From Implied to Spot Volatilities

Valdo Durrleman

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Operations Research and Financial Engineering

June 2004

c© Copyright by Valdo Durrleman, 2004.

All Rights Reserved

Abstract

Given the quote price of a call or put option, the Black-Scholes implied volatility

is the unique volatility parameter to be put into Black-Scholes formula to give the

same price as the option quote price. This dissertation is concerned with the link

between the implied volatility and the actual volatility of the underlying stock. Such

a link is of particular practical interest since it relates the fundamental quantity for

pricing financial derivatives (the actual volatility of the underlying stock), which is

not observable, to directly observable quantities such as implied volatilities.

The link that we derive in chapter 2 is a link between the dynamics of the two

quantities. So far these quantities were mostly studied at a given time whereas we

work at the level of processes. This is the main result of the dissertation. In chapter

1, we shall first review current practical problems in option pricing. Our aim there

is twofold. First, we want to show that from a practical point of view, studying

dynamics is very natural. Second, we shall identify two practical issues to which we

shall propose answers in chapter 3.

Although the main motivation of this dissertation comes from contemporary issues

in the study of financial markets, chapter 2 also gives a solution to an inverse problem

in the mathematical sense. One wishes to recover the structure of a stochastic process

from a family of conditional expectations over its distribution.

Besides the main result, this dissertation makes the following contributions. It

brings new insights about implied volatility dynamics. In particular, it was observed

that its motion was extremely ‘rigid’ in the sense that the motion of a specific point

determines the entire surface dynamics. This statement will be made more precise.

Second, it provides with simple closed form approximations for implied volatilities.

These approximations avoid having to compute expectations to get option prices.

Third, this dissertation gives qualitative and quantitative understanding of common

models used in practice.

iii

Acknowledgements

I would like to thank my advisor, Professor Carmona. His trust and support helped

me a lot, especially during the final steps of completing my work in the Fall of 2003.

I am greatly indebted to my teachers, Professors Cınlar and El Karoui. To all of

them, I express my deepest gratitude. They have taught me what I know about

mathematical finance and stochastic processes, and more importantly, shared with

me their enthusiasm for mathematics and its applications.

I owe many warm thanks to Professor Dayanık for being my dissertation reader

and to Professor Sircar for having agreed to be a part of the committee of examiners.

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1 Implied volatility: a dynamical point of view 1

1.1 The Black, Scholes, and Merton paradigm . . . . . . . . . . . . . . . 1

1.1.1 The smile effect . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Two practical issues . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 What information does the implied volatility surface contain? 6

1.1.4 Model choice: the Dupire, Derman, and Kani approach . . . . 7

1.1.5 The calibration procedure . . . . . . . . . . . . . . . . . . . . 8

1.2 Smile dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.1 Static hedging of a down-and-in call option . . . . . . . . . . . 12

1.2.2 Deterministic models: sticky and floating smiles . . . . . . . . 13

1.2.3 Hedging a book of options . . . . . . . . . . . . . . . . . . . . 14

1.2.4 Stochastic models for implied volatility smiles . . . . . . . . . 17

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 From implied to spot volatilities 21

2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Boundary behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

v

2.3 Implied volatility dynamics . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.1 V dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.2 X dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Recovering the spot volatility . . . . . . . . . . . . . . . . . . . . . . 39

2.5 Recovering the spot volatility dynamics . . . . . . . . . . . . . . . . . 47

2.6 Concluding remarks about our assumptions . . . . . . . . . . . . . . 53

3 From spot to implied volatilities and applications 58

3.1 From spot to implied volatilities . . . . . . . . . . . . . . . . . . . . . 58

3.1.1 Black-Scholes deterministic volatility model . . . . . . . . . . 60

3.1.2 Heston stochastic volatility model . . . . . . . . . . . . . . . . 60

3.1.3 Dupire local volatility model . . . . . . . . . . . . . . . . . . . 61

3.1.4 SABR stochastic volatility model . . . . . . . . . . . . . . . . 62

3.2 Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Stochastic models for implied volatilities . . . . . . . . . . . . . . . . 65

3.4 Computing implied volatilities . . . . . . . . . . . . . . . . . . . . . . 66

3.4.1 First order approximations . . . . . . . . . . . . . . . . . . . . 66

3.4.2 Higher order approximations . . . . . . . . . . . . . . . . . . . 67

3.5 Static hedging of a barrier option when the spot is Markov . . . . . . 70

3.5.1 Static hedging with a constant volatility . . . . . . . . . . . . 71

3.5.2 Static hedging in Dupire local volatility model . . . . . . . . . 72

4 Conclusions 75

A Semimartingales with spatial parameters and generalized Ito for-

mula 77

A.1 Semimartingales with spatial parameters . . . . . . . . . . . . . . . . 77

A.2 Generalized Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vi

List of Figures

1.1 Calibrated smiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Smile dynamics: CEV vs. Heston . . . . . . . . . . . . . . . . . . . . 11

1.3 Smile dynamics: Quadratic vs. SABR . . . . . . . . . . . . . . . . . . 15

3.1 Static hedge ratios for down-and-in call options . . . . . . . . . . . . 73

vii

Chapter 1

Implied volatility: a dynamical

point of view

In this first chapter, we present some of the contemporary issues faced by actors in

financial markets. We shall put emphasis on two problems: model calibration to

market data on the one hand and risk management of a book of options on the other

hand. In so doing, we shall review common practice to tackle these problems as well

as their flaws. In the second part of this chapter, we shall advocate a dynamical point

of view. Such a point of view is very natural in practice. With this first chapter our

hope is to justify from a practical perspective chapters 2 and 3.

1.1 The Black, Scholes, and Merton paradigm

In their seminal works, Black and Scholes (1973) and Merton (1973) show how to

price options on a stock. A common option is a European call option; it is defined by

a date T called date of maturity, a positive number K called strike or exercise price,

and it gives the right to its owner to acquire at time T one unit of the stock at the unit

price K. Black and Scholes (1973) came up with the now well known Black-Scholes

formula which gives the price of such an option as a function of the stock price, K,

1

T , the short rate of interest, and the volatility of the stock. Only this last parameter

is not directly observable but it can be estimated from time series data.

European call options as we just described them are now actively traded and their

price is set by supply and demand in the marketplace in very much the same way

stocks set their prices. At a first glance, the works of Black and Scholes (1973) and

Merton (1973) on how to price an option seem now irrelevant: the market in doing it

for us! Such is however not the case. First, there exist more complex options, often

called exotic options, a simple example that we shall consider later is a barrier option.

These are not actively traded and therefore need to be priced. More importantly the

work of Black and Scholes (1973) and Merton (1973) is still of utmost importance

because of the paradigm they proposed to price options.

Their fundamental contribution was to see that the risk contained in an option

(the uncertainty about stock prices in the future) could be exactly synthesized in a

self financing portfolio. In other words, they provide a ‘trick’ to change risks in the

future into portfolio strategies of stocks and bonds. These portfolios have simply to

be rebalanced continuously and in a precise way but without injecting cash. This

means that the seller of an option will exactly meet his or her obligations at maturity

by simply holding at each time a certain quantities of stocks and bonds. This is true

in every state of the world, whatever happens to the stock! The fair price of the

option ought then to be the initial cost of such a replicating strategy.

As we shall see in the next paragraph, the hypotheses under which the Black-

Scholes formula was established are wrong. However, the idea of dynamically hedging

the risk is still the main methodology to price options.

1.1.1 The smile effect

The Black-Scholes formula in Black and Scholes (1973) and Merton (1973) is obtained

under the assumption that the stock can be modeled as a geometric Brownian motion

2

with constant volatility. Since option quote prices are available and the only unknown

parameter is the volatility, if the Black-Scholes assumption were true, we could find

the volatility of the stock by simply inverting Black-Scholes formula. A volatility thus

obtained is called implied.

In particular, for options with different maturities and different strikes but written

on the same stock, one should find the same implied volatility, i.e., the volatility of

the stock which is unique. Such is not the case. At a given maturity, options with

different strikes trade at different implied volatilities. When plotted against strikes,

implied volatilities exhibit a smile or a skew effect.

The Black-Scholes model and its pricing formula are wrong. As often quoted, the

implied volatility is the wrong number to put in the wrong formula to obtain the right

price. It comes at first as a surprise to see this apparently irrelevant number being

constantly used by traders. Why should it deserve so much attention?

A first answer is pointed out in Lee (2002). “(. . . ) it is helpful to regard the

Black-Scholes implied volatility as a language in which to express an option price.

Use of this language does not entail any belief that volatility is actually constant.

A relevant analogy is the quotation of a discount bond price by giving its yield to

maturity, which is the interest rate such that the observed bond price is recovered

by the usual constant interest rate bond pricing formula. In no way does the use

or study of bond yields entail a belief that interest rates are actually constant. As

yield to maturity is just an alternative way of expressing a bond price, so is implied

volatility just an alternative way of expressing an option price.

The language of implied volatility is, moreover, a useful alternative to raw prices.

It gives a metric by which option prices can be compared across different strikes,

maturities, underlyings, and observation times; and by which market prices can be

compared to assessments of fair value. It is a standard in industry, to the extent that

traders quote option prices in “vol” points, and exchanges update implied volatility

3

indices in real time.”

In chapter 2, we shall give another answer to this apparent puzzle by showing

that the implied volatility smile dynamics for short maturities and the spot volatility

dynamics are intimately related. We shall soon see that spot volatilities are crucial

in practice.

1.1.2 Two practical issues

Before we continue the discussion of implied volatilities, we introduce some of the

problems we would like to solve. We shall focus on two issues.

First we need a method to price exotic options. The simplest example of such

options are barrier options that give the same rights as European options under the

additional constraint that the spot go/do not go below/above a prespecified barrier

level. These options will be priced using the Black-Scholes paradigm, i.e., their price

will be the initial value of a self financing portfolio that synthesizes their risks if such

a portfolio exists. Following the line of reasoning of Black and Scholes (1973) and

Merton (1973), one must start off with a model for the spot process. Whereas Black

and Scholes (1973) and Merton (1973) had lots of degrees of freedom to choose their

model for the spot price, we are now constrained by the fact that our model candidate

must give back the price (or, equivalently, the implied volatilities) of actively traded

options. It is therefore crucial to understand what the implied volatility smile tells

us about the spot process before we can model it any further.

Before assessing this question, we would like to mention a second problem faced

by traders: how should one manage the risk of a book of options?

Dynamically hedging a single option is rather well understood. We are guided

here by El Karoui et al. (1998) and Rebonato (1999). By buying a European call

option and selling its replicating portfolio, a trader is in fact ‘trading the volatility.’

He or she enters such a deal only if he or she thinks the European call option is cheap

4

or, equivalently, that its implied volatility is low. Indeed, his or her profit at option’s

maturity precisely depends on the future realization of the implied volatility during

the option’s life. To see why this is so, one can compute over a small time interval

∆t, the profit or loss of the portfolio (option − replicating portfolio). It is given by:

P & L =1

2ΓS2

((∆S

S

)2

− Σ2∆t

)

where Σ is the implied volatility at which the option was purchased and (∆S/S)2 is

the stock volatility experienced during the small time interval ∆t. Γ is the Black-

Scholes gamma of the option, i.e., the second derivative of the value function with

respect to the current value of the spot. In the case of a European call option, it is

always positive. One sees on the formula above that the trader wants the stock to

‘vibrate’ as much as possible, i.e., (∆S/S)2 to be as large as possible. In particular, if

the realized volatility is always above the implied volatility at which the option was

purchased, the trader makes a profit in all states of the world. This example sheds

some light on terminology like ‘being long/short or trading the volatility,’ ‘being

long/short or trading the gamma,’ etc.

Put it another way, El Karoui et al. (1998) show that using the replicating strategy

proposed by Black and Scholes with the implied volatility makes sense even if the spot

is not Markov. From this example, one should also keep in mind that traders use

nowadays the Black, Scholes, and Merton paradigm to dissociate the risk in the spot

(delta risk) from that in the volatility (vega risk.)

Dynamic hedging of a portfolio consisting of several options with different strikes

and different maturities is more complicated. In particular, the gamma of such a

book does not have a constant sign. As pointed out in Avellaneda et al. (1995) in the

case of a call spread option whose payoff has a mixed convexity, no hedge based on a

Black-Scholes delta can superhedge the claim even when the unknown spot volatility

5

stays within a band. In that context, one needs a model to compute the risks. Such

a model will provide a way of aggregating the risks of options with different strikes

and maturities.

This second practical issue, like the pricing of exotic options, shows the importance

of a model consistent with the observed implied volatility smile. The next paragraph

is about its informational content.

1.1.3 What information does the implied volatility surface

contain?

If the Black-Scholes model is wrong, how can we assess the main issues faced by the

actors in option markets? We now come back to the question of knowing what the

smile tells us about the spot process. As we have explained, this is the very first

question if one wants to build a model for the spot process.

If we keep the Black-Scholes paradigm in mind, the price of an option is the initial

cost of a self financing replicating strategy. If such a replication is possible, i.e., if the

market is complete, this initial cost can be represented as the expected value of the

option payoff under the risk-neutral measure. If we denote the price of European call

option with maturity T and strike price K at time 0 by C(T,K), this means that

C(T,K) = E(ST −K)+ (1.1)

where S is the risk-neutral spot price process. We shall assume for simplicity that

interest rates are zero.

Breeden and Litzenberger (1978) showed that the knowledge C(T,K) for all K

at a given T is equivalent to the knowledge of the risk-neutral distribution of ST .

This is seen by (formally) differentiating the above equality. It gives the cumulative

6

risk-neutral distribution function of the spot at time T :

P ST ≤ K = 1 + ∂KC(T,K).

Observing at a given day option prices of all strikes for a specific maturity date

gives the marginal distribution of the spot process for this date. If in addition we

can also observe prices for all maturities, the marginal distributions of the process

are completely known. However, it is important to notice that this leaves many

possibilities about the full risk-neutral distribution.

1.1.4 Model choice: the Dupire, Derman, and Kani approach

In view of the questions of paragraph 1.1.2, one needs to propose models to explain

smiles. Different ideas have been proposed. Instead of a constant volatility, one can

posit a volatility that is a function of the spot process itself. This way, the spot

process is a Markov process solution of a stochastic differential equation. A more

general approach consists in a volatility being a stochastic process on its own. In

such a case the spot process alone is no longer Markov and such models are often

called (fully) stochastic volatility models. Finally, people have further proposed to

introduce jumps in the spot process. This last idea can of course be combined with

any of the previous ones.

In this section, we shall focus on the first idea, i.e, on the case where the spot is

a continuous Markov process. A first systematic approach to construct a spot model

consistent with the observed implied volatility smile was given by Dupire (1994) and

Derman and Kani (1994). A very detailed exposition of this method with emphasis

on its numerical implementation can be found in Rebonato (1999).

As explained in the previous section, observed prices of European options only

give information about the marginal distributions of the spot process. Dupire (1994)

7

and Derman and Kani (1994) show that under the assumption that the spot process

is continuous and Markov, the spot process distribution is completely specified. More

precisely, let us suppose that the volatility of the spot is a deterministic function

of the current value of the spot and possibly time, say, σ(t, S). Dupire (1994) in

particular says that we can find back this function if we observe the entire implied

volatility surface (function of T and K) at a given date. His result reads in terms of

prices:

σ(t, S) =

√2∂TC(t, S)

S2∂2KKC(t, S)

. (1.2)

where C is the function defined in (1.1).

This method is very appealing from a theoretical point of view. Let alone its

numerical difficulties, it has a main shortcoming. It is its dependence on the date

at which the volatility function σ is computed. This calibration procedure, may give

very different answers at very close dates. This means that traders must change

their model every day. This is enough to loose confidence in such a method. Also,

empirical tests show that spot processes are not Markov. We shall further discuss the

calibration problem in the next paragraph.

One should notice that formula (1.2) gives the spot volatility from prices (or,

equivalently, from implied volatilities.) Chapter 2 is therefore very much in the same

spirit as this result.

