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Financial Engineering PDF

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Introduction to the Handbook of Financial Engineering JohnR.Birge GraduateSchoolofBusiness,UniversityofChicago,USA VadimLinetsky DepartmentofIndustrialEngineeringandManagementSciences,NorthwesternUniversity, USA Financial engineering (FE) is an interdisciplinary field focusing on applica- tions of mathematical and statistical modeling and computational technology to problems in the financial services industry. According to the report by the National Academy of Engineering (2003),1 “Financial services are the foun- dation of a modern economy. They provide mechanisms for assigning value, exchanging payment, and determining and distributing risk, and they provide theessentialunderpinningsofglobaleconomicactivity.Theindustryprovides thewherewithalforthecapitalinvestmentthatdrivesinnovationandproduc- tivity growth throughout the economy.” Important areas of FE include math- ematicalmodelingofmarketandcreditrisk,pricingandhedgingofderivative securitiesusedtomanagerisk,assetallocationandportfoliomanagement. Market risk is a risk of adverse changes in prices or rates, such as interest rates, foreign exchange rates, stock prices, and commodity and energy prices. Credit risk is a risk of default on a bond, loan, lease, pension or any other type of financial obligation. Modern derivatives markets can be viewed as a global marketplace for financial risks. The function of derivative markets is to facilitate financial risk transfer from risk reducers (hedgers) to risk takers (investors). Organizations wishing to reduce their risk exposure to a particu- lartype of financial risk, such as the risk of increasing commodity and energy pricesthatwillmakefutureproductionmoreexpensiveortheriskofincreas- ing interest rates that will make future financing more expensive, can offset thoserisksbyenteringintofinancialcontractsthatactasinsurance,protecting the company against adverse market events. While the hedger comes to the derivativesmarkettoreduceitsrisk,thecounterpartywhotakestheotherside 1NationalAcademyofEngineering,TheImpactofAcademicResearchonIndustrialPerformance, NationalAcademiesPress,Washington,DC,2003,http://www.nap.edu/books/0309089735/html. 3 4 J.R.BirgeandV.Linetsky ofthecontractcomestothemarkettoinvestinthatriskandexpectstobead- equately compensated for taking the risk. We can thus talk about buying and sellingfinancialrisks. Global derivatives markets have experienced remarkable growth over the past several decades. According to a recent survey by the Bank for Interna- tionalSettlementinBasel(2006),2 theaggregatesizeoftheglobalderivatives marketswentfromabout$50trillioninnotionalamountsin1995to$343tril- lioninnotionalamountsbytheendof2005($283trillioninnotionalamounts inover-the-counterderivativescontractsand$58trillioninfuturesandoptions traded on derivatives exchanges worldwide). Major segments of the global derivatives markets include interest rate derivatives, currency derivatives, eq- uityderivatives,commodityandenergyderivatives,andcreditderivatives. A derivative is a financial contract between two parties that specifies con- ditions, in particular, dates and the resulting values of underlying variables, under which payments or payoffs are to be made between parties (payments can be either in the form of cash or delivery of some specified asset). Call andputoptionsarebasicexamplesofderivativesusedtomanagemarketrisk. Acall option is a contract that gives its holder the right to buy somespecified quantityofanunderlyingasset(forexample,afixednumberofsharesofstock ofaparticularcompanyorafixedamountofacommodity)atapredetermined price(calledthestrikeprice)onorbeforeaspecifieddateinthefuture(option expiration).Aputoptionisacontractthatgivesitsholdertherighttosellsome specified quantity of an underlying asset at a predetermined price on or be- foreexpiration.Theholderoftheoptioncontractlocksinthepriceforfuture purchase (in the