Matematik simulation and monte carlo with applications in finance and mcmc phần 8 pot

> GeometricBrownian:=procT,n,seed,mu,sigma,x0 local h,sh,X,P,j,z,mh; # # Procedure generates a list [{[j*h,Xj*h],j=0..n}] where Xj*h is the position at time jh of the geometric Brownian

xð1Þe xþ1=xð Þ=2 on supportð0; 1Þ using a proposal density proportional

to xð1Þex=2 for selected values of  and  The envelope has beenoptimized by setting

¼22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

arbitrary ratio of uniforms generators

The procedure ‘ratioaccep’ is used to compute the acceptance probability for aratio of uniforms generator for a p.d.f proportional to h(x) The parametersare:

h¼ a procedure supplied by the user,

½x1; x2 ¼ the connected support of h,

xinit¼ a suggested start point for the numerical maximization of h(x) over

xinit1¼ a suggested start point for the numerical maximization of x2hðxÞ

Use ‘ratioaccep’ to compute the acceptance probabilities and uþ; þ;  for

a density proportional to 1= 1ð þ x2=3Þ2on supportð1; 1Þ

3p

The warning arises because the initial point chosen for the maximization of hover the entire support happens to be the maximizing point.

This is a polar implementation of the Box–Mu¨ller generator, as described inSection 4.1 Note the warning to set i and X2to global variables and to set i to

‘false’ on the first call

> STDNORM:=proc( ) local ij,U1,U2,S,B,X1;global i,X2;

#

# PROCEDURE GENERATES A STANDARD RANDOM NORMAL

# DEVIATE, USING THE POLAR BOX MULLER METHOD

# SET i to ’false’ ON FIRST CALL Note that i and

# X2 are global variables and have to be declared as

# such in any procedure that calls STDNORM

if type(i,boolean)then

if (i) theni:=not(i);X2;

elsefor ij from 1 to infinity doU1:=evalf(2.0*rand()/10^12)-1.0;

> beta:=proc(a,b) local z,r1,r2,w,rho,m,la;

if a>0 and b>0 thenrho:=a/b;m:=min(a,b);if m<=1 then la:=m elsela:=sqrt((2*a*b-a-b)/(a+b-2)) end if;

(a+b)))<0 then break end if;

4.3 Student’s t distribution

This procedure generates a Student’s t variate with n degrees of freedom using

the method described in Section 4.7 If n=1 then a Cauchy variate is delivered

This procedure generates a variate from a p.d.f that is proportional to

x1eðxþ1=xÞ=2 on support ð0; 1Þ where 0 <  and 0 <  The method is

else ERROR("beta must be positive") end if;

else ERROR("lambda must be positive") end if;

if p>0.5 then n-x else x end if;

else ERROR("p should belong to[0,1]") end if;

else ERROR("n should be a positive integer") end if;

kis a positive real It uses unstored inversion as mentioned in Section 4.11 It

is fast providing that kp=q is not too large In that case the Poisson gamma

method described in Section 4.11 is better

> negbinom:=proc(k,p) local q,r,y,x;

else ERROR("p should belong to[0,1]") end if;

else ERROR("k should be positive") end if;

Appendix 5: Variance reduction

The procedure ‘theta1_2’ samples m unit negative exponential variates, andraises each to the power of 0.9 The sample mean of the latter is an estimate

of ð1:9Þ

> restart;with(stats):

h

> theta1_2:=proc(m) local j,r,x,y,u;

for j from 1 to m do;

print("standard

error"=evalf(describe[standarddeviation[1]](u)/sqrt(m)));u;

Now replace r by 1 r and theta1_hat by theta2_hat in the print statement

of procedure ‘theta1_2’ and run the simulation again with the same seed

The procedure ‘theta_combined’ does the two jobs above in one run, giving

an estimate and estimated standard error using primary and antitheticvariates

2

6

> theta_combined:=proc(m) local j,r,x,y,u,r1,z;

for j from 1 to m do;

