Stochastic Stochastic Volatility and Volatility and

Change of Time: Change of Time: OverviewOverview

Anatoliy SwishchukAnatoliy SwishchukAnatoliy SwishchukAnatoliy SwishchukDept of Mathematics & Statistics Dept of Mathematics & Statistics

University of Calgary, Calgary, AB, CanadaUniversity of Calgary, Calgary, AB, Canada

Modern Stochastics: Theory and Applications IIModern Stochastics: Theory and Applications IIKiev, UkraineKiev, Ukraine

September 7September 7--11,201011,2010Dedicated to: A. Skorokhod, V. S. Korolyuk and I. Dedicated to: A. Skorokhod, V. S. Korolyuk and I.

KovalenkoKovalenko

The research is supported by NSERC/MITACS

OutlineOutline

Volatility: Types Volatility: Types

Stochastic Volatility (SV):1Stochastic Volatility (SV):1--& Multi& Multi--FactorFactor

Change of Time (CT)Change of Time (CT)

Relationship: SV & CTRelationship: SV & CTRelationship: SV & CTRelationship: SV & CT

Numerical ExampleNumerical Example

ProblemsProblems

VolatilityVolatility

VolatilityVolatility is the standard deviation of the is the standard deviation of the change in value of a financial instrument change in value of a financial instrument with specific time horizonwith specific time horizon

It is often used to quantify the It is often used to quantify the riskrisk of the of the It is often used to quantify the It is often used to quantify the riskrisk of the of the instrument over that time periodinstrument over that time period

The higher volatility, the riskier the The higher volatility, the riskier the securitysecurity

Types of VolatilitiesTypes of VolatilitiesHistorical VHistorical V: : standard deviationstandard deviation (uses (uses historical (daily, weekly, monthly, historical (daily, weekly, monthly, quarterly, yearly)) price data to empirically quarterly, yearly)) price data to empirically measure the volatility of a market or measure the volatility of a market or instrument in the pastinstrument in the past

Implied VImplied V: volatility : volatility implied by the market implied by the market priceprice of the option based on an option of the option based on an option pricing model (smile and skewpricing model (smile and skew--varying varying volatility by strike)volatility by strike)

Volatility SmileVolatility Smile

The models by Black & Scholes (continuousThe models by Black & Scholes (continuous--

time (B,S)time (B,S)--security market,1973) and Cox & security market,1973) and Cox &

Rubinstein (discreteRubinstein (discrete--time (B,S)time (B,S)--security market security market

(binomial tree), 1979) are unable to explain the (binomial tree), 1979) are unable to explain the

negative skewnessnegative skewness and and leptokurticity (fat tail) leptokurticity (fat tail) negative skewnessnegative skewness and and leptokurticity (fat tail) leptokurticity (fat tail)

commonly observed in the stock marketscommonly observed in the stock markets

The famous The famous impliedimplied--volatility smilevolatility smile would not would not

exist under their assumptionsexist under their assumptions

Commodity: CoffeeCommodity: Coffee

Coffee options Coffee options

trade on New trade on New

York's Coffee, York's Coffee, York's Coffee, York's Coffee,

Sugar and Sugar and

Cocoa Cocoa

Exchange Exchange

((CSCECSCE).).

Coffee Call OptionCoffee Call Option

CSCE May CSCE May

2001 coffee 2001 coffee

call option call option call option call option

implied implied

volatilitiesvolatilities as as

of March 12, of March 12,

20012001

Implied Volatility: Volatility SmileImplied Volatility: Volatility Smile

Graph indicates Graph indicates implied volatilities at implied volatilities at various strikes for the various strikes for the May 2001 calls based May 2001 calls based upon their March 12, upon their March 12, upon their March 12, upon their March 12, 2001 settlement 2001 settlement prices. The pattern of prices. The pattern of implied volatilities implied volatilities form a "smile" shape, form a "smile" shape, which is called a which is called a volatility smilevolatility smile..