case of call options) or future sale (in the case of put op- tions),thus eliminating anypriceuncertainty orrisk, atthe costofpayingthe premiumtopurchasetheoption.Thesituationisanalogoustoinsurancecon- tractsthatpaypre-agreedamountsintheeventoffire,flood,caraccident,etc. In financial options, the payments are based on financial market moves (and crediteventsinthecaseofcreditderivatives).Justasintheinsuranceindustry thekeyproblemistodeterminetheinsurancepremiumtochargeforapolicy basedonactuarialassessmentsofeventprobabilities,theoption-pricingprob- lemistodeterminethepremiumoroptionpricebasedonastochasticmodel oftheunderlyingfinancialvariables. Portfolio optimization problems constitute another major class of impor- tantproblemsinfinancialengineering.Portfoliooptimizationproblemsoccur throughout the financial services industry as pension funds, mutual funds, in- surance companies, university and foundation endowments, and individual investorsallfacethefundamentalproblemofallocatingtheircapitalacrossdif- ferentsecuritiesinordertogenerateinvestmentreturnssufficienttoachievea particulargoal,suchasmeetingfuturepensionliabilities.Theseproblemsare 2BankforInternationalSettlementQuarterlyReview,June2006,pp.A103–A108,http://www.bis.org/ publ/qtrpdf/r_qa0606.pdf. IntroductiontotheHandbookofFinancialEngineering 5 often very complex owing to their dynamic and stochastic nature, their high dimensionality,andthecomplexityofreal-worldconstraints. Theremarkablegrowthoffinancialmarketsoverthepastdecadeshasbeen accompanied by an equally remarkable explosion in financial engineering re- search.Thegoalsoffinancialengineeringresearcharetodevelopempirically realisticstochasticmodelsdescribingdynamicsoffinancialriskvariables,such asassetprices,foreignexchangerates,andinterestrates,andtodevelopana- lytical,computationalandstatisticalmethodsandtoolstoimplementthemod- elsandemploythemtoevaluatefinancialproductsusedtomanageriskandto optimallyallocateinvestmentfundstomeetfinancialgoals.Asfinancialmod- elsarestochastic,probabilitytheoryandstochasticprocessesplayacentralrole infinancialengineering.Furthermore,inordertoimplementfinancialmodels, awidevarietyofanalyticalandcomputationaltoolsareused,includingMonte Carlosimulation,numericalPDEmethods,stochasticdynamicprogramming, Fouriermethods,spectralmethods,etc. TheHandbookisorganizedinsixparts:Introduction,DerivativeSecurities: ModelsandMethods,InterestRateandCreditRiskModelsandDerivatives, Incomplete Markets, Risk Management, and Portfolio Optimization. This di- visionissomewhatartificial,asmanychaptersareequallyrelevantforseveral orevenalloftheseareas.Nevertheless,thisstructureprovidesanoverviewof themainareasofthefieldoffinancialengineering. A working knowledge of probability theory and stochastic processes is a prerequisite to reading many of the chapters in the Handbook. Karatzas and Shreve (1991) and Revuz and Yor (1999) are standard references on Brownian motion and continuous martingales. Jacod and Shiryaev (2002) and Protter (2005) are standard references on semimartingale processes with jumps. Shreve (2004) and Klebaner (2005) provide excellent introductions to stochastic calculus for finance at a less demanding technical level. For the financial background at the practical level, excellent overviews of deriv- atives markets and financial risk management can be found in Hull (2005) andMcDonald(2005).KeytextsonassetpricingtheoryincludeBjork(2004), Duffie(2001),Jeanblancetal.