The procedure ‘impbeta’ estimates Pfa <Pn

i¼1Xig where fXig are i.i.d betadistributed with shape parameters  and , both greater than one The

importance sampling density is gðxÞ ¼Qn

i¼1xð  1Þi on supportð0; 1Þnwhere

 <  A good choice is ¼ 1= logðn=aÞ and a should satisfy n e1=  a

std_dev:=sqrt(n*alpha*beta/(alpha+beta+1)/(alpha+beta)^2);print("alpha"=alpha,"beta"=beta,"n"=n,"mean "=mean,"stddev"=std_dev);

if a<evalf(n*exp(-1/alpha)) then print("STOP, use naive montecarlo" )

return end if;

gam:=evalf(1/ln(n/a));

gam1:=1/gam;

b:=(GAMMA(alpha+beta)/GAMMA(alpha)/GAMMA(beta)/gam)^n;s1:=0;s2:=0;

for j from 1 to m do;

‘variance reduction ratio’¼ 2.010769567  107

‘central limit approx’¼ 9.865876451  1010

‘variance reduction ratio’¼ 11.53805257

‘central limit approx’¼ 0.02275013195

‘alpha’¼ 2.5, ‘beta’ ¼ 1.5, ‘n’ ¼ 12, ‘mean’ ¼ 7.500000000, ‘std dev’ ¼ 0.7500000000

‘estimate of probability that sum of’, 12, ‘variates exceeds’, 9, ‘is’, 0.01978833352

‘standard error’¼ 0.0005813529138

‘variance reduction ratio’¼ 11.47834773

‘central limit approx’¼ 0.02275013195

‘seed’¼ 6811357

‘alpha’¼ 2.5, ‘beta’ ¼ 1.5, ‘n’ ¼ 24, ‘mean’ ¼ 15.00000000, ‘std dev’ ¼ 1.060660172

‘estimate of probability that sum of’, 24, ‘variates exceeds’, 18, ‘is’, 0.001761757803

‘standard error’¼ 0.00008414497712

‘variance reduction ratio’¼ 49.67684543

‘central limit approx’¼ 0.002338867496

Z 1 0

> weibullnostrat:=proc(k) local j,r1,r2,w,v,y,sum1,sum2,m;sum1:=0;sum2:=0;

for m from 1 to k do;

end proc:

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5.3.2 Stratified version using a single stratification

variable

First plot

Y ¼ ½ ln ðr 1Þ2=3þ ½ lnðr2Þ2=35=4

against stratified variable X¼ r1r2 This confirms that much of the

variation in Y is accounted for by variation in the conditional expectation

of Y given X Therefore, stratification on X will be effective

> with(stats);

Warning, these names have been redefined: anova, describe,

fit, importdata, random, statevalf, statplots, transform

2

6

> strat:=proc(n) local j,r1,r2,x,a1,a2,y,z;

for j from 1 to n do;

0.80.4

0.2

6543Y

X

correlation :=– 0.8369837074

CV_VRF :=0.29945827350

The following procedure, ‘weibullstrat’, performs k independent tions Each realization comprises n observations over n equiprobablestrata, with exactly one observation per stratum.

realiza-24

> weibullstrat:=proc(n,k) localj,u1,t,x,u2,r2,r1,y,w,v,tprev,sum1,sum2,mean,s1,m;sum1:=0;sum2:=0;

for m from 1 to k do;

end proc:

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> t1:=time();seed:=randomize(639156);weibullstrat(100,200);t2:=time()-t1;

t1:¼ 109.077seed:¼ 639156

‘mean’¼ 2.166441095, ‘std error’ ¼ 0.001321419976

t2:¼ 110.250

26666

estimated_vrr:=(.1321419976e-2/.9127721506e-2)^(-2);estimated_efficiency:=t3*estimated_vrr/t2;

estimated_vrr:=47.71369240estimated_efficiency:=8.871933734

266

The procedure ‘grid’ estimates the same integral using k replicationswhere there are now two stratification variables, r1 and r2 Each replica-tion comprises n2 equiprobable strata on [0, 1]2

> grid:=proc(n,k) local j,r1,r2,w,v,y,m,sum1,sum2,k1,s1,

mean;

sum1:=0;sum2:=0;

for k1 from 1 to k do;

s1:=0;

for j from 1 to n do;

for m from 1 to n do;

Appendix 6: Simulation and

# PROCEDURE GENERATES A STANDARD RANDOM NORMAL

# DEVIATE, USING THE POLAR BOX MULLER METHOD

# SET i to 'false' ON FIRST CALL Note that i and

# X2 are global variables and have to be declared as

# such in any procedure that calls this STDNORM

if type(i,boolean)then

if (i) theni:=not(i);X2;

elsefor ij from 1 to infinity doU1:=evalf(2.0*rand()/10^12)-1.0;

