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1、Chapter 27Martingales and Measures,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,1,Derivatives Dependent on a Single Underlying Variable,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,2,Forming a Riskless Portfolio,Options,Futures,and Othe
2、r Derivatives,8th Edition,Copyright John C.Hull 2012,3,Market 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 l,Options,Futures,and Other Derivatives,8th Ed
3、ition,Copyright John C.Hull 2012,4,Extension 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 2012,5,Martingales(Page 635),A martingale is a stochastic process with zero driftA variable followin
4、g a martingale has the property that its expected future value equals its value today,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,6,Alternative Worlds,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,7,The Equivalent Martingale Measure Res
5、ult(Page 635-36),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,8,Forward Risk Neutrality,We 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
6、is FRN wrt g,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,9,Alternative Choices for the Numeraire Security g,Money Market AccountZero-coupon bond priceAnnuity factor,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,10,Money Market Accountas
7、 the Numeraire,The 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 wor
8、ld where l=0,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,11,Money Market Accountcontinued,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,12,Zero-Coupon Bond Maturing at time T as Numeraire,Options,Futures,and Other Derivatives,8th Editio
9、n,Copyright John C.Hull 2012,13,Forward Prices,In a world that is FRN wrt P(0,T),the expected value of a security at time T is its forward price,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,14,Interest Rates,In a world that is FRN wrt P(0,T2)the expected value of an i
10、nterest rate lasting between times T1 and T2 is the forward interest rate,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,15,Annuity Factor as the Numeraire,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,16,Annuity Factors and Swap Rates,Sup
11、pose 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 2012,17,Extension to Several Independent Factors(Page 640),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,18,Ext
12、ension to Several Independent Factors continued,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,19,Applications,Extension of Blacks model to case where inbterest rates are stochasticValuation of an option to exchange one asset for another,Options,Futures,and Other Deriva
13、tives,8th Edition,Copyright John C.Hull 2012,20,Blacks 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 volatili
14、tyc=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 2012,21,Option to exchange an asset worth U for one worth V,This can be valued by working in a world that is forward risk neutral with respect to UValue is,Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,22,Change of Numeraire(Section 27.8,page 643),Options,Futures,and Other Derivatives,8th Edition,Copyright John C.Hull 2012,23,
