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February 25, 2011 11:42 WSPC/S0219-0249 104-IJTAF SPI-J071

S0219024911006280

International Journal of Theoretical and Applied Finance

Vol. 14, No. 1 (2011) 107–135

c

World Scientiﬁc Publishing Company

DOI: 10.1142/S0219024911006280

TANGENT MODELS AS A MATHEMATICAL

FRAMEWORK FOR DYNAMIC CALIBRATION

REN´

E CARMONA∗and SERGEY NADTOCHIY†

Bendheim Center for Finance, ORFE

Princeton University

Princeton, NJ 08544, USA

∗

rcarmona@princeton.edu

†

sergey.nadtochiy@oxford-man.ox.ac.uk

Received 13 May 2010

Accepted 7 October 2010

Motivated by the desire to integrate repeated calibration procedures into a single

dynamic market model, we introduce the notion of a “tangent model” in an abstract

set up, and we show that this new mathematical paradigm accommodates all the recent

attempts to study consistency and absence of arbitrage in market models. For the sake of

illustration, we concentrate on the case when market quotes provide the prices of Euro-

pean call options for a speciﬁc set of strikes and maturities. While reviewing our recent

results on dynamic local volatility and tangent L´evy models, we present a theory of tan-

gent models unifying these two approaches and construct a new class of tangent L´evy

models, which allows the underlying to have both continuous and pure jump components.

Keywords: Market models; Heath–Jarrow–Morton approach; implied volatility; local

volatility; tangent L´evy models.

1. Introduction

Calibration of a ﬁnancial model is most often understood as a procedure to choose

the model parameters so that the theoretical prices produced by the model match

the market quotes. In most cases, the market quotes span a term structure of

maturities, and by nature, the calibration procedure introduces an extra time-

dependence in the parameters that are calibrated. Introducing such a time depen-

dence in the parameters changes dramatically the interpretation of the original

equations. Indeed, even if these equations were originally introduced to capture the

dynamics (whether they are historical or risk neutral) of the prices or index values

underlying derivatives, the equations with the calibrated parameters lost their inter-

pretations as providing the time evolutions of the underlying prices and indexes.

The purpose of market models is to restore this interpretation, and the notion of

tangent models which we introduce formally in this paper appears as a general

framework to do just that.

107

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108 R. Carmona & S. Nadtochiy

In order to illustrate clearly the point of the matter, we review a standard exam-

ple from interest rate theory used routinely as a justiﬁcation for the introduction

of the HJM approach to ﬁxed income models. If we consider Vasicek’s model for

example

drt=κ(r−rt)dt +σdWt,

because of the linear and Gaussian nature of the process, it is possible to derive

explicit formulas for many derivatives and in particular for the forward and yield

curves. However, the term structure given by these formulas is too rigid, and on

most days, one cannot ﬁnd reasonable values of the 3 parameters κ,rand σgiving

atheoretical forward curve matching, in a satisfactory manner, the forward curve

τ→f(τ) observed on that day. This is a serious shortcoming as, whether it is

for hedging and risk management purposes, or for valuing non-vanilla instruments,

using a model consistent with the market quotes is imperative. Clever people found

a ﬁx to this hindrance: replace the constant parameter rby a deterministic function

of time t→r(t). Indeed, this function being deterministic, the interest rate process

remains Gaussian (at least as long as we do not change the initial condition) and

we can still obtain explicit formulas for the forward curves given by the model.

Moreover, if we choose the time dependent parameter to be given by

r(τ)=f(τ)+κf(τ)−σ2

2κ(1 −e−κτ )(3e−κτ −1)

then the model provides a perfect match to the curve observed on the market, in

the sense that the forward rate with time to maturity τproduced by the model

(1.1) with a time dependent r, is exactly equal to f(τ). Our contention is that even

though it provides a stochastic diﬀerential equation (SDE for short)

drt=κ(r(t)−rt)dt +σdWt,(1.1)

this procedure can be misleading, looking as if this SDE actually relates to the

dynamics of the short interest rate. Indeed, this is not a model in the sense that

when the next day comes along, one has to restart the whole calibration procedure

from scratch, and use equation (1.1) with a diﬀerent function t→r(t). Despite

the fact that its left hand side contains the inﬁnitesimal “drt”, which could leave

us to believe that the time evolution of rtis prescribed by its right hand side,

formula (1.1) does not provides a dynamic model, it is a mere artifact designed to

capture the prices observed on the market: it is what we call a tangent model.

The main goal of this paper is to identify a framework in which dynamic models

for the underlying indexes and the quoted prices can coexist and in which their

consistency can be assessed. Despite its generality, this framework can be used

to oﬀer concrete solutions to practical problems. Case in point, one of the nagging

challenges of quant groups supporting equity trading is to be able to generate Monte

Carlo scenarios of implied volatility surfaces which are consistent with historical

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Tangent Models as a Mathematical Framework for Dynamic Calibration 109

observations while being arbitrage free at the same time. We show in Sec. 4.5 how

tangent L´evy models can be used to construct such simulation models.

The paper is organized as follows. Section 2introduces the notation and the

deﬁnitions used throughout. In particular, the general notion of tangent model is

described and illustrated. Sections 3and 4recast the results of [4–6] in the present

framework of tangent models, and for this reason, they are mostly of a review nature.

Section 5introduces and characterizes the consistency of new tangent models that

combine the features of the diﬀusion tangent models of Sec. 3and the pure jump

tangent models of Sec. 4. These models bear some similarities to those appearing

in a recent technical report [21] where Kallsen and Kr¨uhner study a form of Heath-

Jarrow-Morton approach to dynamic stock option price modeling. However, their

approach does not seem to lead to constructive models like the one proposed in

Sec. 4.5.

2. Tangent Models and Calibration

2.1. Market models for equity derivatives: Problem formulation

We now describe the framework of the paper more precisely. First of all, as it is

done in a typical set-up for a mathematical model, we assume that we are given

a stochastic basis (Ω,F,(Ft)t≥0,Q) and that pricing is linear in the sense that

the time tprices of all contingent claims are given as (conditional) expectations

of discounted payoﬀs under the pricing measure Q, with respect to the market

ﬁltration Ft. We assume, for simplicity, that the discounting factor is one, and unless

otherwise speciﬁed, all stochastic processes are deﬁned on the above stochastic basis

and E≡EQ. Interest rates do not have to be zero for the results of this paper to still

hold. Any positive deterministic function of time would do. However, we refrain from

working in this generality for the sake of notation. We denote by (St)t≥0the true

risk-neutral (stochastic) dynamics of the value of the index or security underlying

the derivatives whose prices are quotedinthemarket.WedenotebyDtthe set

of derivatives available at time t. Naturally, we identify each element of Dtwith

its maturity Tand the payoﬀ h(which may be a function of the entire path of

(St)t∈[0,T ]). We assume that the market for these derivatives is liquid in the sense

that each of them can be bought or sold, in any desired quantity, at the price quoted

in the market. Thus, we denote by Pt(T,h) the market price of a corresponding

derivative at time t, and introduce the set of all market prices

Pt={Pt(T,h)}(T,h)∈Dt

In the most commonly used example, Stis the price at time tof a share and Dt

is the set of European call options for all strikes K>0 and maturities T>tat

time t,havingpriceCt(T, K), so that in this case,

Pt={Ct(T,K)}T>t,K>0(2.1)

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110 R. Carmona & S. Nadtochiy

Our goal is to describe explicitly a large class of time-consistent market models,

i.e. stochastic models (say, SDE’s) giving the joint arbitrage-free time evolution of

Sand P. One would like to start the model from “almost” any initial condition,

typically the set of currently observed market prices, and prescribe “almost” any

dynamics for the model provided it doesn’t contradict the no-arbitrage property.

Of course, the above formulation of the problem is rather idealistic. This explains

our use of the word “almost” whose speciﬁc meaning is diﬀerent for each class of

market models.

The need for ﬁnancial models consistent with the observed option prices has

been exacerbated by the fact that call options have become liquid and provide

reliable price signals to market participants. Stochastic volatility models (e.g. Hull-

White, Heston, etc.) are very popular tools in this respect, namely as a means to

capture this signal. Involving a small number of parameters, they are relatively easy

to implement, and they can capture the smile reasonably well for a given maturity.

However, the ﬁt to the entire term structure of implied volatility is not always

satisfactory as they cannot reproduce market prices for all strikes and maturities.

See for example [15].

