Dynamic portfolio strategies under a fully correlated jump-diffusion process

Abstract

This paper presents the first continuous-time model to feature a flexible dependence structure among jump intensity, stock variance, and stock returns. In particular, it addresses a gap in the financial portfolio optimization literature concerning the non-trivial correlation between stock return variance and the intensity of price jumps. The model permits closed-form representations for the optimal strategy and value functions in an expected utility theory setting. It also produces analytical expressions for the value function associated with relevant suboptimal strategies. Such an analytical setting allows for the first wealth-equivalent utility loss (WEL) analysis of the pitfalls of ignoring the aforementioned dependence. The model and results can be easily extended to the pair intensity-covariance in multi-assets. The WEL analysis is carried out for three different suboptimal classes: tailor-made incomplete markets, misspecifications in the parameters of the model, and time-independent (myopic) strategies. For the numerical section, we focus on the correlation between jump intensity and stock variance, which is assumed to be either zero or one in the existing literature. We demonstrate that simplistic assumptions like perfect dependence or independence could lead to wealth-equivalent losses of up to 61%. Similarly, a failure to hedge these variances and intensity drivers could cause losses of up to 95% (in particular, up to 60% due to the factors driving the dependence).

Appendix: Proofs

Proof of Lemma 2

Let $g^{(j)}$ be in ${{\,\mathrm{C}\,}}^{1,2}(\mathbb {R}_{+}\times \mathbb {R}_{+}\times \mathbb {R}^{d+1}_{+}\times \mathbb {R}_{+})$, with $j=1, \ldots ,2d+3$. Using (8), it can be verified that the price of the jth stock option satisfies the following identity

$$\begin{aligned} \mathcal {L}g^{(j)}&=rg^{(j)}+\eta _{1}\mathbb {b}_{1}^{2} \mathbb {v}^{{{\,\mathrm{T}\,}}}sg^{(j)}_{s}+[\eta _{1}\mathbb {a}_{1} +\mathbb {a}_{2}]{{\,\mathrm{diag}\,}}[\mathbb {v}]\left[ {{\,\mathrm{D}\,}}^{1}_{\mathbb {v}}g^{(j)} \right] ^{{{\,\mathrm{T}\,}}}\nonumber \\&\quad +\eta _{3}\sigma _{2}v^{(d+2)}g^{(j)}_{v^{(d+2)}} +\left[ \lambda -\lambda ^{Q}\right] \left[ \mathbb {b}_{2}\mathbb {v}^{{{\,\mathrm{T}\,}}} +av^{(d+1)}\right] \varDelta g^{(j)}. \end{aligned}$$

(34)

The operator $\mathcal {L}$ is the infinitesimal generator of the process $\left( S_{t},\mathbb {V}^{(1)}_{t},V^{(2)}_{t}\right)$ which is given by

$$\begin{aligned} \mathcal {L} f= & {} \partial _{t}f+s\left[ r+\eta _{1}\mathbb {b}_{1}^{2} \mathbb {v}^{{{\,\mathrm{T}\,}}}-\mu \lambda ^{Q}\left[ \mathbb {b}_{2}\mathbb {v}^{{{\,\mathrm{T}\,}}} +av^{(d+2)}\right] \right] f_{s}+\mathbb {k}_{1}{{\,\mathrm{diag}\,}}\left[ \bar{\vartheta }_{1} -\mathbb {v}\right] {{\,\mathrm{D}\,}}^{1}_{\mathbb {v}}f^{{{\,\mathrm{T}\,}}}\nonumber \\&+\kappa _{2}\left[ \vartheta _{2}-\mathbb {v}\right] f_{v^{(d+2)}} +{{\,\mathrm{tr}\,}}\left[ \tilde{a}{{\,\mathrm{D}\,}}^{2}f\right] +\lambda \left[ \mathbb {b}_{2} \mathbb {v}^{{{\,\mathrm{T}\,}}}+av^{(d+2)}\right] \varDelta f, \end{aligned}$$

