Notebook 4

Question 1

How do we apply importance sampling here?

Answer 1

Question 2

How do we apply Naive Monte Carlo here?

Answer 2

Question 3

What is the Naive Monte Carlo program that works here?

Answer 3

Question 4

How do we apply importance sampling?

Answer 4

Question 5

What is the algorithm for applying importance sampling here?

Answer 5

Question 6

What is the Monte Carlo with importance sampling code?

Answer 6

Question 7

What is the code for Importance sampling of a European call (Using alternative method)?

Answer 7

Question 8

How do we use Naive Monte Carlo for the uniform distribution?

Answer 8

Question 9

How do we model the uniform distribution using Stratified Sampling?

Answer 9

Question 10

Modify the code above to compute the mean of the normal distribution N(1,1) using naive Monte Carlo sampling. Note, you should not draw the Y from the normal distribution; instead, you should use the inverse cdf to compute a normal sample X from the uniform sample Y.

Answer 10

Question 11

Modify the code above to compute the mean of the normal distribution N(1,1)
using stratified sampling. In this case, you must first draw the Y from the uniform distribution, and then use the inverse cdf to compute a normal sample X.

Answer 11

Question 12

What is the function of MC_euro_call()?

Answer 12

Question 13

What is the function of MC_euro_call_delta ()?

Answer 13

Question 14

What is the function of MC_euro_call_ant()?

Answer 14

Question 14

What is the function of MC_euro_call_con()?

Answer 14

Question 15

What is the function of MC_euro_call_imp()?

Answer 15

Question 16

What is the function of MC_euro_call _str?

Use M=10

Answer 16

Question 17

How can we validate the MC variation reduction formulas?

Answer 17

Question 18

What are the two ways to validate the variance of our MC functions?

Answer 18

Question 19

Can you implement the two ways to validate the variance of our MC functions?

Answer 19

Question 20

What is the function of MC_euro_call_delta_pathwise()?

Answer 20

Question 20

What is the function of MC_euro_call_delta_likelihood()?

Answer 20

Question 21

How do I compare my MC_delta methods to validate their accuracy?

Answer 21

Question 22

How do we estimate the sample size required to obtain an absolute error of 0.05 on the call price?

Answer 22

Question 23

Which of the variation reduction methods required a smaller sample size to obtain an absolute error of 0.05 of the call price?

Answer 23

Antithetic = For Error = 0.05, the needed sample size is 2.1e+05 Control Variates = For Error = 0.05 the needed sample size is 8.9e+04 Importance Sampling = For Error = 0.05 the needed sample size is 4.3e+05 Stratified Sampling = For Error = 0.05 the needed sample size is 1e+05

Question 24

Which of the delta methods required a smaller sample size to obtain an absolute error of 0.05 of the delta value?

Answer 24

Finite Difference = For Error = 0.05 the needed sample size is 5.4e+02 Pathwise = For Error = 0.05, the needed sample size is 5.9e+02 Likelihood Ratio = For Error = 0.05 the needed sample size is 4.8e+03

Question 25

What are the imports needed when doing MC for Basket Options?

Answer 25

Question 26

What is the mathematical formula for pricing Basket Options

Answer 26

Question 26

What is the difference about our value of N in these Monte Carlo estimations?

Answer 26

Question 26

How do you derive the Covariance matrix from this using code?

Answer 26

Question 27

How do you derive the correlation matrix p from this covariance matrix?

Answer 27

Question 28

What is the algorithm for completing an MC estimation of a European Call Basket Option Price?

Answer 28

Question 29

What is the function of MC_Basket() used to calculate the price of a European Call Basket Option Price?

Answer 29

Question 30

What is the function of MC_Basket_antithetic() used to calculate the price of a European Call Basket Option Price using the antithetic method of variance reduction?

Answer 30

Question 31

Alternative code for MC_Euro_call_con()?

Answer 31