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Changes in variance, or volatility, over time can be modeled using the approach based on autoregressive conditional heteroscedasticity. Another approach is to model variance as an unobserved stochastic process. Although it is not easy to obtain the exact likelihood function for such stochastic variance models, they tie in closely with developments in finance theory and have certain statistical attractions. This article sets up a multivariate model, discusses its statistical treatment, and shows how it can be modified to capture common movements in volatility in a very natural way. The model is then fitted to daily observations on exchange rates.
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Finance Stoch
DOI 10.1007/s00780-008-0078-4
Local volatility dynamic models
René Carmona ·Sergey Nadtochiy
Received: 10 July 2007 / Accepted: 10 July 2008
© Springer-Verlag 2008
Abstract This paper is concerned with the characterization of arbitrage-free dy-
namic stochastic models for the equity markets when Itô stochastic differential equa-
tions are used to model the dynamics of a set of basic instruments including, but
not limited to, the underliers. We study these market models in the framework of the
HJM philosophy originally articulated for Treasury bond markets. The main thrust
of the paper is to characterize absence of arbitrage by a drift condition and a spot
consistency condition for the coefficients of the local volatility dynamics.
Keywords Implied volatility surface ·Local volatility surface ·Market models ·
Arbitrage-free term structure dynamics ·Heath–Jarrow–Morton theory
JEL Classification C61
Mathematics Subject Classification (2000) 60H10 ·91B24
1 Introduction and notation
Most financial market models introduced for the purpose of pricing and hedging
derivatives concentrate on the dynamics of the underlying stocks, or underlying in-
The results of this paper were presented, starting September 2006, in a number of seminars including
Columbia, Cornell, Kyoto, Santa Barbara, Stanford, Banff, Oxford, etc., and the organizers and
participants are thanked for numerous encouraging discussions.
Partially supported by NSF Grant #180-6024.
R. Carmona ()
Bendheim Center for Finance, ORFE, Princeton University, Princeton, NJ 08544, USA
S. Nadtochiy
Dept. ORFE, Princeton University, Princeton, NJ 08544, USA
R. Carmona, S. Nadtochiy
struments on which the derivatives are written. This is clearly the case in the Black–
Scholes theory where the focus is on the dynamics of the underlying stocks, whether
they are assumed to be given by geometric Brownian motions or more general non-
negative diffusions, or even semimartingales with jumps. In line with the problematic
of the so-called market models, the focus of the present paper is on the simultaneous
dynamics of all the liquidly traded derivative instruments written on the underlying
For the sake of simplicity, we limit ourselves to a single underlying index or stock
on which all the derivatives under consideration are written. Choosing more under-
liers would force the price process to be multivariate and make the notation signifi-
cantly more complicated, unnecessarily obscuring the nature of the results. We denote
by {St}t0the price process underlying the derivative instruments forming the mar-
ket. In order to further simplify the notation, we assume that the discount factor is one
or equivalently that the short interest rate is zero, i.e., rt0, and that the underlying
stock does not pay dividends. These assumptions greatly simplify the notation with-
out affecting the generality of our derivations as long as interest and dividend rates
are deterministic. The case of stochastic interest rates is more challenging.
Except in Sect. 8, we assume that in our idealized market, European call options
of all strikes and maturities are traded, that their prices are observable, and that they
can be bought and sold at these prices in any quantity. We denote by Ct(T , K) the
market price at time tof a European call option of strike Kand maturity T>t.
We assume that today, i.e., on day t=0, all the prices C0(T , K) are observable.
According to the philosophy of market models adopted in this paper, at any given
time t, instead of modeling only the price Stof the underlying asset, we use the set of
call prices {Ct(T , K )}Tt,K0as our fundamental market data. This is partly justified
by the well-documented fact that many observed option price movements cannot be
attributed to changes in Stand partly by the fact that many exotic (path-dependent)
options are hedged (replicated) with portfolios of plain (vanilla) call options.
Note that in our idealized market, we assume that European call options of all
strikes and all maturities are liquidly traded. This assumption is highly unrealis-
tic. In practice, the best one can hope for is, for a finite set of discrete maturities
n, quotes for the prices Ct(Ti,K
ij )of a finite set of call options.
In other words, for each of the maturities Ti, prices of calls are only available for a
finite set Ki1<K
iniof strikes. Absence of static arbitrage in this more
realistic form of the set-up has been considered, starting with the work of Laurent and
Leisen [30], followed by the recent technical reports by Cousot [14] and Buehler [9],
who use the Kellerer [27] theorem, and by the recent work of Davis and Hobson [15],
which relies instead on the Sherman–Stein–Blackwell theorem [6,39,40].
For the sake of convenience, we denote by τ=Ttthe time to maturity of the
option and we denote by ˜
Ct(τ, K ) the price Ct(T , K ) expressed as a function of this
new variable. In other words,
Ct(τ, K ) =Ct(t +τ, K), τ > 0,K>0.
