Optimal investment in multidimensional Markov-modulated affine models

Abstract

In a multidimensional affine framework we consider a portfolio optimization problem with finite horizon, where an investor aims to maximize the expected utility of her terminal wealth. We state a very flexible asset price model that incorporates several risk factors modeled both by diffusion processes and by a Markov chain. Exploiting the affine structure of the model we solve the corresponding Hamilton–Jacobi–Bellman equations explicitly up to an expectation only over the Markov chain or equivalently up to a system of simple ODEs. The relevance of the presented model is illustrated on two examples including a stochastic short rate model with trading in the bond and the stock market, and a multidimensional stochastic volatility and stochastic correlation model. Precise verification results for both examples are provided. Economic interpretations of the models and results complement the theoretical analysis.

Appendix

Proof

(Proposition 1) Consider an arbitrary but fixed point $(t,v,x,\mathbf{e}_{\mathbf{i}})\in [0,T]\times \mathbb {R}_{\ge 0}\times D^X\times \{{\mathbf{e}_{\mathbf{1}}},\ldots ,\mathbf{e}_{\mathbf{I}}\}$ and assume that the wealth of the investor at time t is v, further $X(t)=x$ and ${\mathcal {MC}}(t)=\mathbf{e}_{\mathbf{i}}$. The first statement follows directly from the fact that $\varPhi$ is a martingale and the terminal condition in (11). For the second one we apply the idea from Proposition 2.3 in Kallsen and Muhle-Karbe (2010b): First observe that substitution of the ansatz $\varPhi =\frac{v^{\delta }}{\delta }f$ in (11) leads to the following system of PDEs for f, for all $i\in \{1,\ldots ,I\}$:

$$\begin{aligned}&f_t+f\delta \big (r+{\bar{\pi }}'(\mu -r)\big )+f_x'\mu ^X+\frac{1}{2}Tr\big (f_{xx'}\varSigma ^X(\varSigma ^X)'\big )+f_x'\delta \varSigma ^X\rho \varSigma '{\bar{\pi }}\nonumber \\&\quad +\frac{1}{2}f\delta (\delta -1){\bar{\pi }}'\varSigma \varSigma '{\bar{\pi }}\Big |_{\big (t,x,\mathbf{e}_{\mathbf{i}}\big )}=-\sum _{z=1}^{I}q_{iz}f(t,x,\mathbf{e_z}),\quad f(T,x,\mathbf{e}_{\mathbf{i}})=1, \end{aligned}$$

(26)

where ${\bar{\pi }}(t)=\frac{1}{1-\delta }\Big \{\big (\varSigma \varSigma '\big )^{-1}\lambda +\big (\varSigma '\big )^{-1}\rho '(\varSigma ^X)'\frac{f_x}{f}\Big \}\Big |_{\big (t,x,\mathbf{e}_{\mathbf{i}}\big )}$. Now let $\pi$ be an arbitrary strategy and define $L(\tau ):=\big \{V^{{\bar{\pi }}}(\tau )\big \}^{\delta -1}V^{\pi }(\tau )f\big (\tau ,X(\tau ),{\mathcal {MC}}(\tau )\big )$, for all $\tau \in [t,T]$. Apply the semimartingale Representation (1) of the Markov chain and Itô’s rule to obtain the dynamics of process $\{L(\tau )\}_{\tau \in [t,T]}$:

