diff --git a/PhD-Course/book/Asset Pricing/Asset Pricing Review.md b/PhD-Course/book/Asset Pricing/Asset Pricing Review.md new file mode 100644 index 0000000..b685c6a --- /dev/null +++ b/PhD-Course/book/Asset Pricing/Asset Pricing Review.md @@ -0,0 +1,4973 @@ +--- +orphan: true +--- + +# Overview + +## 0. 总览 +### 0.1 总逻辑图 + +$$ +\left\{ +\begin{aligned} +\text{Individual optimization} +&\Longrightarrow +u'(c)\text{ or }V_w +\Longrightarrow m,\xi,\Lambda +&&\text{(SDF / SPD)},\\ +\text{No arbitrage} +&\Longrightarrow +P=E[mX] +\Longleftrightarrow +\frac{P_t}{B_t}\text{ is }Q\text{-martingale} +&&\text{(pricing)},\\ +\text{Mean-variance} +&\Longrightarrow +m=a+b'R +\Longleftrightarrow +\text{factor pricing / CAPM / MV efficiency} +&&\text{(linear SDF)},\\ +\text{Complete markets} +&\Longrightarrow +\text{unique SDF / unique EMM} +\Longrightarrow +\text{replication} +&&\text{(derivatives)},\\ +\text{Information frictions} +&\Longrightarrow +E[v\mid y],\ \text{filtering},\ \text{belief heterogeneity} +\Longrightarrow +\text{price discovery and liquidity}. +\end{aligned} +\right. +$$ + +### 0.2 解题入口索引：先定位题型 + +$$ +\left\{ +\begin{aligned} +\text{Optimization} +&\Longrightarrow +\text{objective} +\Longrightarrow +\text{FOC} +\Longrightarrow +\text{pricing kernel / policy},\\ +\text{Change of measure} +&\Longrightarrow +\text{density process} +\Longrightarrow +P\text{-drift} +\mapsto +Q\text{-drift},\\ +\text{Differential equations} +&\Longrightarrow +\text{SDE} +\Longrightarrow +\text{PDE / ODE} +\Longrightarrow +\text{boundary conditions},\\ +\text{No arbitrage} +&\Longrightarrow +\text{SDF / SPD / EMM} +\Longrightarrow +\text{martingale pricing},\\ +\text{Information} +&\Longrightarrow +\text{Bayesian updating / filtering} +\Longrightarrow +\text{price discovery},\\ +\text{Equilibrium} +&\Longrightarrow +\text{market clearing} +\Longrightarrow +\text{prices and allocations}. +\end{aligned} +\right. +$$ + +#### 0.2.1 Optimization + FOC 类题 + +$$ +\left\{ +\begin{aligned} +\text{Choose control} +&:\quad c,\theta,\pi,w,x,\\ +\text{Write objective} +&:\quad \max E[U],\quad +\max E[u(c)],\quad +\max\{E[R]-\frac{\gamma}{2}\operatorname{Var}(R)\},\\ +\text{Take FOC} +&:\quad +\frac{\partial \mathcal L}{\partial control}=0,\\ +\text{Solve policy} +&:\quad +control^*=f(\text{prices, beliefs, state variables}). +\end{aligned} +\right. +$$ + +$$ +\begin{aligned} +\text{Consumption FOC} +&\Longrightarrow +m_{t+1} +=\delta\frac{u'(c_{t+1})}{u'(c_t)},\\ +\text{Mean-variance FOC} +&\Longrightarrow +w^* +\propto +\Sigma^{-1}(\mu-R_f1),\\ +\text{HJB FOC} +&\Longrightarrow +u'(c^*)=V_w, +\qquad +\pi^*=-\frac{V_w}{V_{ww}}\Sigma^{-1}(\mu-r1),\\ +\text{Kyle insider FOC} +&\Longrightarrow +x=\frac{v-p_0}{2\lambda}. +\end{aligned} +$$ + +1. Consumption-saving： + +$$ +\begin{aligned} +\max_{\{c_t\}} +E\left[\sum_t\delta^tu(c_t)\right] +&\Longrightarrow +u'(c_t) = +\delta R_f E_t[u'(c_{t+1})]\\ +&\Longrightarrow +1 = +E_t\left[ +\delta\frac{u'(c_{t+1})}{u'(c_t)}R_f +\right]. +\end{aligned} +$$ + +2. Portfolio choice： + +$$ +\begin{aligned} +\max_{\theta} +E_t[u(W_{t+1})], +\qquad +W_{t+1}=W_t+\theta'(X_{t+1}-P_tR_f) +&\Longrightarrow +E_t[u'(W_{t+1})(X_{t+1}-P_tR_f)]=0\\ +&\Longrightarrow +P_t = +E_t\left[ +\frac{u'(W_{t+1})}{R_fE_t[u'(W_{t+1})]} +X_{t+1} +\right]. +\end{aligned} +$$ + +3. Mean-variance： + +$$ +\begin{aligned} +\max_w\left\{ +w'(\mu-R_f1)-\frac{\gamma}{2}w'\Sigma w +\right\} +&\Longrightarrow +\mu-R_f1-\gamma\Sigma w=0\\ +&\Longrightarrow +w^*=\frac{1}{\gamma}\Sigma^{-1}(\mu-R_f1). +\end{aligned} +$$ + +4. Dynamic control / HJB： + +$$ +\begin{aligned} +0&=\max_{\pi,c} +\left\{ +u(c)-\rho V+V_t+V_w[rw+\pi'(\mu-r1)-c] ++\frac12V_{ww}\pi'\Sigma\pi +\right\}\\ +&\Longrightarrow +u'(c^*)=V_w, +\qquad +\pi^*=-\frac{V_w}{V_{ww}}\Sigma^{-1}(\mu-r1). +\end{aligned} +$$ + +5. Insider trading： + +$$ +\begin{aligned} +\max_x\ (v-p(x))x, +\qquad +p(x)=p_0+\lambda x +&\Longrightarrow +v-p_0-2\lambda x=0\\ +&\Longrightarrow +x^*=\frac{v-p_0}{2\lambda}. +\end{aligned} +$$ + +$$ +\boxed{ +\text{objective} +\Longrightarrow +\text{budget / law of motion} +\Longrightarrow +\text{FOC} +\Longrightarrow +\text{policy} +\Longrightarrow +\text{price implication}. +} +$$ + +#### 0.2.2 Pricing kernel / no-arbitrage 类题 + +$$ +\begin{aligned} +P_t +&=E_t[m_{t+1}X_{t+1}] +=E_t\left[\frac{\xi_{t+1}}{\xi_t}X_{t+1}\right] +=B_tE_t^Q\left[\frac{X_{t+1}}{B_{t+1}}\right]. +\end{aligned} +$$ + +| 题目关键词 | 你要写的对象 | 公式 | +| ------------------------ | -------- | ------------------------- | +| SDF | $m$ | $P=E[mX]$ | +| state price | $q_s$ | $P=\sum_s q_sX_s$ | +| SPD | $\xi_t$ | $S_t=E_t[\xi_TX_T/\xi_t]$ | +| EMM | $Q$ | $S_t/B_t=E_t^Q[S_T/B_T]$ | +| risk-neutral probability | $\psi_s$ | $P=R_f^{-1}E^Q[X]$ | + +$$ +\boxed{ +\text{payoff} +\Longrightarrow +\text{choose discount object} +\Longrightarrow +\text{write pricing equation} +\Longrightarrow +\text{normalize if needed}. +} +$$ + +#### 0.2.3 Change of measure 类题 + +$$ +\left\{ +\begin{aligned} +dS_t&=(\mu-q)S_tdt+\sigma S_tdW_t^P,\\ +\theta&=\frac{\mu-r}{\sigma},\\ +dW_t^Q&=dW_t^P+\theta dt. +\end{aligned} +\right. +$$ + +代回： + +$$ +\begin{aligned} +dS_t +&=(\mu-q)S_tdt+\sigma S_t +\left(dW_t^Q-\theta dt\right)\\ +&=(\mu-q-\sigma\theta)S_tdt+\sigma S_tdW_t^Q=(r-q)S_tdt+\sigma S_tdW_t^Q. +\end{aligned} +$$ + +density process： + +$$ +\begin{aligned} +\Lambda_t +&=\frac{dQ}{dP}\bigg|_{\mathcal F_t} = +\exp\left( +-\int_0^t\theta_s\,dW_s^P +-\frac12\int_0^t\theta_s^2ds +\right),\\ +E_t^Q[Y] +&= +E_t^P\left[ +\frac{\Lambda_T}{\Lambda_t}Y +\right]. +\end{aligned} +$$ + +$$ +\boxed{ +P\text{-SDE} +\Longrightarrow +\theta=\text{market price of risk} +\Longrightarrow +dW^Q=dW^P+\theta dt +\Longrightarrow +Q\text{-SDE} +\Longrightarrow +Q\text{-pricing}. +} +$$ + +#### 0.2.4 PDE / ODE / SDE 求解类题 + +- SDE：描述 state variable 怎么动。 +- PDE：价格或 value function 同时依赖时间和状态。 +- ODE：时间齐次、无限期、或者猜 affine / power form 后只剩一个变量。 +- Boundary conditions：终端条件、value matching、smooth pasting、absorbing boundary。 + +$$ +\left\{ +\begin{aligned} +dX_t&=\mu_X(t,X_t)dt+\sigma_X(t,X_t)dW_t^Q,\\ +V(t,x)&=E_t^Q\left[e^{-\int_t^T r_sds}g(X_T)\right],\\ +0&=V_t+\mathcal L^QV-rV, +\qquad +V(T,x)=g(x). +\end{aligned} +\right. +$$ + +$$ +\begin{aligned} +\mathcal L^QV +&= +\mu_XV_x+\frac12\sigma_X^2V_{xx}. +\end{aligned} +$$ + +$$ +\begin{aligned} +\text{European option} +&:\quad +V_t+(r-q)SV_S+\frac12\sigma^2S^2V_{SS}-rV=0,\\ +\text{American option} +&:\quad +\max\{V_t+\mathcal L^QV-rV,\ h-V\}=0,\\ +\text{Zero-coupon bond} +&:\quad +P_t+\mu_rP_r+\frac12\sigma_r^2P_{rr}-rP=0,\\ +\text{Optimal stopping} +&:\quad +\frac12\sigma^2x^2V''+\mu xV'-\beta V=0. +\end{aligned} +$$ + +$$ +\boxed{ +\text{state SDE} +\Longrightarrow +\text{generator} +\Longrightarrow +\text{PDE / ODE} +\Longrightarrow +\text{boundary conditions} +\Longrightarrow +\text{solve or characterize}. +} +$$ + +##### （1）常见随机过程 + +$$ +\left\{ +\begin{aligned} +\text{Stock price} +&:\quad +dS_t=\mu_tS_tdt+\sigma_tS_tdW_t,\\ +\text{Money market account} +&:\quad +dB_t=r_tB_tdt,\\ +\text{Discount factor} +&:\quad +\frac{B_t}{B_T} +=\exp\left(-\int_t^T r_sds\right). +\end{aligned} +\right. +$$ + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +r_t=r +&\implies +B_t=B_0e^{rt}, +\qquad +S_t=S_0\exp\left[\left(\mu-\frac12\sigma^2\right)t+\sigma W_t\right],\\ +r_t\text{ stochastic} +&\implies +B_t=B_0\exp\left(\int_0^t r_sds\right), +\qquad +S_t=S_0\exp\left[ +\int_0^t\left(\mu_s-\frac12\sigma_s^2\right)ds+\int_0^t\sigma_sdW_s +\right]. +\end{aligned} +\right. +\end{aligned} +$$ + +Mean-reverting / OU： + +$$ +\left\{ +\begin{aligned} +dX_t&=\kappa(\theta-X_t)dt+\sigma dW_t,\\ +dX_t&=(a-bX_t)dt+\sigma dW_t, +\qquad b=\kappa,\quad a=\kappa\theta,\\ +\kappa&>0. +\end{aligned} +\right. +$$ + +显式解： + +$$ +\begin{aligned} +d(e^{\kappa t}X_t) +&= +\kappa\theta e^{\kappa t}dt ++\sigma e^{\kappa t}dW_t\\ +\Longrightarrow\quad +X_t +&= +\theta+(X_0-\theta)e^{-\kappa t} ++\sigma\int_0^t e^{-\kappa(t-s)}dW_s. +\end{aligned} +$$ + +条件分布： + +$$ +\boxed{ +X_T\mid \mathcal F_t +\sim +N\left( +\theta+(X_t-\theta)e^{-\kappa(T-t)}, +\frac{\sigma^2}{2\kappa}\left(1-e^{-2\kappa(T-t)}\right) +\right). +} +$$ + +老师常用简化： + +$$ +\begin{aligned} +\kappa=1,\quad \theta=0 +\quad +(\text{equiv. }a=0,\ b=1) +\quad\Longrightarrow\quad +dX_t=-X_tdt+\sigma dW_t, +\end{aligned} +$$ + +所以 + +$$ +\begin{aligned} +X_T +&= +X_te^{-(T-t)} ++\sigma\int_t^T e^{-(T-s)}dW_s,\\ +E_t[X_T]&=X_te^{-(T-t)},\\ +\operatorname{Var}_t(X_T) +&= +\frac{\sigma^2}{2} +\left(1-e^{-2(T-t)}\right). +\end{aligned} +$$ + +Vasicek short rate： + +$$ +\left\{ +\begin{aligned} +dr_t&=\kappa(\theta-r_t)dt+\sigma dW_t^Q,\\ +P(t,T)&=E_t^Q\left[e^{-\int_t^T r_sds}\right] +=A(t,T)e^{-B(t,T)r_t}. +\end{aligned} +\right. +$$ + +Coefficient ODE： + +$$ +\left\{ +\begin{aligned} +B_t(t,T)&=\kappa B(t,T)-1, +\qquad B(T,T)=0,\\ +\frac{A_t(t,T)}{A(t,T)} +&= +\kappa\theta B(t,T)-\frac12\sigma^2B(t,T)^2, +\qquad A(T,T)=1. +\end{aligned} +\right. +$$ + +其中 + +$$ +\boxed{ +B(t,T)=\frac{1-e^{-\kappa(T-t)}}{\kappa}. +} +$$ + +#### 0.2.5 Projection / linear algebra 类题 + +$$ +\left\{ +\begin{aligned} +\min_w\quad &w'\Sigma w,\\ +\text{s.t.}\quad &w'\mu=\mu_p,\qquad 1'w=1 +\end{aligned} +\right. +\Longrightarrow +w=\frac12\Sigma^{-1}(\lambda\mu+\gamma1). +$$ + +线性定价： + +$$ +\begin{aligned} +m=a-b'f +&\Longrightarrow +1=E[mR_i]\\ +&\Longrightarrow +E[R_i]-R_f +=\frac{b'}{E[m]}\operatorname{Cov}(f,R_i)\\ +&\Longrightarrow +E[R_i]-R_f=\beta_i'\lambda_f. +\end{aligned} +$$ + +$$ +\boxed{ +\text{quadratic objective / linear SDF} +\Longrightarrow +\text{linear FOC} +\Longrightarrow +\Sigma^{-1}\text{ projection} +\Longrightarrow +\beta\text{ pricing}. +} +$$ + +#### 0.2.6 Bayesian updating / information 类题 +静态更新： + +$$ +\left\{ +\begin{aligned} +v&\sim N(m_0,\rho_0^{-1}),\\ +s_i&=v+\varepsilon_i,\qquad \varepsilon_i\sim N(0,\rho_i^{-1}) +\end{aligned} +\right. +\Longrightarrow +\left\{ +\begin{aligned} +\rho^{post} +&=\rho_0+\sum_i\rho_i,\\ +E[v\mid s_1,\dots,s_n] +&= +\frac{\rho_0m_0+\sum_i\rho_is_i} +{\rho_0+\sum_i\rho_i}. +\end{aligned} +\right. +$$ + +Kyle pricing： + +$$ +\begin{aligned} +y&=x+u,\\ +p&=E[v\mid y] +=E[v]+\frac{\operatorname{Cov}(v,y)}{\operatorname{Var}(y)}(y-E[y]). +\end{aligned} +$$ + +$$ +\boxed{ +\text{prior} +\Longrightarrow +\text{signal / order flow} +\Longrightarrow +\text{conditional expectation} +\Longrightarrow +\text{price} +\Longrightarrow +\text{trading strategy equilibrium}. +} +$$ + +#### 0.2.7 Equilibrium / market clearing 类题 + +$$ +\left\{ +\begin{aligned} +\text{Individual FOC} +&:\quad +\delta_i^t u_i'(c_{i,t})=\lambda_i\xi_t,\\ +\text{Feasibility} +&:\quad +\sum_i c_{i,t}=e_t,\\ +\text{Market clearing} +&:\quad +\sum_i\theta_{i,t}=\bar\theta. +\end{aligned} +\right. +$$ + +$$ +\boxed{ +\text{agents' problems} +\Longrightarrow +\text{FOCs} +\Longrightarrow +\text{aggregate constraints} +\Longrightarrow +\text{solve allocations} +\Longrightarrow +\text{recover prices / SPD}. +} +$$ + +#### 0.2.8 容易遗漏但常考的操作 +1. Normalization：SDF、state price、risk-neutral probability 之间经常只差一个归一化常数。 + +$$ +\begin{aligned} +q_s&=\pi_sm_s,\qquad +\psi_s=\frac{q_s}{\sum_jq_j},\qquad +R_f=\frac{1}{E[m]}. +\end{aligned} +$$ + +2. Discounting：risk-neutral pricing 不是只换概率，还要用 money market account 折现。 + +$$ +\begin{aligned} +S_t +&=B_tE_t^Q\left[\frac{X_T}{B_T}\right]. +\end{aligned} +$$ + +3. Terminal / boundary conditions：微分方程本身不够，必须配边界条件。 + +$$ +\left\{ +\begin{aligned} +V(T,S)&=g(S),\\ +V(t,S^*)&=h(S^*),\\ +V_S(t,S^*)&=h'(S^*). +\end{aligned} +\right. +$$ + +4. Conditioning：信息题的价格通常不是无条件期望，而是条件期望。 + +$$ +\begin{aligned} +p +&=E[v\mid \mathcal I] +=E[v]+\operatorname{Cov}(v,\mathcal I)\operatorname{Var}(\mathcal I)^{-1} +(\mathcal I-E[\mathcal I]). +\end{aligned} +$$ + +5. Existence / uniqueness：complete markets 对应 unique SDF / unique EMM；incomplete markets 对应 many SDFs / many EMMs。 + +$$ +\begin{aligned} +\text{complete markets} +&\Longleftrightarrow +\text{unique }m +\Longleftrightarrow +\text{unique }Q,\\ +\text{incomplete markets} +&\Longleftrightarrow +\text{many }m +\Longleftrightarrow +\text{many }Q. +\end{aligned} +$$ + +### 0.3 总公式链 + +$$ +\begin{aligned} +\max E\sum_t\delta^tu(c_t) +&\Longrightarrow +m_{t+1}=\delta\frac{u'(c_{t+1})}{u'(c_t)}\\ +&\Longrightarrow +P_t=E_t[m_{t+1}X_{t+1}] +\Longleftrightarrow +1=E_t[m_{t+1}R_{t+1}]\\ +&\Longrightarrow +E[R_i]-R_f=-R_f\operatorname{Cov}(m,R_i)\\ +&\Longrightarrow +\text{linear }m +\Longleftrightarrow +\text{CAPM / factor pricing / MV efficiency}\\ +&\Longrightarrow +\xi_t=\frac{\Lambda_t}{B_t}, +\qquad +S_t=E_t^P\left[\frac{\xi_T}{\xi_t}X_T\right] +=B_tE_t^Q\left[\frac{X_T}{B_T}\right]\\ +&\Longrightarrow +\text{discounted prices are }Q\text{-martingales}\\ +&\Longrightarrow +\text{options, bonds, and all derivatives are priced by the same equation}\\ +&\Longrightarrow +\text{with information frictions, replace unconditional payoff by }E[v\mid\text{market information}]. +\end{aligned} +$$ + +## 1. SDF 是核心对象 +### 1.1 题型 +#### 1.1.1 从个人优化到 SDF +**题目入口**：看到 representative agent、consumption-saving、portfolio choice，先写预算约束和 Euler equation；目标是把最优消费转成 pricing kernel。 + +最小系统和任意资产 $i$ 的 Euler equation： + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +\max_{\{c_t,\theta_t\}} +&\quad E_0\left[\sum_{t=0}^T \delta^t u(c_t)\right], +&&\text{(preferences)},\\ +c_t+\sum_i \theta_{i,t}p_{i,t} +&=e_t+\sum_i \theta_{i,t-1}x_{i,t}, +&&\text{(budget constraint)},\\ +R_{i,t+1} +&=\frac{x_{i,t+1}}{p_{i,t}}, +&&\text{(gross return)}. +\end{aligned} +\right. +&\implies +0=E_t\left[ +\delta u'(c_{t+1})x_{i,t+1} +-u'(c_t)p_{i,t} +\right]\\ +\Longleftrightarrow\quad +p_{i,t} +&=E_t\left[ +\delta\frac{u'(c_{t+1})}{u'(c_t)}x_{i,t+1} +\right]\\ +\Longleftrightarrow\quad +p_{i,t} +&=E_t[m_{t+1}x_{i,t+1}], +\qquad +m_{t+1}:=\delta\frac{u'(c_{t+1})}{u'(c_t)}. +\end{aligned} +$$ + +如果题目给的是 return 而不是 payoff，用 $x_{i,t+1}=p_{i,t}R_{i,t+1}$： + +$$ +\begin{aligned} +p_{i,t} +&=E_t[m_{t+1}p_{i,t}R_{i,t+1}]\\ +\Longleftrightarrow\quad +1 +&=E_t[m_{t+1}R_{i,t+1}]. +\end{aligned} +$$ + + +$$ +\boxed{ +\text{optimal consumption} +\Longrightarrow +\text{marginal utility ratio} +\Longrightarrow +\text{SDF} +\Longrightarrow +1=E_t[mR]. +} +$$ + +#### 1.1.2 从 SDF 到 state price、risk-neutral probability +**题目入口**：看到 finite states、Arrow security、risk-neutral probability，先从 $P=E[mX]$ 展开成状态求和，再归一化 state price。 + +有限状态下，设 physical probability 为 $\pi_s$，state payoff 为 $X_s$： + +$$ +\begin{aligned} +P +&=E[mX]=\sum_s \pi_s m_sX_s=\sum_s q_sX_s, +\qquad q_s:=\pi_sm_s,\\ +P_f +&=\sum_s q_s=E[m],\\ +R_f +&=\frac{1}{P_f}=\frac{1}{E[m]},\\ +\psi_s +&:=\frac{q_s}{\sum_jq_j}=\frac{\pi_sm_s}{E[m]}=R_fq_s,\\ +\sum_s\psi_s&=1,\\ +P +&=\sum_s q_sX_s=\frac{1}{R_f}\sum_s\psi_sX_s=\frac{1}{R_f}E^Q[X]. +\end{aligned} +$$ + + +$$ +\boxed{ +P=E[mX] +\Longleftrightarrow +P=\sum_s q_sX_s +\Longleftrightarrow +P=\frac{1}{R_f}E^Q[X]. +} +$$ + +#### 1.1.3 从 SDF 到 risk premium +**题目入口**：看到 expected return、risk premium、bad times，先从 $1=E[mR_i]$ 展开 covariance。 + +由 $1=E[mR_i]$： + +$$ +\begin{aligned} +1 +&=E[mR_i]=E[m]E[R_i]+\operatorname{Cov}(m,R_i), +\qquad +R_f=\frac{1}{E[m]},\\ +E[R_i]-R_f +&=-\frac{\operatorname{Cov}(m,R_i)}{E[m]}=-R_f\operatorname{Cov}(m,R_i). +\end{aligned} +$$ + +因此： + +$$ +\boxed{ +\text{asset has high premium} +\Longleftrightarrow +\operatorname{Cov}(m,R_i)<0. +} +$$ + +经济含义： + +$$ +\begin{aligned} +\operatorname{Cov}(m,R_i)<0 +&\Longleftrightarrow +R_i\text{ 在 bad times }(m\text{ high})\text{ 表现差}\\ +&\Longleftrightarrow +\text{asset carries systematic risk}\\ +&\Longleftrightarrow +\text{investors require positive risk premium}. +\end{aligned} +$$ + +### 1.2 解法 +#### 1.2.1 静态 SDF：所有价格的共同语言 + +$$ +\begin{aligned} +P +&=E[mX],\\ +1 +&=E[mR],\\ +E[R_i]-R_f +&=-R_f\operatorname{Cov}(m,R_i). +\end{aligned} +$$ + +| 对象 | 含义 | +| --------------------------------- | ------------------ | +| state price $q_s$ | 状态 $s$ 的 Arrow price | +| SDF $m_s$ | 每单位 payoff 的随机折现权重 | +| risk-neutral probability $\psi_s$ | 把状态价格归一化后的概率 | + +$$ +\begin{aligned} +q_s&=\pi_s m_s,\\ +\psi_s&=R_f q_s=\frac{\pi_s m_s}{E[m]},\\ +P&=\sum_s q_sX_s +=E[mX] +=\frac{1}{R_f}E^Q[X]. +\end{aligned} +$$ + +#### 1.2.2 边际效用 SDF：价格来自最优消费 + +$$ +\begin{aligned} +P_t +&=E_t\left[ +\delta\frac{u'(c_{t+1})}{u'(c_t)}X_{t+1} +\right],\\ +m_{t+1} +&=\delta\frac{u'(c_{t+1})}{u'(c_t)}. +\end{aligned} +$$ + +$$ +\begin{aligned} +\text{preferences} +\Longrightarrow +\text{marginal rate of substitution} +\Longrightarrow +\text{SDF} +\Longrightarrow +\text{risk premia}. +\end{aligned} +$$ + +#### 1.2.3 偏好设定与 SDF + +$$ +\left\{ +\begin{aligned} +\text{time-separable expected utility} +&:\quad +m_{t+1}=\delta\frac{u'(c_{t+1})}{u'(c_t)},\\ +\text{CRRA} +&:\quad +u(c)=\frac{c^{1-\gamma}}{1-\gamma} +\implies +m_{t+1} +=\delta\left(\frac{c_{t+1}}{c_t}\right)^{-\gamma}, +\qquad +\text{IES}=\frac{1}{\gamma},\\ +\text{log utility} +&:\quad +u(c)=\log c +\implies +m_{t+1}=\delta\frac{c_t}{c_{t+1}}, +\qquad +\gamma=1,\\ +\text{CARA} +&:\quad +u(c)=-\frac{1}{\alpha}e^{-\alpha c} +\implies +m_{t+1} +=\delta e^{-\alpha(c_{t+1}-c_t)},\\ +\text{mean-variance / quadratic} +&:\quad +\max_w\left\{w'(\mu-R_f1)-\frac{\gamma}{2}w'\Sigma w\right\} +\implies +w^*=\frac1\gamma\Sigma^{-1}(\mu-R_f1),\\ +\text{habit} +&:\quad +u(c_t-H_t) +\implies +m_{t+1} +=\delta\frac{u'(c_{t+1}-H_{t+1})}{u'(c_t-H_t)}. +\end{aligned} +\right. +$$ + +Epstein-Zin preference 把 risk aversion 和 intertemporal substitution 分开： + +$$ +\begin{aligned} +V_t +&= +\left[ +(1-\delta)c_t^{1-\frac1\psi} ++\delta +\left(E_t[V_{t+1}^{1-\gamma}]\right)^{ +\frac{1-\frac1\psi}{1-\gamma}} +\right]^{\frac{1}{1-\frac1\psi}},\\ +\theta +&:=\frac{1-\gamma}{1-\frac1\psi}\\ +\implies\quad +m_{t+1}^{EZ} +&= +\left[ +\delta +\left(\frac{c_{t+1}}{c_t}\right)^{-\frac1\psi} +\right]^{\theta} +R_{w,t+1}^{\theta-1}. +\end{aligned} +$$ + +$$ +\boxed{ +\gamma\text{ controls risk aversion}, +\qquad +\psi\text{ controls IES}, +\qquad +\gamma=\frac1\psi +\implies +m_{t+1}^{EZ} +=\delta\left(\frac{c_{t+1}}{c_t}\right)^{-\gamma}. +} +$$ + +#### 1.2.4 CCAPM lognormal 速查 +看到 consumption growth、CRRA、lognormal return，先把 SDF 写成 consumption growth 的函数，再用 lognormal moment generating function 展开 Euler equation。 + +设 + +$$ +\left\{ +\begin{aligned} +g_{t+1} +&:=\Delta\log c_{t+1} +=\log\frac{c_{t+1}}{c_t},\\ +u(c) +&=\frac{c^{1-\gamma}}{1-\gamma},\\ +m_{t+1} +&=\delta\left(\frac{c_{t+1}}{c_t}\right)^{-\gamma} +=\delta e^{-\gamma g_{t+1}}. +\end{aligned} +\right. +$$ + +若 $g_{t+1}\mid\mathcal F_t\sim N(\mu_g,\sigma_g^2)$，则 risk-free rate 来自 + +$$ +\begin{aligned} +R_{f,t}^{-1} +&=E_t[m_{t+1}] +=\delta E_t[e^{-\gamma g_{t+1}}]\\ +&=\delta +\exp\left( +-\gamma\mu_g+\frac12\gamma^2\sigma_g^2 +\right), +\end{aligned} +$$ + +所以 + +$$ +\boxed{ +\log R_{f,t} = +-\log\delta ++\gamma\mu_g +-\frac12\gamma^2\sigma_g^2. +} +$$ + +若 asset $i$ 的 log return 为 $r_{i,t+1}:=\log R_{i,t+1}$，且 $(r_{i,t+1},g_{t+1})$ joint normal，则 Euler equation 给出 log risk premium： + +$$ +\begin{aligned} +1 +&=E_t[m_{t+1}R_{i,t+1}] +=E_t\left[ +\exp\left(\log\delta-\gamma g_{t+1}+r_{i,t+1}\right) +\right]\\ +\Longleftrightarrow\quad +0 +&= +\log\delta-\gamma\mu_g+\mu_i ++\frac12\operatorname{Var}_t(r_{i,t+1}-\gamma g_{t+1})\\ +&= +\log\delta-\gamma\mu_g+\mu_i ++\frac12\sigma_i^2 ++\frac12\gamma^2\sigma_g^2 +-\gamma\operatorname{Cov}_t(r_{i,t+1},g_{t+1}). +\end{aligned} +$$ + +代入 $\log R_{f,t}$： + +$$ +\boxed{ +\mu_i-\log R_{f,t} ++\frac12\sigma_i^2 = +\gamma\operatorname{Cov}_t(r_{i,t+1},g_{t+1}). +} +$$ + +考试判断： + +$$ +\begin{aligned} +\operatorname{Cov}_t(r_{i,t+1},g_{t+1})>0 +&\Longrightarrow +\text{asset pays high when consumption is high}\\ +&\Longrightarrow +\text{bad hedge against bad times}\\ +&\Longrightarrow +\text{positive risk premium}. +\end{aligned} +$$ + +### 1.3 证明 +#### 1.3.1 Jensen / risk aversion / risk premium +凹效用 $\Longleftrightarrow$ 风险厌恶，且 certainty equivalent 不超过期望财富。 + +$$ +\begin{aligned} +u''(w)\le0 +&\Longrightarrow +u(E[\tilde w])\ge E[u(\tilde w)] +&&\text{(Jensen)},\\ +u(x)&=E[u(\tilde w)], +\qquad +w=E[\tilde w]\\ +&\Longrightarrow +u(w)\ge u(x)\\ +&\Longrightarrow +w\ge x +&&\text{if }u'\!>0,\\ +\pi&:=w-x\ge0. +\end{aligned} +$$ + +#### 1.3.2 Marginal utility SDF +投资者最优消费的一阶条件给出 SDF。 + +$$ +\begin{aligned} +\max_{\theta_i}\ E_t[u(c_t)+\delta u(c_{t+1})] +&\Longrightarrow +0= +E_t\left[ +\delta u'(c_{t+1})x_{i,t+1} +-u'(c_t)p_{i,t} +\right]\\ +&\Longleftrightarrow +p_{i,t} = +E_t\left[ +\delta\frac{u'(c_{t+1})}{u'(c_t)}x_{i,t+1} +\right]\\ +&\Longleftrightarrow +p_{i,t}=E_t[m_{t+1}x_{i,t+1}], +\qquad +m_{t+1}:=\delta\frac{u'(c_{t+1})}{u'(c_t)}. +\end{aligned} +$$ + +#### 1.3.3 SDF、state price、risk-neutral probability +$P=E[mX]$、state price pricing、risk-neutral pricing 是同一件事。 + +$$ +\begin{aligned} +P +&=E[mX]=\sum_s\pi_sm_sX_s=\sum_sq_sX_s, +\qquad q_s:=\pi_sm_s,\\ +P_f +&=\sum_sq_s=E[m], +\qquad +R_f=\frac{1}{E[m]},\\ +\psi_s +&:=\frac{q_s}{\sum_jq_j}=R_fq_s,\\ +P +&=\sum_sq_sX_s=\frac{1}{R_f}\sum_s\psi_sX_s=\frac{1}{R_f}E^Q[X]. +\end{aligned} +$$ + +#### 1.3.4 Risk premium covariance formula +风险溢价由资产收益与 SDF 的协方差决定。 + +$$ +\begin{aligned} +1 +&=E[mR_i]=E[m]E[R_i]+\operatorname{Cov}(m,R_i), +\qquad +R_f=\frac{1}{E[m]}\\ +\Longrightarrow\quad +E[R_i]-R_f +&=-\frac{\operatorname{Cov}(m,R_i)}{E[m]}=-R_f\operatorname{Cov}(m,R_i). +\end{aligned} +$$ + +#### 1.3.5 Hansen-Jagannathan bound +Sharpe ratio 被 SDF volatility 约束。 + +$$ +\begin{aligned} +E[R_i]-R_f +&=-R_f\operatorname{Cov}(m,R_i)\\ +&=-R_f\rho_{m,R_i}\sigma(m)\sigma(R_i),\\ +\left| +\frac{E[R_i]-R_f}{\sigma(R_i)} +\right| +&\le +R_f\sigma(m)\\ +&= +\frac{\sigma(m)}{E[m]}. +\end{aligned} +$$ + +$$ +\boxed{ +\max_i +\left| +\frac{E[R_i]-R_f}{\sigma(R_i)} +\right| +\le +\frac{\sigma(m)}{E[m]}. +} +$$ + +## 2. 无套利把 SDF 变成 martingale +### 2.1 题型 +#### 2.1.1 从 one-period SDF 到 dynamic SPD / EMM +**题目入口**：看到 dynamic pricing、continuous time、martingale measure，先把 one-period SDF 串成 $m_{t,T}$，再写成 SPD $\xi_T/\xi_t$。 + +动态模型中，把 one-period SDF 串起来： + +$$ +\begin{aligned} +m_{t,T} +&:=\prod_{s=t+1}^T m_s,\\ +S_t +&=E_t[m_{t,T}X_T] +=E_t^P\left[\frac{\xi_T}{\xi_t}X_T\right],\\ +\xi_t +&=\frac{\Lambda_t}{B_t}, +\qquad +\Lambda_t:=E_t^P\left[\frac{dQ}{dP}\right],\\ +B_t +&=\exp\left(\int_0^t r_sds\right),\\ +S_t +&=E_t^P\left[\frac{\xi_T}{\xi_t}X_T\right] +=E_t^P\left[\frac{\Lambda_T/B_T}{\Lambda_t/B_t}X_T\right] +=B_tE_t^P\left[\frac{\Lambda_T}{\Lambda_t}\frac{X_T}{B_T}\right] +=B_tE_t^Q\left[\frac{X_T}{B_T}\right]. +\end{aligned} +$$ + +若 $X_T=S_T$，则 discounted price 是 $Q$-martingale： + +$$ +\begin{aligned} +\frac{S_t}{B_t} +&=E_t^Q\left[\frac{S_T}{B_T}\right], +\end{aligned} +$$ + + +$$ +\boxed{ +\text{SDF} +\Longrightarrow +\text{SPD }\xi_t +\Longrightarrow +\text{EMM }Q +\Longrightarrow +\frac{S_t}{B_t}\text{ is a }Q\text{-martingale}. +} +$$ + +### 2.2 解法 +#### 2.2.1 从 one-period 到 dynamic pricing +one-period SDF 是一个随机变量： + +$$ +\begin{aligned} +P_t +&=E_t[m_{t,t+1}X_{t+1}],\\ +1 +&=E_t[m_{t,t+1}R_{t+1}]. +\end{aligned} +$$ + +$$ +\begin{aligned} +S_t +&=E_t\left[\frac{\xi_T}{\xi_t}X_T\right] +=B_tE_t^Q\left[\frac{X_T}{B_T}\right]. +\end{aligned} +$$ + +| 符号 | 角色 | +|---|---| +| $m$ | one-period SDF | +| $\xi_t$ | continuous-time state price density / pricing kernel | +| $\Lambda_t$ | Radon-Nikodym density / change of measure | +| $Q$ | equivalent martingale measure | + +SDF 和 SPD 的关系： + +$$ +\left\{ +\begin{aligned} +m_{t,T} +&=\text{从 }t\text{ 到 }T\text{ 的 stochastic discount factor},\\ +\xi_t +&=\text{time-}t\text{ state price density / pricing kernel process},\\ +m_{t,T} +&=\frac{\xi_T}{\xi_t}, +&&\text{(SDF is the ratio of SPD across time)},\\ +S_t +&=E_t[m_{t,T}X_T] +=E_t\left[\frac{\xi_T}{\xi_t}X_T\right]. +\end{aligned} +\right. +$$ + +所以：SDF 是“两个时点之间”的随机折现因子；SPD 是“每个时点上”的状态价格密度过程。给定 SPD $\xi_t$，任意 $t$ 到 $T$ 的 SDF 都由比值 $\xi_T/\xi_t$ 给出。 + +无套利的三种等价语言： + +$$ +\begin{aligned} +\text{No arbitrage} +&\Longleftrightarrow +\exists\ \xi_t>0 \text{ such that prices can be written by SPD}\\ +&\Longleftrightarrow +\exists\ Q\sim P \text{ such that discounted prices are martingales}\\ +&\Longleftrightarrow +\text{linear pricing rule has no free lunch}. +\end{aligned} +$$ + +更具体地，分两种情况： + +1. **没有中间 dividend** + + $$ + \begin{aligned} + S_t + &=E_t\left[\frac{\xi_T}{\xi_t}S_T\right]\\ + \Longleftrightarrow\quad + \xi_tS_t + &=E_t[\xi_TS_T]\\ + \Longleftrightarrow\quad + \xi_tS_t + &\text{ is a }P\text{-martingale}. + \end{aligned} + $$ + +2. **有中间 dividend flow $D_t$** + + $$ + \begin{aligned} + S_t + &=E_t\left[ + \int_t^T\frac{\xi_s}{\xi_t}D_s\,ds + +\frac{\xi_T}{\xi_t}S_T + \right]\\ + \Longleftrightarrow\quad + \xi_tS_t + &=E_t\left[ + \int_t^T\xi_sD_s\,ds+\xi_TS_T + \right]\\ + \Longleftrightarrow\quad + \xi_tS_t+\int_0^t\xi_sD_s\,ds + &\text{ is a }P\text{-martingale}. + \end{aligned} + $$ + +这就是“无套利把 SDF 变成 martingale”的精确含义：不是 $S_t$ 自己一定是 martingale，而是 **用 state price density 调整后的 cum-dividend value** 是 martingale。 + +#### 2.2.2 SPD 与 EMM 的连接 +令 money market account 为 + +$$ +\begin{aligned} +B_t&=\exp\left(\int_0^t r_sds\right). +\end{aligned} +$$ + +若 $\xi_t$ 是 SPD，则可以分解为： + +$$ +\begin{aligned} +\xi_t +&=\frac{\Lambda_t}{B_t}, +\qquad +\Lambda_t:=\xi_tB_t. +\end{aligned} +$$ + +其中 $\Lambda_t$ 是从 $P$ 到 $Q$ 的 density process： + +$$ +\begin{aligned} +\Lambda_t +&=E_t\left[\frac{dQ}{dP}\right]. +\end{aligned} +$$ + +更完整地说，若 $Q\sim P$，则存在 Radon-Nikodym derivative： + +$$ +\begin{aligned} +Z_T +&:=\frac{dQ}{dP}>0, +\qquad +E^P[Z_T]=1. +\end{aligned} +$$ + +density process 是这个终值 likelihood ratio 的条件期望过程： + +$$ +\begin{aligned} +\Lambda_t +&:=E_t^P[Z_T] +=E_t^P\left[\frac{dQ}{dP}\right]. +\end{aligned} +$$ + +因此 $\Lambda_t$ 是 $P$-martingale： + +$$ +\begin{aligned} +E_s^P[\Lambda_t] +&=E_s^P\!\left[E_t^P[Z_T]\right] +=E_s^P[Z_T] +=\Lambda_s, +\qquad s\le t. +\end{aligned} +$$ + +它能做测度变化，是因为条件 Bayes formula： + +$$ +\begin{aligned} +E_t^Q[Y] +&= +\frac{E_t^P[Z_TY]}{E_t^P[Z_T]}\\ +&= +\frac{1}{\Lambda_t}E_t^P[Z_TY]\\ +&= +E_t^P\left[\frac{\Lambda_T}{\Lambda_t}Y\right], +\qquad +\Lambda_T=Z_T. +\end{aligned} +$$ + +所以在 $P$ 下乘上 $\Lambda_T/\Lambda_t$，等价于换到 $Q$ 下取条件期望。 + +把 SPD 定价式改写： + +$$ +\begin{aligned} +S_t +&=E_t^P\left[\frac{\xi_T}{\xi_t}X_T\right] +=E_t^P\left[\frac{\Lambda_T/B_T}{\Lambda_t/B_t}X_T\right] +=B_tE_t^P\left[\frac{\Lambda_T}{\Lambda_t}\frac{X_T}{B_T}\right] +=B_tE_t^Q\left[\frac{X_T}{B_T}\right]. +\end{aligned} +$$ + +注意：测度变换由 $\Lambda_T/\Lambda_t$ 完成，不是由 $X_T/B_T$ 完成。这里 $\dfrac{X_T}{B_T}$ 只是把 payoff 换成 money-market account 作为 numeraire 后的 discounted payoff： + +$$ +\begin{aligned} +\underbrace{\frac{\Lambda_T}{\Lambda_t}}_{\text{change of measure}} +\cdot +\underbrace{\frac{X_T}{B_T}}_{\text{discounted payoff}} +&\Longrightarrow +E_t^Q\left[\frac{X_T}{B_T}\right]. +\end{aligned} +$$ + +所以： + +$$ +\boxed{ +\text{SPD pricing under }P +\quad\Longleftrightarrow\quad +\text{risk-neutral pricing under }Q. +} +$$ + +如果 $X_T=S_T$，则： + +$$ +\begin{aligned} +\frac{S_t}{B_t} +&=E_t^Q\left[\frac{S_T}{B_T}\right], +\end{aligned} +$$ + +也就是说 discounted price 是 $Q$-martingale。 + +##### （1）做题规则：SDF/SPD vs. risk-neutral pricing +1. **用 SDF / SPD 定价** + + $$ + \begin{aligned} + S_t + &=E_t^P[m_{t,T}X_T] + =E_t^P\left[\frac{\xi_T}{\xi_t}X_T\right]. + \end{aligned} + $$ + + 这里折现已经包含在 $m_{t,T}$ 或 $\xi_T/\xi_t$ 里面；不要再额外乘一个 $e^{-r(T-t)}$，除非你先把 SDF 拆成 $\Lambda/B$。 + +2. **用 risk-neutral pricing** + + $$ + \begin{aligned} + S_t + &=B_tE_t^Q\left[\frac{X_T}{B_T}\right]. + \end{aligned} + $$ + + 如果 $r$ 是常数： + + $$ + \begin{aligned} + S_t + &=e^{-r(T-t)}E_t^Q[X_T]. + \end{aligned} + $$ + + 这里必须显式折现，因为 $Q$ 只改变概率分布，不自动把未来 payoff 变成今天价格。 + +3. **从 physical measure 写成 risk-neutral measure** + + 标准步骤： + + $$ + \left\{ + \begin{aligned} + \text{Step 1: }&\text{写出 }P\text{ 下 dynamics},\\ + \text{Step 2: }&\text{找 market price of risk }\theta_t,\\ + \text{Step 3: }&dW_t^Q=dW_t^P+\theta_tdt,\\ + \text{Step 4: }&\text{把 }dW_t^P=dW_t^Q-\theta_tdt\text{ 代回},\\ + \text{Step 5: }&\text{检查 discounted price 在 }Q\text{ 下是 martingale}. + \end{aligned} + \right. + $$ + + 这里 $\theta_t$ 的精确定义是 **market price of risk**。在一维模型里： + + $$ + \begin{aligned} + \theta_t + &= + \frac{\mu_t-r_t}{\sigma_t}, + \end{aligned} + $$ + + 它等于 instantaneous Sharpe ratio。多维模型里不是一个简单标量 Sharpe ratio，而是解下面的 risk premium equation： + + $$ + \begin{aligned} + \mu_t-r_t1 + &= + \sigma_t\theta_t. + \end{aligned} + $$ + + 所以测度转换的本质是： + + $$ + \boxed{ + \text{risk premium }(\mu-r) + = + \text{exposure }\sigma + \times + \text{price of risk }\theta. + } + $$ + + 然后用 $\theta$ 去平移 Brownian drift，使 risky asset 在 $Q$ 下只赚 risk-free rate。 + +例题：股票在 physical measure $P$ 下满足 + +$$ +\begin{aligned} +dS_t +&=\mu S_tdt+\sigma S_tdW_t^P, +\qquad +dB_t=rB_tdt. +\end{aligned} +$$ + +要求写成 risk-neutral measure $Q$。因为风险资产 excess return 为 $\mu-r$，market price of risk 为 + +$$ +\begin{aligned} +\theta +&=\frac{\mu-r}{\sigma}. +\end{aligned} +$$ + +定义 + +$$ +\begin{aligned} +dW_t^Q +&=dW_t^P+\theta dt +=dW_t^P+\frac{\mu-r}{\sigma}dt. +\end{aligned} +$$ + +于是 + +$$ +\begin{aligned} +dW_t^P +&=dW_t^Q-\frac{\mu-r}{\sigma}dt. +\end{aligned} +$$ + +代回股票 dynamics： + +$$ +\begin{aligned} +dS_t +&=\mu S_tdt+\sigma S_t +\left(dW_t^Q-\frac{\mu-r}{\sigma}dt\right)\\ +&=\mu S_tdt+\sigma S_tdW_t^Q-(\mu-r)S_tdt=rS_tdt+\sigma S_tdW_t^Q. +\end{aligned} +$$ + +因此 $Q$ 下： + +$$ +\boxed{ +dS_t=rS_tdt+\sigma S_tdW_t^Q, +\qquad +S_t=e^{-r(T-t)}E_t^Q[S_T]. +} +$$ + +若 payoff 是 option $X_T=(S_T-K)^+$，则： + +$$ +\boxed{ +C_t=e^{-r(T-t)}E_t^Q[(S_T-K)^+]. +} +$$ + +一般多资产版本： + +$$ +\left\{ +\begin{aligned} +dS_t +&=\operatorname{diag}(S_t)\big(\mu_tdt+\sigma_tdW_t^P\big),\\ +dB_t&=r_tB_tdt,\\ +\mu_t-r_t1&=\sigma_t\theta_t. +\end{aligned} +\right. +$$ + +定义 + +$$ +\begin{aligned} +dW_t^Q +&=dW_t^P+\theta_tdt. +\end{aligned} +$$ + +代回： + +$$ +\begin{aligned} +dS_t +&=\operatorname{diag}(S_t) +\left[ +\mu_tdt+\sigma_t(dW_t^Q-\theta_tdt) +\right]\\ +&=\operatorname{diag}(S_t) +\left[ +(\mu_t-\sigma_t\theta_t)dt+\sigma_tdW_t^Q +\right]\\ +&=\operatorname{diag}(S_t) +\left[ +r_t1\,dt+\sigma_tdW_t^Q +\right]. +\end{aligned} +$$ + +所以风险中性测度的判别标准是： + +$$ +\boxed{ +\text{under }Q,\quad +\text{all traded assets have drift }r_t +\text{ after dividend adjustment}. +} +$$ + +做题时不要说“把 $\theta$ 设成 Sharpe ratio 然后偏移”就结束；完整流程是： + +1. 写出 $P$ 下 drift 和 diffusion。 +2. 用 $\mu-r1=\sigma\theta$ 解 market price of risk。 +3. 定义 $dW^Q=dW^P+\theta dt$。 +4. 代回 dynamics，把 drift 改成 $r$。 +5. 用 $Q$ 下 dynamics 计算 discounted expectation。 + +#### 2.2.3 为什么这来自无套利 +套利的本质是：存在一个 payoff 非负且有正概率严格为正，但今天成本非正。若这样的 payoff 存在，就不可能有严格正的 pricing kernel $\xi$，因为： + +$$ +\begin{aligned} +X_T\ge0,\quad P(X_T>0)>0,\quad \xi_T>0 +&\Longrightarrow +E[\xi_TX_T]>0. +\end{aligned} +$$ + +但套利要求它的价格 $\le0$，与 + +$$ +\begin{aligned} +P_0=E[\xi_TX_T] +\end{aligned} +$$ + +矛盾。因此： + +$$ +\begin{aligned} +\text{no arbitrage} +&\Longrightarrow +\text{strictly positive pricing kernel / SPD}\\ +&\Longrightarrow +\text{SPD-adjusted values are martingales}\\ +&\Longrightarrow +\text{discounted prices are }Q\text{-martingales}. +\end{aligned} +$$ + +反过来，如果存在 $Q\sim P$ 使 discounted gains 都是 martingales，那么任何 zero-cost self-financing strategy 的 discounted wealth 也是 martingale，不能从 $0$ 开始变成非负且有正概率严格为正的终值。这排除了套利。 + +#### 2.2.4 该记住的对应关系 +| 语言 | 公式 | 直觉 | +|---|---|---| +| SDF | $P_t=E_t[m_{t,T}X_T]$ | 用随机折现权重给 payoff 定价 | +| SPD | $S_t=E_t[(\xi_T/\xi_t)X_T]$ | 把 SDF 做成跨时点过程 | +| SPD martingale | $\xi_tS_t+\int_0^t\xi_sD_sds$ is $P$-martingale | 价格加 dividend 后，按状态价格调整为公平游戏 | +| EMM | $S_t/B_t=E_t^Q[S_T/B_T]$ | 换到 $Q$ 下，discounted price 是公平游戏 | +| Risk-neutral pricing | $S_t=B_tE_t^Q[X_T/B_T]$ | 用市场利率折现，再用 $Q$ 下期望定价 | + +#### 2.2.5 Complete market 的意义 + +$$ +\begin{aligned} +\text{complete market} +&\Longleftrightarrow +\text{every payoff can be replicated}\\ +&\Longleftrightarrow +\text{unique SDF / unique EMM}\\ +&\Longrightarrow +\text{derivative price is pinned down uniquely}. +\end{aligned} +$$ + +这条线连接： + +- finite states 下的 market completeness。 +- dynamic trading 下的 replication。 +- Black-Scholes 用 stock + bond 复制 option。 +#### 2.2.6 Complete market rank condition +有限状态下，完备市场不是一句“payoff 都能复制”，而是 payoff matrix 的秩条件。 + +设有 $S$ 个 states、$N$ 个 traded assets。令 payoff matrix 为 + +$$ +\begin{aligned} +X +&= +\begin{pmatrix} +x_{11}&\cdots&x_{1N}\\ +\vdots&\ddots&\vdots\\ +x_{S1}&\cdots&x_{SN} +\end{pmatrix}, +\qquad +p= +\begin{pmatrix} +p_1\\ +\vdots\\ +p_N +\end{pmatrix}. +\end{aligned} +$$ + +其中第 $s$ 行是 state $s$ 下各资产 payoff，第 $j$ 列是资产 $j$ 的 state payoff。任意 contingent claim $z\in\mathbb R^S$ 可复制，当且仅当存在 portfolio $\theta\in\mathbb R^N$： + +$$ +\begin{aligned} +X\theta&=z. +\end{aligned} +$$ + +因此 + +$$ +\boxed{ +\text{complete market} +\Longleftrightarrow +\operatorname{rank}(X)=S. +} +$$ + +State price vector $q\in\mathbb R^S$ 满足 + +$$ +\begin{aligned} +X'q&=p. +\end{aligned} +$$ + +无套利、完备性和 state price 的关系： + +$$ +\left\{ +\begin{aligned} +\text{no arbitrage} +&\Longleftrightarrow +\exists q\gg0\ \text{such that }X'q=p,\\ +\text{complete market} +&\Longleftrightarrow +\operatorname{rank}(X)=S +\Longleftrightarrow +q\text{ is unique},\\ +\text{incomplete market} +&\Longleftrightarrow +\operatorname{rank}(X)0 +\Longleftrightarrow +\text{asset return is too high relative to factor exposure},\\ +\text{negative alpha} +&:\quad +\alpha_i<0 +\Longleftrightarrow +\text{asset return is too low relative to factor exposure}. +\end{aligned} +\right. +$$ + +Time-series factor test： + +$$ +\begin{aligned} +R_{i,t}^e +&:=R_{i,t}-R_{f,t},\\ +R_{i,t}^e +&=\alpha_i+\beta_i'f_t+\varepsilon_{i,t},\\ +E[\varepsilon_{i,t}\mid f_t]&=0. +\end{aligned} +$$ + +若 factors 是 traded excess returns 且模型正确，则 + +$$ +\boxed{ +\alpha_i=0 +\qquad +\text{for all test assets }i. +} +$$ + +Cross-sectional beta pricing： + +$$ +\begin{aligned} +E[R_i^e] +&=\beta_i'\lambda_f+\alpha_i,\\ +\beta_i +&=\operatorname{Cov}(R_i^e,f)\operatorname{Var}(f)^{-1}. +\end{aligned} +$$ + +检验逻辑： + +$$ +\boxed{ +\text{factor model works} +\Longleftrightarrow +\text{factor betas explain average excess returns} +\Longleftrightarrow +\alpha_i=0. +} +$$ + +### 3.3 证明 +#### 3.3.1 Mean-variance frontier +MVF 是二次规划，frontier 是 hyperbola。 + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +\min_w\quad &w'\Sigma w,\\ +\text{s.t.}\quad +&w'\mu=\mu_p,\qquad 1'w=1. +\end{aligned} +\right. +&\implies +\mathcal L +=w'\Sigma w-\lambda(w'\mu-\mu_p)-\gamma(1'w-1),\\ +0&=2\Sigma w-\lambda\mu-\gamma1,\\ +w&=\frac12\Sigma^{-1}(\lambda\mu+\gamma1). +\end{aligned} +$$ + +令 + +$$ +\begin{aligned} +A&=1'\Sigma^{-1}1,\qquad +B=1'\Sigma^{-1}\mu,\qquad +C=\mu'\Sigma^{-1}\mu,\qquad +D=AC-B^2. +\end{aligned} +$$ + +则 + +$$ +\boxed{ +\sigma_p^2 = +\frac{A\mu_p^2-2B\mu_p+C}{D}. +} +$$ + +#### 3.3.2 Tangency portfolio +有 risk-free asset 时，efficient risky portfolio 是 tangency portfolio。 + +$$ +\begin{aligned} +\max_w\quad +\frac{w'(\mu-R_f1)} +{\sqrt{w'\Sigma w}} +&\Longrightarrow +w_T\propto\Sigma^{-1}(\mu-R_f1),\\ +w_T +&= +\frac{\Sigma^{-1}(\mu-R_f1)} +{1'\Sigma^{-1}(\mu-R_f1)}. +\end{aligned} +$$ + +CML： + +$$ +\begin{aligned} +R_p&=(1-\omega)R_f+\omega R_T,\\ +E[R_p]-R_f&=\omega(E[R_T]-R_f),\\ +\sigma_p&=|\omega|\sigma_T,\\ +\Longrightarrow\quad +E[R_p]-R_f +&= +\frac{E[R_T]-R_f}{\sigma_T}\sigma_p. +\end{aligned} +$$ + +#### 3.3.3 Two-fund spanning proof +无 risk-free asset 时，frontier weight 对目标均值 $\mu_p$ 是 affine function，所以两只不同 frontier funds 可以生成任意 frontier portfolio。 + +$$ +\begin{aligned} +\mu_p&=a\mu_a+(1-a)\mu_b +\implies +a=\frac{\mu_p-\mu_b}{\mu_a-\mu_b},\\ +a w(\mu_a)+[1-a]w(\mu_b) +&= +\Sigma^{-1}\left[ +\frac{A\mu_p-B}{D}\mu ++\frac{C-B\mu_p}{D}1 +\right] +=w(\mu_p). +\end{aligned} +$$ + +有 risk-free asset 时，risky allocation 只有一个 Sharpe-maximizing 方向： + +$$ +\begin{aligned} +\max_w\frac{w'(\mu-R_f1)}{\sqrt{w'\Sigma w}} +&\implies +w\propto\Sigma^{-1}(\mu-R_f1) +\implies +w=\omega w_T,\\ +R_p&=(1-\omega)R_f+\omega R_T. +\end{aligned} +$$ + +#### 3.3.4 Linear SDF implies CAPM +$m=a-bR_m$ 推出 CAPM。 + +$$ +\begin{aligned} +1&=E[mR_i] +=E[m]E[R_i]-b\operatorname{Cov}(R_i,R_m),\\ +1&=E[mR_m] +=E[m]E[R_m]-b\operatorname{Var}(R_m). +\end{aligned} +$$ + +因此 + +$$ +\begin{aligned} +E[R_i]-R_f +&=\frac{b}{E[m]}\operatorname{Cov}(R_i,R_m),\\ +E[R_m]-R_f +&=\frac{b}{E[m]}\operatorname{Var}(R_m),\\ +\Longrightarrow\quad +E[R_i]-R_f +&=\frac{\operatorname{Cov}(R_i,R_m)} +{\operatorname{Var}(R_m)} +(E[R_m]-R_f)=\beta_i(E[R_m]-R_f). +\end{aligned} +$$ + +#### 3.3.5 Factor model pricing +线性 factor SDF 给出 beta pricing。 + +$$ +\begin{aligned} +m&=a-b'f,\\ +1&=E[mR_i] +=E[m]E[R_i]-b'\operatorname{Cov}(f,R_i),\\ +E[R_i]-R_f +&=\frac{b'}{E[m]}\operatorname{Cov}(f,R_i). +\end{aligned} +$$ + +若 + +$$ +\begin{aligned} +\beta_i +&=\operatorname{Cov}(R_i,f)\operatorname{Var}(f)^{-1}, +\end{aligned} +$$ + +则可写成 + +$$ +\boxed{ +E[R_i]-R_f=\beta_i'\lambda_f. +} +$$ + +## 4. 动态最优选择的两种方法 +### 4.1 题型 +#### 4.1.1 从 portfolio choice 到 HJB +**题目入口**：看到 continuous-time portfolio choice、consumption-investment、Merton problem，先写 state dynamics，再写 HJB 和 FOC。 + +连续时间投资消费问题： + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +dW_t +&= +\big[rW_t+\pi_t'(\mu-r1)-c_t\big]dt ++\pi_t'\sigma dB_t,\\ +V(t,w) +&= +\max_{\{\pi_s,c_s\}} +E_t\left[ +\int_t^T e^{-\rho(s-t)}u(c_s)ds ++e^{-\rho(T-t)}U(W_T) +\right]. +\end{aligned} +\right. +&\implies +0= +\max_{c,\pi} +\left\{ +u(c) ++V_t ++V_w[rw+\pi'(\mu-r1)-c] ++\frac12V_{ww}\pi'\Sigma\pi +-\rho V +\right\}\\ +&\implies +\left\{ +\begin{aligned} +u'(c) +&=V_w,\\ +0 +&=V_w(\mu-r1)+V_{ww}\Sigma\pi,\\ +\pi^* +&= +-\frac{V_w}{V_{ww}}\Sigma^{-1}(\mu-r1). +\end{aligned} +\right. +\end{aligned} +$$ + +这和静态 mean-variance demand 对应： + +$$ +\begin{aligned} +\pi^* +&= +\underbrace{-\frac{V_w}{wV_{ww}}}_{\text{risk tolerance / wealth}} +w\Sigma^{-1}(\mu-r1). +\end{aligned} +$$ + + +$$ +\boxed{ +\text{dynamic optimization} +\Longrightarrow +V_w\text{ is marginal utility of wealth} +\Longrightarrow +\text{portfolio demand depends on risk tolerance}. +} +$$ + +### 4.2 解法 +#### 4.2.1 Martingale method +martingale method 把动态问题变成静态预算问题： + +$$ +\begin{aligned} +\max_{\{c_t\}}\quad +&E\left[\int_0^T e^{-\rho t}u(c_t)dt+e^{-\rho T}U(W_T)\right]\\ +\text{s.t.}\quad +&E\left[\int_0^T \xi_tc_tdt+\xi_TW_T\right]\le W_0\\ +\implies\quad +e^{-\rho t}u'(c_t) +&=\lambda\xi_t. +\end{aligned} +$$ + +适合：complete markets、消费/终值 payoff 可直接用 SPD 定价。 + +#### 4.2.2 SDF pricing vs martingale method + +$$ +\left\{ +\begin{aligned} +\text{SDF / SPD pricing} +&:\quad +X_T\mapsto P_t=E_t\left[\frac{\xi_T}{\xi_t}X_T\right], +\qquad +\text{price a given payoff},\\ +\text{risk-neutral martingale pricing} +&:\quad +\frac{S_t}{B_t}=E_t^Q\left[\frac{S_T}{B_T}\right], +\qquad +\text{change measure then price},\\ +\text{martingale method} +&:\quad +\max_{\{c_t,W_T\}}E[U] +\quad\text{s.t.}\quad +E\left[\int_0^T\xi_tc_tdt+\xi_TW_T\right]\le W_0\\ +&\implies +u'(c_t)=\lambda e^{\rho t}\xi_t, +\qquad +\text{solve an optimal choice problem}. +\end{aligned} +\right. +$$ + +#### 4.2.3 Dynamic programming / HJB +HJB 保留状态变量，用局部最优化： + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +dW_t +&= +\big[rW_t+\pi_t'(\mu-r1)-c_t\big]dt ++\pi_t'\sigma dB_t,\\ +V(t,w) +&= +\max_{\{\pi_s,c_s\}} +E_t\left[\int_t^T e^{-\rho(s-t)}u(c_s)ds ++e^{-\rho(T-t)}U(W_T)\right]. +\end{aligned} +\right. +&\implies +0=\max_{\pi,c} +\left\{ +u(c)-\rho V ++V_t ++V_w\big[rw+\pi'(\mu-r1)-c\big] ++\frac12V_{ww}\pi'\Sigma\pi +\right\}\\ +&\implies +\left\{ +\begin{aligned} +u'(c^*)&=V_w,\\ +\pi^* +&=-\frac{V_w}{V_{ww}}\Sigma^{-1}(\mu-r1). +\end{aligned} +\right. +\end{aligned} +$$ + +所以动态 portfolio demand 是静态 mean-variance demand 的动态版本： + +$$ +\begin{aligned} +\pi^* +&= +\underbrace{\left(-\frac{V_w}{V_{ww}}\right)}_{\text{risk tolerance in wealth}} +\Sigma^{-1}(\mu-r1). +\end{aligned} +$$ + +适合：状态变量多、约束、labor income、optimal stopping、incomplete market。 + +#### 4.2.4 两种方法的关系 + +$$ +\begin{aligned} +\text{martingale method} +&:\quad \text{global budget + FOC},\\ +\text{HJB} +&:\quad \text{local Bellman equation + verification},\\ +\text{same complete-market solution} +&:\quad u'(c_t)\propto \xi_t. +\end{aligned} +$$ + +#### 4.2.5 考试题：HJB、欧式期权、美式期权与最优卖出时机 +##### （1）题干 +设 physical measure $P$ 下，标的资产价格满足 + +$$ +\begin{aligned} +dS_t +&=(\mu-q)S_tdt+\sigma S_tdW_t^P, +\qquad +r>0,\quad q\ge0,\quad \sigma>0. +\end{aligned} +$$ + +无风险资产满足 $dB_t=rB_tdt$。这里 $\mu$ 表示股票的 total expected return，$q$ 是连续分红率，因此 ex-dividend price 的 physical drift 是 $\mu-q$。若无分红，则 $q=0$。考虑到期日为 $T$、执行价为 $K$ 的欧式期权和美式期权。 + +##### （2）题目 +1. 将股票 dynamics 从 physical measure $P$ 改写成 risk-neutral measure $Q$ 下的 dynamics。 +2. 写出欧式期权价格的 risk-neutral pricing formula，并推导其 Black-Scholes PDE。 +3. 写出美式期权的 optimal stopping 表达式，并说明它和欧式期权的本质区别。 +4. 写出美式期权满足的 variational inequality，并解释 continuation region、exercise region、value matching 和 smooth pasting。 +5. 对不支付红利股票，说明 American call 为什么不应提前行权；American put 为什么可能提前行权。 +6. 考虑一个“最优卖出资产”的 American-style stopping problem：投资者持有一单位资产 $X_t$，若卖出需支付固定交易成本 $K>0$，卖出净收益为 $X_\tau-K$。若 + + $$ + \begin{aligned} + dX_t&=\mu X_tdt+\sigma X_tdW_t, + \qquad \mu<\beta, + \end{aligned} + $$ + + 求最优卖出阈值 $x^*$，并解释什么时候卖出合适。 + +##### （3）解析 +**1. 从 physical measure 改成 risk-neutral measure** + +先列出系统： + +$$ +\left\{ +\begin{aligned} +dS_t&=\mu S_tdt+\sigma S_tdW_t^P, +&&\text{(physical total-return notation, no dividend)},\\ +dS_t&=(\mu-q)S_tdt+\sigma S_tdW_t^P, +&&\text{(ex-dividend price with dividend yield }q\text{)},\\ +dB_t&=rB_tdt, +&&\text{(money market account)},\\ +\theta&=\frac{\mu-r}{\sigma}, +&&\text{(market price of risk)},\\ +dW_t^Q&=dW_t^P+\theta dt. +&&\text{(Brownian drift shift)} +\end{aligned} +\right. +$$ + +因此 + +$$ +\begin{aligned} +dW_t^P +&=dW_t^Q-\theta dt +=dW_t^Q-\frac{\mu-r}{\sigma}dt. +\end{aligned} +$$ + +代回股票 dynamics： + +$$ +\begin{aligned} +dS_t +&=(\mu-q)S_tdt+\sigma S_t +\left(dW_t^Q-\frac{\mu-r}{\sigma}dt\right)\\ +&=(\mu-q)S_tdt+\sigma S_tdW_t^Q-(\mu-r)S_tdt=(r-q)S_tdt+\sigma S_tdW_t^Q. +\end{aligned} +$$ + +所以 ex-dividend stock price 在 $Q$ 下 drift 为 $r-q$： + +$$ +\boxed{ +dS_t=(r-q)S_tdt+\sigma S_tdW_t^Q. +} +$$ + +这里的逻辑是： + +$$ +\begin{aligned} +\text{change of measure} +&\Longleftrightarrow +dW^Q=dW^P+\theta dt,\\ +\text{risk-neutral drift} +&\Longleftrightarrow +\text{total expected return }=r,\\ +\text{ex-dividend price drift} +&=r-q. +\end{aligned} +$$ + +**2. 欧式期权：terminal payoff pricing** + +欧式期权只能在 $T$ 行权，故价格是 fixed terminal payoff 的定价： + +$$ +\begin{aligned} +V^E(t,S) +&= +E_t^Q\left[ +e^{-r(T-t)}g(S_T) +\right], +\end{aligned} +$$ + +其中 call 和 put 的 payoff 分别为 + +$$ +\left\{ +\begin{aligned} +g_C(S_T)&=(S_T-K)^+,\\ +g_P(S_T)&=(K-S_T)^+. +\end{aligned} +\right. +$$ + +由 Feynman-Kac / no-arbitrage PDE： + +$$ +\begin{aligned} +0 +&= +V_t^E ++(r-q)SV_S^E ++\frac12\sigma^2S^2V_{SS}^E +-rV^E,\\ +V^E(T,S)&=g(S). +\end{aligned} +$$ + +欧式期权没有 stopping choice，只有 terminal condition。 + +**3. 美式期权：optimal stopping problem** + +美式期权允许持有人选择任意 stopping time $\tau\in[t,T]$ 行权： + +$$ +\begin{aligned} +V^A(t,S) +&= +\sup_{\tau\in[t,T]} +E_t^Q\left[ +e^{-r(\tau-t)}h(S_\tau) +\right]. +\end{aligned} +$$ + +其中 + +$$ +\left\{ +\begin{aligned} +h_C(S)&=(S-K)^+,\\ +h_P(S)&=(K-S)^+. +\end{aligned} +\right. +$$ + +因此： + +$$ +\boxed{ +\text{European option} +=\text{terminal value problem}, +\qquad +\text{American option} +=\text{optimal stopping / free-boundary problem}. +} +$$ + +**4. 美式期权的 HJB / variational inequality** + +在每个点 $(t,S)$，持有人比较： + +$$ +\left\{ +\begin{aligned} +\text{continue} +&:\quad +V_t+(r-q)SV_S+\frac12\sigma^2S^2V_{SS}-rV=0,\\ +\text{exercise} +&:\quad +V=h(S). +\end{aligned} +\right. +$$ + +合并成 variational inequality： + +$$ +\boxed{ +\max\left\{ +V_t+(r-q)SV_S+\frac12\sigma^2S^2V_{SS}-rV,\; +h(S)-V +\right\}=0. +} +$$ + +分区理解： + +1. **Continuation region** + + $$ + \begin{aligned} + V(t,S)>h(S) + &\Longrightarrow + V_t+(r-q)SV_S+\frac12\sigma^2S^2V_{SS}-rV=0. + \end{aligned} + $$ + +2. **Exercise region** + + $$ + \begin{aligned} + V(t,S)=h(S) + &\Longrightarrow + \text{immediate exercise is optimal}. + \end{aligned} + $$ + +3. **Free boundary：value matching 和 smooth pasting** + + 若边界为 $S^*(t)$，通常满足 + + $$ + \begin{aligned} + V(t,S^*(t))&=h(S^*(t)), + &&\text{(value matching)},\\ + V_S(t,S^*(t))&=h'(S^*(t)), + &&\text{(smooth pasting)}. + \end{aligned} + $$ + + value matching 的含义是：在最优行权边界上，继续持有和立刻行权必须给出同一个价值；否则边界不是最优边界。 + + $$ + \begin{aligned} + V(t,S^*)>h(S^*) + &\Longrightarrow + \text{应该继续，不该行权},\\ + V(t,S^*)0$，提前行权可能有价值，因为行权后持有股票可以获得 dividend，而继续持有 call 拿不到 dividend。比较逻辑： + + $$ + \begin{aligned} + \text{exercise benefit} + &= + \text{dividend captured},\\ + \text{exercise cost} + &= + \text{lost option value} + +\text{early payment of }K. + \end{aligned} + $$ + +3. **American put, even no dividend** + + put 提前行权可能最优，因为行权后可以立刻收到 $K$，获得执行价的时间价值： + + $$ + \begin{aligned} + \text{exercise now} + &:\quad K-S_t,\\ + \text{continue} + &:\quad \text{keep downside insurance}. + \end{aligned} + $$ + + 当 $S_t$ 足够低时，继续等待的额外下行价值变小，而立刻拿到 $K$ 的时间价值变大，因此提前行权可能最优。 + +**6. 最优卖出资产：什么时候卖出合适** + +这是一个 American-style stopping problem。价值函数： + +$$ +\begin{aligned} +V(x) +&= +\sup_{\tau\ge0} +E_x\left[ +e^{-\beta\tau}(X_\tau-K) +\right]. +\end{aligned} +$$ + +HJB / variational inequality： + +$$ +\boxed{ +\max\left\{ +\frac12\sigma^2x^2V''(x)+\mu xV'(x)-\beta V(x),\; +x-K-V(x) +\right\}=0. +} +$$ + +若最优策略是达到阈值 $x^*$ 时卖出，则： + +1. **Continuation region：$x1$： + + $$ + \begin{aligned} + \gamma + &= + \frac{ + -(\mu-\frac12\sigma^2) + +\sqrt{(\mu-\frac12\sigma^2)^2+2\beta\sigma^2} + }{\sigma^2}. + \end{aligned} + $$ + +2. **Stopping region：$x\ge x^*$** + + $$ + \begin{aligned} + V(x)&=x-K. + \end{aligned} + $$ + +3. **Boundary conditions** + + $$ + \left\{ + \begin{aligned} + A(x^*)^\gamma&=x^*-K, + &&\text{(value matching)},\\ + A\gamma(x^*)^{\gamma-1}&=1, + &&\text{(smooth pasting)}. + \end{aligned} + \right. + $$ + + 两式相除： + + $$ + \begin{aligned} + \frac{A\gamma(x^*)^{\gamma-1}} + {A(x^*)^\gamma} + &= + \frac{1}{x^*-K}\\ + \Longleftrightarrow\quad + \frac{\gamma}{x^*} + &= + \frac{1}{x^*-K}\\ + \Longleftrightarrow\quad + \gamma(x^*-K)&=x^*\\ + \Longleftrightarrow\quad + \boxed{ + x^*=\frac{\gamma}{\gamma-1}K. + } + \end{aligned} + $$ + +因此最优卖出规则为： + +$$ +\boxed{ +\tau^* = +\inf\{t\ge0:X_t\ge x^*\}, +\qquad +x^*=\frac{\gamma}{\gamma-1}K. +} +$$ + +经济含义： + +$$ +\begin{aligned} +x0$，并配合 transversality condition 排除爆炸解。 +### 4.3 证明 +#### 4.3.1 HJB derivation +动态规划给出 HJB 和最优控制。 + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +dW_t +&= +\big[rW_t+\pi_t'(\mu-r1)-c_t\big]dt ++\pi_t'\sigma dB_t,\\ +V(t,w) +&= +\max_{\pi,c} +E_t\left[\int_t^T e^{-\rho(s-t)}u(c_s)ds+e^{-\rho(T-t)}U(W_T)\right]. +\end{aligned} +\right. +&\implies +0=\max_{\pi,c} +\left\{ +u(c)-\rho V+V_t ++V_w[rw+\pi'(\mu-r1)-c] ++\frac12V_{ww}\pi'\Sigma\pi +\right\}\\ +&\implies +\left\{ +\begin{aligned} +u'(c^*)&=V_w,\\ +\pi^* +&= +-\frac{V_w}{V_{ww}}\Sigma^{-1}(\mu-r1). +\end{aligned} +\right. +\end{aligned} +$$ + +#### 4.3.2 Optimal selling threshold +卖出阈值 $x^*=\frac{\gamma}{\gamma-1}K$。 + +$$ +\begin{aligned} +V(x) +&= +\sup_{\tau\ge0}E_x[e^{-\beta\tau}(X_\tau-K)],\\ +dX_t&=\mu X_tdt+\sigma X_tdW_t. +\end{aligned} +$$ + +Continuation region： + +$$ +\begin{aligned} +\frac12\sigma^2x^2V''+\mu xV'-\beta V=0. +\end{aligned} +$$ + +猜 $V(x)=Ax^\gamma$： + +$$ +\begin{aligned} +\frac12\sigma^2\gamma(\gamma-1)+\mu\gamma-\beta=0. +\end{aligned} +$$ + +边界条件： + +$$ +\left\{ +\begin{aligned} +A(x^*)^\gamma&=x^*-K,\\ +A\gamma(x^*)^{\gamma-1}&=1. +\end{aligned} +\right. +$$ + +两式相除： + +$$ +\begin{aligned} +\frac{\gamma}{x^*} +&=\frac{1}{x^*-K}\\ +\Longleftrightarrow\quad +x^*&=\frac{\gamma}{\gamma-1}K. +\end{aligned} +$$ + +## 5. 衍生品与债券都是定价方程的应用 +### 5.1 题型 +#### 5.1.1 从 fundamental pricing equation 到 options +**题目入口**：看到 option、derivative payoff、replication，先写 fundamental pricing equation；再根据题目选择 risk-neutral expectation 或 PDE。 + +任何 derivative payoff $X_T=g(S_T)$： + +$$ +\begin{aligned} +V_t +&=E_t\left[\frac{\xi_T}{\xi_t}g(S_T)\right] +=B_tE_t^Q\left[\frac{g(S_T)}{B_T}\right] +=e^{-r(T-t)}E_t^Q[g(S_T)] +&&\text{if }r\text{ is constant}. +\end{aligned} +$$ + +Black-Scholes 中： + +$$ +\left\{ +\begin{aligned} +dS_t&=rS_tdt+\sigma S_tdW_t^Q,\\ +C_t&=e^{-r(T-t)}E_t^Q[(S_T-K)^+]. +\end{aligned} +\right. +$$ + +复制组合给 PDE： + +$$ +\begin{aligned} +0 +&=C_t+rSC_S+\frac12\sigma^2S^2C_{SS}-rC. +\end{aligned} +$$ + +因此： + +$$ +\boxed{ +\text{replication PDE} +\Longleftrightarrow +\text{risk-neutral expectation} +\Longleftrightarrow +\text{SDF pricing}. +} +$$ + +#### 5.1.2 从 fundamental pricing equation 到 bonds +**题目入口**：看到 zero-coupon bond、yield、term structure，先把 bond 当成 payoff 为 $1$ 的资产。 + +zero-coupon bond 的 payoff 是 $1$： + +$$ +\begin{aligned} +P(t,T) +&=E_t\left[\frac{\xi_T}{\xi_t}\right]=E_t^Q\left[e^{-\int_t^T r_sds}\right],\\ +y(t,T) +&=-\frac{1}{T-t}\log P(t,T),\\ +dr_t +&=\kappa(\theta-r_t)dt+\sigma dW_t^Q\\ +\Longrightarrow\quad +P(t,T) +&=E_t^Q\left[ +\exp\left(-\int_t^T r_sds\right) +\right]. +\end{aligned} +$$ + + +$$ +\boxed{ +\text{term structure} +\Longleftrightarrow +\text{choose }r_t\text{ dynamics under }Q +\Longleftrightarrow +\text{price }1\text{ payoff by SDF}. +} +$$ + +### 5.2 解法 +#### 5.2.1 Options +Black-Scholes 可以用三种语言推导： + +| 方法 | 核心 | +|---|---| +| replication / PDE | 用 $\Delta$ hedge 消除 Brownian risk | +| SDF direct pricing | $C_t=E_t[(\xi_T/\xi_t)(S_T-K)^+]$ | +| risk-neutral pricing | $C_t=e^{-r(T-t)}E_t^Q[(S_T-K)^+]$ | + +它们本质相同： + +$$ +\begin{aligned} +\text{complete market} +\Longrightarrow +\text{unique hedge} +\Longrightarrow +\text{unique }Q +\Longrightarrow +\text{same option price}. +\end{aligned} +$$ + +#### 5.2.2 Bonds and term structure +zero-coupon bond price 是特殊 payoff $1$ 的价格： + +$$ +\begin{aligned} +P(t,T) +&=E_t\left[\frac{\xi_T}{\xi_t}\right] +=E_t^Q\left[e^{-\int_t^T r_sds}\right]. +\end{aligned} +$$ + +term structure model 只是给 $r_t$ 一个动态，然后算这个 conditional expectation。 +#### 5.2.3 Yield / forward / duration / Greeks 速查 +Bond 题先把 price 转成 yield / forward，再决定是定义题、久期题还是 short-rate pricing 题。 + +| 对象 | 定义 | 作用 | +|---|---|---| +| zero-coupon price | $P(t,T)$ | payoff $1$ at $T$ 的今天价格 | +| continuously compounded yield | $y(t,T)=-\frac{1}{T-t}\log P(t,T)$ | 把 bond price 写成平均利率 | +| instantaneous forward rate | $f(t,T)=-\partial_T\log P(t,T)$ | $T$ 附近的边际远期利率 | +| duration | $D=-\frac{\partial\log P}{\partial y}$ | price 对 yield 的一阶敏感度 | +| convexity | $\mathcal C=\frac{1}{P}\frac{\partial^2P}{\partial y^2}$ | price 对 yield 的二阶敏感度 | + +三条常用关系： + +$$ +\boxed{ +\begin{aligned} +P(t,T) +&=\exp\left(-(T-t)y(t,T)\right),\\ +P(t,T) +&=\exp\left(-\int_t^T f(t,u)\,du\right),\\ +y(t,T) +&=\frac{1}{T-t}\int_t^T f(t,u)\,du. +\end{aligned} +} +$$ + +价格变动近似： + +$$ +\begin{aligned} +\frac{\Delta P}{P} +&\approx +-D\,\Delta y ++\frac12\mathcal C(\Delta y)^2. +\end{aligned} +$$ + +zero-coupon bond 若用 continuously compounded yield 表示， + +$$ +\begin{aligned} +P(t,T)&=e^{-y(T-t)},\\ +D&=T-t,\\ +\mathcal C&=(T-t)^2. +\end{aligned} +$$ + +Option Greeks 题先从 $V(t,S;\sigma,r,q)$ 的偏导数读 hedge 和风险暴露： + +| Greek | 定义 | 含义 | +|---|---|---| +| Delta | $\Delta=V_S$ | underlying price 一阶暴露；delta hedge 用它 | +| Gamma | $\Gamma=V_{SS}$ | convexity / delta sensitivity | +| Theta | $\Theta=V_t$ | time decay | +| Vega | $\mathcal V=V_\sigma$ | volatility sensitivity | +| Rho | $\rho_r=V_r$ | interest-rate sensitivity | + +Black-Scholes with dividend yield 下，call 的常用 Greeks： + +$$ +\left\{ +\begin{aligned} +\Delta_C +&=e^{-q\tau}\Phi(d_1),\\ +\Gamma_C +&=\frac{e^{-q\tau}\phi(d_1)} +{S\sigma\sqrt{\tau}},\\ +\mathcal V_C +&=S e^{-q\tau}\phi(d_1)\sqrt{\tau}. +\end{aligned} +\right. +$$ + +put 的 delta： + +$$ +\begin{aligned} +\Delta_P +&=e^{-q\tau}\left(\Phi(d_1)-1\right). +\end{aligned} +$$ + +PDE 的经济含义： + +$$ +\boxed{ +\underbrace{V_t}_{\text{time decay}} ++\underbrace{(r-q)SV_S}_{\text{risk-neutral carry}} ++\underbrace{\frac12\sigma^2S^2V_{SS}}_{\text{convexity value}} +-\underbrace{rV}_{\text{financing cost}} +=0. +} +$$ + +### 5.3 证明 +#### 5.3.1 Vasicek affine bond pricing + +$$ +\left\{ +\begin{aligned} +dr_t&=\kappa(\theta-r_t)dt+\sigma dW_t^Q,\\ +P(t,T)&=A(t,T)e^{-B(t,T)r_t},\\ +P(T,T)&=1. +\end{aligned} +\right. +$$ + +Bond PDE： + +$$ +\begin{aligned} +0 +&= +P_t+\kappa(\theta-r)P_r+\frac12\sigma^2P_{rr}-rP. +\end{aligned} +$$ + +Derivatives： + +$$ +\begin{aligned} +P_t&=\left(\frac{A_t}{A}-B_tr\right)P,\\ +P_r&=-BP,\\ +P_{rr}&=B^2P. +\end{aligned} +$$ + +代回并按 $r$ 分组： + +$$ +\begin{aligned} +0 +&= +\frac{A_t}{A}-B_tr-\kappa\theta B+\kappa rB ++\frac12\sigma^2B^2-r\\ +&= +\left(\frac{A_t}{A}-\kappa\theta B+\frac12\sigma^2B^2\right) ++r(-B_t+\kappa B-1). +\end{aligned} +$$ + +系数为零： + +$$ +\left\{ +\begin{aligned} +B_t&=\kappa B-1, +\qquad B(T,T)=0,\\ +\frac{A_t}{A} +&= +\kappa\theta B-\frac12\sigma^2B^2, +\qquad A(T,T)=1. +\end{aligned} +\right. +$$ + +因此： + +$$ +\boxed{ +B(t,T)=\frac{1-e^{-\kappa(T-t)}}{\kappa}. +} +$$ + +#### 5.3.2 European option PDE +risk-neutral pricing 等价于 Black-Scholes PDE。 + +$$ +\begin{aligned} +V^E(t,S) +&=E_t^Q[e^{-r(T-t)}g(S_T)],\\ +dS_t&=(r-q)S_tdt+\sigma S_tdW_t^Q. +\end{aligned} +$$ + +Feynman-Kac： + +$$ +\boxed{ +V_t+(r-q)SV_S+\frac12\sigma^2S^2V_{SS}-rV=0, +\qquad +V(T,S)=g(S). +} +$$ + +#### 5.3.3 Black-Scholes replication proof + +$$ +\left\{ +\begin{aligned} +dS_t&=\mu S_tdt+\sigma S_tdW_t^P,\\ +dB_t&=rB_tdt,\\ +C_t&=C(t,S_t). +\end{aligned} +\right. +$$ + +Ito： + +$$ +\begin{aligned} +dC_t +&= +\left(C_t+\mu S C_S+\frac12\sigma^2S^2C_{SS}\right)dt ++\sigma SC_SdW_t^P. +\end{aligned} +$$ + +复制组合： + +$$ +\left\{ +\begin{aligned} +\Pi_t&=\Delta_tS_t+\alpha_tB_t,\\ +\Pi_t&=C_t,\\ +d\Pi_t&=\Delta_tdS_t+\alpha_tdB_t. +\end{aligned} +\right. +$$ + +匹配 diffusion term： + +$$ +\begin{aligned} +\sigma SC_SdW_t^P +&= +\Delta_t\sigma SdW_t^P\\ +\Longleftrightarrow\quad +\Delta_t&=C_S. +\end{aligned} +$$ + +由 $\alpha_tB_t=C_t-C_SS_t$： + +$$ +\begin{aligned} +d\Pi_t +&=C_S(\mu Sdt+\sigma SdW_t^P) ++r(C-C_SS)dt\\ +&= +\left[rC+(\mu-r)SC_S\right]dt ++\sigma SC_SdW_t^P. +\end{aligned} +$$ + +令 $dC_t=d\Pi_t$： + +$$ +\begin{aligned} +C_t+\mu SC_S+\frac12\sigma^2S^2C_{SS} +&= +rC+(\mu-r)SC_S\\ +\Longleftrightarrow\quad +\boxed{ +C_t+rSC_S+\frac12\sigma^2S^2C_{SS}-rC=0. +} +\end{aligned} +$$ + +#### 5.3.4 Black-Scholes risk-neutral proof + +$$ +\left\{ +\begin{aligned} +dS_t&=rS_tdt+\sigma S_tdW_t^Q,\\ +C_t+rSC_S+\frac12\sigma^2S^2C_{SS}-rC&=0. +\end{aligned} +\right. +$$ + +Ito for discounted price： + +$$ +\begin{aligned} +d(e^{-rt}C_t) +&= +e^{-rt}\left(dC_t-rC_tdt\right)\\ +&= +e^{-rt}\left[ +C_t+rSC_S+\frac12\sigma^2S^2C_{SS}-rC +\right]dt ++e^{-rt}\sigma SC_SdW_t^Q\\ +&= +e^{-rt}\sigma SC_SdW_t^Q. +\end{aligned} +$$ + +所以 $e^{-rt}C_t$ 是 $Q$-martingale： + +$$ +\begin{aligned} +e^{-rt}C_t +&= +E_t^Q[e^{-rT}g(S_T)]\\ +\Longleftrightarrow\quad +\boxed{ +C_t=e^{-r(T-t)}E_t^Q[g(S_T)]. +} +\end{aligned} +$$ + +#### 5.3.5 Black-Scholes closed-form proof +令 $\tau=T-t$。在 $Q$ 下： + +$$ +\begin{aligned} +S_T +&= +S_t\exp\left[ +\left(r-q-\frac12\sigma^2\right)\tau ++\sigma\sqrt{\tau}Z +\right], +\qquad Z\sim N(0,1). +\end{aligned} +$$ + +Call price： + +$$ +\begin{aligned} +C_t +&=e^{-r\tau}E_t^Q[(S_T-K)^+]\\ +&=e^{-r\tau} +\left( +E_t^Q[S_T\mathbf 1_{\{S_T>K\}}] +-KQ(S_T>K) +\right). +\end{aligned} +$$ + +Exercise event： + +$$ +\begin{aligned} +S_T>K +&\Longleftrightarrow +Z> +\frac{\ln(K/S_t)-(r-q-\frac12\sigma^2)\tau} +{\sigma\sqrt{\tau}}\\ +&\Longleftrightarrow +Z>-d_2, +\end{aligned} +$$ + +where + +$$ +\begin{aligned} +d_1&= +\frac{\ln(S_t/K)+(r-q+\frac12\sigma^2)\tau} +{\sigma\sqrt{\tau}},\\ +d_2&=d_1-\sigma\sqrt{\tau}. +\end{aligned} +$$ + +第一项： + +$$ +\begin{aligned} +E_t^Q[S_T\mathbf 1_{\{S_T>K\}}] +&= +S_te^{(r-q)\tau} +E\left[ +e^{-\frac12\sigma^2\tau+\sigma\sqrt{\tau}Z} +\mathbf 1_{\{Z>-d_2\}} +\right]\\ +&= +S_te^{(r-q)\tau} +\int_{-d_2}^{\infty} +e^{-\frac12\sigma^2\tau+\sigma\sqrt{\tau}z} +\phi(z)dz\\ +&= +S_te^{(r-q)\tau} +\int_{-d_2}^{\infty} +\phi(z-\sigma\sqrt{\tau})dz\\ +&= +S_te^{(r-q)\tau} +\Phi(d_2+\sigma\sqrt{\tau})\\ +&= +S_te^{(r-q)\tau}\Phi(d_1). +\end{aligned} +$$ + +第二项： + +$$ +\begin{aligned} +Q(S_T>K) +&= +Q(Z>-d_2) +=\Phi(d_2). +\end{aligned} +$$ + +因此： + +$$ +\boxed{ +C_t = +S_te^{-q\tau}\Phi(d_1) +-Ke^{-r\tau}\Phi(d_2). +} +$$ + +Put price： + +$$ +\boxed{ +P_t = +Ke^{-r\tau}\Phi(-d_2) +-S_te^{-q\tau}\Phi(-d_1). +} +$$ + +#### 5.3.6 Put-call parity proof + +$$ +\begin{aligned} +(S_T-K)^+-(K-S_T)^+ +&=S_T-K. +\end{aligned} +$$ + +No-arbitrage pricing： + +$$ +\begin{aligned} +C_t-P_t +&= +E_t^Q\left[e^{-r\tau}(S_T-K)\right]\\ +&= +e^{-r\tau}E_t^Q[S_T]-Ke^{-r\tau}\\ +&= +S_te^{-q\tau}-Ke^{-r\tau}. +\end{aligned} +$$ + +$$ +\boxed{ +C_t-P_t=S_te^{-q\tau}-Ke^{-r\tau}. +} +$$ + +#### 5.3.7 American option variational inequality +美式期权是 continuation 和 exercise 的逐点最大值。 + +$$ +\begin{aligned} +V^A(t,S) +&= +\sup_{\tau\in[t,T]} +E_t^Q[e^{-r(\tau-t)}h(S_\tau)]. +\end{aligned} +$$ + +因此 + +$$ +\boxed{ +\max\left\{ +V_t+(r-q)SV_S+\frac12\sigma^2S^2V_{SS}-rV,\; +h(S)-V +\right\}=0. +} +$$ + +自由边界： + +$$ +\left\{ +\begin{aligned} +V(t,S^*) &=h(S^*), +&&\text{(value matching)},\\ +V_S(t,S^*)&=h'(S^*). +&&\text{(smooth pasting)} +\end{aligned} +\right. +$$ + +#### 5.3.8 No-dividend American call 不提前行权 +$q=0$ 时 American call = European call。 + +$$ +\begin{aligned} +C^E(t,S) +&=E_t^Q[e^{-r\tau}(S_T-K)^+]\\ +&\ge E_t^Q[e^{-r\tau}(S_T-K)]\\ +&=S_t-Ke^{-r\tau}\\ +&\ge S_t-K. +\end{aligned} +$$ + +$$ +\boxed{ +C^A(t,S)=C^E(t,S). +} +$$ + +#### 5.3.9 Bond pricing equation +zero-coupon bond 是 payoff $1$ 的定价。 + +$$ +\begin{aligned} +P(t,T) +&=E_t^Q\left[e^{-\int_t^T r_sds}\right]. +\end{aligned} +$$ + +若短利率模型为 + +$$ +\begin{aligned} +dr_t&=\mu_r(t,r_t)dt+\sigma_r(t,r_t)dW_t^Q, +\end{aligned} +$$ + +则债券 PDE 为 + +$$ +\boxed{ +P_t+\mu_rP_r+\frac12\sigma_r^2P_{rr}-rP=0, +\qquad +P(T,T)=1. +} +$$ + +## 6. 信息、信念与价格发现 +### 6.1 题型 +#### 6.1.1 从信息到价格发现 +**题目入口**：看到 information、order flow、Kyle、filtering，先把价格写成 conditional expectation。 + +信息模型把定价中的 conditional expectation 展开成 learning problem。 + +Kyle 型模型： + +$$ +\left\{ +\begin{aligned} +v&\sim N(0,\sigma_v^2),\quad u\sim N(0,\sigma_u^2),\\ +x&=\beta v,\\ +y&=x+u=\beta v+u,\\ +p&=E[v\mid y]. +\end{aligned} +\right. +$$ + +joint normal projection： + +$$ +\begin{aligned} +p +&=E[v\mid y] +=E[v]+\frac{\operatorname{Cov}(v,y)}{\operatorname{Var}(y)}(y-E[y]) +=\lambda y, +\qquad +\lambda +=\frac{\beta\sigma_v^2}{\beta^2\sigma_v^2+\sigma_u^2}. +\end{aligned} +$$ + +insider optimality： + +$$ +\begin{aligned} +\max_x x(v-\lambda x) +&\Longrightarrow +x=\frac{v}{2\lambda} +\Longrightarrow +\beta=\frac{1}{2\lambda}. +\end{aligned} +$$ + +equilibrium： + +$$ +\begin{aligned} +\beta&=\frac{1}{2\lambda},\\ +\lambda&=\frac{\beta\sigma_v^2}{\beta^2\sigma_v^2+\sigma_u^2}\\ +\Longrightarrow\quad +\lambda&=\frac{\sigma_v}{2\sigma_u}, +\qquad +\beta=\frac{\sigma_u}{\sigma_v}. +\end{aligned} +$$ + + +$$ +\boxed{ +\text{information} +\Longrightarrow +\text{order flow} +\Longrightarrow +E[v\mid y] +\Longrightarrow +\text{price impact / liquidity}. +} +$$ + +#### 6.1.2 从异质信念到均衡价格 +**题目入口**：看到 heterogeneous beliefs、likelihood ratio、wealth shares，先写每个 agent 的 FOC，再用 belief link 和 market clearing 合并 SPD。 + +不同 agents 对同一 payoff 有不同 belief： + +$$ +\begin{aligned} +\eta_t +&:=\frac{dP^2}{dP^1}\bigg|_{\mathcal F_t},\\ +e^{-\rho_i t}u_i'(c_t^i) +&=y_i\xi_t^i,\\ +\xi_t^2 +&=\eta_t\xi_t^1,\\ +\sum_i c_t^i +&=\delta_t. +\end{aligned} +$$ + +因此均衡价格来自： + +$$ +\left\{ +\begin{aligned} +\text{individual FOC} +&:\ e^{-\rho_i t}u_i'(c_t^i)=y_i\xi_t^i,\\ +\text{belief link} +&:\ \xi_t^2=\eta_t\xi_t^1,\\ +\text{market clearing} +&:\ \sum_i c_t^i=\delta_t,\\ +\text{asset pricing} +&:\ S_t=E_t\left[\frac{\xi_T}{\xi_t}D_T\right]. +\end{aligned} +\right. +$$ + + +$$ +\boxed{ +\text{belief heterogeneity} +\Longrightarrow +\text{different SPD} +\Longrightarrow +\text{wealth-share dynamics} +\Longrightarrow +\text{prices, rates, risk premia}. +} +$$ + +#### 6.1.3 线性设定总表 +看到 linear equilibrium、linear price rule、projection、Kyle、Grossman-Stiglitz 时，先把线性猜测写成系统，再用最优性、投影和 market clearing pin down 待定系数。 + +| 模型 | 线性设定 | 价格规则 / belief | 待定系数 | pin down 条件 | +|---|---|---|---|---| +| Bayesian linear signal | $s=a v+\varepsilon$ | $E[v\mid s]$ | posterior weight | joint normal projection | +| Grossman-Stiglitz | $p=\alpha+\beta s+\gamma x$ | partially revealing price | $\alpha,\beta,\gamma$ | filtering + market clearing | +| One-period Kyle | $x=\beta(v-v_0)$，$y=x+u$ | $p=v_0+\lambda y$ | $\beta,\lambda$ | insider FOC + market maker projection | +| Multi-period Kyle | $x_n=\beta_n(v-p_{n-1})$，$y_n=x_n+u_n$ | $p_n=E[v\mid y_1,\dots,y_n]$ | $\beta_n,\lambda_n,\Sigma_n$ | backward induction + filtering | +| Continuous-time Kyle | $\theta_t=\frac{\tilde v-P_t}{(1-t)\lambda}$，$Y_t=X_t+Z_t$ | $P_t=\bar v+\lambda Y_t$ | $\lambda,\Sigma_t$ | Kalman filtering + terminal revelation | + +统一解题顺序： + +$$ +\boxed{ +\begin{aligned} +\text{Step 1: choose linear conjecture} +&\Longrightarrow +\text{strategy / signal / price rule},\\ +\text{Step 2: compute conditional expectation} +&\Longrightarrow +\text{projection coefficient},\\ +\text{Step 3: impose optimality} +&\Longrightarrow +\text{agent demand or insider strategy},\\ +\text{Step 4: impose equilibrium} +&\Longrightarrow +\text{market clearing / pricing consistency},\\ +\text{Step 5: solve coefficients} +&\Longrightarrow +\beta,\lambda,\Sigma,\alpha,\gamma. +\end{aligned} +} +$$ + +最常用的投影公式： + +$$ +\begin{aligned} +E[v\mid z] +&=E[v] ++\operatorname{Cov}(v,z)\operatorname{Var}(z)^{-1}(z-E[z]),\\ +z&=a v+\varepsilon +\Longrightarrow +\frac{\partial E[v\mid z]}{\partial z} +=\frac{a\operatorname{Var}(v)} +{a^2\operatorname{Var}(v)+\operatorname{Var}(\varepsilon)}. +\end{aligned} +$$ + +Kyle 类题的最短模板： + +$$ +\left\{ +\begin{aligned} +\text{linear strategy} +&:\quad x=\beta(v-v_0),\\ +\text{order flow} +&:\quad y=x+u,\\ +\text{linear price} +&:\quad p=v_0+\lambda y,\\ +\text{projection} +&:\quad +\lambda=\frac{\operatorname{Cov}(v,y)}{\operatorname{Var}(y)},\\ +\text{optimality} +&:\quad +\beta=\frac{1}{2\lambda}. +\end{aligned} +\right. +$$ + +### 6.2 解法 +核心问题不是“资产怎么由 payoff 定价”，而是“市场如何从交易和信号中学习 payoff”。 + +#### 6.2.1 信念更新：precision form +最常用的是正态-正态更新。设未知基本面为 $v$，先验为 + +$$ +\begin{aligned} +v +&\sim N(m_0,\rho_0^{-1}), +\qquad +\rho_0:=\frac{1}{\operatorname{Var}(v)}. +\end{aligned} +$$ + +观察到带噪声信号 + +$$ +\begin{aligned} +s_i +&=v+\varepsilon_i, +\qquad +\varepsilon_i\sim N(0,\rho_i^{-1}), +\qquad +\varepsilon_i\perp v. +\end{aligned} +$$ + +精度写法的核心规则： + +$$ +\boxed{ +\text{posterior precision} = +\text{prior precision} ++ +\text{signal precision}. +} +$$ + +##### （1）一期更新 +观察一个信号 $s_1$： + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +\rho_{1}^{post} +&=\rho_0+\rho_1,\\ +m_1 +&=E[v\mid s_1] = +\frac{\rho_0m_0+\rho_1s_1}{\rho_0+\rho_1}. +\end{aligned} +\right. +\implies +v\mid s_1 +&\sim +N\left( +\frac{\rho_0m_0+\rho_1s_1}{\rho_0+\rho_1}, +\frac{1}{\rho_0+\rho_1} +\right). +\end{aligned} +$$ + +##### （2）两期更新 +再观察 $s_2=v+\varepsilon_2$： + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +\rho_2^{post} +&=\rho_0+\rho_1+\rho_2,\\ +m_2 +&=E[v\mid s_1,s_2]\\ +&= +\frac{\rho_0m_0+\rho_1s_1+\rho_2s_2} +{\rho_0+\rho_1+\rho_2}. +\end{aligned} +\right. +\implies +m_2 +&= +\frac{(\rho_0+\rho_1)m_1+\rho_2s_2} +{(\rho_0+\rho_1)+\rho_2}. +\end{aligned} +$$ + +##### （3）三期更新 +观察 $s_1,s_2,s_3$： + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +\rho_3^{post} +&=\rho_0+\rho_1+\rho_2+\rho_3,\\ +m_3 +&=E[v\mid s_1,s_2,s_3]\\ +&= +\frac{\rho_0m_0+\rho_1s_1+\rho_2s_2+\rho_3s_3} +{\rho_0+\rho_1+\rho_2+\rho_3}. +\end{aligned} +\right. +\implies +m_3 +&= +\frac{\rho_2^{post}m_2+\rho_3s_3} +{\rho_2^{post}+\rho_3}. +\end{aligned} +$$ + +##### （4）n 期更新 +一般地，观察 $s_1,\dots,s_n$： + +$$ +\boxed{ +\begin{aligned} +\left\{ +\begin{aligned} +\rho_n^{post} +&=\rho_0+\sum_{i=1}^n\rho_i,\\ +m_n +&=E[v\mid s_1,\dots,s_n]\\ +&= +\frac{\rho_0m_0+\sum_{i=1}^n\rho_is_i} +{\rho_0+\sum_{i=1}^n\rho_i},\\ +\operatorname{Var}(v\mid s_1,\dots,s_n) +&=\frac{1}{\rho_n^{post}}. +\end{aligned} +\right. +\implies +v\mid s_1,\dots,s_n +&\sim +N\left(m_n,\frac{1}{\rho_n^{post}}\right). +\end{aligned} +} +$$ + +若每期信号精度相同，$\rho_i=\rho_\varepsilon$，则 + +$$ +\begin{aligned} +\rho_n^{post} +&=\rho_0+n\rho_\varepsilon,\\ +m_n +&= +\frac{\rho_0m_0+\rho_\varepsilon\sum_{i=1}^n s_i} +{\rho_0+n\rho_\varepsilon} = +\frac{\rho_0}{\rho_0+n\rho_\varepsilon}m_0 ++ +\frac{n\rho_\varepsilon}{\rho_0+n\rho_\varepsilon}\bar s_n. +\end{aligned} +$$ + +其中 $\bar s_n=\frac1n\sum_{i=1}^n s_i$。 + +##### （5）无限期更新 +若 + +$$ +\begin{aligned} +\sum_{i=1}^{\infty}\rho_i +&=\infty, +\end{aligned} +$$ + +则 + +$$ +\begin{aligned} +\rho_n^{post} +&\to\infty,\\ +\operatorname{Var}(v\mid s_1,\dots,s_n) +&=\frac{1}{\rho_n^{post}} +\to0. +\end{aligned} +$$ + +这表示信息最终完全揭示 $v$。