Asset Pricing：Asset Pricing Formula

State Price Model:
pj=∑s=1Sqsxsjp_j=\sum_{s=1}^Sq_sx_s^j pj=s=1∑Sqsxsj
Stochastic Discount Factor:

suppose {πs}s=1S\{\pi_s\}_{s=1}^S{πs}s=1S is the physical probability distribution of states：
pj=∑s=1Sqsxsj=∑s=1Sπsqsπsxsj=∑s=1Sπsmsxsj=E[mxj]ms≡qsπs→StochasticDiscountFactorp_j=\sum_{s=1}^Sq_sx_s^j=\sum_{s=1}^S\pi_s\dfrac{q_s}{\pi_s}x_s^j=\sum_{s=1}^S\pi_sm_sx_s^j=\mathbb E[mx^j]\\m_s\equiv\dfrac{q_s}{\pi_s}\to Stochastic\ Discount\ Factor pj=s=1∑Sqsxsj=s=1∑Sπsπsqsxsj=s=1∑Sπsmsxsj=E[mxj]ms≡πsqs→Stochastic Discount Factor
Note that：
pj=E[m⋅xj]=E[xj]E[m]+Cov(m,xj)risk−freebond:pb=E[m]=1Rf→Rf=1E[m],m=u′(c1)u′(c0)pj=E[xj]Rf+Cov(m,xj)p_j=\mathbb E[m·x_j]=\mathbb E[x_j]\mathbb E[m]+Cov(m,x_j)\\risk-free\ bond:p_b=\mathbb E[m]=\dfrac{1}{R^f}\to R^f=\dfrac{1}{\mathbb E[m]},m=\dfrac{u'(c_1)}{u'(c_0)}\\p_j=\dfrac{\mathbb E[x^j]}{R^f}+Cov(m,x^j) pj=E[m⋅xj]=E[xj]E[m]+Cov(m,xj)risk−free bond:pb=E[m]=Rf1→Rf=E[m]1,m=u′(c0)u′(c1)pj=RfE[xj]+Cov(m,xj)
Typically, Cov(m,xj)<0Cov(m,x^j)<0Cov(m,xj)<0.

Defining Rj≡xjpj→E[mRj]=1R^j\equiv\dfrac{x^j}{p_j}\to \mathbb E[mR^j]=1Rj≡pjxj→E[mRj]=1

我们有 E[mRf]=1\mathbb E[mR^f]=1E[mRf]=1，所以：
E[m⋅(Rj−Rf)]=0E[m](E[Rj]−Rf)+Cov(m,Rj)=0E[Rj]−Rf=−Cov(m,Rj)E[m]\mathbb E[m·(R^j-R^f)]=0\\\mathbb E[m](\mathbb E[R^j]-R^f)+Cov(m,R^j)=0\\\mathbb E[R^j]-R^f=-\frac{Cov(m,R^j)}{\mathbb E[m]} E[m⋅(Rj−Rf)]=0E[m](E[Rj]−Rf)+Cov(m,Rj)=0E[Rj]−Rf=−E[m]Cov(m,Rj)
Which implies that the Expected Excess return for a generic asset jjj is determined solely by the covariance with the stochastic discount factor. −1E[m]→-\dfrac{1}{\mathbb E[m]}\to−E[m]1→ price of market risk , Cov(m,Rj)→Cov(m,R^j)\toCov(m,Rj)→ Quantity of risk for Asset jjj

Equivalent Martingale Measure:

start with:
pj=∑s=1Sqsxsjp_j=\sum_{s=1}^Sq_sx_s^j pj=s=1∑Sqsxsj
for a riskfree bond we have:
pb=∑s=1Sqs=11+rfp_b=\sum_{s=1}^Sq_s=\frac{1}{1+r^f} pb=s=1∑Sqs=1+rf1
where rfr^frf is the risk-free net return. We have:
pj=∑s=1Sqs∑s=1Sqsxsj∑s=1Sqs=11+rf∑s=1Sqs∑s=1Sqsxsj=11+rf∑s=1Sπ^sxsj=11+rfEQ[xj]whereπ^s≡qs∑s=1Sqsp_j=\sum_{s=1}^Sq_s\sum_{s=1}^S\frac{q_sx_s^j}{\sum_{s=1}^Sq_s}=\frac{1}{1+r^f}\sum_{s=1}^S\frac{q_s}{\sum_{s=1}^Sq_s}x_s^j=\frac{1}{1+r^f}\sum_{s=1}^S\hat\pi_sx_s^j=\frac{1}{1+r^f}\mathbb E^Q[x^j]\\where\ \hat\pi_s\equiv\frac{q_s}{\sum_{s=1}^Sq_s} pj=s=1∑Sqss=1∑S∑s=1Sqsqsxsj=1+rf1s=1∑S∑s=1Sqsqsxsj=1+rf1s=1∑Sπ^sxsj=1+rf1EQ[xj]where π^s≡∑s=1Sqsqs
State-Price Beta Model:

stochastic discount factor:
m∗≡[q1∗π⋮qS∗π]m^*\equiv\left[\begin{matrix}\frac{q_1^*}{\pi}\\\vdots\\\frac{q_S^*}{\pi}\end{matrix}\right] m∗≡⎣⎢⎢⎡πq1∗⋮πqS∗⎦⎥⎥⎤
define its return as R∗=m∗pm∗≡αm∗,α>0R^*=\dfrac{m^*}{p_{m^*}}\equiv\alpha m^*,\alpha>0R∗=pm∗m∗≡αm∗,α>0, we can write:
E[Rj]−Rf=−Cov(R∗,Rj)E[R∗]\mathbb E[R^j]-R^f=-\frac{Cov(R^*,R^j)}{\mathbb E[R^*]} E[Rj]−Rf=−E[R∗]Cov(R∗,Rj)
define βj≡Cov(R∗,Rj)Var(R∗)\beta_j\equiv\dfrac{Cov(R^*,R^j)}{Var(R^*)}βj≡Var(R∗)Cov(R∗,Rj), we have:
E[Rj]−Rf=−βjVar(R∗)E[R∗]\mathbb E[R^j]-R^f=-\beta_j\frac{Var(R^*)}{\mathbb E[R^*]} E[Rj]−Rf=−βjE[R∗]Var(R∗)
and for security x∗x^*x∗, we have β∗=Cov(R∗,R∗)Var(R∗)=1\beta^*=\dfrac{Cov(R^*,R^*)}{Var(R^*)}=1β∗=Var(R∗)Cov(R∗,R∗)=1:
E[R∗]−Rf=−Var(R∗)E[R∗]\mathbb E[R^*]-R^f=-\frac{Var(R^*)}{\mathbb E[R^*]} E[R∗]−Rf=−E[R∗]Var(R∗)
so we have:
E[Rj]−Rf=βj(E[R∗]−Rf)\mathbb E[R^j]-R^f=\beta_j(\mathbb E[R^*]-R^f) E[Rj]−Rf=βj(E[R∗]−Rf)

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Asset Pricing：Asset Pricing Formula

Asset Pricing：Asset Pricing Formula

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