1.1.5 The calibration procedure

In this paragraph, we shall give an example of calibration. For definiteness, let us

assume that the spot process is continuous and Markov and more precisely that it is

a CEV process. The volatility of a CEV (Constant Elasticity of Variance) process is

a simple power law:

σ(t, S) = αSβ−1.

8

α > 0 and 0 ≤ β ≤ 1 are fixed parameters. Today’s stock price is S0 = $100. It

evolves as

dSt = Stσ(t, St)dWt

where W is Brownian motion under the risk-neutral measure. Again, interest rates

are zero for simplicity. Since the volatility is not constant, this model gives rise to

a smile as shown by the solid line in Figure 1.1. The parameter α is chosen so that

today’s spot volatility is 20% and β = 0.1.

Let us imagine a trader. Of course, he or she does not know the form of the spot

volatility, let alone the particular values of α and β. Let us suppose that his or her

favorite model is the Heston model. This model is a stochastic volatility model for

which the spot volatility is a correlated mean reverting square root process. This

means:

dSt = σtStdWt

together with

dσ2t = RevSpd

(RevLvl2 − σ2

t

)dt+ VvolσtdW

∗t

where W and W ∗ are correlated Brownian motions with correlation ρ.

One reason he or she might like this model could be that there are four degrees of

freedom (the reversion speed RevSpd, the mean reversion level RevLvl, the volatility

of the volatility Vvol, and the correlation ρ), allowing to fit many different smile

shapes yet being parsimonious. In Figure 1.1, we see that two possible values of the

parameters can give equally good fit to the observed ‘market data.’

There can be more than one answer to the calibration procedure. This is a general

fact which is not particular to the Heston model. As we argued in paragraph 1.1.3, we

only know the marginal distribution of S from the implied volatility smile. There are

many reasonable models that fit any given implied volatility surface as there are many

processes with specified marginal distributions. More importantly these processes can

9

Figure 1.1: Two calibration results against observed smile for options maturing in3 months. Spot trades at $100. The observed smile is that of a CEV model withα = 12.6 and β = 0.1 for which the volatility of the spot is 20%. The two fittedsmiles are obtained with Heston models, one with RevLvl = 25% and RevSpd = 0.12the other with RevLvl = 35% and RevSpd = 0.06. In both cases, Vvol = 10.9% andρ = −0.7.

have widely different distributions yielding to very different prices for exotic options.

In a recent paper Hagan et al. (2002) show that the dynamics of the smile can also

vary widely for different models.

To illustrate this last point, let us make the following experiment. Suppose that

the stock drops tomorrow to $95. Since the spot is Markov, tomorrow’s spot volatil-

ity can be computed and from the parameters we find 20.9%. Let us assume that

the trader can effectively estimate this value and use it together with the estimated

parameters in Figure 1.1 to price options. Depending of the model choice that he or

she has made he or she would get two different answers as shown in Figure 1.2.

In one case, the smile stays rather close to the ‘market data’ but in the other

case it moves upwards. This means that two equally good fits can produce different

10

Figure 1.2: The three smiles of Figure 1.1 after the spot dropped to $95. The volatilityof the spot is now 20.9%.

dynamical answers. If the trader had chosen the ‘bad’ set of parameters, he or she

would have to recalibrate his or her model to the market data. This should be

sufficient to loose confidence in a model.

To compute the implied volatility smiles of this section, we used the formulas

derived in paragraph 3.4.2.

1.2 Smile dynamics

As shown in Figure 1.1 and 1.2, smiles can have different dynamics. This fact will be

further evidenced in Figure 1.3. In this section, we shall show that the smile dynamics

is very natural to a trader in the context of the two questions raised in paragraph

1.1.2.

11

1.2.1 Static hedging of a down-and-in call option

This example is inspired by a case study in Rebonato (1999) pp. 177–181. We

simplify the situation quite a lot by looking at a single barrier option instead of a

double barrier option.

Static hedging of barrier option was introduced by Bowie and Carr (1994); see

also Carr et al. (1998). By contrast to dynamic hedging where continuous trading in

the stock and the risk-free bank account replicates the contingent claim payoff, static

hedging accomplishes the same task with a portfolio that is discretely rebalanced. In

fact, in the particular example of this paragraph, it is rebalanced once or never.

Let us consider the case of a down-and-in call option. The payoff of such an option

is the usual call payoff (ST − K)+ under the condition that the stock process goes

below the barrier H at some point before maturity T . To make our discussion as

simple as possible, we shall assume that the option is regular, i.e., H < K.

Let us first recall the idea of static hedging when volatility is constant. This idea

can be put on a firm mathematical ground by applying the strong Markov property

(see, for instance, paragraph 3.5.1). Assume that the stock reaches the barrier H

at TH < T . At this very moment, the barrier option is precisely worth a call option

with strike K, i.e., it is worth Call (spot = H, strike = K,mat = T ). Since volatility is

constant, the price is given by Black-Scholes formula. Moreover, a simple computation

shows that,

Call (spot = H, strike = K,mat = T ) =K

HPut

(spot = H, strike =

H2

K,mat = T

).

Therefore, at TH , the option is worthK/H put options with strikeH2/K and maturity

T . The advantage of this representation is that such put options are worthless if the

stock stays above H since their strike is H2/K < H. Whether or not the spot has

reached the barrier during the option’s life, the value of the barrier option is worth

12

K/H put options with strike H2/K and maturity T . This gives the price of the

barrier option as well as its replicating strategy: do nothing until the spot hits the

barrier, then convert put options into the call option. Obligations are met at maturity

in all cases.

In the presence of smile, the same reasoning can be made. There is however a

major difficulty: how do we know the option prices when the spot hits the barrier?

From today’s implied volatility smile we only know the distribution of the spot process

at a fix date conditionally on having its value today. As shown by the example in

Figure 1.1 and 1.2, this leaves many possibilities for its distribution at maturity

conditionally on being at the barrier, let alone the law of the hitting time itself.

Part of the information needed to price and hedge barrier option is exactly that of

the shape of the smile when the spot moves to the barrier. We need to know the

dynamics of the smile.

1.2.2 Deterministic models: sticky and floating smiles

The dynamics of the smile is so crucial in the context of pricing and hedging barrier

options that traders are ready to abandon the route sketched in paragraph 1.1.5.

Although this route is mathematically more rigorous, traders often prefer to specify

directly the shape of the smile when the spot hits the barrier independently of any

given model. There are two common practice in that respect: the sticky smile or the

floating smile as explained, for instance, in Rebonato (1999).

Sticky smiles are smiles that do not move if the underlying moves. Floating smiles

on contrary move exactly with the spot. To illustrate this point, let us go back to the

implied volatility smile in Figure 1.1. We wish to compute the implied volatility of a

call option with maturity 3 months and strike $120 when the spot trades at $90. In a

‘sticky smile world’ the implied volatility for this option is again 18.4%. In a ‘floating

smile world’ it would be 19.2%, which is implied volatility for a call option with strike

13

$108 today. In other words, a trader who believes in a sticky smile world would label

the x-axis in Figure 1.1 with strikes and use the same graph whatever the spot value

is. On the contrary, a trader who believes in a floating smile world would label the

x-axis with moneyness (i.e., with the ratio K/S) and use the same graph whatever

the spot value is. This explains why floating smiles are also called sticky delta smiles

since the Black-Scholes delta is a function of moneyness only.

1.2.3 Hedging a book of options

Let us go back for a moment to the example of paragraph 1.1.5. If the trader had cho-

sen the ‘bad’ set of parameters, he or she would have to recalibrate his or her model

to the market data. As explained in Hagan et al. (2002), in a case a trader is dynam-

ically hedging a book of options, a recalibration procedure leads to a recomputation

of hedges. Non stable hedges can prove extremely costly in practice.

With the formulas derived in paragraph 3.4.2, we reproduce in this paragraph the

dramatic example of Hagan et al. (2002). Let us consider the following two models.

The first one is termed quadratic Markov:

σt = γ + δ(St − S

)2.

This model may explode in finite time because the function S 7→ γ+ δ(S − S

)2does

not satisfy the usual linear growth constraint on the coefficients of a stochastic differ-

ential equation. This should not be of much concern since we can always modify this

function for large values of S to make it grow linearly. We are looking at short ma-

turities and such a modification should not change the conclusions of this qualitative

discussion.

The second model is the SABR model of Hagan et al. (2002). It is a lognormal

14

stochastic volatility model:

dSt = σtStdWt

together with

σt = αtSβ−1t

dαt = ναtdW∗t

where W and W ∗ are correlated Brownian motions with correlation ρ.

The dramatic difference between these two models is shown in Figure 1.3. With

an initial spot value of $100, the two models produces similar smiles. They differ

somewhat for large strikes. However, when the spot drops from $100 to $95 the two

smiles move in opposite directions.

Figure 1.3: Two different smile behaviors for options maturing in 3 months whenthe spot drops from $100 to $95. The quadratic smile was obtained with S = 125,γ = 0.14 and δ = 10−4 for which the volatility of the spot is 20%. The SABR smilewas obtained with α = 0.8, β = 0.7, ν = 64% and ρ = −0.72 for which the volatilityof the spot is also 20%. The two smiles move in opposite directions.

15

The value of a portfolio of options depends explicitly on the values of the under-

lying spot process and of different points of the implied volatility surface, one for

each option. Viewed this way, the problem of hedging a portfolio of options is similar

to that of a portfolio of bonds in fixed income markets. Each bond in the portfolio

involves one or more points of the yield curve. To hedge such a portfolio, a trader

would buy additional instruments so that the overall portfolio is unsensitive to moves

of the yield curve. The most important move to be hedged is typically a parallel shift

of the yield curve. Other important moves can be found by performing a principal

component analysis of the yield curve. In practice to find the hedge ratios one com-

putes the value of today’s portfolio and then its value after bumping the whole yield

curve by a small amount. One buys enough bonds to get the same value before and

after bumping the yield curve.

One would like to do the same in the case of the implied volatility surface. Cont

and Da Fonseca (2001) have performed principal component analysis for implied

volatility surfaces. However, how the methodology for the yield curve must be trans-

lated for the implied volatility surface is not straightforward. Indeed, Heath et al.

(1992) give a way of justifying the methodology for interest rate by providing an ar-

bitrage free framework for the dynamics of the yield curve. But such an approach has

not been carried out yet in the case of implied volatilities. We will review attempts

in this direction in the next paragraph. The practical issue can be summarized as

follows. Suppose for instance as in Figure 1.1 that the at-the-money implied volatility

(i.e., strike $100) is 20% and that the implied volatility with strike $110 is 21%. We

bump the at-the-money implied volatility by 1 pt. By how much should one bump

the other implied volatility to respect no arbitrage conditions?

16

1.2.4 Stochastic models for implied volatility smiles

As explained in the previous paragraph, an appealing idea consists in modeling im-

plied volatilities directly. This approach is very much in the same spirit as the Heath

et al. (1992) approach. It was initiated in the present context by Dupire (1993), Der-

man and Kani (1998), Zhu and Avellaneda (1998), Ledoit and Santa-Clara (1999),

Schonbucher (1999), Carr (2000), Brace et al. (2001). Whether they look at a single

implied volatility or at the entire surface, their idea is to model implied volatilities

directly.

There are many reasons why this approach has not been successful yet. In the case

of implied volatilities there are structural constraints on the surface shape. Any model

must ensure that these constraints are satisfied. More precisely, any smooth function

T 7→ f(T ) can be a valid initial forward curve for any Heath-Jarrow-Morton model

(HJM for short). On the contrary, a smooth option price surface (T,K) 7→ C(T,K)

must be such that

∂TC(T,K) ≥ 0, ∂KC(T,K) ≤ 0, and ∂2KKC(T,K) ≥ 0.

These constraints come directly from (1.1) and they of course translate into con-

straints on the volatility surface. Note that these necessary conditions are not far

from being sufficient in the following sense. In case they hold, there is a spot volatil-

ity process that produces that precise implied volatility smile. A simple candidate is

the Markov spot process of Dupire (1994) and Derman and Kani (1994) of paragraph

1.1.4.

Not only are the admissible surface shapes more complicated but their dynamics

are also poorly understood. The HJM methodology gives us risk free dynamics for

the forward curve f without any reference to the short rate of interest r. In case one

wishes to model the implied volatility directly there seems to be no simple way of

17

disentangling implied volatilities from the corresponding spot volatility.

To be more precise, let us sketch the HJM methodology. Start with a short rate

process r. Zero coupon bonds are securities that pay $1 at maturity T and their price

at time t is denoted by B(t, T ). Risk neutral valuation gives

B(t, T ) = Ee−

∫ Tt rsds

∣∣∣Ft

.

Then, we define the continuously compounded forward rate at time t for maturity T ,

f(t, T ), by

B(t, T ) = e−∫ T

t f(t,s)ds. (1.3)

Then, Heath et al. (1992) derive that f satisfies a stochastic differential equation for

each T :

df(t, T ) =

(γ(t, T )

∫ T

t

γ(t, s)ds

)dt− γ(t, T )dWt (1.4)

where γ(t, T ) is related to the martingale Ee−

∫ T0 rsds

∣∣∣Ft

in the following way. If

we are working in a Brownian filtration, this positive martingale has the martingale

representation property so that we can write it as

Ee−

∫ T0 rsds

∣∣∣Ft

= E

e−

∫ T0 rsds

+

∫ t

0

B(s, T )Γ(s, T )dWs

for some adapted processes Γ(s, T ). Finally, γ(t, T ) = ∂T Γ(t, T ). What is important

to notice is that r has disappeared in (1.4). It can be recovered by proving that in fact

f(t, t) = rt. This remark allows us to use (1.4) for modeling purposes by specifying a

volatility function γ(t, T ) for the forward rate. The dynamics are completely specified

and arbitrage free.

Let us now try to mimic this reasoning in case of implied volatilities. We start off

18

by writing the price of call options as

Ct(T,K) = E

(ST −K)+∣∣Ft

where S is the spot process. We denote its volatility by σ. Then, we define the

implied volatility Σt(T,K) by inversion of the Black-Scholes formula that plays the

same role as the exponential function in the interest rate case at (1.3). The problem is

that the stochastic differential equation for Σt(T,K) is plagued by S and its volatility

σ (see, for instance, Proposition 2.3.1 below or Brace et al. (2001).) There seems to

be no simple way of getting rid of σ. This annoying fact prevents us from describing

the dynamics of the implied volatility surface intrinsically as it was possible for f .

However, we shall show how the Σt(T,K) dynamics relates to that of σt in chapter

2. In fact, we shall see that

Σt(t, St) = σt.

This very much in the same spirit as the equality f(t, t) = rt in the case of interest

rates.

1.3 Outline

We close this first chapter by an outline of the next two chapters. Chapter 2 aims at

answering the following question. How much do we know about the spot volatility

if we can observe sufficiently many prices of options written on that spot? This is

precisely the question of paragraph 1.1.3. We shall tackle this problem from a different

angle in the sense that we will study dynamics instead of focusing on data at a precise

date.

We shall take the point of view sketched in the previous section. We are consider-

ing a market where the primary securities are the spot and liquid options. As already

19

pointed out, there is a lot of information about the spot process in option prices. In

fact, we shall see that under some regularity conditions, we can find back the spot

volatility dynamics.

We want to make very few assumptions on the spot dynamics in order to have the

most general understanding. On the other hand, our results will only be local. Unlike

in the case of Dupire’s result, in the Markov case we will not be able to recover entirely

the deterministic volatility function as in (1.2). Instead, we shall get its local shape,

namely its first and second order derivatives in the space variable at the current value

of the spot and its first derivative in the time variable at the current time.

In chapter 3, we shall present applications of the results in chapter 2. We shall

see that they lead to a very simple understanding of dynamics of volatility surfaces.

We shall also derive closed form approximations for implied volatilities for several

popular spot volatility models. This provides some insights and some answers to the

stochastic modeling of implied volatility surfaces.

Finally, we shall describe a static hedging procedure for barrier options in presence

of smile. The two issues raised in paragraph 1.1.2 will therefore find partial answers

in this last chapter. Whether they are satisfactory from a practical point of view is

left for future work.