Implied Volatility: Volatility SkewImplied Volatility: Volatility Skew

Most derivatives markets Most derivatives markets exhibit persistent patterns of exhibit persistent patterns of volatilities varying by strike. In volatilities varying by strike. In some markets, those patterns some markets, those patterns form a form a smilesmile. In others, such . In others, such as as equity index optionsequity index optionsmarkets, it is more of a markets, it is more of a markets, it is more of a markets, it is more of a skewed curve. This has skewed curve. This has motivated the name motivated the name volatility volatility skewskew. In practice, either the . In practice, either the term "volatility smile" or term "volatility smile" or "volatility skew" (or simply "volatility skew" (or simply skewskew) may be used to refer ) may be used to refer to the general phenomena of to the general phenomena of volatilities varying by strike. volatilities varying by strike.

Implied Volatility: Volatility SurfaceImplied Volatility: Volatility Surface

Another dimension to Another dimension to

the problem of the problem of

volatility skew is that volatility skew is that

of volatilities varying of volatilities varying

by expiration. This is by expiration. This is by expiration. This is by expiration. This is

illustrated for CSCE illustrated for CSCE

coffee options. It coffee options. It

indicates what is indicates what is

known as a known as a volatility volatility

surfacesurface

Types of Volaltilites IITypes of Volaltilites II

JumpJump--Diffusion VolatilityDiffusion Volatility

LevelLevel--Dependent Volatility Dependent Volatility (CEV or Firm (CEV or Firm Model)Model)--function of the spot price alonefunction of the spot price alone

Local Local VV--function of the spot price function of the spot price and timeand timeLocal Local VV--function of the spot price function of the spot price and timeand time(Dupire formulae, 1994)(Dupire formulae, 1994)

Stochastic VStochastic V: volatility is not constant, : volatility is not constant, but a stochastic process (explains but a stochastic process (explains smile smile and skewand skew))

JumpJump--DiffusionDiffusion

In addition to the volatility smile observable from In addition to the volatility smile observable from the implied volatilities of the options, there is the implied volatilities of the options, there is evidence that assumption of a pure diffusion for evidence that assumption of a pure diffusion for the stock return is not accuratethe stock return is not accurate

‘Fat Tails’ have been observed away from the ‘Fat Tails’ have been observed away from the ‘Fat Tails’ have been observed away from the ‘Fat Tails’ have been observed away from the mean of the stock returnmean of the stock return

This phenomenon is called leptokurticity and This phenomenon is called leptokurticity and could be explained in many different wayscould be explained in many different ways

One way to explain smile and leptocurticity is to One way to explain smile and leptocurticity is to introduce a jumpintroduce a jump--diffusion processdiffusion process

JumpJump--Diffusion and Leverage Diffusion and Leverage

EffectEffect

JumpJump--diffusion is not a leveldiffusion is not a level--dependent dependent volatility processvolatility process

Explains leverage effectExplains leverage effect

MertonMerton (1976)(1976) was first to introduce jumps was first to introduce jumps MertonMerton (1976)(1976) was first to introduce jumps was first to introduce jumps in the stock distributionin the stock distribution

Kou (2000)Kou (2000) used the same idea to explain used the same idea to explain both existence of fat tails and the volatility both existence of fat tails and the volatility smilesmile

LevelLevel--Dependent VolatilityDependent Volatility

LevelLevel--dependent V (LDV)dependent V (LDV)--function of spot price alonefunction of spot price alone

Constant Elasticity Variance Constant Elasticity Variance (CEV): Cox (1976, 1996)(CEV): Cox (1976, 1996)

Important feature of the levelImportant feature of the level--dependent volatility: dependent volatility: represents the represents the negative negative represents the represents the negative negative correlation between the stock correlation between the stock price and the volatilityprice and the volatility((leverage effectleverage effect))

LDV by Firm structure model: LDV by Firm structure model: Bensoussan, Crouhy & Galai Bensoussan, Crouhy & Galai (1995)(1995)

Local VolatilityLocal Volatility

Local VolatilityLocal Volatility-- VV--function of the function of the spot pricespot price and time and time

Volatility smile was retrieved from Volatility smile was retrieved from the option pricesthe option prices

Dupire (1994)Dupire (1994)--local volatility local volatility Dupire (1994)Dupire (1994)--local volatility local volatility formula (Vformula (V--call price)call price)

Derman & Kani (1994)Derman & Kani (1994)--used the used the binomial (or trinomial tree) binomial (or trinomial tree) framework instead of the framework instead of the continuous onecontinuous one

