Relative wealth concerns with partial information and heterogeneous priors. (English) Zbl 1537.91282
Summary: We establish a Nash equilibrium for \(N\) agents with the relative wealth performance criteria when the market return is unobservable. We show that the optimal investment strategy under a stochastic return rate model can be characterized by a fully coupled forward-backward stochastic differential equation (FBSDE). We establish the existence and uniqueness results for the class of FBSDEs with stochastic coefficients and solve the utility game under partial information by using deep neural networks. We demonstrate the efficiency and accuracy by a base-case comparison with the semianalytical solution in the linear case. We examined the Sharpe ratios and the variance risk ratios by numerical simulation. We observe that the agent with the most accurate prior estimate is likely to lead the herd. Moreover, the effect of competition on heterogeneous agents varies more with market characteristics compared to that of the homogeneous case.
MSC:
| 91G10 | Portfolio theory |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 91A80 | Applications of game theory |
Keywords:
portfolio allocation; relative wealth concerns; partial information; FBSDE; deep neural networksReferences:
| [1] | Abel, A. B., Asset Prices under Habit Formation and Catching up with the Joneses, Technical report, National Bureau of Economic Research, 1990. |
| [2] | Agarwal, V., Daniel, N. D., and Naik, N. Y., Role of managerial incentives and discretion in hedge fund performance, J. Finance, 64 (2009), pp. 2221-2256. |
| [3] | Antonelli, F., Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), pp. 777-793. · Zbl 0780.60058 |
| [4] | Basak, S., Asset pricing with heterogeneous beliefs, J. Bank. Financ., 29 (2005), pp. 2849-2881. |
| [5] | Bielagk, J., Lionnet, A., and dos Reis, G., Equilibrium pricing under relative performance concerns, SIAM J. Financ. Math., 8 (2017), pp. 435-482. · Zbl 1367.91200 |
| [6] | Björk, T., Davis, M. H., and Landén, C., Optimal investment under partial information, Math. Method. Oper. Res., 71 (2010), pp. 371-399. · Zbl 1189.49053 |
| [7] | Brendle, S., Portfolio selection under incomplete information, Stochastic Process. Appl., 116 (2006), pp. 701-723. · Zbl 1137.91012 |
| [8] | Brown, K. C., Harlow, W. V., and Starks, L. T., Of tournaments and temptations: An analysis of managerial incentives in the mutual fund industry, J. Finance, 51 (1996), pp. 85-110. |
| [9] | Brown, S. J., Goetzmann, W. N., and Park, J., Careers and survival: Competition and risk in the hedge fund and CTA industry, J. Finance, 56 (2001), pp. 1869-1886. |
| [10] | Carmona, R. and Laurière, M., Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games I: The ergodic case, SIAM J. Numer. Anal., 59 (2021), pp. 1455-1485. · Zbl 1479.65013 |
| [11] | Carmona, R. and Laurière, M., Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II -The finite horizon case, Ann. Appl. Probab., 32 (2022), pp. 4065-4105. · Zbl 1505.65243 |
| [12] | Cox, J. C. and Huang, C., Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory, 49 (1989), pp. 33-83. · Zbl 0678.90011 |
| [13] | Cvitanić, J. and Karatzas, I., Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2 (1992), pp. 767-818. · Zbl 0770.90002 |
| [14] | Dai, M., Jin, H., and Liu, H., Illiquidity, position limits, and optimal investment for mutual funds, J. Econ. Theory, 146 (2011), pp. 1598-1630. · Zbl 1247.91170 |
| [15] | Davis, M. H. and Norman, A. R., Portfolio selection with transaction costs, Math. Oper. Res., 15 (1990), pp. 676-713. · Zbl 0717.90007 |
| [16] | DeMarzo, P. M., Kaniel, R., and Kremer, I., Relative wealth concerns and financial bubbles, Rev. Financ. Stud., 21 (2008), pp. 19-50. |
| [17] | Detemple, J. B., Asset pricing in a production economy with incomplete information, J. Finance, 41 (1986), pp. 383-391. |
| [18] | E, W., Han, J., and Jentzen, A., Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), pp. 349-380. · Zbl 1382.65016 |
| [19] | Espinosa, G.-E. and Touzi, N., Optimal investment under relative performance concerns, Math. Finance, 25 (2015), pp. 221-257. · Zbl 1403.91310 |
| [20] | Gennotte, G., Optimal portfolio choice under incomplete information, J. Finance, 41 (1986), pp. 733-746. |
| [21] | Gómez, J.-P., The impact of keeping up with the Joneses behavior on asset prices and portfolio choice, Financ. Res. Lett., 4 (2007), pp. 95-103. |
| [22] | Han, J. and Long, J., Convergence of the deep BSDE method for coupled FBSDEs, Probab. Uncertain. Quant. Risk, 5 (2020), pp. 1-33. · Zbl 1454.60105 |
| [23] | Heimer, R. Z., Peer Pressure: Social Interaction and the Disposition Effect, Rev. Financ. Stud., 29 (2016), pp. 3177-3209. |
| [24] | Hornik, K., Approximation capabilities of multilayer feedforward networks, Neural Netw., 4 (1991), pp. 251-257. |
| [25] | Horst, U., Stationary equilibria in discounted stochastic games with weakly interacting players, Games Econom. Behav., 51 (2005), pp. 83-108. · Zbl 1115.91009 |
