Asset Pricing

Zheng Zhenlong

Ch3. Contingent Claims Markets

18:08

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Zheng ZhenlongBrief introduction

• In the frame of complete market, we look forward to see the equation p=E(mx) more intuitive.

• The structure is as follows: Contingent Claims Risk-Neutral Probabilities Investors Again Risk Sharing State Diagram and Price Function

18:08

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Zheng Zhenlong3.1 Contingent Claims

• A contingent claim is a security that pays one dollar (or one unit of the consumption good) in one state s only tomorrow.

• pc(s) is the price today of the contingent claim. （状态价格）

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Zheng ZhenlongComplete market

p(x)= ( )x(s)s

pc s

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Zheng Zhenlong

( )p(x)= ( ) x(s)

( )s

pc ss

s

pc(s)( )

( )m s

s

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Zheng Zhenlong

Conclusion about discount factor

• Now we can write the bundling equation as an expectation,

• If there are complete contingent claims, a discount factor exists, and it is equal to the contingent claim prices divided by probabilities.

p(x)= ( )m(s)x(s)=E(mx)s

s

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Zheng ZhenlongExpand to infinite space

• In general, we posit states of nature ω that can take continuous (uncountably infinite) values in a space Ω. In this case, the sums become integrals, and we have to use some measure to integrate over Ω. Thus, scaling contingent claims prices by some probability-like object is unavoidable.

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Zheng Zhenlong

3.2 Risk- neutral probabilities

( )( )

( )

( ) ( ) ( ) ( ) ( ),

( )( ) ( ) ( ) ( )

( )

s

s

f

pc ss

pc s

pc s m s s pc s E m

m ss s R m s s

E m

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Zheng ZhenlongRisk- neutral probabilities (2)

• Then we can rewrite the asset pricing formula as:

• We use the notation E* to remind us that the expectation uses the risk neutral probabilities π*instead of the real probabilities π.

( ) ( ) ( ) ( ) ( ) ( )

1 ( )( ) ( )

s s

f f

p x pc s x s m s s x s

E xs x s

R R

——风险中性定价

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Zheng Zhenlong

* ( )( ) ( )

( )

m ss s

E m

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Zheng Zhenlong连续时间

• 在完全市场中，两者的风险源相同。• From (1.35),we have

1.34

p p

f

dt dz

dr dt dz

dpp

（ ）

( ) ( )f pt t

dp D d dpE dt r dt E dt

p p p

——超额收益

可见 是风险价格。

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Zheng Zhenlong风险中性定价

• 在风险中性世界，风险价格必须等于 0 ，即

0

)

,

f

f p

f p p

p

dr dt

dp Dr dt dz

p p

Dr

p

＝。这样，在风险中性世界中，

＝

（ ＋

由于 ＝ －

也就是说，我们只要把价格的偏移率减少 ，并去掉随机贴现因子的扰动项就可以得到风险中性世界的随机过程。

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Zheng Zhenlong3.3 Investors’ choice

• The investor starts with a pile of initial wealth y and a state-contingent income y(s). He purchases contingent claims to each possible state in the second period. His problem is then

, ( )( ) ( ) [ ]

.

( ) ( ) ( ) ( )

c c s s

s s

Maxu c s u c s

st

c pc s c s y pc s y s

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Zheng ZhenlongInvestors’ choice (2)

• Eliminating the Lagrange multiplier λ,

•

• Coupled with p=E(mx), we obtain the consumption-based model again.

( ( ))( ) ( )

( )

( ) ( ( ))( )

( ) ( )

u c spc s s

u c

pc s u c sm s

s u c

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Zheng Zhenlong

marginal rates of substitution

• The investor’s first order conditions say that marginal rates of substitution between states tomorrow equals the relevant price ratio,

边际替代率相对价格比（经概率调整）

1 1

2 2

( ) ( ( ))

( ) ( ( ))

m s u c s

m s u c s

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Zheng Zhenlong

Economics behind this approach to asset pricing (figure 3.1)

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Zheng Zhenlong3.4 Risk Sharing

• In complete markets, the prices are the same for all investors. 如果信息是透明的，每个人都知道客观概率，则 marginal utility growth should be the same for all investors

• If investors have the same homothetic utility function (for example, power utility), then consumption itself should move in lockstep.