$$\begin{aligned} \text {d}L&=\big \{V^{{\bar{\pi }}}\big \}^{\delta -1}V^{\pi }\Big [f_t+f_x'\mu ^X+\frac{1}{2}Tr\Big (f_{xx'}\varSigma ^X(\varSigma ^X)'\Big )+f(\delta -1)(r+{\bar{\pi }}'\lambda )\nonumber \\&\quad +f(r+\pi '\lambda ) +f_x'(\delta -1)\varSigma ^X\rho \varSigma '{\bar{\pi }}+f_x'\varSigma ^X\rho \varSigma '\pi +f(\delta -1)\pi '\varSigma \varSigma '{\bar{\pi }}\nonumber \\&\quad +\frac{1}{2}f(\delta -1)(\delta -2){\bar{\pi }}'\varSigma \varSigma '{\bar{\pi }} +\sum _{i=1}^{I}q_{MC(t)i}f\big (t,X(\tau ),\mathbf{e}_{\mathbf{i}}\big )\Big ]\text {d}\tau \nonumber \\&\quad +\big \{V^{{\bar{\pi }}}\big \}^{\delta -1}V^{\pi }f'_x\varSigma ^X\text {d}W^X+\big \{V^{{\bar{\pi }}}\big \}^{\delta -1}V^{\pi }f\big ((\delta -1){\bar{\pi }}'+\pi '\big )\varSigma \text {d}W^P\nonumber \\&\quad +\big \{V^{{\bar{\pi }}}\big \}^{\delta -1}V^{\pi }\big (f(t,X(t),{\mathbf{e}_{\mathbf{1}}}),\ldots ,f(t,X(t),\mathbf{e}_{\mathbf{I}})\big )\text {d}M\nonumber \\&\,{:=}\,\mu ^L\text {d}\tau +\sigma ^L_1\text {d}W^X+\sigma ^L_2\text {d}W^P+\sigma ^L_3\text {d}M, \end{aligned}$$

where we have missed out the arguments $(\tau ,X(\tau ),{\mathcal {MC}}(\tau ))$ for better readability. By a substitution of Eq. (26) and the definition of ${\bar{\pi }}$ it can be shown that $\mu ^L=0$. It follows that L is a local martingale, as all involved functions are continuous in X, $V^{\pi }$, $V^{{\bar{\pi }}}$, $\pi$ and ${\bar{\pi }}$ for all $\mathbf{e}_{\mathbf{i}}\in \mathcal {E}$. Furthermore, as f is assumed to be positive, process L is positive as well, so it is a supermartingale. Using this together with the concavity of the utility function $U(v)=\frac{v^{\delta }}{\delta }$, the martingale property of $\big \{\varPhi \big (t,V^{{\bar{\pi }}}(t),X(t)\big )\big \}_{t\in [0,T]}$, and $L(T)=1$ we obtain the following inequality:

$$\begin{aligned}&\mathbb {E}\big [U\big (V^{\pi }(T)\big )\big |\mathscr {F}_t\big ] \nonumber \\&\le \mathbb {E}\big [U\big (V^{{\bar{\pi }}}(T)\big )\big |\mathscr {F}_t\big ]+\mathbb {E}\big [U_v\big (V^{{\bar{\pi }}}(T)\big )\big (V^{\pi }(T)-V^{{\bar{\pi }}}(T)\big )\big |\mathscr {F}_t\big ]\nonumber \\&=\mathbb {E}\big [U\big (V^{{\bar{\pi }}}(T)\big )\big |\mathscr {F}_t\big ]+\mathbb {E}\big [L(T)\big |\mathscr {F}_t\big ]-\mathbb {E}\big [\big \{V^{{\bar{\pi }}}(T)\big \}^{\delta }\big |\mathscr {F}_t\big ]\nonumber \\&\le \mathbb {E}\big [U\big (V^{{\bar{\pi }}}(T)\big )\big |\mathscr {F}_t\big ]+L(t)-\delta \mathbb {E}\big [\varPhi \big (T,V^{{\bar{\pi }}}(T),X(T),{\mathcal {MC}}(T)\big )\big |\mathscr {F}_t\big ]\nonumber \\&=\mathbb {E}\big [U\big (V^{{\bar{\pi }}}(T)\big )\big |\mathscr {F}_t\big ]+v^{\delta }f\big (t,x,\mathbf{e}_{\mathbf{i}}\big )-\delta \varPhi \big (t,v,x,\mathbf{e}_{\mathbf{i}}\big )\nonumber \\&=\mathbb {E}\big [U\big (V^{{\bar{\pi }}}(T)\big )\big |\mathscr {F}_t\big ]. \end{aligned}$$

