• Use the inverse transform method to generate samples of a random variable X with the p.d.f

• Suppose that we can generate independent samples of standard normal random variables. (Write the pseudo-codes that) Use the acceptance-rejection method to generate samples of a random variable X with the p.d.f.

where c = 1/1.5413. Moreover, calculate the acceptance probability in the algorithm.

• Let the random variable X have p.d.f

Write the pseudo-codes that generate random variates from f(x), using

(a) the inverse-transform method;
(b) the acceptance-rejection method with the proposal density

• A stock with a current value S(0) = 100 has an expected growth rate of its logarithm of v = 12% and a volatility of that growth rate of σ = 20%.

(a) Find suitable parameters of a binomial lattice representing this stock with a basic elementary period of 3 months. Remark: Either approximated or non-approximated formulas is fine. You may try both to see their difference but please stick with one set of parameters for parts (b) and (c).
(b) Draw the lattice and enter the node values for 1 year.
(c) What are the probabilities of attaining the various final nodes in part (b)?