(2007),andKaratzasandShreve(2001).These monographsalsocontainextensivebibliographies. In Chapter 1 “A Partial Introduction to Financial Asset Pricing Theory,” Robert Jarrow and Philip Protter present a concise introduction to Mathe- matical Finance theory. The reader is first introduced to derivative securities andthefundamentalfinancialconceptofarbitrageinthebinomialframework. Thecoreassetpricingtheoryisthendevelopedinthegeneralsemimartingale framework, assuming prices of risky assets follow semimartingale processes. The general fundamental theorems of asset pricing are formulated and illus- trated on important examples. In particular, the special case when the risky asset price process is a Markov process is treated in detail, the celebrated Black–Scholes–Merton model is derived, and a variety of results on pricing European- and American-style options and more complex derivative securi- 6 J.R.BirgeandV.Linetsky tiesarepresented.ThischaptersummarizesthecoreofMathematicalFinance theoryandisanessentialreading. PartII “Derivative Securities:Models andMethods,” contains chapterson a range of topics in derivatives modeling and pricing. The first three chap- ters survey several important classes of stochastic models used in derivatives modeling.InChapter2“Jump-DiffusionModels,”StevenKousurveysrecent developments in option pricing in jump-diffusion models. The chapter dis- cussesempiricalevidenceofjumpsinfinancialvariablesandsurveysanalytical and numerical methods for the pricing of European, American, barrier, and lookback options in jump-diffusion models, with particular attention given to thejump-diffusionmodelwithadouble-exponentialjumpsizedistributiondue toitsanalyticaltractability. InChapter3“ModelingFinancialSecurityReturnsUsingLevyProcesses,” Liuren Wu surveys a class of models based on time-changed Levy processes. Applying stochastic time changes to Levy processes randomizes the clock on which the process runs, thus generating stochastic volatility. If the character- istic exponent of the underlying Levy process and the Laplace transform of the time change process are known in closed form, then the pricing of op- tions can be accomplished by inverting the Fourier transform, which can be done efficiently using the fast Fourier transform (FFT) algorithm. The com- bination of this analytical and computational tractability and the richness of possibleprocessbehaviors(continuousdynamics,aswellasjumpsoffiniteac- tivity or infinite activity) make this class of models attractive for a wide range offinancialengineeringapplications.Thischaptersurveysboththetheoryand empiricalresults. InChapter4“PricingwithWishartRiskFactors,”ChristianGourierouxand RazvanSufanasurveyassetpricingbasedonriskfactorsthatfollowaWishart process. The class of Wishart models can be thought of as multi-factor exten- sionsofaffinestochastic volatility models,whichmodelastochastic variance- covariance matrix as a matrix-valued stochastic process. As for the standard affine processes, the conditional Laplace transforms can be derived in closed form for Wishart processes. This chapter surveys Wishart processes and their applications to building a wide range of multi-variate models of asset prices with stochastic volatilities and correlations, multi-factor interest rate models, andcreditriskmodels,bothindiscreteandincontinuoustime. In Chapter 5 “Volatility,” Federico Bandi and Jeff Russell survey the state of the literature on estimating asset price volatility. They provide a unified framework to understand recent advances in volatility estimation by virtue of microstructurenoisecontaminatedassetpricedataandtransactioncostevalu- ation.Theemphasisisonrecentlyproposedidentificationproceduresthatrely onassetpricedatasampledathigh frequency.Volatility isthe keyfactorthat determines option prices, and, as such, better understanding of volatility is of keyinterestinoptionspricing. In Chapter 6 “Spectral Methods in Derivatives Pricing,” Vadim Linetsky surveys a problem of valuing a (possibly defaultable) derivative asset contin- IntroductiontotheHandbookofFinancialEngineering 7 gent on the underlying economic state modeled asa Markov process.To gain analytical and computational tractability both in order to estimate the model from empirical data and to compute the prices of derivative assets, financial modelsinapplicationsareoftenMarkovian.Inapplications,itisimportantto haveatoolkitofanalyticallytractableMarkovprocesseswithknowntransition semigroupsthatleadtoclosed-formexpressionsforpricesofderivativeassets. The spectral expansion method