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AN INFINITE DIMENSIONAL STOCHASTIC ANALYSIS
APPROACH TO LOCAL VOLATILITY DYNAMIC MODELS
RENÉ CARMONA* AND SERGEY NADTOCHIY
Abstract. The difficult problem of the characterization of arbitrage free
dynamic stochastic models for the equity markets was recently given a new
life by the introduction of market models based on the dynamics of the local
volatility. Typically, market models are based on Itô stochastic differential
equations modeling the dynamics of a set of basic instruments including, but
not limited to, the option underliers. These market models are usually recast
in the framework of the HJM philosophy originally articulated for Treasury
bond markets. In this paper we streamline some of the recent results on
the local volatility dynamics by employing an infinite dimensional stochastic
analysis approach as advocated by the pioneering work of L. Gross and his
students.
1. Introduction and Notation
The difficult problem of the characterization of arbitrage free dynamic stochastic
models for the equity markets was recently given a new life in [3] by the introduction of market models based on the dynamics of the local volatility surface.
Market models are typically based on the dynamics of a set of basic instruments
including, but not limited to, the option underliers. These dynamics are usually
given by a continuum of Itô’s stochastic differential equations, and the first order
of business is to check that such a large set of degrees of freedom in the model
specification does not introduce arbitrage opportunities which would render the
model practically unacceptable.
Market models originated in the groundbreaking original work of Heath, Jarrow
and Morton [11] in the case of Treasury bond markets. These authors modeled the
dynamics of the instantaneous forward interest rates and derived a no-arbitrage
condition in the form of a drift condition. This approach was extended to other
fixed income markets and more recently to credit markets. The reader interested
in the HJM approach to market models is referred to the recent review article
[2]. However, despite the fact that they were the object of the first success of
the mathematical theory of option pricing, the equity markets have offered the
strongest resistance to the characterization of no-arbitrage in dynamic models.
This state of affair is due to the desire to accommodate the common practice of
using the Black-Scholes implied volatility to code the information contained in
the prices of derivative instruments. Indeed while defining stochastic dynamics
2000 Mathematics Subject Classification. Primary 91B24.
Key words and phrases. Local Volatility, Market models, Abstract Wiener space.
* Research partially supported by NSF.
1
2
RENÉ CARMONA* AND SERGEY NADTOCHIY
for the implied volatility surface is rather natural, see also fr example [5, 6, 9],
deriving no-arbitrage conditions is highly technical and could be only done in
specific particular cases [8, 14, 15, 16].
In the present paper, we streamline some of the recent results on local volatility
dynamics by employing an infinite dimensional stochastic analysis approach as
advocated by the pioneering work of L. Gross and his students.
One of the main technical results of [3] is the semi-martingale property of call
option prices corresponding to a local volatility surface which evolves over time
according to a set of Itô’s stochastic differential equations. We denote by Ct (T, K)
the price at time t of a European call option with maturity T ≥ t and strike K > 0.
For each fixed t > 0 we have
2
Ct (T, K),
t<T
∂T Ct (T, K) = 12 a2t (T, K)K 2 ∂K
(1.1)
+
Ct (t, K) = (St − K)
To be more specific, if for each maturity T > 0 and strike K > 0, we have
dat (T, K) = αt (T, K)dt + βt (T, K) · dWt ,
(1.2)
the result we revisit here says that the solution of the Dupire PDE (1.1) is a semimartingale whenever the second order term coefficient a2t (T, K) has for each T > 0
and K > 0, a stochastic Itô’s differential of the form (1.2).
The goal of this paper is to simplify the proof of this result, while at the same
time extending it to the case of infinitely many driving Wiener processes Wt .
Our new proof uses the general framework of infinite dimensional analysis. It
streamlines the main argument and gets rid of a good number of technical lemmas
proved in [3].
The theoretical results from functional analysis and infinite dimensional stochastic analysis which are needed in this paper can be found in Kuo’s original
Lecture Notes in Mathematics [13], and in the more recent book by Carmona
and Tehranchi [4]. Already, this book was dedicated to Leonard Gross for his
groundbreaking work on abstract Wiener spaces and the depth of his contribution to infinite dimensional stochastic analysis. Contributing the present paper
to a volume in the honor of his 70th birthday is a modest way to show our deep
gratitude.
2. Solutions of the Pricing Equations
As explained in the introduction, we denote by Ct (T, K) the price at time t
of a European Call option with trike K and maturity T . It is a random variable
measurable with respect to the σ-field Ft of the natural filtration of a Wiener
process W = {Wt }t . Throughout the paper, we use the notation τ = T − t for
the time to maturity, and we find it convenient to use the notation x for the
log-moneyness x = log(K/S).
2.1. Pricing PDEs. We will find convenient to use the notation
C̃t (τ, x) :=
1
Ct (t + τ, St ex ),
St
τ > 0, x ∈ R.
LOCAL VOLATILITY DYNAMIC MODELS
3
for call prices and
1 2
1 2
∂ 2 − ∂x , Dx∗ :=
∂ 2 + ∂x
2 x
2 x
for partial differential operators which we use throughout the paper. Then, if we
consider the local volatility a2t (T, K) as given, and introduce the notation
Dx :=
ã2t (τ, x) = a2t (t + τ, St ex ),
then we can conclude that the call price C̃t (., .) satisfies the following initial-value
problem
∂τ w = ã2t (τ, x)Dx w(τ, x)
w(0, x) = (1 − ex )+ .
