Arbitrage-free market models for interest rate options and ......(0) Fundamentally, these models should be arbitrage-free. (1) Any initial option price data from the market can be

Technical report, IDE1022 , November 1, 2010

Master’s Thesis in Financial Mathematics

Anastasia Ellanskaya, Hui Ye

School of Information Science, Computer and Electrical EngineeringHalmstad University

Arbitrage-free market modelsfor interest rate options and

future options: the multi-strikecase

Arbitrage-free market models forinterest rate options and futureoptions: the multi-strike case

Anastasia Ellanskaya, Hui Ye

Halmstad University

Project Report IDE1022

Master’s thesis in Financial Mathematics, 15 ECTS credits

Supervisor: Prof. Mikhail Nechaev

Examiner: Prof. Ljudmila A. Bordag

External referee: Prof. Vladimir Roubtsov

November 1, 2010

Department of Mathematics, Physics and Electrical engineeringSchool of Information Science, Computer and Electrical Engineering

Halmstad University

Preface

The investors concern the most, when it comes to the financial derivativesmarket, is a fair and reasonable pricing of instruments. People interpret thisas the underlying fundamental properties that should be possessed by thepricing models. They can be described mainly as the following: the modelsshould be arbitrage-free; the model should be able to reproduce any ini-tial option price data collected in the market; the model should incorporatecharacteristic features of joint dynamics of stock and options. One of suchmodels is constructed and verified to be possessing the above properties byMartin Schweizer and Johannes Wissel in their paper [10] published in 2008.Inspired by their work, we take one step forward in this direction and ex-tend the original model to be applicable for pricing options on interest rateindexes and options on futures. We draw a periodic end to our research bydemonstrating the practical use of Schweizer-Wissel model in dealing withinvestment portfolios.During our work, we get some tremendous help from our professors, andwe would like to thank all of them sincerely. It is a honor to be workingunder the supervision of Professor Mikhail L. Nechaev. We appreciate manyhelpful hints and discussions from him. We are also grateful for valuablecomments and help with some technical difficulties from Professor LjudmilaA. Bordag. We also would like to thank our external referees and otherprofessors from Financial Mathematics group for their sincere, honest andinstructive comments and advices, from which we benefitted a lot. We arereally looking forward to working under their supervisions again some day.

ii

Abstract

This work mainly studies modeling and existence issues for martingalemodels of option markets with one stock and a collection of European calloptions for one fixed maturity and infinetely many strikes. In particular,we study Dupire’s and Schweizer-Wissel’s models, especially the latter one.These two types of models have two completely different pricing approachs,one of which is martingale approach (in Dupire’s model), and other one is amarket approach (in Schweizer-Wissel’s model). After arguing that Dupire’smodel suffers from the several lacks comparing to Schweizer-Wissel’s model,we extend the latter one to get the variations for the case of options oninterest rate indexes and futures options. Our models are based on the newlyintroduced definitions of local implied volatilities and a price level proposedby Schweizer and Wissel. We get explicit expressions of option prices asfunctions of the local implied volatilities and the price levels in our variationsof models. Afterwards, the absence of the dynamic arbitrage in the market forsuch models can be described in terms of the drift restrictions on the models’coefficients. Finally we demonstrate the application of such models by asimple example of an investment portfolio to show how Schweizer-Wissel’smodel works generally.

iii

iv

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Chapter review . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Literature Review 32.1 The Martingale models . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Dupire’s model . . . . . . . . . . . . . . . . . . . . . . 52.2 The market models . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 The Schweizer-Wissel model . . . . . . . . . . . . . . . 92.3 Conclusions and comments . . . . . . . . . . . . . . . . . . . . 16

3 The extensions of the Schweizer-Wissel model 193.1 An option on interest rate indexes . . . . . . . . . . . . . . . . 20

3.1.1 The new parameterization . . . . . . . . . . . . . . . . 223.1.2 The arbitrage-free dynamics of the local implied volatil-

ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 An option on Futures . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 The new parameterization . . . . . . . . . . . . . . . . 363.2.2 The arbitrage-free dynamics of the local implied volatil-

ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Applications 53

5 Conclusions 59

Notation 63

Appendix 65

Bibliography 67

v

vi

Chapter 1

Introduction

1.1 Motivation

Since the famous Black-Scholes model was published by Fischer Black andMyron Scholes in 1973, the evolution of pricing derivative investment instru-ments has taken place. While being widely used in the financial instrumentspricing, there are still some defects found during its application in the realfinancial market. Especially after the “Black Tuesday” of year 1987, a newphenomenon was found in the structure of options prices. Before crisis, it iscommonly believed that the volatility of the option should not be dependingon its strike and expiration. Then the fact that the observed volatility surfacebecame skewed was contradicted to the classic understanding of the volatil-ity. Since the Black-Scholes model can not account for this phenomenon,which had been called the volatility smile, traders had to invent some newtricks to incorporate this effect into pricing models. One of the most popularsolutions is to connect this effect to the notion of Stochastic Local Volatil-ity, which was presented in the paper “Pricing with a smile” by B. Dupire,1994.

The idea of the Stochastic Local Volatility was a real improvement of thepricing models, but still it was not consistent enough with the structure ofthe option prices. Therefore, other pricing models were proposed for thewhole option series pricing instead of describing an evolution of a singleoption. One of such models was introduced by M. Schweizer and J. Wissel intheir paper “Arbitrage-free market models for option prices: the multi-strikecase”. Our study here is mainly based on the Schweizer-Wissel model andtakes one step forward in the proposed direction.

1

2 Chapter 1. Introduction

1.2 Objectives

Our main objective is to get the possible extensions of the Schweizer-Wisselmodel, for the cases of two other derivatives, which are an option on futuresand an option on interest rate indexes. Moreover, our by-product here ishopefully doing some practical financial analyses of the investment portfoliosbased on the two variations of the Schweizer-Wissel model.

1.3 The Chapter review

Chapter 1, Introduction. This chapter intends to presents the readers withsome general ideas about the study background of introducing Schweizer-Wissel’s model, as well as our motivation and objectives of this study.

Chapter 2, Literature Review. In this chapter, we shortly introduce twomain pricing approaches and models. They are the traditional martingaleapproaches and the market models. After reviewing some main literatureconcerning this subject, we discuss our ideas of these two different approachesand models, and describe them more precisely in two specific models cho-sen from each approach’s category, the Dupire’s model and the Schweizer-Wissel’s model.

Chapter 3, The extensions of Schweizer-Wissel’s model. After an introduc-tion of two underlying assets, i.e. futures and interest rate indexes which ourextensions of Schweizer-Wissel’s model are based on, we derive the extentionsof the original Schweizer-Wissel’s model in both of the scenarios, focusing onthe new parameterizations, i.e. definitions and basic properties, and thus,present our results including the arbitrage-free dynamics of the local impliedvolatilities.

Chapter 4, Applications. We start this chapter with demonstrations of somesimple investment portfolios, to show how Schweizer-Wissel model can bepossibly used in applications. We end it by verifying our assertment thatunder some special circumstances, Schweizer-Wissel’s model can be unifiedwith the Black-Scholes model which provides us the consistency betweenpricing models.

Chapter 5, Conclusions. Finally, in the last chapter, we complete the study bydrawing some conclusions of our work, summarizing what we have achievedin this area and proposing some thoughts about a further possible researchdirection.

Chapter 2

The Literature Review

In this chapter, we describe the two main approaches to the option pricingand review them by the comparison of the Dupire model and the Schweizer–Wissel model.

Our ultimate goal of the financial instrument pricing, as started out in [2], isto establish a framework for the pricing and hedging of derivatives (possiblyexotic ones) in an arbitrage-free way, using all the liquid tradables as potentialhedging instruments.

In order to achieve this goal, the class of models for bond, stock and optionsshould at least possess the following features:

(0) Fundamentally, these models should be arbitrage-free.

(1) Any initial option price data from the market can be reproduced by thesemodels; this is called perfect calibration or smile-consistency.

(2) The empirically observed stylized facts from the market time series, i.e.characteristic features of the joint dynamics of the stock and options, can beincorporated in these models. This implies that the explicit expressions forthe option price processes and their dynamics should be available.

After a thorough study of literature concerning the option pricing, one caneasily categorize all approaches into two basic classes. The overwhelming ma-jority of the literature uses the martingale approach, where one specifies thedynamics of the underlying S under some pricing (i.e. martingale) measureQ and defines the option prices Ct(K,T ) by

Ct(K,T ) := EQ[(ST −K)+|Ft],

3

4 Chapter 2. The Literature Review

where Ft is a sub-σ-field of F and (Ω,F , P ) is a stochastic basis.

Obviously, the assumption (0) holds, and for so-called smile-consistent mod-els, a perfect fit as described in the assumption (1) is available. However, thisapproach is still not perfect in the sense that the assumption (2) is usuallynot feasible, or comes with a loss with assumption (1) to some extent.

An alternative approach is the use of market models where one specifies thejoint dynamics of all tradable assets, stock and options, for instance. Thisgives (1) and (2) instantly by construction, but the absence of the arbitrage-free requirement in the assumption (0) still remains to be satisfied. As ex-plained in paragraph 2.2.1, the absence of the dynamic arbitrage correspondsto drift conditions for the joint dynamics of the stock S and the option pricesC(K,T ), and additionally the absence of the static arbitrage enforces a num-ber of relations between the various C(K,T ) and S, which means the statespace is constrained as well.

Thus, to obtain such a tractable model, one must therefore reparameterizethe tradables in such a way that the parameterizing processes have a simplestate space and yet still capture all the static arbitrage constraints.

Generally speaking, most of the literature with actual results on the arbitrage-free market models for option prices, can be summarized in terms of thefamilies K and T . For the case K = K, T = T of one single call optionavailable for trade, there are both an existence result and some explicit ex-amples for models. For models with K = K, T = (0,∞) (one fixed strike,all maturities), the drift restrictions are well known, but the existence of themodels has been proved only very recently. The other extreme K = (0,∞),T = T (all strikes, one fixed maturity) is the focus of the Schweizer–Wisselmodel. It is more difficult and has (to the best of our knowledge) no pre-cursors in the terms of the parameterization or results. Finally, the caseK = (0,∞), T = (0,∞) of the full surface of strikes and maturities is stillopen despite the recent work by Carmona and Nadtochiy [1].

2.1 The Martingale models

The overwhleming majority of the models for the option pricing uses themartingale approach, where the dynamics of the underlying asset S undersome martingale measure Q and option prices are specified by

Arbitrage-free market models for IRO and future options 5

Ct(K,T ) := EQ[(ST −K)+

∣∣Ft], 0 ≤ t ≤ T,

for K ∈ K ⊆ (0,∞) , and T ∈ T ⊆ (0,∞) . These models satisfy the as-sumption (0) by construction. The calibration to the known market optionprices in the assumption (1) is admissable in the stochastic volatility modelsor in models with jumps. But the calibration is restricted by the requirementthat there only exists a finite amount of parameters to be fitted. Most of thefamous models which achieved such a perfect fit for the whole option surfaceC0(K,T ) for K = (0,∞) , T = (0,∞) are the smile-consistent models, forinstance, the Local Volatility Model of Dupire [2], and other local volatil-ity models, like the model developed by Carmona and Nadtochiy [1]. Butthese models have one mutual fundamental defect that the joint dynamicsof the option prices C(K,T ) and the stock prices S are not available. As wementioned earlier, in the beginning of this chapter, an option pricing modelshould possess one of the fundamental properties such that empirically ob-served stylized facts from the market time series can be incorporated in themodel. This leads us to the requirements that explicit expressions for theoption price processes and their dynamics can be attained by some regularprocedures. Otherwise we will not be able to express drift restrictions byterms of the coefficients of the joint dynamics of the price and volatility’sevolutions.

2.1.1 Dupire’s model

Dupire’s model was a breakthrough in 1994 for the option pricing theory.This model is such a type that in it we can avoid the complex calculationslike in stochastic volatility models and difficulties by the fitting parametersto the current prices of options.

The main idea of Dupire’s model is the following that under risk-neutralitythere is a unique diffusion process which is consistent with stock prices’ dis-tributions. The corresponding coefficient σL(S, t) of the unique diffusionprocess is known as local volatility function which is also in consistence withthe current option prices. Indeed, the local volatility models do not repre-sent a separate class of models, the basic idea of them is mainly to simplifyassumptions so that practitioners can price exotic options consistently withthe known prices of vanilla options.