The preferred solution for over 15 years has been based on the so-called local

volatility models introduced by Dupire in [14]. It says that if the true model for

the risk neutral dynamics of the underlying is given by an equation of the form

dSt=σtdWt.(recall that we assume zero interest rate for the sake of simplicity), and

if we assume that the function C(T,K) giving the price of an European call options

with maturity Tand strike Kis smooth, then the stochastic process ˜

Ssolving the

equation

˜

St=S0+t

0

˜

Su˜a(u, ˜

Su)dWu,

with

˜a2(T,K):= 2∂

∂T C(T,K)

K2∂2

∂K2C(T,K),(2.2)

produces at time t=0,thesame exact call prices C(T,K)! In other words, for all

T>0andK>0, we have E(˜

ST−K)+=C(T,K). The function (T, K)→˜a2(T,K)

so deﬁned is called the local volatility. For the sake of illustration, we computed and

plotted the graph of this function in the case of the two most popular stochastic

volatility models mentioned earlier, the Heston and the Hull-White models. These

plots are given in Fig. 1. In the terminology which we develop below, the artiﬁcial

ﬁnancial model given by the process ( ˜

St)t≥0, introduced for the sole purpose of

reproducing the prices of options at time zero (in other words, the result of cal-

ibration at time zero), is said to be tangent to the true model (St)t≥0at t=0.

Together with the simple interest rate model reviewed in the introduction, this dis-

cussion of Dupire’s approach provides the second example of a SDE introduced for

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Tangent Models as a Mathematical Framework for Dynamic Calibration 111

(a) (b)

Fig. 1. Local volatility surfaces for the Heston (a) and Hull-White (b) models as functions of the

time to maturity τ=T−tand log-moneyness log(K/S).

the sole purpose of capturing the prices quoted on the market. We now formalize

this concept in a set of mathematical deﬁnitions.

One of the major problems with calibration is its frequency: stochastic volatility

models have diﬀerent “optimal” parameters most every day, and the local volatility

surface calibrated on a daily basis changes as well. In order to incorporate these

changes in a model, we focus on the “daily” capture of the price signals given by

the market through the quotes of the liquidly traded derivatives.

2.2. Examples of the sets of derivatives

The theoretical framework of this paper was inspired by earlier works on the original

market models which pioneered the analysis of joint dynamics for a large class of

derivatives written on a common underlying index. Most appropriate references

(given the spirit of the present paper) include [9] for the HJM approach to bond

markets, [28]and[16] for the BGM approach to the LIBOR markets, [1]forthe

markets of variance swaps, and [32]and[35] for the markets of synthetic CDOs and

credit portfolios. See also [2]forareview.

However, for the sake of deﬁniteness and notation, we restrict the discussion of

this paper to the models used for the markets of equity derivatives. The following

list is a sample of examples which can be found in the existing literature, and for

which the above formalism applies:

•P

t={St,C

t(T,K); T>t}for some ﬁxed K>0 — considered by Schoenbucher

in [31];

•P

t={St,C

t(T); T>t}where Ct(T) represents the price at time tof a European

call option when the hockey-stick function x→(x−K)+is replaced by a ﬁxed

convex payoﬀ function — considered by Jacod and Protter in [20] and Schweizer

andWisselin[34];

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112 R. Carmona & S. Nadtochiy

•P

t={St,C

t(T,K); K>0}for some ﬁxed T>t— considered by Schweizer and

Wissel in [33];

•P

t={St,C

t(Ti,K

j); i=1,...,m, j =1,...,n}— considered by Schweizer and

Wissel in [33];

•P

t={St,C

t(T,K); T>t,K>0}— considered by Cont et al. in [11]and

Carmona and Nadtochiy in [5].

For the most part of this paper we concentrate on the last example where the prices

of the liquidly traded instruments are:

Pt={St,C

t(T,K); T∈(t, ¯

T],K>0},(2.3)

where we assume, in addition, that both maturity Tand calendar time tare bounded

above by some ﬁnite ¯

T>0. Notice that such a set Ptis inﬁnite (even of continuum

power), even though the set Ptis ﬁnite in practice. This abstraction is standard in

the ﬁnancial mathematic and engineering literature.

2.3. Tangent models

Recall that we use the notation Tand hfor the typical maturity and payoﬀ function

of a derivative in Dt(hmay be path dependent) and Pt(T,h) for its price at time

t≥0. Each process (Pt(T,h)) is adapted and, due to our standing assumption of

risk-neutrality, we have, almost surely

Pt(T,h)=E(h((Su)u∈[0,T ])|F

t)

Motivated by Dupire’s result of exact static calibration, we say that the stochastic

model given by an auxiliary stochastic pro cess ( ˜

Su)u≥0deﬁned on a (possibly dif-

ferent) stochastic basis ( ˜

Ω,˜

F,˜

P)isDt-tangent to the true model (or just tangent

when no ambiguity is possible) at time tfor a given ω∈Ω, if

∀(T,h)∈D

tPt,ω(T,h)=E˜

P(h(¯

St,ω)),(2.4)

where

¯

St,ω =(¯

St,ω

u,˜ω)u∈[0,T ],˜ω∈˜

Ωand ¯

St,ω

u,˜ω=1u≤tSu,ω +1u>t ˜

Su−t,˜ω,

and the expectation in (2.4) is computed over ˜

Ωfortand ωﬁxed. The payoﬀ

appearing in the above expectation is computed over a path which coincides with

the path of the underlying index Sup to time tand with the path of the tan-

gent process ˜

Safter that time. The expression of ¯

St,ω used in (2.4)isinvolved

only because we allow the payoﬀ hto depend upon the entire path of the under-

lying index. However, in all particular applications we discuss below, we deal with

payoﬀs that depend only upon ST, for some maturity T, and in that case we can

simply change the maturities of the payoﬀs from Tto T−t, and use ˜

Sinstead of

¯

St,ω in (2.4).

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Tangent Models as a Mathematical Framework for Dynamic Calibration 113

We want to think of the above notion of tangent model as an analog of the

notion of tangent vector in classical diﬀerential geometry: the two models are tan-

gent in the sense that, locally, at a ﬁxed point in time, they produce the same

prices of derivatives in a chosen family. Recall that tangent vectors in diﬀerential

geometry are often used as a convenient way to describe the time dynamics. In the

same way, we hope that the tangent models introduced above will help in a better

understanding of market models.

2.4. Code books

Let us assume that the martingale models considered for the underlying index can

be parameterized explicitly, say in the form:

M={M(θ)}θ∈Θ,

and that Pθ(h), the price at time t= 0 of a claim with payoﬀ function hin the

model M(θ), is fairly easy to compute. If, in addition, the relation

θ→{Pθ(h)}h∈D0,(2.5)

is invertible, we obtain a one-to-one correspondence between a set of prices for

the derivatives in D0and the parameter space Θ. When this is the case, we also

assume that this one-to-one correspondence can be extended to hold at each time

t. More precisely, at each time t>0 the derivatives we consider can be viewed as

contingent claims depending on the future evolution of the underlying (since the

past is known), hence we deﬁne the “eﬀective” maturity and payoﬀ at time tby

τ=T−tand ˜

h(( ˜

Su)u∈(0,T −t]):=h((Su)u∈[0,t](˜

Su−t)u∈(t,T ])

respectively. In the above we used “” to denote the concatenation of paths. Thus,

given time tand the evolution of the underlying (Su)u∈[0,t]up to time t,foreach

pair (T,h)∈D

tthere is a unique corresponding pair (τ, ˜

h). Therefore we deﬁne ˜

Dt,

the set of target derivative contracts expressed in the “centered” (around current

time) variables, via

˜

Dt:= {(τ,˜

h)|(T,h)∈D

t}

The models M(θ) are now viewed as the models for ˜

S, which are used to compute

Pθ(τ, ˜

h), the time zero prices of derivatives in ˜

Dt. Hence, we assume that at each

time tthere exists a one-to-one mapping

Θθ↔P

θ

t:= {Pθ(τ,˜

h)}(τ,˜

h)∈˜

Dt(2.6)

Then we call the set Θ a code-book and the above bijective correspondence a code.

Recall that set ˜

Dtcontains the same derivatives as Dt, but in the new time coordi-

nates: with the current moment of time tbeing the origin. Hence, the existence of

bijection (2.6) means that at each moment of time there exists a model M(θ)such

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114 R. Carmona & S. Nadtochiy

that the market price Pt(T,h) of any contract in Dtcoincides with the time zero

model-implied price of a corresponding contract in ˜

Dt. Then the above mapping

allows us to think of the set of market prices Ptin terms of its code value θ∈Θ.

We can reformulate the notion of a tangent model in terms of code-books in the

following way: if at time tfor a given ω∈Ωthere exists a code value θt,ω ∈Θwhich

reproduces the market prices (i.e. Pt=Pθt,ω

t),then the model M(θt,ω)is tangent

to the true model in the sense of (2.4).

When the set Θ is simple enough (for example an open subset of a linear space),

the construction of market models reduces to putting in motion the initial code θ0,

which captures the initial prices of the liquidly traded derivatives, and obtaining

(θt) (whenever possible, we drop the dependence upon “ω” in our notation, as most

probabilists do). One can then go from the code-book space to the original domain

by computing the resulting derivatives prices for any future time tin the model

M(θt). Code-books, as more convenient representations of derivatives prices, have

been used by practitioners for a very long time: the examples include yield curve in

the Treasury bond market, implied term structure of default probabilities for CDO

tranches and implied volatility for the European options, etc.

