Kernel estimation of Greek weights by parameter
randomization
R. Elie

J.-D. FermanianN. Touzi

ENSAE & CREST

BNP-Paribas

CREST & Imperial College

elie@ensae.fr

jean-david.fermanian@uk.bnpparibas.com

touzi@ensae.fr

February 9, 2007

Abstract
A Greek weight associated to a parameterized random variable Z(λ) is a randomvariable π such that ∇λ E [φ (Z(λ))] = E [φ (Z(λ)) π] for any function φ. The importance of the set of Greek weights for the purpose of Monte Carlo simulations has beenhighlighted in the recent literature. Our main concern in this paper is to devise methods which produce the optimal weight, which is well-known to be given by the score,in a general context where the density of Z(λ) is not explicitly known. To do this,
we randomize the parameter λ by introducing an a priori distribution, and we useclassical kernel estimation techniques in order to estimate the score function. By an
integration by parts argument on the limit of this first kernel estimator, we define analternative simpler kernel-based estimator which turns out to be closely related to thepartial gradient of the kernel-based estimator of E[φ(Z(λ))].
Similarly to the finite differences technique, and unlike the so-called Malliavin
method, our estimators are biased but their implementation does not require any advanced mathematical calculation. We provide an asymptotic analysis of the meansquared error of these estimators, as well as their asymptotic distributions. For a
discontinuous payoff function, the kernel estimators outperforms the classical finite differences one in terms of the asymptotic rate of convergence. This result is confirmed
by our numerical experiments.

Key words: Greek weights, Monte Carlo simulation, Non-parametric regression.
MSC 1991 subject classifications: Primary 11K45, 62G08; secondary 60G07.

Kernel estimation of Greek weights by parameter randomization

1

Introduction

Let λ be some given parameter in Rd , and define the function
V φ (λ) := E [φ (Z(λ))] ,where Z(.) is a parameterized random variable with values in Rn and φ : Rn → R is ameasurable function. In many applications, we are interested in the numerical computationof the function V φ (λ) for some parameter λ0 , together with the sensitivities of V φ withrespect to the parameter λ.
In particular, in the financial literature, V φ represents the no-arbitrage price of a contingent claim, defined by the payoff φ (Z(λ)), in the context of a complete market with prices
measured in terms of the price of the non-risky asset (so that the model is reduced to thezero-interest rate situation). The sensitivities of V φ with respect to the parameter λ arecalled Greeks, and are widely used by the practitioners in their profit and loss analysis. In
the context of the Black-Scholes model, the derivative of the option price with respect to thecurrent underlying asset price is the so-called Delta, and represents the number of shares ofrisky asset to be held at each time in order to realize a dynamic perfect hedge of the option.
The Gamma is the second derivative of the option price, with respect to the underlying
asset price. It is an indicator of the variation of the hedging portfolio. Another important
Greek is the so-called Vega (although not a Greek letter !) which is the derivative of theoption price with respect to the volatility coefficient (see e.g. Hull [11], for more details).
We observe that the case of Bermudean options (i.e. American options with finite possible
exercise times) is included in our framework by taking φ(Z(λ)) as the value of the option
at the first possible exercise time. The case of American options (with a continuous set of
exercise times) can be covered by a limit argument, but requires a small time asymptotic
analysis in order to control for the stability of the variance, see [7].
Given a numerical scheme for the computation of the function V φ , the first natural idea
for the numerical computation of the Greeks is the finite differences approximation of thecorresponding derivative. In addition to the generic standard error on the numerical computation of the expectation, this approximation leads to a biased estimator at a finite distanceand appears to be inefficient for discontinuous payoff functions φ. We refer to L’Ecuyer
and Perron [6], Detemple, Garcia and Rindisbacher [4] or Milstein and Tretyakov [13] fora theoretical analysis of the rate of convergence of this estimator. Two direct methods
for computing the Greeks have been presented by Broadie and Glasserman [2] : (i) the
pathwise method, which consists in differentiating the random variable φ (Z(λ)) inside theexpectation operator, and (ii) the likelihood ratio method which reports the differentiationon the distribution of Z(λ). The first method requires the computation of the gradientof the payoff function φ, which is a serious limitation in practice as φ is typically highly
complicated or even not differentiable, see also Giles and Glasserman [5] for further developments in this direction. As for the second method (ii), it was (apparently) restricted to1Kernel estimation of Greek weights by parameter randomization

the very special cases where the distribution of Z(λ) is known explicitly. This difficulty was
overcome by Fournié, Lasry, Lebuchoux, Lions and Touzi [8] who exploited the Malliavin
integration-by-parts formula to show that, for smooth random variables Z(.),∇λ E[φ(Z(λ))] = E[φ(Z(λ))π] ,(1.1)

where π, the so-called Greek weight, is a random variable independent of the pay-off function
φ. A quick overview of the notion of Greek weights is reported in Section 2. Further
developments of the results of [8] were obtained by Gobet and Kohatsu-Higa [10]. The
comparison of the above different methods is available in the survey paper of Kohatsu-Higaand Montero [12].
An important observation is that the set of Greek weights which satisfy (1.1) is a convex
set of random variables. By an easy variance reduction argument, it is easily seen that
the score π ∗ := ∇λ ln f λ0 , Z(λ0 ) minimizes Var [φ(Z(λ))π], whenever the density f (λ, z)of the random variable Z(λ) exists and is sufficiently smooth. In general, the use of the
Malliavin calculus does not lead to this optimal Greek weight, except in trivial cases where
the density f (λ, z) is explicitly known, which corresponds to the case covered by [2].
The main purpose of this paper is to focus on the use of the optimal Greek weight in orderto estimate the corresponding Greek by the Monte Carlo method. To do this, our main idea
is to randomize the parameter λ and to re-write V φ as a regression function :V φ (λ) := E [φ(Z(Λ))|Λ = λ] ,where Z(Λ) is a random variable with density ϕ(λ, z) := ℓ(λ0 − λ)f (λ, z), and ℓ(λ0 − .)is some given randomizing distribution on the parameter λ around λ0 . In other words,
the random variable Z(Λ)|Λ = λ has the same distribution as the random variable Z(λ)
defined by the density f (λ, z). We next assume that our observations consist of a family{(Λi , Zi ), 1 ≤ i ≤ N } of independent pairs (Λi , Zi ) drawn in the density ϕ, and we define
various kernel estimators of the Greek∇λ E[φ(Z(λ))]|λ=λ0= E φ Z(λ0 ) s λ0 , Z(λ0 ) ,(1.2)

where s(λ, z) := ∇λ ln f (λ, z) is the score function. The first natural idea is to notice that