20

Chapter 2

From implied to spot volatilities

The main results of this chapter are Theorems 2.4.4 and 2.5.1. The first theorem

gives back the value of the current spot volatility from the observed implied volatility

smile. The second theorem gives its dynamics across time via a stochastic differen-

tial equation. These theorems are of the inverse type. Indeed, we start off with a

spot volatility process and we are interested in recovering it from actually observ-

able quantities such as implied volatilities. We obtained these theorems by a careful

study of option prices near expiry. The main finding of this chapter is that asymp-

totic behavior of option prices near the money give all information about the spot

process1.

The idea of the proof is to observe that implied volatilities are constrained by

no arbitrage arguments to have a very special behavior near expiry. Namely, when

multiplied by the square root of the remaining time to maturity, they go to zero as the

time to maturity approaches zero. We then draw the consequences of that particular

behavior on the dynamics of the implied volatility surface. This dynamics is given by

an application of Ito formula.

1In a recent paper, Carr and Wu (2003) build an econometric test to tell from data of optionprices near expiry whether the underlying spot process is a purely continuous process, a pure jumpprocess or a combination of both. Although we restrict ourselves to the continuous case, it shouldbe noted that on a general level our idea bears some similarities with theirs.

21

To study these questions we found the language of semimartingales with spatial

parameters particularly appropriate. We follow closely the definitions and theorems

of Kunita (1990). In order to have a self-contained dissertation, these results are

recalled in appendix A. The other techniques are basic real analysis techniques and

standard stochastic calculus results for finite dimensional Brownian motion.

2.1 Framework

Let (Ω,H,P) be a probability space with a n-dimensional Wiener process (W t)t≥0

on it. We shall use boldface letters for vectors; W i will denote the i-th component

of W . The filtration generated by the Wiener process has been augmented as usual

and is denoted by (Ft)t≥0. All processes considered in this chapter are adapted with

respect to that filtration. We shall assume that probability measure P is risk-neutral,

that is, discounted price processes are localmartingales if there is no arbitrage in the

market.

We first define the risk-neutral stock process S. As is well known, there is no loss

of generality in assuming that interest rates are zero. We are given a spot volatility

process σ, which is jointly measurable, adapted to the filtration (Ft)t≥0, and satisfies

the following integrability condition:

∀t ≥ 0

∫ t

0

σ2sds <∞ a.s. (2.1)

Since σ satisfies (2.1), we can define S to be the following stochastic exponential

St = S0 exp

(∫ t

0

σsdW1s −

1

2

∫ t

0

σ2sds

).

In other words, S is a typical positive localmartingale in a Brownian filtration. We

shall say that it is a global localmartingale because it is well defined for all t ≥ 0.

22

Let us stress that, whereas S is only driven by the first noise, σ is adapted to the

entire filtration (Ft)t≥0 generated by (W t)t≥0. In financial terms, we do not assume

completeness of the market. We make the following basic assumption:

Assumption 1. St is integrable for all t ≥ 0 and σt(ω) > 0, Leb×P a.e. (t, ω).

Our main and fundamental assumption is that liquid options are marked-to-

market. By liquid options, we mean options with short maturities and strikes near

the money. More precisely, let x > 0 and y > 0 be fixed throughout the rest of the

chapter, at any given time t, we are only considering the continuum of options whose

strike prices K lie in the open interval (Ste−y, Ste

y) and whose maturities T lie in the

open interval (t, t+ x).

For (T,K) ∈ (0, x)× (S0e−y, S0e

y), i.e., for an option with strike K and maturity

T which is liquid today (t = 0), let us define

τ(T,K) = inf

t > 0 :

∣∣∣∣ln(St

K

)∣∣∣∣ ≥ y

∧ T.

τ(T,K) is the time at which this option stops being liquid because either it has

expired or the value of the stock has gone too far away from its strike.

Lemma 2.1.1. τ(T,K) is an accessible lower semicontinuous stopping time.

Proof. Indeed, for (T,K) fixed, τ(T,K) is a stopping time. Next, for a.e. ω, (T,K) 7→

τ(T,K) is a positive lower semicontinuous function. Finally, the sequence of random

fields

τn(T,K) = inf

t > 0 :

∣∣∣∣ln(St

K

)∣∣∣∣ ≥ y(1− 2−n)

∧ T

is such that τn(T,K) < τ(T,K) because S has continuous paths, and such that

τn(T,K) ↑ τ(T,K) as n goes to infinity.

Let (T,K) fixed, on t < τ(T,K), Ct(T,K) denotes the price at time t of the

option struck at K with maturity T . Since we assume it is marked-to-market, no

23

arbitrage arguments yields that it has to be a localmartingale. We know its terminal

value and this gives the well known risk neutral representation:

Ct(T,K) = E(ST −K)+ |Ft

. (2.2)

Ct(T,K) has monotonic and convex properties as a function of T and K. These

properties are well known and easy to establish from (2.2). We recall them in the

following lemma.

Lemma 2.1.2. Let T1 ≤ T2 and K be fixed. On t < τ(T1, K) ∩ t < τ(T2, K) =

t < τ(T1, K), we have

Ct(T1, K) ≤ Ct(T2, K) a.s.

Let K1 ≤ K2 and T be fixed. On t < τ(T,K1) ∩ t < τ(T,K2), we have

Ct(T,K1) ≥ Ct(T,K2) a.s.

Let 0 ≤ λ ≤ 1, K1, K2 and T be fixed. On t < τ(T,K1) ∩ t < τ(T,K2), we have

Ct(T, λK1 + (1− λ)K2) ≤ λCt(T,K1) + (1− λ)Ct(T,K2) a.s.

We wish to study Ct(T,K) as a function of (T,K) for every t and almost every

ω. Since Ct(T,K) is given to us as a conditional expectation, it is only defined off an

exceptional set depending on (T,K). We need to construct a modification of Ct(T,K)

that is continuous in (T,K) for every t and almost every ω.

Lemma 2.1.3. There exists a modification of C(T,K) (also denoted C(T,K)) as a

continuous process such that Ct(T,K) is continuous in (T,K) for every t and almost

every ω.

24

Proof. Let t ≥ 0. Let (Tn)n≥0 and (Km)m≥0 be two enumerations of the positive

rational numbers. Ct(Tn, Km) is defined off a set of measure zero that depends neither

on n nor on m for t < Tn < t+ x and Ste−y < Km < Ste

y. Thanks to the elementary

inequality |a+ − b+| ≤ |a− b|, we have

|Ct(Tn′ , Km)− Ct(Tn, Km)| ≤ E∣∣STn′

− STn

∣∣∣∣Ft

and

|Ct(Tn′ , Km′)− Ct(Tn′ , Km)| ≤ |Km′ −Km| .

Therefore,

|Ct(Tn′ , Km′)− Ct(Tn, Km)| ≤ E∣∣STn′

− STn

∣∣∣∣Ft

+ |Km′ −Km| .

Let us now extend Ct(·, ·) at an arbitrary point (T,K) of (t, t + x) × (Ste−y, Ste

y).

Take two sequences of rational numbers such that Tn → T and Km → K. Let

ε > 0, by Lebesgue dominated convergence theorem, there is an integer N , such that

E |STn − ST || Ft < ε/3 whenever n ≥ N . Obviously, there is also an integer M

such that |Km′ −Km| < ε/3 whenever m′,m ≥M . This shows that

n, n′ ≥ N, m,m′ ≥M =⇒ |Ct(Tn′ , Km′)− Ct(Tn, Km)| ≤ ε.

The sequence (Ct(Tn, Km))n,m is Cauchy and it therefore converges to a limit that

we call Ct(T,K). One checks as usual that this limit does not depend on the chosen

sequences and that the resulting function defined on (t, t+ x)× (Ste−y, Ste

y) is indeed

continuous.

We shall make a further assumption about the regularity of Ct(T,K) as a function

of (T,K). We need a couple of definitions.

25

We first define seminorms and function spaces. Let O be a domain (i.e., an open

and connected set) in Rd for some d ≥ 0. Let m be a positive integer; we denote

by Cm(O; Re), for some e ≥ 0, the set of all maps f : O → Re that are m-times

continuously differentiable on O. Let K ⊂ O; we define the seminorms ‖f‖m:K for

m ≥ 0 and f ∈ Cm(O; Re) by

‖f‖m:K =∑

0≤|α|≤m

supx∈K

|∂αx f(x)| .

|·| denotes the Euclidean norm in Rd or in Re and ∂α the usual multi-indexed differ-

ential operator. As usual, for a multi-index α = (α1, . . . , αk), |α| = α1 + · · ·+ αk.

If m = 0, C0(O; Re) denotes the set of all continuous maps f : O → Re with the

corresponding family of seminorms.

Let m be a positive integer and δ ∈ (0, 1]; we denote by Cm+δ(O; Re), the set of

all maps f : O → Re which are m-times continuously differentiable and whose m-th

derivatives are Holder continuous of order δ. Let K ⊂ O; we define the seminorms

‖f‖m+δ:K for m ≥ 0, δ ∈ (0, 1] and f ∈ Cm+δ(O; Re) by

‖f‖m+δ:K =∑

0≤|α|≤m

supx∈K

|∂αx f(x)|+

∑|α|=m

supx 6=x′∈K

|∂αx f(x)− ∂α

x f(x′)||x− x′|δ

.

The families of seminorms ‖ ·‖m:K and ‖ ·‖m+δ:K where K ranges over the compact

sets in O make the sets Cm(O; Re) and Cm+δ(O; Re) into Frechet spaces. These

spaces are easily seen to be separable. As topological spaces they are also measurable

spaces when endowed with their Borel σ-algebras.

Let F (x, t);x ∈ O, t ≥ 0 be a real valued random field with double parameter

x ∈ O and t ≥ 0, i.e., a random variable taking values in RO×R+ when this set is

endowed with the smallest σ-algebra such that the coordinate mappings are measur-

able. If F (x, t, ω) is a continuous function of x for almost every ω and all t, we can

26

regard F (·, t) as a stochastic process with values in C0(O; R) or a C0-valued process.

If F (x, t, ω) is a m-times continuously differentiable in x and all its m-th derivatives

are Holder continuous of order δ for almost every ω and all t, we can regard F (·, t)

as a stochastic process with values in Cm+δ(O; R) or a Cm+δ-valued process. In case

where F (x, t) is a continuous process with values in Cm+δ it is called a continuous

Cm+δ-process.

We now give the definition of a Cm+δ-localmartingale.

Definition 2.1.4 (Cf. Theorem 3.1.2 p. 75 of Kunita (1990)). Let m ≥ 1 and

0 < δ ≤ 1. Let M(x, t), x ∈ O be a family of continuous localmartingales with joint

quadratic variation A(x, y, t) = 〈M(x, t);M(y, t)〉 such that M(x, 0) is Cm+δ. It is

called a Cm+δ-localmartingale if

• it is a continuous Cm+δ-process

• for each α with |α| ≤ m, ∂αxM(x, t), x ∈ O is a family of continuous local-

martingales with joint quadratic variation ∂αx∂

αyA(x, y, t).

To prove that a family of continuous localmartingales is a Cm+δ-localmartingale

we shall use Theorem A.1.1 in appendix A.

Before we can proceed any further, we need to define local Cm+δ-localmartingales.

Let F (t, x), x ∈ O and t ∈ [0, τ(x)) where τ is an accessible lower semicontinuous

stopping time in the sense of Lemma 2.1.1.

Assume that F is continuous in (x, t). We set Ot(ω) = x : τ(x) > t. It is an

open subset of O a.s. since τ is lower semicontinuous a.s. Then F (x, t) defines a

mapping from Ot(ω) into R for each ω. F is said to be a local Cm+δ-process if for

almost every ω, the map F (·, t, ω) : Ot(ω) → R is a Cm+δ-function for every t.

Let us now fix u ≥ 0. Then, for all t ≤ u, Ot(ω) ⊃ Ou(ω) a.s. Let F be a

local Cm+δ-process. If for every u ≥ 0 and almost every ω, the map t 7→ F (·, t, ω)

27

from [0, u] into Cm+δ (Ou(ω); R) is continuous, then F is said to be a continuous local

Cm+δ-process.

We now take the sequence of stopping times associated with τ as in Lemma

2.1.1 and look at the stopped process F τn(x)(x, t) = F (x, t ∧ τn(x)), x ∈ O and all

t ≥ 0. These are global processes. A continuous local Cm+δ-process is called a local

Cm+δ-localmartingale if for every n and each α with |α| ≤ m, the stopped processes

(∂αxF )τn(x) (x, t), x ∈ O is a family of continuous localmartingales.

Let us now formulate our assumption on C. We take O = (0, x)× (S0e−y, S0e

y).

Assumption 2. C is a local C4+δ-localmartingale on [0, τ) for some 0 < δ ≤ 1.

Moreover, on t < τ(T,K), Ct(T,K) > (St −K)+ Leb×P a.e.

Let us stress that Ct(T,K) > (St −K)+, i.e., we assume that liquid options have

strictly positive time values.

The first part of Assumption 2 implies in particular that Ct(T,K) is a continuous

local C4+δ-process. Given the price of an option (i.e., Ct(T,K)), the Black-Scholes

implied volatility Σt(T,K) is by definition the unique volatility parameter for which

the Black-Scholes formula recovers the option price. In fact, we found it easier to

study the implied variance Vt(T,K). It is simply the usual Black-Scholes implied

volatility squared × time-to-maturity, i.e,

Vt(T,K) = Σt(T,K)2 × (T − t).

Hence, Vt(T,K) is the unique solution to the following equation on t < τ(T,K)

KBS(St/K,

√Vt(T,K)

)= Ct(T,K). (2.3)

28

BS is the normalized Black-Scholes formula2

BS(u, v) = uΦ

(lnu

v+v

2

)− Φ

(lnu

v− v

2

)

It is defined for (u, v) ∈ R∗+ × R∗

+. It is a C∞ function on R∗+ × R∗

+ and, for every

u ∈ R∗+, it is a C∞-diffeomorphism from R∗

+ onto itself. We can and will extend it as

a continuous function from R∗+ × R+ onto R+ by setting BS(u, 0) = (u − 1)+. This

extension is still denoted by BS.

By the implicit mapping theorem, we can solve equation (2.3) on any neighborhood

of (T,K), where t < T < t+ x and Ste−y < K < Ste

y. Vt is also a local C4+δ-process

by the same theorem. Moreover it inherits from Ct the a.s. continuity in t as a process

with values in C4+δ and the a.s. continuity in (t, T,K).

Dynamics for C and V will be further studied in Section 2.3.

2.2 Boundary behavior

In this section, we study the behavior of implied volatilities just before maturity. This

behavior is intimately related to that of the corresponding option prices.

In absence of arbitrage, option prices are continuous in maturity. Mathematically,

this is just Lebesgue dominated convergence theorem applied to the representation

of option prices in terms of risk-neutral expectations. Indeed, since ST is integrable

and continuous as a function of T , we have on t < τ(T,K),

limT ′↓t

E(ST ′ −K)+|Ft

= (St −K)+ a.s.

2We use the notation ϕ(x) and Φ(x) for the density and the cumulative distribution function ofthe standard Gaussian distribution, i.e.,

ϕ(x) =1√2π

e−x2/2 and Φ(x) =1√2π

∫ x

−∞e−u2/2du.

29

Taking the limit makes sense since t < τ(T,K) ⊂ t < τ(T ′, K) for T ′ ≤ T .

Proposition 2.2.1. For (T,K) fixed and on t < τ(T,K) a.e.,

limT↓t

Ct(T,K) = (St −K)+ ⇐⇒ limT↓t

Vt(T,K) = 0 (2.4)

and

Vt(T,K) > 0.

Proof. Recall equation (2.3)

KBS(St/K,

√Vt(T,K)

)= Ct(T,K).

The claim therefore reduces to showing that BS(u, v) > (u− 1)+ ⇐⇒ v > 0. Note

that

BS(u, v) = E(

uevZ−v2/2 − 1)+

for some standard Gaussian random variable Z. By Jensen inequality,

BS(u, v) ≥ (u− 1)+.

