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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.

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Fig. 1

Notes

  1. 1.

    Merton (1971) and Black and Scholes (1973).

  2. 2.

    LeRoy and Porter (1981), Shiller (1981) and Madan and Seneta (1990).

  3. 3.

    Merton (1976), Jones (1984), Naik and Lee (1990), Bates (1991, 2000), Heston (1993), Madan and Seneta (1990), Madan et al. (1998), Barndorff-Nielsen (1998), Eberlein et al. (1998), Duffie et al. (2000), Barndorff-Nielsen and Shephard (2001) and Carr et al. (2002); see also, Bates (1996), Bakshi et al. (1997) and Carr et al. (2003) argue that stochastic volatility is needed to explain option prices at long maturities while jumps are needed to simultaneously explain option prices at short maturity.

  4. 4.

    See studies by Madan and Seneta (1990), Madan et al. (1998), Hirsa and Madan (2004), Fiorani et al. (2010), Madan and Milne (1991), Semeraro (2008), Madan and Wang (2009) and Marfe (2009).

  5. 5.

    See studies such as Almendral and Oosterlee (2007), Asmussen et al. (2007), Ballotta and Kyriakou (2014), Poirot and Tankov (2006) and Wang et al. (2007).

  6. 6.

    Xue et al. (2014) develop a new non-parametric test using wavelets to detect jumps in high frequency financial data. They find that only 20% of jumps occur in US equities between 9:30 AM and 4:00 PM, which suggests that most jumps occur during after-hours trading, when markets suffer from poor liquidity.

  7. 7.

    Other studies on the importance of higher moments in financial applications include Jondeau and Rockinger (2004), Conrad et al. (2010), Yang and Hung (2010) and Kousse et al. (2011).

  8. 8.

    These proxies include Tobin’s Q, firm size, stock return volatility, bid-ask spread (or a component), analysts earnings forecast dispersion, the proportion of intangible assets, debt rating, and accounting accrual quality (Lee and Masulis 2009). Easley and O’Hara (1992) suggest the Probability of Informed Trading (PIN). Johnson and So (2017) put forward the Multi-market measure of information asymmetry (MIA) as a proxy for information asymmetry that uses information from both equity and options markets.

  9. 9.

    Based on the Efficient Market Hypothesis (EMH), the stock price reflects the information of the investor. Although each investor observes the same traded stock price, they have different beliefs as to why it is at its current level. This belief is reflected by the information bank/filtration of the investor. However, in this case, the filtration (\(\mathcal {H}^0_t\)), drift (\(\mu _t^0\)) and driving Brownian motion \(B^0_t\) of the uninformed investor is obtained by contracting the larger filtration \(\mathcal {H}^1_t\) of the informed investor (see Hitsuda 1968 and Guasoni 2006).

References

  1. Aït-Sahalia, Y.: Disentangling diffusion from jumps. J Financ Econ 74, 487–528 (2004)

    Article  Google Scholar 

  2. Almendral, A., Oosterlee, C.W.: Accurate evaluation of European and American options under the CGMY Process. SIAM J Sci Comput 29, 93–117 (2007)

    Article  Google Scholar 

  3. Asmussen, S., Madan, D., Pistorius, M.: Pricing equity default swaps under an approximation to the CGMY Lévy model. J Comput Finance 11(1), 79–93 (2007)

    Article  Google Scholar 

  4. Bakshi, G., Madan, D.: A theory of volatility spread. Manag Sci 52, 1945–1956 (2006)

    Article  Google Scholar 

  5. Bakshi, G., Cao, C., Chen, Z.: Empirical performance of alternative option pricing models. J Finance 52, 2003–2049 (1997)

    Article  Google Scholar 

  6. Ballotta, L., Kyriakou, I.: Monte Carlo simulation of the CGMY process and option pricing. J Futur Mark 34(12), 1095–1121 (2014)

    Article  Google Scholar 

  7. Bates, D.: The Crash of ’87: Was it expected? The evidence from options markets. J Finance 46, 1009–1044 (1991)

    Article  Google Scholar 

  8. Bates, D.: Jumps and stochastic volatility: exchange rate processes implicit in deutschemark options. Rev Financ Stud 9, 69–108 (1996)

    Article  Google Scholar 

  9. Bates, D.: Post -’87 crash fears in S&P 500 futures options. J Econom 94, 181–238 (2000)

    Article  Google Scholar 

  10. Barndorff-Nielsen, O.E.: Processes of normal inverse Gaussian type. Finance Stoch 2, 41–68 (1998)

    Article  Google Scholar 

  11. Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Ornstein-Uhlenbeck based models and some of their uses in financial economics. J R Stat Soc B 63, 167–241 (2001)

