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Asset Pricing:State Price and Risk-Neutral Probability

热度:89   发布时间:2023-11-25 01:01:48.0

Asset Pricing:State Price and Risk-Neutral Probability

State Price

完备市场下: q s = q ( e s ) q_s=q(e_s) qs?=q(es?)

z = ∑ s = 1 S z s e s , q ( z ) = ∑ s = 1 S z s q ( e s ) = q ? z z=\sum_{s=1}^Sz_se_s,q(z)=\sum_{s=1}^Sz_sq(e_s)=q·z z=s=1S?zs?es?,q(z)=s=1S?zs?q(es?)=q?z

不完备市场下:
q s ≡ Q ( e s ) q_s\equiv Q(e_s) qs?Q(es?)
If Q ≥ 0 , q s ≥ 0 Q\geq0,q_s\geq0 Q0,qs?0 ; if Q > 0 , q s > 0 Q>0,q_s>0 Q>0,qs?>0

对每个未定权益 z ∈ R S , z = ∑ s z s e s z\in R^S,z=\sum_sz_se_s zRS,z=s?zs?es?,所以有:
Q ( z ) = Q ( ∑ s z s e s ) = ∑ s z s Q ( e s ) = ∑ s z s q s = q ? z Q(z)=Q(\sum_sz_se_s)=\sum_sz_sQ(e_s)=\sum_sz_sq_s=q·z Q(z)=Q(s?zs?es?)=s?zs?Q(es?)=s?zs?qs?=q?z
因为 Q ( x j ) = p j Q(x_j)=p_j Q(xj?)=pj?,所以有:
p j = q x j p = X q p_j=qx_j\\p=Xq pj?=qxj?p=Xq
状态价格是含 S S S 个未知量 q s q_s qs? J J J 个方程组的解。

if market is complete , J ≥ S , r a n k ( X ) = S → ? J\geq S,rank(X)=S\to\exist JS,rank(X)=S? 唯一的 q > > 0 q>>0 q>>0?

if market is incomplete , r a n k ( X ) < S rank(X)<S rank(X)<S , multiple q > > 0 q>>0 q>>0 is possible.

Theorem : p = X q p=Xq p=Xq??? has solution q>>0 iff ? Q ( z ) > 0 \exist Q(z)>0 ?Q(z)>0??? ; ? q > > 0 , ? Q > 0 , s . t . ? z ∈ R S , Q ( z ) = q z \forall q>>0,\exist Q>0,s.t.\ \forall z\in R^S,Q(z)=qz ?q>>0,?Q>0,s.t. ?zRS,Q(z)=qz

proof:上述等式已经证明,与严格为正的估值泛函相关的状态价格是等式 p = X q p=Xq p=Xq 的一个解。下面说明估值泛函的存在性。

suppose q > > 0 q>>0 q>>0? 是 p = X q p=Xq p=Xq? 的解,then Q ( z ) = q ? z Q(z)=q·z Q(z)=q?z? is linear and strictly positive. Let z ∈ M , ? h , z = h X , Q ( z ) = q z = q ? h ? X = h ? X ? q = p h = q ( z ) z\in M,\exist h,z=hX,Q(z)=qz=q·h·X=h·X·q=ph=q(z) zM,?h,z=hX,Q(z)=qz=q?h?X=h?X?q=ph=q(z)? , 即 Q Q Q M M M 上的收益定价泛函一致 .So Q ( z ) Q(z) Q(z)? is strcitly positive valuation functional.

Fundamental Theorem of Finance : NA iff ? q > > 0 \exist q>>0 ?q>>0 , NSA iff ? q > 0 \exist q>0 ?q>0

Example:

x 1 = ( 1 , 1 , 1 ) , p 1 = 1 / 2 ; x 2 = ( 1 , 2 , 4 ) , p 2 = 1 x_1=(1,1,1),p_1=1/2;x_2=(1,2,4),p_2=1 x1?=(1,1,1),p1?=1/2;x2?=(1,2,4),p2?=1 . Does Market exclude Arbitrage?

→ q 1 + q 2 + q 3 = 1 / 2 , q 1 + 2 q 2 + 4 q 3 = 1 → q 1 = 2 q 3 , q 2 = 1 / 2 ? 3 q 3 \to q_1+q_2+q_3=1/2,q_1+2q_2+4q_3=1\to q_1=2q_3,q_2=1/2-3q_3 q1?+q2?+q3?=1/2,q1?+2q2?+4q3?=1q1?=2q3?,q2?=1/2?3q3?

q 3 > 0 , 2 q 3 > 0 , 1 / 2 ? 3 q 3 > 0 → 0 < q 3 < 1 / 6 q_3>0,2q_3>0,1/2-3q_3>0\to 0<q_3<1/6 q3?>0,2q3?>0,1/2?3q3?>00<q3?<1/6??。此时NA。

0 ≤ q 3 ≤ 1 / 6 0\leq q_3\leq1/6 0q3?1/6,此时NSA。

Farkas-Stiemke Cemma :

suppose y , a ∈ R m , b ∈ R n , Y ∈ R m × n y,a\in R^m,b\in R^n,Y\in R^{m\times n} y,aRm,bRn,YRm×n

Theorem ( Farkas ): 不存在 a ∈ R m a\in R^m aRm,满足 a Y ≥ 0 , a y < 0 aY\geq0,ay<0 aY0,ay<0 的充要条件是当且仅当 ? b ∈ R n , s . t . y = Y b , b ≥ 0 \exist b\in R^n,s.t.\ y=Yb,b\geq0 ?bRn,s.t. y=Yb,b0

Theorem ( Stiemke ): 不存在 a ∈ R m a\in R^m aRm?,满足 a Y ≥ 0 , a y ≤ 0 aY\geq0,ay\leq0 aY0,ay0?,其中或者 a Y > 0 aY>0 aY>0?,或者 a y < 0 ay<0 ay<0? 的充要条件是当且仅当 ? b ∈ R n , s . t . y = Y b , b > 0 \exist b\in R^n,s.t.\ y=Yb,b>0 ?bRn,s.t. y=Yb,b>0??

