138 F. D’Ippoliti et al.
The expressionf(X)
g′p(x)
gp(x)is an unbiased estimator ofα′(p)and the quantityg
′p(x)
gp(x)is
called score function. Note that this latter does not depend onf(X)and that the Greek
for each option is computed according to which quantity is considered a parameter
in the expression ofg.
Consider a discrete knock-out barrier option whose payoff is given by (11).
From (14), it follows that the LR estimator for the option Greeks are given by the
product of f(X)and the score function. The score function is determined by using
the key idea of CMC method: by appropriately conditioning on the paths generated
by the variance and jump processes, the evolution of the asset priceSis a log-normal
random variable (see (10)), hence its conditional density is
g(x)=1
xσi√
tiφ(di(x)), (15)whereσiis defined in (8),φ(·)is the standard normal density function and
di(x)=log(
x
S(ti− 1 )βi)
−(r−^12 σ^2 i)tiσi√
ti. (16)
Now, to find the estimator of delta and gamma, i.e., the first and the second derivative
with respect to the price of the underlying asset, respectively, we letp=Sin (14)
and compute the derivative ofginS( 0 ). After some algebra, we have
(
∂g(x)
∂S
)
S=S( 0 )=
di(x)φ(di(x))
xS( 0 )σ^2 iti. (17)
Dividing this latter byg(x)and evaluating the expression atx=S(t 1 ),wehavethe
following score function for the LR delta estimator
d 1
S( 0 )σ 1√
t 1, (18)
wherediis defined in (16), andσiin (8). The case of the LR gamma estimator is
analogous. The estimator of delta is given by
e−rT(S(T)−K)+ (^1) {max 1 ≤i≤MS(ti)<H}
(
d 1
S( 0 )σ 1√
t 1)
, (19)
and the estimator of gamma is
e−rT(S(T)−K)+ (^1) {max 1 ≤i≤MS(ti)<H}
(
d 12 −d 1 σ 1√
t 1 − 1
S^2 ( 0 )σ^21 t 1)
. (20)
To compute the estimator of rho, it is sufficient to compute the derivative ofgwith
respect tor,
e−rT(S(T)−K)+ (^1) {max 1 ≤i≤MS(ti)<H}
(
−T+
∑M
i= 1di√
ti
σi