It is understood that derivatives of an expectation $E [\phi(S(T)) | S(0) = x]$ with respect to $x$ can be expressed as $E [\phi(S(T)) \pi | S(0) = x]$, where $S(T)$ is a stochastic variable at time $T$ and $\pi$ is a stochastic weighting function (weight) independent of the form of $\phi$. Derivatives of expectations of this form are encountered in various fields of knowledge. We establish two results for weights of higher order derivatives under the dynamics given by (\ref{dynamics}). Specifically, we derive and solve a recursive relationship for generating weights. This results in a tractable formula for weights of any order.price sensitivities, greeks, malliavin calculus

Robert de Rozario

English

Research Papers in Economics

On Higher Derivatives of Expectations
Robert de Rozario
School of Banking and Finance, University of NSW, Sydney, Australia.
Abstract
It is understood that derivatives of an expectation E[Á(S(T))jS(0) = x] with respect
to x can be expressed as E[Á(S(T))¼jS(0) = x], where S(T) is a stochastic variable
at time T and ¼ is a stochastic weighting function (weight) independent of the form
of Á. Derivatives of expectations of this form are encountered in various ¯elds of
knowledge. We establish two results for weights of higher order derivatives under
the dynamics given by (1). Speci¯cally, we derive and solve a recursive relationship
for generating weights. This results in a tractable formula for weights of any order.
Key words: Price Sensitivities, Greeks, Malliavin Calculus.
1 Introductory Remarks
Consider a real valued process fS(t);0 · t · Tg of which dynamics are
described by the stochastic di®erential equation with an accompanying strong
solution
dS(t)
S(t)
= ¹dt + ¾dW(t); S(t) = xexp
µ
(¹ ¡
1
2
¾
2)t + ¾W(t)
¶
; (1)
for S(0) = x > 0;, and associated tangent process fY (t);0 · t · Tg described
by
dY (t)
Y (t)
= ¹dt + ¾dW(t); Y (0) = 1: (2)
where fW(t);0 · t · Tg is a one-dimensional Brownian motion, and coe±-
cients ¹ and ¾ are constant. Consider an expectation u(x) = E[Á(S(T))jS(0) =
Email address: rob.rozario@unsw.edu.au (Robert de Rozario).
1 I am grateful to Galuh Pandita for improving the clarity of my writing.
Preprint submitted to Applied Mathematics Letters 8 August 2003x]. It is well known that derivatives with respect to the initial condition of the
form @n
xu(x) can be expressed as E[Á(S(T))¼njS(0) = x], where ¼n is a stochas-
tic weighting function (weight), see for example [4]. A surprising result is that
the weights are independent of Á, and in fact only depend on the dynamics of
the underlying variable. This has practical importance, since Á depends on the
problem under consideration while the weights do not. This means that the
weights provide a problem independent, general representation of derivatives
for many applications.
In this article we establish two results on the weighting functions for deriva-
tives of the form @n
xu(x) under the dynamics given by (1). Speci¯cally, we
derive and solve a recursive relationship for generating weighting functions,
providing a tractable formula for functions of any order. Furthermore, we
relate the derivatives considered to derivatives of the drift and di®usion coef-
¯cients, comment on e±ciently evaluating these expectations, uniqueness and
optimality.
2 Mathematical Preliminaries
We approach this problem using Malliavin Calculus, appealing to a number
of results in the following analysis that we brie°y outline in this section 2 . Let S
be the set of stochastic functions of the form G = f
³R T
0 h1(t)dW(t);:::;
R T
0 hn(t)dW(t)
´
,
where f belongs to the set of in¯nitely di®erentiable functions with all par-
tial derivatives of at most polynomial growth, and h1;:::;hn belongs to the
set of in¯nitely di®erentiable functions with bounded partial derivatives. The
Malliavin derivative is de¯ned as
DtG =
n X
i=1
rif
0
