基于随机优化的量化风险管理的一些研究

doi:10.15960/j.cnki.issn.1007-6093.2025.03.007

摘要/Abstract

摘要：

风险管理在不确定性环境决策中常常起着重要作用。在量化风险管理中,评估和优化风险指标需要高效的计算技术和可靠的理论保证。本文介绍量化风险管理的几个主题,并回顾关于这些主题的一些研究和进展。我们考虑几个风险指标并研究涉及这些指标的决策模型,主要关注相关的计算技术和理论性质。我们说明随机优化作为一种强大的工具,可以用来有效处理这些问题。

关键词: 随机优化, 量化风险管理, 风险测度, 计算技术, 统计性质

Abstract:

Risk management often plays an important role in decision making under uncertainty. In quantitative risk management, assessing and optimizing risk metrics requires efficient computing techniques and reliable theoretical guarantees. In this paper, we introduce several topics on quantitative risk management and review some of the recent studies and advancements on the topics. We consider several risk metrics and study decision models that involve the metrics, with a main focus on the related computing techniques and theoretical properties. We show that stochastic optimization, as a powerful tool, can be leveraged to effectively address these problems.

Key words: stochastic optimization, quantitative risk management, risk measure, computing technique, statistical property

中图分类号:

O221

胡照林. 基于随机优化的量化风险管理的一些研究[J]. 运筹学学报（中英文）, 2025, 29(3): 135-159.

Zhaolin HU. Some studies on stochastic optimization based quantitative risk management[J]. Operations Research Transactions, 2025, 29(3): 135-159.