The inequality is strict if v > 0 because Gaussian measure has full support on R.

Vt(T,K) > 0 is a consequence of the second part of Assumption 2.

The proposition above is the first key ingredient of this chapter. The second

ingredient is the implied volatility dynamics (more precisely, the implied variance

dynamics). This is the content of the next section.

30

2.3 Implied volatility dynamics

In this section, we derive dynamics equations for V . We shall also introduce a new

surface X which is V in the variables ‘time-to-maturity’ and ‘log-moneyness’, i.e.,

Xt(x, y) = Vt(t+ x, Ste−y).

2.3.1 V dynamics

We are going to derive dynamics for V from that of C. Since liquid options are

marked-to-market, we have already argued that they are local localmartingales, i.e.,

they are localmartingales as long as they are actively traded, or more mathematically,

they are localmartingale on an open stochastic interval. These local localmartingales

are adapted to a Brownian filtration, they have the martingale representation prop-

erty:

dCt(T,K) = H t(T,K) · dW t on t < τ(T,K).

where · denotes the usual scalar product of Rn. Indeed, take the sequence of stopping

times of Lemma 2.1.1, for every n ≥ 0, Cτn(T,K)t (T,K) = Ct∧τn(T,K)(T,K) is a contin-

uous global localmartingale. The classical martingale representation theorem gives a

unique process Hnt (T,K) such that

dCτn(T,K)t (T,K) = Hn

t (T,K) · dW t.

We let H t(T,K) = Hnt (T,K) on t < τn(T,K). Since τn(T,K) ↑ τ(T,K) this gives

a well defined process on t < τ(T,K) for which the martingale representation for

C(T,K) holds.

From Proposition 2.2.1, Vt(T,K) > 0 on t < τ(T,K); then√Vt(T,K)ϕ(d1) > 0

on t < τ(T,K), where d1 = ln(St/K)√Vt(T,K)

+

√Vt(T,K)

2. Moreover, σtSt > 0, we can

31

therefore introduce a new continuous local process ξ(T,K) on t < τ(T,K) by

ξt(T,K) =1√

Vt(T,K)ϕ(d1)

(H t(T,K)

σtSt

− Φ(d1)e

)(2.5)

where e is the first vector of the canonical basis of Rn. Then, the martingale repre-

sentation takes the form:

dCt(T,K) = σtSt

(Φ(d1)e +

√Vt(T,K)ϕ(d1)ξt(T,K)

)· dW t on t < τ(T,K).

This way of writing H t(T,K) will prove easier to handle when it comes to study

implied variance. It also has a financial interpretation. As is well known, martingale

representations are directly related to hedging portfolios. To hedge an option, one

buys ∆ = Φ(d1) stocks, whose dynamics is given by the first term above. The second

term is a correction term proportional to the vega that takes into account the fact

that the spot volatility is not constant.

We need to make a further assumption. First some notation, if K denotes a

compact subset of O = (0, x)× (S0e−y, S0e

y), we let τK to be the stopping time

τK = inf(T,K)∈K

τ(T,K).

Assumption 3. For every t ≥ 0 and almost every ω, the mapping (T,K) 7→

H t(T,K) is in C4+δ ((t, t+ x)× (Ste−y, Ste

y); Rn) and

∀t ≥ 0

∫ t∧τK

0

‖Hs‖24+δ:K ds <∞ a.s. for every K ⊂ O.

Two comments are in order. First, Assumption 3 implies the first part of As-

sumption 2. This is a consequence of Theorem A.1.4. Indeed, by that theorem, if

Assumption 3 holds for some 0 < δ ≤ 1 then the first part of Assumption 2 holds

32

for any ε < δ. Second, if Assumption 3 holds true, the same properties hold true for

σξ, i.e., for every t ≥ 0 and almost every ω, the mapping (T,K) 7→ ξt(T,K) is in

C4+δ ((t, t+ x)× (Ste−y, Ste

y); Rn) and

∀t ≥ 0

∫ t∧τK

0

‖σsξs‖24+δ:K ds <∞ a.s. for every K ⊂ O.

This is easily seen from (2.5) and the fact that S is a continuous process and that V

is a continuous C4+δ-process.

Ito formula shows that for fixed (T,K), Vt(T,K) is a local semimartingale on

t < τ(T,K).

Proposition 2.3.1. On t < τ(T,K) a.e.,

dVt(T,K) =

−σ2t

(1− 2 ln

St

Kξ1t + ln2 St

K|ξt|

2 − Vt |ξt|2 − 1

4V 2

t |ξt|2 + Vtξ

1t

)(T,K)dt

+2σtVt(T,K)ξt(T,K) · dW t.

Proof. Since Ct(T,K) = KBS(St/K,

√Vt(T,K)

), it suffices to show that we find

back the right dynamics for Ct(T,K) if we assume that Vt(T,K) has the dynamics

stated in the proposition.

KdBS(St/K,

√Vt(T,K)

)= K

(∂uBS

σtSt

KdW 1

t + ∂vBS

(dVt

2√Vt

− d 〈V 〉t8V

3/2t

)

+1

2∂2

uuBSσ2

tS2t

K2dt+ ∂2

uvBS2σ2

t Vtξ1t St

2√VtK

dt+1

2∂2

vvBS4σ2

t V2t |ξt|

2

4Vt

dt

)

=σ2

tStϕ(d1)

2√Vt

(−1− y2

t |ξt|2 + 2ytξ

1t + Vt |ξt|

2 +V 2

t |ξt|2

4− Vtξ

1t − Vt |ξt|

2 + 1

−2√Vtξ

1t

(yt√Vt

−√Vt

2

)+ Vt |ξt|

2

(y2

t

Vt

− Vt

4

))dt

+σtStΦ(d1)dW1t + σtSt

√Vtϕ(d1)ξt · dW t,

33

where we used the short-hand notation yt = ln(St/K). The drift vanishes identically,

which proves the proposition.

We now wish to study the regularity of V as a function of (T,K). We shall need

the seminorms ‖g‖∼m+δ:K, defined for functions g from O×O into Re that are m-times

differentiable in x and in y separately and whose m-th derivatives in x and in y are

Holder continuous of order δ. This space is denoted Cm+δ. We define seminorms on

that space by

‖g‖∼m+δ:K =∑

0≤|α|≤m

supx,y∈K

∣∣∂αx∂

αy g(x, y)

∣∣+∑|α|=m

supx 6=x′,y 6=y′∈K

∣∣∂αx∂

αy g(x, y)− ∂α

x∂αy g(x

′, y)− ∂αx∂

αy g(x, y

′) + ∂αx∂

αy g(x

′, y′)∣∣

|x− x′|δ |y − y′|δ.

Let us now give the definition of a Cm+δ-semimartingale.

Definition 2.3.2 (Cf. Kunita (1990) p. 84). A family of continuous semimartin-

gales F (x, t), x ∈ O, decomposed as F (x, t) = B(x, t) + M(x, t) where M(x, t) is a

continuous localmartingale and B(x, t) is a continuous process of bounded variation,

is said to be a Cm+δ-semimartingale if M(x, t) is a continuous Cm+δ-localmartingale

in the sense of Definition 2.1.4 and B(x, t) is a continuous Cm+δ-process such that

∂αxB(x, t), x ∈ O, |α| ≤ m are all processes of bounded variation.

Proposition 2.3.3. V is a local C4+ε-semimartingale on [0, τ) for every ε < δ.

Proof. For any K compact subset of O, Vt∧τK is a global process. It suffices to prove

that the martingale part and the bounded variation parts have local characteristics

with the correct integrability properties (see, Definition 2.1.4 above and Theorem

A.1.1.) For the martingale part, it reads

∫ t∧τK

0

‖ 〈V 〉s ‖∼4+δ:Kds <∞ a.s. for any compact K ⊂ O and t ≥ 0.

34

In our case,

d 〈V (T,K);V (T ′, K ′)〉t = 4σ2t Vt(T,K)Vt(T

′, K ′)ξt(T,K) · ξt(T′, K ′)dt.

By Assumption 3 and the fact that the stopped process V τK is a continuous C4+δ-

processes,

∫ t∧τK

0

‖ 〈V 〉s ‖∼4+δ:Kds ≤ M

∫ t∧τK

0

‖Vs‖24+δ:K‖σsξs‖2

4+δ:Kds

≤ M sup0≤s≤t∧τK

‖Vs‖24+δ:K

∫ t∧τK

0

‖σsξs‖24+δ:Kds <∞

where M is a constant. By Theorem A.1.1, the localmartingale part is a Cm+ε-

localmartingale for any ε < δ. The corresponding condition for the drift ft(T,K) of

Vt(T,K) reads ∫ t∧τK

0

‖fs‖4+δ:Kds <∞.

It holds for the same reason. Further, by Lebesgue dominated convergence theorem,

it shows that for |α| ≤ 4, ∂α(T,K)

∫ t∧τK0

fs(T,K)ds exists and that

∂α(T,K)

∫ t∧τK

0

fs(T,K)ds =

∫ t∧τK

0

∂α(T,K)fs(T,K)ds.

The derivatives of the bounded variation part of V are also local processes of bounded

variation.

2.3.2 X dynamics

Let us let D = (0, x) × (−y, y). Let us define for each t, any (x, y) ∈ D and a.e. ω,

the following quantities:

Xt(x, y) = Vt(t+ x, Ste−y) and ηt(x, y) = ξt(t+ x, Ste

−y)

35

where ξ was defined at (2.5). Note thatX, thus defined, is a global process. Moreover,

by composition of smooth functions, X and η are continuous C4+δ-processes. Also,

ση satisfies the properties of Assumption 3, i.e., for every t ≥ 0 and almost every ω,

(x, y) 7→ ηt(x, y) ∈ C4+δ (D; Rn)

and

∀t ≥ 0

∫ t

0

‖σsηs‖24+δ:K ds <∞ a.s. for every K ⊂ D.

By application of the generalized Ito formula (Theorem A.2.1), for each (x, y) ∈ D,

X(x, y) is a continuous semimartingale. Its decomposition is given in the following

proposition. We can also compute the dynamics of its derivatives ∂xXt, ∂yXt, ∂2xxXt

and ∂2xyXt.

Proposition 2.3.4. For each (x, y) ∈ D,

dXt(x, y) = Dt(x, y)dt+ σt (2Xtηt − ∂yXte) (x, y) · dW t (2.6)

and ∂xXt(x, y), ∂yXt(x, y), ∂2xxXt(x, y) and ∂2

xyXt(x, y) have modifications that are

semimartingales; they satisfy

d∂xXt(x, y) = ∂xDt(x, y)dt+ σt

(2∂x(Xtηt)− ∂2

xyXte)(x, y) · dW t (2.7)

d∂yXt(x, y) = ∂yDt(x, y)dt+ σt

(2∂y(Xtηt)− ∂2

yyXte)(x, y) · dW t (2.8)

d∂2xxXt(x, y) = ∂2

xxDt(x, y)dt+ σt

(2∂2

xx(Xtηt)− ∂3xxyXte

)(x, y) · dW t (2.9)

d∂2xyXt(x, y) = ∂2

xyDt(x, y)dt+ σt

(2∂2

xy(Xtηt)− ∂3xyyXte

)(x, y) · dW t (2.10)

36

where the drift Dt is

Dt(x, y) = ∂xXt(x, y)− σ2t

(1− 1

2∂2

yyXt −1

2∂yXt + 2∂y

(Xtη

1t

)− 2yη1

t + y2 |ηt|2

−Xt |ηt|2 − 1

4X2

t |ηt|2 +Xtη

1t

)(x, y).

It is important to notice that after that change of variable, S does not appear any

more in our equations. It plays a role only through its volatility σ.

Proof. Because V is a continuous C2-process and a C1-semimartingale with the cor-

rect integrability conditions, we can apply the Generalized Ito Formula (Theorem

A.2.1):

dVt(t+x, Stey) = dVt+∂TVtdt+e

y∂KVtdSt+1

2e2y∂2

KKVtd 〈S〉t+d⟨2σey∂K

(V ξ1

);S⟩

t.

Using the following formulas to switch to the (x, y) variables:

∂xXt(x, y) = ∂TVt(t+ x, Ste−y), ∂yXt(x, y) = −Ste

−y∂KVt(t+ x, Ste−y)

and

∂2yyXt(x, y) = Ste

−y∂KVt(t+ x, Ste−y) + S2

t e−2y∂2

KKVt(t+ x, Ste−y),

we get the dynamics of X. Theorem A.1.4 gives us the semimartingale decomposition

of its derivatives. Indeed, let us first look at the differentiability of the localmartingale

part. Because X is a continuous C4+δ-process and ση satisfies the second property

of Assumption 3,

∫ t

0

‖σs (2Xsηs − ∂yXse)‖24+δ:K ds <∞ a.s. for any compact K ⊂ D and t ≥ 0.

By Theorem A.1.4, the localmartingale part therefore has a modification of a C4+ε-

37

localmartingale for ε < δ and it satisfies

∂α(x,y)

∫ t

0

σs (2Xsηs − ∂yXse) (x, y) · dW s =

∫ t

0

σs∂α(x,y) (2Xsηs − ∂yXse) (x, y) · dW s

a.s., for |α| ≤ 4 and every t ≥ 0. The differentiability of the drift Dt is a consequence

of Lebesgue dominated convergence theorem. Since

∫ t

0

‖Ds‖4+δ:K ds <∞ a.s. for any compact K ⊂ D and t ≥ 0,

we have

∂α(x,y)

∫ t

0

Ds(x, y)ds =

∫ t

0

∂α(x,y)Ds(x, y)ds a.s.

for |α| ≤ 4 and every t ≥ 0.

Since we are going to make heavy use of these derivatives of Xt, it is good to

remember how they relate to the usual derivatives of Σt with respect to T and K.

We below give that relationship for the most important ones.

Xt(x, y) = xΣ2t (t+ x, Ste

−y)

∂xXt(x, y) =[Σ2

t + x∂T

(Σ2

t

)](t+ x, Ste

−y)

∂2xxXt(x, y) =

[2∂T

(Σ2

t

)+ x∂2

TT

(Σ2

t

)](t+ x, Ste

−y)

∂2xyXt(x, y) = −Ste

−y[∂K

(Σ2

t

)+ x∂2

TK

(Σ2

t

)](t+ x, Ste

−y)

∂3xyyXt(x, y) = Ste

−y[∂K

(Σ2

t

)+ x∂2

TK

(Σ2

t

)](t+ x, Ste

−y)

+ S2t e−2y[∂2

KK

(Σ2

t

)+ x∂3

TKK

(Σ2

t

)](t+ x, Ste

−y).

38

2.4 Recovering the spot volatility

In this section, we put together the two key ingredients Proposition 2.2.1 and Propo-

sition 2.3.4. More precisely, the implied volatility behavior near expiry induces a

strong restriction on its dynamics. We show that under an appropriate assumption,

we can find back the value of the spot volatility process from the behavior of points

near the boundary x = 0. In fact, as we shall see shortly, the quantity of interest is

∂xXt(0, 0). We start off by translating Proposition 2.2.1 into our new variables (x, y).

Proposition 2.4.1. For every y ∈ (−y, y),

limx→0

X(x, y) = 0

in the sense of the uniform convergence on compact intervals in time and in probability

(u.c.p. for short.)

Proof. Let xn ↓ 0. From Proposition 2.2.1, we know that for every t and y, Xt(xn, y) =

Vt(t+xn, Ste−y) goes to 0 for a.e. ω. By Dini theorem, since x 7→ Xt(x, y) is decreasing

(as a result of Lemma 2.1.2) and since the limit is obviously continuous, the pointwise

limit is in fact uniform in t on compact intervals for a.e. ω. Now, take an arbitrary

sequence xn → 0, we want to show that Xt(xn, y) converges to 0 uniformly in t on

compacts and in probability. Take a subsequence xnk, it has a further subsequence

that is decreasing to 0 and the preceding reasoning applies to that subsequence. The

convergence holds in the sense of the uniform convergence on compacts in t and in

probability.