Remark 2.1. Due to the speciﬁc form of our abstract deﬁnition of a tangent model,

we can identify any such model with the law of the underlying process it produces,

as opposed to the general case when a ﬁnancial model is deﬁned by the pair: “under-

lying process” and “market ﬁltration”. In the same way, by model M(θ) we will

understand a speciﬁc distribution of the process ˜

Sused instantaneously as a proxy

for the underlying index. In this respect, the construction of consistent stochastic

dynamics for tangent models is not without similarities with the foundations of

Knight’s prediction process [22].

We now deﬁne two important classes of tangent models and we review their

main properties in the following two sections.

2.5. Tangent diﬀusion models

We say that a tangent model is a tangent diﬀusion model if at any given time, the

tangent process ˜

Sis a possibly inhomogeneous diﬀusion process. More precisely, we

shall assume that the process ˜

Sis of the form

˜

St=s+t

0

˜

Su˜a(u, ˜

Su)dBu,

for some initial condition s,local volatility function ˜a(.,.) and a Brownian motion

B.Thelawof ˜

Sis then uniquely determined by (s, ˜a(.,.)), where the surface ˜a

has to satisfy mild regularity assumptions (see [5]and[4] for details). Clearly, the

values at time t= 0 of the underlying index and the call prices in any such model

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Tangent Models as a Mathematical Framework for Dynamic Calibration 115

are given by sand

Cs,˜a(τ, x)=E(˜

Sτ−ex)+,(2.7)

respectively, if we use the notation K=exfor the strike. From Dupire’s formula

(2.2), we can conclude that the above mapping from (s, ˜a) to the couple (“value of

the underlying”, “prices of call options”) is one-to-one, thus producing a code-book.

For a gi ven ω∈Ω, if at time tthereexistsavalueofthecodeθt,ω =(st,ω,˜at,ω),

which reproduces the true market prices of all the call options and the underlying,

then the model given by (st,ω ,˜at,ω ) is a tangent diﬀusion model at time t.Inthat

case st,ω has to coincide with the current value of the underlying index St(ω)and

˜at,ω (.,.) can be viewed as the local volatility surface calibrated (ﬁtted) to match

the observed call prices at time t.

2.6. Tangent L´evy models

We say that a tangent model is a tangent L´evy model if the tangent process ˜

Sis

given by an additive (i.e. a (possibly) time-inhomogeneous L´evy process ). To be

more speciﬁc, a given model is a tangent L´evy model if it is tangent (in the sense

of (2.4)) and the corresponding tangent process ˜

Sis a pure jump additive process

satisfying

˜

St=s+t

0R

˜

Su−(ex−1)[N(dx, du)−η(dx, du)],(2.8)

where N(dx, du)isaPoisson random measure — associated with the jumps of

log( ˜

S) — having an absolutely continuous (deterministic) intensity

η(dx, du)=˜κ(u, x)dxdu.

The law of ˜

Sis then uniquely determined by (s, ˜κ). As before, the values at time

t= 0 of the underlying index and the call prices in any such model are given by s

and

Cs,˜κ(τ, x)=E(˜

Sτ−ex)+(2.9)

respectively. From the analytic representation of (2.9) provided in Sec. 4(and dis-

cussed in more detail in [6]), it is not hard to see that the above mapping from

(s, ˜κ) to (“value of the underlying”, “prices of the call options”) is one-to-one, thus

producing a code-book.

As before, for a given ω∈Ω, if at time tthereexistsavalueofthecode

θt,ω =(st,ω ,˜κt,ω), which reproduces the market prices of all the European call

options and the value of the underlying index, then the model given by (st,ω,˜κt,ω)

is a tangent L´evy model at time t.

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116 R. Carmona & S. Nadtochiy

2.7. Time-consistency of calibration

It is important to remember that our standing assumption is that the prices of

all contingent claims are given by conditional expectations in the true (unknown)

model. Therefore, when prescribing the (stochastic) dynamics of the code θt,we

have to make sure that the derivative prices produced by θtat each future time

tare indeed “the market prices”. In other words, they have to coincide with the

corresponding conditional expectations, or, equivalently, M(θt,ω ) has to be tangent

to the true model at each time t, for almost all ω∈Ω. This condition reﬂects

the internal time-consistency of the dynamic calibration, and therefore, we further

refer to it as the consistency of the code dynamics (or simply “consistency”). If the

dynamics of θtare consistent with a true model, then we say that the true model

and (θt)formadynamic tangent model.

3. Dynamic Tangent Diﬀusion Models

In this section we assume that the ﬁltration (Ft)t≥0is Brownian in the sense that

it is generated by a (possibly inﬁnite dimensional) Wiener process, and that the set

Ptof prices of liquidly traded derivatives is given by (2.3).

3.1. The local volatility code book

We capture the prices of all the European call options with the local volatility

˜at(.,.) deﬁned with what is known as Dupire’s formula, which we recalled earlier

in the static case t=0:

˜a2

t(τ,K):= 2∂

∂T Ct(t+τ,K)

K2∂2

∂K2Ct(t+τ,K),(3.1)

where Ct(T,K) is the (true) market price of a call option with strike Kand maturity

Tat time t. As discussed above, this formula deﬁnes a mapping from the surfaces

of call prices to the local volatility functions producing a code-book. We switch to

the log-moneyness x,writing

h(τ,x):=log˜a2(τ,sex) (3.2)

for the logarithm of the square of local volatility. Recall the deﬁnition of Cs,˜a,call

prices produced by local volatility, given by (2.7). Using the normalized call prices

cs,˜a(τ, x)=1

sCs,˜a(τ, log s+x),(3.3)

the analytic representation of the call prices produced by the code value (s, ˜a)takes

the form of the Partial Diﬀerential Equation (PDE)

∂τcs, ˜a(τ,x)=eh(τ,x)Dxcs, ˜a(τ,x),τ>0,x∈R

cs,˜a(τ, x)|τ=0 =(1−ex)+.(3.4)

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Tangent Models as a Mathematical Framework for Dynamic Calibration 117

where we used notation Dxfor the diﬀerential operator Dx=1

2(∂2

x2−∂x). Starting

from a squared local volatility function ˜a2(or equivalently its logarithm h)and

ending with the solution of the above PDE deﬁnes an operator F:h→ cwhich

plays a crucial role in the analysis of tangent diﬀusion models.

Once speciﬁc function spaces are chosen (see Sec. 2.2 of [4] for the deﬁnitions of

the domain and range of F), formula (3.1) and the operator Fprovide a one-to-one

correspondence between call option price surfaces and local volatility surfaces. This

deﬁnes the local volatility code-book for call prices. See also [4]and[5]formore

details.

3.2. Formal deﬁnition of dynamic tangent diﬀusion models

As explained earlier, we assume that a pricing measure has been chosen (it does

not have to be uniquely determined as the “martingale measure”, i.e. we allow

for an incomplete market), and that under the probability structure it deﬁnes, the

underlying index is a martingale as we ignore interest rate and dividend payments

for the sake of simplicity. Consequently, the underlying index value is a martingale

of the form:

dSt=StσtdWt,

for some scalar adapted spot volatility process (σt) and a one-dimensional Wiener

process (Wt)twhich we will identify, without any loss of generality with the ﬁrst

component (B1

t)tof the multidimensional Wiener process (Bt)tgenerating the mar-

ket ﬁltration. In order to specify the dynamics of the code (st,˜at), we notice that

if we want these dynamics to be consistent (see the discussion in Sec. 2.7), we need

to have st=St. Thus we deﬁne the dynamics (time evolution) of the codes by

st=St,dS

t=StσtdB1

t,

˜at(τ, K)=exp

1

2ht(τ,log K/st),dh

t=αtdt +

m

n=1

βn

tdBn

t,(3.5)

where B=(B1,...,Bm)isanm-dimensional Brownian motion (mcould be ∞),

the stochastic processes αand {βn}m

n=1 take values in spaces of functions of τand

x(see Sec. 3 of [4] for the exact deﬁnitions of function spaces for αand β), and σ

is a (scalar) locally square integrable adapted stochastic process, such that Sis a

true martingale.

Atangent diﬀusion model is deﬁned by the dynamics (3.5) in such a way that

for any (T,x)∈(0,¯

T]×Rthe following equality is satisﬁed almost surely for all

t∈[0,T)

Cst,˜at(T−t, x)=E((ST−K)+|F

t),(3.6)

where Cs,˜ais deﬁned in (2.7). Such a constraint is called consistency condition.