E φ Z(λ0 ) s λ0 , Z(λ0 )= E φ (Z(Λ)) s (Λ, Z(Λ)) | Λ = λ0 ,(1.3)

which is a usual regression function. Thus, a two-steps estimation method is proposed :we first perform a kernel-based estimator ŝ of the score function, and then we define akernel regression estimator of the Greek by substituting ŝ to s. In the sequel, the resulting
estimator is referred to as the double kernel-based estimator and is denoted by β̃.
Our next kernel estimator of the Greek is based on a convenient integration-by-parts in(1.2). This leads to a much simpler estimator β̂ which turns out to be closely related to theestimator β̌, obtained by direct differentiation of the classical kernel regression estimator
2

Kernel estimation of Greek weights by parameter randomization
of V φ (λ) = E φ (Z(Λ)) | Λ = λ0 . These two estimators will be referred to as the single
kernel-based estimators.
Let us observe that, unlike the so-called Malliavin Greek technique, our suggested estimators are biased but do not require any advanced mathematical calculation for theirimplementation. These two features are shared with the finite differences method. Also,
intuitively, the randomization of the parameter λ introduces an additional noise which may
imply that our estimators are less accurate than their classical competitors. Our numericalresults show indeed that the Malliavin Greek estimators are by far more accurate even in
the case of an Asian option where the Greek weight is not optimal. Therefore, the main
purpose of this paper is to provide a deep comparison between our estimators and the finite
differences ones.
Our three suggested estimators are defined precisely in Section 3, and their asymptotic
properties are discussed in Section 4. We show that β̂ and β̌ are asymptotically equivalent.
The asymptotic properties of β̃ are derived under stronger conditions on the pay-off function
φ and the kernel functions. The simultaneous choice of the bandwidth, and the number ofobservations is also more restrictive in the latter case.
An important observation is that the two single kernel based estimators coincide if andonly if the randomizing distribution ℓ is a truncated exponential distribution. In this case,by conveniently relating the support of the truncated exponential distribution to the kernel
bandwidth, we observe that the rate of convergence is independent of the dimension of theparameter λ. We next solve the optimal choice of the randomizing distribution within thisclass by minimizing the corresponding mean square error.
Our asymptotic results imply the following main property of the single kernel based estimators: for a discontinuous payoff function φ, the asymptotic rate of convergence of ourestimator is better than the classical finite differences one, whenever the order of the kernel function is larger than some explicit threshold. In the case of a truncated exponential
randomizing distribution, with support related to the kernel bandwidth, the single kernel
based estimator has a better asymptotic rate of convergence whenever the order of the kernel
function is larger than four.
Some numerical results are reported in Section 5. We estimate the delta of an European
and an Asian digital call option. Our experiments show that the Malliavin-based estimators
defined in [8] or [2] are the most efficient, as documented by the previous literature. As predicted by our theoretical asymptotic results, the single-kernel based estimator outperforms
the finite differences one, but this is only observed for a large number of simulations. We
believe that this does not restrict the interest in our new suggested method as this is justa matter of computer power, and the required number of simulations can be significantly
reduced by using variance reduction techniques. For instance, the technique of antithetic
variables applied to the randomizing density appears to be very efficient.

3

Kernel estimation of Greek weights by parameter randomization

2

The Greek weights set

Throughout this paper we consider a complete probability space (Ω, F, P ). Let Z(λ) be somerandom variable, valued in Rn , depending on some finite dimensional parameter λ ∈ Rd ,and setV φ (λ) := E [φ (Z(λ))] for φ ∈ L∞ (Rn , R) .
In order to simplify the presentation, we shall focus our attention on some fixed particularvalue λ0 of λ, and we denote
Z 0 := Z(λ0 ) .
The chief goal of this paper is to devise efficient methods for the computation of the sensitivity parameterβ 0 := ∇λ V φ (λ0 ),for arbitrary functions φ chosen from a suitable large class. We assume that the distributionof Z(λ) is absolutely continuous with respect to the Lebesgue measure, and we denote byf (λ, z) the associated density, i.e.
Z
E [φ(Z(λ))] =φ(z)f (λ, z)dz for all φ ∈ L∞ (Rn , R) .
Under mild smoothness assumptions on the density f , we directly compute that
∇λ V φ (λ0 ) :=∂V φ 0(λ ) = E φ(Z 0 )S 0 ,∂λ

S 0 := s(λ0 , Z 0 ) ,where the function s is independent of φ and is explicitly given bys(λ, z) := ∇λ {ln f (λ, z)} .
This idea was introduced by Broadie and Glasserman [2] in the context of the Black-Scholesmodel where the density f (λ, z) is explicitly known.
We shall always assume that
E S0

2

< ∞.

Under this condition, the set
n
o


W :=
π ∈ L2 (Ω, Rd ) : ∇λ V φ (λ0 ) = E φ(Z 0 )π for all φ ∈ L∞ (Rn , R)
is not empty. From the arbitrariness of φ ∈ L∞ (Rn , R), it is immediately seen thatnoW =π ∈ L2 (Ω, Rd ) : E[π|Z 0 ] = S 0 ,
4

(2.1)

Kernel estimation of Greek weights by parameter randomizationand therefore

Hence





Var φ(Z 0 )π = E φ(Z 0 )2 E[ππ ′ |Z 0 ] − ∇V φ (λ0 )∇V φ (λ0 )′≥ E φ(Z 0 )2 E[π|Z 0 ]E[π|Z 0 ]′ − ∇V φ (λ0 )∇V φ (λ0 )′
h
i


′
= E φ(Z 0 )2 S 0 S 0 − ∇V φ (λ0 )∇V φ (λ0 )′ = Var φ(Z 0 )S 0 .

S 0 ∈ W is a minimizer of Var φ(Z 0 )π , π ∈ W .

Throughout this paper, we call S 0 the optimal Greek weight. When the density functionf (λ, z) is not known, it was suggested in [8] to obtain (inefficient) Greek weights from the set
W by exploiting the integration by-parts-formula from Malliavin calculus. Our main objective is to derive Monte Carlo estimators of the Greek value β 0 , which asymptotically achieve
the minimum variance, by using methods from non-parametric statistics to approximate theabove optimal Greek weight S 0 .