(35)

where $f\in {{\,\mathrm{C}\,}}^{1,2}(\mathbb {R}_{+}\times \mathbb {R}_{+}\times \mathbb {R}^{d+1}_{+}\times \mathbb {R}_{+})$, ${{\,\mathrm{D}\,}}^{2}$ is the matrix operator of the second derivatives with respect $\left( s,\mathbb {v},v^{(d+2)}\right)$, $\mathbb {k}_{1}:=(\kappa _{1,1},\ldots ,\kappa _{1,d+1})$, $\bar{\vartheta }_{1}:=(\vartheta _{1,1},\ldots ,\vartheta _{1,d+1})$ and $\tilde{a}:=\dfrac{1}{2}\tilde{\sigma }\tilde{\sigma }^{{{\,\mathrm{T}\,}}}$, with $\tilde{\sigma }=(\tilde{\sigma }_{i,j})$ a matrix function of size $(d+3)\times (2d+3)$ whose components are given by

$$\begin{aligned} \tilde{\sigma }_{ij}(s,\mathbb {v},v^{(d+2)})= {\left\{ \begin{array}{ll} b_{1,j}s\sqrt{v^{(j)}}, &{}\text {if}\ i=1,\ j=1,\ldots ,d+1,\\ \sigma _{1,i-1}\rho _{1,i-1}\sqrt{v^{(i-1)}}, &{}\text {if}\ i=2,\ldots ,d+2,\ j=i-1,\\ \sigma _{1,i-1}[1-\rho _{1,i-1}^{2}]^{\frac{1}{2}}\sqrt{v^{(i-1)}}, &{}\text {if}\ i=2,\ldots ,d+2,\ j=d+i,\\ \sigma _{2}\sqrt{v^{(d+2)}}, &{}\text {if}\ i=d+3,\ j=2d+3,\\ 0,&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

(36)

Applying Itô formula in $g^{(j)}(t,S_{t},\mathbb {V}^{(1)}_{t},V^{(2)}_{t})$ and using (34)–(36), we see that

$$\begin{aligned} \mathrm {d} g^{(j)}&=\mathcal {L} g^{(j)}\mathrm {d} t+{{\,\mathrm{D}\,}}^{1}g^{(j)}\tilde{\sigma }\mathrm {d} B_{t}^{{{\,\mathrm{T}\,}}}+\varDelta g^{(j)}\left[ \mathrm {d} N_{t}-\lambda Z_{t}\mathrm {d} t\right] \\&=rg^{(j)}\mathrm {d} t+\left[ S_{t}g_{s}\mathbb {b}_{1}+\dfrac{\mathbb {a}_{1}}{\mathbb {b}_{1}} {{\,\mathrm{diag}\,}}\left[ {{\,\mathrm{D}\,}}_{\mathbb {v}}g^{(j)}\right] \right] \nonumber \\&\quad \times \left[ \eta _{1}\left[ \mathbb {b}_{1}{{\,\mathrm{diag}\,}}\left[ \mathbb {V}^{(1)}_{t}\right] \right] ^{{{\,\mathrm{T}\,}}}\mathrm {d} t+\sqrt{{{\,\mathrm{diag}\,}}{\mathbb {V}^{(1)}_{t}}}\mathrm {d} \mathbb {B}^{(1){{\,\mathrm{T}\,}}}_{t}\right] \\&\quad +\dfrac{\mathbb {a}_{2}}{\bar{\eta }_{2}}{{\,\mathrm{diag}\,}}\left[ {{\,\mathrm{D}\,}}_{\mathbb {v}}g^{(j)}\right] \left[ \left[ \bar{\eta }_{2} {{\,\mathrm{diag}\,}}\left[ \mathbb {V}^{(1)}_{t}\right] \right] ^{{{\,\mathrm{T}\,}}}\mathrm {d} t+ \sqrt{{{\,\mathrm{diag}\,}}{\mathbb {V}^{(1)}_{t}}}\mathrm {d} \mathbb {B}^{(2){{\,\mathrm{T}\,}}}_{t}\right] \\&\quad +\sigma _{2}g^{(j)}_{v^{(d+2)}}\left[ \eta _{3}V^{(2)}_{t}\mathrm {d} t+ \sqrt{V^{(2)}_{t}}\mathrm {d} B^{(3)}_{t}\right] \nonumber \\&\quad +\left[ \left[ \lambda -\lambda ^{Q}\right] Z_{t}\mathrm {d} t+\mathrm {d} N_{t}-\lambda Z_{t}\mathrm {d} t\right] \varDelta g^{(j)}, \end{aligned}$$

