Dynamic mean-lpm and mean-cvar portfolio optimization in continuous time(1)
考虑市场上有 n n n项风险资产和1项无风险资产,并且资产可以在连续时间区间 [ 0 , T ] [0, T] [0,T]内被交易. 模型中所有的随机性都用概率空间 ( Ω , F , P , { F t } t ≥ 0 ) (\Omega, \mathcal{F}, \mathbb{P}, \{\mathcal{F}_t\}_{t\geq 0}) (Ω,F,P,{Ft}t≥0)表示,适应域流 F t \mathcal{F}_t Ft的 n n n维布朗运动 W ( t ) = ( W 1 ( t ) , … , W n ( t ) ) ′ W(t)=(W_1(t), \dots, W_n(t))' W(t)=(W1(t),…,Wn(t))′定义在这个概率空间上,对于所有的 i ≠ j i\neq j i=j, W i ( t ) W_i(t) Wi(t)和 W j ( t ) W_j(t) Wj(t)是互相独立的. 定义 L F 2 ( 0 , T ; R n ) \mathcal{L}^2_\mathcal{F}(0, T; \mathbb{R}^n) LF2(0,T;Rn)表示值域为 R n \mathbb{R}^n Rn,且 F t − a d a p t e d \mathcal{F}_t-adapted Ft−adapted平方可积的随机过程.(square integrable stochastic processes)1 定义 L F T 2 ( Ω ; R n ) \mathcal{L}^2_{\mathcal{F}_T}(\Omega; \mathbb{R}^n) LFT2(Ω;Rn)表示值域为 R n \mathbb{R}^n Rn且 F T \mathcal{F}_T FT可测的随机变量 无风险资产的价格过程 S 0 ( t ) S_0(t) S0(t)由以下ODE产生 { d S 0 ( t ) = r ( t ) S 0 ( t ) d t , t ∈ [ 0 , T ] S 0 ( 0 ) = s 0 > 0 (2.1) \begin{cases} dS_0(t)=r(t)S_0(t)dt, t\in[0, T]\\ S_0(0)=s_0>0 \end{cases}\tag{2.1} {dS0(t)=r(t)S0(t)dt,t∈[0,T]S0(0)=s0>0(2.1) 其中 r ( t ) r(t) r(t)是无风险收益率,也是 F t \mathcal{F}_t Ft可测的标量值随机过程. n n n只风险资产的价格由以下SDE方程产生 { d S i ( t ) = S i ( t ) ( μ i ( t ) d t + ∑ j = 1 n σ i j ( t ) d W j ( t ) ) , t ∈ [ 0 , T ] , i = 1 , … , n S i ( 0 ) = s i > 0 , i = 1 , … , n (2.2) \begin{cases} dS_i(t)=S_i(t)(\mu_i(t)dt+\sum\limits_{j=1}^n\sigma_{ij}(t)dW_j(t)), t\in[0, T], i=1, \dots,n\\ S_i(0)=s_i>0, i=1,\dots, n \end{cases}\tag{2.2} ⎩⎨⎧dSi(t)=Si(t)(μi(t)dt+j=1∑nσij(t)dWj(t)),t∈[0,T],i=1,…,nSi(0)=si>0,i=1,…,n(2.2) 其中 μ i \mu_i μi表示增值率(appreciation rate), σ i j \sigma_{ij} σij表示波动率(volatility rate),进一步设置波动率矩阵 σ ( t ) : = { σ i j ( t ) } ∣ i , j = 1 n , n \sigma(t):=\{\sigma_{ij}(t)\}\mid_{i,j=1}^{n, n} σ(t):={σij(t)}∣i,j=1n,n 满足如下非退化条件,对于一些 ϵ > 0 \epsilon>0 ϵ>02 σ ( t ) σ ′ ( t ) ≻ ϵ I , ∀ 0 ≤ t ≤ T , a . s . (2.3) \sigma(t)\sigma'(t)\succ \epsilon I, \forall 0\leq t\leq T, a.s.\tag{2.3} σ(t)σ′(t)≻ϵI,∀0≤t≤T,a.s.