若信号独立同分布且 $s_i=v+\varepsilon_i$，则 + +$$ +\begin{aligned} +\bar s_n +&\to v,\\ +m_n +&= +\frac{\rho_0}{\rho_0+n\rho_\varepsilon}m_0 ++ +\frac{n\rho_\varepsilon}{\rho_0+n\rho_\varepsilon}\bar s_n +\to v. +\end{aligned} +$$ + +##### （6）线性信号的一般形式 +若信号不是 $s_i=v+\varepsilon_i$，而是 + +$$ +\begin{aligned} +s_i +&=a_iv+\varepsilon_i, +\qquad +\varepsilon_i\sim N(0,\rho_{\varepsilon i}^{-1}), +\end{aligned} +$$ + +则每个信号贡献的精度是 $a_i^2\rho_{\varepsilon i}$： + +$$ +\left\{ +\begin{aligned} +\rho_n^{post} +&=\rho_0+\sum_{i=1}^n a_i^2\rho_{\varepsilon i},\\ +m_n +&= +\frac{\rho_0m_0+\sum_{i=1}^n a_i\rho_{\varepsilon i}s_i} +{\rho_0+\sum_{i=1}^n a_i^2\rho_{\varepsilon i}}. +\end{aligned} +\right. +$$ + +直觉： + +$$ +\begin{aligned} +\text{higher precision signal} +&\Longrightarrow +\text{higher weight in posterior mean},\\ +\text{more independent signals} +&\Longrightarrow +\text{posterior variance declines},\\ +\text{infinite precise-enough signals} +&\Longrightarrow +\text{belief converges to truth}. +\end{aligned} +$$ + +#### 6.2.2 Grossman-Stiglitz + +$$ +\left\{ +\begin{aligned} +v&\sim N(\bar v,\sigma_v^2), +&&\text{(asset payoff)},\\ +s&=v+\varepsilon, +&&\text{(private signal)},\\ +p&=\alpha+\beta s+\gamma x, +&&\text{(partially revealing price)},\\ +X_i^* +&=\frac{E[v\mid \mathcal I_i]-p} +{\lambda\operatorname{Var}(v\mid \mathcal I_i)}, +&&\text{(CARA-normal demand)}. +\end{aligned} +\right. +$$ + +信息价值由 certainty equivalent 差给出： + +$$ +\begin{aligned} +\Delta CE +&= +E\left[ +\frac{(E[v\mid s,p]-p)^2} +{2\lambda\operatorname{Var}(v\mid s,p)} - +\frac{(E[v\mid p]-p)^2} +{2\lambda\operatorname{Var}(v\mid p)} +\right],\\ +\text{buy information} +&\Longleftrightarrow +\Delta CE\ge c. +\end{aligned} +$$ + +因此： + +$$ +\begin{aligned} +\text{costly information} +&\Longrightarrow +\text{some agents informed}\\ +&\Longrightarrow +\text{price partially reveals signal}\\ +&\Longrightarrow +\text{information value pinned down by CE gain}. +\end{aligned} +$$ + +#### 6.2.3 Glosten-Milgrom and Kyle +两者都在讲 adverse selection，但交易机制不同： + +| 模型 | 做市商看到什么 | 价格规则 | 核心对象 | +|---|---|---|---| +| Glosten-Milgrom | buy/sell direction | bid/ask quotes | spread | +| Kyle | total order flow $y=x+u$ | $p=E[v\mid y]$ | price impact $\lambda$ | + +共同逻辑： + +$$ +\begin{aligned} +\text{order flow} +\Longrightarrow +\text{Bayesian updating} +\Longrightarrow +\text{price impact / spread} +\Longrightarrow +\text{liquidity cost}. +\end{aligned} +$$ + +##### （1）Kyle model 分类总结 +Kyle 系列模型都围绕同一条链： + +$$ +\boxed{ +\text{private information} +\Longrightarrow +\text{strategic order }x +\Longrightarrow +\text{total order flow }y=x+u +\Longrightarrow +p=E[v\mid y] +\Longrightarrow +\lambda\text{ measures illiquidity}. +} +$$ + +1. **One-period Kyle** + + 模型系统： + + $$ + \left\{ + \begin{aligned} + v&\sim N(v_0,\Sigma_0), + &&\text{(terminal value)},\\ + u&\sim N(0,\sigma^2),\qquad u\perp v, + &&\text{(noise order)},\\ + x&=\beta(v-v_0), + &&\text{(insider linear strategy)},\\ + y&=x+u, + &&\text{(total order flow)},\\ + p&=E[v\mid y]=v_0+\lambda y, + &&\text{(market maker pricing)}. + \end{aligned} + \right. + $$ + + insider optimality： + + $$ + \begin{aligned} + \max_x E[(v-p)x\mid v] + &= + \max_x (v-v_0-\lambda x)x\\ + \Longrightarrow\quad + x + &=\frac{v-v_0}{2\lambda} + =\beta(v-v_0),\\ + \beta&=\frac{1}{2\lambda}. + \end{aligned} + $$ + + market maker regression： + + $$ + \begin{aligned} + \lambda + &= + \frac{\operatorname{Cov}(v,y)}{\operatorname{Var}(y)} + = + \frac{\beta\Sigma_0}{\beta^2\Sigma_0+\sigma^2}. + \end{aligned} + $$ + + 联立求解： + + $$ + \begin{aligned} + \beta&=\frac{1}{2\lambda}, + \qquad + \lambda= + \frac{\beta\Sigma_0}{\beta^2\Sigma_0+\sigma^2}\\ + \Longrightarrow\quad + \boxed{ + \beta=\frac{\sigma}{\sqrt{\Sigma_0}}, + \qquad + \lambda=\frac{\sqrt{\Sigma_0}}{2\sigma}. + } + \end{aligned} + $$ + + 信息揭示： + + $$ + \begin{aligned} +p +&=v_0+\lambda[\beta(v-v_0)+u]=\frac{v+v_0}{2}+\lambda u,\\ + \operatorname{Var}(p) + &=\frac12\Sigma_0. + \end{aligned} + $$ + + 单期模型中价格揭示一半先验不确定性。 + +2. **Two-period / finite-period Kyle** + + 两期模型把交易拆成两次，做市商逐期更新 belief： + + $$ + \left\{ + \begin{aligned} + y_1&=x_1+u_1, + &&u_1\sim N(0,\sigma^2/2),\\ + p_1&=E[v\mid y_1]=v_0+\lambda_1y_1,\\ + y_2&=x_2+u_2, + &&u_2\sim N(0,\sigma^2/2),\\ + p_2&=E[v\mid y_1,y_2]=p_1+\lambda_2y_2. + \end{aligned} + \right. + $$ + + 递推逻辑： + + $$ + \begin{aligned} + \Sigma_0 + &\Longrightarrow + \Sigma_1=\operatorname{Var}(v\mid y_1) + \Longrightarrow + \Sigma_2=\operatorname{Var}(v\mid y_1,y_2),\\ + \lambda_n + &= + \frac{\operatorname{Cov}(v,y_n\mid \mathcal F_{n-1}^y)} + {\operatorname{Var}(y_n\mid \mathcal F_{n-1}^y)},\\ + x_n + &= + \beta_n(v-p_{n-1}). + \end{aligned} + $$ + + 与 one-period 的核心差别： + + $$ + \begin{aligned} + \text{more trading rounds} + &\Longrightarrow + \text{insider can trade more gradually}\\ + &\Longrightarrow + \text{prices reveal information more completely}\\ + &\Longrightarrow + E[\Pi^{(2)}]>E[\Pi^{(1)}]. + \end{aligned} + $$ + + 两期情形： + + $$ + \begin{aligned} + E[\Pi^{(1)}] + &= + \frac{\sigma\sqrt{\Sigma_0}}{2},\\ + E[\Pi^{(2)}] + &\approx + 0.6206\,\sigma\sqrt{\Sigma_0} + > + 0.5\,\sigma\sqrt{\Sigma_0}. + \end{aligned} + $$ + + 即使 $v=v_0$，两期模型中 insider 仍可能有正条件利润，因为第一期交易会影响价格和 belief，第二期还能利用剩余误价。 + +3. **Continuous-time Kyle / Back (1992)** + + 连续时间把有限次交易极限化： + + $$ + \left\{ + \begin{aligned} + \tilde v&\sim N(\bar v,\sigma_v^2),\\ + Z_t&=\sigma_zB_t, + &&\text{(noise order)},\\ + X_t&=\int_0^t\theta_sds, + &&\text{(insider cumulative order)},\\ + Y_t&=X_t+Z_t, + &&\text{(total order flow)},\\ + P_t&=E[\tilde v\mid \mathcal F_t^Y]. + \end{aligned} + \right. + $$ + + 线性均衡猜测： + + $$ + \left\{ + \begin{aligned} + P_t&=\bar v+\lambda Y_t,\\ + \theta_t + &=\frac{(\tilde v-\bar v)/\lambda-Y_t}{1-t} + =\frac{\tilde v-P_t}{(1-t)\lambda}. + \end{aligned} + \right. + $$ + + 做市商过滤： + + $$ + \begin{aligned} + dY_t + &= + \frac{\tilde v-\bar v-\lambda Y_t}{\lambda(1-t)}dt + +\sigma_zdB_t,\\ + P_t&=E[\tilde v\mid\mathcal F_t^Y]=\bar v+\lambda Y_t. + \end{aligned} + $$ + + Kalman filtering pin down： + + $$ + \begin{aligned} + \lambda&=\frac{\sigma_v}{\sigma_z},\\ + \Sigma_t + &:=\operatorname{Var}(\tilde v\mid\mathcal F_t^Y) + = + \sigma_v^2(1-t). + \end{aligned} + $$ + + 关键性质： + + $$ + \begin{aligned} + t\uparrow1 + &\Longrightarrow + \Sigma_t\downarrow0 + \Longrightarrow + P_1=\tilde v, + &&\text{(full revelation)},\\ + P_t + &=E[\tilde v\mid\mathcal F_t^Y] + \Longrightarrow + P_t\text{ is an }\mathcal F_t^Y\text{-martingale}. + \end{aligned} + $$ + +4. **三类 Kyle 的对比** + + | 版本 | 时间 | 价格规则 | 信息揭示 | 核心系数 | + |---|---|---|---|---| + | one-period | 一次交易 | $p=v_0+\lambda y$ | 揭示一半信息 | $\lambda=\sqrt{\Sigma_0}/(2\sigma)$ | + | two-period / finite-period | 多次离散交易 | $p_n=E[v\mid y_1,\dots,y_n]$ | 逐期揭示，利润更高 | $\lambda_n,\beta_n,\Sigma_n$ 递推 | + | continuous-time | $t\in[0,1]$ | $P_t=\bar v+\lambda Y_t$ | $\Sigma_t=\sigma_v^2(1-t)$，终点完全揭示 | $\lambda=\sigma_v/\sigma_z$ | + + 总结： + + $$ + \boxed{ + \text{more trading opportunities} + \Longrightarrow + \text{more gradual insider trading} + \Longrightarrow + \text{more complete price discovery} + \Longrightarrow + \text{higher insider profit}. + } + $$ + +#### 6.2.4 Heterogeneous beliefs +异质信念把 SDF 从“一个代表性边际效用”推广成“不同 agents 的边际效用和 likelihood ratio 的组合”。 + +$$ +\begin{aligned} +\text{different beliefs} +&\Longrightarrow +\text{different likelihood ratios}\\ +&\Longrightarrow +\text{different perceived SPD}\\ +&\Longrightarrow +\text{wealth shares move over time}\\ +&\Longrightarrow +\text{prices, interest rates, risk premia change}. +\end{aligned} +$$ + +### 6.3 证明 +#### 6.3.1 Bayesian updating with precision +后验精度等于先验精度加信号精度。 + +$$ +\begin{aligned} +\left\{ +\begin{aligned} +v&\sim N(m_0,\rho_0^{-1}),\\ +s_i&=v+\varepsilon_i,\qquad \varepsilon_i\sim N(0,\rho_i^{-1}). +\end{aligned} +\right. +\implies +\left\{ +\begin{aligned} +\rho_n^{post} +&=\rho_0+\sum_{i=1}^n\rho_i,\\ +E[v\mid s_1,\dots,s_n] +&= +\frac{\rho_0m_0+\sum_{i=1}^n\rho_is_i} +{\rho_0+\sum_{i=1}^n\rho_i}. +\end{aligned} +\right. +\end{aligned} +$$ + +#### 6.3.2 Kyle one-period equilibrium +$\beta=\sigma/\sqrt{\Sigma_0}$，$\lambda=\sqrt{\Sigma_0}/(2\sigma)$。 + +$$ +\left\{ +\begin{aligned} +v&\sim N(v_0,\Sigma_0),\\ +u&\sim N(0,\sigma^2),\\ +x&=\beta(v-v_0),\\ +y&=x+u,\\ +p&=E[v\mid y]=v_0+\lambda y. +\end{aligned} +\right. +$$ + +Insider optimality： + +$$ +\begin{aligned} +\max_x (v-v_0-\lambda x)x +&\Longrightarrow +x=\frac{v-v_0}{2\lambda} +\Longrightarrow +\beta=\frac{1}{2\lambda}. +\end{aligned} +$$ + +Market maker regression： + +$$ +\begin{aligned} +\lambda +&= +\frac{\operatorname{Cov}(v,y)}{\operatorname{Var}(y)} = +\frac{\beta\Sigma_0}{\beta^2\Sigma_0+\sigma^2}. +\end{aligned} +$$ + +联立： + +$$ +\boxed{ +\beta=\frac{\sigma}{\sqrt{\Sigma_0}}, +\qquad +\lambda=\frac{\sqrt{\Sigma_0}}{2\sigma}. +} +$$ + +#### 6.3.3 Heterogeneous beliefs and SPD ratio +不同信念下的 state price density 通过 likelihood ratio 连接。 + +设两个信念测度 $P^1,P^2$，likelihood ratio 为 + +$$ +\begin{aligned} +\eta_t +&:=\frac{dP^2}{dP^1}\bigg|_{\mathcal F_t}. +\end{aligned} +$$ + +若同一 payoff $X_T$ 的价格在两种信念下必须一致： + +$$ +\begin{aligned} +S_t +&=E_t^{P^1}\left[\frac{\xi_T^1}{\xi_t^1}X_T\right] +=E_t^{P^2}\left[\frac{\xi_T^2}{\xi_t^2}X_T\right]. +\end{aligned} +$$ + +用 Bayes formula： + +$$ +\begin{aligned} +E_t^{P^2}[Y] +&= +E_t^{P^1}\left[\frac{\eta_T}{\eta_t}Y\right]. +\end{aligned} +$$ + +因此 + +$$ +\begin{aligned} +E_t^{P^2}\left[\frac{\xi_T^2}{\xi_t^2}X_T\right] +&= +E_t^{P^1}\left[ +\frac{\eta_T}{\eta_t} +\frac{\xi_T^2}{\xi_t^2} +X_T +\right]. +\end{aligned} +$$ + +要对任意 payoff 定价一致，需要 + +$$ +\boxed{ +\frac{\xi_T^1}{\xi_t^1} = +\frac{\eta_T}{\eta_t} +\frac{\xi_T^2}{\xi_t^2} +} +$$ + +等价地，归一化后可写成 state price density ratio 与 likelihood ratio 对应。 diff --git "a/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/CARA \346\225\210\347\224\250 - \345\270\270\347\273\235\345\257\271\351\243\216\351\231\251\345\216\214\346\201\266.md" "b/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/CARA \346\225\210\347\224\250 - \345\270\270\347\273\235\345\257\271\351\243\216\351\231\251\345\216\214\346\201\266.md" new file mode 100644 index 0000000..e971d51 --- /dev/null +++ "b/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/CARA \346\225\210\347\224\250 - \345\270\270\347\273\235\345\257\271\351\243\216\351\231\251\345\216\214\346\201\266.md" @@ -0,0 +1,31 @@ +--- +orphan: true +--- + +# CARA 效用 - 常绝对风险厌恶 + +导航：[Asset Pricing index](../../../index.md) · 来源：[01_Single_Period_Models](../../01_Single_Period_Models.md) + + +:::{admonition} Definition (Definition) +CARA 效用写作 + +$$ +u(w)=-e^{-\alpha w},\qquad \alpha>0. +$$ + +::: + +## 性质 + +$$ +-\frac{u''(w)}{u'(w)}=\alpha, +$$ + +即绝对风险厌恶不随财富变化。 + +## 经济含义 + +- 没有 wealth effect。 +- 最优风险暴露不依赖初始财富。 +- 与正态分布结合时，常可化成均值—方差问题。 diff --git "a/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/CRRA \346\225\210\347\224\250 - \345\270\270\347\233\270\345\257\271\351\243\216\351\231\251\345\216\214\346\201\266.md" "b/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/CRRA \346\225\210\347\224\250 - \345\270\270\347\233\270\345\257\271\351\243\216\351\231\251\345\216\214\346\201\266.md" new file mode 100644 index 0000000..6de97ea --- /dev/null +++ "b/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/CRRA \346\225\210\347\224\250 - \345\270\270\347\233\270\345\257\271\351\243\216\351\231\251\345\216\214\346\201\266.md" @@ -0,0 +1,46 @@ +--- +orphan: true +--- + +# CRRA 效用 - 常相对风险厌恶 + +导航：[Asset Pricing index](../../../index.md) · 来源：[01_Single_Period_Models](../../01_Single_Period_Models.md) + + +:::{admonition} Definition (Definition) +CRRA 效用写作 + +$$ +u(w)=\frac{w^{1-\rho}}{1-\rho},\qquad \rho\neq 1, +$$ +当 $\rho=1$ 时，退化为 + +$$ +u(w)=\log w. +$$ + +::: + +## 性质 + +$$ +-\frac{wu''(w)}{u'(w)}=\rho, +$$ + +即相对风险厌恶不随财富变化。 + +因此 + +$$ +\alpha(w)=-\frac{u''(w)}{u'(w)}=\frac{\rho}{w}, +\qquad +\frac{d\alpha(w)}{dw}=-\frac{\rho}{w^2}<0, +$$ + +所以 CRRA implies DARA. + +## 经济含义 + +- 对“按财富比例缩放”的风险态度不变。 +- 常用于跨期消费和增长模型。 +- 因为 $\alpha(w)=\rho/w$ 随财富上升而下降，所以它蕴含 DARA（decreasing absolute risk aversion，递减绝对风险厌恶）。 diff --git "a/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/\344\272\214\346\254\241\346\225\210\347\224\250 - Mean-Variance \345\237\272\347\241\200.md" "b/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/\344\272\214\346\254\241\346\225\210\347\224\250 - Mean-Variance \345\237\272\347\241\200.md" new file mode 100644 index 0000000..cffaea5 --- /dev/null +++ "b/PhD-Course/book/Asset Pricing/Theoretical AP/cards/part1/\344\272\214\346\254\241\346\225\210\347\224\250 - Mean-Variance \345\237\272\347\241\200.md" @@ -0,0 +1,52 @@ +--- +orphan: true +--- + +# 二次效用 - Mean-Variance 基础 + +导航：[Asset Pricing index](../../../index.md) · 来源：[01_Single_Period_Models](../../01_Single_Period_Models.md) + + +:::{admonition} Definition (Definition) +二次效用常写作 + +$$ +u(w)=-\frac12(w-\zeta)^2, +$$ +通常要求 $w<\zeta$，以保证边际效用为正。 + +::: + +## 性质 + +若 + +$$ +u(w)=-\frac12(w-\zeta)^2, +$$ + +则 + +$$ +\begin{aligned} +E[u(\tilde w)] +&= -\frac12E[(\tilde w-\zeta)^2] \\ +&= -\frac12\Big(E[\tilde w^2]-2\zeta E[\tilde w]+\zeta^2\Big) \\ +&= -\frac12\Big(\operatorname{Var}(\tilde w)+E[\tilde w]^2-2\zeta E[\tilde w]+\zeta^2\Big) \\ +&= -\frac12\operatorname{Var}(\tilde w)-\frac12\big(E[\tilde w]-\zeta\big)^2. +\end{aligned} +$$ + +故 + +$$ +\max E[u(\tilde w)] +\Longleftrightarrow +\max \Big\{E[\tilde w]-\frac{1}{2}(E[\tilde w]-\zeta)^2-\frac12\operatorname{Var}(\tilde w)\Big\}, +$$ + +因此只要期望收益与方差的组合相同，效用排序就等价于 mean-variance preferences。 + +## 局限 + +它蕴含 IARA（绝对风险厌恶随财富上升），通常被认为不够现实。 