is a powerful approach to generate analytical solutions for Markovian problems. This chapter surveys the spectral method ingeneral,aswellasthoseclassesofMarkovprocessesforwhichthespectral representation can be obtained in closed form, thus generating closed form solutionstoMarkovianderivativepricingproblems. WhenunderlyingfinancialvariablesfollowaMarkovjump-diffusionprocess, thevaluefunctionofaderivativesecuritysatisfiesapartialintegro-differential equation (PIDE) for European-style exercise or a partial integro-differential variationalinequality(PIDVI)forAmerican-styleexercise.UnlesstheMarkov process has a special structure (as discussed in Chapter 6), analytical solu- tions are generally not available, and it is necessary to solve the PIDE or the PIDVInumerically.InChapter7“VariationalMethodsinDerivativesPricing,” Liming Feng, Pavlo Kovalov, Vadim Linetsky and Michael Marcozzi survey a computational method for the valuation of options in jump-diffusion models based on converting the PIDE or PIDVI to a variational (weak) form, dis- cretizingtheweakformulationspatiallybytheGalerkinfiniteelementmethod to obtain a system of ODEs, and integrating the resulting system of ODEs in time. In Chapter 8 “Discrete Path-Dependent Options,” Steven Kou surveys re- centadvancesinthedevelopmentofmethodstopricediscretepath-dependent options, such as discrete barrier and lookback options that sample the mini- mumormaximumoftheassetpriceprocessatdiscretetimeintervals,including discretebarrierandlookbackoptions.Awidearrayofoptionpricingmethods are surveyed, including convolution methods, asymptotic expansion methods, andmethodsbasedonLaplace,HilbertandfastGausstransforms. PartIIIsurveysinterestrateandcreditriskmodelsandderivatives.InChap- ter 9 “Topics in Interest Rate Theory” Tomas Bjork surveys modern interest rate theory. The chapter surveys both the classical material on the Heath– Jarrow–MortonforwardratemodelingframeworkandontheLIBORmarket modelspopularinmarketpractice,aswellasarangeofrecentadvancesinthe interest rate modeling literature, including the geometric interest rate theory (issues of consistency and existence of finite-dimensional state space realiza- tions),andpotentialsandpositiveinterestmodels. Chapters 10 and 11 survey the state-of-the-art in modeling portfolio credit riskandmulti-namecreditderivatives.InChapter10“ComputationalAspects of Credit Risk,” Paul Glasserman surveys modeling and computational issues associated with portfolio credit risk. A particular focus is on the problem of calculatingthelossdistributionofaportfolioofcreditriskyassets,suchascor- porate bonds or loans. The chapter surveys models of dependence, including 8 J.R.BirgeandV.Linetsky structuralcreditriskmodels,copulamodels,themixedPoissonmodel,andas- sociatedcomputationaltechniques,includingrecursiveconvolution,transform inversion, saddlepoint approximation, and importance sampling for Monte Carlosimulation. InChapter11“ValuationofBasketCreditDerivativesintheCreditMigra- tions Environment,” Tomasz Bielecki, Stephane Crepey, Monique Jeanblanc andMarekRutkowskipresentmethodstovalueandhedgebasketcreditderiv- atives (such as collateralized debt obligations (CDO) tranches and nth to de- faultswaps)andportfoliosofcreditriskydebt.Thechapterpresentsmethods formodelingdependentcreditmigrationsofobligorsamongcreditclassesand, inparticular,dependentdefaults.ThefocusisonspecificclassesofMarkovian modelsforwhichcomputationscanbecarriedout. PartIVsurveysincompletemarketstheoryandapplications.Inincomplete markets,dynamichedgingandperfectreplicationofderivativesecuritiesbreak down and derivatives are no longer redundant assets that can be manufac- tured via dynamic trading in the underlying primary securities. In Chapter 12 “Incomplete Markets,” Jeremy Staum surveys, compares and contrasts many proposed approaches to pricing and hedging derivative securities in incom- pletemarkets,fromtheperspectiveofanover-the-counterderivativesmarket makeroperatinginanincompletemarket.Thechapterdiscussesawiderange