(2.3) 投资者具有初始财富 x 0 x_0 x0在时刻0进入市场,并且在时间段 [ 0 , T ] [0, T] [0,T]上连续配置资产,设 x ( t ) x(t) x(t)表示投资者在 t t t时刻的总财富,定义投资组合过程为 π ( t ) = ( π 1 ( t ) , … , π n ( t ) ) ′ \pi(t)=(\pi_1(t), \dots, \pi_n(t))' π(t)=(π1(t),…,πn(t))′ 其中 π ( ⋅ ) ∈ L F 2 ( 0 , T ; R n ) \pi(\cdot)\in\mathcal{L}_\mathcal{F}^2(0, T; \mathbb{R}^n) π(⋅)∈LF2(0,T;Rn),其中 π i ( t ) \pi_i(t) πi(t)表示 t t t时刻在风险资产 i i i上分配的投资额,本文没有考虑投资过程中的交易费用,所以投资者的财富过程 x ( t ) x(t) x(t)满足如下SDE { d x ( t ) = ( r ( t ) x ( t ) + b ( t ) ′ π ( t ) ) d t + π ′ ( t ) σ ( t ) d W ( t ) x ( 0 ) = x 0 (2.4) \begin{cases} dx(t)=\bigg(r(t)x(t)+b(t)'\pi(t)\bigg)dt+\pi'(t)\sigma(t)dW(t)\\ x(0)=x_0 \end{cases}\tag{2.4} ⎩⎨⎧dx(t)=(r(t)x(t)+b(t)′π(t))dt+π′(t)σ(t)dW(t)x(0)=x0(2.4) 其中 b ( t ) b(t) b(t)表示超额收益 b ( t ) : = ( μ 1 ( t ) − r ( t ) μ 2 ( t ) − r ( t ) … μ n ( t ) − r ( t ) ) ′ b(t):=(\mu_1(t)-r(t)\quad \mu_2(t)-r(t) \quad \dots \quad \mu_n(t)-r(t))' b(t):=(μ1(t)−r(t)μ2(t)−r(t)…μn(t)−r(t))′ 建立mean-downside risk portfolio optimization,以下为mean-LPM模型 ( P l p m q ) min π ( ⋅ ) ∈ L F 2 ( 0 , T ; R n ) E [ ( γ − x ( T ) ) + q ] s . t . { E ( x ( T ) ) ≥ d { x ( ⋅ ) , π ( ⋅ ) } satisfies ( 2.4 ) 0 ≤ x ( T ) ≤ B (\mathcal{P}_{lpm}^q)\min\limits_{\pi(\cdot)\in\mathcal{L}^2_\mathcal{F}(0, T; \mathbb{R}^n)} E[(\gamma-x(T))_+^q]\\ s.t. \begin{cases} E(x(T))\geq d\\ \{x(\cdot), \pi(\cdot)\}\quad \text{satisfies} \quad(2.4)\\ 0\leq x(T)\leq B \end{cases} (Plpmq)π(⋅)∈LF2(0,T;Rn)minE[(γ−x(T))+q]s.t.⎩⎪⎨⎪⎧E(x(T))≥d{x(⋅),π(⋅)}satisfies(2.4)0≤x(T)≤B 模型中的符号表如下
SymbolsMeanings d d d在末期投资者希望得到的财富值的期望的下限 B B B末期投资者希望得到财富值的上界 γ ∈ R \gamma\in\mathbb{R} γ∈R给定的基准水平(benchmark level) q q q给定的非负值,表示矩的阶当 q = 0 q=0 q=0时,可以得到 ( γ − x ( T ) ) + 0 = 1 x ( T ) ≤ γ ⇒ E [ ( γ − x ( T ) ) + 0 ] = P ( x ( T ) ≤ γ ) (\gamma-x(T))_+^0=\mathbf{1}_{x(T)\leq \gamma}\Rightarrow\\ E[(\gamma-x(T))_+^0]=\mathbb{P}(x(T)\leq \gamma) (γ−x(T))+0=1x(T)≤γ⇒E[(γ−x(T))+0]=P(x(T)≤γ) it is the disaster probability considered by Roy in his pioneering safety-first principle.