Dupire showed that there is a unique risk-neutral diffusion process whichgenerates distributions of the final spot prices ST for each time T conditional


on some starting spot price S0. The set of all European option prices isknown and it is possible to determine the functional form of the diffusionparameter (local volatility) of the unique risk-neutral diffusion process whichgenerates these prices. In general, the local volatility will be a function ofthe current stock price S0 and this stock price random process is describedby the following equation,

dS = µ (t) dt+ σ (S, t;S0) dZ. (2.1)

Now we demonstrate the derivation of Dupire’s equation lying underneath themodel. Suppose for the given maturity time T and the stock price S0, theset C(S0, K, T );K ∈ (0,∞) of undiscounted option prices with differentstrikes produces the risk-neutral density function ϕ of the final spot price STand is defined by the relationship

C(S0, K, T ) =

∫ ∞K

dStϕ (ST , T ;S0) (ST −K) . (2.2)

Differentiating (2.2) with respect to K gives

∂C

∂K= −

∫ ∞K

dStϕ(ST , T ;S0),

∂2C

∂K2= ϕ(K,T ;S0).

In particular, the pseudo probability densities ϕ(K,T ;S0) = ∂2C∂K2 must satisfy

the Fokker-Planck equation [5], and in the current situation it is consequentlythat

1

2

∂2

∂S2T

(σ2S2

Tϕ)− ∂

∂ST(µSTϕ) =

∂ϕ

∂T.

Now, differentiation (2.2) with respect to T gives

∂C

∂T=

∫ ∞K

dSt

(∂

∂Tϕ(ST , T ;S0)

)(ST −K)

=

∫ ∞K

dSt

(1

2

∂2

∂S2T

(σ2S2

Tϕ)− S ∂

∂ST(µSTϕ)

)(ST −K).

Integrating it by parts twice gives


∂C

∂T=σ2K2

2ϕ+

∫ ∞K

dStµSTϕ =σ2K2

2

∂2C

∂K2+ µ(T )

(C −K ∂C

∂K

). (2.3)

Equation (2.3) is the Dupire equation if the underlying stock has a risk-neutral drift µ(T ). If µ(T ) = r(T ) − D(T ), where r(t) is the risk-free rate,D(t) is the dividend yield and C is the short notation for C(S0, K, T ), thenequation (2.3) acquires a form

∂C

∂T=σ2K2

2

∂2C

∂K2+ (r(T )−D(T ))

(C −K ∂C

∂K

). (2.4)

The forward price of the current stock at time T is given by FT = S0 ×exp(∫ T

0µ(t)dt

)2such that one get the same expression of (2.3) minus the

drift term∂C

∂T=σ2K2

2

∂2C

∂K2,

where C now represents C(FT , K, T ). Inverting this formula gives

σ2(K,T, S0) =∂C∂T

12K2 ∂2C

∂K2

. (2.5)

The right hand side of equation (2.5) can be computed directly from theknown European option prices. This means that the local volatilities definedby equation (2.3) are determined uniquely, meanwhile equation (2.5) canbe represented as a definition of the local volatility function under Dupire’smodel, regardless what kind of the process actually governs the evolution ofthe volatilities.

2.2 The market models

An alternative approach for the options pricing uses the market models, inwhich one can define the arbitrage-free joint dynamics of the stock prices andthe option prices. The market models always satisfy assumptions (1) and (2)by construction, but one other problem remains to be solved is how to showthe absence of arbitrage, i.e. the model has to admit the assumption (0). Inthe interest rate models, it leads to the HJM drift conditions, but in the case


of the options it is much more complicated to ensure the absence of arbitrage.One has to ensure that the option price models have to satisfy not only theHJM drift conditions, but also the static arbitrage bound restrictions on thestate space of the quantities which describe the model. In this sense, theselection of an appropriate parameterization becomes a key issue. Indeed,the absence of the dynamic arbitrage is connected with the drift conditionson the joint dynamics of S and C(K,T ). However, the absence of the staticarbitrage also ensures a number of relations between the various C(K,T )and S. It follows that the state space of the processes C(K,T ) and S isconstrained also. To obtain a well-constructed model, the reparameterizationshould be done in such a way that after parameterizing processes, one get asimple state space which captures all the static arbitrage constraints, as wediscussed above.

Study of some special cases has been done before and can be easily found inthe literature. If the option set comprises only a single call C = C(K,T ) , onehas the static arbitrage bounds (St −K)+ ≤ Ct ≤ St as well as the terminalcondition CT = (ST −K). To define S and C dynamics directly which admitthese constraints on the state space is not an easy procedure. It is muchmore easier to reparameterize the option price C, using the implied volatilityσ via Ct = c(St, K, (T − t)σ2

t ) where c is the Black-Scholes formula definedby (2.6). Then the values of S and σ can take any value in (0,∞). Afterthis procedure, the static arbitrage conditions and the terminal condition arefulfilled. It means that one can proceed to define the joint dynamics of (S, σ).Such market models of implied volatilities for the pricing a single option wasfirstly proposed by Schonbucher [7].

Even in the case of one single option (as simple as it is) the construction ofsuch model is still not completely straightforward: the drifts are essentiallyspecified by the volatilities of S and σ. Additionally, if we take these nonlin-ear drift restrictions, the resulting two-dimensional SDE system for S and σadmits a non-trivial solution.

For the option set consists of not only one single call, the situation becomesmuch more intricate. Indeed, for the resulting model to be an arbitrage-free,several versions of the necessary conditions on the implied volatility dynamicsare described in the literature (one of them was introduced by Schonbucher).But these models didn’t incorporate the case of options with multi-strikesor multi-maturities. It means that the sufficient conditions are not given, inother words, the existence of such models with defined dynamics is an openquestion. The main problem is that in the case of options with differentstrikes and maturities, the realization of the static no-arbitrage conditions has


an unpleasant consequence that quite complicated relations will arise amongthe implied volatilities of options. The illustration of this situation is givenin [10], Chapter 3 in details. The main idea we are trying to point out here isthe existence of a crucial problem that the classical implied volatilities are notappropriate for the modeling call option prices in a multi-strike case or in amulti-maturity case, and they give the wrong parameterization, despite theirimportance in quoting option prices on the market in standard way.

The idea of replacing the implied volatilities by other quantities (such as pa-rameterization of all call option prices in the market models) is not quitenew. In the case of the family with one fixed strike and all maturitiesT > 0, (K = K , T = (0,∞)), Schobucher has specified the forward im-plied volatilities. But in the case of the other extreme of the spectrum, i.e.the case of a family of options with one fixed maturity and all strikes T > 0,it is much more complicated to reparameterize the model because the assetsin the interest rate market models don’t have a “strike structure”.

The breakthrough in this direction was done by Schweizer and Wissel [5].They considered one extreme case K = (0,∞), T = T and constructedarbitrage-free market models for the set of call options, which have one fixedmaturity T and all strikes K > 0. Schweizer and Wissel introduced a newparameterization by specifying the new quantities called local implied volatil-ities of the option prices for the multi-strike case, so that the arbitrage-freedynamic modeling will actually be easier to perform. These quantities haveno comparable predecessors or analogues in the interest rate theory. Thecrucial feature of these parameters is that they have a simple state space andcapture all static arbitrage restrictions. In spite of that Schweizer and Wisselprovided a new parametrization and defined an arbitrage-free dynamic of thelocal implied volatility, but the dynamic arbitrage conditions and existenceresults for the dynamic option models still need to be studied.

2.2.1 The Schweizer-Wissel model

The classical implied volatilities models suffer from one disadvantage. Forthe market models with more than one strike, one stock, several call optionsand squared implied volatilities satisfying the arbitrage-free conditions, it isnot clear how to choose the volatility coefficients in the diffusion processesof the stock price and squared implied volatilities in order to determine thevalue of the drift terms, which implies the absence of arbitrage.

The Schweizer-Wissel model uses market models where the stock and option


price processes are constructed simultaneously, that is why this model fulfillsthe assumption (1) about the perfect calibration and the assumption (2) thatimplies that this model has at the same time joint dynamics for the stock andoptions prices. The main idea of the Schweizer-Wissel model is the choiceof a good parameterization, which should be done in such a way that thestatic arbitrage restrictions do result in a not complicated state space for thequantities which describe the model. Note also that the option prices are notautomatically conditional expectations, as in the martingale approach, andthe absence of the dynamic arbitrage now transfers into the drift conditionson the new modeled quantities to guarantee the local martingale property.Thus Schweizer and Wissel proposed the new concept of the local impliedvolatilities and price levels to overcome difficulty described above. These newquantities can be used to construct, and prove the existence of the arbitrage-free multi-strike market models with the specified volatilities for the optionprices (See Schweizer and Wissel [10], Chapter 4.3 and Chapter 5).

In this chapter we make a brief review of the Schweizer-Wissel work and intro-duce the main definitions and properties underlying the model and describein details the new parametrizaton which provides the joint arbitrage-freedynamics of the stock and option prices.

The new parameterization: definitions and basic properties

Throughout this paragraph, we work with the following setup. Let (Ω,F , P )be a probability space and T > 0 a fixed maturity. Let (St)0≤t≤T be apositive process modeling the (discounted) stock price and (Bt)0≤t≤T be apositive process with BT ≡ 1, P − a.s., modeling the (discounted) price ofa zero-coupon bond with maturity T . Let (Ct(K))0≤t≤T be a non-negativeprocess modeling the price of the European call options on S with one fixedmaturity T > 0 and all strikes K > 0. Also, let c(S,K, γ) be the Black-Scholes function

c(S,K, γ) = SN

(log S

K+ 1

2γ

γ12

)−KN

(log S

K− 1

2γ

γ12

)(γ > 0), (2.6)

c(S,K, 0) = (S −K)+, (2.7)

where N(·) is the standard normal distribution function. The value Ct(K)satisfies


(St −KBt)+ ≤ Ct(K) ≤ St, for all 0 ≤ t ≤ T. (2.8)

According to (2.6)-(2.8), the implied volatility of the price Ct(K) is theunique parameter σt(K) ≥ 0 satisfying the condition

c(St, KBt, (T − t)σ2t (K)) = Ct(K). (2.9)

Equation (2.9) can be rewritten, for every positive numeraire process Mas

c(St/Mt, KBt/Mt, (T − t)σ2t (K)) = Ct(K)/Mt.

We would like to point out that the bond B always uses from now on asa numeraire, and therefore, all price processes B, S, C(K) are denoted asB-discounted price processes, so that B ≡ 1.

By setting Ct(0) = St, the model is specified through the processes C(K), K ≥0 , on the interval [0, T ].Definition 1. A function Γ : [0,∞) → [0,∞) is called a price curve. Aprice curve is called statically arbitrage-free if it is convex and satisfies −1 ≤Γ′+(K) ≤ 0 for all K ≥ 0 .

This definition is motivated by the following proposition.Proposition 1. If the market (where K ≥ 0) does not admit an elementaryarbitrage opportunity, then for each t ∈ [0, T ), the price curve K 7→ Γ(K) :=Ct(K) is statically arbitrage-free.

Proof: See Davis and Hobson [13], Theorem 3.1.

2

Definition 2. An option model C(K), K ≥ 0 is called admissible if the pricecurve Γ(K) := Ct(K) has an absolutely continuous derivative with Γ

′′> 0

for all K > 0, −1 < Γ′(K) < 0 for all K > 0, and limK→∞Γ(K) = 0 for

each t ∈ [0, T ), and if we have CT (K) = (ST −K)+ for all K ≥ 0, P -a.s.

Let N−1 (·) denote the quantile function and n (·) = N′(·) the density func-

tion of the standard normal distribution.

The first and second partial derivatives with respect to the strike of Black-Scholes function c(S,K, (T − t)σ2) are given by

cK(S,K, (T − t)σ2) = −N(d2),


cKK(S,K, (T − t)σ2) = n(d2)1

K

1

σ√T − t

,

with d2 =log( SK )−(T−t)σ2/2

√T−tσ .

Hence the identity is

σ =n(N−1(−cK(S,K, (T − t)σ2)))

KcKK(S,K, (T − t)σ2)√T − t

.

Now a new set of fundamental quantities, which allow a straightforwardparameterization of the admissible option model, can be defined in suchform:Definition 3. Let (Ct(K))0≤t≤T be admissible. The local implied volatilityof the price curve at time t ∈ [0, T ) is the measurable function K 7→ Xt(K)given by

Xt (K) :=1√

T − tKC ′′t (K)

(N−1

(−C ′t (K)

))for a.e. K > 0, (2.10)

and the price level of the price curve at time t ∈ [0, t) for a fixed K0 ∈ (0,∞)is defined by

Yt :=√T − tN−1

(−C ′t (K0)

). (2.11)

Proposition 2. Let X(K) and Y be the local implied volatilities and theprice level of an admissible model C(K), K ≥ 0. Suppose that, for a smallinterval I = [a, b] ⊆ (0,∞) and a fixed t < T , we have Xt(K) = Xt(a) for allK ∈ I. Then there exists a unique pair (xt, zt) ∈ (0,∞)× (0,∞) such that

c(zt, K1, (T − t)x2t )− c(zt, K2, (T − t)x2t ) = Ct(K1)− Ct(K2)

holds for all K1, K2 ∈ I. It is given by

xt = Xt(a),

zt = exp(Xt(a)Yt −Xt(a)

∫ aK0

dhXt(h)h

+ log a+ 12(T − t)Xt(a)2

).

