Optimal demand in a mispriced asymmetric Carr–Geman–Madan–Yor (CGMY) economy

Abstract

We employ a simple numerical scheme to compute optimal portfolios and utilities of informed and uninformed investors in a mispriced Carr–Geman–Madan–Yor (CGMY) Lévy market under information asymmetry using instantaneous centralized moments of returns (ICMR). We also investigate the impact on investors’ demand for stocks and indices at different levels of asymmetric information, mispricing, investment horizon, jump intensity, and volatility. Our simulations not only confirm that uninformed expected demand falls as information asymmetry increases but also offer strong evidence that informed expected demand behaves in a similar manner. In particular, expected demand of informed investors falls whenever information asymmetry exceeds 50%. The investor that demands more of the risky asset maintains that position over the entire investment horizon at each level of mispricing and information asymmetry. The absolute difference in expected demand between the uninformed and informed investors increases with the investment horizon, but decreases with the level of information asymmetry.

Appendices

Appendix A: Optimal portfolios under power utility

We have a similar structure for the optimal portfolio of investors under CARA or power utility, having a constant relative risk aversion, $\rho$ or equivalently, having risk tolerance, $\frac{1}{\rho }$.

$$\begin{aligned} U(x)=\frac{x^{1-\rho }}{1-\rho },\quad \rho >0, \ \rho \ne 1. \end{aligned}$$

(19)

In this case, the optimal portfolio for each investor is

$$\begin{aligned} \pi ^{i,\,*}= \arg \max _{\pi } \mathbf {E}U({{V}_T^{i,\,\pi } }), \end{aligned}$$

(20)

where the wealth of the ith investor is ${V}_t^{i,\,\pi }$ and the budget constraint is given by Eq. (8):

$$\begin{aligned} \frac{dV^i_t}{V^i_{t-}} =(\pi _t^{i} \sigma _t \theta ^i_t + r_t)dt + \pi _t^{i} \sigma _t dB^i_t +\int _{\mathbb {R}} \pi _t^{i}(e^{x}-1) N(dt, dx). \end{aligned}$$

Using the Hamilton–Jacobi–Bellman (HJB) methodology, it can be shown that the optimal portfolio for each investor is given by the implicit equation

$$\begin{aligned} \rho \sigma ^2_t \pi _t^{i,\,*}= & {} b^{i}_t - r_t + \int _{\mathbb {R}} \left[ \left( 1 + \pi _t^{i,\,*}(e^{x}-1)\right) ^{-\rho } - 1\right] (e^{x}-1) v(dx) \nonumber \\= & {} \mu ^{i}_t - r_t + \int _{\mathbb {R}} \left( 1 + \pi _t^{i,\,*}(e^{x}-1)\right) ^{-\rho }(e^{x}-1) v(dx), \end{aligned}$$

(21)

where $b^{i}_t= \mu ^{i}_t + \int _{\mathbb {R}}(e^{x}-1)v(dx)$ is total returns due to the diffusion and jump components. In other words,

$$\begin{aligned} \pi _t^{i,\,*}= \frac{\theta ^{i}_t}{\rho \sigma _t}+ \frac{G'(\pi _t^{i,\,*})}{ \rho \sigma ^2_t} = \frac{1}{\rho \sigma ^2_t} \left[ \mu ^{i}_t - r_t + G'(\pi _t^{i,\,*}) \right] , \end{aligned}$$

(22)

where

$$\begin{aligned} G(\pi )= & {} \int _{\mathbb {R}} U( 1 + \pi (e^{x}-1))v(dx) \end{aligned}$$

(23)

$$\begin{aligned}= & {} \frac{1}{1-\rho }\int _{\mathbb {R}} \left[ 1 + \pi (e^{x}-1)\right] ^{1-\rho } v(dx), \end{aligned}$$

(24)

and, as usual, $\mu _t^i = \mu _t+ \upsilon ^i_t\sigma _t$, $\theta ^i_t = \frac{\mu _t^i-r_t}{\sigma _t}$ are Sharpe ratios, and $1 + \pi (e^{x}-1)$ is the return on the portfolio when a log-jump of x occurs in the the stock price.