参考文献 55

1	GlassermanP.Monte Carlo Methods in Financial Engineering[M].New York:Springer,2004.
2	RuszczyńskiA,ShapiroA.Optimization of convex risk functions[J].Mathematics of Operations Research,2006,31(3):433-452. doi: 10.1287/moor.1050.0186
3	ShapiroA,DentchevaD,RuszczyńskiA.Lectures on Stochastic Programming: Modeling and Theory[M].Philadelphia:SIAM,2014.
4	JorionP.Value at Risk[M].New York:McGraw-Hill,2006.
5	RockafellarR T,UryasevS.Optimization of conditional value-at-risk[J].The Journal of Risk,2000,2(3):21-41. doi: 10.21314/JOR.2000.038
6	HongL J,HuZ,LiuG.Monte Carlo methods for value-at-risk and conditional value-at-risk: A review[J].ACM Transactions on Modeling and Computer Simulation,2014,24(4):Article 22.
7	ArtznerP,DelbaenF,EberJ M,et al.Coherent measures of risk[J].Mathematical Finance,1999,9(3):203-228. doi: 10.1111/1467-9965.00068
8	FÖllmerH,SchiedA.Convex measures of risk and trading constraints[J].Finance and Stochastics,2002,6,429-447. doi: 10.1007/s007800200072
9	FrittelliM,GianinE R.Putting order in risk measures[J].Journal of Banking and Finance,2002,26(7):1473-1486. doi: 10.1016/S0378-4266(02)00270-4
10	CharnesA,CooperW W,SymondsG H.Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil[J].Management Science,1958,4(3):235-263. doi: 10.1287/mnsc.4.3.235
11	Ben-TalA,TeboulleM.Expected utility, penalty functions, and duality in stochastic nonlinear programming[J].Management Science,1986,32(11):1445-1466. doi: 10.1287/mnsc.32.11.1445
12	Ben-Tal,A,TeboulleM.An old-new concept of convex risk measures: The optimized certainty equivalent[J].Mathematical Finance,2007,17(3):449-476. doi: 10.1111/j.1467-9965.2007.00311.x
13	Hamm A M, Salfeld T, Weber S. Stochastic root finding for optimized certainty equivalents[C]//Proceedings of the 2013 Winter Simulation Conference, 2013: 922-932.
14	RockafellarR T,UryasevS.The fundamental risk quadrangle in risk management, optimization and statistical estimation[J].Surveys in Operations Research and Management Science,2013,18(1):33-53.
15	RockafellarR T,RoysetJ O.Measures of residual risk with connections to regression, risk tracking, surrogate models, and ambiguity[J].SIAM Journal on Optimization,2015,25(2):1179-1208. doi: 10.1137/151003271
16	DunkelJ,WeberS.Stochastic root finding and efficient estimation of convex risk measures[J].Operations Research,2010,58(5):1505-1521. doi: 10.1287/opre.1090.0784
17	TrindadeA A,UryasevS,ShapiroA,et al.Financial prediction with constrained tail risk[J].Journal of Banking and Finance,2007,31(11):3524-3538. doi: 10.1016/j.jbankfin.2007.04.014
18	DurrettR.Probability: Theory and Examples[M].Cambridge:Cambridge University Press,2019.
19	HuberP J,RonchettiE M.Robust Statistics[M].New Jersey:John Wiley & Sons,2011.
20	HongL J,LiuG.Simulating sensitivities of conditional value-at-risk[J].Management Science,2009,55(2):281-293. doi: 10.1287/mnsc.1080.0901
21	GlynnP W,FanL,FuM C,et al.Central limit theorems for estimated functions at estimated points[J].Operations Research,2020,68(5):1557-1563. doi: 10.1287/opre.2019.1922
22	KrokhmalP A.Higher moment coherent risk measures[J].Quantitative Finance,2007,7(4):373-387. doi: 10.1080/14697680701458307
23	AlexanderS,ColemanT F,LiY.Minimizing CVaR and VaR for a portfolio of derivatives[J].Journal of Banking and Finance,2006,30(2):583-605. doi: 10.1016/j.jbankfin.2005.04.012
24	HuZ,ZhangD.Utility-based shortfall risk: Efficient computations via Monte Carlo[J].Naval Research Logistics,2018,65(5):378-392. doi: 10.1002/nav.21814
25	Hegde V, Menon A S, Prashanth L A, et al. Online estimation and optimization of utility-based shortfall risk[J/OL].[2025-08-05]. Mathematics of Operations Research. https://doi.org/10.1287/moor.2022.0266.
26	LuedtkeJ,AhmedS.A sample approximation approach for optimization with probabilistic constraints[J].SIAM Journal on Optimization,2008,19(2):674-699. doi: 10.1137/070702928
27	PagnoncelliB K,AhmedS,ShapiroA.Sample average approximation method for chance constrained programming: Theory and applications[J].Journal of Optimization Theory and Applications,2009,142,399-416. doi: 10.1007/s10957-009-9523-6
28	CalafioreG,CampiM C.Uncertain convex programs: Randomized solutions and confidence levels[J].Mathematical Programming,2005,102,25-46. doi: 10.1007/s10107-003-0499-y
29	CalafioreG,CampiM C.The scenario approach to robust control design[J].IEEE Transactions on Automatic Control,2006,51(5):742-753. doi: 10.1109/TAC.2006.875041
