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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

e-mail: rcarmona@princeton.edu

S. Nadtochiy

Dept. ORFE, Princeton University, Princeton, NJ 08544, USA

e-mail: snadtoch@princeton.edu

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

stocks.

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}t≥0the 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., rt≡0, 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 )}T≥t,K≥0as 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

T1<T

2<···<T

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

i2<···<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=T−tthe 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 t≥0, we have

˜

Ct(τ, K ) =E(St+τ−K)+Ft=EQt(St+τ−K)+,

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-

tion,

˜

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-

els.

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

dSt=StσdW

t,S

0=s0

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

BS(St,τ,σ,K)=StΦ(d1)−KΦ(d2)(1.1)

with

d1=logMt+τσ2/2

σ√τ,d

2=logMt−τσ2/2

σ√τ,

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 (St−K)+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-

calities.

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 K→0. 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}t≥0and of an m-dimensional Wiener process

B=(B1,...,Bm)such that FW=FBand

dSt=StσtdB1

t,S

0=s0.

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 t≥0, if we

assume that ˜

Ct(τ, K ) is differentiable in τand twice differentiable in K, we deﬁne

the local volatility ˜at(τ , K) by

˜a2

t(τ, K ) =2∂τ˜

Ct(τ, K )

K2∂2

KK ˜

Ct(τ, K ) ,τ>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

dSt

St=a(t,St)dB

t.

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

∂τ˜

C(τ,K) =1

2K2˜a2(τ , K)∂ 2

KK ˜

C(τ,K), τ > 0,K>0,

˜

Ct(0,K)=(St−K)+.(1.3)

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

2˜a2

tτ,ex∂2

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

ˆ

C(τ,x) =˜

Cτ,exand ˆa(τ, x) =˜aτ,ex,τ≥0,x∈R.

With this notation, the equation for option prices becomes

∂τˆ

Ct(τ, x ) =1

2ˆa2

t(τ, x )(∂2

xx ˆ

Ct(τ, x ) −∂xˆ

Ct(τ, x )), τ > 0,x∈R,

ˆ

Ct(0,x)=(St−ex)+.

(2.2)

We now postulate the market dynamics: under Qour model is given by the system of

stochastic differential equations

dSt=StσtdB1

t,S

0,

dˆa2

t(τ, x ) =ˆa2

t(τ, x )[ˆαt(τ, x ) dt +ˆ

βt(τ, x ) ·dBt],ˆa2

0(τ, x ),

where, for all ﬁxed τ>0 and x∈R, the processes ˆα(τ,x) ={ˆαt(τ , x)}tand

ˆ

β(τ,x) ={ˆ

β1

t(τ,x),..., ˆ

βm

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

t≥0, it holds that E{t

0S2

uσ2

udu}<∞.

2. Almost surely, the random ﬁelds ˆαand ˆ

βhave continuous derivatives up to or-

der seven in (τ, x ) for all τ>0and x∈R, 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 t≥0, we have

sup

τ∈[0,T ],x∈R,

u∈[0,t]e−|x|

0≤j+k≤7∂j

τj∂k

xkˆαu(ε +τ, x)+∂j

τj∂k

xkˆ

βi

u(ε+τ,x)<∞a.s.

(b) For any T>0and t≥0, we have

sup

τ∈[0,T ],x∈R,u∈[0,t]1

|ˆa2

u(τ, x )|+

6

k=0∂k

xkˆa2

u(τ, x )<∞a.s.

(c) For any T>0,i=1,...,m, and t≥0, there exists δ∈(0,1]for which the

supremum

sup

τ=τ∈[0,T ],

x∈R,u∈[0,t]2

k=0

e−|x||∂k

xkˆ

βi

u(τ ,x)−∂k

xkˆ

βi

u(τ ,x)|+|∂k

xkˆa2

u(τ ,x)−∂k

xkˆa2

u(τ ,x)|

|τ−τ|δ

is ﬁnite almost surely.