状态价格和取值的界

对于未定权益 z ∈ R S , z ? M z\in R^S,z\notin M zRS,z/?M
q l ( z ) = max ? h { p h : z ≥ h X } q u ( z ) = min ? h { p h : z ≤ h X } π ∈ [ q l ( u ) , q u ( z ) ] , Q ( z ) = q ( z ) + λ π q_l(z)=\max_h\{ph:z\geq hX\}\\q_u(z)=\min_h\{ph:z\leq hX\}\\\pi\in[q_l(u),q_u(z)],Q(z)=q(z)+\lambda\pi ql?(z)=hmax?{ ph:zhX}qu?(z)=hmin?{ ph:zhX}π[ql?(u),qu?(z)],Q(z)=q(z)+λπ
p = X q , p h = X q h = q z p=Xq,ph=Xqh=qz p=Xq,ph=Xqh=qz
q l ( z ) = min ? q > 0 { q z : p = X q } q u ( z ) = max ? q > 0 { q z : p = X q } q_l(z)=\min_{q>0}\{qz:p=Xq\}\\q_u(z)=\max_{q>0}\{qz:p=Xq\} ql?(z)=q>0min?{ qz:p=Xq}qu?(z)=q>0max?{ qz:p=Xq}
Risk-Neutral Probability

Risk-Free Asset : X = ( 1 , 1 , ? , 1 ) X=(1,1,\cdots,1) X=(1,1,?,1)

Risk-Free return: r ˉ = 1 P b = 1 ∑ s = 1 S q s x s = 1 ∑ s = 1 S q s \bar r=\dfrac{1}{P_b}=\dfrac{1}{\sum_{s=1}^Sq_sx_s}=\dfrac{1}{\sum_{s=1}^Sq_s} rˉ=Pb?1?=s=1S?qs?xs?1?=s=1S?qs?1?

假设证券价格无套利(强套利),具有严格正回报无风险收益 r ˉ \bar r rˉ 属于资产张成空间。令 q q q 是严格为正(为正)的状态价格向量,对每个 s s s,定义:
π ^ s ≡ r ˉ q s = q s ∑ s = 1 S q s ∈ ( 0 , 1 ) ∑ s = 1 S π ^ s = 1 \hat\pi_s\equiv\bar rq_s=\dfrac{q_s}{\sum_{s=1}^Sq_s}\in(0,1)\\\sum_{s=1}^S\hat\pi_s=1 π^s?rˉqs?=s=1S?qs?qs??(0,1)s=1S?π^s?=1
π ^ s → \hat\pi_s\to π^s?? risk-neutral probability
p j = ∑ s = 1 S q s x s j = ∑ s = 1 S q s ∑ s = 1 S q s ∑ s = 1 S q s x s j = 1 r ˉ ∑ s = 1 S q s ∑ s = 1 S q s x s j = 1 r ˉ ∑ s = 1 S π ^ s x s j = 1 r ˉ E ? ( x j ) r ˉ = E ? ( r j ) Q ( z ) = q z = ∑ s q s z s = 1 r ˉ ∑ s π ^ s z s = 1 r ˉ E ? ( z ) p_j=\sum_{s=1}^Sq_sx_s^j=\sum_{s=1}^Sq_s\sum_{s=1}^S\dfrac{q_s}{\sum_{s=1}^Sq_s}x_s^j\\=\dfrac{1}{\bar r}\sum_{s=1}^S\dfrac{q_s}{\sum_{s=1}^Sq_s}x_s^j=\dfrac{1}{\bar r}\sum_{s=1}^S\hat\pi_sx_s^j=\dfrac{1}{\bar r}E^*(x^j)\\\bar r=E^*(r_j)\\Q(z)=qz=\sum_sq_sz_s=\frac{1}{\bar r}\sum_s\hat\pi_sz_s=\frac{1}{\bar r}E^*(z) pj?=s=1S?qs?xsj?=s=1S?qs?s=1S?s=1S?qs?qs??xsj?=rˉ1?s=1S?s=1S?qs?qs??xsj?=rˉ1?s=1S?π^s?xsj?=rˉ1?E?(xj)rˉ=E?(rj?)Q(z)=qz=s?qs?zs?=rˉ1?s?π^s?zs?=rˉ1?E?(z)
可以将未定权益取值的上界和下界表示为:
q u ( z ) = 1 r ˉ max ? π ^ E ? ( z ) q l ( z ) = 1 r ˉ min ? π ^ E ? ( z ) q_u(z)=\frac{1}{\bar r}\max_{\hat\pi}E^*(z)\\q_l(z)=\frac{1}{\bar r}\min_{\hat\pi}E^*(z) qu?(z)=rˉ1?π^max?E?(z)ql?(z)=rˉ1?π^min?E?(z)

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