Proof: See Schweizer and Wissel [10], Proposition 4.4.

2


This property justifies the “therminoligy” local implied volatility by demon-strating that there exists such a unique implied volatility parameter xt insidethe third argument (T − t)x2t of Black-Scholes formula for all call options,that the differences Ct(K1)− Ct(K2) of the call option prices are consistentwith strikes K1, K2 ∈ I = [a, b]. It means that, with the same impliedvolatility parameter xt and the same “implied stock price” zt, the result inProposition 2 holds only locally.

It remaims to give the exact form for the option prices under the new param-eters, using the one-to-one corresponding relation between the local impliedvolatilities and the price levels. It is shown in the following theorem.Theorem 1. Let X (K), Y be the local implied volatilities and the price levelof the admissible model C (K). Then

Ct(K) =

∫ ∞K

N

(Yt −

∫ kK0

dhXt(h)h√

T − t

)dk, K ∈ [0,∞), (2.12)

C′

t(K) = −N

(Yt −

∫ kK0

dhXt(h)h√

T − t

), K ∈ [0,∞), (2.13)

C′′

t (K) = −n

(Yt −

∫ kK0

dhXt(h)h√

T − t

)1

Xt (K)K√T − t

, K ∈ [0,∞). (2.14)

Conversely, for the continuous adapted processes X(K) > 0, Y on [0, T ] forwhich the right-hand side of (2.12) is finite P-a.s., define Ct(K) by (2.12).Then C(K), K ≥ 0 is an admissible model having the local implied volatilitiesX(K) and the price level Y .

Proof: See Schweizer and Wissel [10], Theorem 4.6.

2

We also present, for the finiteness of the integral in (2.12), a sufficient crite-rion which is a condition on X(K).

Proposition 3. If there exists K1 > 0 such that Xt(K) ≤√

12lnKT−t for a.e.

K ≥ K1 , then the outer integral in (2.12) is finite.

Proof: See Schweizer and Wissel [10], Proposition 4.7.

2


The arbitrage-free dynamics of the local implied volatilities

Since Schweizer and Wissel have introduced a new parametrization and de-fined the explicit formulas for the prices of options with the strikes K > 0and one maturity T , it is possible to derive an arbitrage-free dynamics ofthe local implied volatilities. To achieve this goal, we need to add someadditional assumptions.

Let W be an m-dimensional Brownian motion on (Ω,F , P ), F = (Ft)0≤t≤Tthe P -augmented filtration generated by W , and F = FT . Suppose thatthere is a positive process Xt(K) for a.e. K > 0, satisfying the conditions inProposition 3, and a real valued process Yt with P -dynamics

dXt(K) = ut(K)Xt(K)dt+ vt(K)Xt(K)dWt, 0 ≤ t ≤ T, (2.15)

dYt = βtdt+ γtdWt, 0 ≤ t ≤ T, (2.16)

where β, u(K) ∈ L1loc(R), and γ, v(K) ∈ L2

loc(Rm) for a.e. K > 0 and u, v

are uniformly bounded in ω, t, K such that the initial local implied volatilitycurve satisfies

∫ KK0

dhX0(h)2

< ∞ for all K > 0. The process Ct(K), K ≥ 0

is defined by Theorem 1, Xt(K) and Yt are the local implied volatilities andthe price level of the option prices Ct(K), K ≥ 0. Note that St = Ct(0) andthe values Xt(0) for defining Ct(0) via (2.12) are not necessary. Now the aimis to show the existence of a common equivalent local martingale measurefor the set of the option prices C(K) for K ∈ (0,∞).Theorem 2. (a) If there exists a common equivalent local martingale mea-sure Q for all C(K) (K ≥ 0), then there exists a market price of the riskprocess b ∈ L2

loc(Rm) such that the drift restrictions (2.17), (2.18) (for a.e.

K > 0) hold for a.e. t ∈ [0, T ], P -a.s.

βt =1

2

YtT − t

(|γt|2 − 1

)− γtbt, (2.17)

ut =1

T − t

[1

2− 1

2

∣∣∣∣γt +

∫ K

K0

vt(h)

Xt(h)hdh

∣∣∣∣2+

(Yt −

∫ K

K0

dh

Xt(h)h

)(γt +

∫ K

K0

vt(h)

Xt(h)hdh

)vt(K)

]+ |vt(K)|2 − vt(K)bt, (2.18)


(b) Conversely, suppose that the coefficients β, γ, u(K) and v(K) satisfy,as functions of Yt and Xt(K), relations (2.17), (2.18) (for a.e. K > 0)for a.e. t ∈ [0, T ], P -a.s. for some bounded (uniformly bounded in t, ω)process b ∈ L2

loc(Rm). Also suppose that there exists a family of the continuous

adapted processes X(K) > 0, Y satisfying the system (2.15) (for a.e. K > 0)and (2.16). Then there exists a common equivalent local martingale measureQ on FT for C(K), (K ≥ 0). One such measure is given by

dQ

dP:= E

(∫bdW

)T

, (2.19)

where E is the stochastic exponential.

(c) In the situation of (a) or (b), the dynamics of C(K) under Q are givenby

dCt(K) =

∫ ∞K

n

(Yt −

∫ kK0

dhXt(h)h√

T − t

)1√T − t

×(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dkdWt,

(2.20)

for K ≥ 0 and a Q-Brownian motion W = W −∫bsds.

Proof: See Schweizer and Wissel [10], Theorem 4.12.

2

Note that the free input parameters are the market price of the risk processb and the volatilities of the state variables Y and X(K), such as γ and thefamily of the processes v(K) for all K. Since St = Ct(0), then the volatilityσt of the stock price process

dSt = σt · St · dWt

can be easily derived from (2.12) and (2.20) as

σt =

∫ ∞0

n

(Yt −

∫ kK0

dhXt(h)h√

T − t

)1√T − t

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk

×

(∫ ∞0

N

(Yt −

∫ kK0

dhXt(h)h√

T − t

)dk

)−1.


This implies that if γ or vt(K) are random, then for the stock price S can beobtained a model with a certain (quite specific) stochastic volatility and itimplies also that a class of the arbitrage-free local implied volatility modelscan be constructed like Schweizer and Wissel did in [10], Chapter 5.

2.3 Conclusions and comments

In this chapter we make a parallel comparison between two models describedabove.

The discrepancies between Dupire’s model and Schweizer-Wissel’s model andthe motivations for the development of the latter one will become more clearerafter a slight review of the multiple sources of research including the one doneby Carmona and Nadtochiy [1], which is one of the most successful studiesin this local implied volatilities’ direction.

The main attribute of the Dupire’s model is the abibility to be suitablefor any initial option price surface which fulfills the static arbitrage bounds.Despite this significant property, the fact that it is a one-factor model reducesthe usage of itself, because of consideration of one-factor model can notincorporate multiple sources of randomness into stochastic processes. Thus,it leads to the unrealistic price dynamics which is ensured by the absenceof the possibility to recalibrate the option price surface since all sources ofuncertainty are reduced to the one comes from the stock price evolutionS(t), which is driven by a one-dimensional fractional Brownian motion asdemonstrated in (2.1). To overcome this drawback Carmona and Nadtochiy[1] tried to incorporate additional stochastic factors into a local volatilitymodel. In the initial setup for their model, they also changed coordinatesfrom strike price to the log-strike price and assumed that the dynamics is

dS(t) = S(t)σ(t)dW 1(t),

dα2x,τ (t) = (αx,τ (t)dt+ βx,τ (t)dW (t)) ,

for a multi-dimensional Brownian motion W under the risk-neutral measure,where αx,τ (t) = σexp(x),τ+t(t). According to the Carmona and Nadtochiy[1],the local volatility surface is given by

σK,T (t) :=

√2

K2

∂TCK,T (t)

∂KKCK,T (t). (2.21)


This expression coincides with the expression (2.5) of the local volatility inthe Dupire’s model and it is expected that this local volatility surface in prin-ciple can be a market-observable quantity. Under the regularity conditionson σK,T (t), the unique classical solution of the PDE (2.21) for all K > 0 andT > t with the initial condition CK,T (t) = (S(t)−K)+ for all K > 0 andT = t are the option prices CK,T (t).

As for Schweizer-Wissel’s model, the absence of the dynamic arbitrage in thismodel leads to the drift restrictions on αx,τ (Therem 1 in [1]).

Although Carmona and Nadtochiy did eliminate the vulnerabilities of Dupire’smodel regarding the limitation of the usage under a one-dimensional frac-tional Brownian motion model, they couldn’t express the drift restrictionsindependently from the volatility coefficients of the option prices because ofthe option prices CK,T are not given explicitly.

Moreover, in contrast to the local volatility model, the Schweizer-Wissel’smodel has an explicit form of solutions for the option prices (2.12) in theterms of the new parameters X(K) and Y .

So, in conclusion, it is obviously true that Schweizer-Wissel’s model has agreat advantage over all other models because without loss in other proper-ties, it represents the drift restrictions in a closed form as shown in (2.17)and (2.18) and also allows to provide the joint dynamics for the option pricesand the stock prices.


Chapter 3

The extensions of theSchweizer-Wissel model

In our work, we studied both Dupire’s and Schweizer-Wissel’s model. Theyare two different models for a stock S and a set of the European call optionswith all strikes K > 0 and one fixed maturity T . In the previous chapter, wehave already shown the main differences between them, made some commentson the main ideas, analyzed their advantages and disadvantages. We verifiedthat the Schweizer-Wissel model admits all basic assumptions (0)-(2), whichshould be satisfied by a well-constructed option pricing model. Now letus explore the possible extensions of Schweizer-Wissel’s model applicable toother types of options.

In this chapter we provide an extension of this original model to the cases ofan option on interest rate indexes and an option on futures.

As the Schweizer-Wissel approach based on the introduction of a new param-eterization allows us to provide an explicit expression for the option prices,so, consequently, our new extensions of this model will be based on this ideaof parameterization too.

In Chapter 3.1, we discuss an option on interest rate indexes and proceedto introduce the basic definitions and properties using the concept of a newparameterization, and finally derive the arbitrage-free dynamics of the localimplied volatilities. In Chapter 3.2, we follow the similar procedure in thecase of an option on futures.

19

20 Chapter 3. The extensions of the Schweizer-Wissel model

3.1 An option on interest rate indexes

Options on interest indexes, for example, options on the 3-month LIBORindexes, can be regarded as options on interest rate futures. To be exact,an interest rate cap is a derivative in which the buyer receives payments atthe end of each period if the interest rate exceeds the agreed strike price.An example of a cap would be an agreement to receive a payment for eachmonth the LIBOR exceeds 2.5 percent.

The interest rate cap can be analyzed as a series of European call options orcaplets, which exist for each period of the cap agreement.

A cap payoff on the rate L struck at K is N · α · max(L − K, 0), where αstands for the day count fraction, and N is the principal for entering suchoption contracts. An interest rate floor is a series of European put optionson the floorlets on a specified reference rate, usually LIBOR. The buyer ofthe interest rate floor will receive an amount of payments if the interestrate falls below the agreed strike price. An example of a floor would be anagreement to receive a payment for each month in which the LIBOR fallsbelow 2.5 percent. Correspondingly, the floor payoff on the rate L struck atK is N · α ·max(K − L, 0), where α stands for the day count fraction, andN will be the principal for entering such option contracts.

Now, let’s consider the statistic behavior of the interest rate r. It’s reasonableto trust that the interest rate changes over time, and at the time interest rateis affected by multiple macroeconomic factors, such as an inflation rate, themonetary reasons, the international forces and so on. Thus we suppose thestochastic behavior of the interest rate is given by

dr = σdW + µdt,

where σ stands for the volatility of the interest rate, and µ stands for the driftof the indexes, for instance, the drift of 3-month LIBOR, or U.S. 3-monthtreasury bill indexes, and Wt would be the Brownian motion which affectsthe interest rate indexes stochastically.

Since options on the 3-month LIBOR or the 3-month treasury bill rate canbe regarded as options on the interest rate futures, which is a ”zero-cost”security. It follows that the modified Black-Scholes equation for the so-called“zero-cost” security can be expressed in the following way

∂V

∂t+

1

2

∂2V

∂S2σ2S2 − rV = 0,


where V = V (S, t) is the price of the option, r is the risk-free rate, and S isthe price of the underlying asset. This equation was solved by Fischer Blackto give the price of options on ”zero-cost” securities.