4. For any T>0,c>0, and t≥0, we have

sup

τ∈(0,T ],x∈R,u∈[0,t]

1

k=0τk

2e−c(x−log Su)2/τ ∂k

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. 3–5.

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 x∈R, 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

t≥0 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

2∂2

xx −∂xand Lx:= ˆa2

t(τ, x )Dx,

(2.2) becomes

∂τˆ

C(τ,x) =Lxˆ

C, τ > 0,x∈R,

ˆ

C(0,x) =(S −ex)+.

(3.1)

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

F(x)=x

−∞ 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

ˆαε

t(τ, x ) := ˆαt(ε +τ,x)

and, naturally,

ˆ

βε

t(τ, x ) := ˆ

βt(ε +τ,x), ˆaε

t(τ, x ) := ˆat(ε +τ,x), Lε

x:=ˆaε

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

∂τˆ

Cε=Lε

xˆ

Cε,τ>0,x∈R,

ˆ

Cε(0,x)=Fε(S −ex).

(3.2)

R. Carmona, S. Nadtochiy

The purpose of the next lemma is to prove that, as a process in t, the solution is a

semimartingale.

Lemma 3.1 For i=1,...,m,there exist predictable processes {ˆμε

t}tand {ˆνε,i

t}t

with values in C1,2(R+×R)such that for any (t,τ,x),

ˆ

Cε

t(τ, x ) =ˆ

Cε

0(τ, x ) +t

0ˆμε

u(τ, x ) du +t

0ˆνε

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

Cε

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 {ˆμt(τ , x)}tand

{ˆνi

t(τ, x )}twith values in C1,2(R++ ×R)such that for any (t,τ,x),

ˆ

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 K⊂R,there exists a positive

sequence τn→0such that,Q-almost surely,we have

lim

n→∞Kt

0ˆνu(τn,x)−ˆνu(0,x)

·dBudx

+Kˆ

Ct(τn,x)−St−ex+2dx=0.

The proof of this result is given in the ﬁrst part of Appendix B.

Proposition 3.4 Q-almost surely,for any t≥0and h∈C∞

0(R),we have

lim

τ→0Rh(x) ˆμt(τ , x) d x =1

2Stσ2

th(logSt).

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,x∈R:

Drift restriction

ˆαt+ˆ

βt·Dxˆνt

Dxˆ

Ct=∂τlog ˆa2

t.(4.1)

Spot volatility speciﬁcation

ˆa2

t(0,logSt)=σ2

t.(4.2)

Remarks

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

⎧

⎪

⎨

⎪

⎩

∂τˆνi

t=Lxˆνi

t+ˆ

βi

tLxˆ

Ct,τ>0,x∈R,i=1,...,m,

ˆν1

t(0,x)=Stσt1(−∞,log St](x),

ˆνi

t(0,x)=0,i=2,...,m,

(4.3)

which can be efﬁciently solved numerically, provided that we know ˆa2

tand ˆ

βt.

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

ˆα(τ,x)

+m

i=1ˆ

βi(τ, x ) τ

0Rˆa2(u, y ) ˆ

βi(u, y) ˆq(0,log S;u, y)Dyˆq(u,y;τ, x)dy du

ˆq(0,logS;τ,x)

+Sσ ˆ

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 x∈R, the process {ˆ

Ct(T −t,x)}t∈[0,T ) is a semimartingale with the de-

composition

dˆ

Ct(T −t,x) =ˆμt(T −t,x) −∂τˆ

Ct(T −t,x)dt +ˆνt(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 x∈R:

1. For al l t<T,we have,Q-almost surely,

∂Tt

0ˆνu(T −u, x ) ·dBu=t

0∂τˆνu(T −u, x ) ·dBu,

Dxt

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

surely,

∂Tt

0ˆvu(T −u, x ) du =t

0∂τˆvu(T −u, x ) du,

Dxt

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

Cstays

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 x∈R. Then

dˆ

Ct(T −t,x) =ˆνt(T −t, x) ·dBt

and, by the generalized Itô formula,

dˆa2

t(T −t,x)=ˆa2

t(T −t,x)ˆαt(T −t,x)−∂τˆa2

t(T −t,x)dt

+ˆa2

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.