3Kernel estimation and optimal Greek weight

3.1

Randomization of the parameter

The main idea of this paper is to randomize the parameter λ in order to estimate the Greek
by the classical kernel estimation technique. This randomization can be exploited fromtwo viewpoints. First, one can use it in order to estimate the optimal Greek weight, i.e.
the score function. An alternative viewpoint is to take advantage of the smoothness of therandomizing distribution in order to obtain an integration by parts formula similar to theMalliavin integration by parts technique. This technique is well known in the non-parametricstatistics litterature, see eg [1].
Let ℓ : Rd −→ R be some given probability density function, with support containing theorigin in its interior, and set
ϕ(λ, z) := ℓ(λ0 − λ) f (λ, z) for λ ∈ Rd and z ∈ Rn ,where λ0 is the parameter of interest. We consider a sequence(Λi , Zi )1≤i≤Nof N independent r.v. with distribution ϕ(λ, z) ,(3.1)

so that, for any i ≤ N , ℓ(λ0 − .) is the density of Λi and f (Λi , .) is the conditional densityof Z i given Λi .
Remark 3.1 Notice that the simulation of (Λi , Zi )i≥1 can be performed easily even in cases
where the density ϕ can not be written explicitly. This applies typically to the case whereZ(λ) = XT (λ), for some integer T , where {Xt (λ), t ∈ N} is a Markov chain with given
5Kernel estimation of Greek weights by parameter randomizationtransition density. Then, for a given value of λ, the simulation of Z is easily feasible byusual methods. However the marginal distribution of Z(λ) is typically very complicated sothat it is useless for the numerical computation of the score function s(λ, z).
In this section, we provide various estimation methods of β based on non-parametric kernel
methods. We then introduce the kernel functionZd
K : R −→ R with
K = 1,
whose precise properties will be detailed at the beginning of section 4.

3.2

A first kernel estimator of the Greek

The main idea is that the optimal weight S 0 requires a priori the knowledge of the probability
density function f (λ, z) and the associated score function s(λ, z). Indeed, if these functions
were explicitly known, then a natural non-parametric estimator of the Greek β using theobservations (3.1) is :
N

β̄N

:=

X
1
φ(Zi ) s(Λi , Zi ) Kℓ(0) N hd
i=1

 0

λ − Λi.
h(3.2)

Although s is not explicitly known in our applications of interest, one could approximate it
by means of an additional kernel estimator based on another kernel function K defined on
Rn . We therefore introduce our first kernel-based estimator of β :
N

β̃N

:=

X
1
φ(Zi ) ŝ−i
N (Λi , Zi ) Kℓ(0) N hd
i=1

where s−i
N is an approximation of s given by


∂
1
ŝ−i
(λ,
z)
:=
ln
N
0
 ℓ(λ − λ) (N − 1)hd+n
∂λ
−i

=

ϕˆλ
∇ℓ(λ − λ0 )(λ, z) +,−iϕ̂ℓ(λ − λ0 )NXK

j=1,j6=i 0
λ − Λi,hλ − Λj
h



H

(3.3)






z − Zj

h

(3.4)

and
−i

ϕ̂ (λ, z) :=

ϕˆλ

−i

h−d−n
N −1

N
X

j=1,j6=iKλ − Λjhh−d−n−1(λ, z) := ∇λ ϕ̂ (λ, z) =
N −1
−i



H

N
X



j=1,j6=iz − Zj
h

∇K





(3.5)

,

λ − ΛjhHz − Zjh. (3.6)

From a practical point of view, this estimator displays two drawbacks. First, its expression
involves a product of two (possibly multidimensional) kernels K and H. Thus, it suffers
6Kernel estimation of Greek weights by parameter randomizationfrom the so-called ”curse of dimensionality”. Moreover, its calculation is time-consuming.
In the subsequent subsections, we introduce two alternative kernel estimators of β, which
involve a single kernel function and a single summation.
From a theoretical point of view, we shall see that this estimator achieves the same rate ofconvergence as the two following ones but requires more stringent conditions, and involves
heavy calculations.

3.3

A simpler kernel estimator of the Greek

We continue our discussion under the condition thatthe kernel function K has compact support.

(3.7)

We still consider the natural estimator given by (3.2). For fixed h > 0, it follows from thelaw of large numbers that

 0

 
λ −Λ
1
, P − a.s.
(3.8)β̄N −→ E β̄N =E φ(Z)s(Λ, Z) K
N →∞ℓ(0)hdhwhere (Λ, Z) is a random variable with distribution ϕ(λ, z). Recalling the definition of s,and integrating by parts with respect to the variables λ1 , . . . , λd , we see that for h > 0
sufficiently small :
 0

Z
 
1
λ −λ
E β̄N =φ(z)Kℓ(λ0 − λ)∇λ f (λ, z) dλdzℓ(0)hd
h

 0

 0


Z
h−d−1
λ −λ
λ − λ ∇ℓ 0=
φ(z) ∇K+ hK(λ − λ) ϕ(λ, z) dλ dz
ℓ(0)
h
h
ℓ


 0

 0


1
λ −Λ
λ − Λ ∇ℓ 0=
E φ(Z) ∇K+ hK(λ − Λ),ℓ(0)hd+1hhℓwhere we used (3.7). This suggests the following simpler kernel estimator of the Greek β :
β̂N

:=


 0


 0

N
X
1
λ − Λiλ − Λi ∇ℓ 0
φ(Z
)
∇K
−
Λ
)
.(3.9)+hK(λiiℓ(0)N hd+1
h
h
ℓ
i=1

The asymptotic properties of β̂N will be provided in Section 4.

3.4

Differentiating the kernel estimator of the priceWe next start out from the natural kernel estimator of the price V φ (λ) :NV̂Nφ (λ)

X
1:=
φ(Zi )K
N hd ℓ(λ0 − λ)i=1



λ − Λih.

Differentiating V̂Nφ (λ) with respect to λ, we obtain our final kernel estimator of the Greek:

 0

 0


N
X
1
λ − Λiλ − Λi ∇ℓβ̌N :=
φ(Zi ) ∇K
+ hK
(0) .
ℓ(0)N hd+1
h
h
ℓ
i=1

7

(3.10)

Kernel estimation of Greek weights by parameter randomization

Observe that our two estimators β̂N and β̌N are closely related by :
N

β̌N

X
1
= β̂N +
φ(Zi )Kdℓ(0)N h
i=1

 0


λ − Λi
∇ℓ
∇ℓ 0
(0) −
(λ − Λi ) .
hℓℓ

In particular,β̌N = β̂N whenever ℓ : l 7→ ea0 +a1 ·l 1B (l) is a truncated exponential distribution, (3.11)
for some parameters a0 ∈ R, a1 ∈ Rd and some subset B of Rd containing the origin in its
interior.
The asymptotic properties of this third estimator will also be provided in Section 4.