Remark 7

Note that if there is no jump, we obtain the random Merton optimal portfolio under asymmetric information

$$\begin{aligned} \pi _t^{i,\,*}= \frac{\theta ^{i}_t}{\rho \sigma ^2_t} = \frac{\mu _t^i-r_t}{\rho \sigma ^2_t}= \frac{\mu _t-r_t}{\rho \sigma ^2_t} + \frac{\upsilon ^i_t }{\rho \sigma _t} . \end{aligned}$$

Moreover, if in addition, there is no asymmetric information, we get the usual Merton optimal portfolio

$$\begin{aligned} \pi _t^{Mer}= \frac{\mu _t-r_t}{\rho \sigma ^2_t}. \end{aligned}$$

Observe that under asymmetric information, the Merton optimal is strictly random with expected value being the deterministic symmetric Merton portfolio.

Remark 8

Observe from Eq. (22), that we recover the formula for the optimal portfolio in Theorem 1 when utility is logarithmic, with a constant relative risk aversion of 1. This corresponds to a limiting power utility

$$\begin{aligned} U(x)=\frac{x^{1-\rho }-1}{1-\rho },\quad \rho >0, \ \rho \ne 1. \end{aligned}$$

(25)

as $\rho \rightarrow 1$.

Remark 9

The inclusion of CARA utility, $- \,\frac{1}{\rho } e^{\rho x}$, is omitted as its analysis follows similarly.

Appendix B: Proofs of lemmas, propositions and theorems

Proof of Lemma 1

$\int _{0+}^{\infty }e^{-a\,x}\frac{dx}{x^{1+y}}=\int _{0+}^{\infty } x^{-y}\,e^{-a\,x}\frac{dx}{x}=a^y \int _{0+}^{\infty } (a\,x)^{-y}\,e^{-a\,x}\frac{dx}{x}=a^y\,\Gamma (-y)$, by (12). $\square$

Proof of Lemma 2

Applying Lemma 3, we get

$$\begin{aligned} \frac{K(s)}{C}= & {} \int _{-\infty }^{0-}(e^{sx}-1)e^{G\,x}\frac{dx}{(-x)^{1+y}} + \int _{0+}^{\infty }(e^{sx}-1)e^{-M\,x}\frac{dx}{x^{1+y}}\\= & {} \int _{0+}^{\infty }(e^{-sx}-1)e^{-G\,x}\frac{dx}{x^{1+y}} + \int _{0+}^{\infty }(e^{sx}-1)e^{-M\,x}\frac{dx}{x^{1+y}}\\= & {} \int _{0+}^{\infty }(e^{-(G+s)\,x}-e^{-G\,x})\frac{dx}{x^{1+y}} + \int _{0+}^{\infty }(e^{-(M-s)\,x}-e^{-M\,x})\frac{dx}{x^{1+y}}\\= & {} \Gamma (-Y)( (G+s)^Y-G^Y) + \Gamma (-Y)( (M-s)^Y - M^Y), \end{aligned}$$

whence $K(s)=C\,\Gamma (-Y)[(M-s)^Y-M^Y + (G+s)^Y-G^Y].$$\square$

Proof of Lemma 3

In each case, we assume that $M>k$, for $k=1, 2, 3,4$.

• (I) By Corollary 1, $M_1=C\,\Gamma (-Y)[\phi (M-1)- \phi (M) + \phi (G+1)- \phi (G)]=C\,\Gamma (-Y)[\Delta \phi (G)- \Delta \phi (M-1)]$.