ˆa2

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

dˆa2

t(T −t,x) ·Dxˆ

Ct(T −t,x)t

=ˆa2

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, x∈R, and t<T. This yields (4.1).

Now, ﬁx some test function h∈C∞

0(R)and denote by (y) the local time of S

at y. Then we have

supp(h)t

0ˆμu(τ , x) d u −Λt(ex)dx

=supp(h)ˆ

Cu(τ, x ) −Su−ex+u=t

u=0−t

0ˆνu(τ , x) −ˆνu(0,x)

·dBudx.

From Proposition 3.3 we conclude that there exists a sequence τn0 such that

almost surely

lim

n→∞supp(h) t

0ˆμu(τn,x)du−Λtexdx =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

ˆμu(τn,x)=∂τˆ

Cu(τn,x)=ˆa2

u(τn,x)D

xˆ

Cu(τn,x), (4.7)

and from (4.6) and (4.7) we have, almost surely for any tand h∈C∞

0(R),

0=lim

n→∞Rh(x)Λt(ex)−t

0ˆa2

u(τn,x)D

xˆ

Cu(τn,x)du

dx

=Rh(x)Λtexdx −lim

n→∞t

0Rh(x) ˆa2

u(τn,x)D

xˆ

Cu(τn,x)dxdu

.

Now, using the deﬁnition of local time and dominated convergence, we deduce that

the last expression is equal to

1

2t

0h(logSu)Suσ2

udu −t

0lim

n→∞Rh(x) ˆa2

u(τn,x)D

xˆ

Cu(τn,x)dxdu

=1

2t

0h(logSu)Suσ2

u−h(logSu)Suˆa2

u(0,logSu)du,

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 h∈C∞

0(R), we conclude that

ˆa2

t(0,logSt)=σ2

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

obtain

∂τˆvt(τ , x) =1

2ˆa2

t(τ, x )Dxˆvt(τ , x), τ > 0,x∈R,(4.8)

which holds almost surely for all t>0. Since

ˆvt(τ , x) =ˆμt(τ , x) −∂τˆ

Ct(τ, x ),

for any test function h∈C∞

0(R),wehave

Rh(x) ˆvt(τ, x ) dx =Rh(x) ˆμt(τ , x) d x −Rh(x)∂τˆ

Ct(τ, x ) dx .

As τ→0, the ﬁrst term in the above right-hand side converges to 1

2Stσ2

th(logSt)

because of Proposition 3.4. Moreover,

Rh(x)∂τˆ

Ct(τ, x ) dx =1

2Rh(x) ˆa2

t(τ, x )exˆqt(0,log St;τ, x) dx

→1

2Sth(logSt)ˆa2

t(0,logSt)

=1

2Stσ2

th(logSt).

This proves that, almost surely for any t,ˆvt(τ, x ), as function of x, converges weakly

to0asτ→0. Then, applying the maximum principle to (4.8) (see, for example, [3]),

we conclude that, for any τ>0 and x∈R,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

x∈R. But since 0 ≤ˆ

Ct≤St, it is a true martingale. This completes the proof of the

theorem.

Implementation

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 )}t≥0.

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

contracts.

We implemented numerically the above program, specifying a deterministic ˆ

β,

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 )}t≥0is identically zero, i.e., ˆ

β≡0. 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-

comes

ˆαt(τ , x) =∂τlog ˆa2

t(τ, x ).

Then we have

d

dt ˆa2

t(τ, x ) =ˆa2

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,

ˆa2

t(τ, x ) =ˆa2

0(τ +t,x),

from which we conclude, using the consistency condition (4.2), that

σt=ˆa0(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

β2

t(τ, x ) become ﬂat (i.e., ˆa2

tand ˆ

β2

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

βt2

2(x −logSt)2

τˆa2

t−1−1

4τˆa2

t.