In several instances in the following proofs, we shall use the two following ele-

mentary lemmas. The first lemma is a classical result about continuity of quadratic

variations. See, for instance, Theorem 2.2.15 p. 53 in Kunita (1990).

39

Lemma 2.4.2 (Cf. Theorem 2.2.15 p. 53 in Kunita (1990)). Let Mn, n ≥ 0,

be a sequence of localmartingales. It converges to M u.c.p. if and only if 〈Mn −M〉t

converges to 0 in probability for every t ≥ 0.

The second lemma says that if a sequence of semimartingales goes to 0 u.c.p., the

sequences of its bounded variation parts and of its martingale parts go separately to

0 u.c.p.

Lemma 2.4.3. Suppose Y (x, y) for (x, y) ∈ D is a C0-semimartingale with decom-

position

Yt(x, y) =

∫ t

0

as(x, y)ds+

∫ t

0

bs(x, y) · dW s.

Suppose further that for every y ∈ (−y, y),

limx→0

Y (x, y) = 0 u.c.p.

Then, for every y ∈ (−y, y),

limx→0

∫ ·

0

as(x, y)ds = limx→0

∫ ·

0

bs(x, y) · dW s = 0 u.c.p.

Proof. See appendix, section A.3

We now state the fundamental assumption that will allow us to derive the dynam-

ics at x = 0.

Assumption 4. For every y ∈ (−y, y), ∂xX(x, y), ∂yX(x, y), ∂2xxX(x, y), ∂2

xyX(x, y),

∂2yyX(x, y), ∂3

xxyX(x, y), and ∂3xyyX(x, y) have a limit when x→ 0 u.c.p. Their limits

are denoted by ∂xX(0, y), etc. Moreover,

∂yXt(0, y) = ∂2yyXt(0, y) = 0 and ∂xXt(0, y) > 0 Leb×P a.e.

We also require ∂xXt(0, y) to be a C2 function of y and its derivatives to be limits of

40

the derivatives above, i.e.,

d

dy∂xXt(0, y) = ∂xyXt(0, y) and

d2

dy2∂xXt(0, y) = ∂xyyXt(0, y) Leb×P a.e.

Finally, we assume that 〈∂xXt(0, y)〉t is a.s. absolutely continuous in t and that its

density with respect to Lebesgue measure is continuous in y Leb×P a.e.

Thanks to Assumption 4, we can prove our first result, below. The case y = 0

is particularly interesting. It says that the spot volatility value is recovered from

the implied volatility surface. This has been known for some time in the financial

literature, see for example, Ledoit and Santa-Clara (1999), Carr (2000), and Brace

et al. (2001). Rigorous proofs were given in Berestycki et al. (2002) and Berestycki

et al. (2003) in a different context.

Theorem 2.4.4. The behavior of the implied volatility at expiry is given as follows.

For every y ∈ (−y, y),

∂xXt(0, y) = σ2t

(1− 2yη1

t (0, y) + y2 |ηt(0, y)|2) Leb×P a.e.

To prove this theorem, we need the following two lemmas. They will allow us to

compute important limits.

Lemma 2.4.5. For every y ∈ (−y, y), each of the following processes: σ (Xη) (x, y),

σ∂x (Xη) (x, y), σ∂y (Xη) (x, y), σ∂2xy (Xη) (x, y), and σ∂2

xx (Xη) (x, y) have a limit

in L2(0, t) for every t ≥ 0 and in probability when x→ 0. The limits are denoted by

σ (Xη) (0, y), etc. Moreover,

(Xtηt) (0, y) = ∂y (Xtηt) (0, y) = 0 Leb×P a.e.

Proof. Let y ∈ (−y, y) be fixed. From Proposition 2.4.1, equation (2.6) and Lemma

41

2.4.3, we deduce that

limx→0

∫ t

0

σs (2Xsηs − ∂yXse) (x, y) · dW s = 0 u.c.p.

From Lemma 2.4.2, it implies

∀t ≥ 0 limx→0

∫ t

0

σ2s |2Xsηs − ∂yXse|2 (x, y)ds = 0 in probability.

Note that

∀t ≥ 0

∫ t

0

σ2s (∂yXs(x, y))

2 (x, y)ds ≤ sup0≤s≤t

(∂yXs(x, y))2

∫ t

0

σ2sds

and from this last inequality, using Assumption 4 to get ∂yX(x, y) → 0 u.c.p. and

equation (2.1), we have

∀t ≥ 0 limx→0

∫ t

0

σ2s (∂yXs(x, y))

2 ds = 0 in probability.

Further, ∀t ≥ 0,

∫ t

0

σ2s |2Xsηs|

2 (x, y)ds

≤ 2

∫ t

0

σ2s |2Xsηs − ∂yXse|2 (x, y)ds+ 2

∫ t

0

σ2s (∂yXs)

2 (x, y)ds

and therefore,

∀t ≥ 0 limx→0

∫ t

0

σ2s |2Xsηs|

2 (x, y)ds = 0 in probability. (2.11)

The limit is 0, Leb×P a.e.

Similarly, in view of equation (2.8), using the fact that ∂2yyX(x, y) converges u.c.p.

to 0, we deduce that, ∂y(Xtηt)(x, y) converges to 0 when x→ 0 in L2(0, t) for every

42

t ≥ 0 and in probability.

We repeat the same arguments for equations (2.7), (2.9) and (2.10) to get the

existence of the limits.

Lemma 2.4.6. For every y ∈ (−y, y), ση(x, y), σ∂yη(x, y), and σ (X∂xη) (x, y)

have a limit in L2(0, t) for every t ≥ 0 and in probability when x→ 0. The limits are

denoted by ση(0, y), etc. Moreover, Leb×P a.e.,

ηt(0, y) =∂x(Xtηt)(0, y)

∂xXt(0, y)(2.12)

∂yηt(0, y) =∂2

xy(Xtηt)(0, y)

∂xXt(0, y)−∂2

xyXt(0, y)

∂xXt(0, y)ηt(0, y) (2.13)

(Xt∂xηt) (0, y) = 0 (2.14)

Proof. Let y ∈ (−y, y) be fixed. From Proposition 2.4.1 and Lemma 2.4.5, we deduce

that, possibly along a subsequence (xn),

limn→∞

Xt(xn, y) = 0 and limn→∞

(Xtηt) (xn, y) = 0 Leb×P a.e.

From Assumption 4, we get

limn→∞

∂xXt(xn, y) = ∂xXt(0, y) > 0 Leb×P a.e.

and applying Lemma 2.4.5 again, we get the existence of the limit

limn→∞

∂x (Xtηt) (xn, y) Leb×P a.e.

These four limits above combined with L’Hopital rule gives that

limn→∞

ηt(xn, y) =∂x(Xtηt)(0, y)

∂xXt(0, y)Leb×P a.e.

43

Let us now prove that the convergence holds along that subsequence in L2(0, t) for

every t ≥ 0. Note that from the mean-value theorem,

|ηs(xn, y)|2 =

∣∣∣∣∂x(Xsηs)(x′n, y)

∂xXs(x′n, y)

∣∣∣∣2 Leb×P a.e.

where 0 ≤ x′n ≤ xn. For n large enough and s ∈ [0, t], we get from Assumption 4,

∂xXs(x′n, y) ≥

1

2∂xXs(0, y) ≥

1

2inf

s∈[0,t]∂xXs(0, y) > 0.

Therefore for a.e. s ∈ [0, t],

σ2s |ηs(xn, y)|2 ≤

4σ2s(

infs∈[0,t] ∂xXs(0, y))2 |∂x(Xsηs)(x

′n, y)|

2.

From Lemma 2.4.5, the upper bound converges in L1(0, t), it is therefore uniformly

(in n) integrable (in s). So is σ2s |ηs(xn, y)|2 and we have the conclusion.

To prove the existence of ∂yηt(0, y), we proceed the same way by using Assumption

4 and Lemma 2.4.5 to show that possibly along a subsequence xn,

limn→∞

∂y(Xtηt)(xn, y) = limn→∞

∂yXt(xn, y) = 0 Leb×P a.e.

and that limn→∞ ∂2xy(Xtηt)(xn, y) and limn→∞ ∂2

xy(Xt)(xn, y) exist. Another applica-

tion of L’Hopital rule to

∂yηt(xn, y) =∂y(Xtηt)(xn, y)

Xt(xn, y)− ∂yXt(xn, y)

Xt(xn, y)ηt(xn, y)

yields

limn→∞

∂yηt(xn, y) =∂2

xy(Xtηt)(0, y)

∂xXt(0, y)−∂2

xyXt(0, y)

∂xXt(0, y)ηt(0, y) Leb×P a.e.

44

and we also prove in the same way that the convergence holds in L2(0, t) for every

t ≥ 0.

Finally, to prove that σ (X∂xη) (x, y) goes to 0 in L2(0, t) and in probability, write

it as

σt (Xt∂xηt) (x, y) = σt [∂x(Xtηt)(x, y)− ∂x(Xtηt)(0, y)]

+ σt [∂x(Xtηt)(0, y)− ∂xXt(0, y)ηt(x, y)]− σt [∂xXt(x, y)− ∂xXt(0, y)] ηt(x, y).

The first term goes to 0 in L2(0, t) and in probability by Lemma 2.4.5. We just proved

the convergence to 0 in L2(0, t) and in probability for the second term. As for the

last one,

∫ t

0

σ2s [∂xXs(x, y)− ∂xXs(0, y)]

2 |ηs(x, y)|2 ds

≤ sup0≤s≤t

[∂xXs(x, y)− ∂xXs(0, y)]2

∫ t

0

σ2s |ηs(x, y)|

2 ds.

Using Assumption 4 and the first result of this proposition, we see that the last term

also converges to 0 in L2(0, t) and in probability.

We can now conclude this section with the proof of our first main theorem.

Proof of Theorem 2.4.4. Let y ∈ (−y, y) and t ≥ 0 be fixed. From Proposition 2.4.1,

equation (2.6) and Lemma 2.4.3, we deduce that

limx→0

∫ t

0

Ds(x, y)ds = 0 in probability.

45

Recall the form of the drift Dt and write it as follows:

Dt = ∂xXt − σ2t

(1 +

1

2∂2

yyXt −1

2∂yXt + 2∂y

(Xtη

1t

)− ∂2

yyXt − 2yη1t + y2 |ηt|

2

−Xt |ηt|2 − 1

4X2

t |ηt|2 +Xtη

1t

).

Taking into account Assumption 4, we find

limx→0

∫ t

0

σ2s

(1

2∂2

yyXs −1

2∂yXs

)(x, y)ds = 0 in probability.

Taking into account equation (2.8) and Assumption 4

limx→0

∫ t

0

σ2s

(2∂y(Xsη

1s)− ∂2

yyXs

)(x, y)ds = 0 in probability.

In view of equation (2.11),

limx→0

1

4

∫ t

0

σ2s |Xsηs|

2 (x, y)ds = 0 in probability.

Next, by Cauchy-Schwarz inequality,

∫ t

0

σ2sXsη

1s(x, y)ds ≤

√∫ t

0

σ2sds

√∫ t

0

σ2s |Xsηs|

2 (x, y)ds

and the right-hand side goes to 0 in probability because σ satisfies (2.1) and because

of equation (2.11). Moreover, by Cauchy-Schwarz inequality, again,

∫ t

0

σ2sXs(x, y) |ηs(x, y)|

2 ds ≤

√∫ t

0

σ2s |Xsηs|

2 (x, y)ds

√∫ t

0

σ2s |ηs(x, y)|

2 ds.

The right-hand side converges to 0 in probability as x goes to 0 because of equation

46

(2.11) and Lemma 2.4.6. Again from Lemma 2.4.6 and Assumption 4, we get

limx→0

∫ t

0

∂xXs(x, y)− σ2s

(1− 2yη1

s(x, y) + y2 |ηs(x, y)|2) ds

=

∫ t

0

∂xXs(0, y)− σ2s

(1− 2yη1

s(0, y) + y2 |ηs(0, y)|2) ds in probability.

This last limit is 0 and the claim is proven since t is arbitrary.

2.5 Recovering the spot volatility dynamics

In the previous section, we recovered the value of the spot volatility. Here we are

concerned with its dynamics. This is the second part of our main result. Dynamics

for σ2t are given below in equation (2.15). As we shall see later, equation (2.16) is a

very important joint quadratic variation.

Theorem 2.5.1. The dynamics of σ2t is given by

dσ2t =

[∂2

xxXt(0, 0) +(∂2

xyXt(0, 0))2

+ σ2t

(∂2

xyXt(0, 0) + ∂3xyyXt(0, 0)

)]dt

−2σt

(∂2

xyXt(0, 0)e− σ2t ηt(0, 0)

)· dW t (2.15)

where ηt = ηt − η1t e. The joint quadratic variation between the slope of the smile

near expiry and the noise driving the spot is given by

d⟨∂2

xyX(0, 0);W 1⟩

t= −σt

(3

2∂3

xyyXt(0, 0) +3

4σ2t

(∂2

xyXt(0, 0))2 − σ2

t |ηt(0, 0)|2)

dt.

(2.16)

It comes as a surprise that the general spot volatility dynamics has the Heston

structure, i.e., that of a mean reverting square root process. The coefficients are

however random. More importantly, except for coefficients involving ηt, all coefficients

explicitly depend only on implied volatilities. The non directly observable quantity

47

ηt only appears in the part of the noise that is orthogonal to that of the spot.

The proof of the theorem is postponed after we prove the following three lemmas.

Lemma 2.5.2. ηt(0, y) is a continuous function of y on (−y, y) for almost every

t ≥ 0 and ω.

Proof. In view of equation (2.12), it suffices to show that ∂x(Xtηt)(0, y) is a continuous

function of y. To this end, we first fix y ∈ (−y, y) and prove that

〈∂xX(0, y)〉t =

∫ t

0

σ2s

∣∣2∂x(Xsηt)(0, y)− ∂2xyXs(0, y)e

∣∣2 ds.

From there, we shall use Lemma 2.4.6 and Assumption 4 to infer that ηt(0, y) is a

continuous function of y. Indeed, the integrand above will be continuous in y by the

last part of Assumption 4. From Lemma 2.4.6, ∂x(Xsηt)(0, y) = ∂xXs(0, y)ηt(0, y)

and using Assumption 4 again, we get the conclusion.

To prove the equality above, we take the limit as x → 0 in equation (2.7). Let

t ≥ 0 be fixed. We shall show that

limx→0

∫ t

0

σs

(2∂x (Xsηs)− ∂2

xyXse)(x, y) · dW s

=

∫ t

0

σs

(2∂x (Xsηs)− ∂2

xyXse)(0, y) · dW s

holds in probability. By Lemma 2.4.2, it is sufficient to show that

∫ t

0

σ2s

∣∣2∂x (Xsηs) (x, y)− 2∂x (Xsηs) (0, y)− ∂2xyXs(x, y)e + ∂2

xyXs(0, y)e∣∣2 ds

has limit 0 in probability when x→ 0. It holds true from Lemma 2.4.5.

We get a more precise description of ηt(0, y) in the following lemma.

Lemma 2.5.3. η1t (0, y) is differentiable in y at 0 for almost every t and ω. Moreover,

48

we have Leb×P a.e.,

η1t (0, 0) = − 1

2σ2t

∂2xyXt(0, 0)

∂yη1t (0, 0) = − 1

4σ2t

∂3xyyXt(0, 0) +

1

8σ4t

(∂2

xyXt(0, 0))2

+1

2|ηt(0, 0)|2 ,

where ηt = ηt − η1t e.

Proof. In this proof we fix (t, ω). Theorem 2.4.4 rewrites

∂xXt(0, y) = σ2t

(1− yη1

t (0, y))2

+ σ2t y

2 |ηt(0, y)|2 ,

that is (keeping the only solution continuous in y around 0),

η1t (0, y) =

1

y

(1−

öxXt(0, y)

σ2t

− y2 |ηt(0, y)|2

). (2.17)

Using Assumption 4, we have the following Taylor expansion:

∂xXt(0, y) = ∂xXt(0, 0) + y∂2xyXt(0, 0) +

y2

2∂3

xyyXt(0, 0) + o(y2)

and thanks to Lemma 2.5.2, we also have:

|ηt(0, y)|2 = |ηt(0, 0)|2 + o(1).