For the call option C = C(F, t) with the interest rate indexes as an underlyingasset, its differential equation transforms to the following one

∂C

∂t+

1

2

∂2C

∂C2σ2F 2 − rC = 0,

where r is the risk-free rate and F is for the forward rate of the interest rateindexes. We introduce the discounted function Ft for the time point t, so,the discounted function for the maturity time T is FT . Accordingly, from thebasic definition meaning of the forward rate, at time point t, we get

F =1

αt,T

[FtFT− 1

],

and with the risk-neutrality taken into consideration, the dynamics of F is

dF = σdWt.

Thus, for a caplet, the solution given by Fischer Black becomes

C(t, F ) = FTαt,T [FN(d1)−KN(d2)],

where αt,T is simply a day-count fraction which transforms the number oftrading days into equivalent years, and, the parameters in the cumulativedistribution function for standard normal distribution are

d1 =1

σ√T − t

[log

(F

K

)+σ2(T − t)

2

],

d2 = d1 − σ√T − t.

After inserting the parameters above, and treating F , K and (T − t)σ2 asour new arguments like we introduced before in Chapter 2, we obtain thefollowing form of the value of the caplets

c(F,K, (T − t)σ2) = FTαt,T [FN(d1)−KN(d2)].

When K = 0 holds, naturally we have c(F, 0, (T − t)σ2) = FTαt,TF.


3.1.1 The new parameterization

In this section we provide the new parameterization of the local impliedvolatilities and the price level for the market model of the interest rate in-dexes options (as options) and the interest rate indexes forward rates (as aprice evolution), then introduce the underlying definitions and basic proper-ties for our model. This new parameterization allows us in the future researchto define an arbitrage-free joint dynamics of the option prices and the interestrate indexes. Our method for derivation of these new parameters is based onSchweizer-Wissel’s approach, and instead of the treating stock as an under-lying asset we take the interest rate indexes and the set of the European calloptions on the interest rate indexes to replace the set of vanilla Europeancall options.

Throughout this chapter, we work with the following setup.

Let (Ω,F , P ) be a probability space and T > 0 be a fixed maturity. Let(Ft)0≤t≤T be a positive process modeling the forward rate of the interest

rate indexes F and (Bt)0≤t≤T a positive process, with Bt = B0e−∫ Tt r(τ)dτ ,

B0 = 1, modeling the (discounted) bond price evolution. For K > 0, let(Ct(K))0≤t≤T be a nonnegative process modeling the price of the Europeancall options on the interest rate indexes F with one fixed maturity T > 0and all strikes K > 0. By setting Ct(0) = FTαt,TF , the model is specifiedthrough the processes (Ct(K))0≤t≤T , on the interval [0, T ] and the optionmodel Ct(K)0≤t≤T is admissible and statically arbitrage-free.

Now we can introduce a new set of the fundamental quantities which allowsus to provide a straightforward parameterization of the admissible optionmodel. Let N−1 (·) denote the quantile function and n (·) = N

′(·) denote

the density function of the standard normal distribution.

The first and the second partial derivatives with respect to the strike priceof the Black-Scholes formula c(F,K, (T − t)σ2) are given by

cK(F,K, (T − t)σ2) = −αt,TFTN(d2),

cKK(F,K, (T − t)σ2) = αt,TFTn(d2)1

K

1

σ√T − t

,

where d1 = 1σ√T−t

[log(FK

)− σ2(T−t)

2

], F = 1

αt,T

(FtFT− 1). So, we can easily

get the identity


σ =αt,TFTn(N−1(− 1

αt,TFTcK(F,K, (T − t)σ2)))

KcKK(F,K, (T − t)σ2)√T − t

.

Now we define the new parameters.Definition 4. Let (Ct(K))0≤t≤T be admissible. The local implied volatilityof the price curve at time t ∈ [0, T ) is the measurable function K 7→ Xt(K)given by

Xt (K) :=1√


(N−1

(− 1

αt,TFTC′

t (K)

))for a.e. K > 0,

(3.1)and the price level of the price curve at time t ∈ [0, t) for a fixed constantK0 ∈ (0,∞) is defined by

Yt :=√T − tN−1

(− 1

αt,TFTC′

t (K0)

). (3.2)

The terminology the “local implied volatillity” in the case of an option onthe interest rate indexes is justified by the following results.Proposition 4. Let X(K) and Y be the local implied volatilities and theprice level of an admissible model C(K), K ≥ 0. Suppose that, for a smallinterval I = [a, b] ⊆ (0,∞) and fixed t < T , we have Xt(K) = Xt(a) for allK ∈ I. Then there exists a unique pair (xt, zt) ∈ (0,∞)× (0,∞) such that

c(zt, K1, (T − t)x2t )− c(zt, K2, (T − t)x2t ) = Ct(K1)− Ct(K2) (3.3)


xt = αt,TFTXt(a),


∫ aK0

dhXt(h)h

+ log a+ 12(T − t) (αt,TFTXt(a))2

).

Proof: Proof is analogous to the proof of the Proposition 8 in paragraph3.2.1, for the extension of the Schweizer-Wissel model with the future con-tracts as its underlying asset.

2

By this proposition, there exists such an unique implied volatility parameterxt in the Black-Scholes formula for all call options on the interest rate indexes,


that the prices’ difference Ct(K1)−Ct(K2) is consistent with the strike pricesK1, K2 ∈ I = [a, b]. It means that Proposition 4 holds locally only.

Thus we find the exact form for the option prices under the new parame-ters, using the one-to-one corresponding relation between the local impliedvolatilities and the price level, as it is shown by the following.Theorem 3. Let X (K), Y be the local implied volatilities and the price levelof the admissible model C (K). Then

Ct(K) =

∫ ∞K

αt,TFTN

(Yt −

∫ kK0

dhXt(h)h

αt,TFT√T − t

)dk, K ∈ (0,∞) , (3.4)

C′

t(K) = −αt,TFTN

(Yt −

∫ kK0

dhXt(h)h

αt,TFT√T − t

), K ∈ (0,∞) , (3.5)

C′′

t (K) = −n

(Yt −

∫ kK0

dhXt(h)h

αt,TFT√T − t

)× 1

Xt (K)Kαt,TFT√T − t

, K ∈ (0,∞) . (3.6)

Conversely, for the continuous adapted processes X(K) > 0, Y on [0, T ]for which the right-hand side of (3.4) is finite P-a.s., define Ct(K) by (3.4).Then C(K), K ≥ 0 is an admissible model having the local implied volatilitiesX(K) and the price level Y .

Proof: The local implied volatility and the price level in this case are definedby formulas (3.1) and (3.2) correspondingly. So, we have to find an explicitform of Ct(K), C

′t(K), C

′′t (K) expressed by n(·), N(·), Xt(K) and Yt.

It is reasonable to assume that the option value Ct(K) has the following form

Ct(K) =

∫ +∞

K

f(k)dk,

where f(k) is a differentiable function, and for the contingent claim Ct(K)has the form

Ct(K) =

∫ +∞

−∞g(k)dk,


where g(k) can be regarded as a regular density function,

g(k) =

f(k), k ≥ K,

0,−k < K.

Thus from (3.2), we get a so-called initial condition

C′

t(K)|K=K0 = −αt,TFTN(

Yt√T − t

)= f(K0). (3.7)

From (3.1) we get the following expression for Xt(K)

Xt(K) = − 1√T − tKf ′(K)

n

(N−1

(1

αt,TFTf(K)

))

= − 1√T − tKf ′(K)

n

(N−1

(1

αt,TFT

∫ K

−∞f′(k)dk

)).

Using this expression we obtain

N

(N−1

(1

αt,TFT

∫ K

−∞f′(k)dk

))=

1

αt,TFT

∫ K

−∞f′(k)dk =

1

αt,TFTf(K).

So if we denote N−1(

1αt,TFT

∫ K−∞ f

′(k)dk

)as Φ(K), we get the following

deduction.

Because

dN(Φ(K))

dK= n(Φ(K))

dΦ(K)

dK,

anddN(Φ(K))

dK=

1

αt,TFT

df(K)

dK=

1

αt,TFTf′(K),

then it follows that

1

αt,TFTf′(K) = n

(N−1

(1

αt,TFT

∫ K

−∞f′(k)dk

)) d(N−1

(1

αt,TFT

∫ K−∞ f

′(k)dk

))dK

.


It means that

1αt,TFT

f′(K)

n(N−1

(1

αt,TFT

∫ K−∞ f

′(k)dk)) =

d(N−1

(1

αt,TFT

∫ K−∞ f

′(k)dk

))dK

,

and we have that

f′(K)

n(N−1

(1

αt,TFT

∫ K−∞ f

′(k)dk)) = − 1√

T − tXt(K)K.

According to these formulas we obtain

d(N−1

(1

αt,TFT

∫ K−∞ f

′(k)dk

))dK

= − 1√T − tXt(K)K

1

αt,TFT,

from which we have that

d

(N−1

(1

αt,TFT

∫ K

−∞f′(k)dk

))= −

1Xt(K)K

αt,TFT√T − t

dK,

then consequently

N−1(

1

αt,TFT

∫ K

−∞f′(k)dk

)= −

∫ 1Xt(K)K

αt,TFT√T − t

dK. (3.8)

Adding (3.8) to the initial condition (3.7), we obtain

−αt,TFTN(−∫

dhαt,T FTXt(h)h√

T−t

)= −

∫ K−∞ f

′(k)dk = −f(k) = C

′t(K),

−αt,TFTN(

Ytαt,TFT

√T−t

)= −f(K0) = C

′t(K)|K=K0 = C

′t(K0).

By solving this system we obtain the unique solution of the system above

C′

t(K) = −αt,TFTN

(Yt −

∫ KK0

dhXt(h)h

αt,TFT√T − t

).

After differentiation and integration of C′t(K) with respect to K, we also get

C′′

t = −n

(Yt −

∫ KK0

dhXt(h)h

αt,TFT√T − t

)1

Xt(K)K√T − t

,


Ct(K) =

∫ ∞k

αt,TFTN

(Yt −

∫ KK0

dhXt(h)h

αt,TFT√T − t

)dk.

2

The integral in (3.4) is finite. This property is ensured by the followingproposition (the suffcient condition).


12lnKT−t for a.e.


Proof: Proof is analogous to the proof of Proposition 9 in paragraph 3.2.1,for the extension of the Schweizer-Wissel model with the future contracts asits underlying asset.

2

In order to present the images of the new parameters more precisely, we ex-press the local implied volatilities and the price level via the classical impliedvolatility σt(K).

Proposition 6. Define γt(K) = (T − t)σ2t (K) and d2 =

log(Ft/K)− 12γt√

γt(K). The

local implied volatility and the price level can be represented by followingexpressions

Yt =√T − tN−1

(αt,TFT

(N (d2(t,K0))

− 1

2n (d2(t,K0))

√γt(K0)

K0

σ2t (K0)

d

dKσ2t (K0)

)), (3.9)


Xt(K) = σt(K)n

(N−1

(αt,TFT

(N(d2(t,K))

− 1

2n (d2(t,K))

√γt(K)

K

σ2t (K)

d

dKσ2t (K)

)))× αt,TFTn (d2(t,K))

[1 +

(log

(FtK

)+

1

2γt(K)

)K

σ2t (K)

d

dKσ2t (K) +

1

4

(log2

(FtK

)− γt(K)

−1

4γ2t (K)

)·(

K

σ2t (K)

d

dKσ2t (K)

)2

+1

2γt(K)

K2

σ2t (K)

d2

dK2σ2t (K)

]−1. (3.10)

Proof: Proof is analogous to the proof of Proposition 10 in Chapter 3.2.1for the extension of the Schweizer-Wissel model with the future contracts asits underlying asset.

2

3.1.2 The arbitrage-free dynamics of the local impliedvolatilities

In this paragraph, we derive the joint dynamics of option’s price levels and thelocal implied volatilities, under the arbitrage-free condition. First of all, wedo some preparation work for the deduction, which is similar to the originalSchweizer-Wissel model. Let W be an m-dimensional Brownian motion onthe probability space (Ω,F , P ), and F = (Ft)0≤t≤T be the P -augmentedfiltration generated by W , and F = FT . Suppose that there exist thepositive processes Xt(K) for a.e. K > 0, satisfying the condition describedin Proposition 5, and a real valued process Yt with the P -dynamics

dXt(K) = ut(K)Xt(K)dt+ vt(K)Xt(K)dWt, 0 ≤ t ≤ T (3.11)

dYt = βtdt+ γtdWt, 0 ≤ t ≤ T (3.12)



loc(Rm) for a.e. K. Further, we

suppose u, v are uniformly bounded in ω, t, K and that the initial localimplied volatility curve satisfies∫ K

K0

dh

X0(h)2<∞,∀K > 0.

