An infinite dimensional stochastic analysis approach to local volatility dynamic models

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.

Full text

AN INFINITE DIMENSIONAL STOCHASTIC ANALYSIS
APPROACH TO LOCAL VOLATILITY DYNAMIC MODELS
R. CARMONA AND S. 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 Itˆo 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 [2] by the intro-
duction 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 Itˆo’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
[1]. 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 affairs 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 for the
2000 Mathematics Subject Classification. Primary 91B24.
Key words and phrases. Volatility, market models, abstract Wiener space.
109
Serials Publications
www.serialspublications.com
Communications on Stochastic Analysis
Vol. 2, No. 1 (2008) 109-123

110 R. CARMONA AND S. NADTOCHIY
implied volatility surface is rather natural (see, for example [4, 5, 8]), deriving no-
arbitrage conditions is highly technical and could be only done in specific particular
cases [7, 14, 16, 15].
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 [2] is the semi-martingale property of call
option prices corresponding to a local volatility surface which evolves over time
according to a set of Itˆo’s stochastic differential equations. We denote by Ct(T, K )
the price at time tof a European call option with maturity T≥tand strike K > 0.
For each fixed t > 0 we have
½∂TCt(T, K) = 1
2a2
t(T, K)K2∂2
KCt(T, K), t < T
Ct(t, K) = (St−K)+(1.1)
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 semi-
martingale whenever the second order term coefficient a2
t(T, K) has for each T > 0
and K > 0, a stochastic Itˆo’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 [2]. The theoretical results from functional analysis and infinite dimen-
sional 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 Car-
mona and Tehranchi [3]. Already, this book was dedicated to Leonard Gross for
his groundbreaking work on abstract Wiener spaces and the depth of his contri-
bution 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 Kand maturity T. It is a random variable
measurable with respect to the σ-field Ftof the natural filtration of a Wiener
process W={Wt}t. Throughout the paper, we use the notation τ=T−tfor
the time to maturity, and we find it convenient to use the notation xfor the
log-moneyness x= log(K/S).
2.1. Pricing PDEs. We will find convenient to use the notation
˜
Ct(τ, x) := 1
St
Ct(t+τ, Stex), τ > 0, x ∈R.
for call prices and
Dx:= 1
2¡∂2
x2−∂x¢, D∗
x:= 1
2¡∂2
x2+∂x¢

LOCAL VOLATILITY DYNAMIC MODELS 111
for partial differential operators which we use throughout the paper. Then, if we
consider the local volatility a2
t(T, K) as given, and introduce the notation
˜a2
t(τ, x) = a2
t(t+τ, Stex),
then we can conclude that the call price ˜
Ct(., .) satisfies the following initial-value
problem
½∂τw= ˜a2
t(τ, x)Dxw(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(˜a2;τ, x;u, y), with τ > u, the fundamental solution of the forward
partial differential equation (PDE for short) in (2.1) with coefficient ˜a2. Similarly,
we introduce q(˜a2;u, y;τ, x), with u<τ, the fundamental solution of the backward
equation
∂uw=−˜a2(u, y)Dyw(u, y),(2.2)
which is, in a sense, dual to (2.1). We will sometimes drop the argument ˜a2of
the fundamental solutions pand q, when the coefficient ˜a2is assumed to stay the
same. Notice that, if wis the solution of (2.1), we have
Dxw(τ, x) = 1
2exq(0,0; τ, x).(2.3)
This equality will be used later in the paper.
2.2. Fr´echet Differentiability. For each fixed ¯τ > 0 and integers k, m ≥1, and
for any smooth function (τ, x)→f(τ, x) defined in the strip S= [0,¯τ]×R, we
define the norm
kfkCk,m(S)= sup
(τ,x)∈S 

k
X
i=0 ¯¯∂i
τif(τ, x)¯¯+
m
X
j=1 ¯¯¯∂j
xjf(τ, x)¯¯¯
.
Next we denote by ˜
Bthe space of functions fon Swhich are continuously dif-
ferentiable in the first argument and five times continuously differentiable in the
second argument, and for which the norm k.kC1,5(S)is finite. We subsequently
denote k.k˜
B:= k.kC1,5(S).
Now we fix ¯ε > 0, and we define the strip S¯εby S¯ε= [¯ε, ¯τ]×R. We then define
˜
W¯ε=C1,2(S¯ε) and the mapping
F¯ε:˜
B→˜
W¯ε,
where, for any h∈˜
B, the image F¯ε(h) is the restriction to S¯εof the solution
of (2.1) with ehin lieu of the coefficient ˜a2. Notice that eh∈˜
B, and that it is
bounded away from zero, implying that F¯ε(h) is well defined.
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.

