Description
Problem A: Asian Option Pricing using Monte Carlo Control Variate.
The payoff of an arithmetic Asian call option is:
1
N + 1
X
N
i=0
Sti − K
!
+
.
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:
Y
N
i=0
Sti
! 1
N+1
− K
+
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.
We also assume the asset follows the standard log-normal/geometric Brownian motion model:
S(∆t) = S(0)e
((µ− σ2
2
)∆t+(σ
√
∆t))
(a) The price of a geometric Asian option in the Black-Scholes model is given
by:
Pg = e
−rT
S0e
ρT N(d1) − KN(d2)
where:
ρ =
1
2
r −
1
2
σ
2 + ˆσ
2
σˆ := σ
s
2N + 1
6(N + 1)
such that ˆσ is adjusted sigma and N is the total number of trading days
(T ∗ 252).
d1 :=
1
√
Tσˆ
ln
S0
K
+
ρ +
1
2
σˆ
2
T
d2 :=
1
√
Tσˆ
ln
S0
K
+
ρ −
1
2
σˆ
2
T
Use the above formula to price this geometric Asian call option.