Then, we define the process Ct(K), K ≥ 0 by Theorem 3, Xt(K) andYt are the local implied volatilities and the price level of the option pricesCt(K), K ≥ 0, respectively. We notice that

F = Ct(0), so for defining Ct(0) by (3.4), the values of Xt(0) are not neces-sarily needed here.

After all preparations here, it becomes more clear that if we have some con-ditions or restrictions (in our case here, the drift restrictions) imposed on thejoint dynamics of the price level and the local implied volatilities, then thereexists a common equivalent martingale measure for C(K),∀K > 0. Our aimhere is to show the existence of such equivalence between the existence of acommon local martingale measure for C(K),∀K > 0 and the drift restric-tions of joint dynamics of the price levels and the local implied volatilities.We prove our arguments more precisely in the following theorem.Theorem 4. (a) If there exists a common equivalent local martingale mea-sure Q for all C(K) (K ≥ 0) then there exists a market price of the riskprocess b ∈ L2

loc (Rm) such that (3.13), (3.14) (for a.e. K > 0) hold for a.e.t ∈ [0, T ], P − a.s.

βt =1

2

YtT − t

(|γt|2

αt,T− 1

)−Yt ·

d(αt,T )dt

αt,T− bt · γt, (3.13)

ut(k) =1

T − t

1

2−

∣∣∣γt +∫ KK0

vthXt(h)·hdh

∣∣∣22(α2

t,T · F 2T )

+

(γt +

1

2

∫ K

K0

vt(h)

Xt(h) · hdh

)Yt −

∫ KK0

dhXt(h)·h

α2t,T · F 2

T

vt(K)

]

+ v2t (K)− vt(K) · bt +

dαt,Tdt

αt,T. (3.14)


(b) Conversely, suppose that the coefficients β, γ, u(K) and v(K) satisfy, asfunctions of Yt and Xt(K), relations (3.13), (3.14) (for a.e. K > 0) for a.e.t ∈ [0, T ], P−a.s. for some bounded (uniformly in t, ω) process b ∈ L2

loc(Rm).

Also suppose that there exists a family of the continuous adapted processesX(K) > 0, Y satisfying the system (3.11) (for a.e. K > 0) and (3.12).Then there exists a common equivalent local martingale measure Q on FTfor C(K), (K ≥ 0). One such measure is given by

dQ

dP:= E

(∫b′dW

)T

, (3.15)

where E is the stochastic exponential, and b′ is determined by (3.17).

(c) In the situation of (a) or (b), the dynamics of C(K) under Q are givenby

dCt(K) =

∫ ∞K

n

(Yt −

∫ kK0

dhXt(h)h√

T − t · αt,T · FT

)1√T − t

×(γt +

∫ k

K0

vt(h)dh

Xt(h)

)dk · dWt, (3.16)

for K ≥ 0 and a Q-Brownian motion W = W −∫b′sds, and

b′t = −dαt,Tdt

α2t,T · FT

·N

(Yt −

∫ kK0

dhXt(h)h√


)·

γt +∫ kK0

vt(h)dhXt(h)h

√T − t · n

(Yt−

∫ kK0

dhXt(h)h√

T−t·αt,T ·FT

) + bt.

(3.17)Remark 1. Note that the free input parameters are the market price of therisk process b and the volatilities of the state variables Y and X(K), suchas γ and the family of processes v(K) for all K. From the static boundsFTαt,TF = Ct(0) then the volatility σt of the forward rate process of theinterest rate indexes dF = σtdWt can be easily derived from (3.4) and (3.16)by

σt =

∫ ∞0

n

(Yt −

∫ kK0

dhXt(h)h√


)1√T − t

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk


×

[∫ ∞0

N

(Yt −

∫ kK0

dhXt(h)h√


)dk

]−1.

Given all these free input parameters, we get a model for the forward rateprocess F with an actual specific expression of the stochastic volatility.Proposition 7. Let Zt(K) := Yt −

∫ kK0

dhXt(h)h

. Under P the dynamics of

Ct(K) for each K ≥ 0 is given by

dCt(K) =

∫ ∞K

[n

(Zt(k)√


)1√


×[

1

2

Zt(k)

T − tαt,T · FT

1−

∣∣∣γt +∫ kK0

vt(h)Xt(h)h

dh∣∣∣2

α2t,T · F 2

T

+ αt,T · FT ·

√T − t

(βt −

∫ k

K0

v2t (h)− ut(h)

Xt(h)hdh

)+√T − t · dαt,T

dt

]

+

dαt,Tdt·∣∣∣γt +

∫ kK0

vt(h)Xt(h)

dh∣∣∣2

(T − t) · α2t,T · FT

·N(

Zt(k)√T − t · αt,T · FT

)]dk · dt

+

∫ ∞K

n(

Zt(k)√T−t·e−r(T−t)

)√T − t

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk · dWt. (3.18)

Proof: In order to apply the Ito lemma to the integrand inside the integralof equation (3.4) firstly we do the following.

Because of the continuity of Zt(k), we get

dZt(K) = dYt − d(∫ k

K0

dh

Xt(h) · h

),

then by using the Fubini theorem ([6], Theorem 4.65), this expression can besimplied to

dZt(K) = dYt −∫ k

K0

d

(1

Xt(h)

)dh

h. (3.19)

Next, we apply the Ito lemma to 1Xt(h)

with respect to the arguments Xt(h)

and t. We obtain the stochastic differential d(

1Xt(h)

), and after some calcu-


lations we obtain the following expression

d

(1

Xt(h)

)=

1

Xt(h)[(v2t (h)− ut(h))dt− vt(h)dWt]. (3.20)

After inserting equation (3.20) into (3.19), we obtain the following formula

dZt(k) =

(βt −

∫ k

K0

v2t (h)− ut(h)

Xt(h) · hdh

)dt

+

(γt +

∫ k

K0

vt(h)

Xt(h) · hdh

)dWt, (3.21)

d 〈Z(k)〉t =

∣∣∣∣γt +

∫ k

K0

vt(h)

Xt(h) · hdh

∣∣∣∣2 · dt. (3.22)

Then we define

Mt(k) := αt,T · FT ·N(


),

thus we have

Ct(K) =

∫ ∞K

Mt(k)dt,

and by the Fubini theorem

d (Ct(K)) = d

(∫ ∞K

Mt(k)dt

)=

∫ ∞K

dMt(k) · dt. (3.23)

Again, we apply the Ito lemma to

Mt(k) = F

(t,


),

and obtain the following expression


dMt(k) =

[dαt,Tdt

FT ·N(


)+

1

2αt,T · FT · n

(Zt(k)√


)·(− Zt(k)√


)]d

⟨Zt(k)√


⟩t

+ αt,T · FT · n(


)d

(Zt(k)√


). (3.24)

Finally, we insert equation (3.21) and (3.22) into equation (3.24). After astraight forward computation, obtain that

dMt(k) =

[n

(Zt(k)√


)1√


(1

2

Zt(k)

T − tαt,T · FT

·(

1−

∣∣∣γt +∫ kK0

vt(h)Xt(h)h

dh∣∣∣2

α2t,T · F 2

T

))+αt,T ·FT ·

√T − t

(βt −

∫ k

K0

v2t (h)− ut(h)

Xt(h)hdh

)

+√T − t · dαt,T

dt+

dαt,Tdt·∣∣∣γt +

∫ kK0

vt(h)Xt(h)

dh∣∣∣2

(T − t) · α2t,T · FT

N

(Zt(k)√


)]dt

+

∫ ∞K

n(


)√T − t

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dWt (3.25)

By using (3.23) we get exactly the same expression of dCt(K) as in (3.18).Thus proposition has been proved.

2

Now we can prove Theorem 4.

Proof: (a) Since the filtration F is generated by W , and by Ito’s represen-tation theorem we have that

E

[dQ

dP|Ft]

= E(∫

b′dW

)t

for some process b′ ∈ L2

loc(Rm), and


W: = W −∫b′

tdt

is a Q-Bownian motion by Girsanov’s theorem.

By Proposition 7, it yields that

dCt(K) =

∫ ∞K

n

(Zt(k)√

T − t · e−r(T−t)

)· 1√

T − t· µt(k)dk · dt

+

∫ ∞K

n

(Zt(k)√

T − te−r(T−t)

)· 1√

T − t

·(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk · dWt, (3.26)

where

µt(k) =Zt(k)

2(T − t)

1−

∣∣∣γt +∫ kK0

vt(h)Xt(h)·hdh

∣∣∣2α2t,T · F 2

T

+ βt −∫ k

K0

v2t (h)− ut(h)

Xt(h) · hdh

+

(γt +

∫ k

K0

vt(h)

Xt(h) · hdh

)· bt +

Zt(k)dt,Tαt,T

,

for k > 0.

We take the expectations on both sides of (3.26). Because of C(K) are localQ-martingales for all K > 0, by the Fubini theorem we have P -a.s., for a.e.t

µt(k) = 0, ∀k > 0. (3.27)

Let k tends to K0 in (3.27), we obtain an expression to βt, (3.13). Bydifferentiating µt(k) with respect to k, then taking the values in k = K, weobtain the expression of ut(K) (3.14).

(b) Define

dQ

dP:= E

(∫b′dW

)T


on FT ; then W := W−∫b′tdt is a Q-Brownian motion on [0, T ] by Girsanov’s

theorem. To verify this, we plug in (3.13), (3.14), as well as dWt = dWt+btdtinto Proposition 10. We obtain (c) under (b). Now it easily follows from (c)that C(K) for all K ≥ 0 are Q local martingales on [0, T ].

(c) Together with the proof of (b) above, we have already proved the asser-tation (c) under (b). Under the assumption (a), the assertation (c) followsfrom (3.26) and (3.27).

2

3.2 An option on Futures

In Finance, a futures contract is a standardized contract between two partiesto buy or sell a specified asset of the standardized quantity and quality at aspecified future date at a price agreed today (the forward price).

Associated by the future contract pricing model, Ft = Ster(T−t), where the

future contract’s price Ft and the spot price St are driven by the same randomfactor. Here r is the interest rate, t is the current time and T is the maturity.With the assumptions of logarithmic payoff function f(St) = log(St) and theabsence of the dynamic arbitrage, the random walk of the stock spot pricecan be represented by the following:

dSt = µtStdt+ σtStdWt,

where Wt is an m-dimensional Brownian motion. Initially, we have S0 = s0.Thus, with some transformations and calculations, one can easily get thatthe stochastic differential equation of the future contract price is

dFt = (µt − r)Ftdt+ σtFtdWt,

where µ is the drift of the stochastic process and σ is the volatility of the stockprice. Under the risk-neutrality assumption, we have the random process ofthe future price simplified to

dFt = σtFtdWt.

Correspondingly, we obtain the adapted Black-Scholes equation for the calloption on the futures C = C(F, t) such that

∂C

∂t+

1

2

∂2C

∂F 2σ2F 2 − rC = 0.


Thus, the solution of this partial differential equation becomes

C(F, t) = e−r(T−t) (FN (d1)−KN (d2)) ,

where parameters in the cumulative distribution function for the standardnormal distribution are

d1,2 =log(FK

)± 1

2(T − t)σ2

σ√

(T − t).

After treating F , K and (T − t)σ2 as our new arguments similar to theprocedure which we introduced in Chapter 2, we obtain the following formof the value of the call options

c(F,K, (T − t)σ2

)= e−r(T−t) (FN (d1)−KN (d2)) ,

and from the static bounds,

c(F,K, 0) = (F −K)+.

3.2.1 The new parameterization

In this paragraph, again, we first provide the reparameterization procedurefor the local implied volatilities and the price level of the market model foran option on futures and introduce the underlying definitions and the basicproperties. As in the previous chapter, our method for a derivation of thesenew parameters based on Schweizer-Wissel’s approach, but in this chapterwe work with the forward price of the future contracts as our underlyingasset and the set of European call options on futures.

Throughout this paragraph, we work with the following setup.

Let (Ω,F , P ) be a probability space and T > 0 be a fixed maturity. Let(Ft)0≤t≤T be a positive process modeling the price of the future contractsand (Bt)0≤t≤T with B0 = 1 be a discounted bond price process. For K > 0let (Ct(K))0≤t≤T be a non-negative process modeling the price evolution ofa European call option on the futures F with one fixed maturity T > 0 andall strikes K > 0. So the payoff at the time point T , is (FT − K)+, K ∈(0,∞). By setting Ct(0) = St, the model is specified through the process


Ct(K)0≤t≤T on the interval [0, T ] and this option model is admissible andstatically arbitrage-free.