112 R. CARMONA AND S. NADTOCHIY
Proposition 2.1. The mapping F¯ε:˜
B→˜
W¯εdefined above is twice continuously
Fr´echet differentiable and for any h, h0, h00 ∈˜
B, we have
F¯ε0(h)[h0](τ , x) = 1
2Zτ
0ZR
h0(u, y)eh(u,y)+yp(eh;τ , x;u, y)q(eh; 0,0; u, y)dydu,
and
F¯ε00 (h)[h0, h00](τ , x) = 1
2Zτ
0ZR
h0(u, y)eh(u,y)+y·
·µZτ
uZR
p(eh;τ, x;v , z)eh(v,z)h00(v , z)Dzp(eh;v, z;u, y )dzdv¶q(eh; 0,0; u, y)
−p(eh;τ, x;u, y )µZu
0ZR
q(eh; 0,0; v, z)eh(v,z)h00 (v, z )Dzq(eh;v, z;u, y)dzdv¶¸dydu
Proof. Our proof is based on a systematic use of uniform estimates on the funda-
mental 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
estimate holds
¯¯¯∂m
xm∂k
ykΓ(τ, x;u, y )¯¯¯≤C
|τ−u|(1+m+k)/2exp µ−c(x−y)2
|τ−u|¶,(2.4)
and consequently
¯¯¯∂i
xi∂k
ykΓ(τ, x +y;u, y)¯¯¯≤C
|τ−u|(1+i)/2exp µ−cx2
|τ−u|¶,(2.5)
for 0 ≤k+m≤4, i= 0,1, τ6=u∈[0,¯τ] and x, y ∈R. Here, the constants cand
Cdepend only upon the lower bound of ˜a2(τ , x) and the norm k˜a2kC1,5(S), where
˜a2is the coefficient in the PDEs (2.1) and (2.2).
Inequality (2.4) is derived on pp. 251-261 of [9]. The comments on the depen-
dence of constants cand Con ˜a2are given in [12].
Fix h∈˜
B. Estimate (2.4) holds for p(eh+h0;τ, x;u, y) and q(eh+h0;τ , x;u, y),
uniformly over h0varying in a neighborhood of zero, say U(0) ⊂˜
B. 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 ˜
Gs:
Definition 2.2. We say that a family of functions Γ = {Γ(λ;., .;., .)}λ∈Λbelongs
to ˜
Gs(Λ) if, for each λ∈Λ, the function Γ(λ;τ, x;u, y) is defined for all 0 ≤u <
τ≤¯τ,x, y ∈R, and:
(1) Γ is stimes 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;