Therefore, expanding (2.17), we get:

η1t (0, y) = −

∂2xyXt(0, 0)

2σ2t

− y

(∂3

xyyXt(0, 0)

4σ2t

−(∂2

xyXt(0, 0))2

8σ4t

− |ηt(0, 0)|2

2

)+ o(y).

The lemma is proved.

This last lemma gives us useful limits.

49

Lemma 2.5.4. We have Leb×P a.e.,

∂x

(Xtη

1t

)(0, 0) = −1

2∂2

xyXt(0, 0)

∂2xy

(Xtη

1t

)(0, 0) = −1

4∂3

xyyXt(0, 0)− 3

8σ2t

(∂2

xyXt(0, 0))2

+σ2

t

2|ηt(0, 0)|2 .

Proof. Combining equation (2.12), Lemma 2.5.3, and Theorem 2.4.4, we find

∂x

(Xtη

1t

)(0, 0) = ∂xXt(0, 0)η1

t (0, 0) = σ2t η

1t (0, 0) = −1

2∂2

xyXt(0, 0).

Combining (2.13), Lemma 2.5.3, and Theorem 2.4.4, we find

∂2xy

(Xtη

1t

)(0, 0) = σ2

t ∂yη1t (0, 0) + ∂2

xyXt(0, 0)η1t (0, 0)

= −1

4∂3

xyyXt(0, 0)− 3

8σ2t

(∂2

xyXt(0, 0))2

+σ2

t

2|ηt(0, 0)|2 .

We can now prove Theorem 2.5.1.

Proof of Theorem 2.5.1. To get the equation (2.15), we let y = 0 and take the limit

as x→ 0 in equation (2.7). Let t ≥ 0 be fixed. For the localmartingale part, we need

to show that

limx→0

∫ t

0

σs

(2∂x (Xsηs)− ∂2

xyXse)(x, 0) · dW s

=

∫ t

0

σs

(2∂x (Xsηs)− ∂2

xyXse)(0, 0) · dW s

holds in probability. We already proved it in the proof of Lemma 2.5.2. To compute

the limit, use equation (2.12) and Lemma 2.5.4 to get

2∂x (Xsηs) (0, 0)− ∂2xyXs(0, 0)e = −2∂2

xyXs(0, 0)e + 2σ2s ηs(0, 0).

50

Next recall the form of the bounded variation part of ∂xX(x, 0):

∂xDs(x, 0) = ∂2xxXs(x, 0)− σ2

s

(−1

2∂3

xyyXs −1

2∂2

xyXs + 2∂xy

(Xsη

1s

)−∂x

(Xs |ηs|

2)− 1

2Xsηs∂x (Xsηs) + ∂x

(Xsη

1s

))(x, 0).

It follows from Assumption 4 and Lemma 2.4.5 that all terms but ∂x

(Xs |ηs|

2) are

converging in L1(0, t) and in probability as x → 0. For this last term, another

application of L’Hopital rule gives that

limx→0

σ2s∂x

(Xs |ηs|

2) (x, 0) = σ4s |ηs(0, 0)|2 Leb×P a.e. (2.18)

To show that the convergence in (2.18) holds in L1(0, t) and in probability, note that

by the triangle inequality

∫ t

0

σ2s

∣∣∂x

(Xs |ηs|

2) (x, 0)− σ2s |ηs(0, 0)|2

∣∣ ds≤∫ t

0

σ2s

∣∣∂xXs(x, 0) |ηs(x, 0)|2 − σ2s |ηs(0, 0)|2

∣∣ ds+2

∫ t

0

σ2s |Xs (∂xηs) · ηs| (x, 0)ds.

(2.19)

The first term in the right-hand side of (2.19) is bounded by

∫ t

0

σ2s

∣∣∂xXs(x, 0) |ηs(x, 0)|2 − σ2s |ηs(0, 0)|2

∣∣ ds≤∫ t

0

σ2s |∂xXs(x, 0)| |ηs(x, 0)− ηs(0, 0)|2 ds+

∫ t

0

σ2s

∣∣∂xXs(x, 0)− σ2s

∣∣ |ηs(0, 0)|2 ds.

The first term in the right-hand side converges to 0 in probability by Assumption 4

51

and Lemma 2.4.6 since the following inequality holds:

∫ t

0

σ2s |∂xXs(x, 0)| |ηs(x, 0)− ηs(0, 0)|2 ds

≤ sup0≤s≤t

|∂xXs(x, 0)|∫ t

0

σ2s |ηs(x, 0)− ηs(0, 0)|2 ds.

The second term in the right-hand side above also converges to 0 in probability by

Theorem 2.4.4 and Lemma 2.4.6 together with the following inequality:

∫ t

0

σ2s

∣∣∂xXs(x, 0)− σ2s

∣∣ |ηs(0, 0)|2 ds ≤ sup0≤s≤t

∣∣∂xXs(x, 0)− σ2s

∣∣ ∫ t

0

σ2s |ηs(0, 0)|2 ds

Let us now prove that the second term in the right-hand side of (2.19) converges to

0 in probability. By Cauchy-Schwarz inequality,

∫ t

0

σ2s |Xs (∂xηs) · ηs| (x, 0)ds ≤

√∫ t

0

σ2s |Xs (∂xηs) (x, 0)|2 ds

√∫ t

0

σ2s |ηs(x, 0)|2 ds,

therefore, by Lemma 2.4.6 the second term in the right-hand side converges and it is

therefore enough to prove that the first term converges to 0. It is a consequence of

Lemma 2.4.6 and equation (2.14). This shows that the convergence in (2.18) holds in

L1(0, t) and in probability.

We can now compute the value of the limit.

limx→0

σ2s

(−1

2∂3

xyyXs −1

2∂2

xyXs + 2∂xy

(Xsη

1s

)− ∂x

(Xs |ηs|

2)− 1

2Xsηs∂x (Xsηs)

+∂x

(Xsη

1s

))(x, 0)

= σ2s

(−1

2∂3

xyyXs(0, 0)− 1

2∂2

xyXs(0, 0)− 1

2∂3

xyyXs(0, 0)− 3

4σ2s

(∂2

xyXs(0, 0))2

+σ2s |ηs(0, 0)|2 − σ2

s |ηs(0, 0)|2 − 1

2∂2

xyXs(0, 0)

)= σ2

s

(−∂3

xyyXs(0, 0)− ∂2xyXs(0, 0)− 1

σ2s

(∂2

xyXs(0, 0))2)

Leb×P a.e.

52

We used Lemma 2.5.4 and the fact that (see statement of Lemma 2.5.3 for the defi-

nition of η)

|ηs(0, 0)|2 = |ηs(0, 0)|2 +(η1

s(0, 0))2.

To get equation (2.16), we proceed in the same way, we show that

limx→0

∫ t

0

σs

(2∂2

xy(Xsηs)− ∂3xyyXse

)(x, 0) · dW t

=

∫ t

0

σs

(2∂2

xy(Xsηs)− ∂3xyyXse

)(0, 0) · dW t

holds in probability. By Lemma 2.4.2, it is sufficient to show that

∫ t

0

σ2s

∣∣2∂2xy (Xsηs) (x, 0)− 2∂2

xy (Xsηs) (0, 0)− ∂3xyyXs(x, 0)e + ∂3

xyyXs(0, 0)e∣∣2 ds

has limit 0 in probability when x → 0. It holds true as before. Therefore, using

Lemma 2.5.4 to compute the value of the limit, we find

d⟨∂2

xyX(0, 0);W 1⟩

t= d

⟨∫ ·

0

σs

(2∂2

xy (Xsηs) + ∂3xyyXse

)(0, 0) · dW s;W

1

⟩t

= σt

(−3

2∂3

xyyXt(0, 0)− 3

4σ2t

(∂2

xyXt(0, 0))2

+ σ2t |ηt(0, 0)|2

)dt.

The proof of the theorem is complete.

2.6 Concluding remarks about our assumptions

Even to derive our first equation in Theorem 2.4.4 for y = 0, namely,

limx→0

∂xXt(x, 0) = σ2t ,

53

we seem to need an additional assumption to Assumptions 1, 2, and 3. Indeed, under

these last assumptions, we proved that ∂xXt(x, 0) was a semimartingale for x > 0. It

is therefore a process measurable with respect to the optional σ-algebra. A pointwise

limit of optional processes is optional. On the other hand, σ can merely be a progres-

sively measurable process, or even a measurable adapted process, provided it satisfies

(2.1). Even when the filtration is generated by the Wiener process and augmented

as usual, there exist progressively measurable processes that are not optional and the

equality above cannot hold.

If additional assumptions seem to be necessary, one can ask whether they are not

too strong and if they are satisfied for models encountered in practice.

Let us take the following very general stochastic volatility model where St = S0eZt :

dZt = −1

2f 2(t,Y t, Zt)dt+ f(t,Y t, Zt)dW

1s

dY t = θ(t,Y t)dt+ ν(t,Y t) · dW t.

θ is the drift vector of size m and ν is the diffusion matrix of size m×n. The vector Y

of size m models the factors driving the volatility of the stock. We used the logarithm

Zt of the stock price to avoid degeneracy.

Let us now assume that f , θ and ν are C∞ functions in all their variables with

globally bounded derivatives in their space variables y and z. Assume that there

exists λ > 0 such that

1

λ≤ f(t, z, y) ≤ λ

for all (t, y, z). Note that since f is bounded, the exponential localmartingale St =

S0eZt satisfies the Novikov condition and is therefore a martingale. In particular it is

integrable and Assumption 1 is fulfilled.

Assume also that the diffusion is non degenerate in the sense that the diffusion

54

matrix, where νi denotes the i-th column of ν,

f(t, z, y) 0 · · · 0

ν1(t, y) ν2(t, y) · · · νn(t, y)

has full rank for all (t, y, z). It is then well known that the transition semigroup for

the diffusion (Z,Y ) has a density with respect to the Lebesgue measure of Rm+1.

Moreover, this transition density p(t, z, y; t′, z′, y′) is C∞ as a function of all its vari-

ables. Let us write the option prices as follows (Ft denotes the history of (Z,Y ) up

to time t)

Ct(T,K) = E0,Z0,Y 0

(eZT −K

)+∣∣∣Ft

= Et,Zt,Y t

(eZT −K

)+

As it is customary in the theory of Markov processes, we put superscripts to proba-

bility measures and expectations to specify from where the diffusion starts. To check

Assumption 2, we need to study the differentiability properties of C as a function of

(T,K). It is enough to study that of Ct(T, k) = Ct(T, ek). With this density, we can

rewrite option prices as

Ct(T, k) =

∫Rm+1

dy′dz′(ez′ − ek

)+

p(t, Zt,Y t;T, z′,y′)

=

∫Rm

dy′∫ +∞

k

dz′(ez′ − ek

)p(t, Zt,Y t;T, z

′,y′)

=

∫Rm

dy′∫ +∞

0

dz′(ez′ − ek

)p(t, Zt,Y t;T, z

′,y′)

−∫

Rm

dy′∫ k

0

dz′(ez′ − ek

)p(t, Zt,Y t;T, z

′,y′)

=

∫Rm

dy′∫ +∞

0

dz′(ez′ − ek

)p(t, Zt,Y t;T, z

′,y′)

−∫

Rm

dy′∫ 1

0

dz′ k(ekz′ − ek

)p(t, Zt,Y t;T, kz

′,y′).

Since eZT is integrable, it is clear on the last expression that Lebesgue derivation

55

theorem applies and that C is a local C∞-process on [0, T ). It is also clear that

for every ω, Ct(T,K) is jointly continuous in its three variables. The first part of

Assumption 2 holds in that model. The fact that Ct(T,K) > (St−K)+ follows from

the fact that the density is positive on Rm+1 and from Jensen inequality, exactly as

in the proof of Proposition 2.2.1.

Let us now turn ourselves to Assumption 3. It is concerned with H t(T,K) which is

the process of the martingale representation theorem. This process can be computed

from the Malliavin derivative of the random variable ST . Indeed, let ψn be a sequence

of positive smooth functions that decreases pointwise to the function x 7→ x+ and

such that the ψ′n are bounded and converge pointwise to x 7→ 1x≥0. We are about

to use the Clark-Ocone theorem which can be found, for instance, in the primer of

Malliavin calculus for finance in Fournie et al. (1999). When applied to the smooth

(in the Malliavin sense) random variable ψn(ST −K) it gives

E ψn(ST −K)| Ft = E ψn(ST −K)+

∫ t

0

E ψ′n(ST −K)DsST | Fs · dW s a.s.

DtST denotes the Malliavin derivative of the smooth random variable ST . By taking

limits in the above equality, we find

E

(ST −K)+∣∣Ft

= E

(ST −K)+

+

∫ t

0

E1ST≥KDsST

∣∣Fs

· dW s a.s.

and by uniqueness of the martingale representation,

H t(T,K) = E1ST≥KDtST

∣∣Ft

a.s.

To check that (T,K) 7→ H t(T,K) has the regularity and integrability in order to

satisfy Assumption 3, we use the fact that the law of (ST ,DtST ) conditioned by Ft

has a smooth density. One can then proceed in exactly the same way as we did before

56

to check Assumption 2.

It remains to prove that Assumption 4 holds true. At this stage, we do not have a

result to prove that it holds for the class of spot processes considered in this section.

We can only rely on results of Berestycki et al. (2002) and Berestycki et al. (2003).

In the later reference, the authors adopt the same framework as in this section and

they show that the implied volatility Σt(x, y) admits a Taylor expansion at any order

in x at x = 0. In other words,

Σt(x, y) = Σ0t (y) + xΣ1

t (y) +x2

2Σ2

t (y) +Ot(x2)

This is almost what we need but not quite. For instance, such an expansion does not

imply that ∂yΣt(x, y) converges to ∂yΣ0t (y) as x→ 0.

57

Chapter 3

From spot to implied volatilities

and applications

In this chapter, we present some possible applications of Theorem 2.5.1. First we

show that we can easily get qualitative behavior for a wide class of spot volatility

models. Second, we show how to derive approximate closed form formulas for option

pricing. These formulas are used to produce hedges for barrier options in Markov

models.

3.1 From spot to implied volatilities

Our tool is the following theorem. It is a converse to Theorem 2.5.1. Theorem 2.5.1

was already a solution to an inverse problem, one can therefore genuinely question

the point of this new theorem. Its interest is that it links spot volatility to implied

volatilities in a straightforward way. There is no need to compute the law of ST , to

integrate over it to get option prices and then, to invert Black-Scholes formula. We

can go directly from spot volatilities to implied volatilities. The price to pay for that

simple result is to work locally around the money and short maturities.

In the following theorem and in the remainder of this chapter, W and W are two

58

independent real valued Wiener processes on some probability space.

Theorem 3.1.1. Let us suppose that we are given adapted processes a, a, b, c such that

there exists a strictly positive solution to the following stochastic differential equation:

dσ2t =

(bt + a2

t + σ2t (at + ct)

)dt− 2σt

(atdWt + atdWt

)(3.1)

with a given initial condition σ0 and such that

d 〈a;W 〉t = −σt

(3ct2

+3a2

t

4σ2t

− a2t

σ2t

)dt. (3.2)

Further let

St = S0 exp

(∫ t

0

σsdWs −1

2

∫ t

0

σ2sds

)(3.3)

be the stock process and Xt(x, y) be the corresponding implied volatility. If Assump-

tions 1, 2, 3, and 4 are satisfied, then

∂2xyXt(0, 0) = at, ∂2

xxXt(0, 0) = bt, ∂3xyyXt(0, 0) = ct.