Now we introduce a new set of the fundamental quantities which allows usto make a straightforward parameterization of the admissible option model.Let N−1 (·) denote the quantile function and n (·) = N

′(·) denote the density

function of the standard normal distribution.

The first and the second partial derivatives with respect to the strike priceof the Black-Scholes formula for c(F,K, (T − t)σ2) are given by

cK(F,K, (T − t)r, (T − t)σ2) = −e−r(T−t)N(d2),

cKK(F,K, (T − t)σ2) = e−r(T−t)n(d2)1

K

1

σ√T − t

with

d2 =log(FK

)− σ2(T−t)

2

σ√T − t

.

Thus we obtain the identity

σ =e−r(T−t)n

(N−1

(−er(T−t)cK (F,K, (T − t)σ2)

))KcKK (F,K, (T − t)σ2)

√T − t

. (3.28)

Now we can proceed to define the new parameters.Definition 5. Let (Ct(K))0≤t≤T be admissible. The local implied volatilityof the price curve at time t ∈ [0, T ) is the measurable function K 7→ Xt(K)given by

Xt (K) :=1√


(N−1

(−er(T−t)C ′t (K)

)), for a.e. K > 0,

(3.29)and the price level of the price curve at time t ∈ [0, t) for a fixed constantK0 ∈ (0,∞) is defined by

Yt :=√T − tN−1

(−er(T−t)C ′t (K0)

), (3.30)

To justify the terminology ”the local implied volatility” in this case for anoption on futures, we have the following supporting arguments.


Proposition 8. Let X(K) and Y be the local implied volatilities and theprice level of an admissible model C(K), K ≥ 0. Suppose that, for a smallinterval I = [a, b] ⊆ (0,∞) and fixed t < T , we have Xt(K) = Xt(a) for allK ∈ I. Then there exist a unique pair (xt, zt) ∈ (0,∞)× (0,∞) such that

c(zt, K1, (T − t)x2t )− c(zt, K2, (T − t)x2t ) = Ct(K1)− Ct(K2) (3.31)


xt = e−r(T−t)Xt(a),


∫ aK0

dhXt(h)h

+ log a+ 12(T − t)e−2r(T−t)Xt(a)2

).

The proof of the theorem (5) is included at the end of this section. By thisproposition, there exists such a unique implied volatility parameter xt in thethird argument ((T − t)x2t ) of the Black-Scholes formula for all call optionson futures, that the prices’ difference Ct(K1)−Ct(K2) is consistent with thestrike prices K1, K2 ∈ I = [a, b]. It means that Proposition 8 merely holdslocally.

The exact form of the option price in the framework of the new parameterswhere we use the one-to-one corresponding relation between the local im-plied volatilities and the price level, the price of the option is defined by thefollowing theorem.Theorem 5. Let X (K), Y be the local implied volatilities and the price levelof the admissible model C (K). Then

Ct(K) =

∫ ∞K

e−r(T−t)N

(Yt −

∫ kK0

dhXt(h)h

e−r(T−t)√T − t

)dk, K ∈ (0,∞) , (3.32)

C′

t(K) = −e−r(T−t)N

(Yt −

∫ kK0

dhXt(h)h


), K ∈ (0,∞) , (3.33)

C′′

t (K) = −n

(Yt −

∫ kK0

dhXt(h)h


)× 1

Xt (K)Ke−r(T−t)√T − t

, K ∈ (0,∞) . (3.34)


Conversely, for the continuous adapted processes X(K) > 0, Y on [0, T ] forwhich the right-hand side of (3.32) is finite P-a.s., define Ct(K) by (3.32).Then C(K), K ≥ 0 is an admissible model having the local implied volatilitiesX(K) and the price level Y .

Proof: The local implied volatility and the price level are defined by formu-las (3.29) and (3.30) correspondingly. So, we have to find an explicit form ofCt(K), C

′t(K), C

′′t (K) expressed by n(·), N(·), Xt(K) and Yt.

It is reasonable to assume that the option value Ct(K) has the following form

Ct(K) =

∫ +∞

K

f(k)dk,

where f(k) is a differentiable function, and Ct(K) is the price of the contin-gent claim.

Thus from (3.30) we get a so-called initial condition that

C′

t(K)|K=K0 = −e−r(T−t)N(

Yt√T − t

)= f(K0). (3.35)

From (3.29), we get the following expression for Xt(K)

Xt(K) = − 1√T − tKf ′(K)

n(N−1(er(T−t)∫ K

−∞f′(k)dk)), (3.36)

since we know that

N

(N−1

(er(T−t)

∫ K

−∞f′(k)dk

))= er(T−t)

∫ K

−∞f′(k)dk = er(T−t)f(K).

So if we denote

N−1(er(T−t)

∫ K

−∞f′(k)dk

)as Φ(K), we will get the following deduction.

Because we have

dN(Φ(K))

dK= n(Φ(K))

dΦ(K)

dK


anddN(Φ(K))

dK= er(T−t)

df(K)

dK= er(T−t)f

′(K),

then it follows that

er(T−t)f′(K) = n

(N−1

(er(T−t)

∫ K

−∞f′(k)dk

)) d(N−1

(er(T−t)

∫ K−∞ f

′(k)dk

))dK

,

it means that

er(T−t)f′(K)

n(N−1

(er(T−t)

∫ K−∞ f

′(k)dk)) =

d(N−1

(er(T−t)

∫ K−∞ f

′(k)dk

))dK

. (3.37)

Considering the other identity (3.36), which can also be represented as thefollowing identity

f′(K)

n(N−1

(er(T−t)

∫ K−∞ f

′(k)dk)) = − 1√

T − tXt(K)K.

According to these two identities (3.36) and (3.37) we obtain

d(N−1

(er(T−t)

∫ K−∞ f

′(k)dk

))dK

= − er(T−t)√T − tXt(K)K

,

from which we have that

d

(N−1

(er(T−t)

∫ K

−∞f′(k)dk

))= −

1Xt(K)K


dK,

then consequently

N−1(er(T−t)

∫ K

−∞f′(k)dk

)= −

∫ 1Xt(K)K


dK. (3.38)

Adding (3.38) to the initial condition (3.35) that we got at the beginning,we obtain the following system

−e−r(T−t)N

(−∫

dhXt(h)h

e−r(T−t)√T−t

)= −

∫ K−∞ f

′(k)dk = −f(k) = C

′t(K),

−e−r(T−t)N(

Yter(T−t)

√T−t

)= −f(K0) = C

′t(K)|K=K0 = C

′t(K0).


We obtain the unique solution of the system above in the form

C′

t(K) = −e−r(T−t)N

(Yt −

∫ KK0

dhXt(h)h


).

After a proper differentiation and integration of C′t(K) with respect to K we

also get

C′′

t = −n

(Yt −

∫ KK0

dhXt(h)h


)1

Xt(K)K√T − t

,

and

Ct(K) =

∫ ∞k

e−r(T−t)N

(Yt −

∫ KK0

dhXt(h)h


)dk.

Thus, the theorem is proved.

2

The integral in (3.32) is finite, and this is ensured by the following propositionof suffcient condition.


12logKT−t for a.e.


Proof: Assume K1 > K0. For a sufficiently large k we have

∫ k

K0

dh

Xt(h)h≥∫ k

K1

dh

2h√

log h

√8(T − t),

Yt −∫ kK0

dhXt(h)h


≤ −√k√

8−√K1

√8

e−r(T−t)+

Yt


≤ −er(T−t)√

4 log k.

Note that the function N(·) is the cumulative function of the normal distri-bution and n(·) = N

′(·) the density function of the normal distribution, then

we obtain

N

(Yt −

∫ kK0

dhXt(h)h


)≤ N

(−er(T−t)

√4 log k

)≤ n

(−er(T−t)

√4 log k

)=

1√2πe−2e2r(T−t) log k =

1√2π

1

k2exp(2r(T−t)).

It means that the integral is finite (P -a.s., for each t).


2

In order to present the images of the new parameters more precisely, we ex-press the local implied volatilities and the price level via the classical impliedvolatility σt(K).

Proposition 10. Define γt(K) = (T−t)σ2t (K) and d2 =

log(Ft/K)− 12γt(K)√

γt(K). The

local implied volatility and the price level can be represented by the followingexpressions

Yt =√T − tN−1

(e−r(T−t)

(N (d2(t,K0))

−K0

√γt(K0)n (d2(t,K0))

2σ2t (K0)

d

dKσ2t (K0)

)), (3.39)

Xt(K) = σt(K)n

(N−1

(e−r(T−t)N (d2(t,K))

− 1

2n (d2(t,K))

√γt(K)

K

σ2t (K)

d

dKσ2t (K)

))× er(T−t)

n (d2(t,K))

[1 +

(log

(FtK

)+

1

2γt(K)

)K

σ2t (K)

d

dKσ2t (K) +

1

4

(log2

(FtK

)−γt(K)− 1

4γ2t (K)

)·(

K

σ2t (K)

d

dKσ2t (K)

)2

+1

2γt(K)

K2

σ2t (K)

d2

dK2σ2t (K)

]−1. (3.40)

Proof: By definition the implied volatility of the price Ct(K) is a uniqueparameter σ2

t such that c (Ft, K, (T − t)σ2t ) = Ct(K) and correspondingly

C′

t(K) = cK(Ft, K, γt(K)) + cγ(Ft, K, γt(K))(T − t) d

dKσt(K),

C′′

t = cKK(Ft, K, γt(K)) + 2cKγ(Ft, K, γt(K))(T − t) d

dKσt(K)


+cγγ(Ft, K, γt(K))

((T − t) d

dKσt(K)

)2

+cγ(Ft, K, γt(K))(T − t) d2

dK2σ2t (K).

According to the settings of our model, the partial derivatives of the functionc(F,K, γ(K)) are

C′

t(K) = e−r(T−t)

(N (d2(t,K0))−

1

2n (d2(t,K0))

K0 ·√γt(K0)

σ2t (K0)

d

dKσ2t (K0)

),

(3.41)

C′′

t (K) =e−r(T−t)

kn (d2(t,K))

1√γt(K)

×[1 +

(log

(FtK

)+

1

2γt(K)

)K

σ2t (K)

d

dKσ2t (K)

+1

4

(log2

(FtK

)− γt(K)− 1

4γ2t (K)

)·(

K

σ2t (K)

d

dKσ2t (K)

)2

+1

2γt(K)

K2

σ2t (K)

d2

dK2σ2t (K)

]. (3.42)

Thus if we insert (3.41) and (3.42) into the formulas (3.29) and (3.30) of Ytand Xt(K), we obtain (3.45), (3.40) correspondingly.

2

It remains to prove Proposition 8.

Proof: By (3.32) for any K1, K2 ∈ I, we have

Ct(K1)− Ct(K2) = e−r(T−t)∫ K2

K1

N

(Yt −

∫ kK0

dhXt(h)h


)dk

= e−r(T−t)∫ K2

K1

N

(Yt −

∫ aK0

dhXt(h)h

+ logaXt(a)

− logkXt(a)


)dk.

For any xt > 0, zt > 0 we have


d

dKc(zt, K, (T − t)x2t ) = −e−r(T−t)N

(log (zt/K)− 1

2(T − t)x2t

xt√T − t

),

for all K, hence

c(zt, K1, (T − t)x2t )− c(zt, K2, (T − t)x2t )

= e−r(T−t)∫ K2

K1

N

(log ztxt− 1

2(T − t)xt − log k

xt

xt√T − t

)dk. (3.43)

According to (3.31), we get that

zt = exp

(Xt(a)Yt −Xt(a)

∫ a

K0

dh

Xt (h)h+ log a+

1

2e−2r(T−t)(T − t)X2

t (a)

)and

xt = e−r(T−t)Xt(a).

In order to check the uniqueness of (xt, zt) we use (3.29), then (3.31), andfinally (3.28), and we obtain for K ∈ I

e−r(T−t)Xt(a) = e−r(T−t)Xt(K) =e−r(T−t)√


(N−1

(−er(T−t)C ′t (K)

))

=e−r(T−t)n(N−1(−er(T−t)cK(F,K, (T − t)x2t )))

KcKK(F,K, (T − t)x2t )√T − t

= xt.

Moreover, the left-hand side of the equation (3.34) is strictly increasing in ztand the uniqueness of zt follows immediately.

2


3.2.2 The arbitrage-free dynamics of the local impliedvolatilities

In this paragraph, we again derive the joint dynamics of the option pricelevel and the local implied volatilities under the arbitrage-free condition aswe have already done in the case of the interest rate options. For this purposewe resume some setup.