Local Volatility: DrawbacksLocal Volatility: Drawbacks

The LV models are very elegant and theoretically The LV models are very elegant and theoretically

soundsound

However, they present in practice many However, they present in practice many stability stability

issuesissuesissuesissues

They are They are illill--posed inversion problemsposed inversion problems and are and are

extremely extremely sensitive to the input datasensitive to the input data

This might introduce This might introduce arbitrage opportunities arbitrage opportunities

and in some cases negative probabilities or and in some cases negative probabilities or

variancesvariances

Stochastic Volatility (SV)Stochastic Volatility (SV)

SVSV is the main concept used in the fields is the main concept used in the fields of financial economics and mathematical of financial economics and mathematical finance to deal with the endemic timefinance to deal with the endemic time--varying volatility and covarying volatility and co--dependence found dependence found in financial marketsin financial marketsin financial marketsin financial markets

Such dependence has been known for a Such dependence has been known for a long time, early comments include long time, early comments include Mandelbrot (1963)Mandelbrot (1963) and and Officer (1973)Officer (1973)

Stochastic VolatilityStochastic Volatility

The aim with a stochastic volatility model: The aim with a stochastic volatility model:

volatility appears not to be constantvolatility appears not to be constant and indeed and indeed

varies, at least in part, randomly. The idea is to varies, at least in part, randomly. The idea is to

make the make the volatility itself a stochastic processvolatility itself a stochastic process..

Stochastic volatilityStochastic volatility models are useful because models are useful because

they explain in a selfthey explain in a self--consistent way why it is consistent way why it is

that options with different strikes and expirations that options with different strikes and expirations

have different Blackhave different Black--Scholes implied volatilities Scholes implied volatilities

(the (the volatility smilevolatility smile))

Two Approaches to Introduce SVTwo Approaches to Introduce SV

One approachOne approach--to to

change the clock time change the clock time

t to a random time t to a random time

T(t) (change of time)T(t) (change of time)

Another approachAnother approach--

change constant change constant

volatility into a volatility into a

positive stochastic positive stochastic

processprocess

Stochastic Volatilities: ContinuousStochastic Volatilities: Continuous-- Time Time ModelsModels

OrnsteinOrnstein--Uhlenbeck Uhlenbeck ProcessProcess

Hull & White (1987)Hull & White (1987)(GBM, positive)(GBM, positive)

Wiggins (1987)Wiggins (1987)(GBM, positive)(GBM, positive)(GBM, positive)(GBM, positive)

Scott (1989)Scott (1989)(OU, mean(OU, mean--reverting, positive)reverting, positive)

Stein & Stein (1991)Stein & Stein (1991)(OU, mean(OU, mean--reverting, negative)reverting, negative)

Heston (1993)Heston (1993)(mean(mean--reverting, semireverting, semi--analytical analytical

pricing formulae)pricing formulae)

Stochastic Volatilities: ContinuousStochastic Volatilities: Continuous-- Time Time Models IIModels II

Heston & Nandi (1997)Heston & Nandi (1997)--showed that OU process showed that OU process corresponds to a special corresponds to a special case of the GARCH case of the GARCH modelmodel

Another popular process Another popular process Another popular process Another popular process is the continuousis the continuous--time time GARCH(1,1) process, GARCH(1,1) process, developed by Engle developed by Engle (1982) and Bollerslev (1982) and Bollerslev (1986) in discrete (1986) in discrete frameworkframework

Stochastic Volatilities: DiscreteStochastic Volatilities: Discrete--Time Time ModelsModels

Even though continuous time models Even though continuous time models provide the natural framework for an provide the natural framework for an analysis of option pricing, analysis of option pricing, discrete time discrete time modelsmodels are ideal for the are ideal for the statistical and statistical and descriptive analysisdescriptive analysis of the patterns of of the patterns of descriptive analysisdescriptive analysis of the patterns of of the patterns of daily price changesdaily price changes

Volatility ClusteringVolatility Clustering: there are periods of : there are periods of high and low variance (‘large changes tend high and low variance (‘large changes tend to be followed by small changes’ to be followed by small changes’ ((MandelbrotMandelbrot))))--led to use of GARCH led to use of GARCH modelsmodels

DiscreteDiscrete--Time SV Models: Two Main Time SV Models: Two Main

ClassesClasses

The first classThe first class, the , the autoregressive random autoregressive random variance (ARV) or stochastic variance variance (ARV) or stochastic variance modelsmodels, is a discrete time approximation to the , is a discrete time approximation to the continuous time diffusion models that we continuous time diffusion models that we outlined aboveoutlined aboveoutlined aboveoutlined above

The second classThe second class is the is the autoregressive autoregressive conditional heteroskedastic (ARCH)conditional heteroskedastic (ARCH) models models introduced by Engle (1982), and its descendents introduced by Engle (1982), and its descendents (GARCH (Bolerslev (1986)), NARCH, NGARCH (GARCH (Bolerslev (1986)), NARCH, NGARCH (Duan,1996), LGARCH, EGARCH, GJR(Duan,1996), LGARCH, EGARCH, GJR--GARCH, etc.)GARCH, etc.)