where $B:=\left( \mathbb {B}^{(1)},\mathbb {B}^{(2)},B^{(3)}\right)$ and ${{\,\mathrm{D}\,}}^{1}$ is the gradient with respect to $\left( s,\mathbb {v},v^{(d+2)}\right)$. $\square$

Proof of Proposition 5

Suppose that a solution f to the HJB equation (14) has the form of Eq. (16), i.e., $f=J$. Take first and second derivatives to $f(t,w,\mathbb {v},v^{(d+2)})$. Now, applying these in (14) and dividing by $\gamma f$, the following identity can be verified

$$\begin{aligned}&\max _{\tilde{\theta }}\biggr \{-\left[ h^{\prime }+H^{(1)\prime }\mathbb {v}^{{{\,\mathrm{T}\,}}}+H^{(2)\prime }v_{(d+2)}\right] +\mathbb {k}_{1}{{\,\mathrm{diag}\,}}\left[ \bar{\vartheta }_{1}-\mathbb {v}\right] \left[ H^{(1)}\right] ^{{{\,\mathrm{T}\,}}}+\kappa _{2}\left[ \vartheta _{2}-v^{(d+2)} \right] H^{(2)}\nonumber \\&\quad +\dfrac{\lambda \left[ \mathbb {b}_{2}\mathbb {v}^{{{\,\mathrm{T}\,}}} +av^{(d+2)}\right] }{\gamma }\left[ \left[ 1+\mu \theta ^{N}\right] ^{1-\gamma } \exp \left[ \gamma \left[ H^{(1)}\bar{\mu }_{1}^{{{\,\mathrm{T}\,}}}+\mu _{2}H^{(2)}\right] \right] -1\right] \nonumber \\&\quad +\dfrac{1-\gamma }{\gamma }\Big [r-\mu \lambda ^{Q}\left[ \mathbb {b}_{2} \mathbb {v}^{{{\,\mathrm{T}\,}}}+av^{(d+2)}\right] \theta ^{N}+\eta _{1} \bar{\theta }^{(1)}\left[ \mathbb {b}_{1}{{\,\mathrm{diag}\,}}\left[ \mathbb {v}\right] \right] ^{{{\,\mathrm{T}\,}}}+\bar{\theta }^{(2)}[\bar{\eta }_{2} {{\,\mathrm{diag}\,}}[\mathbb {v}]]^{{{\,\mathrm{T}\,}}}\nonumber \\&\quad +\eta _{3}v^{(d+2)}\theta ^{(3)}\Big ]+\dfrac{\gamma }{2} \left[ \bar{\sigma }_{1}^{2}{{\,\mathrm{diag}\,}}[\mathbb {v}]\left[ [H^{(1)}]^{2} \right] ^{{{\,\mathrm{T}\,}}}+\sigma _{2}^{2}v^{(d+2)}\left[ H^{(2)}\right] ^{2}\right] \nonumber \\&\quad -\dfrac{[1-\gamma ]}{2}[\bar{\theta }^{(1)}{{\,\mathrm{diag}\,}}[\mathbb {v}] \bar{\theta }^{(1){{\,\mathrm{T}\,}}}+\bar{\theta }^{(2)}{{\,\mathrm{diag}\,}}[\mathbb {v}] \bar{\theta }^{(2){{\,\mathrm{T}\,}}}+[\theta ^{(3)}]^{2}v^{(d+2)}] \nonumber \\&\quad +[1-\gamma ]\biggr [\biggr [\dfrac{\mathbb {a}_{1}}{\mathbb {b}_{1}}{{\,\mathrm{diag}\,}}[\bar{\theta }^{(1)}]+\dfrac{\mathbb {a}_{2}}{\bar{\eta }_{2}}{{\,\mathrm{diag}\,}}[\bar{\theta }^{(2)}]\biggl ] {{\,\mathrm{diag}\,}}[\mathbb {v}]\left[ H^{(1)}\right] ^{{{\,\mathrm{T}\,}}}+\sigma _{2}v^{(d+2)} \theta ^{(3)}H^{(2)}\biggl ]\biggl \}=0. \end{aligned}$$