    Article  Google Scholar 

  12. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J Polit Econ 81, 637–654 (1973)

    Article  Google Scholar 

  13. Buckley, W., Long, H.: A discontinuous mispricing model under asymmetric information. Eur J Oper Res 243, 944–955 (2015)

    Article  Google Scholar 

  14. Buckley, W.S., Brown, G.O., Marshall, M.: A mispricing model of stocks under asymmetric information. Eur J Oper Res 221(3), 584–592 (2012)

    Article  Google Scholar 

  15. Buckley, W., Long, H., Perera, S.: A jump model for Fads in asset prices under asymmetric information. Eur J Oper Res 236, 200–208 (2014)

    Article  Google Scholar 

  16. Buckley, W., Long, H., Perera, S.: Link between symmetric and asymmetric optimal portfolios in Lévy markets. Math Finance Lett 6, 1–12 (2015)

    Google Scholar 

  17. Buckley, W., Long, H., Marshall, M.: Numerical approximations of optimal portfolios in mispriced Kou asymmetric markets. Eur J Oper Res 252, 676–686 (2016)

    Article  Google Scholar 

  18. Carr, P., Geman, H., Madan, D., Yor, M.: The fine structure of asset returns: an empirical investigation. J Bus 76, 305–332 (2002)

    Article  Google Scholar 

  19. Carr, P., Geman, H., Madan, D., Yor, M.: Stochastic volatility for Lévy processes. Math Finance 13, 345–382 (2003)

    Article  Google Scholar 

  20. Chabi-Yo, F.: Conditioning information and variance bounds on pricing kernels with higher moments: theory and evidence. Rev Financ Stud 21, 181–232 (2008)

    Article  Google Scholar 

  21. Conrad, J., Dittmar, R.F., Ghysels, E.: Ex Ante Skewness and Expected Stock Returns. Working Paper, University of North Carolina at Chapel Hill (2010)

  22. Cvitanic, J., Polimenis, V., Zapatero, F.: Optimal portfolio allocation with higher moments. Ann Finance 4(1), 1–28 (2008)

    Article  Google Scholar 

  23. Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)

    Article  Google Scholar 

  24. Easley, D., O’Hara, M.: Time and the process of security price adjustment. J Finance 47, 577–606 (1992)

    Article  Google Scholar 

  25. Easley, D., O’Hara, M.: Information and the cost of capital. J Finance 59, 1553–1583 (2004)

    Article  Google Scholar 

  26. Eberlein, E., Keller, U., Prause, K.: New insights into smile, mispricing and value at risk. J Bus 71, 371–406 (1998)

    Article  Google Scholar 

  27. Fiorani, F., Luciano, E., Semeraro, P.: Single and joint default in a structural model with purely discontinuous asset prices. Quant Finance 10(3), 249–263 (2010)

    Article  Google Scholar 

  28. Guasoni, P.: Asymmetric information in Fads models. Finance Stoch 10, 159–177 (2006)

    Article  Google Scholar 

  29. Harvey, C.R., Liechy, J.C., Liechy, M.W., Muller, P.: Portfolio selection with higher moments. Quant Finance 10(5), 469–485 (2010)

    Article  Google Scholar 

  30. Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6, 327–343 (1993)

    Article  Google Scholar 

  31. Hirsa, A., Madan, D.: Pricing American options under variance gamma. J Comput Finance 7, 63–80 (2004)

    Article  Google Scholar 

  32. Johnson, T., So, E.: A simple multimarket measure of information asymmetry. Manag Sci (2017). https://doi.org/10.1287/mnsc.2016.2608

    Article  Google Scholar 

  33. Jondeau, E., Rockinger, M.: Optimal portfolio allocation under higher moments. In: EFMA 2004 Basel Meeting Paper (2004)

  34. Jones, E.P.: Option arbitrage and strategy with large price changes. J Financ Econ 13, 91–113 (1984)

    Article  Google Scholar 

  35. Kelly, B., Ljungqvist, A.: Testing asymmetric information asset pricing models. Rev Financ Stud 28(5), 1366–1413 (2012)

    Article  Google Scholar 

  36. Kousse, K., Berrada, T., Hugonnier, J.: On Some Approximations of Dynamic Optimal Portfolios Policies. Working Paper, University of Geneva (2011)

  37. Lee, G., Masulis, R.W.: Seasoned equity offerings: quality of accounting information and expected flotation costs. J Financ Econ 92(3), 443–469 (2009)

