# 成本、约束与稳健决策 由精确费用预算推导不交易带，再用整股佣金枚举检验连续近似的边界。 Entry: zh-qt23 | Node: QT23 | Language: zh | Editorial revision: 2026-09-21 ## Teaching instructions 请以具有微积分、线性代数与基本概率背景的高年级本科至研究生为对象，围绕“成本、约束与稳健决策”完成一次学习。先实际取得并读完 required_readings 的完整指定单元，记录实际版本/定位/假设/支持范围；目录、摘要或入口不抵扣阅读。可先读取本篇所有必读以备课，但不要把它们全变成读者额外作业。来源打不开时先尝试同版作者可读入口；仍缺承重单元应说明具体缺口，不伪称已读，也不要求无关新批准。先让读者用调仓前财富写预算，完整推导左右导数与不交易带；再枚举q37/38并拆净期望、风险罚项、目标、现金。最后改最低现金使w0不可行，要求用可行方向而非套内点公式。 先让读者尝试，再根据本文完整解析反馈；引用范围内解释，不把模拟当市场事实、不把最优目标当收益保证。只在读者选择分支时启用 optional_readings。完成后用一项新输入迁移检验，明确其能独立完成什么。 Before substantive teaching, actually retrieve every required reading unit for the selected scope. Read its complete designated section, including necessary assumptions, tables and footnotes. A working URL or an editorial access date is not a runtime reading receipt. Record the actual version, location, scope and what it supports. If unavailable, use a previously verified equivalent source; if the required unit remains unavailable, identify that gap rather than teach it from memory. Start runtime_reading_log empty. Once reading is complete, use a substantive diagnostic or follow the reader's request for direct explanation. Advance one complete reasoning task at a time; skip mastered basics. Distinguish original facts, supplied teaching assumptions and inference. ## Required readings and runtime protocol ```json { "export_mode": "public", "required_readings": [ { "source_id": "QTDE-CVX", "access": { "kind": "pdf_full_text", "uri": "https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "§4.2.2–4.2.3 pp.138–140；Markowitz子节pp.155–156", "scope": "凸可行集最优性、预算与费用子单元", "purpose": "核严格凹直接证明，不补造KKT依赖" }, "version": "2004", "supports": "凸可行集最优性、预算与费用子单元", "title": "Convex Optimization", "authors": [ "Stephen Boyd", "Lieven Vandenberghe" ] }, { "source_id": "QTDE-MARKOWITZ", "access": { "kind": "pdf_full_text", "uri": "https://web.stanford.edu/~boyd/papers/pdf/markowitz.pdf", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "§2.1–2.4 pp.13–18；§3.4–3.5 pp.21–24；§4.1–4.3 pp.24–28", "scope": "交易约束、成本预测、盒集及非凸实施完整采用单元", "purpose": "分清论文近似与本站精确预算；区间半径不是已估置信界" }, "version": "manuscript January 5, 2024; journal July 2024 separately identified", "supports": "交易约束、成本预测、盒集及非凸实施完整采用单元", "title": "Markowitz Portfolio Construction at Seventy", "authors": [ "Stephen Boyd", "Kasper Johansson", "Ronald Kahn", "Philipp Schiele", "Thomas Schmelzer" ] }, { "source_id": "QTDE-IBKR", "access": { "kind": "html_full_text", "uri": "https://www.interactivebrokers.com/en/pricing/commissions-stocks.php?re=amer", "verified_access_at": "2026-09-21" }, "required_unit": { "locator": "US Fixed table、Fixed Examples、minimum/maximum以及相关适用脚注完整单元", "scope": "普通整股单笔佣金组件", "purpose": "确认规则日期、上限优先与费用排除" }, "version": "official page accessed 2026-09-21; not an assumed effective date", "supports": "普通整股单笔佣金组件", "title": "US stocks IBKR Pro Fixed commission component", "authors": [ "Interactive Brokers" ] } ], "optional_readings": [], "runtime_reading_log": [], "supplied_inputs": { "input_identity": "QT-DE-adopted-inputs-20260921-v1", "config": { "content_version": "2026-09-21-QT-DE-review-v2", "input_version": "QT-DE-adopted-inputs-20260921-v1", "dataset": { "file": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/BusEq-value-weighted-monthly-199001-202512.csv", "origin_input_version": "QT-C-inputs-20260921-v1", "transport_sha256": "72dd0f35d9437d268971de8553827a5c7a4f85ebcf9712c0c1b85ea2ec7cfa74", "source_url": "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/30_Industry_Portfolios_CSV.zip", "source_details_url": "https://mba.tuck.dartmouth.edu/pages/Faculty/ken.french/Data_Library/det_30_ind_port.html", "regime_url": "https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html", "vintage": "202607 