期權,期貨及其衍生品第27彈ppt課件

1、Chapter 27Martingales and MeasuresOptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 20191Derivatives Dependent on a Single Underlying VariableOptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 20192 dz dtd dz dtd dzsdtmd22221111. Suppose and price

2、s withon dependent sderivative two Imagine process the follows that security) traded a ofprice they necessaril (not variable a Consider21Forming a Riskless PortfolioOptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 20193t= )()()(211221212111221122derivative 2nd the of and

3、derivative 1st the of + of consisting portfolio riskless a up set can WeMarket 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 O

4、ther Derivatives, 8th Edition, Copyright John C. Hull 20194 or :gives This :riskless is portfolio the Since2211121221rrr r t=r 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 201

5、95 then withvariables underlying several on depends If r dz dtdfniiiniii11Martingales (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 Edi

6、tion, Copyright John C. Hull 20196Alternative WorldsOptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 20197dzfdtfrdfdzfdtrfdf)( is risk of price market the where worlda In worldneutral-risk ltraditiona the InThe Equivalent Martingale Measure Result (Page 635-36) Options, F

7、utures, and Other Derivatives, 8th Edition, Copyright John C. Hull 20198fgfg pricessecurity derivative allfor martingale a is that shows lemma sIto then ,security a of volatility the to equal set weIfForward Risk NeutralityWe will refer to a world where the market price of risk is the volatility of

8、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 20199fgEfggTT00Alternative Choices for the Numeraire Security gMoney Market AccountZero-coupon bond priceAnn

9、uity factorOptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 201910 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

10、 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 201911Money Market Accountcontinued Options, Futures, and Other Derivatives, 8th Edition, Cop

11、yright John C. Hull 201912 worldneutral-risk ltraditiona the in nsexpectatio denotes wherebecomesequation the ,= and 1= SinceEfeEfgfEgfeggTrdtTTgrdtTTT000000Zero-Coupon Bond Maturing at time T as Numeraire price bond the wrtFRN is that worlda in nsexpectatio denotes and price bond coupon-zero the is

12、 ),( wherebecomesequation TheTTTTTgETPfETPfgfEgf0), 0(000Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 201913Forward 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 Edi

13、tion, Copyright John C. Hull 201914Interest 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 201915Annuity Factor as the Numeraire)()

14、0(000TAfEAfgfEgfTATTgbecomesequation TheOptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 201916Annuity 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, Cop

15、yright John C. Hull 201917Extension to Several Independent Factors (Page 640)Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 201918miiigmiigimiiifmiifimiiigmiiifdztgtdttgttrtdgdztftdttfttrtdfdztgtdttgtrtdgdztftdttftrtdf1,1,1,1,1,1,)()()()()()()()()()()()()()()()()()()()(

16、)()(consistent internally are that worldsother For worldneutral-risk ltraditiona the InExtension to Several Independent Factors continuedOptions, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 201919hold. results the of rest the and martingalea is case, factor-one the in As wher

17、e worldas wrtFRN is that worlda define Wegfgigi,ApplicationsExtension 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 201920Blacks Model (page 641)By worki

18、ng 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 201921TTKFdTTKF