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1. what is  an Antithetic Variate? Explain the technique and how it reduces the standard  error of the estimate.

Answer:  An Antithetic Variate is a simulation trial involves calculating two values of  the derivative. The first valueimage.png  is  calculated in the usual way; the second value image.png is  calculated by changing the sign of all the random samples from standard normal  distributions. 

The  method of antithetic variables provides variance reduction by simulating m  i.i.d. pairs image.png such  that image.png and  image.png have  the same distribution as , and  image.png<0.  For example, if X has a symmetric distribution (such as normal or student-t),  then one can choose image.png. These  antithetic pairs can be used to estimate  by  image.png, which  is unbiased and has variance

image.png 

 

2. What  is a low discrepancy sequence?

Answer:  A low discrepancy sequence (also called a quasi-random sequence) is a sequence  of representative samples from a probability distribution. A low discrepancy  sampling procedure is flexible. The samples are taken in such a way that we are  always “filling in” gaps between existing samples. At each stage of the  simulation, the sampled points are roughly evenly spaced throughout the  probability space.

a  low discrepancy sequence can have the desirable property that they lead to the  standard error of an estimate being proportional to 1/M rather than , where  M is the sample size.

 

3.  Explain the Cholesky Decomposition technique and how to generate multivariate  normal random variable.

Answer:  Cholesky Decomposition technique is a method of sampling from multivariable  distribution.

Consider  the situation where we require n correlated samples normal distributions with  the correlation between sample  and  sample j being  image.png. We  first sample n independent variables image.png, from  univariate standardized normal distributions. The required samples, image.png, are  then defined as follows:

image.png 

And  so on. We choose the coefficients image.png so  that the correlations and variances are correct. This can be done step by step  as follows. Set image.png choose  image.png so  that image.png; choose  image.png so  that image.png choose  image.pngso  that image.pngchoose  image.png so  that image.png choose  image.png so  that image.png and  so on. This procedure is known as the Cholesky Decomposition Technique, and we  generate the multivariate normal random variable image.png,  where  image.png

And  we have image.png

 is  a lower triangular matrix with real and positive diagonal entries.

 

 

4.  Using R and the code to provide in the readings, you will value the following  European Call and Put options and plot the convergence versus Black-Scholes from  the code that you wrote for Assignment 1. To plot  the convergence, you use a range of steps and plot the MCS values Black-Scholes  e.g. use

European  Call:

S=100,  X=100, r=0.05, q=0, sigma=0.2, T=1

European  Put:

S=100,  X=100, r=0.05, q=0, sigma=0.2, T=1

Answer:  as image.png

Using  R, we get the following result:

nsimulations

Monte  Carlo Euro Call

Monte  Carlo Euro Put

100

9.807764

5.422017

1000

10.864832

5.672406

5000

10.614599

5.631602

10000

10.543076

5.614285

20000

10.416220

5.613068

50000

10.423037

5.568663

By  BSM equation,

    We  can calculate the Euro Call price and Euro Put price:

image.png

Where

 

image.png 

We  can get image.png.

let  x-axis be log(nsimulation), we have the figure shown below.

image.png

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