This switch in notation has the clear advantage of casting the data at each time tinto
a parametric surface over the range {(τ, K );τ>0,K>0}which does not change
with t. We work in a linear pricing system. In other words, we assume that the market
Local volatility dynamics
prices by expectation in the sense that the prices of the liquid instruments are given
by expectations of the present values of their cashflows. So saying that Qis a pricing
measure used by the market implies that for each time t0, we have
Ct(τ, K ) =E(St+τK)+Ft=EQt(St+τK)+,
where we denote by Qta regular version of the conditional probability of Qwith
respect to Ft. For each τ>0, we denote by ˜μt,t+τthe distribution of St+τfor the
conditional distribution Qt.ItisanFt-measurable random measure. With this nota-
Ct(τ, K ) =
0(x K)+˜μt,t+τ(dx),
and for each fixed τ>0, the knowledge of the prices ˜
Ct(τ, K ) for all strikes K>0
completely determines the distribution ˜μt,t+τon [0,).
Notice that we do not assume the uniqueness of the pricing measure Q. In other
words, our analysis holds in the case of incomplete models as well as complete mod-
Notation convention In order to help with the readability of the paper, we use the no-
tation without tildes or hats for all the quantities expressed in terms of the variables T
and K. But we shall add a tilde when a value is expressed in terms of the variables τ
and K, and a hat when the strike is given in terms of the variable x=log K.
Implied volatility code-book In the classical Black–Scholes theory, the dynamics of
the underlying asset are given by the stochastic differential equation
for some univariate Wiener process {Wt}and some positive constant σ. In this case,
the price ˜
Ct(τ, K ) of a call option is given by the Black–Scholes formula
where Mt=St/K is the moneyness of the option and where we use the notation Φfor
the cumulative distribution function of the standard normal distribution. The Black–
Scholes price is an increasing function of the parameter σwhen all the other para-
meters are held fixed. As a consequence, for every real number C(think of such a
number as a quoted price for a call option with time to maturity τand strike K)in
the interval between (StK)+and St, there exists a unique number σfor which
Ct(τ, K ) =C. This unique value of σobtained by inverting the Black–Scholes for-
mula (1.1) is known as the implied volatility, and we shall denote it by ˜
Σt(τ, K ).
R. Carmona, S. Nadtochiy
This quantity is extremely important as it is used by most if not all market partici-
pants as the currency in which the option prices are quoted. For each time t>0, the
one-to-one correspondence
Ct(τ, K );τ>0,K > 0˜
Σt(τ, K );τ>0,K > 0
offers a code-book translating without loss all the information contained in the call
prices in terms of implied volatilities. We call it the implied volatility code-book. The
mathematical analysis of this surface is based on a subtle mixture of empirical facts
and arbitrage theories, and it is rather technical in nature. The literature on the subject
is vast and it cannot be done justice in a few references. Choosing a few samples for
their relevance to the present discussion, we invite the interested reader to consult
[11,19,20,31,32], and the references therein to get a better sense of these techni-
Valuation and risk management of complex option positions require models for
the time evolution of implied volatility surfaces. [12] and [13] are early examples
of attempts to go beyond static models, but despite the fact that they consider only
a cross section of the surface (say for Kfixed), the works of Schönbucher [35] and
Schweizer and Wissel [37] are more in the spirit of the market model approach which
we advocate in this paper.
At any given time t, absence of (static) arbitrage imposes conditions on the surface
of call option prices: the surface {˜
Ct(τ, K )}τ,K should be increasing in τ, nonincreas-
ing and convex in K, it should converge to 0 as K→∞and recover the underlying
price Stfor zero strike as K0. Because of the one-to-one correspondence be-
tween call prices and implied volatilities, these conditions can be expressed in terms
of properties of the implied volatility surface {˜
Σt(τ, K )}τ,K. However inverting the
Black–Scholes formula (1.1) is not simple, and these conditions become rather tech-
nical. Moreover, when defining a market model by a set of stochastic differential
equations, the first point of the check list is to make sure that the properties guaran-
teeing the absence of static arbitrage are preserved throughout the time evolution. So
even before we consider the problem of absence of dynamic arbitrage, this first point
on the check list gives us enough reasons to search for another way to capture the
information contained in the surface of call option prices.
Throughout this paper we assume that the filtration F=FWis generated by a
multidimensional Brownian motion W=(W 1,...,Wm). We assume that pricing is
done by expectation with respect to some pricing measure Q, which, however, does
not have to be unique. Since it is tradable, the underlying {St}must be a local martin-
gale, and the martingale representation theorem for Brownian filtrations gives the ex-
istence of an adapted scalar process {σt}t0and of an m-dimensional Wiener process
B=(B1,...,Bm)such that FW=FBand
In the absence of restrictive assumptions on the particular form of the spot volatil-
ity σt, pricing by computing expectations of discounted payoffs is usually imprac-
tical. An alternative in the spirit of the market models would be to postulate the
dynamics of the set of option prices explicitly, for example using a system of Itô
Local volatility dynamics
stochastic differential equations. However, doing so is likely to introduce arbitrage
opportunities and identifying these would require much more than a mere modicum
of care. As explained earlier, in order to rule out static arbitrages, at any given time,
the solution of the system of stochastic differential equations should give a surface
increasing in the variable τ, convex and decreasing in the variable K, and satisfying
specific boundary conditions. It seems to be quite difficult to identify stochastic dy-
namics preserving these properties, let alone ruling out dynamic arbitrages! A model
based on the simultaneous dynamics of all the call option prices was considered in a
recent paper [25] of Jacod and Protter. These authors work in the more general set-
ting of jump processes, and they study the problem of the completion of a market by
added derivative instruments. In so doing, they derive conditions very similar to our
spot consistency and drift conditions. See also [37] for a discussion of some of the
difficulties associated with the simultaneous dynamics of all the call prices.