SV With Delay: ContinuousSV With Delay: Continuous--Time GARCH Time GARCH Model with DelayModel with Delay

(Kazmerchuk, Swishchuk, Wu (2002))(Kazmerchuk, Swishchuk, Wu (2002))

General Class of SV ModelsGeneral Class of SV Models(C.(C.--O. Ewald, R. Poulsen, K.R. SchenkO. Ewald, R. Poulsen, K.R. Schenk--Hoppe (2006))Hoppe (2006))

Specification of General SV ModelsSpecification of General SV Models

HestonHeston--Like ModelLike Model(J. (J. GatheralGatheral (2005), Merrill Lynch)(2005), Merrill Lynch)

MultiMulti--Factor SV ModelsFactor SV ModelsOneOne--Factor SV ModelsFactor SV Models ((all aboveall above--mentionedmentioned): ): 1) incorporate the leverage between returns and 1) incorporate the leverage between returns and volatility and volatility and

2) reproduce the ‘skew’ of implied volatility2) reproduce the ‘skew’ of implied volatility

However, it However, it fails to match either the high fails to match either the high conditional kurtosis of returnsconditional kurtosis of returns ((Chernov Chernov et.et. alal. . (2003))(2003)) or or the full term structure of implied the full term structure of implied conditional kurtosis of returnsconditional kurtosis of returns ((Chernov Chernov et.et. alal. . (2003))(2003)) or or the full term structure of implied the full term structure of implied volatility surfacevolatility surface (Cont&Tankov (2004))(Cont&Tankov (2004))

Adding jump componentsAdding jump components in returns and/or in returns and/or volatility process, or considering volatility process, or considering multimulti--factor SVfactor SVmodels models are two primary generalizations of oneare two primary generalizations of one--factor SV modelsfactor SV models

MultiMulti--Factor SV ModelFactor SV Model(J.(J.--P. Fouque, C.P. Fouque, C.--H. Han (2005))H. Han (2005))

MultiMulti--Factor SV ModelsFactor SV Models

Chernov et al. (2003):Chernov et al. (2003): used efficient method of moments used efficient method of moments to obtain comparable empiricalto obtain comparable empirical--ofof--fit from affine jumpfit from affine jump--diffusion mousiondels & twodiffusion mousiondels & two--factor SV family modelsfactor SV family models

Molina et al. (2003):Molina et al. (2003): used a Markov Chain Monte Carlo used a Markov Chain Monte Carlo method to find strong evidence of twomethod to find strong evidence of two--factor SV models factor SV models with wellwith well--separated time scales in foreign exchange dataseparated time scales in foreign exchange data

Cont &Tankov (2004):Cont &Tankov (2004): found that jumpfound that jump--diffusion models diffusion models have a fairly good fit to the implied volatility surfacehave a fairly good fit to the implied volatility surfacehave a fairly good fit to the implied volatility surfacehave a fairly good fit to the implied volatility surface

Fouque et al. (2000):Fouque et al. (2000): found that twofound that two--factor SV models factor SV models provide a better fit to the term structure of implied volatility provide a better fit to the term structure of implied volatility than onethan one--factor SV models by capturing the behavior at factor SV models by capturing the behavior at short and long maturitiesshort and long maturities

Swishchuk (2006):Swishchuk (2006): introduced twointroduced two--factor and threefactor and three--factor factor SV models with delay (incorporating meanSV models with delay (incorporating mean--reverting level reverting level as a random process (GBM, OU, Pilipovich or continuousas a random process (GBM, OU, Pilipovich or continuous--time GARCH(1,1) model))time GARCH(1,1) model))

Advantages and Disadvantages of Advantages and Disadvantages of MultiMulti--Factor SV ModelsFactor SV Models