Our proof is complete. $\square$

Proof

(Proposition 3) We will show the verification result for the corresponding time-dependent model and apply Proposition 2. As in Sect. 3 we obtain the following candidate for the value function in the time-dependent model: $\varPhi ^m(t,v,x)=\frac{v^{\delta }}{\delta }\exp \Big \{\int _t^Tw(s,m(s))\text {d}s\Big \}\exp \{B(t)x\}.$ Obviously it is conitnuous and it can be shown by direct substitution that it solves the HJB equation for the time-dependent model (see 14). The candidate for the optimal portfolio is:

$$\begin{aligned} {\bar{\pi }}^m(t)=\frac{1}{(1-\delta )}\left( \begin{array}{c}\frac{\lambda _1(m(t))-\rho _{12}(m(t))\lambda _2(m(t))}{\chi A_2(T_1-t)(1-\rho _{12}^2(m(t)))}-\frac{B(t)}{A_2(T_1-t)}\\ \frac{\lambda _2(m(t))-\rho _{12}(m(t))\lambda _1(m(t))}{\nu (m(t))(1-\rho _{12}^2(m(t)))}\end{array}\right) . \end{aligned}$$

What remains to be demonstrated is that $\{\varPhi ^m\big (t,V^{m,{\bar{\pi }}^m}(t),X^m(t)\big )\}_{t\in [0,T]}$ is a martingale. Itô’s formula and Eq. (11) show that it has zero drift. So, we can show that it is a square integrable martingale by applying Th. III.6.4 from Jacod and Shiryaev (2003). To this aim we need to show:

$$\begin{aligned} \mathbb {E}\Big [\int _{0}^{T}\Big (\varPhi ^m_v(t)\varSigma ^V_1(t)-\varPhi ^m_x(t)\chi \Big )^2\text {d}t\Big ]<\infty \text { and } \mathbb {E}\Big [\int _{0}^{T}\Big (\varPhi ^m_v(t)\varSigma ^V_2(t)\Big )^2\text {d}t\Big ]<\infty , \end{aligned}$$

where we omitted the arguments of the functions to improve readability. Recalling the definition of $\varPhi ^m(t)$, $\varSigma ^V_1(t)$ and $\varSigma ^V_2(t)$, and the fact that B(t), $w(t,\mathbf{e}_{\mathbf{i}})$, $A_2(T_1-t)$ and ${\bar{\pi }}^m(t)$ are bounded on [0, T], it becomes clear that it suffices to show the following inequality:

$$\begin{aligned} A:=\mathbb {E}\Big [\int _{0}^{T}(V^{m,{\bar{\pi }}^m}(t))^{2\delta }\exp \big \{2B(t)X^m(t)\big \}\text {d}t\Big ]<\infty . \end{aligned}$$

(27)

Inserting the solution for process $V^{m,{\bar{\pi }}^m}$ and the relation $W^P_1=-W^X$ in (27) leads to:

$$\begin{aligned} A&=v_0^{2\delta }\mathbb {E}\left[ \int _{0}^{T}\exp \left\{ 2\delta \int _0^tX^m(s)+{\bar{\pi }}^m_1(s)\lambda _1\chi A_2(T_1-s)+{\bar{\pi }}^m_2(s)\lambda _2\nu \right. \right. \nonumber \\&\quad -\frac{1}{2}\big [\chi A_2(T_1-s){\bar{\pi }}^m_1(s)+\bar{\pi }^m_2(s)\rho _{12}\nu \big ]^2-\frac{1}{2}\big [{\bar{\pi }}^m_2(s)\sqrt{1-\rho ^2_{12}}\nu \big ]^2\text {d}s\nonumber \\&\quad +2\delta \int _0^t-{\bar{\pi }}^m_2(s)\rho _{12}\nu -\chi A_2(T_1-s){\bar{\pi }}^m_1(s)\text {d}W^X(s)\nonumber \\&\left. \left. \quad +2\delta \int _0^t{\bar{\pi }}^m_2(s)\nu \sqrt{1-\rho _{12}^2}\text {d}W^P_2(s)+2B(t)X^m(t)\right\} \text {d}t\right] . \end{aligned}$$