ofmethods,includingindifferencepricing,gooddealbounds,marginalpricing, andminimum-distancepricingmeasures. In Chapter 13 “Option Pricing: Real and Risk-Neutral Distributions,” George Constantinides, Jens Jackwerth, and Stylianos Perrakis examine the pricingofoptionsinincompleteandimperfectmarketsinwhichdynamictrad- ing breaks down either because the market is incomplete or because it is im- perfectduetotradingcosts,orboth.Marketincompletenessrenderstherisk- neutral probability measure nonunique and allows one to determine option prices only within some lower and upper bounds. Moreover, in the presence of trading costs, the dynamic replicating strategy does not exist. The authors examinemodificationsofthetheoryrequiredtoaccommodateincompleteness andtrading costs, survey testable implications ofthe theory foroption prices, andsurveyempiricalevidenceinequityoptionsmarkets. In Chapter 14 “Total Risk Minimization Using Monte Carlo Simulation,” ThomasColeman,YuyingLi,andMaria-CristinaPatronstudyoptionshedging strategiesinincompletemarkets.Whileinanincompletemarketitisgenerally impossible to replicate an option exactly, total risk minimization chooses an optimalself-financingstrategythatbestapproximatestheoptionpayoffbyits terminal value. Total risk minimization is a computationally challenging dy- namicstochasticprogrammingproblem.Thischapterpresentscomputational approachestotacklethisproblem. In Chapter 15 “Queueing Theoretic Approaches to Financial Price Fluc- tuations,” Erhan Bayraktar, Ulrich Horst, and Ronnie Sircar survey recent research on agent-based market microstructure models. These models of fi- nancial prices are based on queueing-type models of order flows and are ca- IntroductiontotheHandbookofFinancialEngineering 9 pable of explaining many stylized features of empirical data, such as herding behavior,volatilityclustering,andfattailedreturndistributions.Inparticular, the chapter examines models of investor inertia, providing a link with behav- ioralfinance. Part V “Risk Management” contains chapters concerned with risk mea- surement and its application to capital allocation, liquidity risk, and actuarial risk. In Chapter 16 “Economic Credit Capital Allocation and Risk Contribu- tions,” Helmut Mausser and Dan Rosen provide a practical overview of risk measurementandmanagementprocess,andinparticularthemeasurementof economiccapital(EC)contributionsandtheirapplicationtocapitalallocation. ECactsasabufferforfinancialinstitutionstoabsorblargeunexpectedlosses, therebyprotectingdepositorsandotherclaimholders.OncetheamountofEC has been determined, it must be allocated among the various components of theportfolio(e.g.,businessunits,obligors,individualtransactions).Thischap- terprovidesanoverviewoftheprocessofriskmeasurement,itsstatisticaland computational challenges, and its application to the process of risk manage- mentandcapitalallocationforfinancialinstitutions. In Chapter 17 “Liquidity Risk and Option Pricing Theory,” Robert Jarrow and Phillip Protter survey recent research advances in modeling liquidity risk andincludingitintoassetpricingtheory.Classicalassetpricingtheoryassumes that investors’ trades have no impact on the prices paid or received. In real- ity, there is a quantity impact on prices. The authors show how to extend the classicalarbitragepricingtheoryand,inparticular,thefundamentaltheorems of asset pricing, to include liquidity risk. This is accomplished by studying an economy where the security’s price depends on the trade size. An analysis of thetheoryandapplicationstomarketdataarepresented. In Chapter 18 “Financial Engineering: Applications in Insurance,” Phelim BoyleandMaryHardyprovideanintroductiontotheinsurancearea,theold- est branch of risk management, and survey financial engineering applications in insurance. The authors compare the actuarial and financial engineering approaches to risk assessment and focus on the life insurance applications in particular. Life insurance products often include an embedded investment