Please note: for the problems using data we are not interested in all the minor steps performed when analyzing the data
Problem | Points | Score |
A | 100 | |
B | 100 | |
C | 100 | |
Bonus | 20 | |
Total | 300 |
The payoff of an arithmetic Asian call option is:
Its value may be computed using straight Monte Carlo simulations. However, in order to obtain a small standard error, the number of simulations must be very high. To solve this computationally extensive problem, we will use the payoff of a geometric Asian call option as the control variate:
The idea is to use the known analytic price of the geometric Asian and the distance between MC simulations to obtain an approximate for the analytical formula for the arithmetic Asian price.
In this problem we consider r = 3%, σ = 0.3, S0 = 100, and assume the goal
is to price an arithmetic Asian call option with strike K = 100 and maturity
T = 5.
(a)The price of a geometric Asian option in the Black-Scholes model is given by:
where:
such that σˆ is adjusted sigma and N is the total number of trading days
(b)Implement a Monte Carlo scheme to price an arithmetic Asian call option (P sim). Use M = 1, 000, 000 simulations. Record the answer, a confidence intervaland the time it takes to obtain the
(c)Implement a Monte Carlo scheme to price a geometric Asian Call option (Psim).
(d)Using M = 10, 000 simulations and the same exact random variablescreate:
Finally calculate b∗ such that:
Note that b∗ is actually the slope of a regression line Y = a + bX + ε. Please
also record the price of the arithmetic P sim and the geometric P sim.
(e)Calculate the error of pricing for the geometric Asian: Eg= Pg − P sim
(f)Calculatethe modified arithmetic option price (Pa∗) as:
Pa∗ = P sim − b∗Eg