期权期货及其衍生品第27弹.pptx

Chapter 27Martingales and MeasuresOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20121Derivatives Dependent on a Single Underlying VariableOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20122Forming a Riskless PortfolioOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20123Market Price of Risk(Page 632)This shows that(m r)/s is the same for all derivatives dependent on the same underlying variable,qWe refer to(m r)/s as the market price of risk for q and denote it by lOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20124Extension of the Analysisto Several Underlying Variables(Equations 27.12 and 27.13,page 634)Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20125Martingales(Page 635)A martingale is a stochastic process with zero driftA variable following a martingale has the property that its expected future value equals its value todayOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20126Alternative WorldsOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20127The Equivalent Martingale Measure Result(Page 635-36)Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20128Forward Risk NeutralityWe will refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral with respect to g.If Eg denotes expectations in a world that is FRN wrt gOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 20129Alternative Choices for the Numeraire Security gMoney Market AccountZero-coupon bond priceAnnuity factorOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201210 Money Market Accountas the NumeraireThe money market account is an account that starts at$1 and is always invested at the short-term risk-free interest rateThe process for the value of the account isdg=rg dtThis has zero volatility.Using the money market account as the numeraire leads to the traditional risk-neutral world where l=0Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201211Money Market Accountcontinued Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201212Zero-Coupon Bond Maturing at time T as Numeraire Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201213Forward PricesIn a world that is FRN wrt P(0,T),the expected value of a security at time T is its forward priceOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201214Interest RatesIn a world that is FRN wrt P(0,T2)the expected value of an interest rate lasting between times T1 and T2 is the forward interest rateOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201215Annuity Factor as the NumeraireOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201216Annuity Factors and Swap RatesSuppose that s(t)is the swap rate corresponding to the annuity factor A.Then:s(t)=EAs(T)Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201217Extension to Several Independent Factors(Page 640)Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201218Extension to Several Independent Factors continuedOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201219ApplicationsExtension of Blacks model to case where inbterest rates are stochasticValuation of an option to exchange one asset for anotherOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201220Blacks Model(page 641)By working in a world that is forward risk neutral with respect to a P(0,T)it can be seen that Blacks model is true when interest rates are stochastic providing the forward price of the underlying asset is has a constant volatilityc=P(0,T)F0N(d1)KN(d2)p=P(0,T)KN(d2)F0N(d1)Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201221Option to exchange an asset worth U for one worth VThis can be valued by working in a world that is forward risk neutral with respect to UValue isOptions,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201222Change of Numeraire(Section 27.8,page 643)Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 201223