4

Asymptotic results

We now compare the estimators defined in the previous section from the viewpoint of their
asymptotic distributions. The heavy asymptotic analysis of the double kernel-based estimator is reported in [7], where it is shown to have the same asymptotic rate of convergenceas the single kernel-based estimators, under more stringent conditions. Since the practicalimplementation of the double kernel-based estimator is in addition more time consuming,the discussion of this section will focus on the single kernel-based estimators.
We shall first show that the two single kernel-based estimators have equal asymptotic
rates of convergence. Then, we derive the optimal choice of the number of simulations
N and the bandwidth h of the kernel function K, by using the classical mean squareerror minimization criterion. We next specialize the discussion to the case of a truncatedexponential randomizing distribution (3.11) with support defined by B := [−ε, ε]d . In thissetting, we observe that the rate of convergence of the kernel estimator is independent of
the dimension of the parameter λ for some convenient choice of ε in terms of the bandwidthh. Finally, we discuss the optimal choice of the randomizing density ℓ within the class oftruncated exponential distribution, and we provide a quasi-explicit characterization of the
optimal truncated exponential randomizing distribution in the sense of the mean squareerror criterion.
Before stating our results, we recall that the order of the kernel function K is defined as thesmallest non zero integer p such that there exist some integers (j1 , . . . , jp ), jk ∈ {1, . . . , d},such thatZ
lα1 . . . lαr K(l)dl = 0 for 0 < r < p, αk ∈ {1, . . . , d}, andZ
lj1 . . . ljp K(l)dl 6= 0.
Typically, if K is the product of d even univariate kernels, then it is of order p = 2 (atleast). The regularity hypothesis on the kernel function K will be the following.
8

Kernel estimation of Greek weights by parameter randomization

Assumption K The kernel function K : Rd → R is C 1 , compactly supported, satisfies
R
K = 1, and is of order p ≥ 2.
In the subsequent subsections, we shall use the notationpξK[ψ](λ, z) :=

(−1)p
p!

d
X

j1 ,...,jp =1

Z
lj1 . . . ljp K(l)dl ∇pλj ...λj ψ(λ, z) ,
1

(4.1)

p

for every smooth function ψ defined on Rd × Rn . We shall also denote A⊗ := AA′ for every
matrix A.

4.1

Asymptotic results for the single kernel-based estimators

Our first result requires some regularity conditions on the density functions f and ℓ.
Assumption R1 For every z, the functions f (·, z) and ℓ are p + 1 times differentiable,and for every integer i ≤ p, the function λ 7−→ ∇iλ ℓ(λ0 − λ)∇λ f (λ, z) is continuous at
λ0 uniformly with respects to z ∈ S, for some subset S s.t. Supp(φ) ⊂ int(S).
Proposition 4.1 Under Assumptions K and R1, as N → ∞ and h → 0, the bias and thevariance of β̂N satisfy
h i
h i
Σ
E β̂N − β ∼ C1 hp and Var β̂N ∼,(4.2)
N hd+2
where

1
C1 :=
ℓ(0)

Z


p 
ℓ(λ0 − .) fλ (λ0 , z) φ(z) dz
ξKE[φ2 (Z 0 )]and Σ :=ℓ(0)Z

∇K ⊗ . (4.3)

h i Proof. By definition of β̂N , we have E β̂N = E β̄N . By (3.8), this provides
h iψ(h) := E β̂N==
Clearly, ψ(0) =

Z

 0

Z
1
λ −λ
0
φ(z)ℓ(λ − λ)∇λ f (λ, z)Kdλ dzℓ(0)hdhZ
1φ(z)ℓ(hl) fλ (λ0 − hl, z)K(l)dl dz .
ℓ(0)φ(z)fλ (λ0 , z)dz=

β. Moreover, since K has compact support, it

follows from Assumption R1 that the function ψ is p times differentiable at zero, with
derivatives obtained by differentiating inside the integral sign, so that its i−th iterated
derivative denoted ψ (i) (0) are given by
(−1)i
ℓ(0)

d
X

j1 ,...,ji =1

Z

lj1 . . . lji K(l) dl

 Z

φ(z)

h

i

∇iλj ,...,λj {ℓ(λ0 − .) fλ }1i0(λ , z) dz

for every 1 ≤ i ≤ p. Since p is the order of K, observe that ψ (i) (0) = 0 for every 1 ≤ i < p,so that a Taylor expansion of ψ provides the first part of the proposition.
9

Kernel estimation of Greek weights by parameter randomization

As for the variance, we directly compute that(v1 − v2⊗ ),N h2d+2 ℓ(0)2h i
Var β̂N
=
where
v1
v2

 0

 0

⊗ #
λ −Λ
λ − Λ ∇ℓ 0
+ hK
(λ − Λ):= E φ(Z) ∇K
,
h
h
ℓ


 0

 0


λ −Λ
λ − Λ ∇ℓ 0
+ hK
(λ − Λ):= E φ(Z) ∇K
.
h
h
ℓ
"

2



By a similar argument as in the first part of this proof, we compute that

⊗
Z
∇ℓ
φ2 (z) ∇K (l) + hK (l)(hl)ℓ(hl)f (λ0 − hl, z)dl dz
v1 = hd
ℓ
Z



d
⊗
∇K(l) dl E φ2 (Z 0 ) .
∼ h ℓ(0)
The required result follows by observing that v2 = O hd+1 .

2

We are now ready for our first main result.

Theorem 4.1 (i) Let the conditions of Proposition 4.1 hold, and assume that
(4.4)
h −→ 0 and N hd+2 −→ ∞ as N → ∞ .
√Then, with Σ as in (4.3), we have N hd+2 β̂N − E[β̂N ] −→ N (0, Σ) in distribution.
(ii) In addition to the above conditions, assume that
N hd+2+2p −→ 0 as N → ∞ .
√Then the bias vanishes and N hd+2 β̂N − β −→ N (0, Σ) in distribution.

(4.5)

Proof. We shall prove this result by verifying the Lyapounov conditions (see e.g. Billingsley(1968), p44). Let a be an arbitrary vector in Rd , and define, for every i = 1, . . . , N ,

 0

 0


1
λ − Λiλ − Λi ∇ℓ 0
N
Yi:=
φ(Zi ) ∇K+ hK(λ − Λi )N hd+1 ℓ(0)hhℓXiN := a′ YiN − E[YiN ] .

In view of Proposition 4.1, the only condition which remains to check in order to verify the
Lyapounov conditions is the existence of δ > 2 such that
"N
#
N
X
1 X
N δ
2
N
sup δ
E[|Xi | ] < +∞ where σN := Var
Xi .
(4.6)
N σN i=1
i=110

Kernel estimation of Greek weights by parameter randomization

In order to prove (4.6), we start by observing from (4.2) that
Z
1
2
2
0
E[φ (Z )] |a′ ∇K(l)|2 dl .
σN ∼N hd+2 ℓ(0)We next estimate by the Minkowski inequality and (4.2) thatXiN δ ≤=≤

a′ YiN δ + a′ E[YiN ]1a′ YiN δ +a′ E[β̂N ]
N
 0 
 0 

Pd
λ −Λ
λ −Λ ∇i ℓ 0
∇
K
+
hK
(λ
−
Λ)
φ(Z)a
i
i
i=1
h
h
ℓ
N hd+1 ℓ(0)≤ Const1
hd/δ
+N hd+1 N

!