This type of model was ﬁrst proposed by Derman and Kani in [12] and stud-

ied mathematically by Carmona and Nadtochiy in [5]and[4]. Notice that the

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118 R. Carmona & S. Nadtochiy

consistency condition deﬁned by (3.6) is rather implicit and makes it very hard

to construct dynamic tangent diﬀusion models explicitly. Therefore, the main goal

of the following subsection is to express the consistency condition (3.6)intermsof

the input parameters of the model: σ, α and β.

3.3. Consistency of dynamic tangent diﬀusion models

The above question turns out to be equivalent to obtaining a necessary and suﬃcient

conditions for the call prices CSt,˜atproduced by the code-book to be martingales.

Starting from Itˆo’s dynamics for h(or equivalently ˜a), an inﬁnite dimensional

version of Itˆo’s formula shows that call prices are semi-martingales, and being able

to compute their drifts should lead to consistency conditions merely stating that the

call prices are martingales (i.e. setting the drifts to zero, since the local martingale

property is enough in this case). Clearly, this reasoning depends upon proving that

the mapping provided by the operator Fis twice Fr´echet diﬀerentiable. This strategy

for the analysis of no-arbitrage was used in [5], and a more transparent proof fo the

Fr´echet-diﬀerentiability is presented in [4], whose main result we state below after

we agree to denote by p(h) the fundamental solution of the forward PDE

∂τw(τ,x)=eh(τ,x)Dxw(τ,x)

and by q(h) the fundamental solution of the dual (backward) PDE

∂τw(τ,x)=−eh(τ,x)Dxw(τ,x)

It is proven in [4] that once the proper function spaces are chosen, the operator

F(acting on appropriate domain ˜

B, deﬁned in Sec. 2.2 of [4]) is twice continuously

Frech´et-diﬀerentiable, and that for any h, h,h

∈˜

B,wehave

F(h)[h]=1

2K[p(h),h

eh,q(h)],

and

F(h)[h,h

]= 1

2(K[I[p(h),h

eh,p(h)],h

eh,q(h)]

+K[p(h),h

eh,J[q(h),h

eh,q(h)]])

where the operators I,J,andKare deﬁned by

•I[Γ2,f,Γ1](τ2,x

2;τ1,x

1)

:= τ2

τ1R

Γ2(τ2,x

2;u, y)f(u, y)DyΓ1(u, y ;τ1,x

1)dydu,

•J[Γ2,f,Γ1](τ2,x

2;τ1,x

1)

:= τ2

τ1R

DyΓ2(τ2,x

2;u, y)f(u, y)Γ1(u, y ;τ1,x

1)dydu,

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Tangent Models as a Mathematical Framework for Dynamic Calibration 119

•K[Γ2,f,Γ1](τ2,x

2;τ1,x

1)

:= τ2

τ1R

Γ2(τ2,x

2;u, y)eyf(u, y)Γ1(u, y ;τ1,x

1)dydu.

Finally, as it is shown in [4]and[5], if we use the notations D∗

x:= 1

2(∂2

x+∂x),

L(ht):=logq(ht), and Lfor the Fr´echet derivative of L, and provided that Sis a

martingale and processes αand βare chosen to take values in appropriate spaces

(again, see Sec. 3 of [4] for the deﬁnitions of appropriate spaces), then we have

adynamic tangent diﬀusion model if and only if the following two conditions are

satisﬁed:

(1) Drift restriction:

αt=∂τht−σ2

tD∗

xht+σt∂xβ1−1

2(β1

t+σt∂xht)2−1

2

∞

n=2

βn2

t

−(β1

t−σt∂xht)(L(ht)[β1

t]−σt∂xL(ht)) −

∞

n=2

βn

tL(ht)[βn

t] (3.7)

(2) Spot volatility speciﬁcation:

2logσt=ht(0,0).(3.8)

From the form of the above drift condition (3.7) and the spot volatility speciﬁ-

cation condition (3.8), it looks like βis a free parameter whose choice completely

determines both αand σ. And the following strategy appears as a natural method of

constructing dynamic tangent diﬀusion models: choose a vector of random processes

βand deﬁne process has the solution of the following SDE

dht=α(ht,β

t)dt +βt·dBt,(3.9)

where α(ht,β

t) is given by the right hand side of (3.7). Having the dynamics of h,

we obtain the time evolution of σ(via (3.8)) and, therefore, S. However, studying

equation (3.9) is extremely diﬃcult due to the complicated structure of the drift

condition (3.7), and in particular the operator Linvolved in it. Therefore, the

problem of existence of the solution to the above SDE is still open. In addition, to

the best of our knowledge, the only explicit example of βtand htwhich produce a

tractable expression for the drift in the right hand side of (3.7) is the “ﬂat” case:

βn

t(.,.)≡const and ht(.,.)≡const. However, as discussed at the end of Sec. 6

in [5], any regular enough stochastic volatility model falls within the framework of

dynamic local volatility and, therefore, gives an implicit example of αand βthat

satisfy condition (3.7). This set of examples, of course, is not satisfactory since the

way such (classical) models are constructed does not agree with the market model

philosophy (discussed in the introduction) and, hence, produces very rigid dynamics

of local volatility.

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120 R. Carmona & S. Nadtochiy

Even though we don’t have a formal proof of the existence of the solution to (3.9)

for any admissible β, we have everything needed in order to approximate the solution

(assuming it exists) with an explicit Euler scheme: given the input, h0and β,make

the step from htto ht+∆tby “freezing” the coeﬃcients of the equation, α(ht,β

t)

and βt. This method allows one to simulate (approximately, due to the numerical

error of the Euler method) future arbitrage-free evolution of h(and hence the call

prices) by choosing its diﬀusion coeﬃcient β. The Euler scheme itself is guaranteed

to work, in the sense that it will always produce future values of ht(.,.), however,

in order for these values to make sense, they need to satisfy the conditions imposed

on the code-book values: in other words, ht(.,.) has to be a regular enough surface

so that one could compute the corresponding call prices via (2.7). The regularity of

ht(.,.), simulated via the above Euler scheme, can be violated if the drift α(ht,β

t)

does not always produce a regular enough surface. It turns out that, in order to

make sure that the right hand side of drift restriction (3.7) is regular enough at

time t, one has to choose βt(τ,x) satisfying certain additional restrictions as τ→0,

i.e. βis not a completely free parameter (see Sec. 5.2 of [5] for precise conditions

βt(0,x) has to satisfy).

3.4. When shouldn’t local volatility models be used?

Leaving aside the problem of existence of a solution to (3.9), another, more funda-

mental, question is the applicability of the diﬀusion-based code-book: “Given a set

of call option prices,when can we use the local volatility as a (static)code-book?”

A classical result of Gy¨ongy [18] shows that this is possible if the true underlying

Sis an Itˆo process satisfying some mild regularity conditions. However, if the true

underlying dynamics have a non-trivial jump component, the local volatility func-

tion ˜a(T,K) given by Dupire’s equation (2.2) will be singular as T0. To see this,

recall that for all K=S0, the denominator of the right hand side of (2.2)converges

to zero as T0. Indeed, the second derivative of the call price with respect to strike

is given by the density of the marginal distribution of the underlying index at time

Twhenever this density exists. To conclude, it is enough to notice (and this can be

done by an application of the Itˆo’s formula, or using (5.3) in the case of exponential

L´evy processes) that, in the presence of jumps, the T-derivative of call prices does

not necessarily vanish as T0, which yields the explosion mentioned above. In fact,

one can detect (at least in theory) the presence of jumps in the underlying (or the

lack of thereof) by observing the short-maturity behavior of the implied volatility:

it also explodes when the underlying has a non-trivial jumps component. In addi-

tion, at-the-money short-maturity behavior of the implied volatility may allow us

to test for the presence of continuous component as in the pure jump models, at-

the-money implied volatility vanishes as T0. The detailed discussion of the above

can be found in [10,17,27,30] and references therein.

Our work [6] on tangent L´evy models was a natural attempt to depart from

the assumption that Sis an Itˆo process, and introduce jumps in its dynamics. The

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Tangent Models as a Mathematical Framework for Dynamic Calibration 121

natural question: “What is the right substitute for the local volatility code-book in

this case? ” is addressed in the next section.

4. Dynamic Tangent L´evy Models

Using processes with jumps in ﬁnancial modeling goes back to the pioneering work

of Merton [26]. Fitting option prices with L´evy-based models has also a long history.

At the risk of missing important contributions, we mention for example the series

of works by Carr, Geman, Madan, Yor and Seneta between 1990 and 2005 [7,9,25]

on models with jumps of inﬁnite activity, such as the Variance Gamma (VG) and

CGMY models, and the easy to use double exponential model of Kou [23]. Still in

the static case at time t=0,Carret al. noticed in their 2004 paper [8] that Dupire’s

local volatility can be interpreted as an St-dependent time change. On this ground,

they introduced Local L´evy models which they deﬁned as Markovian time changes

of a L´evy process. However, following their approach to deﬁne a code-book would

lead to the same level of complexity in the formulation of the consistency of the

models. For this reason, we chose to deﬁne the code-book in a diﬀerent way — via

the tangent Le´evy models (see the deﬁnition in Sec. 2.6).