LOCAL VOLATILITY DYNAMIC MODELS 113
(2) for any g∈C1
0(R) and all λ∈Λ,
lim
u→τZR
Γ(λ;τ, x;u, y )g(y)dy =cig(x),
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 functions Γ is said to belong to class Gs(Λ), for
some integer s≥0, if it belongs to ˜
Gs(Λ), and, in addition, satisfies the following:
if s≥2, then Γ is continuously differentiable in τ, and, for all λ∈Λ,
∂τΓ(λ;τ, x;u, y ) =
2
X
i=0
fi(λ;τ, x)∂i
xi˜
Γi(λ;τ, x;u, y ),
where each ˜
Γi∈˜
Gs, and each kfi(λ;., .)kC1,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) ⊂˜
B, and therefore drop the argument Λ of the class Gs.
Notice that the families of fundamental solutions
np(eh+h0;., .;., .)oh0∈U(0) and nq(eh+h0;., .;., .)oh0∈U(0)
belong to G4.
We now derive some important properties of the classes of functions introduced
above. Let us consider Γ1,Γ2∈˜
Gswith s≥2, let us fix integers i, k, j, m satisfying
0≤i+k+j+m≤s+ 1,(i+k)∨(j+m)≤s
and let f∈C1,(i+k−1)+∨(j+m−1)+(S). Then, for all λ1, λ2∈Λ, x1, x2∈R, and
0≤τ1< τ2≤¯τ, we have:
Zτ2
τ1¯¯¯¯¯ZR
∂i+k
∂xi
2∂ykΓ2(λ2;τ2, x2;u, y)f(u, y)∂j+m
∂xj
1∂ymΓ1(λ1;u, y;τ1, x1)dy¯¯¯¯¯
du
=Zτ1+τ2
2
τ1¯¯¯¯¯ZR
∂m∧1
∂ym∧1µ∂
∂x1
+∂
∂y −∂
∂y ¶j
Γ1(λ1;u, y;τ1, x1)·
∂m−m∧1
∂ym−m∧1·f(u, y)∂i+k
∂xi
2∂ykΓ2(λ2;τ2, x2;u, y)¸dy¯¯¯¯
du
+Zτ2
τ1+τ2
2¯¯¯¯¯ZR
∂k∧1
∂yk∧1µ∂
∂x2
+∂
∂y −∂
∂y ¶i
Γ2(λ2;τ2, x2;u, y)·
∂k−k∧1
∂yk−k∧1"f(u, y)∂j+m
∂xj
1∂ymΓ1(λ1;u, y;τ1, x1)#dy¯¯¯¯¯
du

114 R. CARMONA AND S. NADTOCHIY
≤c1kfkZτ1+τ2
2
τ1ZR
1
(u−τ1)1+(j+m)∧1
2
exp µ−c2
y2
u−τ1¶·(2.6)
1
(τ2−τ1)1+i+k+(j+m−1)+
2
exp µ−c2
(x2−x1−y)2
τ2−u¶dydu
+c3kfkZτ2
τ1+τ2
2ZR
1
(τ2−u)1+(i+k)∧1
2
exp µ−c2
y2
τ2−u¶·
1
(τ2−τ1)1+j+m+(i+k−1)+
2
exp µ−c2
(x1−x2−y)2
u−τ1¶dydu
=c4kfk(τ2−τ1)1−i−j−k−m
2exp µ−c2
(x2−x1)2
τ2−τ1¶.
To derive the above inequality, we integrated by parts in yand applied the
Gaussian estimates (2.4), (2.5), then used Lemma 3 on p.15 of [9] to compute
integrals of the form
Zτ2
τ1ZR
1
(τ2−u)α(u−τ1)βexp µ−c(x2−y)2
τ2−u¶exp µ−c(y−x1)2
u−τ1¶dydu.
Now, fix some s≥2, choose some Γ1,Γ2∈˜
Gsand any family of functions
©f(λ;., .)∈C1,s−1(S)ªλ∈Λ, and define
I[Γ2, f, Γ1](λ;τ2, x2;τ1, x1)
:= Zτ2
τ1ZR
Γ2(λ;τ2, x2;u, y)f(λ, u, y)DyΓ1(λ;u, y ;τ1, x1)dydu.
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∈˜
Gs−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 Γ3in (x1, x2), and estimate
(2.4) follow from (2.6), after we integrate by parts in the definition of I. To
obtain inequality (2.5), we only need to make a shift of the integration variable
and proceed as in (2.6).
We now verify the second condition of Definition 2.2. Pick some g∈C1
0(R),
and, assuming that s≥2, proceed as follows
¯¯¯¯ZR
g(x1)Γ3(λ;τ2, x2;τ1, x1)dx1¯¯¯¯
=¯¯¯¯ZR
g(x1)Zτ2
τ1ZR
Γ2(λ;τ2, x2;u, y)f(λ;u, y)
kf(λ)k·
(∂y+∂x1−∂x1) (∂y+∂x1−∂x1−1) Γ1(λ;u, y;τ1, x1)dydudx1|