δ

+ O



1
N



,

where the last estimate is obtained by a Taylor expansion with respect to the h variable, inthe neighborhood of the origin, following the method used in the proof of Proposition 4.1.
Hence,N
1 X h N δi
Const
hd
E
|X
|
(N hd+2 )δ/2 ≤
,
≤
Const
N
i
δ
d+1
δ
(N h )(N hd )(δ−2)/2
σN i=1PN
and condition (4.6) is satisfied when N hd → ∞, as assumed in (4.4). Therefore, N
i=1 Xi2 . By the arbitrariness ofis asymptotically gaussian, with a variance matrix given by σNa ∈ Rd , the required result follows from the Cramer-Wold device, see e.g. Theorem 25.5
p405 in [3].
2
We next turn to the estimator β̌ which was defined as the gradient, with respect to λ, ofthe kernel based estimator V̂Nφ (λ) of the function VNφ (λ). The asymptotic properties of thisestimator are obtained by following the techniques of the previous proofs and require thefollowing regularity condition on the densities f and ℓ.
Assumption R2 For every z, the functions f (·, z) and ℓ are p + 1 times differentiable,and for every integer integer i ≤ p, the function λ 7−→ ∇iλ ℓ(λ0 − λ)f (λ, z) is continuous
at λ0 uniformly with respects to z ∈ S, for some subset S s.t. Supp(φ) ⊂ int(S).
Proposition 4.2 Under Assumptions K and R2, as N → ∞ and h → 0, the bias and thevariance of β̌N satisfy
E[β̌N ] − β ∼ C2 hp

Var[β̌N ] ∼

and

Σ
,
N hd+2where Σ is given by (4.3), and
C2 :=

1
ℓ(0)

Z 

∇ℓ
p
p
ξK
[ϕλ ] +
(0) ξK
[ϕ]
ℓ

11



(λ0 , z) φ(z) dz .

Kernel estimation of Greek weights by parameter randomization

Proof. The proof is essentially similar to that of Proposition 4.1. Recall that the estimatorsβ̌N and β̂N are related by :
N

β̌N

X
1
= β̂N +
φ(Zi )Kℓ(0)N hd
i=1

 0


λ − Λi
∇ℓ
∇ℓ 0
(0) −
(λ − Λi ) .
h
ℓ
ℓ

(4.7)

We start by analyzing the bias term. Recall from the proof of Proposition 4.1 that :Z
h i1
φ(z)ℓ(hl)fλ (λ0 − hl, z)K (l) dl dz .
E β̂N
=
ℓ(0)
We then deduce from (4.7) that :


Z
 
∇ℓ
1
0
0
E β̌N =
φ(z) ϕλ (λ − hl, z) +(0)ϕ(λ − hl, z) K(l) dl dz .
ℓ(0)ℓ

We now observe that Assumption R2 allows to derive an expansion in the h variable of theabove expression, near the origin, up to the order p. The coefficients of the expansion are
obtained by simple differentiation inside the integral sign. Finally, since p is the order of
the kernel K, it is easily seen that the coefficients of hi , i < p, in this expansion vanish, andthe only non-zero coefficient is that of hp .
The variance of β̌N is also treated by the same argument as in the proof of Proposition 4.1,
and the dominant term in the expansion of the variance is easily seen to be the same as inthat proof.
2
Proposition 4.2 says that β̂N and β̌N have the same asymptotic variance, and the ordersof their asymptotic biases are the same. Our next result states that these two estimators
have exactly the same asymptotic distribution.
Theorem 4.2 (i) Let the conditions of Proposition 4.2 hold, and assume further that (4.4)√
holds. Then, with Σ as in (4.3), we have N hd+2 β̌N − E[β̌N ] −→ N (0, Σ) in distribution.
(ii) Let (4.5) hold, in addition to the above conditions. Then the bias vanishes and :√N hd+2 β̌N − β −→ N (0, Σ)
in distribution .
N →∞Proof. Define the sequence
YiN

:=

 0


 0

λ − Λiλ − Λi ∇ℓφ(Zi ) ∇K
+ hK
(0) ,
N hd+1 ℓ(0)
h
h
ℓ
1



and follow the lines of the proof of Theorem 4.1.

4.2

2

Optimal choice of N and h

The two single kernel-based estimators β̂N and β̌N have similar asymptotic properties. Theyboth have a bias of order hp , a variance of order 1/(N hd+2 ) and a convergence in distribution√
at the rate N hd+2 . Therefore, the determination methods of the optimal N and h will
12

Kernel estimation of Greek weights by parameter randomization

be similar for both of them, and we only detailed calculations for the estimator β̂N . Let
the conditions of Proposition 4.1 hold. Then (4.2) holds, and we calculate an asymptotic
equivalent for the mean square error between β̂N and β
h
i
Tr(Σ)
MSE(β̂N ) := E |β̂N − β|2 ∼+ h2p |C1 |2 .
N hd+2
Minimizing the MSE in h, we get the asymptotically optimal bandwidth selector :(d + 2)T r(Σ) 1/(d+2p+2).
ĥ =2p|C1 |2 N

(4.8)

Note that ĥ is of order N −1/(d+2p+2) , leading to an MSE of order N −2p/(d+2p+2) . Similarly,
the asymptotically optimal bandwidth selector for β̌N is(d + 2)T r(Σ) 1/(d+2p+2)
.
(4.9)
ȟ =
2p|C2 |2 N
These results imply an asymptotic theoretical choices for h relative to N , but we may stillencounter difficulties in the numerical calculation of h. Even if the optimal order of h were
known, we still need to evaluate the associated constant coefficients. From our empiricalexperiments, we observed that the accuracy of our estimators depends heavily on the choiceof the bandwidth h, as usual in kernel estimation.

4.3

The case of a uniform randomizing distribution

We first study further the case where the randomizing density is uniform on the sphere of
Rd centered at 0 with radius ǫ :ℓ(l) 7→11(l) .
(2ǫ)d [−ǫ,ǫ]Observe that this is a particular example from the truncated exponential distributions (3.11)
for which the single kernel density estimators coincide : 0N(2ǫ)d Xλ − Λiβ̂N = β̌N =φ(Zi )∇K.
N hd+1hi=1Without loss of generality, we assume that the kernel K has support on [−1, 1]d . We firstrewrite Assumption R1 in the setting of this section.
Assumption R3 For every z, the function f (·, z) is p + 1 times differentiable, and forevery integer i ≤ p + 1, the function λ 7−→ ∇iλ f (λ, z) is continuous at λ0 uniformly withrespects to z ∈ S, for some subset S s.t. Supp(φ) ⊂ int(S).
Proposition 4.3 Let Assumptions K and R3 hold. Then, as N → ∞, h → 0 and ǫ → 0
with ǫ ≥ h, we haveh i
h i
E β̂N − β ∼ Cu hp and Var β̂N ∼ N −1 h−d−2 ǫd Σu ,(4.10)
13

Kernel estimation of Greek weights by parameter randomization

where
Cu :=

Z

p
ξK
[fλ ] (λ0 , z) φ(z) dz and Σu := 2d E[φ2 (Z 0 )]Z

∇K ⊗ .