4.1. The L´evymeasurecodebook

Formula (2.9) deﬁning the notation Cs, ˜κ(τ,x) for the European call prices in pure

jump exponential additive models can be used, together with the speciﬁcation of

“s” as the current value of the underlying, to establish a code-book, and as it was

demonstrated above, in order to construct a dynamic tangent model we only need to

prescribe the dynamics of the code value (s, ˜κ)andmakesuretheyareconsistent.

However, in order to study consistency of the code-book dynamics, we need to

have a convenient analytic representation of the cod e : the associated transform

between call prices and (s, ˜κ). With this goal in mind, we introduce the Partial

Integral Diﬀerential Equation (PIDE) representation of the call prices in pure jump

exponential additive models:

∂τCs,˜κ(τ, x)=R

ψ(˜κ(τ,·); x−y)DyCs,˜κ(τ, y)dy

Cs,˜κ(τ, x)|τ=0 =(s−ex)+,

(4.1)

where the double exponential tail function ψis deﬁned by

ψ(f;x)=

x

−∞

(ex−ez)f(z)dz x < 0

∞

x

(ez−ex)f(z)dz x > 0.

(4.2)

Clearly, the presence of convolutions and constant coeﬃcient diﬀerential operators

in (5.3) are screaming for the use of Fourier transform.

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122 R. Carmona & S. Nadtochiy

4.2. Fourier transform

Unfortunately, the setup is not Fourier transform friendly as the initial condition

in (5.3) is not an integrable function! In order to overcome this diﬃculty, we work

with derivatives. Before taking Fourier transform (we use a “hat” for functions in

Fourier space), we diﬀerentiate both sides of the PIDE (5.3) using the notation

∆s,˜κ(τ, x)=−∂xCs,˜κ(τ,x).

∂τˆ

∆s,˜κ(τ, ξ)=−(4π2ξ2+2πiξ)ˆ

ψ(˜κ(τ,·),ξ)ˆ

∆s,˜κ(τ, ξ)

ˆ

∆s,˜κ(τ, ξ)|τ=0 =exp{log s(1 −2πiξ)}

1−2πiξ

(4.3)

The above equation gives us a mapping:

Cs,˜κ→ˆ

∆s,˜κ→ˆ

ψ→˜κ.

Conversely, in order to go from ˜κand sto call prices we only need to solve the

evolution equation in Fourier domain (4.3), and obtain ˆ

∆s,˜κin closed form. We

recover ∆s,˜κ(T,x)=−∂xCs,˜κ(T,x) by inverting the Fourier transform. A plain

integration gives

Cs,˜κ(τ, x)=slim

λ→+∞R

e2πiξλ −e2πiξ(x−log s)

2πiξ(1 −2πiξ)

·exp−2π(2πξ2+iξ)τ

0

ˆ

ψ(˜κ(u, ·),ξ)dudξ

providing the required inverse mapping: ˜κ→ˆ

ψ→Cs,˜κ.

4.3. Formal deﬁnition of dynamic tangent L´evy models and

consistency results

In this case we assume that the set of liquid derivatives consists of call options

with all possible strikes and with maturities not exceeding some ﬁxed ¯

T>0. As

in the case of tangent diﬀusion models we need to put the code value (s, ˜κ)in

motion by constructing a pair of stochastic processes (st,˜κt)t∈[0,¯

T]under the pricing

measure. As before, we would like to keep the true model for the dynamics of

the underlying index as general as possible while keeping the computations at a

reasonable level of complexity. In this section, we assume that under the pricing

measure, the underlying index Sis a positive pure jump martingale given by

St=S0+t

0R

Su−(ex−1)(M(dx, du)−Ku(x)dxdu) (4.4)

for some (unknown) integer valued random measure Mwhose predictable compen-

sator is absolutely continuous, i.e. of the form Ku,ω (x)dxdu for some stochastic

process (Ku) with values in the Banach space B0constructed in Sec. 3.1 of [6].

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Tangent Models as a Mathematical Framework for Dynamic Calibration 123

It may seem too restrictive to assume that the underlying process has no con-

tinuous martingale component and that the compensator of Mis absolutely con-

tinuous. These assumptions are dictated by our choice of the code-book, which is

based on pure jump processes without ﬁxed points of discontinuity. Indeed, as we

explained earlier, the short-maturity properties of call prices produced by pure jump

models are incompatible with the presence of a continuous component in the under-

lying dynamics. Nevertheless, we propose an extension of the present code-book in

the next section, and as a result, allow for slightly more general dynamics of the

underlying.

If we want the model corresponding to (st,˜κt) to be almost surely tangent to the

true model at time t(in other words, if we want (st,ω,˜κt,ω )tobeacodevaluethat

reproduces the market prices at time t, for almost all ω∈Ω), then sthas to coincide

with St, and its dynamics must be given by (4.4). Therefore, the only additional

process whose time-evolution we need to specify is (˜κt)t∈[0,¯

T]. In this case, it is more

convenient to use the time-of-maturity Tinstead of the time-to-maturity τ,sowe

introduce the L´evy density κt(T,x) deﬁned by

κt(T,x)=˜κt(T−t, x),

and we specify its dynamics by an equation of the form

κt=κ0+t

0

αudu +

m

n=1 t

0

βn

udBn

u(4.5)

where B=(B1,...,Bm)isanm-dimensional Brownian motion (mcan be ∞), α

is a progressively measurable integrable stochastic process with values in a Banach

space Bdeﬁned in Sec. 3.1 of [6], and β=(β1,...,βm) is a vector of progressively

measurable square integrable stochastic processes taking values in a Hilbert space

Hdeﬁned also in Sec. 3.1 of [6].

Notice again, that the dynamics of κtcould, in principle, include jumps. How-

ever, we chose to restrict our framework to the continuous evolution of κin order

to keep the results and their derivations more transparent.

Thus, a dynamic tangent L´evy model is deﬁned by the pair of equations (4.4)

and (4.5), given that such dynamics are consistent, or in other words, given that

for any (T,x)∈(0,¯

T]×Rthe following equality is satisﬁed almost surely for all

t∈[0,T)

CSt,˜κt(T−t, x)=E((ST−K)+|F

t)

As in Sec. 3, the above formulation of the consistency condition is not very

convenient. It is important to characterize the consistency of code-book dynamics,

(4.4)and(4.5), explicitly in terms of the input parameters: α,βand K. Such an

explicit formulation of the consistency condition is one of the main results of [6],

and it is given in Theorem 12 of the above mentioned paper. In order to state this

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124 R. Carmona & S. Nadtochiy

result we introduce the notation:

¯

βn

t(T,x):=T

t∧T

βn

t(u, x)du, Ψ(f;x)=−exsign(x)∞

x

f(y)dy. (4.6)

Assuming that Sis a true martingale, κ≥0andβsatisﬁes the regularit y

assumptions RA1–RA4 given in Sec. 3.2 of [6], the code-book dynamics given by

(4.4)and(4.5) are consistent if and only if the following conditions are satisﬁed:

(1) Drift restriction:

αt(T,x)=−e−x

m

n=1 R

∂2

y2Ψ( ¯

βn

t(T); y)

×[Ψ(βn

t(T); x−y)−(1 −y∂x)Ψ(βn

t(T); x)]

−2∂yΨ( ¯

βn

t(T); y)[Ψ(βn

t(T); x−y)−Ψ(βn

t(T); x)]

+Ψ( ¯

βn

t(T); y)Ψ(βn

t(T); x−y)dy, (4.7)

(2) Compensator speciﬁcation:Kt(x)=κt(t, x).

4.4. Model speciﬁcation and existence result

Denoting by ρthe weight function

ρ(x):=e−λ|x|(|x|−1−δ∨1),

with some λ>1andδ∈(0,1), and switching from κtto ˇκtgiven by ˇκt(T,x)=

κt(T,x)/ρ(x), we can easily force ˇκtto take values in a more convenient space of

continuous functions, in which its maximal and minimal values can be controlled.

Introducing the weighted drift ˇαt=αt/ρ, weighted diﬀusion terms {ˇ

βn

t=βn

t/ρ}m

n=1

(which take values in corresponding function spaces, ˜

Band ˜

H,deﬁnedinSec.5.1

of [6]) and the stopping time

τ0=inft≥0: inf

T∈[t, ¯

T],x∈R

ˇκt(T,x)≤0,

(τ0is predictable and ˇκt∧τ0is nonnegative), we can specify the model as follows:

•Assume that the market ﬁltration supports a Brownian motion {Bn}m

n=1 and an

independent Poisson random measure Nwith compensator ρ(x)dxdt.

