Chp.4 The Discount Factor

Main Contents The Relationship between Law of One Price and Existence of Discount Factor; The Relationship Between No Arbitrage and Existence of Positive Discount Factor; An Alternative Formula to Compute the Discount Factor in Discrete and Continuous Time.

4.1 law of one price and Existence of a Discount factor

Assumptions A1:(Portfolio formation): for any real a and b. Remark: It’s an important and restrictive simplifying assumption. short sales constraints, leverage limitations, and so on. A2:(Law of one price, Linearity): Remark: if the payoff of asset A is the same as that 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。 08:20

Theorem 1 Given 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 .

Geometric Proof 1 一价定律＝线性价格函数。 线性价格函数 等价线如下图所示。假设支付空间是二维的。 线性价格函数 等价线如下图所示。假设支付空间是二维的。 根据p＝0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 ） Price=2 Price=1(return) x* Price=0(excess return) x2 x1

Geometric Proof 2  

Algebraic Proof  

Other discount factors  

Theorem 2 The existence of a discount factor implies the law of one price Proof: if x+y=z,and there is a discount factor, then p(x+y)=E(m(x+y))=E(mz)=p(z)

4.2 No Arbitrage and Positive Discount Factors

Definition: No arbitrage D1:Every payoff x that is always nonnegative (almost surely), and positive with some positive probability, has positive price. D2:If x>=y almost surely and x>y with positive probability, then p(x)>p(y).

Theorem3: m>0 imply No arbitrage Proof: For x>=0 and in some states x>0. Because m>0(positive in every state). P=E(mx)>0

No arbitrage implies a m>0 直觉： 无套利意味着正象限payoff的价格严格为正.p=0线将正负 价格区域分割开来。为了使正负价格区域不交叉，等价线 必须经过0和第2、4象限，因此m必须经过0指向第一象限 。

Theorem4:No arbitrage implies a m>0，可以对回报空间的任何x定价  

Other discount factors The theorem says that a positive m exists, but it does not say every m must be positive. In incomplete market, even x* need not be positive. m>0 X* X

Arbitrage-free extension of prices Each particular choice of m>0 induces an arbitrage-free extension of prices on X to all contingent claims p=2 p=1 由于Ox*m与OBA相似， 所以x*×OA=OB×m m B o A X* X

No arbitrage and the law of one price No arbitrage is more strict than the law of one price. No arbitrage implies the law of one price, but not vice versa.

Why no arbitrage is more strict than law of one price? Law of one price implies the same payoff has the same price, but does not consider the situation of different payoffs. For example, if payoff A>payoff B in any case, under the law of one price, p(A)<p(B) may hold. This implies arbitrage opportunity. No arbitrage implies positive payoff has positive price, which includes the law of one price.

4.3 an alternative formula, and x* in continuous time

Alternative fromula Proof:

Alternative formula(2) If a risk-free rate is traded, and the payoff space consists solely of excess returns(p=0), then we have:

X* in continuous time Similarly, we can get Proof:

Other discount factors in continuous time plus orthogonal noise will also act as a discount factor:

重要结论（1） 在完全市场中，m只有一个，且严格为正。 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。 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，但可以确定价格区间。

重要结论（2） 在完全市场中，等价鞅测度是唯一的。概率测度可以 自由转换。而在不完全世界中，等价鞅测度不是唯一 的。 在不完全世界，给定风险中性概率和任一资产（包括 无风险资产）的价格，就可以求出在这种测度下任意 新产品的价格。因为知道风险中性概率就知道所有状 态价格之比，再利用已知价格就可算出所有状态价格， 从而为所有资产定价。风险中性概率决定相对价格， 该证券价格决定绝对价格。 不完全世界中，即使我们知道现实的概率，和N（N 小于S）种证券价格，我们仍无法对任意新产品定价。