6.2 Geometric Brownian motion

Let fBðtÞg denote a standard Brownian motion The procedure

‘Geometric-Brownian’ generates the position, XðtÞ, of a Brownian motion

dX

X ¼  dt þ  d

at times 0, h, 2h, .,nh where nh¼ T The solution to the stochastic

differ-ential equation is XðtÞ ¼ XðsÞ eð 2 =2ÞðtsÞþBðtsÞ Suppose XðsÞ ¼ xðsÞ

Since EðXðtÞÞ ¼ xðsÞeðtsÞ; is interpreted as the expected growth rate

> GeometricBrownian:=proc(T,n,seed,mu,sigma,x0) local

h,sh,X,P,j,z,mh;

#

# Procedure generates a list [{[j*h,X(j*h)],j=0 n}] where

X(j*h) is the position at time jh of the geometric Brownian

motion(with expected growth rate, mu, and volatility, sigma)

TITLE("Three independent realizations of a geometric

Brownian motion with expected \n growth rate of 0.1,

volatility 0.3, initial price

Three independent realizations of a geometric Brownian motion with expected

growth rate of 0.1, volatility 0.3, initial price = 100 300

250 200 150 100 50

t X(t)

Independent Geometric Brownian motions over 10 years with expected growth rate

of 0.15 p.a., volatility 0.02, 0.04, 0.08 p.a., initial price = 100 pence

Using the built-in ‘blackscholes’ procedure (part of Maple’s finance package),find the price of a European call option on a share that is currently priced at

£100, has volatility 20 % per annum (this means that the standard deviation

on the return in 1 year is 0.2), and where the option has 103 trading days till

expiry (252 trading days in a year) The strike price is £97 The risk-free interest

rate is 5 % Assume no dividends

p ffiffiffi2p

p

 0:0007936507936 ffiffiffiffiffiffiffiffi

103

p ffiffiffiffiffiffiffiffi252

The procedure ‘BS’ below simulates m independent payoffs together with their

antithetic counterparts; x0 is the current asset price at time t, and te is the

exercise time of the option

The procedure ‘hedge’ returns the cost of writing and hedging a call option,where the hedging is performed nnþ 1 times The asset earns interest at rate

rf The risk-free interest rate is r and the (unknown) expected growth rate is .Print statements are currently suppressed using ‘#’ Removing ‘#’ will showthe priceðxÞ of the asset, the D, the change in D since the last hedge, and thecumulative borrowings to finance the hedge, all at each hedging instant.Remember to load the procedure ‘STDNORM’ in Appendix 6.1

> hedge:=proc(K,r,rf,sigma,T,n,x0,mu) locala1,a2,a3,a4,a5,a6,h,x,d,delta,delta_prev,c,j,z,cost,xprev;global i,X2;

delta_prev:=delta*a6;# The holding in assets changes due

to the interest earned on them at rate rf

The procedure ‘effic’ calls ‘hedge’ replic times It stores the costs in the list p

and then prints the resulting mean and standard deviation of hedging cost

together with a histogram If the option were continuously hedged, then the

standard deviation would be zero, and the hedging cost would always equal

the Black–Scholes price The current price of the asset is £680 and the strike

price is £700 The risk-free interest rate is 5% per annum and the asset earns

interest continuously at the rate of 3% per annum The volatility is 0.1 per

annum and the exercise time is 0.5 years from now

print("mean cost of hedging"=e1, "std dev of cost"=e2,

"number of hedges"=nn+1, "number of contracts"=replic);statplots[histogram](p);

‘mean cost of hedging’=21.53800165, ‘std dev of cost’=22.17558076,

‘number of hedges’=2, ‘number of contracts’=10000

‘mean cost of hedging’=17.26850774, ‘std dev of cost’=13.21884756,

‘number of hedges’=3, ‘number of contracts’=10000

‘mean cost of hedging’=16.02928370, ‘std dev of cost’=10.25258825,

‘number of hedges’=4, ‘number of contracts’=10000

‘mean cost of hedging’=15.30498281, ‘std dev of cost’=8.665076040,

‘number of hedges’=5, ‘number of contracts’=10000

‘mean cost of hedging’=13.99661047, ‘std dev of cost’=4.750293651,

‘number of hedges’=13, ‘number of contracts’=10000

‘mean cost of hedging’=13.40525657, ‘std dev of cost’=1.452085739,

‘number of hedges’=127, ‘number of contracts’=10000

The cost of continuous hedging is the Black–Scholes cost, which is computed

in the procedure ‘bscurrency’