30	De FariasD P,Van RoyB.On constraint sampling in the linear programming approach to approximate dynamic programming[J].Mathematics of Operations Research,2004,29(3):462-478. doi: 10.1287/moor.1040.0094
31	KüçükyavuzS,JiangR.Chance-constrained optimization under limited distributional information: A review of reformulations based on sampling and distributional robustness[J].EURO Journal on Computational Optimization,2022,10,100030. doi: 10.1016/j.ejco.2022.100030
32	HenrionR,MÖllerA.A gradient formula for linear chance constraints under Gaussian distribution[J].Mathematics of Operations Research,2012,37(3):475-488. doi: 10.1287/moor.1120.0544
33	van AckooijW,HenrionR.Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions[J].SIAM Journal on Optimization,2014,24(4):1864-1889. doi: 10.1137/130922689
34	HongL J.Estimating quantile sensitivities[J].Operations Research,2009,57(1):118-130. doi: 10.1287/opre.1080.0531
35	HongL J,JiangG.Gradient and hessian of joint probability function with applications on chance-constrained programs[J].Journal of the Operations Research Society of China,2017,5,431-455. doi: 10.1007/s40305-017-0154-6
36	Feng G, Liu G. Conditional Monte Carlo: A change-of-variables approach[EB/OL].[2025-08-05]. https://arxiv.org/abs/1603.06378.
37	HongL J,YangY,ZhangL.Sequential convex approximations to joint chance constrained programs: A Monte Carlo approach[J].Operations Research,2011,59(3):617-630. doi: 10.1287/opre.1100.0910
38	HuZ,HongL J,ZhangL.A smooth Monte Carlo approach to joint chance constrained program[J].IIE Transactions,2013,45(7):716-735. doi: 10.1080/0740817X.2012.745205
39	ShanF,ZhangL,XiaoX.A smoothing function approach to joint chance-constrained programs[J].Journal of Optimization Theory and Applications,2014,163,181-199. doi: 10.1007/s10957-013-0513-3
40	CuiY,LiuJ,PangJ S.Nonconvex and nonsmooth approaches for affine chance-constrained stochastic programs[J].Set-Valued and Variational Analysis,2022,30(3):1149-1211. doi: 10.1007/s11228-022-00639-y
41	Peña-OrdieresA,LuedtkeJ,WächterA.Solving chance-constrained problems via a smooth sample-based nonlinear approximation[J].SIAM Journal on Optimization,2020,30(3):2221-2250. doi: 10.1137/19M1261985
42	HuZ,SunW,ZhuS.Chance constrained programs with Gaussian mixture models[J].IISE Transactions,2022,54(12):1117-1130. doi: 10.1080/24725854.2021.2001608
43	WeiJ,HuZ,LuoJ.Appointment scheduling optimization with chance constraints in a singleserver consultation system[J].Systems Engineering-Theory & Practice,2024,44(10):3400-3417.
44	WeiJ,HuZ,LuoJ,et al.Enhanced branch-and-bound algorithm for chance constrained programs with Gaussian mixture models[J].Annals of Operations Research,2024,338(2):1283-1315.
45	Pang X, Zhu S, Hu Z. Chance constrained program with quadratic randomness: A unified approach based on Gaussian mixture distribution[EB/OL].[2025-07-06]. arXiv:2303.00555v1.
46	GordyM B,JunejaS.Nested simulation in portfolio risk measurement[J].Management Science,2010,56(9):1658-1673.
47	BroadieM,DuY,MoallemiC C.Risk estimation via regression[J].Operations Research,2015,63(5):1077-1097. doi: 10.1287/opre.2015.1419
48	HongL J,JunejaS,LiuG.Kernel smoothing for nested estimation with application to portfolio risk measurement[J].Operations Research,2017,65(3):657-673. doi: 10.1287/opre.2017.1591
49	ZhangK,LiuG,WangS.Bootstrap-based budget allocation for nested simulation[J].Operations Research,2022,70(2):1128-1142. doi: 10.1287/opre.2020.2071
50	WangW,WangY,ZhangX.Smooth nested simulation: Bridging cubic and square root convergence rates in high dimensions[J].Management Science,2024,70(2):9031-9057.
51	Liu G, Zhang K. A tutorial on nested simulation[C]//Proceedings of the 2024 Winter Simulation Conference, 2024: 1-15.
52	HuZ,HongL J.Robust simulation with likelihood-ratio constrained input uncertainty[J].INFORMS Journal on Computing,2022,34(4):2350-2367. doi: 10.1287/ijoc.2022.1169
53	KuhnD,ShafieeS,WiesemannW.Distributionally robust optimization[J].Acta Numerica,2025,34,579-804. doi: 10.1017/S0962492924000084
54	ZhuS,FukushimaM.Worst-case conditional value-at-risk with application to robust portfolio management[J].Operations Research,2009,57(5):1155-1168.
55	GuoS,XuH.Distributionally robust shortfall risk optimization model and its approximation[J].Mathematical Programming,2019,174(1):473-498.

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