(2.1)
We will introduce more notation later in the paper, but for the time being we
denote by p(ã2 ; τ, x; u, y), with τ > u, the fundamental solution of the forward
partial differential equation (PDE for short) in (2.1) with coefficient ã2 . Similarly,
we introduce q(ã2 ; u, y; τ, x), with u < τ , the fundamental solution of the backward
equation
∂u w = −ã2 (u, y)Dy w(u, y),
(2.2)
which is, in a sense, dual to (2.1). We will sometimes drop the argument ã2 of
the fundamental solutions p and q, when the coefficient ã2 is assumed to stay the
same. Notice that, if w is the solution of (2.1), we have
1 x
e q(0, 0; τ, x).
2
This equality will be used later in the paper.
Dx w(τ, x) =
(2.3)
2.2. Fréchet Differentiability. For each fixed τ̄ > 0 and integers k, m ≥ 1, and
for any smooth function (τ, x) ,→ f (τ, x) defined in the strip S = [0, τ̄ ] × R, we
define the norm


k
m X
i
X
j
∂ i f (τ, x) +
kf kC k,m (S) = sup 
∂xj f (τ, x) .
τ
(τ,x)∈S
i=0
j=1
Next we denote by B̃ the space of functions f on S which are continuously differentiable in the first argument and five times continuously differentiable in the
second argument, and for which the norm k.kC 1,5 (S) is finite. We subsequently
denote k.kB̃ := k.kC 1,5 (S) .
Now we fix ε̄ > 0, and we define the strip Sε̄ by Sε̄ = [ε̄, τ̄ ] × R. We then define
W̃ε̄ = C 1,2 (Sε̄ ) and the mapping
Fε̄ : B̃ ,→ W̃ε̄ ,
where, for any h ∈ B̃, the image Fε̄ (h) is the restriction to Sε̄ of the solution
of (2.1) with eh in lieu of the coefficient ã2 . Notice that eh ∈ B̃, and that it is
bounded away from zero, implying that Fε̄ (h) is well defined.
4
RENÉ CARMONA* AND SERGEY NADTOCHIY
We are ready to state and prove the main functional analytic result of the paper.
This result is technical in nature, but it should be viewed as the work horse for
the paper.
Proposition 2.1. The mapping Fε̄ : B̃ ,→ W̃ε̄ defined above is twice continuously
Fréchet differentiable and for any h, h0 , h00 ∈ B̃, we have
Z Z
1 τ
h0 (u, y)eh(u,y)+y p(eh ; τ, x; u, y)q(eh ; 0, 0; u, y)dydu,
Fε̄ 0 (h)[h0 ](τ, x) =
2 0 R
and
Z Z
1 τ
Fε̄ (h)[h , h ](τ, x) =
h0 (u, y)eh(u,y)+y ·
2 0 R
Z τ Z
h
h(v,z) 00
h
p(e ; τ, x; v, z)e
h (v, z)Dz p(e ; v, z; u, y)dzdv q(eh ; 0, 0; u, y)
00
0
u
00
R
− p(eh ; τ, x; u, y) ·
Z u Z
h
h(v,z) 00
h
q(e ; 0, 0; v, z)e
h (v, z)Dz q(e ; v, z; u, y)dzdv dydu
0
R
Proof. Our proof is based on a systematic use of uniform estimates on the fundamental solutions of the parabolic equations (2.1) and (2.2), and their derivatives.
These estimates are known as Gaussian estimates. Typically, they hold when the
second order coefficients are uniformly bounded together with a certain number
of its derivatives. As a preamble to the technical details of the proof, we first
state the Gaussian estimates on the fundamental solutions that we will use in this
paper.
If Γ denotes the fundamental solution of (2.1) or (2.2), then the following estimates hold
C
(x − y)2
m k
exp −c
,
(2.4)
∂xm ∂yk Γ(τ, x; u, y) ≤
|τ − u|
|τ − u|(1+m+k)/2
and consequently
i k
∂xi ∂yk Γ(τ, x + y; u, y) ≤
C
x2
exp −c
,
|τ − u|
|τ − u|(1+i)/2
(2.5)
for 0 ≤ k + m ≤ 4, i = 0, 1, τ 6= u ∈ [0, τ̄ ] and x, y ∈ R. Here, the constants c and
C depend only upon the lower bound of ã2 (τ, x) and the norm kã2 kC 1,5 (S) , where
ã2 is the coefficient in the PDEs (2.1) and (2.2).
Inequality (2.4) is derived on pp. 251-261 of [10]. The comments on the dependence of constants c and C on ã2 are given in [12].
0
0
Let us fix h ∈ B̃. Estimate (2.4) holds for p(eh+h ; τ, x; u, y) and q(eh+h ; τ, x; u, y),
uniformly over h0 varying in a neighborhood of zero, say U (0) ⊂ B̃. In the following
we consider only h0 ∈ U (0).