MultiMulti--Factor SV models do not admit in general Factor SV models do not admit in general

explicit solutions for option pricesexplicit solutions for option prices

But have direct implications on hedgesBut have direct implications on hedges

Comparison: class of Comparison: class of jumpjump--diffusion modelsdiffusion models

(Bates (1996))(Bates (1996)) enjoys enjoys closedclosed--form solutionsform solutions for for (Bates (1996))(Bates (1996)) enjoys enjoys closedclosed--form solutionsform solutions for for

option prices option prices but the complexity of hedging but the complexity of hedging

strategies increases due to jumpsstrategies increases due to jumps

There is no strong empirical evidence to judge There is no strong empirical evidence to judge

the overwhelming position between jumpthe overwhelming position between jump--

diffusion models and multidiffusion models and multi--factor SV modelsfactor SV models

Other Generalization of SVMOther Generalization of SVM

Allow jumps into the volatility SDE Allow jumps into the volatility SDE ((Bates Bates

(1996), Barndorff(1996), Barndorff--Nielsen & Shephard (2001Nielsen & Shephard (2001), ), Eraker, Eraker, Johannes & Polson (2003), Nicolato & Venardos Johannes & Polson (2003), Nicolato & Venardos (2003)(2003)--affine class (Duffie, Pan & Singleton (2000)))affine class (Duffie, Pan & Singleton (2000)))

Discrete and continuousDiscrete and continuous--time long time long Discrete and continuousDiscrete and continuous--time long time long memory SV memory SV ((Breidt, Crato & Lima (1998), Harvey Breidt, Crato & Lima (1998), Harvey

(1998), Comte & Renault (1998), Compte, Coutin & (1998), Comte & Renault (1998), Compte, Coutin & Renault (2003), BarndorffRenault (2003), Barndorff--Nielsen (2001Nielsen (2001))))

Multivariate models: introducing volatility clustering into Multivariate models: introducing volatility clustering into

traditional factor models (traditional factor models (Diebold & Nerlove (1989))Diebold & Nerlove (1989))

Change of Time: Definition and ExamplesChange of Time: Definition and Examples

Change of TimeChange of Time--change time from t to a nonchange time from t to a non--negative process T(t) with nonnegative process T(t) with non--decreasing sample decreasing sample pathspaths

Example1 Example1 ((SubordinatorSubordinator): X(t) and T(t)>0 are ): X(t) and T(t)>0 are some processes, then X(T(t)) is subordinated to some processes, then X(T(t)) is subordinated to X(t); T(t) is change of timeX(t); T(t) is change of time

Example 2Example 2 ((TimeTime--Changed Brownian MotionChanged Brownian Motion): ): Example 2Example 2 ((TimeTime--Changed Brownian MotionChanged Brownian Motion): ): M(t)=B(T(t)), B(t)M(t)=B(T(t)), B(t)--Brownian motionBrownian motion

Example 3Example 3 ((Product ProcessProduct Process):):

Interpretation of CTInterpretation of CT

If M(t) is a If M(t) is a martingalemartingale (another (another namename-- fair gamefair game process)process)Then Then M(t)=B(T(t))M(t)=B(T(t)) (Dambis(Dambis--DubinsDubins--Schwartz Theorem)Schwartz Theorem)TimeTime--change is the change is the quadratic quadratic variationvariation process T(t)=[M(t)]process T(t)=[M(t)]Then M(t) can be written as a Then M(t) can be written as a Then M(t) can be written as a Then M(t) can be written as a SV processSV process (martingale (martingale representation theorem, Doob representation theorem, Doob (1953))(1953))This implies that This implies that timetime--changed changed BMs are BMs are canonical in canonical in continuous sample path continuous sample path price processesprice processes and and SVMs SVMs are special cases of this are special cases of this classclass

TimeTime--Changed Brownian Motion by Changed Brownian Motion by BochnerBochner

BochnerBochner (1949) (‘Diffusion Equation and (1949) (‘Diffusion Equation and Stochastic Process’, Proc. N.A.S. USA, v. Stochastic Process’, Proc. N.A.S. USA, v. 35)35)--introduced the notion of change of introduced the notion of change of time (CT) (timetime (CT) (time--changed Brownian motion)changed Brownian motion)time (CT) (timetime (CT) (time--changed Brownian motion)changed Brownian motion)