(4.11)

Proof. The proof is similar to the one of proposition 4.1. Denoting by 1d the vector of Rd
with unit component, we rewrite
!
 0

Z
Z λ0 +ǫ1d
h i
1
λ −λ
E β̂N
=
φ(z)
∇K
f (λ, z) dλ dz
hd+1 Rnhλ0 −ǫ1d
!
Z
Z
1
=
φ(z)
∇K(u)f (λ0 − uh, z) du dz.
h Rn[− ǫ , ǫ ]dh hSince ǫ ≥ h and K is supported on [−1, 1]d , we may replace in our last term the integrationon [− hǫ , hǫ ]d by an integration on Rd , which is necessary to get the convergence of ourestimator to β 0 . Then, as in the proof of proposition 4.1, an integration by parts followed
by Taylor expansions gives us the expected equivalent of the bias. The same argument
applies for the computation of the variance of β̂N .
2Sending ǫ to zero, we obtain the same asymptotic properties as in Proposition 4.1, as longas ǫ ≥ h. Therefore, the asymptotic optimal ǫ is simply the bandwidth h. The kernel-based
u , associated with this optimal uniform density ℓ is then given by
estimator β̂N
 0

N
2d X
λ − Λiuβ̂N :=
φ(Zi )∇K,(4.12)Nhhi=1

and satisfies

h i
h i
u
u
E β̂N
− β ∼ Cu hp and Var β̂N∼ N −1 h−2 Σu ,(4.13)

with Cu and Σu defined in (4.11). Minimizing the corresponding mean square error, we
obtain the optimal bandwidth

 1
2p+2
T rΣu
u
h :=
.
(4.14)
2
p|Cu | N
As in the study of the previous estimators, we also obtain a central limit theorem for theu.
estimator β̂N

Theorem 4.3 (i) Let the conditions of Proposition 4.3 hold in the particular case whereǫ = h, and assume further thath −→ 0 and N h2 −→ ∞ as N → ∞ .
(4.15)

√u − E[β̂ u ] −→ N (0, Σ ) in distribution.
Then, with Σu as in (4.11), we have N h2 β̂NuN2p+2(ii) If in addition N h→ 0, then the bias vanishes and :√u
N h2 β̂N− β −→ N (0, Σu )in distribution.

A remarkable feature of the above asymptotic result is that the rate of convergence isindependent of the dimension d of the parameter λ0 .
14Kernel estimation of Greek weights by parameter randomization

4.4

The case of a truncated exponential randomizing distribution

In this subsection, we specialize the discussion to the one-dimensional case, and we consider
a truncated exponential randomizing distribution:
eθl
1
(l) ,
eθǫ − e−θǫ [−ǫ,ǫ]with the parameter θ ∈ R, so that the two single kernel estimators associated to this density
coincide:

 0

 0

N
X
1
λ − Λiλ − Λiβ̌N = β̂N =φ(Zi ) ∇K+ θhK.
ℓ(0)N hd+1hhℓ(l) := θi=1Using the same line of arguments as in Proposition 4.3 , we see that, under Assumptions K
and R3, as N → ∞, h → 0 and ǫ → 0 with ǫ ≥ h, we haveh i
h i
E β̂N − β ∼ Ce hp and Var β̂N ∼ N −1 h−3 ǫ Σe ,(4.16)

where Σe := Σu defined in (4.11) and
Z
X
 Z

p+1 
(−1)p
p
p
k
0
Ce :=
u K(u)du∇λ f (λ , z) φ(z) dz (−θ)p−k+1 (4.17)
.
p!
k−1
k=1

Again, the asymptotic optimal ǫ is simply the bandwidth h and the kernel-based estimatore , associated with this optimal exponential density is given by
β̂N

 0

 0

N
eθh − e−θh X
λ − Λiλ − Λieβ̂N :=
φ(Zi ) ∇K+ θhK.
(4.18)
θN h2
h
h
i=1

The optimal bandwidth is obtained by minimizing the corresponding mean square error:

 1
2p+2
T rΣe
e
h :=
,
(4.19)
p|Ce |2 N
which leads to the following MSE :
 
 1
− p 
− p
e
MSE β̂N= 2(p + 1)p p+1 |Ce |2 (T rΣe )p p+1 N p+1 .

(4.20)e can be derived.
As in Theorem 4.3, a central limit theorem for the estimator β̂N

Remark 4.1 From the asymptotic viewpoint, the estimators based on the truncated exponential randomizing density differ by their bias, as the constants Ce depends on θ whilethe variance Σe = Σu is independent of θ. The optimal truncated exponential randomizing
density is then obtained by minimizing the squared bias, defined by the polynomial function Ce2 , with respect to θ. In our numerical experiments of Section 5, this minimization is
performed by classical Newton-Raphson iterations.
Remark 4.2 Notice that, in both cases, the choice of the radius ǫ of ℓ depends on the
kernel function K only through its support. For instance, if supp(K) = [−M, M ]d , thenthe optimal radius is ǫ = M h.
15

Kernel estimation of Greek weights by parameter randomization

4.5

Comparison with the finite differences estimators

We first start by recalling the finite differences estimators. For ease of presentation, we let
d = 1. The finite differences estimator of the parameter β 0 := ∇λ E[φ(Z(λ0 ))] is based on
the finite differences approximation of the gradient∇λ E[φ(Z(λ0 ))] ∼E[φ(Z(λ0 + αε))] − E[φ(Z(λ0 − (1 − α)ε))],ε

where ε > 0 is a "small" parameter, and α ∈ [0, 1]. The values α = 0, 0.5 and 1 correspond respectively to the backward, centered and forward finite difference. The above finite
difference approximation suggests the following finite differences estimator of β :
FD
β̂N

=

N



1 X  i 0
φ Z (λ + αε) − φ Z i (λ0 − (1 − α)ε) .
Nεi=1

The asymptotic properties of these estimators were first studied by L’Ecuyer and Perron(1994). In the case where λ 7→ φ[Z(λ)] ∈ Cb3 (Rd ), when N → ∞ and ε → 0 with N 1/4 ε → 0,
they obtained a parametric rate of convergence :√  FD1
N β̂N − β −→ N (0, Σα ) in distribution,
for α = 0 , and 1 .
N →∞2When the payoff function φ has a countable number of discontinuities, Detemple, Garcia
and Rindisbacher (2005) obtained the following central limit theorems :1For α = ,2
For α = 0, 1 ,when N 1/5 ε → 0 ,N

2/5when N 1/3 ε → 0 , N 1/3




FD
β̂N
−β
FD
β̂N
−β




−→ N (0, Σα ) in distribution.

N →∞−→ N (0, Σα ) in distribution.