By inserting the following two equations for process $X^m$:

$$\begin{aligned} \int _0^tX^m(s)\text {d}s&=\frac{1}{\kappa }\big (X^m(0)-X^m(t)+\kappa \theta t+\chi W^X(t)\big )\\ X^m(t)&=X^m(0)e^{-\kappa t}+\theta (1-e^{-\kappa t})+\int _0^t\chi e^{\kappa (s-t)}\text {d}W^X(s), \end{aligned}$$

and exploiting the boundedness of ${\bar{\pi }}^m(t)$ and $A_2(T_1-t)$ on [0, T], we can show the boundedness of A. Our proof is complete. $\square$

Proof

(Proposition 4) As in the proof of Proposition 3 we will first find the solution in the corresponding time-dependent model and then we will apply Proposition 2 to obtain the verification result for the model with Markov switching.

Following the steps from Sect. 3 it is easily shown that the solution to the HJB equation (14) for the time-dependent model is given by:

$$\begin{aligned} \varPhi ^m(t,v,x)&=\frac{v^{\delta }}{\delta }\xi ^m(t)\exp \{B_1(t)x_1+B_2(t)x_2\}, \end{aligned}$$

(28)

where $\xi ^m(t)=\exp \Big \{\int _t^Tw(s,m(s))\text {d}s\Big \}$, for function w as in (22), and functions $B_j$, for $j=1,2$ satisfying the following ODEs:

$$\begin{aligned} \frac{\partial }{\partial t}B_j+B_j\left[ \frac{\delta }{1-\delta } \chi _j\rho _jc_j-\kappa _j\right] +\frac{1}{2}\big (B_j\big )^2\frac{\chi _j^2}{\vartheta _j}+\frac{1}{2}\frac{\delta }{1-\delta }c_j^2=0,\quad B_j(T)=0. \end{aligned}$$

(29)

Note that Eq. (29) corresponds to Eq. (15), where we have substituted the model specifications from (19). The conditions in (20) allow us to apply Prop. 5.1 from Kraft (2005) and conclude that functions $B_j$, for $j=1,2$ are as in Eq. (21). Note that $\varPhi ^m$ is continuous, piecewise sufficiently differentiable and satisfies the HJB PDE with the following portfolio strategy:

$$\begin{aligned} {\bar{\pi }}^m(t)=\frac{1}{1-\delta }\big (A'(m(t))\big )^{-1}\left\{ \left( \begin{array}{c}\frac{c_1\sqrt{X^{m}_1(t)}}{\sigma _1(X^m_1(t))}\\ \frac{c_2\sqrt{X^m_2(t)}}{\sigma _2(X^m_2(t))}\end{array}\right) +\left( \begin{array}{c}\frac{\rho _1\chi _1\sqrt{X^m_1(t)}B_1(t)}{\sigma _1(X^m_1(t))}\\ \frac{\rho _2\chi _2\sqrt{X^m_2(t)}B_2(t)}{\sigma _2(X^m_2(t))}\end{array}\right) \right\} . \end{aligned}$$

What remains to be proved is that $\{\varPhi ^m\big (t,V^{m,{{\bar{\pi }}}^m}(t),X^m(t)\big )\}_{t\in [0,T]}$ is a martingale. We will show this by applying Cor. 3.4 from Kallsen and Muhle-Karbe (2010a). First substitute the solution for $V^{m,{{\bar{\pi }}}^m}(t)$ as well as the dynamics for $B_j(t)X_j^m(t)$ together with the definition of $\mu$, $\sigma$, $\mu ^X$, $\varSigma ^X$ and ${\bar{\pi }}^m$ in Eq. (28) to obtain the following expression for $\varPhi ^m$:

$$\begin{aligned}&\varPhi ^m\big (t,V^{m,{{\bar{\pi }}}^m}(t),X^m(t)\big )\nonumber \\&\quad =\varPhi ^m\big (0,V^{m,{{\bar{\pi }}}^m}(0),X^m(0)\big )\nonumber \\&\qquad \cdot \exp \Bigg \{\int _0^t\sum _{j=1}^{2}X_j^m(s)\Bigg [\frac{\partial }{\partial t}B_j(s)-\kappa _jB_j(s)+\frac{\delta }{1-\delta }\big (c_j^2+B_j(s)\chi _j\rho _jc_j\big )\nonumber \\&\qquad -\frac{1}{2}\frac{\delta }{(1-\delta )^2}\big (c_j+B_j(s)\chi _j\rho _j\big )^2\Bigg ]\text {d}s+\sum _{j=1}^{2}\int _0^tB_j(s)\chi _j\sqrt{X^m_j(s)}\text {d}W^X_j(s)\nonumber \\&\qquad +\sum _{j=1}^{2}\int _0^t\frac{\delta }{1-\delta }\sqrt{X_j^{m}(s)}\big (c_j+B_j(s)\chi _j\rho _j\big )\text {d}W^P_j(s)\Bigg \}\nonumber \\&\quad =:\varPhi ^m\big (0,V^{m,{{\bar{\pi }}}^m}(0),X^m(0)\big )\exp \Bigg \{\int _0^t\mu ^L\big (s,X^{m}(s)\big )\text {d}s\nonumber \\&\qquad +\sum _{j=1}^{2}\int _0^t\varSigma ^L_j\big (s,X^{m}(s)\big )\text {d}W^X_j+\sum _{j=1}^{2}\int _0^t\varSigma ^L_{(2+j)}\big (s,X^{m}(s)\big )\text {d}W^P_j\Bigg \}\nonumber \\&\quad =:\varPhi ^m\big (0,V^{m,{{\bar{\pi }}}^m}(0),X^m(0)\big )\exp \big \{L(t)\big \}. \end{aligned}$$

Now consider process $Z:=(X^m_1,X^m_2,L)$ and calculate its semimartingale characteristics $\mu ^Z(t,x)$ and $\varGamma ^Z(t,x)$:

$$\begin{aligned} \mu ^Z_j(t,x)&=\mu ^X_j(x,m(t)),\quad \mu ^Z_3(t,x)=\mu ^L(t,x)\\ \varGamma ^Z_{jj}(t,x)&=\chi _j^2x_j,\quad \varGamma ^Z_{12}(t,x)=\varGamma ^Z_{21}(t,x)=0\\ \varGamma ^Z_{33}(t,x)&=\sum _{j=1}^{2}x_j\Big [\big (B_j(t)\big )^2\chi ^2_j+\frac{\delta ^2}{(1-\delta )^2}\big (c_j+B_j\chi _j\rho _j\big )^2\nonumber \\&\quad +2\frac{\delta }{1-\delta }B_j(s)\chi _j\rho _j\big (c_j+B_j\chi _j\rho _j\big )\Big ]\\ \varGamma ^Z_{j3}(t,x)&=\varGamma ^Z_{3j}(t,x)=\chi _jx_j\Big [B_j(t)\chi _j+\frac{\delta }{1-\delta }\rho _j\big (c_j+B_j(t)\chi _j\rho _j\big )\Big ], \end{aligned}$$

for $j=1,2$. Finally, compute:

$$\begin{aligned} \mu ^Z_3+\frac{1}{2}\varGamma ^Z_{33}&=\sum _{j=1}^{2}x_j\Bigg [\frac{\partial }{\partial t}B_j+B_j\big [\frac{\delta }{1-\delta }\chi _j\rho _jc_j-\kappa _j\big ]\nonumber \\&\quad +\frac{1}{2}\big (B_j\big )^2\frac{\chi _j^2}{\vartheta _j}+\frac{1}{2}\frac{\delta }{1-\delta }c_j^2\Bigg ]=0. \end{aligned}$$

Thus, we can apply Cor. 3.4 from Kallsen and Muhle-Karbe (2010a) and conclude that $\big \{\exp \{L(t)\}\big \}_{t\in [0,T]}$ and hence also $\big \{\varPhi ^m\big (t,V^{m,{{\bar{\pi }}}^m}(t),X^m(t)\big )\}\big \}_{t\in [0,T]}$ are martingales. So, by Remark 2 $\varPhi ^m$ is indeed the value function in the time-dependent model and ${\bar{\pi }}^m$ is the optimal investment strategy. Application of Proposition 2 completes the proof. $\square$