Asset Pricing：Asset Pricing Formula
State Price Model:
p j = ∑ s = 1 S q s x s j p_j=\sum_{s=1}^Sq_sx_s^j pj?=s=1∑S?qs?xsj?
Stochastic Discount Factor:
suppose { π s } s = 1 S \{\pi_s\}_{s=1}^S {
πs?}s=1S??? is the physical probability distribution of states：
p j = ∑ s = 1 S q s x s j = ∑ s = 1 S π s q s π s x s j = ∑ s = 1 S π s m s x s j = E [ m x j ] m s ≡ q s π s → S t o c h a s t i c D i s c o u n t F a c t o r p_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∑S?qs?xsj?=s=1∑S?πs?πs?qs??xsj?=s=1∑S?πs?ms?xsj?=E[mxj]ms?≡πs?qs??→Stochastic Discount Factor
Note that：
p j = E [ m ? x j ] = E [ x j ] E [ m ] + C o v ( m , x j ) r i s k ? f r e e b o n d : p b = E [ m ] = 1 R f → R f = 1 E [ m ] , m = u ′ ( c 1 ) u ′ ( c 0 ) p j = E [ x j ] R f + C o v ( m , x j ) 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, C o v ( m , x j ) < 0 Cov(m,x^j)<0 Cov(m,xj)<0.
Defining R j ≡ x j p j → E [ m R j ] = 1 R^j\equiv\dfrac{x^j}{p_j}\to \mathbb E[mR^j]=1 Rj≡pj?xj?→E[mRj]=1
我们有 E [ m R f ] = 1 \mathbb E[mR^f]=1 E[mRf]=1，所以：
E [ m ? ( R j ? R f ) ] = 0 E [ m ] ( E [ R j ] ? R f ) + C o v ( m , R j ) = 0 E [ R j ] ? R f = ? C o v ( m , R j ) 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 j j j?? is determined solely by the covariance with the stochastic discount factor. ? 1 E [ m ] → -\dfrac{1}{\mathbb E[m]}\to ?E[m]1?→?? price of market risk , C o v ( m , R j ) → Cov(m,R^j)\to Cov(m,Rj)→?? Quantity of risk for Asset j j j
Equivalent Martingale Measure:
start with:
p j = ∑ s = 1 S q s x s j p_j=\sum_{s=1}^Sq_sx_s^j pj?=s=1∑S?qs?xsj?
for a riskfree bond we have:
p b = ∑ s = 1 S q s = 1 1 + r f p_b=\sum_{s=1}^Sq_s=\frac{1}{1+r^f} pb?=s=1∑S?qs?=1+rf1?
where r f r^f rf is the risk-free net return. We have:
p j = ∑ s = 1 S q s ∑ s = 1 S q s x s j ∑ s = 1 S q s = 1 1 + r f ∑ s = 1 S q s ∑ s = 1 S q s x s j = 1 1 + r f ∑ s = 1 S π ^ s x s j = 1 1 + r f E Q [ x j ] w h e r e π ^ s ≡ q s ∑ s = 1 S q s p_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∑S?qs?s=1∑S?∑s=1S?qs?qs?xsj??=1+rf1?s=1∑S?∑s=1S?qs?qs??xsj?=1+rf1?s=1∑S?π^s?xsj?=1+rf1?EQ[xj]where π^s?≡∑s=1S?qs?qs??
State-Price Beta Model:
stochastic discount factor:
m ? ≡ [ q 1 ? π ? q S ? 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 ? p m ? ≡ α m ? , α > 0 R^*=\dfrac{m^*}{p_{m^*}}\equiv\alpha m^*,\alpha>0 R?=pm??m??≡αm?,α>0?, we can write:
E [ R j ] ? R f = ? C o v ( R ? , R j ) 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 β j ≡ C o v ( R ? , R j ) V a r ( R ? ) \beta_j\equiv\dfrac{Cov(R^*,R^j)}{Var(R^*)} βj?≡Var(R?)Cov(R?,Rj)?, we have:
E [ R j ] ? R f = ? β j V a r ( R ? ) E [ R ? ] \mathbb E[R^j]-R^f=-\beta_j\frac{Var(R^*)}{\mathbb E[R^*]} E[Rj]?Rf=?βj?E[R?]Var(R?)?
and for security x ? x^* x?, we have β ? = C o v ( R ? , R ? ) V a r ( R ? ) = 1 \beta^*=\dfrac{Cov(R^*,R^*)}{Var(R^*)}=1 β?=Var(R?)Cov(R?,R?)?=1:
E [ R ? ] ? R f = ? V a r ( 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 [ R j ] ? R f = β j ( E [ R ? ] ? R f ) \mathbb E[R^j]-R^f=\beta_j(\mathbb E[R^*]-R^f) E[Rj]?Rf=βj?(E[R?]?Rf)