Equivalently, one could encode all the option prices by the implied volatility sur-
face Σt, whose definition was recalled earlier, and model its dynamics. But again,
isolating tractable conditions characterizing the absence of arbitrage is very technical
and cumbersome. See, for example, [35] and [37] for a discussion of the particular
case when Kis fixed.
For all these reasons, we choose to code the market information (i.e., the option
prices) using the so-called local volatility surface. For each fixed time t0, if we
assume that ˜
Ct(τ, K ) is differentiable in τand twice differentiable in K, we define
the local volatility ˜at(τ , K) by
t(τ, K ) =2τ˜
Ct(τ, K )
KK ˜
Ct(τ, K ) >0,K>0.(1.2)
In this form, the definition was first given in 1994 by Dupire [18], who considered
the case t=0, assumed that the underlying was Markovian, and used the theory of
the backward Kolmogorov equation to show that the underlier necessarily satisfies a
stochastic differential equation of the form
Dupire’s idea was to deduce the dynamics of the underlier from a snapshot at time
t=0 of the option prices quoted on the market and use these dynamics to price exotic
derivatives on the same underlying. This very same idea appeared approximately at
the same time in a work of Derman and Kani in the context of tree models [16]. These
groundbreaking works had a tremendous impact on the life on equity desks, where
pricing and hedging of exotic options take place, and on the research on the mathe-
matical foundations of hedging and pricing of these options. However, this approach
is not without shortcomings. First, the Markovian property of the underlier is highly
unrealistic. Also, the local volatility surface is changing with time, so in general it
should be viewed as a function-valued random process at}t. In some sense, this is
the starting point of the present paper.
If we turn definition (1.2) around, we see that any local volatility surface
a(τ,K)}τ,K satisfying some mild assumptions (say positivity and continuity for ex-
R. Carmona, S. Nadtochiy
ample) gives a set of option prices as a solution of the initial-value problem
C(τ,K) =1
2K2˜a2(τ , K)∂ 2
KK ˜
C(τ,K), τ > 0,K>0,
It is important to stress the fact that the four properties which we articulated for the
absence of static arbitrage are, in the local volatility framework, encapsulated in the
nonnegativity of the diffusion coefficient (τ , K) → ˜a2
t(τ, K ).
As we already mentioned, Schönbucher [35] and Schweizer and Wissel [37]have
argued that the implied volatility was not the right code-book for equity market mod-
els, and in the simpler case of a cross-section, they work instead with the term struc-
ture of volatility for a fixed option. Our point of view is to follow the spirit of the
approach advocated by Schönbucher [36] in the case of credit portfolios. In this ap-
proach, using the Breeden–Litzenberger trick, the market prices are put in correspon-
dence with a set of marginal distributions, which is in turn captured by a Markov
martingale having these distributions as marginals. Defining a dynamic model is
then done by prescribing a time evolution for this Markov martingale. In the case of
the CDO markets, this Markov martingale is a finite-state nonhomogeneous Markov
process with values in the set of possible levels of loss, while here it is a nonhomoge-
neous diffusion in the strike variable. Note that the classical HJM model is included in
this approach, the state space of the Markov martingale being a singleton! The reader
interested in more information on this approach is referred to the survey article [10].
It seems that the terminology local volatility is due to Derman and Kani [17].
In a paper mostly known for its analysis of tree models, the authors discuss infor-
mally arbitrage-free dynamic models for the local volatility. In the present paper, we
develop the program they outlined by providing rigorous proofs, new analytic deriva-
tions, and numerical examples. Incidently, our no-arbitrage conditions are slightly
different from theirs.
We conclude this introduction with a short summary of the contents of the paper.
In Sect. 2we define the time evolution of the (stochastic) volatility surface at(·,·)by
means of a family of Itô stochastic differential equations. Recall that for each time t,
the data encapsulated in the surface at(·,·)give the option prices Ctby solving
the initial value problem (1.3). Section 3addresses the problem of the probabilistic
characterization of these solutions. Naturally, the individual call prices Ct(T , K ) are
expected to be semimartingales in t. Strangely enough, we could not find a simple
proof of this plain fact. Our proof is rather lengthy and technical, and for this rea-
son, the details are postponed to a couple of appendices at the end of the paper. The
dynamics of the local volatility imply specific dynamics for the set of option prices.
One of the thrusts of the paper is to derive necessary and sufficient conditions which
guarantee that these dynamics do not produce arbitrage opportunities. We derive such
conditions in Sect. 4. These conditions turn out to be analogous to the drift restriction
and short rate specification of the classical Heath–Jarrow–Morton theory (see [10]
and [16] for an analogue in a similar setup). The following Sect. 5characterizes the
Markov spot models as those with bounded variation local volatility dynamics. Next,
Sect. 6provides a new expression for the local volatilitysurfaces of stochastic volatil-
ity models. This expression is especially well suited to Monte Carlo computations.
