Content uploaded by Rene A. Carmona

Author content

All content in this area was uploaded by Rene A. Carmona

Content may be subject to copyright.

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]×˜

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]×˜

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]×˜

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]×˜

W¯ε×R.

Consider any (t0, w0, y0)∈[0, T −¯ε]×˜

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]×˜

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]×˜

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].

References

1. Carmona, R. HJM: A uniﬁed approach to dynamic models for ﬁxed income, credit and equity

markets, Paris - Princeton Lecutues in Mathematical Finance, 2005 (R. Carmona et al., ed.),

Lecture Notes in Mathematics,1919, Springer Verlag, 2007, pp. 3–45.

2. Carmona, R and Nadtochiy, S.: Local volatility dynamic models, Finance and Stochastics

(2008), (to appear).

3. Carmona, R. and Tehranchi, M.: Interest rate models: an inﬁnite dimensional stochastic

analysis perspective, Springer Verlag, 2006.

4. Cont, R and da Fonseca, J.: Dynamics of implied volatility surfaces, Quantitative Finance

2(2002), 45–60.

5. Cont, R., da Fonseca, J., and Durrleman, V.: Stochastic Models of Implied Volatility Sur-

faces, Economic Notes 31 (2002)

6. Cr´epey, S.: Calibration of the local volatility in a generalized Black–Scholes model using

Tikhonov regularization, SIAM Journal on Mathematical Analysis 34 (2003) 1183–1206.

7. Derman, E. and Kani, I.: Stochastic implied trees: Arbitrage pricing with stochastic term

and strike structure of volatility, International Journal of Theoretical and Applied Finance

1(1998) 61–110.

8. Durrleman, V.: From implied to spot volatility, Tech. report, Stanford University, April 2005.

9. Friedman, A.: Partial diﬀerential equations of parabolic type, Prentice Hall Inc., 1964.

10. Gross, L.: Abstract Wiener spaces, in: Proc. 5th Berkeley Symp. Math. Stat. and Probab.

2, part 1 (1965) 31–42, University of California Press, Berkeley.

11. Jarrow, R., Heath, D., and Morton, A.: Bond pricing and the term structure of interest

rates: a new methodology for contingent claims valuation, Econometrica 60 (1992) 77–105.

12. Kouritzin, M. A.: Averaging for fundamental solutions of parabolic equations, Journal of

Diﬀerential Equations 136 (1997) 35–75.

13. Kuo, H.-H.: Gaussian measures in Banach spaces, Lecture Notes in Mathematics 463,

Springer-Verlag, Berlin, 1975.

14. Sch¨onbucher, P.: A market model for stochastic implied volatility, Phil. Trans. of the Royal

Society, Series A 357 (1999) 2071–2092.

15. Schweizer, M. and Wissel, J.: Arbitrage-free market models for option prices: The multi-

strike case. Technical report (2007).

16. Schweizer, M. and Wissel, J.: Term structures of implied volatilities: Absence of arbitrage

and existence results. Mathematical Finance 18 (2008) 77–114

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