•Denote by {(tn,x

n)}∞

n=1 the atoms of N.ThenmeasureM(recall (4.4)) can be

deﬁned by its atoms

{(tn,W[ˇκtn(tn,·)](xn))}∞

n=1,

for some deterministic mapping f(·)→ W[f](·), so that it has the desired com-

pensator ρ(x)ˇκt(t, x)dxdt, and therefore, the compensator speciﬁcation is satisﬁed.

An explicit expression for Wis given in Sec. 5 of [6].

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Tangent Models as a Mathematical Framework for Dynamic Calibration 125

•Rewrite the right hand side of drift restriction (4.7)using ˇ

βinstead of β,and

denote the resulting quadratic operator by Qˇ

βt(T,x). Construct ˇκtby integrating

“Qˇ

βtdt +ˇ

βt·dBt”, and stop it at τ0. Such ˇκwill satisfy the drift restriction and

the nonnegativity property.

In addition, βt=ρˇ

βtsatisﬁes the regularity assumptions RA1–RA4 in Sec. 3.2

of [6] due to the choice of state space for ˇ

βt(the Hilbert space ˜

Hdeﬁned in Sec. 5.1

of [6]).

•If we also choose ˇ

βto be independent of N, we can guarantee that S, produced

by (4.4) and the above choice of M, is a true martingale. Thus, the above speci-

ﬁcation allows to determine the model uniquely through N,Band ˇ

β.

As a result we obtain the following class of code-book dynamics:

St=S0+t

0R

Su−(exp(W[ˇκu(u, ·)](x)) −1)(N(dx, du)−ρ(x)dxdu),

˜κt(τ, x)=ρ(x)ˇκt(t+τ, x),ˇκt=ˇκ0+t

0

Qˇ

βu1u≤τ0du +

m

n=1 t

0

ˇ

βn

u1u≤τ0dBn

u

(4.8)

Theorem 2 in [6] states that for any square integrable stochastic process ˇ

βthe

above system has a unique solution, and if, in addition, ˇ

βis independent of N,then

the resulting processes (St)t∈[0,¯

T]and (˜κt)t∈[0,¯

T]are consistent, and, therefore, form

adynamic tangent L´evy model.

This “local existence” result, albeit limited (the presence of stopping time τ0and

the independence assumption should eventually be relaxed, as it is demonstrated by

the example that follows), provides a method for construction of the future evolution

of the code value, starting from any given one. In practice, it means that, if we are

able to calibrate a model from the chosen space of pure jump exponential additive

models to the currently observed option prices, we can use the above result to

generate a large family of dynamic stochastic models for the future joint evolution of

the option prices (or, equivalently, the implied volatility surface) and the underlying.

4.5. Example of a dynamic tangent L´evy model

The following tangent L´evy model was proposed in [6]. Its analysis and implemen-

tation on real market data is being carried out in [3]. Here we outline the main

steps of the analysis to illustrate the versatility of the model, and the fact that it

does provide an answer to the nagging question of the Monte Carlo simulation of

arbitrage free time evolutions of implied volatility surfaces.

•Choose m=1,and ˇ

βt(T,x)=γtC(x),

•Let γt=γ(ˇκt,t):= σ

(infT∈[t, ¯

T],x∈Rˇκt(T,x)∧), for some σ, > 0,

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126 R. Carmona & S. Nadtochiy

•and C(x)=e−λ|x|(|x|∧1)1+δ˜

C(x), for some λ>0,δ ∈(0,1) and some bounded

absolutely continuous function ˜

C, with bounded derivative, such that

R

(ex−1)e−(λ+λ)|x|(|x|∧1)−δ˜

C(x)dx =0,

and

R

e−(λ+λ)|x|(|x|∧1)−δ˜

C(x)dx =0

•Then

dˇκt(T,x)=γ2(ˇκt,t)(T−t∧T)A(x)dt +γ(ˇκt,t)C(x)dBt,(4.9)

where Ais obtained from Cvia the “drift restriction”, which in this case (due to

the properties of ˜

Cpresented above) takes its simplest form, namely:

A(x)=−1

ρ(x)R

ρ(y)C(y)ρ(x−y)C(x−y)dy

Please, see Sec. 6 of [6] for the derivation of the above formulae.

It is worth mentioning that, as shown in Proposition 17 of [6], the process ˇκ

deﬁned by (4.9) always stays positive. In addition, as discussed in [6], the above

example can be extended to diﬀusion coeﬃcients of the form γtC(T,x), and, of

course, one can consider ˇ

βn(·,·)’s given by functions “C” of diﬀerent shapes. These

functions, {Cn}, would correspond to diﬀerent Brownian motions and can be esti-

mated, for example, via the analysis in principal components (or an alternative

statistical method) of the time series of ˜κt(·,·), ﬁtted to the historical call prices on

dates tof a recent past.

5. Extension of Dynamic Tangent L´evy Models

Notice that the dynamic tangent L´evy models introduced above do not allow for

a continuous martingale component in the evolution of the underlying. This is a

direct consequence of our choice of the space of tangent models: by being pure jump

martingales, they force the evolution of the underlying index to have pure jump

dynamics since short time asymptotic properties of the marginal distributions of

pure jump processes are incompatible with the presence of continuous martingale

component (recall the discussion in Sec. 3.4). In this section we consider an extension

of the space of tangent L´evy models introduced above, which includes underlying

processes with nontrivial continuous martingale components.

In the deﬁnition of tangent L´evy models given in Sec. 2.6, we now allow the

tangent processes ˜

Stobegivenbyanequationoftheform

˜

St=s+t

0

˜

Σ(u)d˜

Bu+t

0R

˜

Su−(ex−1)[N(dx, du)−˜κ(u, x)dxdu],(5.1)

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Tangent Models as a Mathematical Framework for Dynamic Calibration 127

for a one-dimensional Brownian motion ˜

Band an independent Poisson random

measure Nwhose compensator we denote ˜κ(u, x)dxdu. The class of such models

is then parameterized by (s, ˜

Σ(·),˜κ(.,.)). As before, we introduce the call prices

produced by (s, ˜

Σ,˜κ)

Cs, ˜

Σ,˜κ(τ,x)=E(˜

Sτ−ex)+,(5.2)

and derive their analytic representation via the following PIDE:

∂τCs, ˜

Σ,˜κ(τ,x)=1

2˜

Σ(τ)DxCs, ˜

Σ,˜κ(T,x)+R

ψ(˜κ(τ,·); x−y)DyCs, ˜

Σ,˜κ(τ,y)dy

Cs,˜κ(τ, x)|τ=0 =(s−ex)+,

(5.3)

where Dx=∂2

x2−∂xand ψis deﬁned in (4.2). Analogous to the case of pure jump

L´evy code-book, we introduce ∆s, ˜

Σ,˜κ(τ,x)=−∂xCs,˜

Σ,˜κ(τ,x), and ˆ

∆s, ˜

Σ,˜κ(τ,ξ)as

the Fourier transform of ∆s, ˜

Σ,˜κ(τ,·). Then we can rewrite (4.3) in the present setup

(with one additional term on the right hand side of the equation) and obtain

ˆ

∆s, ˜

Σ,˜κ(τ,ξ)=e(1−2πiξ)logs

1−2πiξ exp−2π(2πξ2+iξ)τ

0

1

2˜

Σ2

t(u)+ ˆ

ψ(˜κt(u, ·); ξ)du,

(5.4)

where ˆ

ψis the Fourier transform of ψ.

Given s, we obtain the desired one-to-one correspondence:

Cs, ˜

Σ,˜κ↔ˆ

∆s, ˜

Σ,˜κ↔(˜

Σ,ˆ

ψ)→(˜

Σ,˜κ).

Finally, we choose a stochastic motion in the code-book, producing the following

dynamics of the code value:

st=St,S

t=S0+t

0

SuσudB1

u+t

0R

Su−(ex−1)(M(dx, du)−Ku(x)dxdu),

˜κt(τ, x)=κt(t+τ,x),κ

t=κ0+t

0

αudu +

m

n=1t

0

βn

udBn

u,

˜

Σt(τ)=Σ

t(t+τ),Σt=Σ

0+t

0

µudu +

m

n=1t

0

νn

udBn

u,

(5.5)

where B=(B1,...,Bm) is a multidimensional Brownian motion, Mis an inte-

ger valued random measure with predictable compensator Ku,ω(x)dxdu;(Kt)t∈[0,¯

T]

is a predictable integrable stochastic process with values in the Banach space

B0;(αt)t∈[0,¯

T]and (µt)t∈[0,¯

T]are progressively measurable integrable stochastic

processes with values in Banach spaces Band C([0,¯

T]) respectively; (βn)t∈[0,¯

T]

and (νn)t∈[0,¯

T]are progressively measurable square integrable stochastic processes

February 25, 2011 11:42 WSPC/S0219-0249 104-IJTAF SPI-J071

S0219024911006280

128 R. Carmona & S. Nadtochiy

taking values in the Hilbert spaces Hand W1,2([0,¯

T]) respectively. Recall that

C([0,¯

T]) is the space of continuous functions on [0,¯

T], equipped with “sup” norm,

and W1,2([0,¯

T]) is the space of absolutely continuous functions on [0,¯

T]withsquare

integrable derivatives. Spaces B0,Band Hare constructed in Sec. 3.1 of [6], and

their deﬁnitions are presented in Appendix A. The following result characterizes

the consistency of the above dynamics.