Local volatility dynamics
We use intuition based on formulae derived in Sect. 6to define a parametric family
of local volatility surfaces in Sect. 7. Even though we believe that, like in the case
of the classical HJM models for the bond markets, generic parametric families of lo-
cal volatility surfaces will not be consistent with the restrictions of the no-arbitrage
dynamics (see, for example, [22] for the classical case), we illustrate the potential
usefulness of such parametric families in the analysis of real data. A short Sect. 8
discusses hedging issues and Sect. 9concludes.
Most of the results of the paper were announced in [10], where the interested
reader will find a nontechnical introduction to dynamic market models in the form of
a survey.
2 Model setup
In this section we mostly work with the variables τand x=logK. With this choice,
the partial differential equation (1.3) becomes uniformly parabolic,
τu(τ, x ) =1
xxu(τ , x) xu(τ , x).(2.1)
We know that this initial-value problem is well posed and has a classical fundamental
solution. We perform the same change of variables in the option prices by defining
the function (τ, x ) → ˆ
C(τ,x) by
C(τ,x) =˜
Cτ,exand ˆa(τ, x) aτ,ex0,xR.
With this notation, the equation for option prices becomes
Ct(τ, x ) =1
t(τ, x )(∂2
xx ˆ
Ct(τ, x ) xˆ
Ct(τ, x )), τ > 0,xR,
We now postulate the market dynamics: under Qour model is given by the system of
stochastic differential equations
t(τ, x ) a2
t(τ, x )αt(τ, x ) dt +ˆ
βt(τ, x ) ·dBt],ˆa2
0(τ, x ),
where, for all fixed τ>0 and xR, the processes ˆα(τ,x) ={ˆαt(τ , x)}tand
β(τ,x) ={ˆ
t(τ,x),..., ˆ
t(τ, x )}t, as well as the process σ={σt}t, are adapted
to the filtration F=FB. In addition, we require the following regularity assumptions:
1. The spot volatility process σhas almost surely continuous paths, and for any
t0, it holds that E{t
2. Almost surely, the random fields ˆαand ˆ
βhave continuous derivatives up to or-
der seven in (τ, x ) for all τ>0and xR, and each of these derivatives is a
continuous process in t.
R. Carmona, S. Nadtochiy
3. (a) For any T>0,ε>0,i=1,...,m, and t0, we have
τ∈[0,T ],xR,
xkˆαu +τ, x)+j
(b) For any T>0and t0, we have
τ∈[0,T ],xR,u∈[0,t]1
u(τ, x )|+
u(τ, x )<a.s.
(c) For any T>0,i=1,...,m, and t0, there exists δ(0,1]for which the
τ=τ∈[0,T ],
is finite almost surely.
4. For any T>0,c>0, and t0, we have
τ(0,T ],xR,u∈[0,t]
2ec(xlog Su)2k
xkˆαu(τ , x)<a.s.
The first assumption guarantees that, under Q,Sis a continuous martingale with
strictly positive paths. It also plays an important role in the proof of Proposi-
tion 3.3. Assumptions (2)and (3.a), together with Exercise 3.1.5 on p. 78 of Kunita’s
book [29], imply that ˆa2has a modification that is smooth enough in (τ , x), so that the
expressions in (3.b) and (3.c) make sense. The inequality in (3.b) guarantees the exis-
tence of the fundamental solution of (2.2), as a well-behaved random process in t(in
particular, it allows us to apply Gaussian estimates on the derivatives of fundamental
solutions). Also, together with (3.a), it is used to show the convergence of the Euler
schemes appearing in the proof of Lemma 3.1. Finally, assumptions (3.c) and (4) were
introduced to control the possible singularities of ˆα,τˆa2,τˆ
β, and their x-derivatives
at τ=0. For example, α(τ,x), given by the drift restriction (4.1), can potentially be
singular at τ0, as demonstrated by the example in Sect. 5.2. And, in order to al-
low for such singularities, we introduce the weight function exp(c(x logSt)2/τ )
in assumption (4).
Conditions (1)–(4) above are assumed to hold throughout Sects. 35.
According to the model specified above by a continuum of Itô stochastic dif-
ferentials, for simulation purposes, we have to specify the stochastic spot volatility
process σ, the random fields ˆαand ˆ
β, infer the values of S0and ˆa2
0from observed
prices, and only then simulate the sample paths of the underlying stock price Sand
its local volatility surface ˆa(which in turn will give us the sample paths of option
prices). However, the main result of the paper is that if we want to avoid arbitrage,
the only parameter to be specified is β; all the other specifications will follow.
Local volatility dynamics
3 Semimartingale representation of the option prices
The goal of this section is to show that for all fixed τ>0 and xR, the option price
process ˆ
Ct(τ, x ), defined for each fixed tas the solution in τand xof the initial value
problem (2.2), is a semimartingale in t. This yields the same result for ˜
Ct(τ, K ) for
t0 and Ct(T , K ) for t∈[0,T). These results are based on two auxiliary lemmas
and two technical propositions. These results are believed to hold in greater general-
ity and under weaker conditions than the smoothness assumptions made above. How-
ever, their proofs are quite lengthy and technical. So in order not to disrupt the flow
of the paper, we state them without proof, postponing the gory details to appendices
at the end of the paper.