N →∞In the general case d ≥ 1, the finite differences estimators are defined componentwise, andtherefore, the rate of convergence is not affected by the dimension d of the parameter λ0 .
The main objective of this paragraph is to provide an asymptotic comparison of the singlekernel based estimator with the finite differences one. The key point of our single-kernel
based estimators is that the differentiation with respect to the parameter λ is reported on
the density of Z(λ) so that our asymptotic results do not involve the regularity of the pay-off
function φ. For any pay-off function φ, and when N hd+2p+2 −→ 0, we derived in Theorems
4.1 and 4.2 that√
N hd+2 β̂N − β−→ N (0, Σ) in distribution,
N →∞where p is the order of the kernel function. Minimizing the corresponding MSE, we obtained in Section 4.2 an optimal h of order N −1/(d+2p+2) which, of course, almost satisfies
the condition required in the convergence in distribution. Therefore, taking a bandwidth
h of order N −1/(d+2p+2)−2δ/(d+2) with δ > 0 sufficiently small leads to a convergence indistribution at rate N r with r := p/(d + 2p + 2) − δ > 0. Therefore, the single-kernel based16

Kernel estimation of Greek weights by parameter randomization

estimators, with kernel of order p > 2d + 4 and δ sufficiently small, achieve a convergence
rate of order r > 2/5. Hence, they outperform all the finite differences estimators in thecase of discontinuous payoffs.
Notice that, by taking kernel functions of order p sufficiently large, we can obtain a√convergence rate in distribution as close as desired to the parametric rate N .
Remark 4.3 Consider the optimized kernel estimators β̂nu and β̂ne , based on uniform or
exponential density ℓ on the sphere with radius h, as derived in section 4.3. Then, for√N h2p+2 → 0, we obtain a rate of convergence of N h2 . Therefore, in order to outperformthe finite differences estimators of a Greek associated to a discontinuous payoff function φ,
one just needs to use a kernel function of order p > 4.

5

Numerical results

In this section, we present some numerical results obtained in the Black-Scholes model :



σ2
x
St := x exp r −t + σWt , t ≥ 0, x > 0 ,2where W is a standard Brownian motion on (Ω, F, P) with values in R, and r ∈ R, σ > 0
are two given constants. We focus on the estimation of the so-called Delta :β := ∇x E[φ(Z x )] ,RT
where Z x = STx for an European option and Z x = 0 Stx dt for an Asian option. As in theprevious sections, we denote by f (x, .) the density of Z x .
We simulate independent observations Xi distributed in the (optimal) exponential randomizing distribution ℓ on the sphere centered at S0 = x with radius h, as derived in section
e is therefore given by (4.18).
4.4. The single-kernel based estimator β̂N

5.1

Computation of the optimal bandwidth

As the "bumping" parameter ǫ for the finite differences estimator, the bandwidth in kernel
estimation needs to be chosen carefully. The asymptotic results of Section 4 provide theexpression of the asymptotic optimal bandwidth. For the truncated exponential randomizing
distribution, we obtainehR
where Σe = 2 E[φ2 (Z x )] (∇K)2 and
Ce :=

(−1)p
p!

Z

p

u K(u)du=ΣepCe2 N

X
p+1 
k=1



1
2p+2

,

 

k
x
p
x ∇x f (x, Z )E φ(Z )(−θ)p−k+1
k−1
f (x, Z x )17Kernel estimation of Greek weights by parameter randomization

Given a kernel function K, the coefficient Σe can be estimated by a standard Monte Carloprocedure. We next focus on the estimation of the parameterkx
x ∇x f (x, Z )Ek := E φ(Z ).
f (x, Z x )for a given k ∈ {1, . . . , p + 1}.
2(i) Let Z x = STx = x eY , where Y has a normal distribution with mean m := (r − σ2 )Tand variance Σ := σ 2 T . Then, it is easily checked that :" k#Xf (x, z)∇kx f (x, z) =aki d(x, z)i
xk
i=0

where

d(x, z) :=ln z − ln x − m
,
Σ

(5.1)

and the coefficients (aji )(i,j)∈{0,...,k}2 are given by
a0i = 1{i=0} ,aj+1= aji−1 − j aji −
i

i+1 j
a ,
Σ i+1

(5.2)

with the convention aji = 0 for i < 0 and i > j. Hence :
"
!#
k
X
1
x
k
x i
Ek =
E φ(Z )ai d(x, Z )
,
xk
i=0

and this parameter can be estimated by a straightforward Monte Carlo procedure.
(ii) In practice, the distribution function is unknown, and the calculation of the previous
paragraph can not be used to estimate Ek . We suggest to mimic the same principle as theusual Silverman’s rule-of-thumb in kernel estimation (see Scott (1992) e.g.) : let m̂ and Σ̂ˆ z)
be two given estimates of the mean the variance ln(Z x /x), respectively, and define d(x,jand (âi )(i,j)∈{0,...,k}2 by substituting (m̂, Σ̂) to (m, Σ) in (5.1)-(5.2); then the coefficient Ek
is approximated by
"
!#
k
X
1
ˆ Z x )iÊk =E φ(Z x )âki d(x,.
xp
i=0Once the coefficients Ek estimated for 1 ≤ k ≤ p + 1, the parameter θ is chosen through a
numerical minimization, see Remark 4.1. In the particular case of an uniform randomizing
distribution (θ = 0), remark that only the estimation of Ep+1 is necessary.
Therefore, the numerical procedure is divided in three steps: first, we estimate the terms
detailed in the previous subsection Σe , Ek , m̂ and Σ̂e through a Monte Carlo procedure withvery few simulations. Then, we calibrate the parameter θ by minimization and we deduce
the exponential optimal theoretical bandwidth . Finally we estimate the delta of the option
by means of a single-kernel based estimator with the estimated bandwidth.
Remark 5.1 The numerical effort dedicated to the calculation of the optimal bandwidthparameter h is also encountered in the classical finite differences method, as the optimalbumping parameter ǫ involves some a priori numerical simulations.
18

Kernel estimation of Greek weights by parameter randomization

5.2

Numerical comparison of the estimators

We present here numerical results obtained for the estimation of the delta of an European
and an Asian at-the-money digital calls, i.e. with a payoff of the form φ(s) = 1s>K . Since
this payoff function is discontinuous, the results of Section 4.5 show that the single-kernel
based estimator achieves a better rate of convergence than the finite differences estimators,
whenever the kernel has order p > 4. The main object of this section is to verify theempirical validity of these asymptotic results.
In order to compare their behavior, each estimator has been computed 200 times andtheir empirical distributions have been approximated by classical smoothing non parametricestimation.
Our numerical experiments are performed with the following values of the parameters :S0 = 120,r = 0,

σ = 0.2,

T = 1,and K = 120 .