We now extend the properties of the fundamental solutions to a larger class of
functions. For each integer s ≥ 0 we introduce the space G̃ s :
LOCAL VOLATILITY DYNAMIC MODELS
5
Definition 2.2. We say that a family of functions Γ = {Γ(λ; ., .; ., .)}λ∈Λ belongs
to G̃ s (Λ) if, for each λ ∈ Λ, the function Γ(λ; τ, x; u, y) is defined for all 0 ≤ u <
τ ≤ τ̄ , x, y ∈ R, and:
(1) Γ is s times differentiable in (x, y), and its derivatives are jointly continuous
in (τ, x; u, y), moreover, Γ satisfies estimates (2.4), for 0 ≤ k + m ≤ s,
uniformly over λ ∈ Λ;
(2) for any g ∈ C01 (R) and all λ ∈ Λ,
Z
lim
Γ(λ; τ, x; u, y)g(y)dy = ci g(x),
u→τ
R
where the ci ’s are real constants which depend only on Γ.
We will need another class of functions:
Definition 2.3. The family of function Γ is said to belong to class G s (Λ), for
some integer s ≥ 0, if it belongs to G̃ s (Λ), and, in addition, satisfies the following:
if s ≥ 2, then Γ is continuously differentiable in τ , and, for all λ ∈ Λ,
∂τ Γ(λ; τ, x; u, y) =
2
X
fi (λ; τ, x)∂xi i Γ̃i (λ; τ, x; u, y),
i=0
where each Γ̃i ∈ G̃ s , and each kfi (λ; ., .)kC 1,s−2 (S) is bounded over λ ∈ Λ.
For the most part of this proof we assume that the functions are parameterized
by the set Λ = U (0) ⊂ B̃, and therefore drop the argument Λ of the class G s .
Notice that the families of fundamental solutions
n
o
0
p(eh+h ; ., .; ., .)
h0 ∈U (0)
,
and
n
o
0
q(eh+h ; ., .; ., .)
h0 ∈U (0)
belong to G 4 .
We now derive some important properties of the classes of functions introduced above. Let us consider Γ1 , Γ2 ∈ G̃ s with s ≥ 2, let us fix integers i, k, j, m
satisfying 0 ≤ i + k + j + m ≤ s + 1 and (i + k) ∨ (j + m) ≤ s, and let
+
+
f ∈ C 1,(i+k−1) ∨(j+m−1) (S). Then, for all λ1 , λ2 ∈ Λ, x1 , x2 ∈ R, and 0 ≤
6
RENÉ CARMONA* AND SERGEY NADTOCHIY
τ1 < τ2 ≤ τ̄ , we have:
Z τ2 Z
∂ i+k
∂ j+m
Γ
(λ
;
τ
,
x
;
u,
y)f
(u,
y)
Γ
(λ
;
u,
y;
τ
,
x
)dy
du
2
1 1
1
1
i ∂y k 2 2 2
j
m
∂x
∂y
∂x
R
τ1
2
1
τ
+τ
Z 1 2 2 Z
j
∂ m∧1
∂
∂
∂
=
Γ1 (λ1 ; u, y; τ1 , x1 )·
+
−
R ∂y m∧1 ∂x1
∂y ∂y
τ1
∂ m−m∧1
∂ i+k
f (u, y) i k Γ2 (λ2 ; τ2 , x2 ; u, y) dy du
m−m∧1
∂y
∂x2 ∂y
i
Z τ2 Z
k∧1
∂
∂
∂
∂
+
Γ2 (λ2 ; τ2 , x2 ; u, y)·
+
−
k∧1
τ1 +τ2 ∂x2
∂y ∂y
R ∂y
2
"
# ∂ k−k∧1
∂ j+m
f
(u,
y)
dy
Γ
(λ
;
u,
y;
τ
,
x
)
du
1
1
1
1
∂y k−k∧1
∂xj1 ∂y m
Z τ1 +τ
2 Z
2
1
y2
≤ c1 kf k
exp −c2
·
(2.6)
1+(j+m)∧1
u − τ1
2
R (u − τ1 )
τ1
(x2 − x1 − y)2
1
exp
−c
dydu
2
1+i+k+(j+m−1)+
τ2 − u
2
(τ2 − τ1 )
Z τ2 Z
1
y2
exp
−c
·
+ c3 kf k
2
1+(i+k)∧1
τ1 +τ2
τ2 − u
2
R (τ2 − u)
2
1
(x1 − x2 − y)2
exp
−c
dydu
2
1+j+m+(i+k−1)+
u − τ1
2
(τ2 − τ1 )
1−i−j−k−m
(x2 − x1 )2
2
exp −c2
.
= c4 kf k(τ2 − τ1 )
τ 2 − τ1
To derive the above inequality, we integrated by parts in y and applied the
Gaussian estimates (2.4), (??), then used Lemma 3 on p.15 of [10] to compute
integrals of the form
Z τ2 Z
(x2 − y)2
1
(y − x1 )2
exp
−c
exp
−c
dydu.
α
β
τ2 − u
u − τ1
τ1
R (τ2 − u) (u − τ1 )
s
Now, fix some s ≥ 2,
choose some Γ1 , Γ2 ∈ G̃ and any family of functions
f (λ; ., .) ∈ C 1,s−1 (S) λ∈Λ , and define
I[Γ2 , f, Γ1 ](λ; τ2 , x2 ; τ1 , x1 )
Z τ2 Z
:=
Γ2 (λ; τ2 , x2 ; u, y)f (λ, u, y)Dy Γ1 (λ; u, y; τ1 , x1 )dydu.