Using the notation
Dx:= 1
xx xand Lx:= ˆa2
t(τ, x )Dx,
(2.2) becomes
C(τ,x) =Lxˆ
C, τ > 0,xR,
C(0,x) =(S ex)+.
Our first step is to smooth the initial condition with an approximate identity. We
introduce ϕ(x) C
0(R)such that
1. ϕ(x) 0.
2. ϕ(x) =0for|x|≥1.
3. Rϕ(u) du =1.
4. 1
−∞ x
−∞ ϕ(y)dy dx =1.
Then we set
−∞ y
−∞ ϕ(z)dz dy,
and for each ε>0, we define Fε(x) by Fε(x ) =εF (x/ε), so that F0(x) =x+.Next,
we need to get rid of the possible singularities of ˆα. Consider
t(τ, x ) := ˆαt+τ,x)
and, naturally,
t(τ, x ) := ˆ
βt+τ,x), ˆaε
t(τ, x ) := ˆat+τ,x), Lε
t(τ, x )2Dx.
Clearly, the new processes ˆaε,ˆαε, and ˆ
βεsatisfy all the assumptions made in the
previous section. In addition, ˆαεhas no singularity at τ0. Now, consider the solu-
tion ˆ
Cεof the initial-value problem
Cε(0,x)=Fε(S ex).
R. Carmona, S. Nadtochiy
The purpose of the next lemma is to prove that, as a process in t, the solution is a
Lemma 3.1 For i=1,...,m,there exist predictable processes με
t}tand νε,i
with values in C1,2(R+×R)such that for any (t,τ,x),
t(τ, x ) =ˆ
0(τ, x ) +t
u(τ, x ) du +t
u(τ, x ) ·dBu
holds Q-a.s.
The proof of this result is given in the first part of Appendix A. Once we know that
the prices ˆ
t(τ, x ) of options with smoothed payoffs and bounded-drift local volatil-
ity are semimartingales, we show that, when ε0, the semimartingale property is
preserved. Here and further in the paper, R++ stands for (0,).
Lemma 3.2 For i=1,...,m,there exist predictable processes μt(τ , x)}tand
t(τ, x )}twith values in C1,2(R++ ×R)such that for any (t,τ,x),
Ct(τ, x ) =ˆ
C0(τ, x ) +t
0ˆμu(τ , x) d u +t
0ˆνu(τ , x) ·dBu
holds Q-a.s.
The proof is given in the second part of Appendix A. We conclude this section
with two more technical results, which will be needed in the proof of the main result
of the paper.
Proposition 3.3 For any fixed t>0and compact KR,there exists a positive
sequence τn0such that,Q-almost surely,we have
The proof of this result is given in the first part of Appendix B.
Proposition 3.4 Q-almost surely,for any t0and hC
0(R),we have
τ0Rh(x) ˆμt(τ , x) d x =1
The proof of this result is given in the second part of Appendix B.
Local volatility dynamics
4 Absence of arbitrage
Since the stock price process is already assumed to be a local martingale under the
pricing measure, in the present context, absence of arbitrage merely means that for
each fixed strike K>0 and for each fixed date of maturity T>0, the processes of
option prices
Ct(T , K )t∈[0,T ) =ˆ
Ct(T t,log K)t∈[0,T )
are local martingales. The following theorem addresses this problem. It can be viewed
as the main contribution of the paper.
Theorem 4.1 Absence of arbitrage,in the sense that Sand all the Ct(T , K)t∈[0,T )
are martingales under Q,is equivalent to the following conditions satisfied a.s.for
all t>0,τ>0,xR:
Drift restriction
Ct=τlog ˆa2
Spot volatility specification
1. Recall that .
0ˆνu·dBuis the local martingale component in the semimartingale
decomposition of the call price process given in Lemma 3.2. Also, as follows
from the proof of this lemma, ˆνtsatisfies the initial-value problem
t(0,x)=Stσt1(−∞,log St](x),
which can be efficiently solved numerically, provided that we know ˆa2
tand ˆ
2. We can also rewrite (4.1) in the way it was originally touted by Derman and Kani
in [17]. If we plug into (4.1) the representation of ˆν(τ,x) in terms of ˆpand ˆq(the
fundamental solutions of (2.1) and (9.14), respectively), we obtain
βi(τ, x ) τ
0Rˆa2(u, y ) ˆ
βi(u, y) ˆq(0,log S;u, y)Dyˆq(u,y;τ, x)dy du
β1(τ, x )∂Sˆq(0,logS;τ, x)
ˆq(0,logS;τ,x) =τlog ˆa2(τ, x ).
This is essentially, after changing variables from the couple (“maturity,” “strike”)
used by Derman and Kani in [17], to the couple (“time-to-maturity”, “log-strike”)
R. Carmona, S. Nadtochiy
used here, the drift condition proposed by Derman and Kani. However, the third
term of the above left-hand side seems to be missing in [17].
