Chp.4 The Discount Factor

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Chp.4 The Discount Factor
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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.

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4.1 law of one price and Existence of a Discount
factor
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Assumptions

• A1(Portfolio formation)
• for any real a and b.
• Remark Its 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
  Aprice of B. happy meal theorem. It rules out
  bid/ask spreads.???????

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Theorem 1

• Given free portfolio formation A1, and the law of
  one price A2, there exists a unique payoff
  such that p(x)E(xx) for all
  
• .

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Geometric Proof 1

• ???????????
• ?????? ????????????????????
• ??p0?????x?????(??)(??????
  ,??????????? )
• 
  Price2
• 
  Price1(return)
• x
• Price0(excess return)

x2
x1
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Geometric Proof 2

•  

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Algebraic Proof

•  

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Other discount factors

•  

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Theorem 2

• The existence of a discount factor implies the
  law of one price
• Proof if xyz,and there is a discount factor,
  then p(xy)E(m(xy))E(mz)p(z)

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4.2 No Arbitrage and Positive Discount Factors
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Definition No arbitrage

• D1Every payoff x that is always nonnegative
  (almost surely), and positive with some positive
  probability, has positive price.
• D2If xgty almost surely and xgty with positive
  probability, then p(x)gtp(y).

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Theorem3 mgt0 imply No arbitrage

• Proof
• For xgt0 and in some states xgt0.
• Because mgt0(positive in every state).
• PE(mx)gt0

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No arbitrage implies a mgt0

• ??
• ?????????payoff???????.p0????????????????????????
  ?,???????0??2?4??,??m????0???????

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Theorem4No arbitrage implies a mgt0,??????????x??

•  

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•  

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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.

mgt0
X
X
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Arbitrage-free extension of prices

• Each particular choice of mgt0 induces an
  arbitrage-free extension of prices on X to all
  contingent claims

p2
p1
??Oxm?OBA??, ??xOAOBm
m

B
o
A
X
X
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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.

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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
  Agtpayoff B in any case, under the law of one
  price, p(A)ltp(B) may hold. This implies arbitrage
  opportunity.
• No arbitrage implies positive payoff has positive
  price, which includes the law of one price.

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4.3 an alternative formula, and x in continuous
time
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Alternative fromula

• Proof

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Alternative formula(2)

• If a risk-free rate is traded, and the payoff
  space consists solely of excess returns(p0),
  then we have

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X in continuous time

• Similarly, we can get
• Proof

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Other discount factors in continuous time

• plus orthogonal noise will also act as a
  discount factor

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????(1)

• ??????,m????,??????
• ???????,???????????,m??,????m??????,?????m?????
• ???????,???(??????????????)??????????????????,????
  ??????,??????????

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????(2)

• ??????,?????????????????????????????,???????????
• ??????,?????????????(???????)???,?????????????????
  ????????????????????????,?????????????????,???????
  ???????????????,????????????
• ??????,???????????,?N(N??S)?????,??????????????

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