BochnerBochner (1955) (‘Harmonic Analysis and (1955) (‘Harmonic Analysis and the Theory of Probability’, UCLA Press, the Theory of Probability’, UCLA Press, 176)176)--further development of CTfurther development of CT

Change of Time and Embedding Change of Time and Embedding

ProblemProblemThe change of time method is closely associated The change of time method is closely associated with the embedding problem: to embed a with the embedding problem: to embed a process X(t) in Brownian motion is to find a process X(t) in Brownian motion is to find a Wiener process W(t) and an increasing family of Wiener process W(t) and an increasing family of stopping times T(t) such that W(T(t)) has the stopping times T(t) such that W(T(t)) has the stopping times T(t) such that W(T(t)) has the stopping times T(t) such that W(T(t)) has the same joint distributions as X(t). The embedding same joint distributions as X(t). The embedding problem was first treated by problem was first treated by SkorokhodSkorokhod, 1961, 1961, , who showed that the sum of any sequence of who showed that the sum of any sequence of independent random variables with mean zero independent random variables with mean zero and finite variance could be embedded in and finite variance could be embedded in Brownian motion using stopping times.Brownian motion using stopping times.

Change of Time: First Intro into Financial Change of Time: First Intro into Financial EconomicsEconomics

Clark (1973)Clark (1973) (‘(‘A A

‘Subordinated Stochastic ‘Subordinated Stochastic

Process Model with Fixed Process Model with Fixed

Variance for Speculative Variance for Speculative

PricesPrices’, Econometrica, ’, Econometrica,

He wrote down a model He wrote down a model

for the logfor the log--priceprice M as M as

M(t)=B(T(t)), M(t)=B(T(t)), PricesPrices’, Econometrica, ’, Econometrica,

41, 13541, 135--156)156)--introduced introduced

Bochner’s (1949) timeBochner’s (1949) time--

changed Brownian changed Brownian

motion into financial motion into financial

economicseconomics::

where B(t) is Brownian where B(t) is Brownian

motion, T(t) is timemotion, T(t) is time--

change (B and T are change (B and T are

independent)independent)

Change of Time: Short History. I.Change of Time: Short History. I.

Feller (1966)Feller (1966) (‘(‘An Introduction to Probability An Introduction to Probability

TheoryTheory’, vol. II, NY: Wiley)’, vol. II, NY: Wiley)--introduced subordinated introduced subordinated

processes X(T(t)) with Markov process X(t) and T(t) as a processes X(T(t)) with Markov process X(t) and T(t) as a

process with independent increments (i.e., Poisson process with independent increments (i.e., Poisson

process); T(t) was called process); T(t) was called randomized operational timerandomized operational time

Johnson (1979)Johnson (1979) (‘Option Pricing When the (‘Option Pricing When the Johnson (1979)Johnson (1979) (‘Option Pricing When the (‘Option Pricing When the

Variance Rate is Changing’, working paper, UCLA)Variance Rate is Changing’, working paper, UCLA)--

introduced timeintroduced time--changed SVM in continuous timechanged SVM in continuous time

Johnson & Shanno (1987)Johnson & Shanno (1987) (‘Option Pricing (‘Option Pricing

When the Variance is Changing’, J. of Finan. & Quantit. When the Variance is Changing’, J. of Finan. & Quantit.

Analysis, 22, 143Analysis, 22, 143--151)151)--studied the pricing of options studied the pricing of options

using timeusing time--changing SVMchanging SVM

Change of Time: Short History. II.Change of Time: Short History. II.

Ikeda & WatanabeIkeda & Watanabe (1981) (‘SDEs and Diffusion (1981) (‘SDEs and Diffusion Processes’, NorthProcesses’, North--Holland Publ. Co)Holland Publ. Co)--introduced and introduced and studied CTM for the solution of SDEsstudied CTM for the solution of SDEs

BarndorffBarndorff--Nielsen, Nicolato & ShephardNielsen, Nicolato & Shephard (2003) (2003) (‘Some recent development in stochastic volatility (‘Some recent development in stochastic volatility modelling’)modelling’)--review and put in context some of their review and put in context some of their modelling’)modelling’)--review and put in context some of their review and put in context some of their recent work on stochastic volatility (SV) modelling, recent work on stochastic volatility (SV) modelling, including the relationship between subordination and SV including the relationship between subordination and SV (random time(random time--chronometerchronometer))