LOCAL VOLATILITY DYNAMIC MODELS 115
≤c6ZR
(|g(x1)|+|g0(x1)|)Zτ2
τ1ZR
Γ2(λ;τ2, x2;u, y)|f(λ;u, y)|
kf(λ)k·(2.7)
2
X
i=0
1
X
j=0 ¯¯¯(∂y+∂x1)i∂j
xj
1
Γ1(λ;u, y;τ1, x1)¯¯¯dydudx1
≤c7√τ2−τ1,
which goes to zero as τ1→τ2. We integrated by parts in x1, and applied estimates
(2.4), (2.5) to obtain the above inequality. The interchangeability of integration
and differentiation is justified by (2.6) (just notice that, as it is clear from the
first line of (2.6), the integrals are, sometimes, understood as iterated rather than
double integrals). The above estimate proves that Γ3satisfies the second condition
in Definition 2.2.
Now, assume that, in addition, Γ1and Γ2belong to Gs. We claim that, in this
case, Γ3is in Gs−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∈˜
Gs−1, we obtain the following
∂
∂τ2·Zτ2
τ1ZR
Γ2(λ;τ2, x2;u, y)f(λ;u, y)
kf(λ)kDyΓ1(λ;u, y;τ1, x1)dydu¸
=c6
f(λ;τ2, x2)
kf(λ)kDx2Γ1(λ;τ2, x2;τ1, x1) +
2
X
i=0
fi(λ;τ2, x2)
∂i
xi
2Zτ2
τ1ZR
˜
Γi(λ;τ2, x2;u, y)f(λ;u, y)
kf(λ)kDyΓ1(λ;u, y;τ1, x1)dydu
where each fi(λ;., .) is in C1,s−2(S), and the ˜
Γi’s belong to ˜
Gs. The above decom-
position completes the proof of the claim: Γ3∈ Gs−1.
It is easy to see, integrating by parts, that the operator Jdefined by
J[Γ2, f, Γ1](λ;τ2, x2;τ1, x1)
:= Zτ2
τ1ZR
DyΓ2(λ;τ2, x2;u, y)f(λ, u, y)Γ1(λ;u, y ;τ1, x1)dydu
has the same properties as I.
Similarly, for any ©f(λ;., .)∈C1,2(S)ªλ∈Λ, and Γ1,Γ2∈ G2, we define the
function K[Γ2, f, Γ1] by:
K[Γ2, f, Γ1](λ;τ2, x2;τ1, x1)
:= Zτ2
τ1ZR
Γ2(λ;τ2, x2;u, y)ey−x1f(λ, u, y)Γ1(λ;u, y ;τ1, x1)dydu,
and, using (2.6) and (2.7), we obtain the estimate:
|∂τ2K|+
2
X
j=0 ¯¯¯∂j
xj
2
K¯¯¯≤c8kf(λ)k(τ2−τ1)−3/2exp µ−c9
(x2−x1)2
τ2−τ1¶,(2.8)
where the constants c8,c9depend on Γ1and Γ2, but not on λ.

116 R. CARMONA AND S. NADTOCHIY
We now proceed with the proof of the proposition. Writing the initial value
problem (2.1) twice, first with eh, and then with eh+h0, and subtracting one from
another, we can, formally, apply the Feynman-Kac formula and obtain
F¯ε(h+h0)(τ , x) = F¯ε(h)(τ , x) (2.9)
+1
2Zτ
0ZR
p(eh+h0;τ, x;u, y )eh(u,y)+y(eh0(u,y)−1)q(eh; 0,0; u, y )dydu.
This representation follows from the uniqueness of weak solution of (2.1), see, for
example, [6] for details. Applying the same technique to the fundamental solution
p, we get
∆p(τ, x;u, y ) := p(eh+h0;τ, x;u, y)−p(eh;τ, x;u, y)
=Zτ
uZR
p(eh+h0;τ, x;v , z)eh(v,z)(eh0(v,z)−1)Dzp(eh;v, z;u, y)dzdv (2.10)
=I£©p(eh+h0)ªh0∈U(0),©eh(eh0−1)ªh0∈U(0) ,©p(eh)ªh0∈U(0)¤(h0;τ , x;u, y)
Since all the families of functions considered in this part of the proof are parame-
terized 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 ∆qin a similar way. Next we rewrite (2.10) as
F¯ε(h+h0) = F¯ε(h) + F¯ε0(h)[h0] + r1+r2,
with
r1(τ, x) = 1
2Zτ
0ZR
p(eh+h0;τ, x;u, y )eh(u,y)+y³eh0(u,y)−1−h0(u, y)´·
q(eh; 0,0; u, y)dydu
=1
2Khp(eh+h0), eh(eh0−1−h0), q(eh)i(h0;τ, x; 0,0),
and
r2(τ, x) = 1
2Zτ
0ZR
∆p(τ, x;u, y )eh(u,y)+yh0(u, y)·
q(eh; 0,0; u, y)dydu
=1
2KhIhp(eh+h0, eh(eh0−1), p(eh)i, ehh0, q(eh)i(h0;τ , x; 0,0).
Because of the properties of the operator Iderived earlier, it is easy to see that
the function Ihp(eh+h0), eh(eh0−1), p(eh)ibelongs to (eh0−1) · G3. Therefore,
using estimate (2.8), we have immediately that for i= 1,2,
krik˜
W¯
ε≤c10kh0k2
˜
B
and this implies that F¯εis Fr´echet differentiable, with Fr´echet derivative as given
in the statement of the proposition. The fact that F¯ε0(h)[.] is bounded on the unit
ball of ˜
Bfollows, again, from (2.8).
We now compute the Fr´echet derivative of F¯ε0(.) using the same technique as
in the first part of the proof.