Let W be an m-dimensional Brownian motion on probability space (Ω,F , P ),and F = (Ft)0≤t≤T be the P -augmented filtration generated by W , andF = FT . The P-dynamics of the positive processes Xt(K) for a.e. K > 0and a real valued process Yt are

dXt(K) = ut(K)Xt(K)dt+ vt(K)Xt(K)dWt, 0 ≤ t ≤ T, (3.44)

dYt = βtdt+ γtdWt, 0 ≤ t ≤ T, (3.45)


loc(Rm) for a.e. K. Further we

suppose u, v are uniformly bounded in ω, t, K and that the initial localimplied volatility curve satisfies the condition∫ K

K0

dh

X0(h)2<∞, ∀K > 0.

The process Ct(K), K ≥ 0 is defined by Theorem 5, Xt(K) and Yt arethe local implied volatilities and the price level of the option price processesCt(K), K ≥ 0, respectively. We notice that Ft = Ct(0), so for defining Ct(0)by (3.28) the values of Xt(0) are not necessarily needed here.

Thus in order to demonstrate the arbitrage-free dynamics of the option pricelevel and the local implied volatilities, as we have already done in paragraph3.1.2, it is enough to show the existence of the equivalence between theexistence of a common local martingale measure for C(K), ∀K > 0 and thedrift restrictions of the joint dynamics of the price levels and the local impliedvolatilities. This argument is formulated and proved more precisely in thefollowing theorem.Theorem 6. (a) If there exists a common equivalent local martingale mea-sure Q for all C(K) (K ≥ 0), then there exists a market price of the riskprocess b ∈ L2

loc(Rm) such that (3.46), (3.47) (for a.e. K > 0) hold for a.e.

t ∈ [0, T ], P − a.s.


βt =1

2

YtT − t

e−r(T−t)

[|γt|2

e−2r(T−t)− 1

]− r · Yt · e−r(T−t) − btγt, (3.46)

ut(K) =1

er(T−t)(T − t)

[1

2− 1

2e2r(T−t)

∣∣∣∣γt +

∫ K

K0

vt(h)dh

Xt(h)h

∣∣∣∣2+e4r(T−t)

(Yt −

∫ K

K0

dh

Xt(h)h

)(γt +

∫ K

K0

vt(h)

Xt(h)hdh

)vt(k)

]+ v2t (K) + re−r(T−t) − btvt(K), (3.47)

(b) Conversely, suppose that the coefficients β, γ, u(K) and v(K) satisfy,as functions of Yt and Xt(K), relations (3.46) and (3.47) (for a.e. K > 0)for a.e. t ∈ [0, T ], P − a.s. for some bounded (uniformly in t, ω) processb ∈ L2

loc(Rm). Also suppose that there exists a family of the continuous

adapted processes X(K) > 0, Y satisfying the system (3.44) (for a.e. K > 0)and (3.45). Then there exists a common equivalent local martingale measureQ on FT for C(K), (K ≥ 0). One such measure is given by

dQ

dP:= E

(∫b′dW

)T

, (3.48)

where E is the stochastic exponential, and b′

is determined by (3.50).

(c) Under the conditions (a) or (b), the dynamics of C(K) under Q are givenby

dCt(K) =

∫ ∞K

n

(Yt −

∫ kK0

dhXt(h)h√

T − t · e−r(T−t)

)1√

T − t · e−r(T−t)

×(γt +

∫ k

K0

vt(h)dh

Xt(h)

)dkdWt, (3.49)

for K ≥ 0 and a Q-Brownian motion W = W −∫b′sds, and

b′t = −r ·N

(Yt −

∫ kK0

dhXt(h)h√

T − t · e−r(T−t)

)γt +

∫ kK0

vt(h)dhXt(h)h

√T − tn

(Yt−

∫ kK0

dhXt(h)h√

T−t·e−r(T−t)

) + bt. (3.50)


Remark 2. Note that the free input parameters are the market price of therisk process b, the volatilities of the state variables Y and X(K), such as γand the family of the processes v(K) for all K. Since St = Ct(0), the volatilityσt of the forward price process of the future contracts dFt = σtFtdWt can beeasily derived from (3.32) and (3.49) in the form

σt =

∫ ∞0

n

(Yt −

∫ kK0

dhXt(h)h√

T − t · e−r(T−t)

)1√

T − t · e−r(T−t)

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk

×

(∫ ∞0

e−r(T−t) ·N

(Yt −

∫ kK0

dhXt(h)h√

T − t · e−r(T−t)

)dk

)−1.

Given all these free input parameters, we obtain a model for the forwardprice process F with an actual specific expression of the stochastic volatility,no matter whether this volatility is Markovian and depends on γ, v(K) ornot.Proposition 11. Let Zt(K) := Yt −

∫ kK0

dhXt(h)h

. Under P the dynamics of

Ct(K) for each K ≥ are given by

dCt(K) =

∫ ∞K

[n

(Zt(k)√

T − t · e−r(T−t)

)1√

T − t · e−r(T−t)

(1

2

Zt(k)

T − te−r(T−t)

·(1−

∣∣∣γt +∫ kK0

vt(h)Xt(h)h

dh∣∣∣2

e−2r(T−t))

+ βt −∫ k

K0

v2t (h)− ut(h)

Xt(h)hdh

)+rZt(k)√T − t

·n(

Zt(k)√T − t · e−r(T−t)

)+r·N

(Zt(k)√

T − t · e−r(T−t)

)×

∣∣∣γt +∫ kK0

vt(h)Xt(h)h

dh∣∣∣2

(T − t) · e−r(T−t)

]dk·dt

+

∫ ∞K

n(


)√T − te−r(T−t)

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk · dWt. (3.51)

Proof: In order to apply the Ito lemma to the integrand inside the integralin (3.32), we do the following. Using the continuity of Zt(k) we get

dZt(K) = dYt − d(∫ k

K0

dh

Xt(h) · h

),


then by using of the Fubini theorem([6], Theorem 4.65), we simplify thisexpression to

dZt(K) = dYt −∫ k

K0

d(1

Xt(h))dh

h. (3.52)

Next, we apply the Ito lemma to 1Xt(h)

with respect to the arguments Xt(h)

and t, get the stochastic differential d(

1Xt(h)

). After that, we obtain the

following expression

d

(1

Xt(h)

)=

1

Xt(h)

[(v2t (h)− ut(h))dt− vt(h)dWt

]. (3.53)

Inserting equation (3.53) into (3.52), we obtain the following

dZt(k) =

(βt −

∫ k

K0

v2t (h)− ut(h)

Xt(h)hdh

)dt

+

(γt +

∫ k

K0

vt(h)

Xt(h) · hdh

)dWt, (3.54)

d 〈Z(k)〉t =

∣∣∣∣γt +

∫ k

K0

vt(h)

Xt(h) · hdh

∣∣∣∣2 dt. (3.55)

Then we define

Mt(k) := e−r(T−t) ·N(


),

thus we have

Ct(K) =

∫ ∞K

Mt(k)dt,

and accordingly dCt(k) = d(∫∞

KMt(k)dt

)=∫∞KdMt(k) · dt (again by the

Fubini theorem).

We apply the Ito lemma to


Mt(k) = F

(t,


),

and obtain the following expression

dMt(k) =

[re−r(T−t)N

(Zt(k)√

T − t · e−r(T−t)

)+

1

2e−r(T−t)n

(Zt(k)√

T − t · e−r(T−t)

)·(− Zt(k)√

T − t · e−r(T−t)

)]×⟨


⟩t

+ e−r(T−t)n

(Zt(k)√

T − t · e−r(T−t)

)d

(Zt(k)√

T − t · e−r(T−t)

). (3.56)

Finally, we insert equation (3.54) and (3.55) into equation (3.56), after astraight forward computing, obtain that

dMt(k) =

∫ ∞K

[n

(Zt(k)√

T − t · e−r(T−t)

)1√

T − t · e−r(T−t)

×(

1

2

Zt(k)

T − te−r(T−t)

(1−

∣∣∣γt +∫ kK0

vt(h)Xt(h)h

dh∣∣∣2

e−2r(T−t)

)+ βt −

∫ k

K0

v2t (h)− ut(h)

Xt(h)hdh

)+rZt(k)√T − t

· n(


)

+ r ·N(


) ∣∣∣γt +∫ kK0

vt(h)Xt(h)h

dh∣∣∣2

(T − t) · e−r(T−t)

]dt

+

∫ ∞K

n(


)√T − t · e−r(T−t)

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dWt. (3.57)

Using the relation between dMt(k) and dCt(K) that Ct(K) =∫∞KMt(k)dt, we

obtain exactly the same expression of dCt(K) as the one in the proposition’sdescriptions.

2


Now we are ready to prove Theorem 6.

Proof: (a) Since the filtration F is generated by W , and by Ito’s represen-tation theorem, we obtain that

E

[dQ

dP|Ft]

= E(∫

b′dW

)t

,

for some process b ∈ L2loc (Rm), and

W: = W −∫b′

sds

is a Q-Bownian motion by Girsanov’s theorem.

By Proposition 11 it yields that

dCt(K) =

∫ ∞K

n(Zt(k)

√T − t · e−r(T−t)

) 1√T − t · e−r(T−t)

µt(k) · dk · dt

+

∫ ∞K

n(


)√T − t · e−r(T−t)

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk · dWt, (3.58)

where

µt(k) :=1

2

Zt(k)

T − te−r(T−t)

1−

∣∣∣γt +∫ kK0

vt(h)Xt(h)h

dh∣∣∣2

e−2r(T−t)

+ βt

−∫ k

K0

v2t (h)− ut(h)

Xt(h)hdh+ rZt(k) · e−r(T−t) + bt

(γt +

∫ k

K0

vt(h)

Xt(h)dh

).

We take the expectations on both sides of (3.58) (Q is a martingale measureby Fubini theorem) we must have that µt(k) = 0 for a.e. k, also providedthat µt(k) is continuous in k, then

µt(k) = 0, ∀k > 0. (3.59)

Let k tends to K0 in (3.59), we obtain expression for βt, (3.46). By differen-tiating µt(k) with respect to k, then taking values in k = K, we obtain theexpression of ut(K), (3.47).


(b) DefinedQ

dP:= E

(∫b′dW

)T

on FT ; then W := W−∫b′tdt is a Q-Brownian motion on [0, T ] by Girsanov’s

theorem. To verify this, we can plug in (3.46) and (3.47), as well as dWt =dWt + btdt into Proposition 11, we obtain clause (c) under the condition (b).It now easily follows from (c) that C(K) for all K ≥ 0 are Q local martingaleson [0, T ].

(c) Together with the proof of (b) above, we have already proved the asser-tation (c) under (b). Under the assumption (a), the assertation (c) followsfrom (3.58) and (3.59).

2


Chapter 4

Applications

In this chapter we mainly discuss the applications of the Schweizer-Wisselmodel.

Before we start our discussions in this Chapter, we firstly impose some as-sumptions on the market. Let the market’s assets consist of one bond, onestock and a series of call options.

Nextly, let us consider the portfolio. Assuming that an investor has aninvestment portfolio consisting of a series of n call options with differentstrike prices, and that the portion of the investment of each option dependson its stike price K. We denote this weight as δ(K), then as for the value ofthe portfolio π, obtain the following expression

π =n∑i=1

δ(Ki) · Ct(Ki).

Usually, investors are more concerned about the dynamics of the portfolio,by using the results of Theorem 2 in Chapter 2 and equation (2.20). We havethe expression of the portfolio dynamics given by

dπ = d

(n∑i=1

δ(Ki) · Ct(Ki)

)=

n∑i=1

δ(Ki) · dCt(Ki)

=n∑i=1

δ(Ki)

∫ ∞Ki

n

(Yt −

∫ kK0

dhXt(h)h√

T − t

)1√T − t

(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk ·dWt,

53

54 Chapter 4. Applications

for Ki ≥ 0, i = 1, .., n and a Q-Brownian motion W = W −∫bsds under the

martingale measure Q.

Now let us illustrate the applications of the Schweizer-Wissel model in detailsby a simple example of an investment portfolio.

Suppose we have two call options, C1 and C2, as our investment objectiveshere. Two call options have strike prices K1 and K2 (K1 > K2), respectively.Our investment strategy is taking the long position in the option with a lowerstrike price, and shorting the other one. So, by construction, our portfolio πis π = C2 − C1.

Again, using the option dynamics expression (2.20), we have the portfoliodynamics given by

dπ =

∫ K1

K2

n

(Yt −

∫ kK0

dhXt(h)h√

T − t

)· 1√

T − t

·(γt +

∫ k

K0

vt(h)

Xt(h)hdh

)dk · dWt. (4.1)

By simple plotting, we get the pay-off diagram (Figure 4).