Before we give a complete proof of the main theorem, we need to state another
auxiliary result. First, from Lemmas 3.1 and 3.2 above, with the help of the gener-
alized Itô rule (see, for example, Theorem 3.3.1 of [29]), we conclude that, for any
T>0 and xR, the process {ˆ
Ct(T t,x)}t∈[0,T ) is a semimartingale with the de-
Ct(T t,x) =ˆμt(T t,x) τˆ
Ct(T t,x)dt νt(T t,x) ·dBt.(4.4)
We denote its drift by ˆvt(τ, x ) := ˆμt(τ , x) τˆ
Ct(τ, x ). Now we are ready to formu-
late the following proposition.
Proposition 4.2 For any T>0and xR:
1. For al l t<T,we have,Q-almost surely,
0ˆνu(T u, x ) ·dBu=t
0τˆνu(T u, x ) ·dBu,
0ˆνu(T u, x ) ·dBu=t
0Dxˆνu(T u, x ) ·dBu.
2. If the drift restriction (4.1)is satisfied,then for all t<T,we have,Q-almost
0ˆvu(T u, x ) du =t
0τˆvu(T u, x ) du,
0ˆvu(T u, x ) du =t
0Dxˆvu(T u, x ) du.
The proof of this result is given in the third part of Appendix B.
Proof of Theorem 4.1 First, we notice that (4.1) is well defined since Dxˆ
positive because of the maximum principle applied to the fundamental solution of a
uniformly parabolic initial value problem.
Proof of the “only if” part. Assume that {ˆ
Ct(T t,x)}t∈[0,T ) is a martingale for
every fixed T>0 and xR. Then
Ct(T t,x) νt(T t, x) ·dBt
and, by the generalized Itô formula,
t(T t,x)=ˆa2
t(T t,x)ˆαt(T t,x)τˆa2
t(T t,x)dt
t(T t,x) ˆ
βt(T t,x) ·dBt.
Local volatility dynamics
Recall that, because of the defining relationship between local volatility and call
prices, we have a.s.
t(T t,x) ·Dxˆ
Ct(T t,x) =τˆ
Ct(T t,x). (4.5)
Applying the first part of Proposition 4.2, we conclude that the drift on the right-hand
side of (4.5) is zero. Writing that the left-hand side has zero drift as well, we get
t(T t,x) ·Dxˆ
Ct(T t,x)t
t(T t,x)ˆαt(T t,x)τˆa2
t(T t,x)Dxˆ
Ct(T t,x)dt
∂t ˆa2
t(T t,x),Dxˆ
Ct(T t,x)dt +(···)·dBt,
which holds for any fixed T>0, xR, and t<T. This yields (4.1).
Now, fix some test function hC
0(R)and denote by (y) the local time of S
at y. Then we have
0ˆμu(τ , x) d u Λt(ex)dx
Cu(τ, x ) Suex+u=t
0ˆνu(τ , x) −ˆνu(0,x)
From Proposition 3.3 we conclude that there exists a sequence τn0 such that
almost surely
n→∞supp(h) t
0ˆμun,x)duΛtexdx =0.(4.6)
Recall that, since (ˆ
Ct(T t,x))t∈[0,T ) is a local martingale for any T>0, its drift is
zero. So, using (4.4), we can conclude that
Cun,x), (4.7)
and from (4.6) and (4.7) we have, almost surely for any tand hC
=Rh(x)Λtexdx lim
0Rh(x) ˆa2
Now, using the definition of local time and dominated convergence, we deduce that
the last expression is equal to
udu t
n→∞Rh(x) ˆa2
R. Carmona, S. Nadtochiy
which holds for any t>0. The above integrand is a.s. continuous, therefore, it is
identically equal to zero. And since this is true for any hC
0(R), we conclude that
t, which completes the proof of (4.2).
Proof of the “if” part. Using the second part of Proposition 4.2 and differenti-
ating (4.5) with respect to t(just like it was done in the first part of the proof), we
τˆvt(τ , x) =1
t(τ, x )Dxˆvt(τ , x), τ > 0,xR,(4.8)
which holds almost surely for all t>0. Since
ˆvt(τ , x) μt(τ , x) τˆ
Ct(τ, x ),
for any test function hC
Rh(x) ˆvt(τ, x ) dx =Rh(x) ˆμt(τ , x) d x Rh(x)∂τˆ
Ct(τ, x ) dx .
As τ0, the first term in the above right-hand side converges to 1
because of Proposition 3.4. Moreover,
Ct(τ, x ) dx =1
2Rh(x) ˆa2
t(τ, x )exˆqt(0,log St;τ, x) dx
This proves that, almost surely for any t,ˆvt(τ, x ), as function of x, converges weakly
to0asτ0. Then, applying the maximum principle to (4.8) (see, for example, [3]),
we conclude that, for any τ>0 and xR,wehaveˆvt(τ, x ) =0.
This implies that (ˆ
Ct(T t,x))t∈[0,T ) is a local martingale in tfor any T>0 and
xR. But since 0 ˆ
CtSt, it is a true martingale. This completes the proof of the
We are now in a position to give the specifics of the Monte Carlo pricing algorithm
mentioned earlier:
1. The last theorem shows that the only free parameter in the model is {ˆ
βt(τ, K )}t0.