LOCAL VOLATILITY DYNAMIC MODELS 117
We fix h∈˜
Band we consider families of functions parameterized by (h0, h00)∈
Λ := ˜
B × U(0). We redefine ∆p, using h00 instead of h0in (2.10). Then we have
¡F¯ε0(h+h00 )−F¯ε0(h)¢[h0](τ , x) =
1
2Zτ
0ZR
h0(u, y)eh(u,y)+y∆p(τ , x;u, y)q(eh; 0,0; u, y)dydu
+1
2Zτ
0ZR
h0(u, y)eh(u,y)+yp(eh;τ , x;u, y)∆q(0,0; u, y)dydu (2.11)
+1
2Zτ
0ZR
h0(u, y)eh(u,y)+y∆p(τ , x;u, y)∆q(0,0; u, y)dydu
+1
2Zτ
0ZR
h0(u, y)eh(u,y)+y³eh00 (u,y)−1´p(eh+h00 ;τ, x;u, y )·
q(eh+h00 ; 0,0; u, y)dydu
Next, we decompose the first integral in (2.11)
Zτ
0ZR
h0(u, y)eh(u,y)+y∆p(τ , x;u, y)q(eh; 0,0; u, y)dydu
=K£I£p(eh), h00eh, p(eh)¤, ehh0, q(eh)¤(h0, h00;τ , x; 0,0)
+KhIhp(eh), eh(eh00 −1−h00), p(eh)i, ehh0, q(eh)i(h0, h00;τ , x; 0,0)
+KhIhIhp(eh+h00 ), eh(eh00 −1), p(eh)i, eh(eh00 −1), p(eh)i,
ehh0, 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 F¯ε00 . Using the properties of the operator I, we conclude
that
Ihp(eh), eh(eh00 −1−h00), p(eh)i
+IhIhp(eh+h00 ), eh(eh00 −1), p(eh)i, eh(eh00 −1), p(eh)i=kh0k˜
Bkh00k2
˜
BΓ,
where Γ ∈ G2. Therefore, using estimate (2.8), we conclude that the k.k˜
W¯εnorms
of the last two terms in the right hand side of (2.12) are bounded by a constant
times kh0k˜
Bkh00k2
˜
B.
A similar decomposition holds true for the second integral in the right hand side
of (2.11) provided the operator Iis replaced by J. Moreover, the k.k˜
W¯ε- norms
of last two integrals in (2.11) are also bounded by a constant times kh0k˜
Bkh00k2
˜
B:
to see this, recall (2.10) and write its analog for ∆q, then apply the properties
operators Iand J, and use estimate (2.8). This yields the existence of F¯ε00(h), as
given in the proposition.
To show the continuity of the second derivative, fix any h0and h00 in ˜
Band
consider any ∆h∈U(0). We only show the continuity of the first component of
F¯ε00 (.)[h0, h00] at h, uniformly over h0and h00 in a bounded set. The proof for the
second component is the same. We introduce the difference