component, andthus requirethe mergingofactuarialandfinancial riskman- agementtoolsofanalysis. PartVIisdevotedtoportfoliooptimization.InChapter19“DynamicPort- folioChoiceandRiskAversion,”CostisSkiadassurveysoptimalconsumption andportfoliochoicetheory,withtheemphasisonthemodelingofriskaversion given a stochastic investment opportunity set. Dynamic portfolio choice the- ory was pioneered in Merton’s seminal work, who assumed that the investor maximizes time-additive expected utility and approached the problem using theHamilton–Jacobi–Bellmanequationofoptimalcontroltheory.Thischap- ter presents a modern exposition of dynamic portfolio choice theory from a moreadvancedperspectiveofrecursiveutility.Themathematicaltoolsinclude backwardstochasticdifferentialequations(BSDE)andforward–backwardsto- chasticdifferentialequations(FBSDE). 10 J.R.BirgeandV.Linetsky InChapter20“OptimizationMethodsinDynamicPortfolioManagement,” John Birge describes optimization algorithms and approximations that apply to dynamic discrete-time portfolio models including consumption-investment problems,asset-liabilitymanagement,anddynamichedgingpolicydesign.The chapter develops an overall structure to the many methods that have been proposedbyinterpretingthemintermsoftheformofapproximationusedto obtaintractablemodelsandsolutions.Thechapterincludestherelevantalgo- rithmsassociatedwiththeapproximationsandtherolethatportfolioproblem structureplaysinenablingefficientimplementation. In Chapter 21 “Simulation Methods for Optimal Portfolios,” Jerome De- temple, Rene Garcia and Marcel Rindisbacher survey and compare Monte Carlo simulation methods that have recently been proposed for the compu- tation of optimal portfolio policies. Monte Carlo simulation is the approach ofchoiceforhigh-dimensionalproblemswithlargenumberofunderlyingvari- ables. Simulation methods have recently emerged as natural candidates for the numerical implementation of optimal portfolio rules in high-dimensional portfolio choice models. The approaches surveyed include the Monte Carlo Malliavinderivativemethod,theMonteCarlocovariationmethod,theMonte Carloregressionmethod, andthe Monte Carlofinite differencemethod. The mathematicaltoolsincludeMalliavin’sstochasticcalculusofvariations,abrief surveyofwhichisincludedinthechapter. In Chapter 22 “Duality Theory and Approximate Dynamic Programming forPricingAmericanOptionsandPortfolioOptimization,”MartinHaughand Leonid Kogan describe how duality and approximate dynamic programming can be applied to construct approximate solutions to American option pric- ing and portfolio optimization problems when the underlying state space is high-dimensional. While it has long been recognized that simulation is an in- dispensabletoolinfinancialengineering,itisonlyrecentlythatsimulationhas begun to play an important role in control problems in financial engineering. This chapter surveys recent advances in applying simulation to solve optimal stoppingandportfoliooptimizationproblems. InChapter23“AssetAllocationwithMultivariateNon-GaussianReturns,” DilipMadanandJu-YiYenconsideraproblemofoptimalinvestmentinassets withnon-Gaussianreturns.Theypresentandbacktestanassetallocationpro- cedurethataccountsforhighermomentsininvestmentreturns.Theprocedure is made computationally efficient by employing a signal processing technique knownasindependentcomponentanalysis(ICA)toidentifylong-tailedinde- pendentcomponentsinthevectorofassetreturns.Themultivariateportfolio allocation problem is then reduced to univariate problems of component in- vestment. They further assume that the ICs follow the variance gamma (VG) Levy processes and build a multivariate VG portfolio and analyze empirical results of the optimal investment strategy in this setting and compare it with theclassicalmean–varianceGaussiansetting. In Chapter 24 “Large Deviation Techniques and Financial Applications,” PhelimBoyle,ShuiFengandWeidongTiansurveyrecentapplicationsoflarge IntroductiontotheHandbookofFinancialEngineering 11 