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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 value is  calculated in the usual way; the second value  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 such  that and have  the same distribution as , and <0.  For example, if X has a symmetric distribution (such as normal or student-t),  then one can choose . These  antithetic pairs can be used to estimate by , which  is unbiased and has variance 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 . We  first sample n independent variables , from  univariate standardized normal distributions. The required samples, , are  then defined as follows: And  so on. We choose the coefficients so  that the correlations and variances are correct. This can be done step by step  as follows. Set choose so  that ; choose so  that choose so  that choose so  that choose so  that and  so on. This procedure is known as the Cholesky Decomposition Technique, and we  generate the multivariate normal random variable ,  where And  we have  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 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: Where We  can get .

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