118 R. CARMONA AND S. NADTOCHIY
∆K:= K£I£p(eh+∆h), h00eh+∆h, p(eh+∆h)¤, eh+∆hh0, q(eh+ ∆h)¤
−K£I£p(eh), h00eh, p(eh)¤, ehh0, q(eh)¤=
K£I£I£p(eh+∆h), eh(e∆h−1), p(eh)¤, h00eh+∆h, p(eh+∆h)¤, eh+∆hh0, q(eh+ ∆h)¤
+K£I£p(eh), h00eh(e∆h−1), p(eh+∆h)¤, eh+∆hh0, q(eh+ ∆h)¤
+K£I£p(eh), h00eh, I £p(eh+∆h), eh(e∆h−1), p(eh)¤)¤, eh+∆hh0, q(eh+ ∆h)¤
+K£I£p(eh), h00eh, p(eh)¤, h0eh(e∆h−1), q(eh+ ∆h)¤
+K£I£p(eh), h00eh, p(eh)¤, h0eh,−J£q(eh+∆h), eh(e∆h−1), q(eh)¤¤.
And, as before, using the properties of I,Jand K, we conclude that
k∆Kk˜
W¯ε≤c11kh0k˜
Bkh00k˜
Bk∆hk˜
B,
which completes the proof of the proposition. ¤
Recall that the price Ct(T, x) at time tof an European call option is given
by w(˜a2;T−t, x + log St), where w(˜a2;., .) is the solution of (2.1). Therefore, in
order to get to the Fr´echet differentiability of the price of a call option from the
above result, we will need to compose F¯εwith another mapping. This justifies the
introduction, for each T∈(¯ε, ¯τ] and x∈Rof the mapping
δT,x : [0, T −¯ε]ט
W¯ε×R→R
defined by
δT,x (t, w, y ) = w(T−t, x +y).
We have:
Proposition 2.4. (1) For each (w, y)∈˜
W¯ε×R,δT ,x (., w, y)is continuously
differentiable, and the partial derivative ∂δT,x /∂t is a continuous func-
tional on [0, T −¯ε]ט
W¯ε×R.
(2) For each t∈[0, T −¯ε],δT ,x(t, ., .)is twice Fr´echet differentiable and for
any w, w0, w00 ∈˜
W¯εand y , y, y00 ∈R, its derivatives satisfy
δ0
T,x (t, w, y )[w0, y0] = w0(T−t, x +y) + y0∂xw(T−t, x +y)
and
δ00
T,x (t, w, y )[(w0, y0),(w00, y00 )] = y00∂xw0(T−t, x +y) + y0∂xw00(T−t, x +y)
+y0y00∂2
x2w(T−t, x +y).
Moreover, δ0
T,x and δ00
T,x are continuous operators from [0, T −¯ε]ט
W¯ε×R
into ˜
W∗
¯ε×Rand L³˜
W¯ε×R,˜
W∗
¯ε×R´respectively.
Proof. Let us fix (w, y)∈˜
W¯ε×R. Then, for any t∈[0, T −¯ε], we have
∂
∂t δT,x (t, w, y ) = −∂τw(T−t, x +y).