τ1
R
We are going to show that
I[Γ2 , f, Γ1 ](λτ2 , x2 ; τ1 , x1 ) = kf (λ)kΓ3 (λ; τ2 , x2 ; τ1 , x1 ),
for some Γ3 ∈ G̃ s−1 .
If, for some λ ∈ Λ, f (λ) ≡ 0, the statement of the claim is obvious. Therefore
we will assume that kf (λ)k > 0. The smoothness of Γ3 in (x1 , x2 ), and estimate
LOCAL VOLATILITY DYNAMIC MODELS
7
(2.4) follow from (2.5), after we integrate by parts in the definition of I. To
obtain inequality (??), we only need to make a shift of the integration variable
and proceed as in (2.5).
We now verify the second condition of Definition 2.2. Pick some g ∈ C01 (R),
and, assuming that s ≥ 2, proceed as follows
Z
g(x1 )Γ3 (λ; τ2 , x2 ; τ1 , x1 )dx1 R
Z
Z τ2 Z
f (λ; u, y)
Γ2 (λ; τ2 , x2 ; u, y)
= g(x1 )
·
kf (λ)k
R
τ1
R
(∂y + ∂x1 − ∂x1 ) (∂y + ∂x1 − ∂x1 − 1) Γ1 (λ; u, y; τ1 , x1 )dydudx1 |
Z
Z τ2 Z
|f (λ; u, y)|
≤ c6 (|g(x1 )| + |g 0 (x1 )|)
Γ2 (λ; τ2 , x2 ; u, y)
·
(2.7)
kf (λ)k
τ1
R
R
2 X
1 X
i j
(∂y + ∂x1 ) ∂xj Γ1 (λ; u, y; τ1 , x1 ) dydudx1
i=0 j=0
√
1
≤ c7 τ2 − τ1 ,
which goes to zero as τ1 → τ2 . We integrated by parts in x1 , and applied estimates
(2.4), (??) to obtain the above inequality. The interchangeability of integration
and differentiation is justified by (2.5) (just notice that, as it is clear from the
first line of (2.5), the integrals are, sometimes, understood as iterated rather than
double integrals). The above estimate proves that Γ3 satisfies the second condition
in Definition 2.2.
Now, assume that, in addition, Γ1 and Γ2 belong to G s . We claim that, in this
case, Γ3 is in G s−1 . We only need to verify the additional property in the Definition
2.3. Assume s − 1 ≥ 2, then, using the expression for the τ2 - derivatives of Γ2 ,
and the fact that Γ3 ∈ G̃ s−1 , we obtain the following
Z τ2 Z
∂
f (λ; u, y)
Dy Γ1 (λ; u, y; τ1 , x1 )dydu
Γ2 (λ; τ2 , x2 ; u, y)
∂τ2 τ1 R
kf (λ)k
2
X
f (λ; τ2 , x2 )
Dx2 Γ1 (λ; τ2 , x2 ; τ1 , x1 ) +
fi (λ; τ2 , x2 )
kf (λ)k
i=0
Z τ2 Z
f (λ; u, y)
Dy Γ1 (λ; u, y; τ1 , x1 )dydu
∂xi i
Γ̃i (λ; τ2 , x2 ; u, y)
2
kf (λ)k
τ1
R
= c6
where each fi (λ; ., .) is in C 1,s−2 (S), and the Γ̃i ’s belong to G̃ s . The above decomposition completes the proof of the claim: Γ3 ∈ G s−1 .
It is easy to see, integrating by parts, that the operator J defined by
J[Γ2 , f, Γ1 ](λ; τ2 , x2 ; τ1 , x1 )
Z τ2 Z
:=
Dy Γ2 (λ; τ2 , x2 ; u, y)f (λ, u, y)Γ1 (λ; u, y; τ1 , x1 )dydu
τ1
R
has the same properties as I.
8
RENÉ CARMONA* AND SERGEY NADTOCHIY
Similarly, for any f (λ; ., .) ∈ C 1,2 (S) λ∈Λ , and Γ1 , Γ2 ∈ G 2 , we define the
function K[Γ2 , f, Γ1 ] by:
K[Γ2 , f, Γ1 ](λ; τ2 , x2 ; τ1 , x1 )
Z τ2 Z
:=
Γ2 (λ; τ2 , x2 ; u, y)ey−x1 f (λ, u, y)Γ1 (λ; u, y; τ1 , x1 )dydu,
τ1
R
and, using (2.5) and (2.6), we obtain the estimate:
|∂τ2 K| +
2 X
(x2 − x1 )2
j ,
∂xj K ≤ c8 kf (λ)k(τ2 − τ1 )−3/2 exp −c9
τ2 − τ1
2
j=0
(2.8)
where the constants c8 , c9 depend on Γ1 and Γ2 , but not on λ.
We now proceed with the proof of the proposition. Writing the initial value
0
problem (2.1) twice, first with eh , and then with eh+h , and subtracting one from
another, we can, formally, apply the Feynman-Kac formula and obtain
Fε̄ (h + h0 )(τ, x) = Fε̄ (h)(τ, x)
(2.9)
Z Z
0
0
1 τ
+
p(eh+h ; τ, x; u, y)eh(u,y)+y (eh (u,y) − 1)q(eh ; 0, 0; u, y)dydu.