Figure 4

Remark 3. Suppose that, we know the dynamic processes of the local im-plied volatility and the price level are of the form as in equation (2.15) andequation (2.16). We notice that the dynamics of the portfolio can be easilyexpressed in terms of the free-input parameters, the market price of the riskprocess b as well as γ and the family of the processes v(K), ∀K > 0, i.e. thevolatilities of the state varibles Y and X(K).


Next, we use this example to show that Schweizer-Wissel’s model is consistentwith the Black-Scholes model under some special conditions.

First, let us consider the Black-Scholes model. We state some reasonableand basic assumptions of the Black-Scholes model. The stochastic process ofthe stock price follows the geometric Brownian motion, risk-neutrality holdsand the volatility of the stock is a constant not depending on the time andoptions’ strike price. Additionally, from the risk-neutral assumption in theSchweizer-Wissel model, we put interest rate r equal to zero. Then we havethat the stock process is

dS = µ · S · dt+ σ · S · dWt.

By applying the Ito lemma to the Black-Scholes formula for the call optionvalue CBS(t, S), then simplifying it by using the Black-Scholes differentialequation, we obtain the random process of the option price in the form

dC =

[(µ− r)S∂C

∂S+ rC

]dt+ σS

∂C

∂SdWt,

which under the risk-neutrality condition and the zero-interest rate conditionfrom the Schweizer-Wissel model, i.e. µ = r = 0 yields

dC = σS∂C

∂SdWt.

Besides that, according to the Black-Scholes formula for the call options,we have C = SN(d1(K)) − KN(d2(K)), and the attendant equality, ∂C

∂S=

N(d1(K)).

Thus, for our portfolio π = C2 − C1, we get the dynamics

dπ = dC2 − dC1 = σS [N (d1(K1))−N (d1(K2))] dWt

= σS

∫ d1(K1)

d1(K2)

1√2πe−

x2

2 · dx · dWt,

where

d1 =ln(

SK0

)+ 1

2σ2(T − t)

σ√T − t

.

We introduce the new variable y to simplify this integration

y = Se−xσ√T−t− 1

2σ2(T−t),


i.e.,

x =ln(Sy

)+ 1

2σ2(T − t)

σ√T − t

,

dx = − dy

yσ√T − t

.

Thus, it yields that

dπ = −σS∫ K1

K2

1√2πe− 1

2

(ln(Sy )+1

2σ2(T−t)

σ√T−t

)2

dy

−yσ√T − t

dWt

= −σS∫ K1

K2

1√2πe− 1

2·(ln(Sy ))

2+( 1

2σ2(T−t))

2

(σ√T−t)2 ·

√y

S· dy

−yσ√T − t

dWt

=

∫ K1

K2

1√2πe− 1

2·(ln(Sy ))

2+( 1

2σ2(T−t))

2

(σ√T−t)2 ·

√S

y· dy√

T − tdWt

=

∫ K1

K2

1√2πe− 1

2·(ln(Sy ))

2+( 1

2σ2(T−t))

2

(σ√T−t)2 · e

− 12·2·

(− 12σ

2(T−t)) ln(Sy )(σ√T−t)2 · dy√

T − tdWt

=

∫ K1

K2

1√2πe− 1

2

[ln(Sy )− 1

2σ2(T−t)

σ√T−t

]2· dy · dWt.

We obtain

dπ =

∫ K1

K2

n

ln(Sy

)− 1

2σ2(T − t)

σ√T − t

· 1√T − t

· dy · dWt. (4.2)

The rest of this Chapter is devoted to the development the similar proce-dure for the Schweizer-Wissel model, and verifying our assertment that theSchweizer-Wissel model is consistent with the Black-Scholes model undersome special conditions.

Because of the volatility σ is a constant here, by equation (2.15) and (2.16)we instantly have

Xt(K) = σ,

Yt =ln(

SK0

)− 1

2σ2(T − t)

σ.


For the dynamics we consequently have

dXt(K) = 0, ut(K) = vt(K) = 0,

dYt(K) =1

σSdS +

1

2σ · dt = dWt +

1

2σdt,

γt = 1, βt =1

2σ.

Insert them into equation (4.1) we obtain

dπ =

∫ K1

K2

n

ln(SK0

)− 1

2σ2(T−t)

σ−

ln(kK0

)σ√

T − t

· 1√T − t

dk · dWt,

dπ =

∫ K1

K2

n

(ln(Sk

)− 1

2σ2(T − t)

σ√T − t

)· 1√

T − t· dk · dWt. (4.3)

Thus equation (4.3) is equivalent to equation (4.2), and our assertment isproved.


Chapter 5

Conclusions

Our study in this master project was devoted to the study of models forone stock S and a set of European call options with one maturity and allstrikes K > 0, and possible extensions for European call options with interestrate indexes and future contracts as their underlying assets. We discussedin Chapter 2 that, there are usually two approaches to model the financialmarket, one of them is the martingale approach, whose corresponding modelis referred to as the stock model, and the other one is the market model. Themartingale approach specifies the option prices as the conditional expecta-tions of the payoff functions under a common equivalent martingale measurefor the stock S, and for most of the time, the parameters of the stock modelare calibrated to a set of the vanilla option prices expressed in terms of theimplied volatilities. Instead of using the traditional martingale approach, wefollowed the way of the Schweizer-Wissel model, which belongs to the mar-ket models category and was firstly introduced by Schweizer and Wissel intheir paper [10]. Different from the former investigations, this new modelconstructs the stock and the option price processes simultaneously, thus wecan have at the same time not only the joint dynamics for the stock and theoptions, but also the perfect calibration to the given set of the initial vanillaoption prices.

However, under these conditions the option prices are no longer automati-cally staying to be the conditional expectations, the absence of the dynamicarbitrage now translates into the drift conditions on the modeled quantitiesto ensure the local martingale property of all the tradables’ price processesin this new model.

As pointed out in their paper [10], a crucial point here for a next step forward

59

60 Chapter 5. Conclusions

in the extensions of Schweizer-Wissel model still remains to be the properparameterization where the static arbitrage restrictions do not already resultin a complicated state space for the quantities describing the model. Andthe existence issue of such a model satisfies this simple state space conditionhas been briefly discussed in Chapter 2.

In our work we presented the actual models for both options on the interestrate indexes and the options on futures. In Chapter 3.1 we discuss the interestrate options and in paragraph 3.1.1 we provide the new parameters for thiskind of options, for instance the local implied volatility and the price level.The Reparameterization has been done in such a way that the static arbitragerestrictions do not have as a consequence to give a complicated state spacefor the quantities describing the model. Besides that, it allows us to getthe explicit formulas for the IRO prices in the multi-strike case in terms ofthe local implied volatility and the price level. Finally we solve the jointdynamics issue easily. After reparameterization the drift restrictions can beexpressed by the explicit functions and the arbitrage-free joint dynamic ofthe IRO prices and the forward rate of the interest rate indexes as well asthe arbitrage-free dynamics of the local implied volatilities become available.This result is justified by Theorem 4 in paragraph 3.1.2. In Chapter 3.2 wedescribe the future options, provide the similar reparameterization procedureas for the IRO options and argue the absence of the dynamic arbitrage.

Since we already have the information about both the option pricing and theasset-option joint dynamics, a natural question will rise is that how to applythis model in reality. We demonstrate by a simple example of the investmentportfolio consisting of the long position in a European call option and theshort position in another European call option with a different strike price, togive the readers a first impression of how the Schweizer-Wissel model worksin general. In addition, we also proved that in some specific scenario, theSchweizer-Wissel model is essentially equivalent to the well-known Black-Scholes model. In other words, under some special conditions, these twomodel are consistent.

Our paper here is certainly just a small step forward in the possible general-ization of the original Schweizer-Wissel model. There are still many questionsremained unresolved. There are several directions for a possible future re-search. Firstly, we can do the skewness-consideration work based on theoriginal Schweizer-Wissel model, considering the limited availability of themarket data of the IRO and the future options. To be exact, we plot thecurves of local implied volatility Xt(K) and Yt, simulate the curves of theclassical implied volatilities σt(K) based on the market data. Then we will


be able to establish an explanation for the difference between their skew-ness with reference to the background assumptions of this model. Secondly,we can try to add an appropriate and a suitable function for the impliedvolatilitiy σt(K) that might be able to give a simple explanation to the com-plex drift-restrictions . Finally, once we extent the model to more differentderivatives, or take some realistic concerns into account, we might encountermore realistic problems, for instance, to calibrate the Schweizer-Wissel modelto market data like Wissel did in [11], Chapter 3.3 for the case of a finitefamily of the European call options with the various strikes and the matu-rities. Hopefully, we can discuss these issues addressed above in our furtherresearch.


Notation

IRO the Interest Rate Options.

α the day-count fraction which transforms the numderof trading days into equivalent years.

β, u(K) the drifts of the state variables Y and X(K).

δ(K) the weight of the portion of investment of each option includedin the portfolio with the different strike prices K and the same maturity T .

γ, v(K) the volatilities of the state variables Y and X(K).

ϕ the risk-neutral density function.

µ the drift of the underlying asset price processes.

(Ω,F , P ) the probability space with the set of outcomes Ω,the sigma algebra F and the probability measure P .

π the investment portfolio.

σ = σ(S, t;S0) the local volatility function in Dupire’s model.

σ the volatility of the underlying asset price process.

σ the classical implied volatility.

b the market price of risk.

(Bt)0≤t≤T the discounted bond price process withBT ≡ 1, P − a.s. (in Chapter 2).

(Bt)0≤t≤T the discounted bond price process with Bt = B0e−∫ Tt r(τ)dτ ,

B0 = 1 (in Chapter 3).

F the forward rate of the interest rate indexes in Chapter 3.1.

Ft the discounted function for the time point t in Chapter 3.1.

FT the discounted function for the time point T in Chapter 3.1.

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Ft the future contract’s price in Chapter 3.2.

(Ft)0≤t≤T the positive process modeling the (discounted) prices ofthe forward rate of the interest rate indexes F in Chapter 3.2.

K the strike price.

K0 the fixed constant from the interval (0,∞).

K the family of the strike prices.

N(·) the standard normal distribution function.

N−1 (·) the quantile function.

n (·) the density function of the standard normal distribution.

Q the martingale measure.

Q the common equivalent local martingale measure forall C(K) (K ≥ 0) in paragraph 3.1.1 and Chapter 4.

S the stock price.

S0 the initial spot price of the asset.

ST the final spot price of the asset.

(St)0≤t≤T the positive process modeling the (discounted) stock price.

T the maturity date of the options.

T the family of the maturity dates of the option.

W the m-dimensional Brownian motion on (Ω,F , P ).

W the Q-Brownian motion and W = W −∫bsds

in paragraph 3.1.1 (the Schweizer-Wissel model) and W = W −∫b′sds

in Chapter 4.

X(K) the local implied volatilities of themodel C(K), K ≥ 0.

Y the price level of the model C(K), K ≥ 0.

Z the one-dimensional Brownian motion in Dupire’s model.

Appendix

The program code for plotting of the diagram of the payoff portfolio consist-ing of one long and one short European call options with the strike pricesK1 > K2, i.e. π = C(K2)− C(K1).

Call[K_, S] := Max[0, S - K];

K1 = 30;

K2 = 20;

payoff = Plot[-Call[K1, S] + Call[K2, S], S, 15, 40,

AxesLabel \rightarrow StockPrice S, PortfolioValue Pi,

PlotLabel \rightarrow Payoff Diagram of Portfolio]

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[2] Bruno Dupire (1994)Pricing with a smile. Risk Publications/Over The Rainbow, 7 (1), 271– 275.

[3] Jim Gatheral, Merill Lynch (2002)Lecture 1: Stochastic volatilility and Local volatil-ity. Case Studies in Financial Modelling Course Notes,Courant Institute of Mathematical Sciences, Fall term.http://www.math.ku.dk/ rolf/teaching/ctff03/Gatheral.1.pdf

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[9] Paul Wilmott, Sam Howison and Jeff Dewynne (1995)The Mathematics Of Financial Derivatives. New York: The Press Syn-dicate of the University of Cambrige, 98 – 102.

[10] Johannes Wissel, Martin Schweizer (2007)Arbitrage-free market models for option prices: The multi-strike case. Preprint, ETH Zurich. http://www.nccr-finrisk.uzh.ch/media/pdf/wp/WP380 D1.pdf

[11] Johannes Wissel (2007)Arbitrage-free market models for option prices. Preprint, ETH Zurich.http://www.nccr-finrisk.uzh.ch/media/pdf/wp/WP428 D1.pdf

[12] Johannes Wissel (2008)Arbitrage-free market models for liquid options. Ph.D Thesis No. 17538,ETH Zurich. http://e-collection.ethbib.ethz.ch/eserv/eth:30171/eth-30171-02.pdf

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