deviationtechniquesinportfoliomanagement(establishingportfolioselection criteria, construction of performance indexes), risk management (estimation oflargecreditportfoliolossesthatoccurinthetailofthedistribution),Monte Carlo simulation to better simulate rare events for risk management and as- set pricing, and incomplete markets models (estimation of the distance of an incomplete model to a benchmark complete model). A brief survey of the mathematicsoflargedeviationsisincludedinthechapter. Anumberofimportanttopicsthathaverecentlybeenextensivelysurveyed elsewhere were not included in the Handbook. Statistical estimation of sto- chastic models in finance is an important area that has received limited at- tentioninthisvolume,withtheexceptionofthefocusedchapteronvolatility. Recent advances in this area are surveyed in the forthcoming Handbook of Financial Econometrics editedbyAit-SahaliaandHansen(2007).Inthe cov- erage of credit risk the Handbook is limited to surveying recent advances in multi-name credit portfolios and derivatives in Chapters 10 and 11, leaving out single-name credit models. The latter have recently been extensively sur- veyed in monographs Bielecki and Rutkowski (2002), Duffie and Singleton (2003), and Lando (2004). The coverage of Monte Carlo simulation meth- ods is limited to applications to multi-name credit portfolios in Chapter 10, to hedging in incomplete markets in Chapter 14, and to portfolio optimiza- tioninChapters21and22.MonteCarlosimulationapplicationsinderivatives valuation have recently been surveyed by Glasserman (2004). Our coverage of risk measurement and risk management is limited to Chapters 16, 17 and 18 on economic capital allocation, liquidity risk, and insurance risk, respec- tively. We refer the reader to the recently published monograph McNeil et al. (2005) for extensive treatments of Value-at-Risk and related topics. Mod- eling energy and commodity markets and derivatives is an important area of financialengineeringnotcoveredintheHandbook.Wereferthereadertothe recent monographs by Eydeland and Wolyniec (2002) and Geman (2005) for extensivesurveysofenergyandcommoditymarkets. References Ait-Sahalia,Y.,Hansen,L.P.(Eds.)(2007).HandbookofFinancialEconometrics.Elsevier,Amsterdam, inpress. Bielecki,T.,Rutkowski,M.(2002).CreditRisk:Modeling,ValuationandHedging.Springer. Bjork,T.(2004).ArbitrageTheoryinContinuousTime,seconded.OxfordUniversityPress,Oxford,UK. Duffie,D.(2001).DynamicAssetPricingTheory,thirded.PrincetonUniversityPress,Princeton,NJ. Duffie,D.,Singleton,K.(2003).CreditRisk.PrincetonUniversityPress,Princeton,NJ. Eydeland,A.,Wolyniec,K.(2002).EnergyandPowerRiskManagement:NewDevelopmentsinModeling, PricingandHedging.JohnWiley&Sons,NewJersey. Jacod,J.,Shiryaev,A.N.(2002).LimitTheoremsforStochasticProcesses,seconded.Springer. Jeanblanc,M.,Yor,M.,Chesney,M.(2007).MathematicalMethodsforFinancialMarkets.Springer,in press. Geman, H. (2005). Commodities and Commodity Derivatives: Modeling and Pricing for Agriculturals, Metals,andEnergy.JohnWiley&Sons,NewJersey. 12 J.R.BirgeandV.Linetsky Glasserman,P.(2004).MonteCarloMethodsinFinancialEngineering.Springer. Hull,J.(2005).Options,Futures,andOtherDerivatives,sixthed.PrenticeHall. Karatzas,I.,Shreve,S.E.(1991).BrownianMotionandStochasticCalculus,seconded.Springer. Karatzas,I.,Shreve,S.E.(2001).MethodsofMathematicalFinance.Springer. Klebaner,F.C.(2005).IntroductiontoStochasticCalculuswithApplications,seconded.ImperialCollege Press. Lando,D.(2004).CreditRiskModeling.PrincetonUniversityPress,Princeton,NJ. McDonald,R.L.(2005).DerivativesMarkets,seconded.Addison–Wesley. McNeil,A.J.,Frey,R.,Embrechts,P.(2005).QuantitativeRiskManagement:Concepts,Techniques,and Tools.PrincetonUniversityPress,Princeton,NJ. Protter,P.E.(2005).StochasticIntegrationandDifferentialEquations,seconded.Springer. Revuz,D.,Yor,M.(1999).ContinuousMartingalesandBrownianMotion,thirded.Springer. Shreve,S.E.(2004).StochasticCalculusforFinanceII:Continuous-timeModels.Springer.

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