It is chosen from intuition and historical observations not already contained in the
set of observed call option prices.
2. We use the current stock price for S0, and the call option prices to deduce the initial
local volatility surface. Many methods have been proposed to do that, ranging
from parametric and non-parametric statistical methods (see, for example, [21]
and the references therein) to smoothing by optimization and partial differential
equations (see, for example, [1]). We revisit this issue later in the paper.
Local volatility dynamics
3. We simulate sample paths of Stand ˆa2
tas follows. Given ˆa2
tand ˆ
βt, we compute ˆαt
from the no-arbitrage condition (4.1) and σtfrom the consistency condition (4.2).
Then we use ˆαtand ˆ
βtto find ˆa2
t+Δt and use σtto find St+Δt byasinglestepina
forward plain Euler’s scheme, and we iterate. Voilà!
This algorithm will give us arbitrage-free dynamics of the option prices together
with the stock. It will allow us to calculate, in a consistent manner and without re-
calibration, the prices of other instruments that use the stock and/or the dynamically
modeled option prices as underlying. This is especially useful for forward-starting
We implemented numerically the above program, specifying a deterministic ˆ
and using an explicit Euler scheme to compute ˆν(·,·)and ˆ
C(·,·), needed to ap-
ply (4.1), as solutions of the corresponding partial differential equations. However,
because of the technicality of the computations, we leave the discussion of the results
of this numerical simulation to a forthcoming paper.
5 Examples
This section provides a couple of simple consequences of the no-arbitrage condition
derived above.
5.1 Bounded variation dynamics and Markovian spot models
Let us first consider the simplest possible model of local volatility stochastic dy-
namics by assuming that {ˆ
βt(τ, x )}t0is identically zero, i.e., ˆ
β0. Obviously, this
corresponds to assuming that the time evolution of the local volatility surface is of
bounded variation. Under this assumption, the no-arbitrage drift condition (4.1) be-
ˆαt(τ , x) =τlog ˆa2
t(τ, x ).
Then we have
dt ˆa2
t(τ, x ) a2
t(τ, x )∂τlog ˆa2
t(τ, x ) =τˆa2
t(τ, x ),
which, for each fixed strike K=ex, is a plain hyperbolic transport equation whose
solutions are given by traveling waves. Consequently,
t(τ, x ) a2
from which we conclude, using the consistency condition (4.2), that
σta0(t , log St).
This above derivation proves the following aside:
Proposition 5.1 The local volatility surface is a process of bounded variation if and
only if it is the shift of a fixed deterministic surface along the space-time underlying
sample path,and the underlying stock process is Markovian.
R. Carmona, S. Nadtochiy
5.2 Case of flat surfaces
Let us now assume that the local volatility surface is only driven by the second com-
ponent of the Brownian motion and that from some instant of time ton, ˆa2
t(τ, x ) and
its diffusion term ˆ
t(τ, x ) become flat (i.e., ˆa2
tand ˆ
tare constant functions of τ
and xfrom some value of tonward). Then we can solve (2.2) and (4.3) explicitly,
and the no-arbitrage drift condition (4.1) gives for the drift the expression
ˆαt(τ , x) =−ˆ
2(x logSt)2
This is another instance of the fact that in the absence of arbitrage, the drift ˆαis
determined by the volatility structure ˆ
β. More generally, we are interested in the reg-
ularity of ˆαgiven by the drift restriction (4.1)asτ0. Let us assume, for example,
that ˆa2
t(·,·), and ˆ
t(·,·)are smooth enough as functions of (τ, x ), but not
necessarily flat. Lemma 3.2 and Proposition 4.2 imply that t
Cu·dBuis the local-
martingale part of log Dxˆ
C. Recall that
C(τ,x) =1
where ˆqis the fundamental solution of (9.14). The following claims (leading to (5.1))
should be understood as reasonable conjectures since we do not provide them with
rigorous proofs.
One would expect that, based on Varadhan’s large deviations theory (see [42]),
2τDxˆν.(τ , x)
C.(τ, x) tδ2
as τ0, where δ2
t(x, y ) is the square geodesic distance on a one-dimensional man-
ifold, with
δ2(x, y ) =y
Applying the generalized Itô formula, we obtain
=(. . .) d t
log Stˆat(0,z) ˆ
Thus, from the drift restriction (4.1) (recalling that ˆa2(τ, x ) is Hölder-continuous at
τ=0) we have the following asymptotic expression for ˆαt(τ, x ) as τ0:
ˆαt(τ , x) =δt(log St,x)
log Stˆat(0,z) ˆ
Local volatility dynamics
If we want local volatility to have a continuous drift ˆα, then the main term in the
above expansion should be zero. Thus, we obtain the integral equation
log Stˆat(0,z) ˆ
log Stˆat(0,z) ˆ
We see that if ˆ
β1(0,x) and ˆ
β2(0,x) conspire cleverly, then ˆα(τ, x) does not have to
be singular at τ=0. This is the case for many diffusion-based stochastic volatility
models discussed in Sect. 6.
5.3 Random scaling
Here we consider the case of a local volatility surface obtained by randomly scaling
a fixed deterministic surface.
