spreads不考慮流動性

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.不考慮流動性。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一價定律線性價格函數(shù)。n線性價格函數(shù) 等價線如下圖所示。假設(shè)支付空間是二維的。n根據(jù)p0等價線可知x*與之正交。（存在）（注意我們定義 ，因此求內(nèi)積時要乘以概率 ）n Price=2 n n Price=1(return) n x*n Price=0(excess return)x1x2()E XYX YGeometric Proof 2n用x*為p=1等價線上的任一證券X1定價可確定X*的長度。即：n給定任意證券X2，將它與0連線（或延長線），與p1等價線相交于X1。即x2=ax1.從圖上可以看出，用x*定價可得p(x2)=ap(x1),符合一價定律。*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對于任意的證券組合ax,我們用x*來定價得：n由于x*對于任意證券都一樣，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由一價定律可知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證明：由于無套利蘊含著一價定律，也就意味著存在隨機折現(xiàn)因子，故僅需證明m為正的。n聯(lián)合(-p(x),x) 形成s+1維空間 中的向量。令M表示所有的數(shù)對(-p(x),x) 構(gòu)成的集合。n由一價定律，M仍是一個線性空間。n無套利意味著M的元素（ s+1維向

10、量）不能夠全部由正的分量組成。如果x是正的，那么- p(x),一定為負(fù)（無套利保證的）。這樣，超平面M就與正的向量空間 只相交于原點。1sR1sR( ), );Mp x x xXn這樣就存在一個函數(shù)F： 使得對于(-p(x),x) M的點 F(-p,x)=0 ，并且 除原點外的(-p(x),x) 的點F(-p,x)0 （由超平面分離定理保證的）。n由于可以采用向量的內(nèi)積來表示任何的線性函數(shù)，并且存在向量（1，m）使得n由于對所有(-p(x),x) 0的點F(-p,x)都是正的，所以m必須是正的。n在連續(xù)的情況下，可以由凸集分離定理和Riesz表示定理同樣得到結(jié)論。(, )(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與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 假設(shè)：Other discount factors in continuous timen plus orthogonal noise will also act as a discount factor:*;()0;()0.dddw E dwE dzdw重要結(jié)論（1）n在完全市場中，m只有一個，且嚴(yán)格為正。n在不完全市場中，即使處于無套利均衡狀態(tài)，m很多，其中有的m可能完全為負(fù)，但肯定有的m完全為正。n在不完全市場中，新產(chǎn)品（只要不是原有產(chǎn)品的線性復(fù)制品）可以使市場趨于完全。但若沒有其他信息，該產(chǎn)品就無法準(zhǔn)確定價，但可以確定價格