(37)

It is easy to realize that the solution $\tilde{\theta }^{*}$ to the maximizing problem in (37) is determined by (17). Substituting (17) in (37) and grouping terms multiplied by $\mathbb {v}$ and $v^{(d+2)}$, respectively, we have that

$$\begin{aligned}&-h^{\prime }+\dfrac{1-\gamma }{\gamma }r+\mathbb {k}_{1}{{\,\mathrm{diag}\,}}\left[ \bar{\vartheta }_{1}\right] \left[ H^{(1)}\right] ^{{{\,\mathrm{T}\,}}}+\kappa _{2}\vartheta _{2}H^{(2)}+\frac{\gamma }{2}\bar{\sigma }_{1}^{2}{{\,\mathrm{diag}\,}}\left[ \left[ H^{(1)}\right] ^{2}\right] \nonumber \\&\quad +\biggr [-H^{(1)\prime }+\biggr [[1-\gamma ]\biggr [\dfrac{\mathbb {a}_{1}}{\mathbb {b}_{1}} {{\,\mathrm{diag}\,}}\left[ \bar{\theta }^{(1)*}\right] +\dfrac{\mathbb {a}_{2}}{\bar{\eta }_{2}}{{\,\mathrm{diag}\,}}\left[ \bar{\theta }^{(2)*}\right] \biggl ] -\mathbb {k}_{1}\biggl ]{{\,\mathrm{diag}\,}}\left[ H^{(1)}\right] \nonumber \\&\quad +\dfrac{1}{\gamma }\left[ \lambda \left[ \left[ 1+\mu \theta ^{N*} \right] ^{1-\gamma }\exp \left[ \gamma \left[ H^{(1)}\bar{\mu }_{1}^{{{\,\mathrm{T}\,}}} +\mu _{2}H^{(2)}\right] \right] -1\right] -[1-\gamma ]\mu \lambda ^{Q} \theta ^{N*}\right] \mathbb {b}_{2}\nonumber \\&\quad +\frac{1-\gamma }{\gamma }\left[ \eta _{1}\mathbb {b}_{1}{{\,\mathrm{diag}\,}}\left[ \bar{\theta }^{(1)*}\right] +\bar{\eta }_{2}{{\,\mathrm{diag}\,}}\left[ \bar{\theta }^{(2)*}\right] \right] -\dfrac{[1-\gamma ]}{2} \left[ \left[ \bar{\theta }^{(1)*}\right] ^{2}+\left[ \bar{\theta }^{(2)*} \right] ^{2}\right] \biggl ]\mathbb {v}^{{{\,\mathrm{T}\,}}} \nonumber \\&\quad +v^{(d+2)}\biggr [-H^{(2)\prime }+\left[ [1-\gamma ]\sigma _{2} \theta ^{(3)*}-\kappa _{2}\right] H^{(2)}+\dfrac{\gamma \sigma _{2}^{2}}{2} \left[ H^{(2)}\right] ^{2}\nonumber \\&\quad +\dfrac{a}{\gamma }\left[ \lambda \left[ \left[ 1+\mu \theta ^{N*} \right] ^{1-\gamma }\exp \left[ \gamma \left[ H^{(1)}\bar{\mu }_{1}^{{{\,\mathrm{T}\,}}} +\mu _{2}H^{(2)}\right] \right] -1\right] -[1-\gamma ]\mu \lambda ^{Q} \theta ^{N*}\right] \nonumber \\&\quad +\dfrac{1-\gamma }{\gamma }\eta _{3}\theta ^{(3)*}-\dfrac{[1-\gamma ]}{2}\left[ \theta ^{(3)*}\right] ^{2} \biggl ]=0. \end{aligned}$$

