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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.JELClassi?cation:G12,G13,C63?e.tzavalis@qmul.ac.uk?s.j.wang@qmul.ac.uk11IntroductionPricingAmericanoptionsisoneofthemostdi?cultproblemsinoptionpricingliterature.Thedi?cultystems
6、fromthefactthat,unlikeaEuropean,anAmer-icancall(orput)optionhasnoexplicitclosedformsolution.ThishappensbecausetheoptimalboundaryabovewhichtheAmericancalloptionwillbeexercisedisunknownandpartoftheoptionpricesolution.Therefore,e?ortshavebeenconcentratedondevelopingnumeri
7、calapproximationschemeswhichcanpricetheAmericanoptionsaccuratelyandfasterthanthelatticeorsim-ulationbasedmethods,whicharetimeconsumingandcomputationallymoredemanding.TheseschemesarebasedonintegralrepresentationsoftheAmer-icanoptionevaluationformulaortheyexploittheparti
8、aldi?erentialequationsatis?edbytheoptionprices.1TheexistingapproximationschemesforpricingAmericancall(orput)op-tionsi