• (II) If $M>2$, then $K_2$ exists and $M_2=K_2 - 2K_1$. Thus

  $$\begin{aligned} M_2= & {} C\,\Gamma (-Y)[\phi (M-2)- \phi (M) + \phi (G+2)- \phi (G)\\&{} -2(\phi (M-1)- \phi (M) + \phi (G+1)- \phi (G) )]\\= & {} C\,\Gamma (-Y)[\phi (G+2)-2\phi (G+1)-\phi (G)+(\phi (M)-2\phi (M-1)\\&-\phi (M-2)]\\= & {} C\,\Gamma (-Y)[\Delta \phi (G+1)-\Delta \phi (G)+ \Delta \phi (M-1) -\Delta \phi (M-2)]\\= & {} C\,\Gamma (-Y)[\Delta ^2 \phi (G) + \Delta ^2 \phi (M-2)]\\= & {} C\,\Gamma (-Y)[\Delta ^k \phi (G) + (-1)^k\,\Delta ^2 \phi (M-k)],\,\,k=2. \end{aligned}$$

• (III) If $M>3$, then $K_3$ exits and $M_3 = K_3 -3K_2 + 3K_1$. Thus

  $$\begin{aligned} M_3= & {} C\,\Gamma (-Y)[\phi (G+3)-3\phi (G+2)+3\phi (G+1)-\phi (G)\\&{} -( \phi (M)-3\phi (M-1)+3\phi (M-2)-\phi (M-3)]\\= & {} C\,\Gamma (-Y)[\Delta ^3 \phi (G) -\Delta ^3 \phi (M-3)]\\= & {} C\,\Gamma (-Y)[\Delta ^3 \phi (G)+ (-1)^3\Delta ^3 \phi (M-3)]\\= & {} C\,\Gamma (-Y)[\Delta ^k \phi (G)+ (-1)^k\Delta ^k \phi (M-k)],\,\,k=3. \end{aligned}$$

• (IV) If $M>4$, then $K_4$ exists and $M_4 = K_4 +6K_2 -4K_3 - 4K_1$. Thus

  $$\begin{aligned} M_4= & {} C\,\Gamma (-Y)[(\phi (G+4)-4\phi (G+3)+6\phi (G+2)-4\phi (G+1)+\phi (G)\\&{} + (\phi (M)-4\phi (M-1)+6\phi (M-2)-4\phi (M-3)+\phi (M-4))]\\= & {} C\,\Gamma (-Y)[\Delta ^4 \phi (G) + \Delta ^4 \phi (M-4)]\\= & {} C\,\Gamma (-Y)[\Delta ^k \phi (G) + \Delta ^k \phi (M-k)], \,\;k=4. \end{aligned}$$

$\square$

Proof of Proposition 1

The proof for the optimal portfolio and expected logarithmic utility follow directly from Theorem 1 by replacing $\pi _t$ by $\pi _t^{(k, i)}$.

To prove the accuracy of the approximation, let $s=\pi ^{(k)}$. By suppressing superscripts and subscripts in Theorem 1, we get $\pi = \frac{\theta }{\sigma }+ \frac{G'(\pi )}{\sigma ^2}$,    $\pi \in [0,\,1]$ and $s = \frac{\theta }{\sigma }+ \frac{G_k'(s)}{\sigma ^2}$,    $s \in [0,\,1]$, where $G_k'(s) = G'(s) - R_k'(s)$ and $R_k(\pi )=\frac{1}{(k+1)!} \pi ^{k+1} G^{k+1}(\psi _{\pi })$ is the remainder/error term in the approximation of $G(\pi )$ by $G_k(\pi )$. It follows from above that

$$\begin{aligned} \pi - s = \frac{G'(s) - G_k'(s)}{\sigma ^2}=\frac{G'(\pi ) - G'(s) + R'_k(s) }{\sigma ^2} = \frac{G''(\alpha _{\pi })(\pi -s) + R'_k(s) }{\sigma ^2} \end{aligned}$$

for some $\alpha _{\pi }$ between $\pi$ and s. Thus $(\pi - s)( 1- \frac{G''(\alpha _{\pi })}{\sigma ^2}) = \frac{R'_k(s) }{\sigma ^2}$ and $(\pi - s)\left( 1 + \frac{|G''(\alpha _{\pi })|}{\sigma ^2} \right) = \frac{R'_k(s) }{\sigma ^2}$ since $G '' (\pi )$ is always negative. Therefore