We use the following polynomial kernel functions of order 2, 4 and 6, respectively, withsupport on [−1, 1] :K2 (u) =K4 (u) =K6 (u) =3(1 − u2 ) ,415(1 − u2 )(3 − 7u2 ) ,32
105(1 − u2 )(33u4 − 30u2 + 5) .
256

From the viewpoint of computing time, kernel based or finite differences estimations withthe same number of simulations are comparable. All the numerical tests have been realized
in Visual C++ on a Pentium 4 xeon 3 GHz processor with 1 Gb of RAM.
European Digital Call Option In the context of the Black-Scholes model, it was observed by [8] that the optimal weight for European options can be obtained by means of
the Malliavin integration by part formula, and coincides with the likelihood estimator introduced by [2]. Therefore, we are not hoping to compete with the the Malliavin-based Monte
Carlo estimator.
From our numerical experiments, we observed that the gain from using kernel estimators
based on an exponential rather than a uniform randomizing distribution ℓ was very poor,
especially when the order of the kernel function increases. From a numerical viewpoint, thegain obtained at most counter-balanced the numerical price of the minimization procedure.
The examples presented here are therefore based on a uniform randomization distributionℓ.
The distributions of the different estimators based on N = 106 simulations are reported in
Figure 1. The good performance of the Malliavin estimator is confirmed by our numericalexperiments. However, we observe surprisingly that the three kernel based estimators areless accurate than the centered finite differences one, although their numerical computing
19

Kernel estimation of Greek weights by parameter randomization
K2

0,016

0,0161

K4

0,0162

K6

0,0163

Malliavin0,0164

0,0165FD

0,0166

True value

0,0167

0,01680,0169Figure 1: Delta of an European Digital Call, N = 1 Million

K6

0,016525

0,016529

FD

0,016533

True value

0,016537

0,016541

Figure 2: Delta of an European Digital Call, N = 1 Billion

times are comparable, of the order of 2 seconds. According to Section 4.5, the kernel oforder 6 should perform better than the other ones, but this is not the case here. Actually,
the terms Ce and Σe are such that the constant term of the mean square error increases
20Kernel estimation of Greek weights by parameter randomization

very fast with the variability of K, which naturally increases with its order. For example,
the MSE of the estimator based on the kernel of order 4 is ten times bigger than the one of
the finite differences one, although they have the same rate of convergence. Furthermore,
the optimal bandwidth h increases with the order of the kernel, so that the asymptotic
approximations become less accurate.
In order to further investigate this effect, we increase the number of simulations. Figure
2 shows the distribution of the finite differences estimator and the kernel based estimatorof order 6 based on N = 109 simulations where each simulation takes approximately 30
minutes on our computer. In this case, we observe that the kernel based estimator oforder 6 truly outperforms the finite differences one: its bias and its variance are two times
smaller. This confirms the theoretical asymptotic results obtained in section 4.5. We do
not consider that the high number of simulations required is a serious restriction since it isjust a matter of computer power or time given to the simulation. Furthermore, the good
performance of the kernel based estimators of high order can be observed for a smaller
number of simulation if we use in addition variance reduction technique. For example,
by performing the antithetic variable technique with respect to the randomizing densityℓ, we observe that the kernel based estimator of order 6 outperforms the finite differences
estimator with 6 ∗ 107 simulations, corresponding to a computer time of about 2 minutes.
K2

0,0277

K4

0,0282

K6

Malliavin

0,0287FD

0,0292"True value"

0,0297

Figure 3: Delta of an Asian Digital Call, N = 1 Million

Asian Digital Call Option We next investigate the case of an Asian option, where theMalliavin integration by parts formula does not lead to the optimal weight, see [8]. The

21

Kernel estimation of Greek weights by parameter randomization

distribution of the different estimators based on N = 106 simulations are reported in Figure
3, where the "true value" of the Greek has been approximated by an unbiased Malliavin
estimation with a very large number of simulations. Even if the Malliavin weight is not
optimal, the Malliavin estimator still outperforms the other estimators. As for the European
digital call, the finite differences estimator outperforms the kernel based estimators but one
simply requires more simulations in order to make the kernel estimator of order 6 more
efficient than the finite differences one.
Conclusion (numerical results) Other tests realized with different parameters, payoff
functions or randomizing densities lead to rather similar results. Our kernel based estimatorwith order p > 4 of the delta of a digital option outperforms asymptotically the finite
differences one, but one requires a large number of simulation to verify this fact empirically.
Nevertheless, the high number of simulations required can be significantly reduced by meansof variance reduction techniques. When the density of the underlying is unknown and the
pay-off function is irregular, the Malliavin based estimator is still more efficient than theothers. Nevertheless, in general, Malliavin weights are very difficult to derive analyticallyand this is precisely the advantage of the other estimators which are straightforward to
implement.

22

Kernel estimation of Greek weights by parameter randomization

References[1] Ait-Sahalia, Y. (1996). Non parametric pricing of interest rate derivative securities.,
Econometrica 64, 527-560.
[2] Broadie, M. and Glasserman, P. (1996). Estimating security prices using simulation. Management Science 42, 269-285.
[3] Davidson, J. (1994). Stochastic Limit Theory, Oxford University Press.
[4] Detemple, J., Garcia, R. and Rindisbacher, M. (2005). Asymptotic Propertiesof Monte Carlo Estimators of Derivatives. Management Science 51, 11, 1657-1675.
[5] Giles, M. and Glasserman, P. (2006). Smoking adjoints: fast Monte Carlo Greeks.
Risk January 2006, 92-96.
[6] L’Ecuyer, P., Perron, G. (1994). On the Convergence Rates of IPA and FDC
derivative Estimators. Operations Research 42, 643-656.
[7] Elie, R. (2006). Contrôle stochastique et méthodes numériques en finance mathématique. PhD thesis, Universite Paris Dauphine.
[8] Fournié, E., Lasry, J.M., Lebuchoux, J., Lions, P.L., and Touzi, N. (1999).
Applications of Malliavin Calculus to Monte Carlo Methods in Finance. Finance andStochastics 3, 391-412.
[9] Fournié, E., Lasry, J.M., Lebuchoux, J. and Lions, P.L. (2000). Applicationsof Malliavin Calculus to Monte Carlo Methods in Finance. II. Finance and Stochastics
5, 201-236.
[10] Gobet, E., Kohatsu-Higa, A. (2003). Computation of Greeks for barrier and Lookback options using Malliavin Calculus. Electronic Communications in Probability 8,
51-62.
[11] Hull, J. (2002). Options, futures, and other derivatives. Prentice Hall.
[12] Kohatsu-Higa, A. and Montero, M. (2004). Malliavin Calculus in Finance. Handbook of Computational and Numerical Methods in Finance, Birkhauser, 111-174.
[13] Milstein, G. and Tretyakov, M. (2005). Numerical Analysis of Monte Carlo Evaluation of Greeks by Finite Differences. Journal of Computational Finance 8-3, 1-34.

23