Now, as suggested in the text, perform your own experiments to verify that

(subject to sampling error) the expected cost of writing and discrete hedging the

option is the Black–Scholes price, when the expected growth rate of the

under-lying asset happens to be the same as the risk-free interest rate Of course, there

is still variation in this cost, reflecting the risk from not hedging continuously

The procedure ‘asiannaive’ computes the price of an Asian call option

(arithmetic average) using standard Monte Carlo with no variance

redu-cation The parameters are:

r¼ risk-free interest rate

x0¼ known asset price at time zero

¼ volatility

T ¼ expiry (exercise) time for the option

n¼ number of time periods over which the average is taken

h¼ time increment such that T ¼ nhnpath¼ number of paths simulated

K¼ strike price

> asiannaive:=proc(r,x0,sigma,T,n,npath,K) localR,a1,a2,s1,s2,theta,x,xc,i1,i2,z,h,mean,stderr; globali,X2;

# Computes call price for Asian option

print("point estimate of price"=exp(-r*T)*theta);

print("estimated standard error"=exp(-r*T)*stderr);end proc:

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Now call ‘asiannaive’ for the parameter values given below, ing first to load the procedure STDNORM (Section 6.1) Thesevalues are used in the paper by P Glasserman, P Heidelberger, and

remember-P Shahabuddin (1999), Asymptotically optimal importance samplingand stratification for pricing path-dependent options, MathematicalFinance, 9, 117–52 The present results can be compared with thoseappearing in that paper

266664

> randomize(13753);asiannaive(0.05,50,0.3,1,16,25000,55);

13753

‘K’=55, ‘sigma’=0.3,‘n’=16

‘#paths’=25000

‘point estimate of price’=2.214911961

‘estimated standard error’=0.03004765102

‘point estimate of price’=4.166563741

‘estimated standard error’=0.03986122915

13753

‘K’=45, ‘sigma’=0.3, ‘n’=16

‘# paths’=25000

‘point estimate of price’=7.145200333

‘estimated standard error’=0.04857962659

13753

‘K’=55, ‘sigma’=0.1, ‘n’=16

‘# paths’=25000

‘point estimate of price’=0.2012671276

‘estimated standard error’=0.004621266108

13753

‘K’=50, ‘sigma’=0.1, ‘n’=16

‘# paths’=25000

‘point estimate of price’=1.917809699

‘estimated standard error’=0.01401995095

13753

‘K’=45, ‘sigma’=0.1, ‘n’=16

‘# paths’=25000

‘point estimate of price’=6.048215279

‘estimated standard error’=0.01862584391

The procedure ‘asianimpoststrat’ computes the price of an Asian average

price call option (arithmetic average) using importance sampling and post

stratified sampling Let n be the number of time points over which the

average is calculated The drift is set tob where i¼ ð@=@ZiÞ ln (payoff)

at zi¼ i; i¼ 1; ; n, and where payoff is based on the geometric

average This gives  ¼ ðn  i þ 1Þ where

Pn i¼1i2

Pn i¼1ðn  i þ 1Þzi

 nðn þ 1Þð2n þ 1Þ=6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> asianimppoststrat:=proc(r,x0,sigma,T,t,n,m,npath,K,p,upper) local

R,a1,a2,a3,a4,b3,theta,j,s,f,i2,x,xc,x1,i1,z,h,st,xbar,v,c1,c2,mean,stderr,jj,lambda; global i,X2;

#

# Computes price of an Asian average price call option attime t, with expiry time T, and strike K, using importancesampling combined with post stratification

#

# r=risk-free interest rate

# x0=asset price at time t

# sigma=volatility

# T=exercise time

# T=n*h where the average is taken over times h,2h, ,nh

# m=number of strata

# npath=number of paths in one replication It should be

at least 20*m for post stratification to be efficient

# K=strike price

# p=number of replications

# upper=an upper bound for lambda

#i:=false;

for jj from 1 to p do:

theta:=0;

for j from 1 to m do s[j]:=0;f[j]:=0 end do;

for i2 from 1 to npath do;

f[j]:=f[j]+1; # increment frequency, this stratum;

s[j]:=s[j]+R; # increment sum, this stratum;

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Appendices 263