diff --git a/PhD-Course/book/Corporate Finance/attachments/AharonySwary1980.md b/PhD-Course/book/Corporate Finance/attachments/AharonySwary1980.md new file mode 100644 index 0000000..bdd89a0 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/AharonySwary1980.md @@ -0,0 +1,24 @@ +# Reading Note: Aharony and Swary (1980) + +Back to the main note + +## 1. Core Question +Do dividend announcements contain information about future cash flows, as reflected in announcement-window CARs? + +## 2. Figure / Evidence +![[Pasted image 20260421200909.png]] + +$$ +\begin{aligned} +\Delta d<0 &\Longrightarrow CAR \downarrow \\ +\Delta d>0 &\Longrightarrow CAR \uparrow +\end{aligned} +$$ + +## 3. Interpretation +- dividend decrease：公告前后 $CAR$ 明显下跌，市场把减息视为坏消息。 +- dividend increase：公告前后 $CAR$ 明显上升，市场把加息视为好消息。 +- 这支持 signaling 解释；dividend change 是 management 对 future cash flow 的公开信号，但不等于 dividend change 机械预测 future earnings。 + +## 4. Notes +- Use this card for the slide / table screenshot and any extra annotation. diff --git a/PhD-Course/book/Corporate Finance/attachments/AllenMichaely2003.md b/PhD-Course/book/Corporate Finance/attachments/AllenMichaely2003.md new file mode 100644 index 0000000..58ecad2 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/AllenMichaely2003.md @@ -0,0 +1,24 @@ +# Reading Note: Allen and Michaely (2003) + +Back to the main note + +## 1. Core Question +How much of dividend policy and dividend recipient composition can be explained by tax clientele, and how much is left once transaction costs and trading opportunities are taken into account? + +## 2. Main Point +- Dividend taxation can create different after-tax payoffs across investor types. +- But actual holder composition is not perfectly segmented by tax rate. +- Ex-dividend day trading and transaction costs weaken a clean clientele split. + +## 3. Reading Frame +- Use this card for any figure or slide that illustrates: + - shareholder composition + - tax clientele + - ex-dividend day trading + - transaction-cost frictions + +## 4. Figure Slots + +## 5. Notes +- Keep the main note concise. +- Add screenshots or paper figures here when needed. diff --git a/PhD-Course/book/Corporate Finance/attachments/BakerWurgler2002.md b/PhD-Course/book/Corporate Finance/attachments/BakerWurgler2002.md new file mode 100644 index 0000000..0d1abd9 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/BakerWurgler2002.md @@ -0,0 +1,114 @@ +--- +title: "Market Timing and Capital Structure" +year: 2002 +journal: "Journal of Finance" +author: "Malcolm Baker and Jeffrey Wurgler" +contribution: "提出 market timing theory：公司在高估值时发行 equity、低估值时回购 equity，长期资本结构因此带有历史市场择时的痕迹。" +tags: [CapitalStructure, MarketTiming, EquityIssuance, Repurchase, Empirical] +type: Course +--- + +# Reading Note: Baker and Wurgler (2002) + +## 1. Core Question +这篇文章问的不是“STO 为什么有最优杠杆”或“PO 为什么先内后外”，而是： + +- 公司是否会系统性地在股价高时发股、股价低时回购？ +- 如果会，这种 market timing 会不会长期留在 capital structure 里？ + +## 2. Why This Paper Matters +前面的文献已经给了两类经典解释： + +- **STO / trade-off**: leverage 由税盾和困境成本共同决定 +- **PO / pecking order**: leverage 由内部资金优先和信息不对称决定 + +Baker and Wurgler 增加的是第三条机制： + +- **Market timing**: 资本结构是历史上多次择时发行的累积结果 + +所以你图里的三张 slide 不是重复 STO/PO，而是在补充一个新维度： + +1. leverage 和 market-to-book 的关系为什么是负的 +2. 这种关系为什么更像 equity issuance timing，而不只是 distress +3. repurchase side 是否也有同样的 timing 证据 + +## 3. Main Channels + +### 3.1 Equity issuance timing +当 market-to-book 高时，公司股票价格相对更“贵”： + +- 发行 equity 更划算 +- 发行后 leverage 被动下降 +- 这种下降会积累成长期低杠杆 + +### 3.2 Repurchase timing +当 market-to-book 低或公司被低估时： + +- 公司更容易回购股票 +- 回购会提高 leverage +- 这与 equity issuance 形成对称关系 + +### 3.3 Persistent capital structure effect +如果公司长期在高估值时发股、低估值时回购： + +- 当前 leverage 会和历史 market-to-book 强相关 +- 这解释了为什么资本结构不是只看当期最优，而是包含历史路径依赖 + +## 4. Evidence Moved to Part 1 +这部分正文已经移动到 Part，这里仅保留入口。 + +- 主讲义中的对应部分现在按三张图整理了 SEO、prior return、undervaluation index 的证据 +- 如果需要完整的课内读法和图注解释，以 Part 1 为准 + +## 5. Relation to Other Papers + +### 5.1 Rajan and Zingales (1995) +- RZ95 先证明：国际样本里 leverage 的相关性很稳定 +- 但它并没有把 market-to-book 的负相关完全解释为哪一种理论 +- 在这门课的读法里，RZ95 提供的是 market timing 的前置证据 +- Baker and Wurgler 进一步把这条负相关整理成 historical market timing 的核心经验动机 + +### 5.2 Frank and Goyal (2009) +- FG09 说：大样本里 trade-off 更稳，但 profitability 等变量并不完全符合简单 trade-off +- Baker and Wurgler 则补充：market-to-book 那一块，可能不是 trade-off 或 pecking order，而是 timing 效应 + +### 5.3 SEO and repurchase literatures +这篇不是从零开始，而是把已有 long-run return evidence 组织成同一个解释框架： + +- **Ritter (1991)**, **Loughran and Ritter (1995)**, **Spiess and Affleck-Graves (1995)**: equity issues 之后长期表现偏弱 +- **Ikenberry, Lakonishok and Vermaelen (1995)**: repurchases 之后长期表现偏强 +- **Peyer and Vermaelen (2009)**: open market repurchases 也有类似 undervaluation / long-run return 证据 + +这些论文共同支持的不是 STO 或 PO，而是： + +- 公司在高估值时发股 +- 在低估值时回购 +- 资本结构因此带有历史 timing 的痕迹 + +## 6. Main Conclusion +这篇文章的核心结论可以简化成一句话： + +> capital structure is the cumulative outcome of past attempts to time the equity market. + +也就是说： + +- STO 解释“杠杆为什么有目标” +- PO 解释“为什么融资顺序不同” +- **Baker and Wurgler 解释“为什么历史发行时点会长期留在资本结构里”** + +## 7. Course Use +- 这篇是 market timing 理论的经典入口 +- 和 RZ95、FG09、SEO / repurchase literatures 放在一起看，逻辑最完整 +- 相关课堂笔记：Part 1 papers section + +## 8. Symbol List + +| Symbol / Term | Meaning | +| :--- | :--- | +| leverage | 资本结构指标；公司债务占比 | +| market-to-book | 市值与账面价值比，常被用来衡量估值高低或 market timing 环境 | +| equity issue | 增发 / SEO，发行新股融资 | +| repurchase | 回购，尤其是 open market repurchase | +| cumulative abnormal returns | 长期超额收益 | +| undervaluation index | 低估程度指标；高值通常对应更强的回购后正收益 | +| large equity issuers | 历史上大量发股的公司，Baker-Wurgler 里是关键分组 | diff --git a/PhD-Course/book/Corporate Finance/attachments/FarreMensaMichaelySchmalz2014.md b/PhD-Course/book/Corporate Finance/attachments/FarreMensaMichaelySchmalz2014.md new file mode 100644 index 0000000..a701d72 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/FarreMensaMichaelySchmalz2014.md @@ -0,0 +1,22 @@ +# Reading Note: Farre-Mensa, Michaely and Schmalz (2014) + +Back to the main note + +## 1. Core Question +This review asks what the main empirical determinants of dividend policy are, and how much of payout behavior can be explained by investor demand, taxes, agency frictions, signaling, and lifecycle effects. + +## 2. Main Channels +- investor demand +- taxes +- agency +- signaling +- lifecycle + +## 3. Reading Frame +- The review is useful as a map of payout determinants rather than a single clean identification exercise. +- Dividends and repurchases should be read together when the goal is to understand payout policy. +- The card is intentionally left open for figures and slide screenshots. + +## 4. Notes +- Add any figures from the review or the lecture slides here. +- Keep the main note concise; use this card for images and longer annotations. diff --git a/PhD-Course/book/Corporate Finance/attachments/FrankGoyal2009.md b/PhD-Course/book/Corporate Finance/attachments/FrankGoyal2009.md new file mode 100644 index 0000000..0f4ec05 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/FrankGoyal2009.md @@ -0,0 +1,108 @@ +--- +title: "Capital Structure Decisions: Which Factors Are Reliably Important?" +year: 2009 +journal: "Journal of Finance" +author: "Murray Z. Frank and Vidhan K. Goyal" +contribution: "用大样本公司特征系统检验资本结构决定因素，发现 trade-off theory 的若干预测更稳定，但盈利能力的经验关系常常为负。" +tags: [CapitalStructure, TradeOff, PeckingOrder, MarketTiming, Empirical] +type: Course +--- + +# Reading Note: Frank and Goyal (2009) + +## 1. Core Question +Frank and Goyal 关心的是：哪些 firm characteristics 能稳定预测 leverage，且这些变量的方向更支持哪种资本结构理论。 + +## 2. Empirical Design +- leverage choice: market leverage 或 book leverage；total debt 或 long-term debt +- sample period: 1950-2003 +- main variables: profitability, size, growth, taxes, tangibility, industry, risk, prior abnormal return, macro conditions + +## 2.1 Abbreviations +- **STO** = **static trade-off theory**，静态权衡理论 +- **PO** = **pecking order theory**，融资优序理论 +- 这篇文章的核心就是看：哪些变量的经验方向更像 STO，哪些更像 PO + +## 3. Theory Predictions + +### 3.0 Hypotheses and Channels + +| Hypothesis | Core channel | Intuition for leverage | +| :--- | :--- | :--- | +| STO | tax shield vs. distress cost | debt 带来税盾收益，但杠杆越高，财务困境和破产成本越大，所以存在最优杠杆 | +| PO | information asymmetry / financing deficit | 内部资金最便宜，外部融资有逆向选择成本，所以公司先用 retained earnings，再用 debt，最后才是 equity | + +更具体地说： + +- **STO channel**: 税盾收益 $+$ 破产/困境成本 $+$ 债务相关 agency cost +- **PO channel**: 信息不对称 $+$ 外部融资成本 $+$ 内部资金缺口（financing deficit） + +### 3.1 Variable-by-variable mapping + +![[Pasted image 20260419142554.png|536]] + +| Variable | STO prediction | PO prediction | Main channel | +| :--- | :--- | :--- | :--- | +| profitability | 更高 | 更低 | 税盾价值 vs. internal cash flow | +| size | 更高 | 通常更高或不确定 | default risk / information problem | +| growth | 更低 | 可能更高 | distress cost vs. financing need | +| tangibility | 更高 | 更高 | collateral / information asymmetry | +| risk | 更低 | 更低 | distress cost / equity difficulty | + +### 3.2 Profitability +- trade-off: 盈利越高，tax shield 越有价值，且 distress risk 更低，所以 leverage 应更高 +- pecking order: 盈利越高，internal finance 越充足，所以 leverage 应更低 + +### 3.3 Size +- trade-off: 大公司更分散，default risk 更低，所以 leverage 应更高 +- pecking order: 大公司更成熟，内部留存通常更充足，也更少依赖外部融资 + +### 3.4 Growth +- trade-off: 增长提高 distress cost，也放大 debt-related agency problems，所以 leverage 应更低 +- pecking order: 投资需求更强的公司更容易依赖 external finance，因此 leverage 可能更高 + +### 3.5 Tangibility +- trade-off: tangible assets 便于抵押，降低 distress cost，所以 leverage 应更高 +- pecking order: 有形资产通常意味着更低的信息不对称，也支持更高杠杆 + +### 3.6 Risk +- trade-off: 风险越高，financial distress cost 越大，所以 leverage 应更低 +- pecking order: 风险高的公司更难使用 equity，杠杆也往往更低 + +## 4. Main Finding +这组证据整体更支持 **STO / trade-off theory**，但不是“全支持”： + +| Channel / Theory | 含义 | 这篇论文里的证据 | 结论 | +| :--- | :--- | :--- | :--- | +| **STO / trade-off** | debt 有税盾收益，但也带来 distress cost，所以公司会在两者之间选一个最优杠杆 | 大多数变量方向与 STO 一致；size、tangibility、risk 等变量的方向总体比较稳定 | **主导解释**，是这篇文章最支持的框架 | +| **PO / pecking order** | 内部资金最便宜，外部融资有信息不对称成本，所以公司先 internal funds，再 debt，最后 equity | 一些变量也能解释，但整体不如 STO 稳定；尤其是许多核心结果不如 trade-off 统一 | **次强解释**，有部分支持，但不如 STO 稳 | +| **Market timing** | 公司会根据市场高估/低估或融资窗口来调整发行时点和资本结构 | prior abnormal return、宏观条件等变量没有给出很干净、稳定的识别 | **证据较弱**，这篇设计里没有被清晰识别出来 | + +最关键的一点是 profitability： + +- 按 **STO**，盈利越高，税盾价值越大，leverage 应更高 +- 但数据里 profitability 往往是 **负相关** +- 这说明：简单 STO 不是完全充分的解释，PO 相关机制和其他因素也在起作用 + +## 5. Course Use +- 这篇是资本结构经验研究的 benchmark +- 适合作为后续分类的入口文献 +- 相关课堂笔记：Part 1 papers section + +## 6. Symbol List + +| Symbol / Term | Meaning | +| :--- | :--- | +| leverage | 资本结构指标；公司债务占比 | +| market leverage | debt / market value of assets (or equity) 口径的杠杆 | +| book leverage | debt / book value of assets 口径的杠杆 | +| total debt | 总债务 | +| long-term debt | 长期债务 | +| profitability | 盈利能力，常对应 ROA / cash flow 类指标 | +| size | 公司规模，常用 log(assets) 或 log(sales) | +| growth | 增长机会，常用 market-to-book 或 sales growth | +| tangibility | 资产有形性，常用 fixed assets / total assets | +| risk | 收益波动或资产波动，反映财务困境概率 | +| taxes | 税负或税盾相关变量 | +| prior abnormal return | 先前异常收益，常用于 market timing 代理 | +| macro conditions | 宏观融资环境、利率、信用条件等 | diff --git a/PhD-Course/book/Corporate Finance/attachments/JohnKnyazeva2006.md b/PhD-Course/book/Corporate Finance/attachments/JohnKnyazeva2006.md new file mode 100644 index 0000000..5f7b04c --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/JohnKnyazeva2006.md @@ -0,0 +1,20 @@ +# Reading Note: John and Knyazeva (2006) + +Back to the main note + +## 1. Core Question +How does governance quality affect dividend payout, and does better governance reduce the need for dividends as a substitute mechanism? + +## 2. Main Point +- Better governance can reduce agency conflicts directly. +- When governance is strong, firms may rely less on dividends as a disciplinary device. +- This supports a substitution-style interpretation in which dividend demand falls as internal governance improves. + +## 3. Reading Frame +- Use this card for any figure or slide on governance quality and payout substitution. + +## 4. Figure Slots + +## 5. Notes +- Keep the main note concise. +- Add screenshots or paper figures here when needed. diff --git a/PhD-Course/book/Corporate Finance/attachments/Kalay1982.md b/PhD-Course/book/Corporate Finance/attachments/Kalay1982.md new file mode 100644 index 0000000..516312c --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/Kalay1982.md @@ -0,0 +1,19 @@ +# Reading Note: Kalay (1982) + +Back to the main note + +## 1. Core Question +How far can ex-dividend day price drops deviate from the dividend amount once taxes and transaction costs are taken into account? + +## 2. Main Point +- Ex-date price drops need not equal the cash dividend one-for-one. +- Trading costs and tax wedges create an arbitrage band. + +## 3. Reading Frame +- Use this card for any figure or slide on ex-dividend day pricing and arbitrage bounds. + +## 4. Figure Slots + +## 5. Notes +- Keep the main note concise. +- Add screenshots or paper figures here when needed. diff --git a/PhD-Course/book/Corporate Finance/attachments/LearyMichaely2011.md b/PhD-Course/book/Corporate Finance/attachments/LearyMichaely2011.md new file mode 100644 index 0000000..4736e21 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/LearyMichaely2011.md @@ -0,0 +1,28 @@ +# Reading Note: Leary and Michaely (2011) + +Back to the main note + +## 1. Core Question +This paper