    Article  Google Scholar 

  38. LeRoy, S., Porter, R.: The present value relationship. Econometrica 49(3), 555–574 (1981)

    Article  Google Scholar 

  39. Liu, J., Longstaff, F., Pan, J.: Dynamic asset allocation and event risk. J Finance 58, 231–259 (2003)

    Article  Google Scholar 

  40. Madan, D., Milne, F.: Option pricing with V. G. Martingale components. Math Finance 1(4), 39–55 (1991)

    Article  Google Scholar 

  41. Madan, D., Seneta, E.: The variance gamma (VG) model for share market returns. J Bus 63, 511–524 (1990)

    Article  Google Scholar 

  42. Madan, D.B., Wang, J.: A new multivariate variance Gamma model and multi-asset option pricing. In: AFMathConf 2009 (2009)

  43. Madan, D.B., Yor, M.: Representing the CGMY and Meixner Lévy processes as time changed Brownian motions. J Comput Finance 12, 27–47 (2008)

    Article  Google Scholar 

  44. Madan, D., Carr, P., Chang, E.: The variance gamma process and option pricing. Eur Finance Rev 2, 79–105 (1998)

    Article  Google Scholar 

  45. Marfe, R.: A generalized variance gamma process for financial applications. Quant Finance 12, 75–87 (2009)

    Article  Google Scholar 

  46. Martin, I.W.R.: Consumption-based asset pricing with higher cumulants. Rev Econ Stud 80(2), 745–773 (2013)

    Article  Google Scholar 

  47. Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J Econ Theory 3, 373–413 (1971)

    Article  Google Scholar 

  48. Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J Financ Econ 3, 125–144 (1976)

    Article  Google Scholar 

  49. Naik, V., Lee, M.: General equilibrium pricing of options on the market portfolio with discontinuous returns. Rev Financ Stud 3, 493–522 (1990)

    Article  Google Scholar 

  50. Poirot, J., Tankov, P.: Monte Carlo option pricing for tempered stable (CGMY) processes. Asia Pac Financ Mark 13(4), 327–344 (2006)

    Article  Google Scholar 

  51. Semeraro, P.: A multivariate variance gamma model for financial application. Int J Theor Appl Finance 11, 1–18 (2008)

    Article  Google Scholar 

  52. Shiller, R.: Do stock prices move to much to be justified by subsequent changes in dividends. Am Econ Rev 71, 421–436 (1981)

    Google Scholar 

  53. Wang, I.R., Wan, J.W.L., Forsyth, P.A.: Robust numerical valuation of European and American options under the CGMY process. J Comput Finance 10, 31–69 (2007)

    Article  Google Scholar 

  54. Xue, Y., Gencay, R., Fagan, S.: Jump detection with wavelets for high-frequency financial time series. Quant Finance 14, 1427–1444 (2014)

    Article  Google Scholar 

  55. Yang, C.W., Hung, K.: A generalized Markowitz portfolio selection model with higher moments. Int Rev Account Bank Finance 2, 1–7 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the editor, professor Anne Villamil for her contribution and the anonymous reviewer for his/her helpful comments and suggestions.

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Correspondence to Sandun Perera.

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 \(k=1, 2, 3,4\).

  1. (I)

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

  2. (II)

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

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

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

    $$\begin{aligned} M_3= & {} C\,\Gamma (-Y)[\phi (G+3)-3\phi (G+2)+3\phi (G+1)-\phi (G)\\&{} -( \phi (M)-3\phi (M-1)+3\phi (M-2)-\phi (M-3)]\\= & {} C\,\Gamma (-Y)[\Delta ^3 \phi (G) -\Delta ^3 \phi (M-3)]\\= & {} C\,\Gamma (-Y)[\Delta ^3 \phi (G)+ (-1)^3\Delta ^3 \phi (M-3)]\\= & {} C\,\Gamma (-Y)[\Delta ^k \phi (G)+ (-1)^k\Delta ^k \phi (M-k)],\,\,k=3. \end{aligned}$$
  4. (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

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

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
figure2

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

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
figure3

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

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
figure4

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

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
figure5

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

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Buckley, W., Perera, S. Optimal demand in a mispriced asymmetric Carr–Geman–Madan–Yor (CGMY) economy. Ann Finance 15, 337–368 (2019). https://doi.org/10.1007/s10436-018-0335-2

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Keywords

  • Carr–Geman–Madan–Yor (CGMY)markets
  • Mispricing models under asymmetric information
  • Optimal portfolio
  • Instantaneous centralized moments of returns (ICMR)

JEL Classification

  • G10
  • G11