$$\begin{aligned} |\pi - s| \le \frac{|R'_k(s) |}{\sigma ^2}. \end{aligned}$$

Since $R_k(s)=\frac{1}{(k+1)!} s^{k+1} G^{k+1}(\psi _{s})$ where $0< \psi _{s} < s$, then $0 \le \psi '_{s} \le 1$, and

$$\begin{aligned} |R'_k(s)|= & {} \left| \frac{1}{k!} s^{k} G^{k+1}(\psi _{s}) + \frac{1}{(k+1)!} s^{k+1} G^{k+2}(\psi _{s})\psi '_{s}\right| \\\le & {} \frac{s^{k}}{k!} \left( | G^{k+1}(\psi _{s})| + \frac{s|G^{k+2}(\psi _{s}) |}{(k+1)} \right) \le \frac{s^{k}}{k!} \left( L_k + \frac{s\,L_k}{(k+1)} \right) , \end{aligned}$$

where $L_k= \max _{s \in [0,1] }\{| G^{k+1}(s)|, | G^{k+2}(s)|\}$. Thus,

$$\begin{aligned} |R'_k(s)| \le \frac{L_k(k+2)s^{k}}{(k+1)!}, \end{aligned}$$

which leads to the desired result

$$\begin{aligned} |\pi - s| \le \frac{L_k (k+2)s^{k}}{\sigma ^2 (k+1)!}, \end{aligned}$$

where $L=\max _{s \in [0,1] }\{| G^{k+1}(s)|, | G^{k+2}(s)|\}$. $\square$

Proof of Proposition 2

This follows directly from Theorem 1, and Proposition 1 by replacing $\pi _t$ by $\pi _t^{(k, i)}$ and G(.) by its kth degree approximation, $G_k(.)$. The same applies to the optimal expected utility. $\square$

Proof of Corollary 1

This follows directly from Proposition 2, with $k=1,2,3$. $\square$

Appendix C: Simulations

We present numerical outputs of approximate optimal portfolios (tables) and their expected demand (figures) for informed ($i=1$) and uninformed investors ($i=0$) for various CGMY markets under logarithmic utility. For each investor, we calculate expected demand or expected optimal portfolio using 10,000 simulations of optimal portfolios at each time point. The optimal portfolios for the investors under logarithmic utility and asymmetric information are generated numerically from the equations below using Newton’s method applied to Theorem 5. Asymmetric information ($q^2$) ranges from 0 to 100% in steps of 25%. In each simulation, we assume that the market coefficients $\,r_t,\,\mu _t,\, \sigma ^2_t\,$ are constants. We compare the computed optimal portfolios to the benchmark Merton (1971) optimal $\pi _{mer} (t)=(\mu _t -r_t)/\sigma ^2_t$. The inputs are

$$\begin{aligned} \pi _t^i=\frac{\theta _t}{\sigma _t}+ \frac{\upsilon ^i_t}{\sigma _t} + \frac{G'(\pi _t^i)}{\sigma _t^2}, \end{aligned}$$

where

$$\begin{aligned} \upsilon _t^1 = -\lambda \,q\,\tilde{B}_t^1,\,\,\,\,\,\,\upsilon _t^0 = -\lambda \,\tilde{B}^0_t, \end{aligned}$$

and

$$\begin{aligned} \tilde{B}_t^1= & {} \int _{0}^{t}e^{-\lambda (t-s)}dB^1_{s},\,\,\,\, \tilde{B}^0_t = \int _0^t e^{-\lambda \, (t-s)} (1+\gamma (s))\,dB^0_s,\nonumber \\ \gamma (s)= & {} \frac{q^2}{1+p\,\tanh (\lambda \,p\,s)} - 1,\,\,\,p^2+q^2=1, p\ge 0,\,\, q\ge 0. \end{aligned}$$

$\theta _t=(\mu _t -r_t)/\sigma _t$, is the Sharpe ratio, $\lambda$ is the mean reversion rate of the mispricing process, $B^1_{t}$ and $B^0_t$ are independent standard Brownian motions. $\pi _t^i$ are estimated by $\pi _t^{(i, k)}$, and G is approximated by the kth degree polynomial $G_k(\alpha )$ built from the instantaneous centralized moments of returns $M_j$, where

$$\begin{aligned} G_k(\pi )=\sum _{j=1}^k (-1)^{j-1}\,M_j\,\frac{\pi ^j}{j},\,\,\,\,\,\,M_j = \int _{R}(e^x-1)^j\,v(x)\,dx. \end{aligned}$$

1.1 McDonald’s Corporation: MCD

See Table 4 and Fig. 2.