@
T Z
0
h1(t)dW(t);:::;
T Z
0
hn(t)dW(t)
1
Ahi(t); t 2 [0;T] (3)
where ri denotes the derivative with respect to the i-th argument 3 . The
adjoint of the Malliavin derivative is the Skohorod integral. For processes that
are adapted, as is the process described by (1), the Skohorod integral coincides
with the Ito integral. As a consequence, the Malliavin integration by parts
formula can be expressed in terms of Ito integrals as E
h
G
R T
0 h(t)dW(t)
i
=
E
hR T
0 DtGh(t)dt
i
, where we take G as one dimensional and h the same form
2 For a comprehensive introduction to the subject consult [8] and [9].
3 We also de¯ne the norm on S as kGk1;2 =
¡
E(G2)
¢1=2 + (E(
R T
0 (DtG)2dt))1=2.
Then D1;2 denotes the Banach space that is the completion of S with respect to the
norm k ¢ k1;2.
2as one of hi, i = 1;:::;n. Also note that an analog of the chain rule in
deterministic calculus exists for derivatives in Malliavin calculus.
3 Analysis
In this section we derive our principal results. We are interested in the Malli-
avin derivative of our process fS(t);0 · t · Tg. Taking the derivative directly
we determine that 4
DtS(¿) = S(¿)¾1ft·¿g; ¿ 2 [0;T]: (4)
Consider the expectation u(x) = Ex[Á(S(T))], where we introduce the con-
vention that Ex[¢] ´ E[¢jS(0) = x]. Take the normal derivative of u(x) with
respect to x then
@xu(x) = Ex[Á
0(S(T))Y (T)]; (5)
and interpret the tangent process at T as @xS(T). From (4) notice that
Y (¿)1ft·¿g = DtS(¿)(x¾)¡1 which is true for any t · ¿ · T, therefore for
all 5 t
Y (¿) =
1
¿
¿ Z
0
DtS(¿)(x¾)
¡1dt: (6)
It follows from (5) using the chain rule and performing an integrating by parts
that
@xu(x)=Ex
2
4Á0(S(T))
x¾T
T Z
0
DtS(T)dt
3
5
=Ex
2
4 1
x¾T
T Z
0
Á
0(S(T))DtS(T)dt
3
5
4 Alternatively, we could arrive at (4) by noting that in general DtS(¿) =
Y (¿)Y ¡1(t)¾(S(t))1ft·¿g for ¾(S(t)) ´ ¾S(t) and by direct inspection S(t) = xY (t).
See [8] for details.
5 We could have multiplied by a function prior to integrating, chosen such that R ¿
0 c(s)ds = 1. However, this generality provides no advantage here, and it su±ces
to take c = 1=¿.
3=Ex
2
4 1
x¾T
T Z
0
DtÁ(S(T))dt
3
5
=Ex
"
Á(S(T))
Ã
W(T)
x¾T
!#
; (7)
which is also achievable by an application of the Bismut-Elworthy formula. We
turn our attention to the second derivative. Di®erentiating (7) with respect to
x and making identical manipulations shows that
@
2
xu(x) = Ex
2
4Á(S(T))
x2¾T
0
@ 1
¾T
T Z
0
W(T)dW(t) ¡ W(T)
1
A
3
5: (8)
But notice that the weight involves the di®erence between two iterated Ito
integrals of the form Ik =
R
[0;T]­k(¾T)¡kdW ­k(t) such that ¼2 = (I2 ¡ I1)=x2.
This leads to the following proposition.
Proposition 1 The n-th weight is ¼n = wn=xn where wn is recursively gener-
ated by wn = m(wn¡1)¡(n¡1)wn¡1 for all n = 2;3;:::, with initial condition
w1 = I1, where the linear function m is de¯ned by m(Ik) = Ik+1.
PROOF. Proceed by induction. The trivial case n = 0 corresponds to no
di®erentiation and is avoided. The recursion is veri¯ed for n = 1;2. Assume
that the recursion is valid for n, then show for n + 1
@
n+1
xn+1u(x)=Ex [@x (Á(FT)¼n)]
=Ex
"
@x
Ã
Á(FT)
xn
!
wn
#
=Ex
"Ã
Á0(FT)YT
xn ¡ n
Á(FT)
xn+1
!
wn
#
;
=Ex
2
4Á(S(T))
xn+1
0
@ 1
¾T
T Z
0
wndW(t) ¡ nwn
1
A
3
5
=Ex
"
Á(S(T))
xn+1 (m(wn) ¡ nwn)
#
=Ex
"
Á(S(T))
xn+1 wn+1
#
;
This completes the proof. 2
4In addition, this shows that we may represent wn =
Pn
k=1 an;kIk with constant
coe±cients an;k. So the problem of solving the recursion reduces to solving
for an;k and Ik for all k · n. We ¯rst turn our attention to the coe±cients.
From Proposition 1 it follows by inspection that an;k must satisfy the recursion
relation
an;k =
8
> <
> :
an¡1;k¡1 ¡ (n ¡ 1)an¡1;k; if 1 · k · n,
0; otherwise,
(9)
We observe that this recursion is solved by the Stirling numbers of ¯rst kind
such that an;k = s1(n;k) with closed form solution 6
s1(n;k) =
n¡k X
i=0
n¡k X
j=i
(¡1)
i+j
µ
