12. Generate another 99 prices for the exercise in 11

(d) above, by generating another 99 batches of 500 two-year paths.

(a) Calculate the estimated price ON using all N ¼ 50;000 paths, and show that this is equivalent to simply averaging the 100 batch prices.

(b) Calculate the variance of the 100 batch prices, Var½O500, and use this to estimate the variance of the estimated price in part (a), Var½ON. (Hint: Recall that as a random variable ON is the average of 100 prices.)

(c) With L0ðS0Þ defined as the lattice price obtained in exercise 11(a), and using Var½O500 from part (b), compare for various values of  the proportion of the 100 prices that satisfy jO500  L0ðS0Þj >  to the upper bound for the probability of this event, Var½O500

2 , developed in proposition 8.35.

Assignment Exercises