Theorem 5.1. Assume that (St)t∈[0,¯

T]is a martingale,βsatisﬁes the regular-

ity assumptions RA1–RA4 in Sec. 3.2of [6](see Appendix A for their exact

formulations)and κt(T, x)≥0,almost surely for all t∈[0,¯

T)and almost all

(T,x)∈[t, ¯

T]×R. Then processes (St,˜

Σt,˜κt)t∈[0,¯

T]satisfying (5.5)are consistent,

in the sense that

CSt,˜

Σt,˜κt(T−t, x)=E((ST−ex)+|F

t)almost surely,

for all x∈Rand 0≤t<T ≤¯

T,

if and only if the following conditions hold almost surely for almost every x∈R

and t∈[0,¯

T),and all T∈(t, ¯

T]:

(1) Drift restriction:

αt(T,x)=−e−x

m

n=1 R

∂2

y2Ψ( ¯

βn

t(T,·); y)

×[Ψ(βn

t(T,·); x−y)−(1 −y∂x)Ψ(βn

t(T,·); x)]

−2∂yΨ( ¯

βn

t(T,·); y)Ψ(βn

t(T,·); x−y)−Ψ(βn

t(T,·); x)]

+Ψ(¯

βn

t(T,·); y)Ψ(βn

t(T,·); x−y)dy +σt∂xβ1

t(T,x),

(2) Compensator speciﬁcation:Kt(x)=κt(t, x),

(3) Volatility speciﬁcation:σ2

t=Σ

2

t(t),

(4) Stability of volatility :µ≡0,ν ≡0,

where Ψand ¯

βare deﬁned in (4.6).

Proof. First we prove that consistency of the code-book dynamics (5.5)isequiv-

alent to the local martingale property of ( ˆ

∆St,˜

Σt,˜κt(T−t, ξ))t∈[0,T ), for all ξ∈R

(see (5.4) for the deﬁnition of ˆ

∆s, ˜

Σ,˜κ). The proof of this equivalence is, essentially,

a repetition of the propositions and corollaries from Sec. 4 of [6]andtheﬁrstpart

of the proof of Theorem 12 in the above mentioned paper.

Recall Proposition 6 from Sec. 4 of [6], which states that the code-book dynamics

are consistent if and only if the call prices (CSt,˜

Σt,˜κt(T−t, x))t∈[0,¯

T)produced by the

code values are martingales. It is not hard to see, by essentially repeating the proof

of the proposition, that its statement holds in the present setup. The necessity of the

martingale property is obvious, let’s prove the suﬃciency. Notice that, as it is shown

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Tangent Models as a Mathematical Framework for Dynamic Calibration 129

in [6], ψis a bounded linear operator from B0to L1(R), and since κt(T, ·)B0and

ΣtC([0,¯

T]) are almost surely bounded over t∈[0,T], we conclude that

ˆ

∆St,˜

Σt,˜κt(T−t, ξ)→elog ST−(1−2πiξ)

(1 −2πiξ),

in L2(R) as a function of ξ,astT. Then we invert the fourier transform on both

sides of the above to obtain that ∆St,˜

Σt,˜κt(T−t, x)convergestoex1(−∞,log ST−](x).

Passing to the call prices CSt,˜

Σt,˜κt(T−t, x) via integration we conclude that they

converge to the payoﬀ (ST−ex)+as tT(recall that ST−=STalmost surely).

Thus, we conclude that, by construction of the code and due to the regularity of

the code-book dynamics, the call prices produced by the code values converge to

the right payoﬀs. Therefore, since call price processes (CSt,˜

Σt,˜κt(T−t, x))t∈[0,¯

T)are

uniformly integrable (bounded by St), whenever they are martingales they have to

coincide with the corresponding conditional expectations, which implies consistency

of the model.

Now we need to prove that the martingale property of call prices (CSt,˜

Σt,˜κt(T−

t, x))t∈[0,¯

T)is equivalent to the local martingale property of ( ˆ

∆St,˜

Σt,˜κt(T−

t, ξ))t∈[0,¯

T). This, again, can be done along the lines presented in Sec. 4 of [6]. First,

since CSt,˜

Σt,˜κtis bounded by Stas shown above, the martingale property can indeed

be substituted to the local martingale property. Next, we notice that ˆ

∆St,˜

Σt,˜κt(τ,·)

can be obtained from CSt,˜

Σt,˜κt(τ,·) by a composition of diﬀerentiation and Fourier

transform, which is a linear operator and hence, in principle, should preserve the

local martingale property. However, in order to apply this logic one needs to choose

the right function spaces on which the above linear operator is bounded. A typical

choice would be to embed the above processes into some Banach space such that

the corresponding operator maps this space into itself. This approach turns out to

be quite problematic since in the present case the Fourier transform is understood

in the generalized sense, and there is no standard Banach space it would preserve.

Hence, we consider ( ˆ

∆St,˜

Σt,˜κt(T−t, ·))t∈[0,¯

T)as a process in S∗,andshowthatits

local martingale property in the “weak sense” is equivalent to the local martingale

property of the call prices. Recall that S∗is the topological dual of S,theSchwartz

space of (complex-valued) C∞functions on Rwhose derivatives of all orders decay

at inﬁnity faster than any negative power of |x|. Then any polynomially bounded

Borel function fis an element of S∗since it can be viewed as a continuous functional

on Svia the duality

f,φ=R

f(x)φ(x)dx. (5.6)

This particular choice of the function space is dictated by the fact that both diﬀer-

entiation and Fourier transform map S∗into itself and are invertible on this space.

We then deﬁne the “weak” local martingale property of a stochastic process (Xt)

with values in S∗as the local martingale property of (Xt,φ) for all φ∈S.Itis

February 25, 2011 11:42 WSPC/S0219-0249 104-IJTAF SPI-J071

S0219024911006280

130 R. Carmona & S. Nadtochiy

shown in the ﬁrst part of the proof of Theorem 12 in [6] that the local martingale

property of (CSt,˜

Σt,˜κt(T−t, x))t∈[0,¯

T), for all x∈R, is equivalent to the weak local

martingale property of ( ˆ

∆St,˜

Σt,˜κt(T−t, ·))t∈[0,¯

T).

Thus, we need to characterize the weak local martingale property of

(ˆ

∆St,˜

Σt,˜κt(T−t, ·))t∈[0,¯

T)in terms of the input parameters of the model. Recall

that (5.3) provides an explicit formula for ˆ

∆St,˜

Σt,˜κt(T−t, ξ)intermsof(St,Σt,κ

t).

Then, for ﬁxed (T,ξ), we can apply an inﬁnite dimensional Itˆo’s formula to the pro-

cess ( ˆ

∆St,˜

Σt,˜κt(T−t, ξ))t∈[0,¯

T)(the process itself is one dimensional but the input,

(St,Σt,κ

t) is inﬁnite dimensional) to compute its drift. Notice that

ˆ

∆St,˜

Σt,˜κt(T−t, ξ)= ˆ

∆St,˜κt(T−t, ξ)exp

−π(2πξ2+iξ)T

t

Σ2

t(u)du,

where ˆ

∆s,˜κis deﬁned by (4.3), or more explicitly in equation (13) of [6]. The

semimartingale decomposition of ˆ

∆St,˜κt(T−t, ξ) is provided in Corollary 9 of [6],

therefore we only need to compute the semimartingale decomposition of the addi-

tional factor. Applying Itˆo’s lemma for conditional Banach spaces (see, for example,

Theorem III.5.4 in [24]), we obtain

dexp−π(2πξ2+iξ)T

t

Σ2

t(u)du

=exp

−π(2πξ2+iξ)T

t

Σ2

t(u)duπ(2πξ2+iξ)

·Σ2

t(t)−T

t

2Σt(u)µt(u)+

m

n=1

νn

t(u)2du

+2π(2πξ2+iξ)

m

n=1 T

t

Σt(u)νn

t(u)du2

dt

−

m

n=1 T

t

2Σt(u)νn

t(u)du dBn

t

Combining the above decomposition with Corollary 9 and Proposition 7 of [6]we

apply classical Itˆo’s rule to a product of two processes to obtain

d(ˆ

∆St,˜

Σt,˜κt(T−t, ξ)) = ˆ

∆St,˜

Σt,˜κt(T−t, ξ)2πiξ(1 −2πiξ)

·ˆ

ψ(κt(t, ·)−Kt(·); ξ)+1

2(Σ2

t(t)−σ2

t)−T

t

Σt(u)µt(u)

+1

2

m

n=1

νn

t(u)2du −T

t

ˆ

ψ(αt(u, ·); ξ)du +πiξ(1 −2πiξ)

×

m

n=1 T

t

Σt(u)νn

t(u)+ ˆ

ψ(βn

t(u, ·); ξ)du2

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Tangent Models as a Mathematical Framework for Dynamic Calibration 131

−(1 −2πiξ)σtT

t

Σt(u)ν1

t(u)+ ˆ

ψ(β1

t(u, ·); ξ)dudt

−

m

n=1 T

t

Σt(u)νn

t(u)+ ˆ

ψ(βn

t(u, ·); ξ)du dBn

t

+R

ˆ

∆St−,˜

Σt,˜κt(T−t, ξ)(ex(1−2πiξ)−1)

×[M(dx, dt)−Kt(x)dxdt]

Denote the drift in the right hand side of the above equation by Γt(T,ξ).