Carr, Geman, Madan & Yor (2003) Carr, Geman, Madan & Yor (2003) (‘SV for Levy (‘SV for Levy Processes’, mathematical Finance, vol.13)Processes’, mathematical Finance, vol.13)--used used subordinated processes to construct SV for Levy subordinated processes to construct SV for Levy Processes (T(t)Processes (T(t)--business time)business time)

TimeTime--Changed Models and SVMsChanged Models and SVMs

The probability The probability

literature has literature has

demonstrated that demonstrated that

SVMs and their timeSVMs and their time--

changed BM relatives changed BM relatives

Shephard (2005):Shephard (2005):

Stochastic Volatility, Stochastic Volatility,

working paper, working paper,

University of OxfordUniversity of Oxford

changed BM relatives changed BM relatives

and timeand time--changed changed

models are models are

fundamentalsfundamentals

Shephard (2005):Shephard (2005):

Stochastic Volatility: Stochastic Volatility:

Selected Readings, Selected Readings,

Oxford, Oxford, Oxford Oxford

University PressUniversity Press

Change of Time: Change of Time:

Simplest (Martingale) CaseSimplest (Martingale) Case

Change of Time: Change of Time:

General (Ito Integral) CaseGeneral (Ito Integral) Case

Change of Time: SDE’s CaseChange of Time: SDE’s Case

Geometric Brownian Motion SVMGeometric Brownian Motion SVM

Change of Time MethodChange of Time Method

Solution for GBM EquationSolution for GBM Equation

Using Change of TimeUsing Change of Time

Explicit Expression forExplicit Expression for

MeanMean--Reverting SV ModelReverting SV Model

Solution of MRM by CTMSolution of MRM by CTM

Explicit Expression forExplicit Expression for

Explicit Expression forExplicit Expression for

Comparison: Solution of GBM & MRM)Comparison: Solution of GBM & MRM)

Explicit Expression for S(t)Explicit Expression for S(t)

Heston Model (1993)Heston Model (1993)

Explicit Solution for CIR Process: CTMExplicit Solution for CIR Process: CTM

Comparison: Solutions to the Three Comparison: Solutions to the Three

ModelsModels

-GBM

-MRM

-Heston model

Volatility Swap for Heston Model. I.Volatility Swap for Heston Model. I.

Why Trade Volatility (Variance)?Why Trade Volatility (Variance)?

Statistics on Log Returns of S&P Canada Statistics on Log Returns of S&P Canada Index (Jan 1997Index (Jan 1997--Feb 2002)Feb 2002)

Histograms of LogHistograms of Log--Returns Returns for S&P60 Canada Indexfor S&P60 Canada Index

Convexity AdjustmentConvexity Adjustment

S&P60 Canada Index Volatility SwapS&P60 Canada Index Volatility Swap

Wilmott, Javaheri & Haug (2002) Wilmott, Javaheri & Haug (2002)

ModelModel

Wilmott, Javaheri & HaugWilmott, Javaheri & Haug (GARCH and (GARCH and Volatility Swaps,Wilmott Magazine, 2002) Volatility Swaps,Wilmott Magazine, 2002) ResultResult

-continuous-time GARCH(1,1) model

Wilmott, Javaheri & Haug (2002) Wilmott, Javaheri & Haug (2002)

Volatility SwapVolatility Swap

ComparisonComparisonWilmott, Javaheri&Haug (2002)

Continuous-time GARCH(1,1)

S&P500

Sw (2004), Heston model,

S&P60 Canada Index

ComparisonComparisonWilmott, Javaheri&Haug (2002)

Convexity adjustment, S&P500

Sw (2004), Convexity adjustment,

S&P60 Canada Index

Summary (SV and CT)Summary (SV and CT)

-martingale

1.

GBM Model

Mean-Reverting Model

-martingale

-sum of two martingales

2.

3.

Heston Model

Problems. I.Problems. I.

-explicit expression ?

Problems. IIProblems. II

ContinuousContinuous--Time Time

GARCH SV GARCH SV

Model with DelayModel with Delay

Solution by Solution by

Change of Time Change of Time

Method?Method?

The EndThe End

Thank you for your time and Thank you for your time and Thank you for your time and Thank you for your time and

attention!attention!