LOCAL VOLATILITY DYNAMIC MODELS 119
We first show that this functional is continuous in (t, w, y)∈[0, T −¯ε]ט
W¯ε×R.
Consider any (t0, w0, y0)∈[0, T −¯ε]ט
W¯ε×R, then
|∂τw(T−t, x +y)−∂τw0(T−t0, x +y0)|=
|∂τw(T−t, x +y)−∂τw(T−t0, x +y0)|(2.12)
+|∂τw(T−t0, x +y0)−∂τw0(T−t0, x +y0)|
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−w0k˜
W¯ε. 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 −¯ε]×
˜
W¯ε×Rfixed, and consider (w0, y0)∈U(0) ⊂˜
W¯ε×R, where U(0) is a neighborhood
of zero. Notice that
δT,x (t, w +w0, y +y0)−δT,x (t, w, y) =
w(T−t, x +y+y0)−w(T−t, x +y) + w0(T−t, x +y+y0) =
y0∂xw(T−t, x +y) + ¯
¯o(y0) + w0(T−t, x +y) + y0∂xw0(T−t, x +y+ξy0),
for some ξ∈[0,1], and that
|¯
¯o(y0) + y0∂xw0(T−t, x +y+ξy0)|=¯
¯o³q|y0|2+kw0k2
˜
W¯ε´,
Therefore, we have obtained the expression for ˜
C0
T,x , as given in the proposition.
Now consider (w0, y0)∈˜
W¯ε×Rand (w00 , y00 )∈U(0) ⊂˜
W¯ε×R, the rest of
parameters being fixed. Then:
¡δ0
T,x (t, w +w00, y +y00 )−δ0
T,x (t, w, y )¢[w0, y0] =
w0(T−t, x +y+y00 )−w0(T−t, x +y) + y0∂xw00(T−t, x +y+y00 )
+y0∂xw(T−t, x +y+y00 )−y0∂xw(T−t, x +y) =
y00∂xw0(T−t, x +y)+(y00 )2∂2
x2w0(T−t, x +y+ξy00 ) + y0∂xw00 (T−t, x +y)
+y0y00∂2
x2w00(T−t, x +y+ξ0y00) + y0¡y00 ∂2
x2w(T−t, x +y) + ¯
¯o(y00 )¢,
for some ξ, ξ0∈[0,1]. Again, noticing that
¯¯(y00 )2∂2
x2w0(T−t, x +y+ξy00 ) + y0y00 ∂2
x2w00(T−t, x +y+ξ0y00) + y0¯
¯o(y00)¯¯
≤q|y0|2+kw0k2
˜
W¯ε
¯
¯o³q|y00 |2+kw00 k2
˜
W¯ε´.
we get the desired expression for δ00
T,x .
In order to show the continuity of δ0
T,x and δ00
t,x, we fix (w0, y0),(w00, y00)∈˜
W¯ε×R,
and we prove the continuity of δ0
T,x (., ., .)[w0, h0] and δ00
T,x (., ., .)[(w0, h0),(w00 , h00)]
by, essentially, repeating the argument of (2.13). Finally, notice that the continuity
is uniform over (w0, y0),(w00, y00 ) when they are restricted to a bounded set. ¤
Now, consider the composition of the two operators introduced above. For each
T∈(¯ε, ¯τ] and x∈R, we have
CT,x : [0, T −¯ε]ט
B × R→R
CT,x (t, h, y) = δT ,x (t, F¯ε(h), y)

120 R. CARMONA AND S. NADTOCHIY
As a composition of twice Fr´echet differentiable operators, CT,x (t, ., .) is, clearly
twice Fr´echet differentiable, for each t∈[0, T −¯ε]. Due to the continuity of F¯ε00(.),
δ0
T,x (., ., .) and δ00
T,x (., ., .), the Fr´echet derivatives of CT,x (t, h, y) are also continuous
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 −¯ε], functional CT ,x(t, ., .)is twice Fr´echet
differentiable, such that, for any h, h0, h00 ∈˜
Band y, y, y 00 ∈R, we have
C0
T,x (t, h, y)[h0, y 0] = F¯ε0(h)[h0](T−t, x +y) + y0∂xF¯ε(h)(T−t, x +y),
and
C00
T,x (t, h, y)[(h0, y 0),(h00, y00 )] = F¯ε00 (h)[h0, h00](T−t, x +y)
+y00∂xF¯ε0(h)[h0](T−t, x +y) + y0∂xF¯ε0(h)[h00 ](T−t, x +y)
+y0y00∂2
x2F¯ε(h)(T−t, x +y),
and C0
T,x ,C00
T,x are continuous operators from [0, T −¯ε]ט
B × Rinto ˜
B∗×Rand
L³˜
B × R,˜
B∗×R´respectively.
3. Using Itˆo’s Formula in Infinite Dimension
The purpose of this section is to extend the proof of the semi-martingale prop-
erty given in [2] to the case of infinitely many driving Wiener processes.
We denote by Bthe cylindrical Brownian motion constructed on the canonical
cylindrical Gaussian measure of some separable Hilbert space ˜
H. The reader can
think of ˜
H=l2- the space of square - sumable 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 ˜
B. For each functions fand
gwith enough derivatives square integrables and for each non-negative integers k
and m, we define the scalar product
< f, g > ˜
Wk,m(S)=
k
X
i=0
∂i
τif(0,0)∂i
τig(0,0) +
m
X
j=0
∂j
xjf(0,0)∂j
xjg(0,0)
+ZS5¡∂k
τkf(τ, x)¢5¡∂k
τkg(τ, x)¢+5(∂m
xmf(τ, x)) 5(∂m
xmg(τ, x)) dxdτ.
Now we fix a compact set Kcontained in Sand containing the origin (0,0), and
we consider the space of functions on Swhich 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 < . , . > ˜
W1,5(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 ˜
B. Define by B, the completion of Hin the k.kC1,5(S)norm. Thus,
the pair (H,B) forms a conditional Banach Space.
Clearly, Bis a subspace of ˜
B, and therefore, Proposition 2.5 holds for the
restriction of CT,x to Bas well.