j
i
¶ µ
n ¡ 1 + j
n ¡ k + j
¶ µ
2n ¡ k
n ¡ k ¡ j
¶ (j ¡ i)n¡k+j
j!
;
for n = 1;2;::: and k · n, as noted in [2] (Appendix 4) due to [3]. Further,
we see that iterated Ito integrals of tensor power can be solved by a formula
noted in [9] (Section 1) due to [7]. For our case we have
Ik = (¾
p
T)
¡khk
Ã
W(T)
p
T
!
; hk(y) =
[k=2] X
i=0
yk¡2i
(k ¡ 2i)!
k!
i!(¡2)i; (10)
for k = 0;1;2;:::, where hk are the (modi¯ed) Hermite polynomials, and [¢]
means the integer part of. Together these results imply the following lemma.
Lemma 2 The n-th weighting function for the n-th derivative of u(x) with
respect to x is
¼n = x
¡n
n X
k=1
s1(n;k)(¾
p
T)
¡khk
Ã
W(T)
p
T
!
: (11)
PROOF. The proof follows by direct substitution. 2
4 Concluding Remarks
We remark that these weights are not complicated - both Stirling numbers and
Hermite polynomials are e®ortlessly evaluated in packages like Mathematica.
6 Alternatively, coe±cients can be expressed as an;k = (¡1)n¡k'n¡k(1;:::;n ¡ 1),
where 'n¡k(1;:::;n¡1) denotes the (n¡k)-th symmetric polynomial on 1;:::;n¡1.
We exclude the proof in sake of brevity.
5Also, these weights are not unique. Indeed, there exists a spectrum of weights
which corresponds to di®erent choices of c in (6). However, these weights are
optimal in the sense that they are the ones that minimize the variance of the
resulting expectation. To see this note that (11) proves that the weights are
always a polynomial function of W(T). The strong solution (1) allows us to
express W(T), and thus the weights, in terms of S(T). This means that the
weights are S(T) measurable, and hence optimal, see [5], [1].
By virtue of this representation we can evaluate these expectations e±ciently
and exactly, without the need of computationally expensive simulations. This
is important in \real world" applications. We observe that since we can write
the weights in terms of S(T), the expectation becomes an integral over S(T) 2
[0;1) with respect to the known density. These integrals are easily solved.
We can represent derivatives with respect to the drift and di®usion coe±cients
in terms of the derivatives with respect to the initial condition, see [2]. Ex-
plicitly, we have that @¹u(x) = xT@xu(x) and @¾u(x) = x2¾T@2
xu(x); and we
can derive higher derivatives in an analogous way to the analysis presented.
Also note that, from our ¯rst remark, these weights will also be optimal.
References
[1] E. Benhamou, Optimal Weighting Function for the Computation of the Greeks,
Math. Fin. 1 (2003) 37 - 53.
[2] P. Carr, Deriving Derivatives of Derivative Securities, J. Comp. Fin. 2 (2000)
5 - 29.
[3] M. Comtet, Advanced Combinatorics, (Reidel, Dordrecht, 1974).
[4] E. Fournie, J. Laszry, J. Lebuchoux, P. Lions, and N. Touzi, Applications of
Malliavin Calculus to Monte Carlo Methods in Finance, Fin. Stoch. 3 (1999)
391 - 412.
[5] E. Fournie, J. Laszry, J. Lebuchoux, and P. Lions, Applications of Malliavin
Calculus to Monte Carlo Methods in Finance 2, Fin. Stoch. 5 (2001) 201 - 236.
[6] J. Hull, Options, Futures, and Other Derivatives, (Prentice Hall, 5th ed., 2003).
[7] K. Ito, Multiple Wiener Integral, J. Math. Soc. Japan 3 (1951) 157 - 169.
[8] D. Nualart, Malliavin Calculus and Related Topics, (Springer, 1995).
[9] B. Âksendal, An Introduction to Malliavin Calculus with Applications to
Economics, Norway, University of Oslo (1997).
6

Advanced Combinatorics,

An Introduction to Malliavin Calculus with Applications to

Applications of Malliavin Calculus to Monte Carlo Methods in Finance,

Deriving Derivatives of Derivative Securities,

Malliavin Calculus and Related Topics,

Multiple Wiener Integral,

Optimal Weighting Function for the Computation of the Greeks,

On Higher Derivatives of Expectations

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On Higher Derivatives of Expectations

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