2 0 R
This representation follows from the uniqueness of weak solution of (2.1), see, for
example, [7] for details. Applying the same technique to the fundamental solution
p, we get
0
∆p(τ, x; u, y) := p(eh+h ; τ, x; u, y) − p(eh ; τ, x; u, y)
Z τZ
0
0
=
p(eh+h ; τ, x; v, z)eh(v,z) (eh (v,z) − 1)Dz p(eh ; v, z; u, y)dzdv (2.10)
u
n R
o
n
o
0
0
= I p(eh+h ) 0
, eh (eh − 1) 0
, p(eh ) h0 ∈U (0) (h0 ; τ, x; u, y)
h ∈U (0)
h ∈U (0)
Since all the families of functions considered in this part of the proof are parameterized by h0 ∈ U (0), we use the shorter notation f (h0 ) instead of {f (h0 )}h0 ∈U (0) ,
for the arguments of operator I.
We define ∆q in a similar way. Next we rewrite (2.8) as
Fε̄ (h + h0 ) = Fε̄ (h) + Fε̄ 0 (h)[h0 ] + r1 + r2 ,
with
r1 (τ, x)
=
=
1
2
τ
Z
0
Z
0
0
p(eh+h ; τ, x; u, y)eh(u,y)+y eh (u,y) − 1 − h0 (u, y) ·
R
0
0
1
K p(eh+h ), eh (eh
2
h
q(eh ; 0, 0; u, y)dydu
i
− 1 − h0 ), q(eh ) (h0 ; τ, x; 0, 0),
LOCAL VOLATILITY DYNAMIC MODELS
9
and
r2 (τ, x)
=
=
1
2
Z
0
τ
Z
∆p(τ, x; u, y)eh(u,y)+y h0 (u, y) ·
R
q(eh ; 0, 0; u, y)dydu
h
h
i
i
0
0
1
K I p(eh+h , eh (eh − 1), p(eh ) , eh h0 , q(eh ) (h0 ; τ, x; 0, 0).
2
Because of the
of the operatori I derived earlier, it is easy to see that
h properties
0
h+h0
h h0
the function I p(e
), e (e − 1), p(eh ) belongs to (eh − 1) · G 3 . Therefore,
using estimate (2.7), we have immediately that for i = 1, 2,
kri kW̃ε̄ ≤ c10 kh0 k2B̃
and this implies that Fε̄ is Fréchet differentiable, with Fréchet derivative as given
in the statement of the proposition. The fact that Fε̄ 0 (h)[.] is bounded on the unit
ball of B̃ follows, again, from (2.7).
We now compute the Fréchet derivative of Fε̄ 0 (.) using the same technique as
in the first part of the proof.
We fix h ∈ B̃ and we consider families of functions parameterized by (h0 , h00 ) ∈
Λ := B̃ × U (0). We redefine ∆p, using h00 instead of h0 in (2.9). Then we have
Fε̄ 0 (h + h00 ) − Fε̄ 0 (h) [h0 ](τ, x) =
Z Z
1 τ
h0 (u, y)eh(u,y)+y ∆p(τ, x; u, y)q(eh ; 0, 0; u, y)dydu
2 0 R
Z Z
1 τ
+
h0 (u, y)eh(u,y)+y p(eh ; τ, x; u, y)∆q(0, 0; u, y)dydu
(2.11)
2 0 R
Z Z
1 τ
+
h0 (u, y)eh(u,y)+y ∆p(τ, x; u, y)∆q(0, 0; u, y)dydu
2 0 R
Z Z
00
00
1 τ
+
h0 (u, y)eh(u,y)+y eh (u,y) − 1 p(eh+h ; τ, x; u, y) ·
2 0 R
00
q(eh+h ; 0, 0; u, y)dydu
Next, we decompose the first integral in (2.10)
Z τZ
h0 (u, y)eh(u,y)+y ∆p(τ, x; u, y)q(eh ; 0, 0; u, y)dydu
0
R
= K I p(eh ), h00 eh , p(eh ) , eh h0 , q(eh ) (h0 , h00 ; τ, x; 0, 0)
h h
i
i
00
+ K I p(eh ), eh (eh − 1 − h00 ), p(eh ) , eh h0 , q(eh ) (h0 , h00 ; τ, x; 0, 0)
h h h
i
i
00
00
00
+ K I I p(eh+h ), eh (eh − 1), p(eh ) , eh (eh − 1), p(eh ) ,
eh h0 , q(eh ) (h0 , h00 ; τ, x; 0, 0).
The first term in the right hand side of the above expression is linear in h00 . It is
the first component of Fε̄ 00 . Using the properties of the operator I, we conclude
10
RENÉ CARMONA* AND SERGEY NADTOCHIY
that
h
i
00
I p(eh ), eh (eh − 1 − h00 ), p(eh )
h h
i
i
00
00
00
+ I I p(eh+h ), eh (eh − 1), p(eh ) , eh (eh − 1), p(eh ) = kh0 kB̃ kh00 k2B̃ Γ,
where Γ ∈ G 2 . Therefore, using estimate (2.7), we conclude that the k.kW̃ε̄ norms
of the last two terms in the right hand side of (2.11) are bounded by a constant
times kh0 kB̃ kh00 k2B̃ .