LOCAL VOLATILITY DYNAMIC MODELS 121
For any given real separable Banach space Gwe denote by L(G) the space of all
non-anticipative random processes in G(measurable mappings X: Ω×[0,∞)→ G)
, such that
EZt
0kXuk2
Gdu < ∞,
for all t≥0. Where Gis 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∈L³L2³˜
H,H´´, and we model dynamics
of ht, the logarithm of the squared local volatility at time t, ˜a2
t, by the infinite
dimensional Itˆo’s stochastic differential
dht=αtdt +βtdBt,
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
dlog St=−1
2σ2
tdt +σt< e1, dBt>, log S0,(3.1)
where σis R- valued random process with ERt
0σ2
udu < ∞almost surely, for any
t≥0, and e1∈˜
His a fixed unit vector.
Now, thanks to Proposition 2.5, we can apply Itˆo’s formula (see, for example,
[13], p. 200) to (CT ,x(t, ht,log St))t∈[0,T −¯ε]. We get that for any T∈(¯ε, ¯τ] and
x∈R, we have, almost surely, for all t∈[0, T −¯ε],
CT,x (t, ht,log St) = CT ,x(0, h0,log S0)
+Zt
0µ∂
∂t C0
T,x (u, hu,log Su) + C0
T,x (u, hu,log Su)[αu,−1
2σ2
u]
+1
2Tr ¡(βu, σue1)∗◦C00
T,x (u, hu,log Su)◦(βu, σue1)¢¶du
+Zt
0
C0
T,x (u, hu,log Su)◦(βu, σue1)dBu
where C0
T,x and C00
T,x are given in Proposition 2.5.
Remark 3.1.Since ¯εcan be made as small as we want, the above representation
holds for any T∈(0,¯τ], and all t∈[0, T ). Then, since we choose ¯τas 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}nof ˜
H. Notice that without any loss of generality we can assume that the
first element e1of 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 ˜
H=l2. If we consider that βtis
given by the sequence {βn
t(., .)∈ H}∞
n=1 of its components on the basis vectors,
then we have the following theorem.

122 R. CARMONA AND S. NADTOCHIY
Theorem 3.2. For any T > 0and x∈R, we have, almost surely, for all t∈[0, T ),
CT,x (t, ht,log St) = CT ,x(0, h0,log S0)
+Zt
0·F¯ε0(hu)[αu]−1
2σ2
u∂xF¯ε(hu)−∂τF¯ε(hu) + 1
2σ2
u∂2
x2F¯ε(hu)
+σu∂xF¯ε0(hu)[β1
u] + 1
2
∞
X
n=1
F¯ε00 (hu)[βn
u, βn
u]#(T−u, x + log Su)du
+Zt
0"∞
X
n=1
F¯ε0(hu)[βn
u] + σu∂xF¯ε(hu)#(T−u, x + log Su)dBn
u,
if we use the notation {Bn}nfor the sequence of independent standard one - dimen-
sional Brownian motions Bn
t=< en, Bt>.F¯ε0and F¯ε00 are given in Proposition
2.1.
This is the infinite dimensional version of the semi-martingale result of [2].
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LOCAL VOLATILITY DYNAMIC MODELS 123
R. Carmona: Bendheim Center for Finance, ORFE, Princeton University, Prince-
ton, NJ 08544, USA
E-mail address:rcarmona@princeton.edu
S. Nadtochiy: Bendheim Center for Finance, ORFE, Princeton University, Prince-
ton, NJ 08544, USA
E-mail address:snadtoch@princeton.edu