当 q = 1 q=1 q=1时且 γ = E [ x ( T ) ] \gamma=E[x(T)] γ=E[x(T)],下行风险测度为 E [ ( E [ x ( T ) ] − x ( T ) ) + 1 ] E[(E[x(T)]-x(T))_+^1] E[(E[x(T)]−x(T))+1] 即半绝对离差(semi-absolute deviation). 当 q = 2 q=2 q=2时且 γ = E [ x ( T ) ] \gamma=E[x(T)] γ=E[x(T)],下行风险测度为 E [ ( E [ x ( T ) ] − x ( T ) ) + 2 ] E[(E[x(T)]-x(T))_+^2] E[(E[x(T)]−x(T))+2] 即为半方差(semi-variance) 设置 x ˉ T \bar{x}_T xˉT表示期末财富的safe-level,一种不失合理性(without loss of rationality)的值为 x ˉ T = E [ e ∫ 0 T r ( s ) d s ] x 0 (2.5) \bar{x}_T=E[e^{\int_0^Tr(s)ds}]x_0\tag{2.5} xˉT=E[e∫0Tr(s)ds]x0(2.5) 该值得意义为将所有初期财富全部投资于无风险资产,上界 B B B设置为 B > max { d , x ˉ T , γ } B>\max\{d, \bar{x}_T, \gamma\} B>max{d,xˉT,γ} 本文同时也研究了dynamic mean-CVaR portfolio optimization,定义投资损失函数为 f ( x ( T ) ) : = x ˉ T − x ( T ) f(x(T)):=\bar{x}_T-x(T) f(x(T)):=xˉT−x(T) 定义投资组合的CVaR为 C V a R [ f ( x ( T ) ) ] CVaR[f(x(T))] CVaR[f(x(T))],可以得到mean-CVaR模型如下 ( P c v a r ) min π ( ⋅ ) ∈ L F 2 ( 0 , T ; R n ) C V a R [ f ( x ( T ) ) ] s . t . { E ( x ( T ) ) ≥ d { x ( ⋅ ) , π ( ⋅ ) } satisfies ( 2.4 ) 0 ≤ x ( T ) ≤ B (\mathcal{P}_{cvar}) \min\limits_{\pi(\cdot)\in\mathcal{L}^2_\mathcal{F}(0, T; \mathbb{R}^n)} CVaR[f(x(T))]\\ s.t. \begin{cases} E(x(T))\geq d\\ \{x(\cdot), \pi(\cdot)\}\quad \text{satisfies} \quad(2.4)\\ 0\leq x(T)\leq B \end{cases} (Pcvar)π(⋅)∈LF2(0,T;Rn)minCVaR[f(x(T))]s.t.⎩⎪⎨⎪⎧E(x(T))≥d{x(⋅),π(⋅)}satisfies(2.4)0≤x(T)≤B
本文后面将证明对于越大的上界 B B B,投资组合策略会变得越贪婪(the more aggressive the portfolio policy becomes).
If we let B B B go to infinite, both problems P l p m q \mathcal{P}_{lpm}^q Plpmq and P c v a r \mathcal{P}_{cvar} Pcvar will become ill-posed, i.e., the investor would take an infinite position. From the view point of real applications, any portfolio that generates extremely high level of terminal wealth is not realistic.
本文也证明了末期财富达到上界的概率随着上界水平的提高而单调下降,因此当设置很高的上界时等效于不存在上界,同时模型中加入了非破产约束限制(non-bankruptcy constraint)3, x ( t ) ≥ 0 , t ∈ [ 0 , T ] x(t)\geq 0, t\in [0, T] x(t)≥0,t∈[0,T]
stochastic integral ↩︎
a.s. 表示almost surely, 排除了发生几率为0的事件 ↩︎
continuous_time_mean_variance_portfolio_with_no_bankrupcy ↩︎