spreads不考虑流动性

1、Chp.4 The Discount FactorMain ContentsnThe Relationship between Law of One Price and Existence of Discount Factor;nThe Relationship Between No Arbitrage and Existence of Positive Discount Factor;nAn Alternative Formula to Compute the Discount Factor in Discrete and Continuous Time.4.1 law of one pri

2、ce and Existence of a Discount factor AssumptionsnA1:(Portfolio formation): for any real a,b.nRemark: Its an important and restrictive simplifying assumption. short sales constraints, leverage limitations, and so on.nA2:(Law of one price,Linearity): nRemark: if the payoff of asset A is the same as t

3、hat of asset B in any case, then price of A=price of B. happy meal theorem. It rules out bid/ask spreads.不考虑流动性。1212,x xXaxbxX1212()()()p axbxap xbp xTheorem 1nGiven free portfolio formation A1, and the law of one price A2, there exists a unique payoff such that p(x)=E(x*x) for all .xX*xXGeometric P

4、roof 1n一价定律线性价格函数。n线性价格函数 等价线如下图所示。假设支付空间是二维的。n根据p0等价线可知x*与之正交。（存在）（注意我们定义 ，因此求内积时要乘以概率 ）n Price=2 n n Price=1(return) n x*n Price=0(excess return)x1x2()E XYX YGeometric Proof 2n用x*为p=1等价线上的任一证券X1定价可确定X*的长度。即：n给定任意证券X2，将它与0连线（或延长线），与p1等价线相交于X1。即x2=ax1.从图上可以看出，用x*定价可得p(x2)=ap(x1),符合一价定律。*11()11/()pro

5、j x xxxproj x x Algebraic Proof nSuppose the basis payoffs (after pruning redundant rows of x) nThen we want to find a discount factor x* in payoff space，so it must be of the formn对于任意的证券组合ax,我们用x*来定价得：n由于x*对于任意证券都一样，因此是唯一的。12,.,Nxx xx*xc x1*1()()()()()()p a xEc xa xc aE x xp a xcE xxpa pE x x cxp E

6、xxpx由一价定律可知Other discount factorsnThe discount factor in payoff space X is unique.nThere are many other discount factors m not in X. (unless the market is complete).nIf p=E(mx),then p=E(m+e)xfor any e orthogonal to x,E(ex)=0.nAny discount factor m can be represented as m=x*+e,with E(ex)=0.nThe prici

7、ng implication of any discount factor m for a set of payoff X are the same as those of projection m on X.n is called the mimicking portfolio for m.()(|) (|) pE mxEproj m XxE proj m X x( | )proj m XTheorem 2nThe existence of a discount factor implies the law of one pricenProof: if x+y=z,and there is

8、a discount factor, then p(x+y)=E(m(x+y)=E(mz)=p(z)4.2 No Arbitrage and Positive Discount FactorsDefinition: No arbitrage nD1:Every payoff x that is always nonnegative (almost surely), and positive with some positive probability, has positive price.nD2:If x=y almost surely and xy with positive probab

9、ility, then p(x)p(y).Theorem3: m0 imply No arbitrage nProof:qFor X=0 and in some states x0. qBecause m0(positive in every state).qP=E(mx)0Theorem4:No arbitrage implies a m0n证明：由于无套利蕴含着一价定律，也就意味着存在随机折现因子，故仅需证明m为正的。n联合(-p(x),x) 形成s+1维空间 中的向量。令M表示所有的数对(-p(x),x) 构成的集合。n由一价定律，M仍是一个线性空间。n无套利意味着M的元素（ s+1维向

10、量）不能够全部由正的分量组成。如果x是正的，那么- p(x),一定为负（无套利保证的）。这样，超平面M就与正的向量空间 只相交于原点。1sR1sR( ), );Mp x x xXn这样就存在一个函数F： 使得对于(-p(x),x) M的点 F(-p,x)=0 ，并且 除原点外的(-p(x),x) 的点F(-p,x)0 （由超平面分离定理保证的）。n由于可以采用向量的内积来表示任何的线性函数，并且存在向量（1，m）使得n由于对所有(-p(x),x) 0的点F(-p,x)都是正的，所以m必须是正的。n在连续的情况下，可以由凸集分离定理和Riesz表示定理同样得到结论。(, )(1,) (, )()

11、Fp xmp xpm xorpE mx 1sRR1sROther discount factorsnThe theorem says that a positive m exists, but it does not say every m must be positive.nIn incomplete market, even x* need not be positive.Xm0X*Arbitrage-free extension of pricesnEach particular choice of m0 induces an arbitrage-free extension of pr

12、ices on X to all contingent claims. An observed and incomplete set of prices and payoffs can be generated by some complete market and contingent-claims economies if there is no arbitrage.X* mp=1p=2oABX由于Ox*m与OBA相似，所以x*OA=OBmNo arbitrage and the law of one pricenNo arbitrage is more strict than the l

13、aw of one price.nNo arbitrage implies the law of one price, but not vice versa.Why no arbitrage is more strict than law of one price?nLaw of one price implies the same payoff has the same price, but does not consider the situation of different payoffs. For example, if payoff Apayoff B in any case, u

14、nder the law of one price, p(A)p(B) may hold. This implies arbitrage opportunity.nNo arbitrage implies positive payoff has positive price, which includes the law of one price.4.3 an alternative formula, and x* in continuous timeAlternative fromulan n Proof:)()()()(1*xExxExEpxEx*1*1*1() ( ) ( ) ( )(

15、) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )E xxE E xp E x E xx E x xE x E xp E x E xE x E x xE x E xp E x E xp Alternative formula(2)nIf a risk-free rate is traded, then we have:*111()();cov()eeeeffxE RRE RRRR X* in continuous timenSimilarly, we can getnProof: *1*()ffdDr dtrdzp*11,()(),(/),(/) ,(/) (/)ftff

16、tffffdpddtdzr dtdzpdpDDddpEdtr dtr dtEdtppppD prD prD prD pr 假设：Other discount factors in continuous timen plus orthogonal noise will also act as a discount factor:*;()0;()0.dddw E dwE dzdw重要结论（1）n在完全市场中，m只有一个，且严格为正。n在不完全市场中，即使处于无套利均衡状态，m很多，其中有的m可能完全为负，但肯定有的m完全为正。n在不完全市场中，新产品（只要不是原有产品的线性复制品）可以使市场趋于完全。但若没有其他信息，该产品就无法准确定价，但可以确定价格