T = 5。

S（t）= S（0）e（（2
2）t +（
p t））
（a）给出了Black-Scholes模型中亚洲几何期权的价格

Pg = e R·T
S0e TN（d1）KN（d2）

（b）实施蒙特卡洛计划，以定价算术亚洲看涨期权的价格
（Psim a）。使用M = 1; 000; 000个模拟。记录答案，即95％的骗局-

（c）实施蒙特卡洛计划，以定价几何亚洲看涨期权
（Psim g）。
（d）使用M = 10； 000个模拟和相同的精确随机变量将创建：
•数字Xi代表几何亚洲期权的价格

•数字Yi代表算术亚洲期权的价格

b
=

i = 1（Xi X）（Yi Y）

i = 1（Xi X）2

（e）计算亚洲几何期权的定价误差：

（f）计算修改后的算术亚洲期权价格（P
a）为：
P
a = Psim a b

（d）。你观察到什么。

Problem 1. The payo 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 expensive problem, we will use the payo
of a geometric Asian call option as the control variate:

The idea is to use the known analytic price of the geometric Asian option
and the distance between MC simulations to obtain an approximate value for
the arithmetic Asian option.
In this problem we consider r = 3%;  = 0:3; S0 = 100, and assume the goal
is to price an arithmetic Asian call option with strike K = 100 and maturity
T = 5.
We assume the asset follows the standard log-normal/geometric Brownian
motion model:
S(t) = S(0)e(( 2
2 )t+(
pt))
(a) The price of a geometric Asian option in the Black-Scholes model is given
by:
Pg = e rT
S0eTN(d1) KN(d2)

such that ^  is adjusted sigma and N is the total number of trading days
Use the above formula to price this geometric Asian call option.

(b) Implement a Monte Carlo scheme to price an arithmetic Asian call option
(Psim a ). Use M = 1; 000; 000 simulations. Record the answer, a 95% con -
dence interval and the time it takes to obtain the result.
(c) Implement a Monte Carlo scheme to price a geometric Asian Call option
(Psim g ).
(d) Using M = 10; 000 simulations and the same exact random variables create:
• numbers Xi which represent the price of the geometric Asian Option
price for path i
• numbers Yi which represent the price of the arithmetic Asian Option
price for path i
Finally calculate b such that:
b
=
PM
i=1(Xi X)(Yi Y )
PM
i=1(Xi X)2
Note that b is actually the slope of a regression line Y = a+bX+”. Please
also record the price of the arithmetic Psim a and the geometric Psim g .
(e) Calculate the error of pricing for the geometric Asian option:
Eg = Psim g Pg
(f) Calculate the modi ed arithmetic Asian option price (P
a ) as:
P
a = Psim a b
Eg
Compare with the results in (b). Comment. Vary the value of M in part
(d). What do you observe.