Copyright c 2007 by Karl Sigman
1
Estimating sensitivities
When estimating the Greeks, such as the Δ, the general problem involves a random variable
Y
=
Y
(
α
) (such as a discounted payoff) that depends on a parameter
α
of interest (such as initial
price
S
0
, or volitility
σ
, etc.). In addition to estimating the expected value
K
(
α
)
def
=
E
(
Y
(
α
))
(this might be, for example, the price of an option), we wish to estimate the sensitivity of
E
(
Y
)
with respect to
α
, that is, the derivative of
E
(
Y
) with respect to
α
,
K
0
(
α
) =
dE
(
Y
)
dα
= lim
h
→
0
K
(
α
+
h
)

K
(
α
)
h
.
1.1
Samplepath approach
If the mapping
α
→
Y
(
α
) is “nice” enough, we can interchange the order of taking expected
value and derivative,
dE
(
Y
)
dα
=
E
h
dY
dα
i
.
(1)
Under this scenario,
K
0
(
α
) itself is an expected value so we can estimate it by standard
Monte Carlo: Simulate
n
iid copies of
dY
dα
and take the empirical average.
To dispense with the notion that such an interchange as in (1) is always possible (no, it is
not!) one merely need consider the Δ of a digital option with payoff
Y
=
Y
(
S
0
) =
e

rT
I
{
S
(
T
)
>
K
}
, where
S
(
T
) =
S
0
e
X
(
T
)
=
S
0
e
σB
(
T
)+(
r

σ
2
/
2)
t
. (The riskneutral probability is being used
for pricing purposes.) In this case,
dY
dS
0
= 0 since the indicator is a piecewise constant function
of
S
0
, thus
E
h
dY
dS
0
i
= 0. But
E
(
Y
) =
e

rT
P
(
S
(
T
)
> K
) is a nice smooth function of
S
0
>
0,
with a nonzero derivative. (Yes, the sample paths of
Y
are not differentiable (nor continuous
even) at the value of
S
0
for which
S
(
T
) =
K
, but
P
(
S
(
T
) =
K
) = 0, so this point can be
ignored.)
On the other hand, a European call option, with payoff
Y
=
e

rT
(
S
(
T
)

K
)
+
satisfies
dY
dS
0
=
e

rT
e
X
(
T
)
I
{
S
(
T
)
> K
}
which can be rewritten as
dY
dS
0
=
e

rT
S
(
T
)
S
0
I
{
S
(
T
)
> K
}
, and
indeed it can be proved that (1) holds.
The basic condition needed to ensure that the interchange is legitimate is uniform integra
bility of the rvs
{
h

1
(
Y
(
α
+
h
)

Y
(
α
))
}
as
h
↓
0. We present a sufficient condition for this
next.
Proposition 1.1
Suppose that
Y
(
α
)
is
1. wp1, differentiable at the point
α
0
, and satisfies
2. there exists an interval
I
= (
α
0

, α
0
+ )
, (some
>
0
), and a nonnegative rv
B
with
E
(
B
)
<
∞
such that wp1

Y
(
α
1
)

Y
(
α
2
)
 ≤ 
α
1

α
2

B, α
1
, α
2
∈
I.
Then (1) holds.
1
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Proof :
For sufficiently small
h
,

Y
(
α
0
+
h
)

Y
(
α
0
)
 ≤
hB
, wp1, thus

Y
(
α
0
+
h
)

Y
(
α
0
)

h
≤
B,
wp1
,
and the result follows by the dominated convergence theorem (letting
h
→
0).
For the European call option, we have
Y
(
S
0
+
h
)

Y
(
S
0
)
≤
e

rT
he
X
(
T
)
, so the above
proposition applies with
B
=
e

rT
e
X
(
T
)
. For the digital call option,
Y
(
S
0
) is not differentiable
at the point
S
0
where
S
0
e
X
(
T
)
=
K
; it is not even continuous there.
A general rule is that if the mapping
α
→
Y
(
α
) wp1 is continuous at all points, and
differentiable except at most a finite number of points, then the interchange will be valid.
Even if one can justify the interchange (1), it may not be possible to explicitly compute the
derivative
dY
dα
, thus rendering the samplepath approach impractical. So clearly other methods
are needed for estimating
K
(
α
).
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 Spring '07
 sigman
 Derivative, Variance, Probability theory, probability density function, Option style

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