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(·,·),ˆ

β1

t(·,·), and ˆ

β2

t(·,·)are smooth enough as functions of (τ, x ), but not

necessarily ﬂat. Lemma 3.2 and Proposition 4.2 imply that t

0Dxˆνu

Dxˆ

Cu·dBuis the local-

martingale part of log Dxˆ

C. Recall that

Dxˆ

C(τ,x) =1

2exˆq(0,logS;τ,x),

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)

Dxˆ

C.(τ, x) t→δ2

.(logS.,x)

t

as τ→0, where δ2

t(x, y ) is the square geodesic distance on a one-dimensional man-

ifold, with

δ2(x, y ) =y

x

dz

ˆa(0,z)2

.

Applying the generalized Itô formula, we obtain

dδ2

.(logS.,x)

=(. . .) d t

+2δt(logSt,x)

−1

2x

log Stˆat(0,z) ˆ

βt(0,z)dz

·dBt−dB1

t.

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 τ→0:

ˆαt(τ , x) =δt(log St,x)

τ

×1

2x

log Stˆat(0,z) ˆ

βt(0,z)dz

·ˆ

βt(0,x)+ˆ

β1

t(0,x)

+O(1).

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

ˆ

β2

t(0,x)

x

log Stˆat(0,z) ˆ

β2

t(0,z)dz=−ˆ

β1

t(0,x)

2+x

log Stˆat(0,z) ˆ

β1

t(0,z)dz

.(5.1)

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

ˆa2

0and some semimartingale λwith decomposition dλt=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)bye−xDxˆ

Cand

integrate with respect to x, then the second term on the left-hand side disappears since

Re−xDxˆνdx=0. This leaves us with

ˆα0=R∂τlog 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

R∂ττ log ˆa2

0(τ, x ) ˆqt(0,log S;τ, x)dx

+R∂τlog 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

0(0,logSt)+1

2λtˆa2

0(0,logSt)∂2

xx +∂x∂τlog ˆa2

0(0,logSt)=0,

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(1−ρ2dB1

t+ρdB

2

t), S0,

dσt=f(t,σ

t)dt +g(t, σt)dB2

t,σ

0,(6.1)

where {B1

t}t≥0and {B2

t}t≥0are 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

T¯

ST

¯σTe−d2

1(T ,K)

2

E¯

ST

¯σTe−d2

1(T ,K)

2,(6.2)

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

St=S0exp−ρt

0σu

f(u,σ

u)

g(u, σu)du +ρt

0

σu

g(u, σu)dσu

×exprt −1

2t

0σ2

udu +(1−ρ2)t

0σudB1

u.

For each t>0, we deﬁne the quantities Stand σtby

St=S0exp−ρ2

2t

0σ2

udu −ρt

0σu

f(u,σ

u)

g(u, σu)du +ρt

0

σu

g(u, σu)dσu(6.3)

Local volatility dynamics

and

σt=1

tt

0σ2

udu. (6.4)

Both quantities depend only on {B2

u}0≤u≤t. Moreover, conditioning on {B2

u}0≤u≤t,

we obtain

C(T, K) =E(ST−K)+=EBSST,T,1−ρ2¯σT,K,

where the expectation is over the second Brownian motion {B2

t}t≥0and 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) =−rKe−rT EΦd1−√T1−ρ2¯σT

+1

1−ρ2

1

√T

1

2√2πESTσ2

T

¯σT

e−d2

1

2,

∂KC(T,K) =−e−rT EΦd1−√T1−ρ2¯σT,

∂KKC(T, K) =1

√2π1−ρ2√TK2E¯

ST

¯σT

e−d2

1

2,

where d1=d1(T , K ) is deﬁned in the usual way by

d1(T , K ) =log ¯

ST

K+(r +1

2(1−ρ2)¯σ2

T)T

1−ρ2¯σT√T,(6.5)

which in turn gives the desired result since a2(T , K) =2(∂TC+rK∂KC)

K2∂KKC.

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}t≥0), 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 (σt,t

0σ2

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].