mean-variance portfolio selection

The continuous-time mean-variance portfolio selection problem seeks to minimise the variance of terminal wealth $\text{Var}x(T)$ while targeting a given expected return $Ex(T)$, or equivalently to minimise $-Ex(T)+\mu \text{Var}x(T)$ for a risk-aversion parameter $\mu >0$. With $m$ risky stocks whose prices follow geometric Brownian motions and one risk-free bond, the wealth equation is $dx=[rx+{\sum }_i(b_i-r)u_i]dt+{\sum }_{i,j}{\sigma }_{ij}{u}_{i}d{W}_j$, where $u_i(t)$ is the dollar amount invested in stock $i$.

The key difficulty is that the variance $\text{Var}x(T)=Ex(T{)}^2-[Ex(T){]}^2$ contains the squared expectation $[Ex(T){]}^2$, which is not a standard stochastic control objective. Yong and Zhou resolve this by embedding the problem into an auxiliary LQ problem: for each $(\mu ,\lambda )$, minimise $E\{\mu x(T{)}^2-\lambda x(T)\}$. By the concavity argument of Theorem 8.2, any efficient portfolio of the original problem is also optimal for the auxiliary problem with the specific choice $\lambda =1+2\mu E\bar{x}(T)$. The auxiliary problem, after a translation $y=x-\gamma$ with $\gamma =\lambda /(2\mu )$, becomes a standard stochastic LQ problem with $Q=S=R=0$, $G=\mu$, and control appearing only in the diffusion through $D_j$.

Because $R=0$ and the state is scalar, the stochastic Riccati equation $\dot{P}=(\rho -2r)P$, $P(T)=\mu$ reduces to a linear ODE with explicit solution $P(t)=\mu e^{-{\int }_t^T(\rho (s)-2r(s))ds}$, where $\rho (t)=B(t)(\sigma {\sigma }^T{)}^{-1}B(t{)}^T$ is the squared market price of risk. The optimal portfolio is $\bar{u}(t)=-(\sigma {\sigma }^T{)}^{-1}{B}^T\{e^{-{\int }_t^{T}r(s)ds}\gamma -x(t)\}$, a mean-reverting strategy targeting the discounted certainty-equivalent wealth level $\gamma$.

The efficient frontier takes the elegant form $\text{Var}\bar{x}(T)=\frac{e^{-{\int }_0^T\rho (t)dt}}{1-e^{-{\int }_0^T\rho (t)dt}}(E\bar{x}(T)-x_0{e}^{{\int }_0^{T}r(t)dt}{)}^2$, a perfect square that reflects the availability of the risk-free asset.

Key Details

• Embedding trick: The non-standard term $[Ex(T){]}^2$ is handled by parametrising over $\lambda$ and optimising a family of standard LQ problems
• Optimal parameter: ${\lambda }^{\ast }=1+2\mu \alpha x_0/(1-\beta )$ where $\alpha =e^{{\int }_0^T(r-\rho )dt}$ and $\beta =e^{-{\int }_0^T\rho dt}$
• Riccati equation: Reduces to a scalar linear ODE because $R=0$ and the state is one-dimensional
• Efficient frontier: Parabola in the $(E\bar{x}(T),\text{Var}\bar{x}(T))$ plane; zero variance is achieved only by the pure bond portfolio
• Uncertainty cost interpretation: With $R=0$, there is no direct cost on the control; the implicit cost ${\sum }_j{D}_j^{T}P{D}_j=P\sigma {\sigma }^T$ arises entirely from the control’s effect on diffusion volatility
• Connection to XVA: The mean-variance structure is analogous to the risk-return trade-off in Riccati system for XVA hedging, where the hedger balances P&L variance against transaction costs

Textbook References

Stochastic Controls - Hamiltonian Systems and HJB Equations (Yong & Zhou, 1999)

• Definition 8.1 (p. 337): Admissible portfolio, efficient portfolio, and efficient frontier
• Theorem 8.2 (p. 338): Any optimal portfolio of $P(\mu )$ belongs to ${\bigcup }_{\lambda }{H}_A(\mu ,\lambda )$; the optimal ${\lambda }^{\ast }$ satisfies $\lambda =1+2\mu E\bar{x}(T)$
• Theorem 8.3 (p. 341): Explicit efficient frontier formula as a perfect square involving the squared market price of risk $\rho (t)$
• Equations (8.26)—(8.29) (pp. 339—340): Explicit solution of the scalar stochastic Riccati equation and derivation of the optimal feedback portfolio

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