studies what drives dividend smoothing: which firm characteristics are associated with more or less smoothing, and how the cross-sectional patterns relate to signaling, agency frictions, and payout adjustment. + +## 2. Main Channels +- information asymmetry +- signaling +- agency +- payout adjustment costs +- lifecycle / maturity + +## 3. Reading Frame +- Dividend smoothing is not a single clean test of one theory. +- It is a reduced-form outcome of multiple payout frictions. +- Use this card to attach figures from the paper or lecture slides. + +## 4. Figure Slots + +![[attachment/Pasted image 20260502100602.png]] + +![[attachment/Pasted image 20260502100609.png]] + +## 5. Notes +- Add any regression tables, figures, or slide screenshots here. +- Keep the main note concise; use this card for visual evidence and annotations. diff --git a/PhD-Course/book/Corporate Finance/attachments/Ma2019.md b/PhD-Course/book/Corporate Finance/attachments/Ma2019.md new file mode 100644 index 0000000..3536188 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/Ma2019.md @@ -0,0 +1,19 @@ +--- +title: "Nonfinancial Firms as Cross-Market Arbitrageurs" +year: 2019 +journal: "Journal of Finance" +author: "Yueran Ma" +contribution: "展示 nonfinancial firms 会根据 debt 与 equity 的 relative valuation 调整融资 mix，把资本结构当成跨市场套利对象。" +tags: [CapitalStructure, MarketTiming, Arbitrage, Empirical] +type: Course +--- + +# Reading Note: Ma (2019) + +## 1. Core Idea +Nonfinancial firms respond to relative valuation across debt and equity markets. When one side looks cheap or expensive, firms substitute between repurchases and issuance. + +## 2. Course Use +- used in the main note as a market-timing / relative-misvaluation example +- complements the long-run timing literature on SEOs and repurchases + diff --git a/PhD-Course/book/Corporate Finance/attachments/MichaelyVila1986.md b/PhD-Course/book/Corporate Finance/attachments/MichaelyVila1986.md new file mode 100644 index 0000000..e697056 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/MichaelyVila1986.md @@ -0,0 +1,20 @@ +# Reading Note: Michaely and Vila (1986) + +Back to the main note + +## 1. Core Question +How do heterogeneous tax clienteles and short-term arbitrage trading affect ex-dividend day pricing and trading volume? + +## 2. Main Point +- Tax differences can create heterogeneous ex-dividend day preferences. +- Arbitrage trading and transaction costs shape both price drops and volume. +- The ex-date is therefore a joint test of clientele effects and trading frictions. + +## 3. Reading Frame +- Use this card for figures or slides on ex-date trading, price drop, and volume. + +## 4. Figure Slots + +## 5. Notes +- Keep the main note concise. +- Add screenshots or paper figures here when needed. diff --git a/PhD-Course/book/Corporate Finance/attachments/MyersMajluf1984.md b/PhD-Course/book/Corporate Finance/attachments/MyersMajluf1984.md new file mode 100644 index 0000000..6f930b5 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/MyersMajluf1984.md @@ -0,0 +1,19 @@ +--- +title: "Corporate Financing and Investment Decisions When Firms Have Information That Investors Do Not Have" +year: 1984 +journal: "Journal of Financial Economics" +author: "Stewart C. Myers and Nicholas S. Majluf" +contribution: "提出 pecking order：信息不对称使 external financing 变贵，equity 发行最容易触发 adverse selection 和 dilution。" +tags: [CapitalStructure, InformationAsymmetry, PeckingOrder, EquityIssuance] +type: Course +--- + +# Reading Note: Myers and Majluf (1984) + +## 1. Core Idea +Managers know more about firm value than outside investors. As a result, equity issuance can be interpreted as bad news because it may signal overvaluation. + +## 2. Course Use +- canonical reference for information asymmetry and pecking order +- linked to the main note’s dilution example and issuance-announcement effects + diff --git a/PhD-Course/book/Corporate Finance/attachments/PeyerVermaelen2009.md b/PhD-Course/book/Corporate Finance/attachments/PeyerVermaelen2009.md new file mode 100644 index 0000000..96f7408 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/PeyerVermaelen2009.md @@ -0,0 +1,20 @@ +# Reading Note: Peyer and Vermaelen (2009) + +Back to the buyback framework + +## 1. Core Question +Which buyback characteristics predict stronger and more persistent post-buyback abnormal returns? + +## 2. Main Point +- Buyback signal credibility is not binary. +- Cheap-stock features such as low market-to-book, small size, prior price decline, and explicit undervaluation statements are associated with stronger buyback anomalies. +- An undervaluation index can summarize these features and predict post-buyback performance. + +## 3. Reading Frame +- Use this card for any figure or slide on buyback anomalies and long-run return patterns. + +## 4. Figure Slots + +## 5. Notes +- Keep the main note concise. +- Add screenshots or paper figures here when needed. diff --git a/PhD-Course/book/Corporate Finance/attachments/RajanZingales1995.md b/PhD-Course/book/Corporate Finance/attachments/RajanZingales1995.md new file mode 100644 index 0000000..a7429a8 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/RajanZingales1995.md @@ -0,0 +1,106 @@ +--- +title: "What Do We Know About Capital Structure? Some Evidence from International Data" +year: 1995 +journal: "Journal of Finance" +author: "Raghuram G. Rajan and Luigi Zingales" +contribution: "用 G-7 国际样本检验资本结构的跨国稳健性，发现 leverage 的若干经验相关性在各国大体一致，但这些相关性未能完全区分不同理论。" +tags: [CapitalStructure, InternationalEvidence, TradeOff, PeckingOrder, Empirical] +type: Course +--- + +# Reading Note: Rajan and Zingales (1995) + +## 1. Core Question +这篇文章问的是：美国样本里常见的 capital structure regularities，到了其他主要工业化国家是否仍然成立？ + +更进一步，作者想看的是： + +- 这些相关性是否只是美国制度环境下的偶然现象 +- 还是说它们反映了更一般的资本结构机制 + +## 2. Sample and Design +- sample: G-7 countries + - United States + - Japan + - Germany + - France + - Italy + - United Kingdom + - Canada +- dependent variable: leverage + - book leverage + - market leverage +- key regressors: + - tangibility + - market-to-book + - log sales + - profitability + - industry leverage + +## 3. Main Channels + +### 3.1 Trade-off channel +STO 主要关注税盾收益和困境成本： + +- tangibility 高，抵押能力强，debt capacity 更高 +- size 大，diversification 更强，distress risk 更低 +- industry leverage 高，说明同业融资环境和 lender norms 支持更高杠杆 + +### 3.2 Pecking order channel +PO 主要关注信息不对称和内部融资优先： + +- profitability 高，internal funds 更充足，所以外部融资需求更低 +- growth opportunities 高时，公司可能更偏向保留融资弹性，短期杠杆未必更高 + +### 3.3 Market timing / equity issuance channel +市场时点也可能影响 leverage： + +- market-to-book 高，有时意味着股价高、equity issuance 更容易 +- 这会让高估值公司在融资后表现出更低 leverage + +## 4. Main Findings +从跨国样本看，很多美国里的相关性也成立，但作者强调：这些相关性**并不等于**理论识别完成。 + +### 4.1 Stable correlations +- tangibility 和 leverage 正相关 +- size 和 leverage 正相关 +- profitability 和 leverage 负相关 +- market-to-book 和 leverage 负相关 +- industry leverage 很重要 + +### 4.2 Interpretation +- **tangibility / size / industry leverage** 更像是 trade-off 机制在起作用 +- **profitability** 的负相关更接近 pecking order 的直觉 +- **market-to-book** 的负相关在这门课的课件里被读作 **market timing 的支持性证据**，尤其是当这种关系主要由 large equity issuers 驱动时 +- 因为 accounting rules、tax code、bankruptcy code 和金融制度不同，跨国数据能增加辨识力，但也让理论映射更复杂 + +### 4.3 Bottom line +这篇文章的结论不是“某一个理论赢了”，而是： + +- leverage 的很多经验规律在国际样本里很稳 +- 但这些规律背后的理论基础仍然部分 unresolved +- 所以它更像是一个 **cross-country benchmark**，而不是单一理论的终极检验 + +在这门课的讲法里，它还有一个更具体的作用： + +- 它给了 market-to-book 与 leverage 负相关的早期跨国证据 +- 这条证据后来被 Baker and Wurgler (2002) 重新组织成 market timing theory 的核心经验动机 + +## 5. Course Use +- 这篇是国际资本结构证据的经典 benchmark +- 和 Frank and Goyal (2009) 一起看，可以区分“美国样本规律”与“跨国稳健规律” +- 相关课堂笔记：Part 1 papers section + +## 6. Symbol List + +| Symbol / Term | Meaning | +| :--- | :--- | +| leverage | 资本结构指标；文中主要看 book leverage 和 market leverage | +| book leverage | adjusted debt / (adjusted debt + adjusted equity) | +| market leverage | adjusted debt / (adjusted debt + market value of equity) | +| tangibility | 固定资产占比，衡量抵押品能力 | +| market-to-book | 市值与账面价值的比率，常代理 growth opportunities 或 equity timing | +| log sales | 公司规模的对数代理 | +| profitability | 盈利能力，文中常用 EBITDA / assets 一类指标 | +| industry leverage | 行业杠杆，反映同行融资环境和 lender norms | +| G-7 | 七个主要工业化国家样本 | diff --git a/PhD-Course/book/Corporate Finance/attachments/Ross1977.md b/PhD-Course/book/Corporate Finance/attachments/Ross1977.md new file mode 100644 index 0000000..cf7d0b3 --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/Ross1977.md @@ -0,0 +1,23 @@ +--- +title: "The Determination of Financial Structure: The Incentive-Signaling Approach" +year: 1977 +journal: "Bell Journal of Economics" +author: "Stephen A. Ross" +contribution: "提出 leverage can serve as a costly signal: good firms can sustain higher debt because they expect lower bankruptcy cost." +tags: [CapitalStructure, Signaling, Debt] +type: Course +--- + +# Reading Note: Ross (1977) + +## 1. Core Idea +Ross argues that financial structure can convey private information. If managers know firm quality and leverage is costly in distress, higher debt can signal higher quality because only good firms can credibly bear the expected bankruptcy cost. + +## 2. Signaling Logic +- leverage is a **dissipative signal**: credibility comes from a real cost that bad firms cannot mimic cheaply +- low-quality firms avoid high debt because distress cost would be too large +- high-quality firms can choose higher leverage and separate themselves from lemons + +## 3. Course Use +- used as the canonical leverage signaling reference in the main note + diff --git a/PhD-Course/book/Corporate Finance/attachments/Vermaelen1984.md b/PhD-Course/book/Corporate Finance/attachments/Vermaelen1984.md new file mode 100644 index 0000000..63340da --- /dev/null +++ b/PhD-Course/book/Corporate Finance/attachments/Vermaelen1984.md @@ -0,0 +1,23 @@ +--- +title: "Repurchase Tender Offers, Signaling, and Managerial Incentives" +year: 1984 +journal: "Journal of Financial and Quantitative Analysis" +author: "Theo Vermaelen" +contribution: "将 share repurchase 进一步形式化为 signaling device，并强调 repurchase premium 和 managerial incentives 如何支撑信号可信度。" +tags: [CapitalStructure, Signaling, Buyback, Empirical] +type: Course +--- + +# Reading Note: Vermaelen (1984) + +## 1. Core Question +Vermaelen asks how repurchase tender offers can credibly signal undervaluation and how managerial incentives affect the credibility of that signal. + +## 2. Core Logic +- repurchase is a **non-dissipative signal**: the main cost often shows up as wealth transfer through a premium rather than pure deadweight loss +- the signal is more credible when managers have more skin in the game or when the premium / repurchase commitment is costly to mimic + +## 3. Course Use +- used together with the main note to illustrate non-dissipative signaling in buybacks +- complements [Vermaelen (1981)](../cards/Vermaelen-1981-buybacks) on repurchase signaling + diff --git a/PhD-Course/book/_toc.yml b/PhD-Course/book/_toc.yml index 11b72fd..d7c51a4 100644 --- a/PhD-Course/book/_toc.yml +++ b/PhD-Course/book/_toc.yml @@ -4,6 +4,7 @@ parts: - caption: Asset Pricing chapters: - file: Asset Pricing/index + - file: Asset Pricing/Asset Pricing Review - file: Asset Pricing/Empirical AP/01_Volatility_ARCH_GARCH - file: Asset Pricing/Empirical AP/02_Implied_Volatility_VIX_VRP - file: Asset Pricing/Empirical AP/03_Consumption_Based_AP_Puzzles @@ -17,6 +18,10 @@ parts: - file: Asset Pricing/Theoretical AP/02_Dynamic_Asset_Pricing - file: Asset Pricing/Theoretical AP/03_Continuous_Time_Pricing_Options_Term_Structure - file: Asset Pricing/Theoretical AP/04_Information_Beliefs_and_Learning + sections: + - file: Asset Pricing/Theoretical AP/cards/part1/CARA 效用 - 常绝对风险厌恶 + - file: Asset Pricing/Theoretical AP/cards/part1/CRRA 效用 - 常相对风险厌恶 + - file: Asset Pricing/Theoretical AP/cards/part1/二次效用 - Mean-Variance 基础 - caption: Corporate Finance chapters: - file: Corporate Finance/index @@ -24,6 +29,22 @@ parts: - file: Corporate Finance/02_Econometric_Methods_in_Corporate_Finance - file: Corporate Finance/03_Applied_Corporate_Finance_Theory - file: Corporate Finance/04_Special_Topics + sections: + - file: Corporate Finance/attachments/AharonySwary1980 + - file: Corporate Finance/attachments/AllenMichaely2003 + - file: Corporate Finance/attachments/BakerWurgler2002 + - file: Corporate Finance/attachments/FarreMensaMichaelySchmalz2014 + - file: Corporate Finance/attachments/FrankGoyal2009 + - file: Corporate Finance/attachments/JohnKnyazeva2006 + - file: Corporate Finance/attachments/Kalay1982 + - file: Corporate Finance/attachments/LearyMichaely2011 + - file: Corporate Finance/attachments/Ma2019 + - file: Corporate Finance/attachments/MichaelyVila1986 + - file: Corporate Finance/attachments/MyersMajluf1984 + - file: Corporate Finance/attachments/PeyerVermaelen2009 + - file: Corporate Finance/attachments/RajanZingales1995 + - file: Corporate Finance/attachments/Ross1977 + - file: Corporate Finance/attachments/Vermaelen1984 - file: Corporate Finance/theories/00_Theory_Integration sections: - file: Corporate Finance/theories/Common_Mistakes