A similar decomposition holds true for the second integral in the right hand side
of (2.10) provided the operator I is replaced by J. Moreover, the k.kW̃ε̄ - norms
of last two integrals in (2.10) are also bounded by a constant times kh0 kB̃ kh00 k2B̃ :
to see this, recall (2.9) and write its analog for ∆q, then apply the properties
operators I and J, and use estimate (2.7). This yields the existence of Fε̄ 00 (h), as
given in the proposition.
To show the continuity of the second derivative, fix any h0 and h00 in B̃ and
consider any ∆h ∈ U (0). We only show the continuity of the first component of
Fε̄ 00 (.)[h0 , h00 ] at h, uniformly over h0 and h00 in a bounded set. The proof for the
second component is the same. We introduce the difference
∆K := K I p(eh+∆h ), h00 eh+∆h , p(eh+∆h ) , eh+∆h h0 , q(eh + ∆h)
− K I p(eh ), h00 eh , p(eh ) , eh h0 , q(eh ) =
K I I p(eh+∆h ), eh (e∆h − 1), p(eh ) , h00 eh+∆h , p(eh+∆h ) , eh+∆h h0 , q(eh + ∆h)
+ K I p(eh ), h00 eh (e∆h − 1), p(eh+∆h ) , eh+∆h h0 , q(eh + ∆h)
+ K I p(eh ), h00 eh , I p(eh+∆h ), eh (e∆h − 1), p(eh ) ) , eh+∆h h0 , q(eh + ∆h)
+ K I p(eh ), h00 eh , p(eh ) , h0 eh (e∆h − 1), q(eh + ∆h)
+ K I p(eh ), h00 eh , p(eh ) , h0 eh , −J q(eh+∆h ), eh (e∆h − 1), q(eh ) .
And, as before, using the properties of I, J and K, we conclude that
k∆KkW̃ε̄ ≤ c11 kh0 kB̃ kh00 kB̃ k∆hkB̃ ,
which completes the proof of the proposition.
Recall that the price Ct (T, x) at time t of an European call option is given
by w(ã2 ; T − t, x + log St ), where w(ã2 ; ., .) is the solution of (2.1). Therefore, in
order to get to the Fréchet differentiability of the price of a call option from the
above result, we will need to compose Fε̄ with another mapping. This justifies the
introduction, for each T ∈ (ε̄, τ̄ ] and x ∈ R of the mapping
δT,x : [0, T − ε̄] × W̃ε̄ × R ,→ R
defined by
δT,x (t, w, y) = w(T − t, x + y).
We have:
LOCAL VOLATILITY DYNAMIC MODELS
11
Proposition 2.4.
(1) For each (w, y) ∈ W̃ε̄ × R, δT,x (., w, y) is continuously
differentiable, and the partial derivative ∂δT,x /∂t is a continuous functional on [0, T − ε̄] × W̃ε̄ × R.
(2) For each t ∈ [0, T − ε̄], δT,x (t, ., .) is twice Fréchet differentiable and for
any w, w0 , w00 ∈ W̃ε̄ and y, y, y 00 ∈ R, its derivatives satisfy
0
δT,x
(t, w, y)[w0 , y 0 ] = w0 (T − t, x + y) + y 0 ∂x w(T − t, x + y)
and
00
δT,x
(t, w, y)[(w0 , y 0 ), (w00 , y 00 )] = y 00 ∂x w0 (T − t, x + y) + y 0 ∂x w00 (T − t, x + y)
+ y 0 y 00 ∂x22 w(T − t, x + y).
0
00
Moreover, δT,x
and δT,x
are continuous operators from [0, T − ε̄] × W̃ε̄ × R
∗
into W̃ε̄ × R and L W̃ε̄ × R, W̃ε̄∗ × R respectively.
Proof. Let us fix (w, y) ∈ W̃ε̄ × R. Then, for any t ∈ [0, T − ε̄], we have
∂
δT,x (t, w, y) = −∂τ w(T − t, x + y).
∂t
We first show that this functional is continuous in (t, w, y) ∈ [0, T − ε̄] × W̃ε̄ × R.
Consider any (t0 , w0 , y 0 ) ∈ [0, T − ε̄] × W̃ε̄ × R, then
|∂τ w(T − t, x + y) − ∂τ w0 (T − t0 , x + y 0 )| =
|∂τ w(T − t, x + y) − ∂τ w(T − t0 , x + y 0 )|
(2.12)
+ |∂τ w(T − t0 , x + y 0 ) − ∂τ w0 (T − t0 , x + y 0 )|
The first difference in the right hand side above can be made as small as we want
by choosing (t, x) and (t0 , x0 ) close enough. The second difference is bounded by
kw − w0 kW̃ε̄ . This implies continuity of the partial derivative ∂δT,x /∂t, proving
the first statement of the proposition.