Notice that the above decomposition holds almost surely for ξﬁxed. However, since

ˆ

∆St,˜

Σt,˜κt(T−t, ·) is continuous, we conclude that this decomposition holds almost

surely for all ξ∈R. Then we can apply stochastic Fubini’s theorem (see, for exam-

ple, Theorem 65 in [29]) to obtain, for any φ∈S

dˆ

∆St,˜

Σt,˜κt(T−t, ·),φ=Γt(T,·),φdt +dZt,

where Zis a local martingale. The conditions needed for the application of stochastic

Fubini’s theorem can be veriﬁed by repeating the proof of Proposition 10 in [6].

We have shown that the model is consistent if and only if, for any φ∈Sand

T∈(0,¯

T],Γt(T,·),φ= 0 almost surely for almost all t∈[0,T). We can choose a

dense countable subset of Sand recall that Γt(·,·) is continuous to conclude that

consistency is equivalent to: almost surely,Γt(T, ξ)=0for all ξ∈Rand T∈(0,¯

T]

and almost all t∈[0,T). Since ˆ

∆St,˜

Σt,˜κt(T−t, ξ)= 0, we will search for necessary

and suﬃcient conditions for Γt(T,ξ)/ˆ

∆St,˜

Σt,˜κt(T−t, ξ) to be zero, for all T∈(t, ¯

T)

and ξ∈R. Since this expression is absolutely continuous as a function of T∈[t, ¯

T],

it vanishes if and only if its value at T=tis zero and the value of its T-derivative

is zero for all (T,ξ)∈(t, ¯

T)×R. Thus, we obtain a system of two equations:

ˆ

ψ(κt(t, ·)−Kt(·); ξ)+1

2Σ2

t(t)−σ2

t=0,

2πiξ(1 −2πiξ)

m

n=1

(Σt(T)νn

t(T)+ ˆ

ψ(βn

t(T,·); ξ))

×T

t

Σt(u)νn

t(u)+ ˆ

ψ(βn

t(u, ·); ξ)du

−Σt(T)µt(T)−ˆ

ψ(αt(T,·,); ξ)+ 1

2

m

n=1

νn

t(T)2

−(1 −2πiξ)σt(Σt(T)ν1

t(T)+ ˆ

ψ(β1

t(T,·); ξ)) = 0

Now, recall that Fourier transform of an absolutely integrable function con-

verges to zero as the argument goes to inﬁnity. Also notice that multiplication

by “2πiξ” in the Fourier domain corresponds to taking derivative in the orig-

inal domain. Due to the regularity assumptions RA1–RA4 (see Appendix A),

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132 R. Carmona & S. Nadtochiy

∂xψ(βn

t(T,·); x)=Ψ(βn

t(T,·); x) is absolutely integrable in x∈R, therefore,

ξˆ

ψ(βn

t(T,·); ξ)→0, as |ξ|→∞. Using this observation, we can split the above

system into the following parts

κt(t, x)−Kt(x)=0,Σ2

t(t)−σ2

t=0,

m

n=1

Σt(T)νn

t(T)T

t∧T

Σt(u)νn

t(u)du =0,Σt(T)µt(T)−1

2

m

n=1

νn

t(T)2=0,

2πiξ(1 −2πiξ)

m

n=1

ˆ

ψ(βn

t(u, ·); ξ)T

t∧T

ˆ

ψ(βn

t(u, ·); ξ)du

−(1 −2πiξ)σtˆ

ψ(β1

t(T,·); ξ)−ˆ

ψ(αt(T,·); ξ)=0,

which, after inverting the Fourier transform and operator ψ(see the end of the

proofofTheorem12inSec.4of[6]), yields the statement of the theorem.

As one can see, the parameter Σtin the above tangent models cannot change

as a continuous stochastic process in t, and therefore, the spot volatility σthas to

be deterministic. This surprising result can be interpreted as follows: calibrating

exponential additive model to the call option market at each time, assuming that

the parameters of the calibration change continuously, one has to keep the same

continuous quadratic variation component Σ2(·) in order to avoid arbitrage.

6. Conclusions

In this paper we introduce the general formalism of tangent models for construction

of market models for the time evolution of the prices of a speciﬁed set of liquidly

traded derivatives. According to this methodology, a market model is deﬁned by

the choice of a code- book for the prices of the target set of derivatives (the “mar-

ket prices”) and by prescribing statistics of the market prices through a stochastic

process for the code value. The above construction is motivated by the dynamic cal-

ibration frequently used by the practitioners and provides a rigorous mathematical

framework for this phenomenon.

We illustrate the above formalism by a review of recent work based on the

following representations of the call price surface:

•via Local Volatility surface,

•via Tangent L´evy Density.

Each of the above classes of models corresponds to a diﬀerent type of dynamics

of the underlying: continuous in the ﬁrst case and pure jump in the second, while

keeping the semimartingale property. Our description of tangent L´evy models is

complete in the sense that for any admissible value of the free parameter (taking

values in a given linear space), we can construct a unique arbitrage-free model for

the future stochastic evolution of the call price surface.

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Tangent Models as a Mathematical Framework for Dynamic Calibration 133

Finally, the last contribution of this paper is to generalize the class of tangent

L´evy models to include underlying processes with a continuous martingale compo-

nent, and to extend the characterization of the consistency of the model (including

the classical drift condition) to this enlarged class of models going beyond the pure

jump underlying models studied in [6].

Appendix A

Here we recall the deﬁnitions of spaces B0,Band H,aswellastheregularityassump-

tions RA1–RA4, presented in Sec. 3.1 of [6].

Banach space B0is the space of equivalence classes of Borel measurable functions

f:R→Rsatisfying

fB0:= R

(|x|∧1)|x|(1 + ex)|f(x)|dx < ∞.

Banach space Bconsists of absolutely continuous functions f:[0,¯

T]→B

0sat-

isfying

fB:= f(0)B0+¯

T

0

d

du f(u)

B0

du < ∞.

Recall that a Borel function f:[0,¯

T]→B

0is said to be absolutely continuous

if there exists a measurable function g:[0,¯

T]→B

0, such that for any t∈[0,¯

T]

we have

f(t):=f(0) + t

0

g(u)du,

where the above integral is understood as the Bochner integral (see p. 44 in [13]for

a deﬁnition) of a B0-valued function. In such a case, the equivalence class of such

functions gis denoted d

dt f.

Hilbert space H0is deﬁned as the space of equivalence classes of functions

satisfying

f2

H0:= R

|x|4(1 + ex)2|f(x)|2dx < ∞

(the inner product of H0being obtained by polarization), and the Hilbert space H

consists of absolutely continuous functions f:[0,¯

T]→H

0satisfying

f2

H:= f(0)2

H0+¯

T

0

d

du f(u)

2

H0

du < ∞.

Finally, we introduce

In,k

t,ε := sup

T∈[t, ¯

T]

[esssupx∈R\[−ε,ε](ex+1)|∂k

xkβn

t(T,x)|

+R

(ex+1)|x|3(|x|∧1)k−1|∂k

xkβn

t(T,x)|dx],

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S0219024911006280

134 R. Carmona & S. Nadtochiy

whenever the derivatives appearing in right hand side are well deﬁned, and re-

call the

Regularity Assumptions. For ea ch n≤m, almost surely, for almost every t∈

[0,¯

T], we have:

RA1: supT∈[t, ¯

T]1

−1|x||βn

t(T,x)|dx < ∞

RA2: For every T∈[t, ¯

T], the function βn

t(T,·) is absolutely continuous on R\{0}.

RA3: For any ε>0,In,0

t,ε +In,1

t,ε <∞.

RA4: For any T∈[t, ¯

T],R(ex−1)βn

t(T,x)=0.

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