《期权期货及其衍生品第27弹课件》由会员分享，可在线阅读，更多相关《期权期货及其衍生品第27弹课件（23页珍藏版）》请在装配图网上搜索。
1、期权期货及其衍生品第27弹1Chapter 27Martingales and Measures期权期货及其衍生品第27弹2Derivatives Dependent on a Single Underlying Variable dz dtd dz dtd dzsdtmd22221111.Suppose and prices withon dependent sderivative two Imagine process the follows that security)traded a ofprice they necessaril(not variable a Consider21期权
2、期货及其衍生品第27弹3Forming a Riskless Portfoliot=)()()(211221212111221122derivative 2nd the of and derivative 1st the of+of consisting portfolio riskless a up set can We期权期货及其衍生品第27弹4Market Price of Risk(Page 632)This shows that(r)/is the same for all derivatives dependent on the same underlying variable,W
3、e refer to(r)/as the market price of risk for and denote it by l or :gives This :riskless is portfolio the Since2211121221rrr r t=r 期权期货及其衍生品第27弹5Extension of the Analysisto Several Underlying Variables(Equations 27.12 and 27.13,page 634)then withvariables underlying several on depends If r dz dtdfn
4、iiiniiil11期权期货及其衍生品第27弹6Martingales(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 today期权期货及其衍生品第27弹7Alternative Worldsdzfdtfrdfdzfdtrfdfll)(is risk of price market the where worlda In w
5、orldneutral-risk ltraditiona the In期权期货及其衍生品第27弹8The Equivalent Martingale Measure Result(Page 635-36)fgfg pricessecurity derivative allfor martingale a is that shows lemma sIto then,security a of volatility the to equal set weIfl期权期货及其衍生品第27弹9Forward Risk NeutralityWe will refer to a world where th
6、e 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 gfgEfggTT00期权期货及其衍生品第27弹10Alternative Choices for the Numeraire Security gMoney Market AccountZero-coupon bond priceAnnuity factor期权期货及其衍生品第27
7、弹11 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 tra
8、ditional risk-neutral world where l=0期权期货及其衍生品第27弹12Money Market Accountcontinued worldneutral-risk ltraditiona the in nsexpectatio denotes wherebecomesequation the,=and 1=SinceEfeEfgfEgfeggTrdtTTgrdtTTT000000期权期货及其衍生品第27弹13Zero-Coupon Bond Maturing at time T as Numeraire price bond the wrtFRN is th
9、at worlda in nsexpectatio denotes and price bond coupon-zero the is),(wherebecomesequation TheTTTTTgETPfETPfgfEgf0),0(000期权期货及其衍生品第27弹14Forward PricesIn a world that is FRN wrt P(0,T),the expected value of a security at time T is its forward price期权期货及其衍生品第27弹15Interest RatesIn a world that is FRN w
10、rt P(0,T2)the expected value of an interest rate lasting between times T1 and T2 is the forward interest rate期权期货及其衍生品第27弹16Annuity Factor as the Numeraire)()0(000TAfEAfgfEgfTATTgbecomesequation The期权期货及其衍生品第27弹17Annuity Factors and SSuppose that s(t)is the s corresponding to the annuity factor A.Th
11、en:s(t)=EAs(T)期权期货及其衍生品第27弹18Extension to Several Independent Factors(Page 640)llmiiigmiigimiiifmiifimiiigmiiifdztgtdttgttrtdgdztftdttfttrtdfdztgtdttgtrtdgdztftdttftrtdf1,1,1,1,1,1,)()()()()()()()()()()()()()()()()()()()()()(consistent internally are that worldsother For worldneutral-risk ltradition
12、a the In期权期货及其衍生品第27弹19Extension to Several Independent Factors continuedhold.results the of rest the and martingalea is case,factor-one the in As where worldas wrtFRN is that worlda define Wegfgigi,期权期货及其衍生品第27弹20ApplicationsExtension of Blacks model to case where inbterest rates are stochasticValu
13、ation of an option to exchange one asset for another期权期货及其衍生品第27弹21Blacks 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
14、 constant volatilityc=P(0,T)F0N(d1)KN(d2)p=P(0,T)KN(d2)F0N(d1)TTKFdTTKFdFFFF202201)/ln()/ln(期权期货及其衍生品第27弹22Option 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 isVUVUVUqqTddTTqqUVddNUedNVeUV2)2(ln)()(2221220012010期权期货及其衍生品第27弹23Change of Numeraire(Section 27.8,page 643)wvwghwvvhgwvwv and between ncorrelatio the is and of volatility the is of volatility the is whereby increases variable a of drift the,to fromsecurity numeraire the change weWhen,
期权期货及其衍生品第27弹课件
