As part of the quantitative team working at an US investment firm, you are expected to implement Monte Carlo methods to price/hedge exotic options. The option you want to consider is an (European) arithmetic average put option.
       
(a) Given the above statement derive the pathways estimator of the option’s Vega. You may want (but you are not restricted) to consider a Black and Scholes framework. Note, you must explain your calculation.

(b) Write a “pseudo” code for the estimator derived in (a) and use Monte Carlo methods to compute the option’s Vega.

(c) Discuss how Monte Carlo methods can be used to price an arithmetic Asian option and use Monte Carlo to compute the option’s price.

(d) Consider a standard European (Equity) call option and discuss the rate of convergence of a typical Monte Carlo estimator for the option price as opposed to Binomial method ( you may want to provide some numerical examples).