If we use the usual implied volatility Σt(T,K) instead of Xt(x, y), the conclusion

rewrites:

∂KΣt(t, St) = − at

2StΣt(t, St), ∂T Σt(t, St) =

bt4Σt(t, St)

,

∂2KKΣt(t, St) =

1

2S2t Σt(t, St)

(at + ct −

a2t

2Σt(t, St)2

).

Proof. Since Assumptions 1, 2, 3, and 4 are satisfied, σ satisfies equations (2.15) and

(2.16). By uniqueness of the semimartingale decomposition we get the result.

This theorem states that under some regularity assumptions, the behavior of the

implied volatility surface can be directly read off from spot dynamics of the form

(3.1). By behavior of the implied volatility, we mean behavior for options near the

59

money and with short maturities.

We now carry out the following program. Given four different well known models,

we rewrite them in the form (3.1)-(3.3) and therefore derive the dynamics of the smile

around the money and for short maturities.

3.1.1 Black-Scholes deterministic volatility model

The deterministic volatility model of Black and Scholes is of the form (3.1)-(3.3),

when written as

dσ2t = btdt at = at = ct = 0,

with b a deterministic function of time. In such a case, there is no smile (slope and

convexity in K are zero) but there is a well known maturity effect. Here,

Xt(x, y) = xΣ2t (t+ x, Ste

−y) =

∫ t+x

t

σ2sds =

∫ t+x

t

(∫ s

0

brdr

)ds

and we check that, indeed,

∂2xxXt(0, 0) = bt and ∂2

xyXt(0, 0) = ∂3xyyXt(0, 0) = 0.

3.1.2 Heston stochastic volatility model

Beyond Black-Scholes model, Heston model is a popular alternative spot volatility

model. It has the form

dσ2t = κ

(µ− σ2

t

)dt+ εσt

(ρdWt + ρdWt

)(3.4)

where ρ2 + ρ2 = 1. Heston model is almost of the form (3.1). In fact it is the typical

example of equations like (3.1). By matching localmartingale parts in (3.1) and (3.4),

60

we get

at = −ερ2

and at = −ερ2.

Then,

d 〈a;W 〉t = 0

and therefore equation (3.2) gives

ct =ε2

6σ2t

(1− 7ρ2

4

).

By matching bounded variation parts in (3.1) and (3.4), we get

bt = κ(µ− σ2t ) +

ερ

2σ2

t −ε2

6

(1− ρ2

4

).

3.1.3 Dupire local volatility model

Dupire local volatility model is a model for the spot volatility where the latter is a

deterministic function f of time and spot, i.e.,

σ2t = f(t, St)

Its dynamics is given (provided f is smooth enough) by Ito formula:

dσ2t =

(∂tf(t, St) +

1

2σ2

tS2t ∂

2SSf(t, St)

)dt+ σtSt∂Sf(t, St)dWt (3.5)

By matching localmartingale parts in (3.1) and (3.5), we get

at = −1

2St∂Sf(t, St) and at = 0.

61

Then,

d 〈a;W 〉t = −1

2σtSt

(St∂

2SSf(t, St) + ∂Sf(t, St)

)dt

and therefore equation (3.2) gives

ct =1

3S2

t ∂2SSf(t, St) +

1

3St∂Sf(t, St)−

(St∂Sf(t, St))2

8f(t, St).

By matching bounded variation parts in (3.1) and (3.5), we get

bt = ∂tf(t, St) +1

6S2

t f(t, St)∂2SSf(t, St)−

1

8S2

t (∂Sf(t, St))2 +

1

6Stf(t, St)∂Sf(t, St).

3.1.4 SABR stochastic volatility model

The SABR stochastic volatility model was recently introduced in Hagan et al. (2002).

It is specified by three constants, β, ν and ρ and initial conditions σ0 and α0.

σt = αtSβ−1t

dαt = ναt

(ρdWt + ρdWt

)

We first need to derive the dynamics for σ2t .

dσ2t

σ2t

=(ν2 + (β − 1)(2β − 3)σ2

t + 4νρ(β − 1)σt

)dt+2(νρ+(β− 1)σt)dWt +2νρdWt.

(3.6)

By matching localmartingale parts in (3.1) and (3.6), we get

at = −(νρ+ (β − 1)σt)σt and at = −νρσt.

Then,

d 〈a;W 〉t = −σt(νρ+ (β − 1)σt)(νρ+ 2(β − 1)σt)dt

62

and therefore equation (3.2) gives

ct =ν2

6(4− 3ρ2) + νρ(β − 1)σt +

5

6(β − 1)2σ2

t .

By matching bounded variation parts in (3.1) and (3.6), we get

bt =σ2

t

6

(ν2(2− 3ρ2) + 6νρβσt + (β − 1)2σ2

t

).

3.2 Model analysis

Our findings in Theorem 3.1.1 allow to find back basic facts about spot volatility mod-

els. These facts have already been observed by performing Monte Carlo simulations

(see, for example, Zhu and Avellaneda (1998)).

When the correlation between the spot and its volatility is zero, one expects to

find a symmetric smile around the money. This is always the case for short maturities

when Theorem 3.1.1 applies, since in that case

at = ∂xyXt(0, 0) = −2StΣt(t, St)∂KΣt(t, St) = 0.

For positive correlation, i.e., at < 0, the center of the parabola is shifted to the left

since then ∂KΣt(t, St) > 0. For negative correlation, it is shifted to the right.

To study the effect of the vol-vol on the shape of the smile, let us assume that

at = 0, that is, the smile is centered around the money. Increasing the vol-vol means

increasing at. Since in that case

ct = 2S2t Σt(t, St)∂

2KKΣt(t, St) =

2a2t

3σ2t

,

the larger the vol-vol, the larger the convexity of the smile. As is intuitive, for low

63

vol-vol, we get closer to the Black-Scholes model and the smile should flatten out.

Paragraph 1.2.3 showed that smile dynamics were particularly important as far as

hedging is concerned. We recalled the work of Hagan et al. (2002) who compare the

Dupire local volatility model with the SABR stochastic volatility model in terms of

their dynamics in a change of the spot. They are interested in the dynamical relation

between changes in the spot and changes in the shape implied volatility smile. More

precisely, we want to know if an increase in the spot will lead to smile that moves to

the left or to the right.

With our formulas, we can easily compute similar dynamics. For instance one can

ask the question of whether the smile becomes more pronounced after a spot move.

This is seen by computing ∂Sat. As we have explained, the smile’s slope is directly

related to at by the formula

∂KΣt(t, St) = − St

2atΣt(t, St).

Whether the slope increases with S is seen on the sign of the derivative of the quantity

above with respect to S.

A quick look at the formula for at in Dupire model with a power law, i.e. f(t, S) =

α2S2β−2 with β < 1 leads to

∂S∂KΣt(t, St) =4− 3β

α3(β − 1)S3β−3

t < 0.

An increase in the spot makes the smile more pronounced in that model (the slope

increases in absolute value).

64

3.3 Stochastic models for implied volatilities

This short section is devoted to applications of Theorem 3.1.1 to stochastic modeling

of implied volatilities. We argued in chapter 1 that, though a very appealing idea,

constructing an arbitrage free model for the dynamics of the implied volatility smile

was a difficult problem. Here, we propose a way to partially solve that problem.

In view of Theorem 3.1.1, there is a direct relationship between the stochastic

differential equation driving the spot volatility and the local shape of the volatility

smile for short maturities and strikes near the money. It is therefore natural to model

these quantities and build from them a spot volatility model that will give back a

smile with the prespecified local shape. The correspondence between the local shape

of the smile and the coefficients of the stochastic differential equation driving the spot

volatility is recalled in Table 3.1.

smile’s slope at

spot/smile’s slope correlation at

smile’s convexity ctsmile’s term structure bt

Table 3.1: Correspondence between implied volatility smile quantities and the spotvolatility quantities.

Of course, reducing the whole volatility surface to three quantities: its slope and

convexity in the strike direction and its slope in the maturity direction at a single

point is a rather crude approximation. On the other hand, these quantities are rather

natural.

Even with this crude approximation, modeling takes a peculiar form because of

equation (3.2). Let us assume that we are working on probability space with two

independent Brownian motions W and W . We are free to choose the dynamics for a,

b and c but the dynamics of a is constrained by (3.2), i.e., we are free to choose its

drift and the volatility coefficient of W but the volatility coefficient of W is built in.

65

More precisely, given adapted processes a, b, c, µa and νa that model the dynamics

of the smile, a natural stochastic volatility model is given by the following system of

differential equations.

dSt = σtStdWt

dσ2t =

(bt + a2

t + σ2t (at + ct)

)dt− 2σt

(atdWt + atdWt

)dat = µa

t dt− σt

(3ct2

+3a2

t

4σ2t

− a2t

σ2t

)dWt + νa

t dWt

Of course, given general a, b, c, µa and νa, there is no reason why there should be

a solution to this system. Note that we also require the second equation to have a

positive solution. This is however to be expected since the shape of the smile cannot

be arbitrary. There must be no arbitrage constraints on its shape, even when modeled

locally as we are doing here. The arbitrage constraints are hidden in the existence of

a solution to this stochastic differential system.

If such an approach can prove to be useful in practice, it will be important to

understand this system of equations. Such a study is unfortunately left for further

work.

3.4 Computing implied volatilities

This section is devoted to finding closed form approximations for the implied volatility

in models satisfying the hypothesis of Theorem 3.1.1.

3.4.1 First order approximations

For each one of the models of sections 3.4, 3.5, and 3.6, we can use our expressions

for ∂2xxXt(0, 0) and ∂2

xyXt(0, 0) and ∂3xyyXt(0, 0) to find closed form approximations

66

for option prices. Namely, writing the Taylor expansion of Xt around (0, 0) gives

Xt(x, y) = Xt(0, 0) + x∂xXt(0, 0) + y∂yXt(0, 0) +1

2

(x2∂2

xxXt(0, 0)

+2xy∂2xyXt(0, 0) + y2∂2

yyXt(0, 0))

+1

2xy2∂3

xyyXt(0, 0) + . . .

By switching to the (T,K) variables we get:

Σapproxt (T,K) =

√σ2

t + at lnSt

K+ bt

T − t

2+ct2

ln2 St

K(3.7)

In the case of the Dupire local volatility model with a power law (constant elas-

ticity of variance), we compared approximation (3.7) with the formula obtained by

Hagan and Woodward (1999). The relative error with respect to their formula is of

the order of 10−5 for strikes not too far from the money and not too long maturities.

In fact, we can expand both (3.7) and the formula of Hagan and Woodward (1999)

in the log-moneyness and time-to-maturity variables and get the same coefficients.

a, b and c can be computed either by taking f(S) = α2S2(β−1) in section 3.1.3 or by

taking ν = 0 in the formulas of section 3.1.4.

Similarly, in the case of the SABR model, we can compare our approximations

with that of Hagan et al. (2002). Observe the perfect agreement of a, b and c with

equation (3.1a) in Hagan et al. (2002).

3.4.2 Higher order approximations

Hagan and Woodward (1999), Hagan et al. (2002) and Berestycki et al. (2003) provide

with methods to get approximate formulas for the implied volatility at any order. Our

methodology also enables us to do the same and this is the content of this section.

We will not give rigorous proofs since they are in the same spirit as those in chapter 2.

Since we are making computations at a higher order, we must make a corresponding

67

assumption. This means that Assumptions 2, 3, and 4 must take into account higher

order derivatives. We shall refrain from stating them properly. Instead we carry out

the computations formally and give the final result. This last theorem is stated in a

slightly different form from Theorem 3.1.1.

Theorem 3.4.1. Suppose that the spot volatility σ is given by the following system

of stochastic differential equations.

dσ2t = µtdt− 2σt

(atdWt + atdWt

)dµt = ♥dt+ wtdWt +♥dWt

dat = mtdt+ utdWt + utdWt

dat = ♥dt+ vtdWt +♥dWt

dut = ♥dt+ xtdWt +♥dWt

where a, a, u, u, v, w, x, and m are some given continuous adapted processes and ♥

is a generic symbol for a continuous adapted process. Let

bt = µt −a2

t

2− 2a2

t

3− atσ

2t +

2utσt

3,

ct = −2ut

3σt

− a2t

2σ2t

+2a2

t

3σ2t

,

dt =2mt

3− wt

3σt

− xt

2− atµt

3σ2t

+atut

6σt

+atvt

σt

+2ata

2t

3σ2t

+2utσt

3− a2

t

3,

and

et =xt

2σ2t

+2atut

σ3t

− 3atut

2σ3t

− atvt

σ3t

+3a3

t

2σ4t

− 4ata2t

σ4t

.

Then,

∂2xyXt(0, 0) = at, ∂2

xxXt(0, 0) = bt, ∂3xyyXt(0, 0) = ct,

∂3xxyXt(0, 0) = dt, and ∂4

xyyyXt(0, 0) = et.

68

With these formulas, a better approximation for the implied volatility is given by

Σapproxt (T,K) =

√σ2

t + at lnSt

K+ bt

T − t

2+ct2

ln2 St

K+ dt

T − t

2lnSt

K+et

6ln3 St

K.

For any given model of the previous section, we can compute as before the coefficients

d and e.

Heston stochastic volatility model

In the case of Heston stochastic volatility model of section 3.1.2:

dt =κερ

6

(µ

σ2t

+ 1

)− ε2ρ

12

(ρ+

ερ2

σ2t

)et =

ε3ρ

2σ4t

(1− 11

8ρ2

).

Dupire local volatility model

In the case of Dupire local volatility model of section 3.1.3:

dt =Sf ′f

6f− 2Sf ′

3− S3f (3)f

12− S2f ′′f

4+S3f ′′f ′

24− Sf ′f

12+S2f ′2

24

et = −1

4

(S3f (3) + 3S2f ′′ + Sf ′

)+

3

8

(S2f ′′ + Sf ′

) Sf ′f− Sf ′

16

(Sf ′

f

)2

,

where we suppressed the dependence in t and St, ˙ denotes differentiation with respect

to t and ′ denotes differentiation with respect to S.

69

SABR stochastic volatility model

In the case of SABR model of section 3.1.4, this gives

dt =σt

6

(ν3ρ− ν2(2(β − 1) + 3ρ2(β + 1))σt − 2(β − 1)(5β − 1)νρσ2

t

−2(β − 1)3σ3t

)et =

1

2σt

((β − 1)2νρσ2

t + (β − 1)3σ3t − 3ν3ρ(1− ρ2)

).

Our expression for e agrees with formulas of Berestycki et al. (2003) but disagrees

with those in Hagan et al. (2002). The discrepancy between Hagan et al. (2002) and

Berestycki et al. (2003) was already pointed out in Berestycki et al. (2003). As for d

when ρ = 0, it is consistent with Berestycki et al. (2003) who only gives formulas for

the two term expansion in that case. It also agrees with Hagan et al. (2002) but the

formulas are a little bit different when ρ 6= 0.

3.5 Static hedging of a barrier option when the

spot is Markov

In this section, we explain how equations like (3.7) provide a very simple way of

generalizing the classical static hedging of barrier options when the spot process is

Markov. We shall explain at the end of the section why it is more complicated when

the volatility is driven by an additional noise.

Let us consider the case of a down-and-in call option. The payoff of such an option

is the usual call payoff (ST −K)+ under the condition that the stock process go below

the barrier H at some point before maturity T . To make our discussion as simple as

possible, we shall assume that the option is regular, i.e., H < K.

70

3.5.1 Static hedging with a constant volatility

We are going to see that a regular down-and-in call option is equivalent to a certain

number of put options with an appropriate strike.

Recall our heuristic discussion in paragraph 1.2.1. Assume that the stock reaches

the barrier H at TH < T . At this very moment, the barrier option is precisely

worth a call option with strike K, i.e., Call (spot = H, strike = K,mat = T ). Since

the volatility is constant, the price is given by Black-Scholes formula. Moreover, a

simple computation shows that,

Call (spot = H, strike = K,mat = T ) =K

HPut

(spot = H, strike =

H2

K,mat = T

).

