Tangent models as a mathematical framework for dynamic calibration

Abstract

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 European call options for a specific set of strikes and maturities. While reviewing our recent results on dynamic local volatility and tangent Lévy models, we present a theory of tangent models unifying these two approaches and construct a new class of tangent Lévy models, which allows the underlying to have both continuous and pure jump components.

Full text

February 25, 2011 11:42 WSPC/S0219-0249 104-IJTAF SPI-J071
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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
Θθ↔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
0R
˜
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
τ1R
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
τ1R
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
τ1R
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 Lfor 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 T0. To see this,
recall that for all K=S0, the denominator of the right hand side of (2.2)converges
to zero as T0. 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 T0, 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 T0. 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)dudξ
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
0R
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)=−exsign(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=inft≥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
0R
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
0R
˜
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
0R
Su−(ex−1)(M(dx, du)−Ku(x)dxdu),
˜κt(τ, x)=κt(t+τ,x),κ
t=κ0+t
0
αudu +
m

n=1t
0
βn
udBn
u,
˜
Σt(τ)=Σ
t(t+τ),Σt=Σ
0+t
0
µudu +
m

n=1t
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
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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
ΣtC([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,astT. 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 tT(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
dexp−π(2πξ2+iξ)T
t
Σ2
t(u)du
=exp
−π(2πξ2+iξ)T
t
Σ2
t(u)duπ(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)du2
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)du2

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Tangent Models as a Mathematical Framework for Dynamic Calibration 131
−(1 −2πiξ)σtT
t
Σt(u)ν1
t(u)+ ˆ
ψ(β1
t(u, ·); ξ)dudt
−
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
dˆ
∆St,˜
Σt,˜κt(T−t, ·),φ=Γt(T,·),φdt +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
2Σ2
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),

February 25, 2011 11:42 WSPC/S0219-0249 104-IJTAF SPI-J071
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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
fB0:= R
(|x|∧1)|x|(1 + ex)|f(x)|dx < ∞.
Banach space Bconsists of absolutely continuous functions f:[0,¯
T]→B
0sat-
isfying
fB:= 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
f2
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
f2
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],

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