),...,,(

0),...,,(

202120

20212

20211

x x x f

x x x f

x x x f

Where we define the xs to lie in a unit cube.



In effect we are asking what is the proportion of points in this

space which lie inside these inequalities.

For some simple functions it may be possible using

integration to solve this analytically but generally this will not

be possible.

We could think of dividing up the unit interval into say 10sections for each x and then checking each point this would

lead to checking 100,000,000,000,000,000,000 different points.



The monte carlo approach would take the above set of

equations as the DGP,

assume the xs are generated by an independent uniform

distribution over the unit interval.

It would then generate say 5000 random draws from this

distribution.

It would then calculate what proportion of these draws are in

the area defined above.

This would then be an approximation to the size of the area.



A Statistical Example

If we have a symmetric distribution such as the normal or `t’,

we know that the mean and the median should be identical in

terms of their asymptotic properties.

But what about in a small sample?

Intuitively if the sample is very small we might expect themean to do badly because any outlier will be given too much

weight.

In a bigger sample the mean may be a better.

But where is the divide, and is this true.



A Monte Carlo

Draw a set of n observations from a `t’ distribution with ddegrees of freedom.

Calculate the mean and the median.

Repeat this R times.

Tabulate the relative error ratio of the median to the mean



d=3 d=6 d=10

n=10 0.29 0.91 1.03

n=25 0.89 0.93 1.08

n=100 1.00 1.19 1.34

So the median does best for n<=25 and d<=6



Project Appraisal

We can evaluate the likely profitability of a project using

these techniques in the light of many uncertainties using thistechnique.

Build a model of the cash flow of the project which calculates

the net present value as an output.



Draw random realisations of the uncertain factors which

might affect the project.

Repeat many simulations.

Calculate the distribution of the net present value.

This gives the probability of making a profit and a standard

error around this. So we can asses how uncertain we areabout this mean forecast. We can also asses the probability

of making a loss of a particular size.



Generating random numbers

In fact all computer generated random numbers come from a

fixed deterministic sequence which is highly non-linear sothat the numbers appear random. They are therefore called

pseudo-random numbers.

Most computers start with a uniform random number over the

interval 0-1, and then transform this to derive particular

distributions.

Because the numbers are generated from a formulae the set

of numbers depends on an initial value which starts of thewhole sequence. This is called the seed.



GAUSS will normally take the system time as the seed value,

this is pretty much a random event.

Sometimes it is useful to produce identical sequences of random numbers then we can set the seed to an actual value.

This is done with this command

rndseed= 1000; or any other fixed number

Generating Random Numbers in GAUSS



Having either defined a seed or used the default of the

system clock to seed the process we can then generate a

series of uniform random numbers by,

y=rndu(r,c); where r is the number of rows

And c the number of columns

So

y=rndu(5,1); would create a column of 5 uniformrandom numbers



similarly normal (0,1) random numbers are generated by,

y=rndn(r,c);

we can then transform these as we want, so if we want a set

of 50 normal numbers with mean 10 and variance 25

(standard error 5),

x=10+y*5;

y=rndn(50,1);

Other distributions can be created by transforming these

random numbers (see Hendry article).



Increasing efficiency

There are a range of tricks which allow us to increase the

efficiency of a monte carlo.

Antithetic Variates

If we are interested in estimating the mean of a process then

this trick can increase the efficiency of the monte carlo

enormously.

The idea is that we perform the simulations in pairs, the

second one of the pair uses the same absolute values for the

random shocks but with the opposite sign. This guarantees

that we are always using an exactly symmetric distribution.



Control Variates

This is a technique which is problem specific but it really

amounts to using our knowledge of the structure of the

problem to increase the efficiency of the replications.

This will generally involve adding restrictions on the way the

model works to ensure that it exactly follows our prior beliefsin some way.

See the Boyle et al paper.

S t t tith ti l f



So to generate antithetic values for z.

y=rndn(50,1);

x=10+y*5;

i=1;do while i<=2;

if i=1;

z=x;

elseif i>1;z=-x;

endif;

i=i+1;

endo;

Note this does not help if we are interested in assessing the

variance of an outcome.



Moment Matching

This is a generalisation of antithetics. The idea is that we

transform the random numbers in each replication to exactlymatch some of the moments of the underlying distribution.

So to exactly match the zero mean assumption.

y=rndn(50,1);

a=meanc(y);

b=stdc(y);

z=(y-a)*(1/b);

x=10+z*5;

There are also a range of more specialised techniques

surveyed in Boyle.

O ti P i i th h M t C l



Option Pricing through Monte Carlo.

In the case of a standard European call option we can use

Black-Scholes, but we can illustrate the monte carlo

approach to get the same answer.

If S0 is the current stock price, r is the riskless rate of return,

I is the conditional variance of the stock, T is the maturity

date and z are independent standard normal random

numbers. Then for a particular realisation of z the price at T

is,

2

Tz T r

T eS S

))2/1((

0

2



So the price of the option with strike K will be given by

)),0(max( K S E eC T

rT

This can be approximated in a monte carlo by

n

i

n

i T

irT

i K S enC nC 1 1 ),0max(

11



T

S0

mean

Strike price

profit



and if we assumed some other process to generate ST

, or a

variable strike price or a different distribution of shocks etc.

we can still perform the simulation.

** MONTE3 JOB



MONTE3.JOB** A MONTE CARLO SIMULATION

** for derivative pricing with antithetics

period =50;reps=500;

tau=5/12;

reps2=reps*2;

i=1;

m=0;

cls;

print "replications = " reps;

print "Number of periods per year ahead = " period;

print "proportion of a year for duration (tau)= " tau;nobs=trunc(period*tau);



@calcualte period standard error to compund up to 0.15@

c = { 0 };

c[1]=( (0.15^2)/period)^0.5;

print "Standard error of returns";

print c[1];

x=zeros(1,reps2);

y=zeros(1,reps2);

z=zeros(1,reps2);w=zeros(1,reps2);

p=zeros(nobs,1);

rndseed 23456567;

p[1,1]=62;strike=60;

@calculates period rate from an anual rate of 10% @

r=(1+0.10)^(1/period)-1;



do while i<=reps;

e = rndn(nobs,1); @ GENERATE NORMAL ERRORS @

j=1;

do while j<=2;if j <= 1;

k=1;

do while k<nobs;

k=k+1;

p[k,1]=exp( ln(p[k-1,1]) + r +e[k]*c[1] );

endo;

else;

k=1;

do while k<nobs;k=k+1;

p[k,1]=exp(ln(p[k-1,1]) + r - e[k]*c[1] );

endo;

endif ;

This simulates the path of

the asset price



if p[nobs,1]>strike;

callprof=p[nobs,1]-strike;

putprof=0;else;

callprof=0;

putprof=strike-p[nobs,1];

endif;

m=m+1;

x[1,m]=callprof;

y[1,m]=putprof;

z[1,m]=p[nobs,1];

j=j+1;endo;

i=i+1;

endo;

This decides if the

put or call is in profit

and collects up the

profit values



m=meanc(z');

se=stdc(z');

m=meanc(x');

se=stdc(x');

cp=m[1,1]*(1/(1+r))^nobs;

print "call price = " cp;

n=meanc(y');

se=stdc(y');

pp=n[1,1]*(1/(1+r))^nobs;

print "put price = " pp;end;

Takes the mean of allthe call profits,

discounts it back to

today, thats the call

price

Takes the mean of all the put profits,

discounts it back

to today and that’s

the put price

lecture 10 Monte carlo,ppt.ppt

Documents

Transcript of lecture 10 Monte carlo,ppt.ppt