| [26] | Hu, Y., Imkeller, P., and Müller, M., Utility maximization in incomplete markets, Ann. Appl. Probab., 15 (2005), pp. 1691-1712. · Zbl 1083.60048 |
| [27] | Hu, Y. and Peng, S., Solution of forward-backward stochastic differential equations, Probab. Theory Related Fields, 103 (1995), pp. 273-283. · Zbl 0831.60065 |
| [28] | Huré, C., Pham, H., and Warin, X., Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comp., 89 (2020), pp. 1547-1579. · Zbl 1440.60063 |
| [29] | Huré, C., Pham, H., and Warin, X., Some machine learning schemes for high-dimensional nonlinear PDEs, Math. Comp., 89 (2020), pp. 1547-1579. · Zbl 1440.60063 |
| [30] | Karatzas, I., Lehoczky, J. P., and Shreve, S. E., Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control Optim., 25 (1987), pp. 1557-1586. · Zbl 0644.93066 |
| [31] | Karatzas, I. and Xue, X., A note on utility maximization under partial observations, Math. Finance, 1 (1991), pp. 57-70. · Zbl 0900.90051 |
| [32] | Kramkov, D. and Schachermayer, W., The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. Appl. Probab., 9 (1999), pp. 904-950. · Zbl 0967.91017 |
| [33] | Lacker, D. and Zariphopoulou, T., Mean field and n-agent games for optimal investment under relative performance criteria, Math. Finance, 29 (2019), pp. 1003-1038. · Zbl 1433.91158 |
| [34] | Lee, S. and Papanicolaou, A., Pairs trading of two assets with uncertainty in co-integration’s level of mean reversion, Int. J. Theoret. Appl. Finance, 19 (2016), 1650054. · Zbl 1396.91693 |
| [35] | Liang, G. and Zariphopoulou, T., Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE, SIAM J. Financial Math., 8 (2017), pp. 344-372. · Zbl 1391.91148 |
| [36] | Liptser, R. S. and Shiryaev, A. N., Statistics of Random Processes II: Applications, , Springer, New York, 2013. |
| [37] | Ma, J., Protter, P., and Yong, J., Solving forward-backward stochastic differential equations explicitly—A four step scheme, Probab. Theory Related Fields, 98 (1994), pp. 339-359. · Zbl 0794.60056 |
| [38] | Ma, J., Wu, Z., Zhang, D., and Zhang, J., On well-posedness of forward-backward SDEs—A unified approach, Ann. Appl. Probab., 25 (2015), pp. 2168-2214. · Zbl 1319.60132 |
| [39] | Magill, M. J. and Constantinides, G. M., Portfolio selection with transactions costs, J. Econ. Theory, 13 (1976), pp. 245-263. · Zbl 0361.90001 |
| [40] | Mania, M. and Santacroce, M., Exponential utility maximization under partial information, Finance Stoch., 14 (2010), pp. 419-448. · Zbl 1226.91091 |
| [41] | Matoussi, A., Possamaï, D., and Zhou, C., Robust utility maximization in nondominated models with 2BSDE: The uncertain volatility model, Math. Finance, 25 (2015), pp. 258-287. · Zbl 1335.91067 |
| [42] | Merton, R. C., Optimum consumption and portfolio rules in a continuous-time model, in Stochastic Optimization Models in Finance, Elsevier, Amsterdam, 1975, pp. 621-661. |
| [43] | Papanicolaou, A., Backward SDEs for control with partial information, Math. Finance, 29 (2019), pp. 208-248. · Zbl 1458.91196 |
| [44] | Pardoux, E. and Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications, Springer, New York, 1992, pp. 200-217. · Zbl 0766.60079 |
| [45] | Pham, H. and Quenez, M.-C., Optimal portfolio in partially observed stochastic volatility models, Ann. Appl. Probab., 11 (2001), pp. 210-238. · Zbl 1043.91032 |
| [46] | Pliska, S. R., A stochastic calculus model of continuous trading: Optimal portfolios, Math. Oper. Res., 11 (1986), pp. 371-382. · Zbl 1011.91503 |
| [47] | Qiu, Z., Equilibrium-informed trading with relative performance measurement, J. Financ. Quant. Anal., 52 (2017), pp. 2083-2118. |
| [48] | Rieder, U. and Bäuerle, N., Portfolio optimization with unobservable Markov-modulated drift process, J. Appl. Probab., 42 (2005), pp. 362-378. · Zbl 1138.93428 |
| [49] | Rogers, L., Duality in constrained optimal investment and consumption problems: A synthesis, in Paris-Princeton Lectures on Mathematical Finance 2002, Springer, New York, 2003, pp. 95-131. · Zbl 1074.91020 |
| [50] | Rouge, R. and Karoui, N. El, Pricing via utility maximization and entropy, Math. Finance, 10 (2000), pp. 259-276. · Zbl 1052.91512 |
| [51] | Sass, J. and Haussmann, U. G., Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain, Finance Stoch., 8 (2004), pp. 553-577. · Zbl 1063.91040 |
| [52] | Shreve, S. E. and Soner, H. M., Optimal investment and consumption with transaction costs, Ann. Appl. Probab., 4 (1994), pp. 609-692. · Zbl 0813.60051 |
| [53] | Wang, G. and Wu, Z., Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems, J. Math. Anal. Appl., 342 (2008), pp. 1280-1296. · Zbl 1141.93070 |
| [54] | Yong, J., Finding adapted solutions of forward-backward stochastic differential equations: Method of continuation, Probab. Theory Related Fields, 107 (1997), pp. 537-572. · Zbl 0883.60053 |
| [55] | Zariphopoulou, T., Consumption-investment models with constraints, SIAM J. Control Optim., 32 (1994), pp. 59-85. · Zbl 0790.90007 |
| [56] | Zhang, J., The wellposedness of FBSDEs, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 927-940. · Zbl 1132.60315 |
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