Proposition 5.2 If ˆa2
t(τ, x ) =λtˆa2
0(τ, x ) for some fixed deterministic initial surface
0and some semimartingale λwith decomposition t=αtdt +βt·dBt,then ab-
sence of arbitrage,in the sense that Sand all the Ct(T , K)t∈[0,T ) are martingales
under Q,implies that λis a deterministic function of the underlying stock.
In such a case, the setting of the proposition reduces to the simple constant local
volatility model discussed above.
Proof For the proof, we also assume (in addition to the regularity assumptions stated
in Sect. 2) that all τ-derivatives of ˆa2
0(τ, x ), up to order three, are continuous at the
boundary τ=0.
If we multiply both sides of the no-arbitrage drift condition (4.1)byexDxˆ
integrate with respect to x, then the second term on the left-hand side disappears since
RexDxˆνdx=0. This leaves us with
ˆα0=Rτlog ˆa2
0(τ, x ) ˆq(0,logS;τ,x) dx,
and differentiating both sides with respect to τand using the properties of the funda-
mental solution, we obtain
Rττ log ˆa2
0(τ, x ) ˆqt(0,log S;τ, x)dx
+Rτlog ˆa2
0(τ, x )Dxλtˆa2
0(τ, x ) ˆqt(0,log St;τ, x)dx =0.
Integrating by parts the second term and letting τ0, we get
ττ log ˆa2
xx +xτlog ˆa2
which proves that, excluding some degenerate cases, λtis a deterministic function of
the underlying stock St.
R. Carmona, S. Nadtochiy
6 From stochastic to local volatility: The Markovian case
Although it is clear that our framework potentially (if we forget for the moment the
regularity assumptions made in Sect. 2) includes all market models in which Euro-
pean call prices are semimartingales (sufficiently smooth in strike and maturity), it is
interesting to see how the local volatility looks in some of the most popular models
used by practitioners and academics.
We consider stochastic volatility models in which the volatility processes are
Markovian with respect to their own filtrations and given by Itô stochastic differ-
ential equations. Obviously, the Hull–White and Heston models are particular cases.
Our goal is to exhibit generic properties of the local volatility surface in these models.
So we assume that
dSt=Strdt +Stσt(1ρ2dB1
t), S0,
t)dt +g(t, σt)dB2
where {B1
t}t0and {B2
t}t0are independent Brownian motions, ρ∈[1,+1], and
f(t,x) and g(t,x) satisfy the usual conditions which guarantee the existence and
uniqueness of a positive solution to the above system. Notice that, contradictory to
the convention used in this paper, we featured the interest rate rin the model. This is
to emphasize its role (or lack thereof) in the formula we derive in this section.
Notice also that, for the purpose of this section, we can limit ourselves to the case
t=0, and doing so we are back to the static case considered originally by Dupire and
Derman and Kani, and we can use τor Tinterchangeably, ignoring the tildes and hats
on the coefficients. The goal of this section is to state and prove a new expression for
the local volatility of Markovian stochastic volatility models.
Proposition 6.1 In a stochastic volatility model (6.1), the local volatility surface is
given at time t=0by the formula
a2(T , K ) =Eσ2
1(T ,K)
1(T ,K)
where ¯
ST,¯σT,and d1(T , x ) are defined in formulae (6.3), (6.4), and (6.5)below.
Proof Solving the system (6.1)forSt, we get
g(u, σu)du +ρt
g(u, σu)u
×exprt 1
udu +(1ρ2)t
For each t>0, we define the quantities Stand σtby
udu ρt
g(u, σu)du +ρt
g(u, σu)u(6.3)
Local volatility dynamics
udu. (6.4)
Both quantities depend only on {B2
u}0ut. Moreover, conditioning on {B2
we obtain
C(T, K) =E(STK)+=EBSST,T,1ρ2¯σT,K,
where the expectation is over the second Brownian motion {B2
t}t0and where the
Black–Scholes call price BS(S,T,σ,K) was defined in (1.1). Itô’s rule, repeated
differentiations, and Fubini’s theorem give
TC(T,K) =−rKerT EΦd1T1ρ2¯σT
KC(T,K) =−erT EΦd1T1ρ2¯σT,
KKC(T, K) =1
where d1=d1(T , K ) is defined in the usual way by
d1(T , K ) =log ¯
K+(r +1
which in turn gives the desired result since a2(T , K) =2(∂TC+rK∂KC)
Notice that formula (6.2) still contains expectations. However, these expectations
are only over the paths of the stochastic volatility σt(i.e., over the second Brownian
motion {B2
t}t0), and in order to implement this formula in Monte Carlo computa-
tions, we only need to simulate the (Markovian) paths of σt. But the main strength of
formula (6.2) is to have got rid of the singularities in the numerator and denominator.
Such singularities would appear automatically if we were to try to use straightforward
Monte Carlo simulations. Finally, we notice that the expectations appearing in for-
mula (6.2) can be computed explicitly if the joint distribution of t,t
udu) (or its
Laplace transform) is known. This is, for example, the case in Heston and Hull–White
models. For small τ, the behavior of local volatility in these model is discussed, for
example, in [5] and [24].