1 1( ) ( )

( ) ( )

i ji jt t

i jt t

u c u c

u c u c

1 1i jt ti jt t

c c

c c

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Zheng ZhenlongRisk Sharing (2)

• It means that shocks to consumption are perfectly correlated across individuals.

• It doesn’t say that expected consumption growth should be equal; it says that consumption growth should be equal ex post.

• In a complete contingent claims market, all investors share all risks, so when any shock hits, it hits us all equally (after insurance payments).

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Zheng ZhenlongPareto-optimal risk sharing

• Suppose a social planner wished to maximize everyone’s utility given the available resources. For example, with two investors i and j, he would maximize

max ( ) ( )

. .

t i t ji t j t

t t

i j at t t

E u c E u c

s t c c c

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Zheng Zhenlong

Pareto-optimal risk sharing(2)

• first order condition

• The same risk sharing that we see in a complete market.

• This simple fact has profound implications:It shows you why only aggregate shocks should matter for risk prices. Any idiosyncratic income risk will be equally shared, and so 1/N of it becomes an aggregate shock. Then the stochastic discount factors m that determine asset prices are no longer affected by truly idiosyncratic risks. Much of this sense that only aggregate shocks matter stays with us in incomplete markets as well.

( ) ( )i i j jt tu c u c

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Zheng ZhenlongSub-markets for risk sharing:

• Insurance market• bond market• stock market

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Reasons for individual consumptions not move in lockstep• The real economy does not yet have complete

markets or full risk sharing.• Different utility functions• Different value of individual impatient

coefficients.

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Zheng Zhenlong

3.5 State Diagram and Price Function

'[ (1) (2) ( )]pc pc pc pc S '[ (1) (2) ( )]x x x x S

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Zheng Zhenlong图 7 状态价格与回报

P=0(超额收益率 )

Rf

P=2

P=1（收益率）

状态 1回报

状态 2回报

1/Rf

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The contingent claims price vector pc points in to the positive orthant

( ( ))( )

( )

u c sm s

u c

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Zheng Zhenlongpc线的斜率

• 在经概率调整后的状态偏好中性世界中， pc(1)=pc(2),因此 pc 线是 45 度线。

• 在现实生活中，投资者对不同状态的偏好不同。投资者越爱好（经概率调整）某种状态的 PAYOFF ，该状态的 PC就越高。在上图中， pc(1)<pc(2), 所以 pc 线较陡。

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Zheng ZhenlongPC线的长度

• 对无风险资产定价可知： pc(1) Rf ＋ pc(2) Rf ＝ 1, 可得： pc(1)＋ pc(2)=1/(Rf)

• 可见， pc 向量一定在 0 （ 1/Rf ）（ 1/Rf ）这个三角形中。

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The set of payoffs with any given price lie on a (hyper)plane perpendicular to the contingent claim price vector

• Since the price of the payoff x must be given by its contingent claim value,

• Interpreting pc and x as vectors, this means that the price is given by the inner product of the contingent claim price and the payoff.

• If two vectors are orthogonal —— if they point out from the origin at right angles to each other —— then their inner product is zero.

• The set of all zero price payoffs must lie on a plane orthogonal to the contingent claims price vector, as shown in figure 7.

( ) ( ) ( )s

p x pc s x s

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( ) ( ) ( ) ( | )s

p x pc s x s pc x pc proj x pc

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Zheng Zhenlong

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Zheng Zhenlong

• Planes of constant price move out linearly, and the origin x= 0 must have a price of zero.

• If payoff y= 2x, then its price is twice the price of x

( ) ( ) ( ) ( )2 ( ) 2 ( )s s

p y pc s y s pc s x s p x

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Zheng Zhenlong

( ) ( ) ( )p ax by ap x bp y

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Zheng Zhenlong

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Zheng Zhenlong