Let us now compute the derivatives of δT,x . We will keep (t, w, y) ∈ [0, T − ε̄] ×
W̃ε̄ ×R fixed, and consider (w0 , y 0 ) ∈ U (0) ⊂ W̃ε̄ ×R, where U (0) is a neighborhood
of zero. Notice that
δT,x (t, w + w0 , y + y 0 ) − δT,x (t, w, y) =
w(T − t, x + y + y 0 ) − w(T − t, x + y) + w0 (T − t, x + y + y 0 ) =
y 0 ∂x w(T − t, x + y) + ō¯(y 0 ) + w0 (T − t, x + y) + y 0 ∂x w0 (T − t, x + y + ξy 0 ),
for some ξ ∈ [0, 1], and that
|ō¯(y 0 ) + y 0 ∂x w0 (T − t, x + y + ξy 0 )| = ō¯
q
|y 0 |2 + kw0 k2W̃ ,
ε̄
Therefore, we have obtained the expression for C̃0T,x , as given in the proposition.
12
RENÉ CARMONA* AND SERGEY NADTOCHIY
Now consider (w0 , y 0 ) ∈ W̃ε̄ × R and (w00 , y 00 ) ∈ U (0) ⊂ W̃ε̄ × R, the rest of
parameters being fixed. Then:
0
0
δT,x
(t, w + w00 , y + y 00 ) − δT,x
(t, w, y) [w0 , y 0 ] =
w0 (T − t, x + y + y 00 ) − w0 (T − t, x + y) + y 0 ∂x w00 (T − t, x + y + y 00 )
+ y 0 ∂x w(T − t, x + y + y 00 ) − y 0 ∂x w(T − t, x + y) =
y 00 ∂x w0 (T − t, x + y) + (y 00 )2 ∂x22 w0 (T − t, x + y + ξy 00 ) + y 0 ∂x w00 (T − t, x + y)
+ y 0 y 00 ∂x22 w00 (T − t, x + y + ξ 0 y 00 ) + y 0 y 00 ∂x22 w(T − t, x + y) + ō¯(y 00 ) ,
for some ξ, ξ 0 ∈ [0, 1]. Again, noticing that
00 2 2 0
(y ) ∂ 2 w (T − t, x + y + ξy 00 ) + y 0 y 00 ∂ 22 w00 (T − t, x + y + ξ 0 y 00 ) + y 0 ō¯(y 00 )
x
x
q
q
≤ |y 0 |2 + kw0 k2W̃ ō¯
|y 00 |2 + kw00 k2W̃ .
ε̄
we get the desired expression for
ε̄
00
.
δT,x
00
0
, we fix (w0 , y 0 ), (w00 , y 00 ) ∈ W̃ε̄ ×R,
and δt,x
In order to show the continuity of δT,x
0
0
00
0
(., ., .)[(w0 , h0 ), (w00 , h00 )]
and we prove the continuity of δT,x (., ., .)[w , h ] and δT,x
by, essentially, repeating the argument of (??). Finally, notice that the continuity
is uniform over (w0 , y 0 ), (w00 , y 00 ) when they are restricted to a bounded set.
Now, consider the composition of the two operators introduced above. For each
T ∈ (ε̄, τ̄ ] and x ∈ R, we have
CT,x : [0, T − ε̄] × B̃ × R → R
CT,x (t, h, y) = δT,x (t, Fε̄ (h), y)
As a composition of twice Fréchet differentiable operators, CT,x (t, ., .) is, clearly
twice Fréchet differentiable, for each t ∈ [0, T − ε̄]. Due to the continuity of Fε̄ 00 (.),
00
0
(., ., .), the Fréchet derivatives of CT,x (t, h, y) are also continuous
(., ., .) and δT,x
δT,x
in (t, h, y). Finally, CT,x , clearly, satisfies the first statement of Proposition 2.4.
Thus, applying the chain rule we obtain the following
Proposition 2.5. For each t ∈ [0, T − ε̄], functional CT,x (t, ., .) is twice Fréchet
differentiable, such that, for any h, h0 , h00 ∈ B̃ and y, y, y 00 ∈ R, we have
C0T,x (t, h, y)[h0 , y 0 ] = Fε̄ 0 (h)[h0 ](T − t, x + y) + y 0 ∂x Fε̄ (h)(T − t, x + y),
and
C00T,x (t, h, y)[(h0 , y 0 ), (h00 , y 00 )] = Fε̄ 00 (h)[h0 , h00 ](T − t, x + y)
+ y 00 ∂x Fε̄ 0 (h)[h0 ](T − t, x + y) + y 0 ∂x Fε̄ 0 (h)[h00 ](T − t, x + y)
+ y 0 y 00 ∂x22 Fε̄ (h)(T − t, x + y),
and C0T,x , C00T,x are continuous operators from [0, T − ε̄] × B̃ × R into B̃ ∗ × R and
L B̃ × R, B̃ ∗ × R respectively.
LOCAL VOLATILITY DYNAMIC MODELS
13
3. Using Itô’s Formula in Infinite Dimension
The purpose of this section is to extend the proof of the semi-martingale property given in [3] to the case of infinitely many driving Wiener processes.
We denote by B the cylindrical Brownian motion constructed on the canonical
cylindrical Gaussian measure of some separable Hilbert space H̃. The reader can
think of H̃ = l2 - the space of square - summable sequences but the specific form
of this Hilbert space is totally irrelevant for what we are about to do.