(38)

Notice that Eq. (38) is equivalent to solving the generalized Riccati system in (18). By Theorem 2.14 of Keller-Ressel and Mayerhofer (2015), it follows that (18) has unique solutions which are given by h, $H^{(1)}=(H^{(1,1)},\ldots ,H^{(1,d+1)})$ and $H^{(2)}$. $\square$

Proof of Proposition 6

Let $f^{\varPi }$ be as the right side of (20). We shall prove that $f^{\varPi }$ satisfies (19). To see that $J^{\varPi }=f^{\varPi }$, where $J^{\varPi }$ is given by (13), we remit the reader to Escobar et al. (2015). Notice that (19) is equivalent to

$$\begin{aligned}&-h_{\varPi }^{\prime }+\dfrac{1-\gamma }{\gamma }r+\mathbb {k}_{1} {{\,\mathrm{diag}\,}}[\bar{\vartheta }_{1}]H_{\varPi }^{(1){{\,\mathrm{T}\,}}}+\kappa _{2}\vartheta _{2} H_{\varPi }^{(2)} \\&\quad +\biggr [-H_{\varPi }^{(1)\prime }+\biggr [[1-\gamma ] \biggr [\dfrac{\mathbb {a}_{1}}{\mathbb {b}_{1}}{{\,\mathrm{diag}\,}}[\bar{\theta }^{(1)}] +\dfrac{\mathbb {a}_{2}}{\bar{\eta }_{2}}{{\,\mathrm{diag}\,}}[\bar{\theta }^{(2)}]\biggl ] -\mathbb {k}_{1}\biggl ]{{\,\mathrm{diag}\,}}[H_{\varPi }^{(1)}]\\&\quad +\frac{\gamma }{2}\bar{\sigma }_{1}^{2}{{\,\mathrm{diag}\,}}[[H_{\varPi }^{(1)}]^{2}] +\dfrac{1}{\gamma }[\lambda [[1+\mu \theta ^{N}]^{1-\gamma } \exp [\gamma [H_{\varPi }^{(1)}\bar{\mu }_{1}^{{{\,\mathrm{T}\,}}}+\mu _{2}H_{\varPi }^{(2)}]]-1]\\&\quad -[1-\gamma ]\mu \lambda ^{Q}\theta ^{N}]\mathbb {b}_{2}\\&\quad +\frac{1-\gamma }{\gamma }[\eta _{1}\mathbb {b}_{1} {{\,\mathrm{diag}\,}}[\bar{\theta }^{(1)}]+\bar{\eta }_{2}{{\,\mathrm{diag}\,}}[\bar{\theta }^{(2)}]] -\dfrac{[1-\gamma ]}{2}[[\bar{\theta }^{(1)}]^{2}+[\bar{\theta }^{(2)}]^{2}] \biggl ]\mathbb {v}^{{{\,\mathrm{T}\,}}}\\&\quad +v^{(d+2)}\biggr [-H_{\varPi }^{(2)\prime }+[[1-\gamma ]\sigma _{2} \theta ^{(3)}-\kappa _{2}]H_{\varPi }^{(2)}+\dfrac{\gamma \sigma _{2}^{2}}{2} [H_{\varPi }^{(2)}]^{2}\\&\quad +\dfrac{a}{\gamma }[\lambda [[1+\mu \theta ^{N}]^{1-\gamma } \exp [\gamma [H_{\varPi }^{(1)}\bar{\mu }_{1}^{{{\,\mathrm{T}\,}}}+\mu _{2}H_{\varPi }^{(2)}]] -1]-[1-\gamma ]\mu \lambda ^{Q}\theta ^{N}]\\&\quad +\dfrac{1-\gamma }{\gamma }\eta _{3}\theta ^{(3)}-\dfrac{[1 -\gamma ]}{2}[\theta ^{(3)}]^{2} \biggl ]=0. \end{aligned}$$

From here we see that $f^{\varPi }$ is a solution of (19) if $h_{\varPi },H^{(1)}_{\varPi }, H^{(2)}_{\varPi }$ are solutions to the Riccati system given in (21). $\square$