Table 4 Optimal portfolios ($\pi ^i_t$) for MCD as a function of asymmetric information ($q^2$) and investment horizon (t) for one simulation of MCD stock

Fig. 2

The plots show optimal expected optimal demand/portfolios for MCD (McDonald stock) as a function of investment horizon and asymmetric information, $q^2$. At each level of information asymmetry, optimal expected demand falls for each investor over the investment horizon. However, the informed investor always maintains a higher optimal expected demand relative to the uninformed investor. In addition, the excess optimal expected demand of the informed investor decreases as information asymmetry increases. For MSFT, our simulations not only confirm the findings of Kelly and Ljungqvist (2012) and Easley and O’Hara (2004), that uniformed expected demand falls as information asymmetry increases, but that this is also true for informed expected demand

1.2 KBW Nasdaq Bank Index: BKX

See Table 5 and Fig. 3.

Table 5 Optimal portfolios ($\pi ^i_t$) for BKX (Bank Index) as a function of asymmetric information ($q^2$) and investment horizon (t) for one simulation of BKX index

Fig. 3

The plots show expected optimal demand/portfolios for BKX (Bank Index) as a function of investment horizon and asymmetric information, $q^2$. At each level of information asymmetry, optimal expected demand falls for each investor over the investment horizon. However, the informed investor always maintains a higher optimal expected demand relative to the uninformed investor. In addition, the excess optimal expected demand of the informed investor decreases as information asymmetry increases. For MSFT, our simulations not only confirm the findings of Kelly and Ljungqvist (2012) and Easley and O’Hara (2004), that uniformed expected demand falls as information asymmetry increases, but that this is also true for informed expected demand

1.3 NYSE Arca Pharmaceutical Index: DRG

See Table 6 and Fig. 4.

Table 6 Optimal portfolios ($\pi ^i_t$) for DRG as a function of asymmetric information ($q^2$) and investment horizon (t) for one simulation of DRG index

Fig. 4

The plots show expected optimal demand/portfolios for DRG (Pharmaceutical Index) as a function of investment horizon and asymmetric information, $q^2$ based on 10,000 simulations of the DRG index. At each level of information asymmetry, optimal expected demand increases with investment horizon, but decreases with information asymmetry. However, the informed investor always maintains a higher optimal expected demand relative to the uninformed investor. In addition, the excess optimal expected demand of the informed investor increases as information asymmetry increases. Our simulations confirm the findings of Kelly and Ljungqvist (2012) and Easley and O’Hara (2004), that uniformed expected demand falls as information asymmetry increases, but that this is also generally true for informed expected demand

1.4 S&P 500 Index: SPX

See Table 7 and Fig. 5.

Table 7 Optimal portfolios ($\pi ^i_t$) for SPX (S & P 500 Index) as a function of asymmetric information ($q^2$) and investment horizon (t) for one simulation of S & P index

Fig. 5

The plots show expected optimal demand/portfolios for SPX (S & P 500 Index) as a function of investment horizon and asymmetric information, $q^2$ based on 10,000 simulations. At each level of information asymmetry, optimal expected demand increases with investment horizon, but decreases with information asymmetry. However, the informed investor always maintains a lower optimal expected demand relative to the uninformed investor. In addition, the excess optimal expected demand of the informed investor increases as information asymmetry increases. Our simulations confirm the findings of Kelly and Ljungqvist (2012) and Easley and O’Hara (2004), that uniformed expected demand falls as information asymmetry increases, but that this is also generally true for informed expected demand