Therefore, at TH , the option is worthK/H put options with strikeH2/K and maturity

T . The advantage of this representation is that such put options are worthless if the

stock stays above H since their strike is H2/K < H. In either case, the value of the

barrier option is worth K/H put options with strike H2/K and maturity T .

To justify this result, we apply the strong Markov property at the stopping time

TH = inft ≥ 0 : St ≤ H.

Indeed, the price is:

E0,S01TH≤T(ST −K)+

= E0,S0

1TH≤TETH ,H

(ST−TH

−K)+

.

The last expression rewrites

E0,S0

1TH≤TETH ,H

(K ′ − ST−TH

)+ ETH ,H (ST−TH

−K)+ETH ,H (K ′ − ST−TH

)+

(3.8)

71

for any K ′. As explained above, choosing K ′ = H2/K leads to

ETH ,H

(H2

K− ST−TH

)+

=H

KETH ,K

(H − ST−TH

)+=

H

KETH ,H

(ST−TH

−K)+ ,so that the price is

K

HE0,S0

1TH≤TETH ,H

(H2

K− ST−TH

)+

=K

HE0,S0

1TH≤T

(H2

K− ST

)+

by another application of the strong Markov property. We can drop the indicator

function in the last expectation since

H2

K− ST ≥ 0

⊂ TH ≤ T .

3.5.2 Static hedging in Dupire local volatility model

We now try to generalize the previous result in the case of Dupire local volatility

model. If we follow the lines of the preceding reasoning, we see that we must know

the price of a call option with maturity T and strike K at time TH . Knowing the smile

at time 0 gives no information whatsoever on the smile at time TH when the spot

trades at H. However, in a Markov model, we can use the formula (3.7). Coefficients

at, bt, and ct are functions of St only. This means that we have a simple way of

computing prices at any date in the future.

Now look at (3.8). In the Black-Scholes model, we were able to find K ′ such that

the ratio

ETH ,H (ST−TH−K)+

ETH ,H (K ′ − ST−TH)+

did not depend on TH . This meant that we could pull it out of the expectation.

In a general local volatility model, there is little hope that such a cancellation might

72

occur. However we can look for the value K ′ for which the derivative of the ratio with

respect to TH vanishes. This is numerically extremely easy since, as we argued before,

we have closed form approximations for option prices. Once we have computed K ′ in

a particular model, we can check whether the dependence in TH is indeed weak.

We performed this experiment in the case of the Dupire local volatility model with

a power law (constant elasticity of variance), i.e.,

σt = αSβ−1t .

We computed the optimal ratios for different values of β in Figure 3.1. The case

β = 1 is the Black-Scholes model, for which the ratio is independent of time. We see

that even when β 6= 1, these ratios are amazingly constant over time.

Figure 3.1: Number of puts to statically hedge the down-and-in call option in aconstant elasticity of variance model. α is set such that σ0 = 20% in all cases. Theother parameters are S0 = 100, H = 90, K = 110 and T = 1.

These results have to be taken with a grain of salt, in particular one should

remember that we are using approximate formulas obtained for short maturities.

73

Whether these results are really as good as they look is left for future investigation.

With this proviso in mind, static hedging can be achieved in exactly the same way as

for the Black-Scholes model. Number of puts and their strike can be computed in a

very straightforward way. Some values can be found in Table 3.2.

β average ratio K ′ lower price upper price1 1.222 73.636 0.572 0.572

0.8 1.183 73.085 0.583 0.5830.6 1.143 72.495 0.594 0.5940.4 1.104 71.861 0.605 0.6060.2 1.064 71.179 0.618 0.6180 1.024 70.441 0.630 0.632

Table 3.2: Prices of down-and-in call option in a constant elasticity of variance model.The value of α is set such that σ0 = 20% in all cases. The other parameters areS0 = 100, H = 90, K = 110 and T = 1.

The situation is more involved when the model is non Markov. Indeed, to compute

the smile when the stock touches the barrier, we must not only know the value of

the stock but also the value of its volatility. One could however consider models that

are Markov in both variables: the stock and its volatility (like the Heston and SABR

models.) In such cases, the strong Markov still applies to the joint process. It remains

to see if we can find a value K ′ such that the ratio

ETH ,H,σTH (ST−TH−K)+

ETH ,H,σTH (K ′ − ST−TH)+

depends only mildly on both TH and σTH. This is left for future research. One could

as well be interested in other barrier options and more complicated products.

74

Chapter 4

Conclusions

This dissertation showed how to recover the spot volatility by observing the dynamics

of implied volatilities with short maturities and strike around the money. We first

motivated such a result in chapter 1 by a survey of some of today’s problems faced

in the practice of risk management. We then solved this inverse problem under a

certain set of assumptions in chapter 2. Most of these assumptions were shown to be

satisfied by the popular models often proposed for spot processes.

The present work has interesting parallels to the recent work Berestycki et al.

(2003)1. In this paper, the authors use PDE and large deviations techniques (Varad-

han’s formula) to study the behavior of the implied volatility near expiry. They

then derive a way of computing implied volatilities in stochastic volatility models by

computing a geodesic distance associated with the diffusion operator.

Independently and using probabilistic arguments, this dissertation extends their

work in two directions. First, its scope is not limited to diffusion processes. Of

course, we are aware of the fact that checking hypotheses for processes that are not

diffusions might be very difficult. Second and more importantly from a practical

point of view, the link that we found between spot and implied volatilities works in

both directions. Namely, by observing the dynamics of implied volatilities, one can

1I am grateful to N. El Karoui for pointing this reference to me.

75

find the spot volatility dynamics but one can also start with a spot volatility model

and compute approximate implied volatilities in a simple way. We explored some

applications of this last direction in chapter 3.

Paragraph 3.4.2 makes it clear that the link between spot and implied volatilities

is in fact a relationship between the coefficients of the Taylor expansion of the implied

volatility and the coefficients of a certain ‘chaos expansion’ of the spot volatility. By

‘chaos expansion’ we simply mean an expansion by means of iterated integrals. This

suggests a new angle of attack of this problem, which is potentially more elegant. We

unfortunately have to leave this idea for further research.

76

Appendix A

Semimartingales with spatial

parameters and generalized Ito

formula

In this appendix, we recall definitions and properties of semimartingales with spatial

parameters that we used in chapter 2. We follow Kunita (1990) very closely but other

references are possible, for instance, Carmona and Nualart (1990).

A.1 Semimartingales with spatial parameters

We first recall the definitions of seminorms and function spaces. We first define

seminorms and function spaces. Let D be a domain (i.e., an open and connected set)

in Rd for some d ≥ 0. Let m be a positive integer; we denote by Cm(D; Re), for some

e ≥ 0, the set of all maps f : D → Re that are m-times continuously differentiable on

D. Let K ⊂ D; we define the seminorms ‖f‖m:K for m ≥ 0 and f ∈ Cm(D; Re) by

‖f‖m:K =∑

0≤|α|≤m

supx∈K

|∂αx f(x)| .

77

|·| denotes the Euclidean norm in Rd or in Re and ∂α the usual multi-indexed differ-

ential operator. As usual, for a multi-index α = (α1, . . . , αk), |α| = α1 + · · ·+ αk.

If m = 0, C0(D; Re) denotes the set of all continuous maps f : D → Re with the

corresponding family of seminorms.

Let m be a positive integer and δ ∈ (0, 1]; we denote by Cm+δ(D; Re), the set of

all maps f : D → Re which are m-times continuously differentiable and whose m-th

derivatives are Holder continuous of order δ. Let K ⊂ D; we define the seminorms

‖f‖m+δ:K for m ≥ 0, δ ∈ (0, 1] and f ∈ Cm+δ(D; Re) by

‖f‖m+δ:K =∑

0≤|α|≤m

supx∈K

|∂αx f(x)|+

∑|α|=m

supx 6=x′∈K

|∂αx f(x)− ∂α

x f(x′)||x− x′|δ

.

We shall also need the seminorms ‖g‖∼m+δ:K, defined for functions g from D × D

into Re that are m-times differentiable in x and in y separately and whose m-th

derivatives in x and in y are Holder continuous of order δ. This space is denoted

Cm+δ. We define seminorms on that space by

‖g‖∼m+δ:K =∑

0≤|α|≤m

supx,y∈K

∣∣∂αx∂

αy g(x, y)

∣∣+∑|α|=m

supx 6=x′,y 6=y′∈K

∣∣∂αx∂

αy g(x, y)− ∂α

x∂αy g(x

′, y)− ∂αx∂

αy g(x, y

′) + ∂αx∂

αy g(x

′, y′)∣∣

|x− x′|δ |y − y′|δ.

The families of seminorms ‖ ·‖m:K and ‖ ·‖m+δ:K where K ranges over the compact

sets in D make the sets Cm(D; Re) and Cm+δ(D; Re) into Frechet spaces. Correspond-

ingly, the family of seminorms ‖ · ‖∼m+δ:K where K ranges over the compact sets in

D ×D makes the set Cm+δ(D ×D; Re) into a Frechet space. These spaces are easily

seen to be separable. As topological spaces they are also measurable spaces when

endowed with their Borel σ-algebras.

Let F (x, t);x ∈ D, t ≥ 0 be a family of real valued processes with parameter

x ∈ D. By this we mean a random variable taking values in RD×R+ when this

78

set is endowed with the smallest σ-algebra such that the coordinate mappings are

measurable. If F (x, t, ω) is a continuous function of x for almost every ω and all t,

we can regard F (·, t) as a stochastic process with values in C0(D; R) or a C0-valued

process. If F (x, t, ω) is a m-times continuously differentiable in x and all its m-th

derivatives are Holder continuous of order δ for almost every ω and all t, we can

regard F (·, t) as a stochastic process with values in Cm+δ(D; R) or a Cm+δ-valued

process. In case where F (x, t) is a continuous process with values in Cm+δ it is called

a continuous Cm+δ-process.

Cm-processes, Cm+δ-processes, continuous Cm-processes, and continuous Cm+δ-

processes are defined similarly.

Theorem A.1.1 (Cf. Theorem 3.1.2 p. 75 of Kunita (1990)). Let M(x, t),

x ∈ D be a family of continuous localmartingales such that M(x, 0) = 0. Assume

that its joint quadratic variation has a modification A(x, y, t) of a continuous Cm+δ-

process for some m ≥ 1 and 0 < δ ≤ 1. Then, M(x, t) has a modification of a

continuous Cm+ε-process for any ε < δ. Furthermore for each α with |α| ≤ m,

∂αxM(x, t), x ∈ D is a family of continuous localmartingales with joint quadratic

variation ∂αx∂

αyA(x, y, t).

Definition A.1.2. We shall call the random field M(x, t) with the property of theo-

rem A.1.1 a continuous localmartingale with values in Cm+δ or a continuous Cm+δ-

localmartingale.

The next theorem is a converse to theorem A.1.1.

Theorem A.1.3 (Cf. Theorem 3.1.3 p. 76 of Kunita (1990)). Let M(x, t),

x ∈ D and N(y, t), y ∈ D be continuous localmartingales with values in Cm+δ for

some m ≥ 1 and 0 < δ ≤ 1. Then the joint quadratic variation has a modification of

a Cm+ε-process for any ε < δ. Furthermore the modification satisfies

∂αx∂

βy 〈M(x, t);N(y, t)〉 =

⟨∂α

xM(x, t); ∂βyN(y, t)

⟩a.s.

79

for any t and |α|, |β| ≤ m.

Finally, the following theorem is an application of theorem A.1.1.

Theorem A.1.4 (Cf. Exercise 3.1.5 p. 78 of Kunita (1990)). Let Mt be a

continuous localmartingale and f(x, t), x ∈ D be a predictable process with values in

Cm+δ for some m ≥ 0 and 0 < δ ≤ 1 such that

∫ T

0

‖f(s)‖2m+δ:K d 〈M〉s <∞ a.s. for all compact K ⊂ D.

Then M(x, t) =∫ t

0f(x, s)dMs has a modification of a Cm+ε-localmartingale for any

ε < δ. Further it satisfies

∂αx

∫ t

0

f(x, s)dMs =

∫ t

0

∂αx f(x, s)dMs for |α| ≤ m.

Proof. The joint quadratic variation is

A(x, y, t) =

∫ t

0

f(x, s)f(y, s)d 〈M〉s

Moreover, if K is a compact subset containing x and y

|∂αx f(x, s)|

∣∣∂βy f(y, s)

∣∣ ≤ ‖f(·, s)‖2m+δ:K

for any |α|, |β| ≤ m. The right-hand side is integrable and invoking Lebesgue deriva-

tion theorem we get that A(·, ·, t) is in Cm+δ and

∂αx∂

βyA(x, y, t) =

∫ t

0

∂αx f(x, s)∂β

y f(y, s)d 〈M〉s .

The proof is completed by applying A.1.1.

Let F (x, t), x ∈ D be a family of continuous semimartingale decomposed as

80

F (x, t) = B(x, t)+M(x, t) where M(x, t) is a continuous localmartingale and B(x, t)

is a continuous process of bounded variation.

Definition A.1.5 (Cf. Kunita (1990) p. 84). A family of continuous semimartin-

gale F (x, t), x ∈ D is said to be a Cm+δ-semimartingale if M(x, t) is a continuous

Cm+δ-localmartingale in the sense of Definition A.1.2 and B(x, t) is a continuous

Cm+δ-process such that ∂αxB(x, t), x ∈ D, |α| ≤ m are all processes of bounded vari-

ation.

A.2 Generalized Ito formula

The generalized Ito formula is sometimes called Ito-Wentzell formula because A. D.

Wentzell seems to be the first to contribute to that problem in 1965. In the early

1980’s and in connection with the study of stochastic flows, several authors worked

on the topic, including, J.-M. Bismut, H. Kunita, and A.-S. Sznitman.

Theorem A.2.1 (Cf. Theorem 3.3.1 p. 92 of Kunita (1990)). Let Ft(x),

x ∈ D be a family of continuous semimartingale decomposed as

Ft(x) =

∫ t

0

bs(x)ds+

∫ t

0

σs(x) · dW s

Suppose it is a continuous C2-process and a continuous C1-semimartingale such that

∫ T

0

‖σs(·) · σs(·)‖∼1:K + ‖bs(·)‖1:K ds <∞ for any compact K ⊂ D.

Let gt be a continuous semimartingale with values in D. Then Ft(gt) is a semimartin-

81

gale and satisfies for t ≤ T

Ft(gt)− F0(g0) =

∫ t

0

bs(gs)ds+

∫ t

0

σs(gs) · dW s +d∑

i=1

∫ t

0

∂xiFs(gs)σis(gs)ds

+1

2

d∑i,j=1

∫ t

0

∂2xixjFs(gs)d

⟨gi

s; gjs

⟩+

d∑i=1

⟨∫ t

0

∂xiσs(gs) · dW s; git

⟩.

A.3 Convergence

We give here a proof of Lemma 2.4.3. First recall it.

Lemma A.3.1. Suppose Y (x, y) for (x, y) ∈ D is a C0-semimartingale with decom-

position

Yt(x, y) =

∫ t

0

as(x, y)ds+

∫ t

0

bs(x, y) · dW s.

Suppose further that for every y ∈ (−y, y),

limx→0

Y (x, y) = 0 u.c.p.

Then for every y ∈ (−y, y),

limx→0

∫ ·

0

as(x, y)ds = limx→0

∫ ·

0

bs(x, y) · dW s = 0 u.c.p.

Proof. Let us fix y ∈ (−y, y). Define for every x ∈ (0, x)

Yt(x, y) =

∫ t

0

as(x, y)ds−∫ t

0

bs(x, y) · dW s =

∫ t

0

as(x, y)ds+

∫ t

0

bs(x, y) · dW s.

W = −W is also Brownian motion. The random fields Yt(x, y) and Yt(x, y) for t ≥ 0

and x ∈ (0, x) have the same law. Therefore, limQ3x→0 Y (x, y) = limQ3x→0 Y (x, y) = 0

u.c.p. Since Y is continuous in x for x ∈ (0, x), limx→0 Y (x, y) = 0 u.c.p. The same

holds true for 12(Y + Y ) and 1

2(Y − Y ).

82

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