The first step is to construct a Hilbert subspace of B̃. For each functions f and
g with enough derivatives square integrables and for each non-negative integers k
and m, we define the scalar product
< f, g >W̃k,m (S) =
k
X
∂τi i f (0, 0)∂τi i g(0, 0) +
i=0
Z
+
S
m
X
∂xj j f (0, 0)∂xj j g(0, 0)
j=0
5 ∂τkk f (τ, x) 5 ∂τkk g(τ, x) + 5 (∂xmm f (τ, x)) 5 (∂xmm g(τ, x)) dxdτ.
Now we fix a compact set K contained in S and containing the origin (0, 0), and
we consider the space of functions on S which are constant outside K, namely
whose derivatives vanish outside K. For the sake of definiteness we will choose
K = [0, τ ] × [−M, M ] for a positive (large) number M . Equipped with the scalar
product < . , . >W̃1,5 (S) , defined above, this space of functions (more precisely of
equivalence classes of functions) is a Hilbert space which we denote H. It is clearly
contained in B̃. Define by B, the completion of H in the k.kC 1,5 (S) norm. Thus,
the pair (H, B) forms a conditional Banach Space.
Clearly, B is a subspace of B̃, and therefore, Proposition 2.5 holds for the
restriction of CT,x to B as well.
For any given real separable Banach space G we denote by L (G) the space of all
non-anticipative random processes in G (measurable mappings X : Ω×[0, ∞) → G)
, such that
Z t
E
kXu k2G du < ∞,
0
for all t ≥ 0. Where G is a Banach space. Also, we denote by L2 (H) the space of
all Hilbert-Schmidt operators on H.
Next, we choose α ∈ L (B) and β ∈ L L2 H̃, H , and we model dynamics
of ht , the logarithm of the squared local volatility at time t, ã2t , by the infinite
dimensional Itô’s stochastic differential
dht = αt dt + βt dBt ,
which together with an initial condition h0 ∈ B, defines a random process in B.
Also, we assume the following dynamics for the logarithm of the underlying
1
d log St = − σt2 dt + σt < e1 , dBt >, log S0 ,
2
(3.1)
14
RENÉ CARMONA* AND SERGEY NADTOCHIY
where σ is R - valued random process with E
t ≥ 0, and e1 ∈ H̃ is a fixed unit vector.
Rt
0
σu2 du < ∞ almost surely, for any
Now, thanks to Proposition 2.5, we can apply Itô’s formula (see, for example,
[13], p. 200) to (CT,x (t, ht , log St ))t∈[0,T −ε̄] . We get that for any T ∈ (ε̄, τ̄ ] and
x ∈ R, we have, almost surely, for all t ∈ [0, T − ε̄],
CT,x (t, ht , log St ) = CT,x (0, h0 , log S0 )
Z t
∂ 0
1
+
CT,x (u, hu , log Su ) + C0T,x (u, hu , log Su )[αu , − σu2 ]
∂t
2
0
1
∗
00
+ Tr (βu , σu e1 ) ◦ CT,x (u, hu , log Su ) ◦ (βu , σu e1 ) du
2
Z t
+
C0T,x (u, hu , log Su ) ◦ (βu , σu e1 )dBu
0
where C0T,x and C00T,x are given in Proposition 2.5.
Remark 3.1. Since ε̄ can be made as small as we want, the above representation
holds for any T ∈ (0, τ̄ ], and all t ∈ [0, T ). Then, since we choose τ̄ as large as we
want, the above representation holds for any T > 0, and all t ∈ [0, T ).
We now restate the above result after choosing a complete orthonormal basis
{en }n of H̃. Notice that without any loss of generality we can assume that the
first element e1 of this basis is in fact the unit vector entering the equation for
the dynamics (3.1) of the logarithm of the underlying spot price. As it should be
clear, fixing a basis is essentially assuming that H̃ = l2 . If we consider that βt
∞
is given by the sequence {βtn (., .) ∈ H}n=1 of its components on the basis vectors
then we have:
Then, we have
Theorem 3.2. For any T > 0 and x ∈ R, we have, almost surely, for all t ∈ [0, T ),
CT,x (t, ht , log St ) = CT,x (0, h0 , log S0 )
Z t
1
1
+
Fε̄ 0 (hu )[αu ] − σu2 ∂x Fε̄ (hu ) − ∂τ Fε̄ (hu ) + σu2 ∂x22 Fε̄ (hu )
2
2
0
#
∞
1 X 00
+σu ∂x Fε̄ 0 (hu )[βu1 ] +
Fε̄ (hu )[βun , βun ] (T − u, x + log St )du
2 n=1
#
Z t "X
∞
0
n
+
Fε̄ (hu )[βu ] + σu ∂x Fε̄ (hu ) (T − u, x + log Su )dBun ,
0
n=1
if we use the notation {B n }n for the sequence of independent standard one - dimensional Brownian motions Btn =< en , Bt >. Fε̄ 0 and Fε̄ 00 are given in Proposition
2.1.
This is the infinite dimensional version of the semi-martingale result of [3].
LOCAL VOLATILITY DYNAMIC MODELS
15
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René Carmona: Bendheim Center for Finance, ORFE, Princeton University, Princeton, NJ 08544, USA
E-mail address: [email protected]
URL: http://www.princeton.edu/∼rcarmona
Sergey Nadtochiy: Department ORFE, Princeton University, Princeton, NJ 08544,
USA
E-mail address: [email protected]
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