Pricing American Options under Stochastic Volatility .pdf

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1、DepartmentofEconomicsPricingAmericanOptionsunderStochasticVolatility:ANewMethodUsingChebyshevPolynomialstoApproximatetheEarlyExerciseBoundaryEliasTzavalisandShijunWangWorkingPaperNo.488February2003ISSN1473-0278PricingAmericanOptionsunderstochasticvolatility:Anewmethodu

2、singChebyshevpolynomialstoapproximatetheEarlyExerciseBoundaryEliasTzavalis∗andShijunWang†DepartmentofEconomics,QueenMary,UniversityofLondonLondonE14NS,UKThisversionFebruary,2003AbstractThispaperpresentsanewnumericalmethodforpricingAmericancalloptionswhenthevolatilityof

3、thepriceoftheunderlyingstockisstochastic.Byexploitingalog-linearrelationshipoftheoptimalexer-ciseboundarywithrespecttovolatilitychanges,wederiveanintegralrepresentationofanAmericancallpriceandtheearlyexercisepremiumwhichholdsunderstochasticvolatility.Thisrepresentation

4、isusedtode-velopanumericalmethodforpricingtheAmericanoptionsbasedonanapproximationoftheoptimalexerciseboundarybyChebyshevpolyno-mials.NumericalresultsshowthatournumericalapproachcanquicklyandaccuratelypriceAmericancalloptionsbothunderstochasticand/orconstantvolatility.

5、Keywords:Americancalloption,stochasticvolatility,earlyexerciseboundary,Chebyshevpolynomials.JELClassification:G12,G13,C63∗e.tzavalis@qmul.ac.uk†s.j.wang@qmul.ac.uk11IntroductionPricingAmericanoptionsisoneofthemostdifficultproblemsinoptionpricingliterature.Thedifficultystems

6、fromthefactthat,unlikeaEuropean,anAmer-icancall(orput)optionhasnoexplicitclosedformsolution.ThishappensbecausetheoptimalboundaryabovewhichtheAmericancalloptionwillbeexercisedisunknownandpartoftheoptionpricesolution.Therefore,effortshavebeenconcentratedondevelopingnumeri

7、calapproximationschemeswhichcanpricetheAmericanoptionsaccuratelyandfasterthanthelatticeorsim-ulationbasedmethods,whicharetimeconsumingandcomputationallymoredemanding.TheseschemesarebasedonintegralrepresentationsoftheAmer-icanoptionevaluationformulaortheyexploittheparti

8、aldifferentialequationsatisfiedbytheoptionprices.1TheexistingapproximationschemesforpricingAmericancall(orput)op-tionsi