CRSP database; current CIZ reconstruction", "download_date": "2026-09-21", "block": "Average Value Weighted Returns -- Monthly", "column": "BusEq", "sample": [ "199001", "202512" ], "n": 432, "source_lines": [ 775, 1206 ], "missing_removed": 0, "missing_codes": [ -99.99, -999 ], "raw_unit": "percent", "internal_unit": "monthly decimal return", "display_errors": "百分点/月", "annualized": false, "identity": "单一当前重建快照；不是FIZ/CIZ时间拼接，不是逐月当时可见版本。" }, "forecast": { "experiment_id": "EXP-QT-D-FORECAST-01", "p_grid": [ 1, 3, 12 ], "lambda_grid": [ 0.0, 0.1, 1.0, 10.0 ], "default_candidate": "p1-l10", "target": "y_t=r_t", "features": "(r_(t-1),...,r_(t-p))", "warmup": [ "199001", "199012" ], "initial_train": [ "199101", "200912", 228 ], "validation": [ "201001", "201912", 120 ], "historical_evaluation": [ "202001", "202512", 72 ], "train_start_index": 12, "objective": "RSS/n + lambda*||beta||^2", "matrix_penalty": "n*lambda*I", "standardization_ddof": 0, "intercept_penalized": false, "selection": "minimum validation SSE including zero and expanding-mean baselines", "tie_rule": "regression grid: smaller p then smaller lambda", "frozen_hyperparameters_after_validation": true, "refit": "每原点用新成熟标签扩展重拟合；不再次选择p或lambda", "availability": "教学月末可知收益假设；French历史真实发布和接收时点未重建", "future_scaling_control": { "p": 12, "lambdas": [ 0.0, 0.1 ], "scope": "validation only", "identity": "故意信息越界对照" }, "default_clock_month": "201001", "default_clock_lambda": 0.1 }, "selection": { "experiment_id": "EXP-QT-D-SELECT-01", "B": 5000, "m_grid": [ 1, 10, 100, 500 ], "default_m": 100, "seed_all_null": 2201, "seed_mixed": 2202, "rng": "numpy.random.Generator(PCG64)", "n_development": 120, "n_holdout": 120, "sigma": 0.05, "alpha": 0.05, "hypothesis": "事前单侧H0:mu=0 versus H1:mu>0; sigma known", "pvalue": "0.5*erfc(Z/sqrt(2))", "draw_order": "先取(5000,500)开发Z矩阵，再取独立保留Z矩阵；M为共同500列前缀。mixed另重置2202。", "bh_adopted_comparison": "strict <", "source_pilot_comparison": "<=", "empty_BH": "k=0, no rejections, FDP=0", "mixed": { "m": 100, "signals": 10, "signal_mu": 0.01, "null_mu": 0.0 }, "identity": "直接模拟正态样本均值的Z分布；不含逐月路径，不是真实策略面板", "equal_example": { "p": [ 0.025, 0.06 ], "q": 0.05 }, "manual_example": { "p": [ 0.006, 0.012, 0.601, 0.756, 0.918 ], "q": 0.05 } }, "cost": { "experiment_id": "EXP-QT23-COST-01", "rf": 0, "continuous_default": { "mu": 0.015, "rho": 0, "sigma": 0.1, "gamma": 2, "w0": 0.5, "kappa": 0.002, "cap": 0.8, "cash_min": 0.2 }, "integer_default": { "mu": 0.002, "rho": 0, "sigma": 0.02, "gamma": 10, "wealth": 10000, "initial_shares": 0, "mid": 100, "ask": 100.05, "cap": 80, "cash_min": 2000, "mode": "spread_and_commission" }, "fee": { "id": "QTDE-IBKR", "url": "https://www.interactivebrokers.com/en/pricing/commissions-stocks.php?re=amer", "accessed_at": "2026-09-21", "scope": "IBKR Pro US Fixed ordinary whole-share single-order commission component only", "formula": "q=0:0; otherwise min(0.01*execution_price*abs(q),max(1,0.005*abs(q)))" }, "identity": "收益、波动、风险系数、报价、价差、容量为教学设定；佣金是有日期的实际规则组件", "end_value": "期末中价标记；没有退出委托，非往返净利", "excluded_costs": [ "监管/交易所/清算等其他收费", "税", "市场冲击", "持仓融资费", "退出交易费", "零股及改单最低费规则" ] }, "clocks": { "view_id": "EXP-QT-D-CLOCK-01", "shared_experiment": "EXP-QT-D-FORECAST-01", "BEA": [ { "period": "2020Q2", "vintage": "advance", "release_at": "2020-07-30T08:30:00-04:00", "received_at": null, "value_percent_SAAR": -32.9 }, { "period": "2020Q2", "vintage": "second", "release_at": "2020-08-27T08:30:00-04:00", "received_at": null, "value_percent_SAAR": -31.7 } ], "h3_task": { "formula": "prod(1+r_(s+j),j=1..3)-1", "cutoff": "202003", "latest_mature_origin": "201912", "performance_run": false } }, "scale_examples": { "raw_x": [ -1, 1 ], "y": [ -1, 1 ], "lambda": 1.0, "default_feature_scale": 10.0, "response_scale": 100.0 }, "source_identities": { "d_pilot": { "file": "qt-d-experiment-pilot.py", "sha256": "b8a23b01294d543b3b74c6a153168529db74ab7c84837ea3724a55f5071fd3fb", "read_scope": "1–193" }, "ef_pilot": { "file": "qt-ef-calculations.py", "sha256": "28eff60cf87400805af145fabd22d4fe17e2c25bac3708cd9e71506d939acb20", "adopted_scope": "commission, continuous_case, integer_case, decision_calculation" }, "approval": { "file": "qt-de-draft-approved.md", "read_scope": "1–7", "sha256": "80e92333d057772de9463d786a9be435c6dc1208744bf828d5eb2a4d2a5e9a48" }, "plan_review": { "file": "qt-def-plan-review.md", "read_scope": "1–20", "sha256": "56fddfa847ca0f83ec4822652c506f847e9da4e45f7223c039b31428f3aedfd4" } } }, "static_equivalent_html": "

rf=0，费用期初现金扣除。连续默认w0=.5, μ=.015, σ=.1, γ=2, κ=.002, u=.8, cmin=.2。收益/报价/容量是教学假设；佣金是2026-09-21官方组件。

c=1−x−κ|x−w0|；J=(μ−ρ)x−γσ²x²/2−κ|x−w0|。w0是可行内点时，不交易 iff |μ−ρ−γσ²w0|≤κ。
默认连续量值
x0.650000000
cost0.000300000
cash0.349700000
post_cost_wealth0.999700000
post_weight_risky0.650195059
nominal_expected_net_return0.009450000
risk_penalty0.004225000
objective0.005225000

ρ=.004则x=.5；cmin=.8时w0不可行，只能沿可行方向。真实规则组件comm=min(.01p|q|,max(1,.005|q|))，q=0不下单费用零。

整股情景全部最优q目标USD最大可行q
default[37, 38]1.81279
no_cost[50]5.00080
spread_only[37, 38]2.81279
low_mu[0]0.00079
box0006[22, 23]0.01279
cap10[10]0.30010
cash9000[9]0.1889
默认q37账本USD或明确标注
spread1.85
commission1.0
cash6297.15
nominal_expected_net_pnl4.55
risk_penalty_dollars2.738
objective_dollars1.812
terminal_mid_wealth[9930.55, 10078.55]
returns_prewealth[-0.006945, 0.007855]

q37/38的目标同为1.812，不代表风险和净期望相同；q80现金1995不可行。风险罚项不是现金收费。期末仅中价标记，不含退出费用。盒集半径不是统计置信区间。

", "files": { "frozen_csv": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/BusEq-value-weighted-monthly-199001-202512.csv", "complete_outputs": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/results.json", "reproduction": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/compute/reproduce.py", "interactive_engine": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/engine.js", "selection_arrays": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/selection-arrays.npz", "shared_inputs": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/shared_inputs.json", "sources": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/sources.json" }, "algorithm_identity": "forecast/selection使用真实计算的冻结网格及PCG64结果；成本和人工边界可重算。完整reader在body_markdown中仅一份。", "reading_base": "公开同源包；按完整 URL 取得冻结输入与复算文件。来源网站后续更新不覆盖本份 202607 快照。", "cost_results_for_teaching": { "continuous": { "default": { "feasible": true, "input": { "mu": { "exact": "3/200", "decimal": 0.015 }, "rho": { "exact": "0", "decimal": 0.0 }, "cap": { "exact": "4/5", "decimal": 0.8 }, "cash_min": { "exact": "1/5", "decimal": 0.2 }, "kappa": { "exact": "1/500", "decimal": 0.002 }, "w0": { "exact": "1/2", "decimal": 0.5 }, "sigma": { "exact": "1/10", "decimal": 0.1 }, "gamma": { "exact": "2", "decimal": 2.0 }, "rf": { "exact": "0", "decimal": 0.0 } }, "upper": { "exact": "267/334", "decimal": 0.7994011976047904 }, "x": { "exact": "13/20", "decimal": 0.65 }, "cost": { "exact": "3/10000", "decimal": 0.0003 }, "cash": { "exact": "3497/10000", "decimal": 0.3497 }, "post_cost_wealth": { "exact": "9997/10000", "decimal": 0.9997 }, "post_weight_risky": { "exact": "500/769", "decimal": 0.6501950585175552 }, "post_weight_cash": { "exact": "269/769", "decimal": 0.34980494148244473 }, "nominal_expected_net_return": { "exact": "189/20000", "decimal": 0.00945 }, "worst_expected_net_return": { "exact": "189/20000", "decimal": 0.00945 }, "risk_penalty": { "exact": "169/40000", "decimal": 0.004225 }, "objective": { "exact": "209/40000", "decimal": 0.005225 }, "no_trade_mu_band": [ { "exact": "1/125", "decimal": 0.008 }, { "exact": "3/250", "decimal": 0.012 } ], "w0_feasible_interior": true }, "cap06": { "feasible": true, "input": { "mu": { "exact": "3/200", "decimal": 0.015 }, "rho": { "exact": "0", "decimal": 0.0 }, "cap": { "exact": "3/5", "decimal": 0.6 }, "cash_min": { "exact": "1/5", "decimal": 0.2 }, "kappa": { "exact": "1/500", "decimal": 0.002 }, "w0": { "exact": "1/2", "decimal": 0.5 }, "sigma": { "exact": "1/10", "decimal": 0.1 }, "gamma": { "exact": "2", "decimal": 2.0 }, "rf": { "exact": "0", "decimal": 0.0 } }, "upper": { "exact": "3/5", "decimal": 0.6 }, "x": { "exact": "3/5", "decimal": 0.6 }, "cost": { "exact": "1/5000", "decimal": 0.0002 }, "cash": { "exact": "1999/5000", "decimal": 0.3998 }, "post_cost_wealth": { "exact": "4999/5000", "decimal": 0.9998 }, "post_weight_risky": { "exact": "3000/4999", "decimal": 0.6001200240048009 }, "post_weight_cash": { "exact": "1999/4999", "decimal": 0.39987997599519903 }, "nominal_expected_net_return": { "exact": "11/1250", "decimal": 0.0088 }, "worst_expected_net_return": { "exact": "11/1250", "decimal": 0.0088 }, "risk_penalty": { "exact": "9/2500", "decimal": 0.0036 }, "objective": { "exact": "13/2500", "decimal": 0.0052 }, "no_trade_mu_band": [ { "exact": "1/125", "decimal": 0.008 }, { "exact": "3/250", "decimal": 0.012 } ], "w0_feasible_interior": true }, "box004": { "feasible": true, "input": { "mu": { "exact": "3/200", "decimal": 0.015 }, "rho": { "exact": "1/250", "decimal": 0.004 }, "cap": { "exact": "4/5", "decimal": 0.8 }, "cash_min": { "exact": "1/5", "decimal": 0.2 }, "kappa": { "exact": "1/500", "decimal": 0.002 }, "w0": { "exact": "1/2", "decimal": 0.5 }, "sigma": { "exact": "1/10", "decimal": 0.1 }, "gamma": { "exact": "2", "decimal": 2.0 }, "rf": { "exact": "0", "decimal": 0.0 } }, "upper": { "exact": "267/334", "decimal": 0.7994011976047904 }, "x": { "exact": "1/2", "decimal": 0.5 }, "cost": { "exact": "0", "decimal": 0.0 }, "cash": { "exact": "1/2", "decimal": 0.5 }, "post_cost_wealth": { "exact": "1", "decimal": 1.0 }, "post_weight_risky": { "exact": "1/2", "decimal": 0.5 }, "post_weight_cash": { "exact": "1/2", "decimal": 0.5 }, "nominal_expected_net_return": { "exact": "3/400", "decimal": 0.0075 }, "worst_expected_net_return": { "exact": "11/2000", "decimal": 0.0055 }, "risk_penalty": { "exact": "1/400", "decimal": 0.0025 }, "objective": { "exact": "3/1000", "decimal": 0.003 }, "no_trade_mu_band": [ { "exact": "1/125", "decimal": 0.008 }, { "exact": "3/250", "decimal": 0.012 } ], "w0_feasible_interior": true }, "w0_infeasible": { "feasible": true, "input": { "mu": { "exact": "3/200", "decimal": 0.015 }, "rho": { "exact": "0", "decimal": 0.0 }, "cap": { "exact": "4/5", "decimal": 0.8 }, "cash_min": { "exact": "4/5", "decimal": 0.8 }, "kappa": { "exact": "1/500", "decimal": 0.002 }, "w0": { "exact": "1/2", "decimal": 0.5 }, "sigma": { "exact": "1/10", "decimal": 0.1 }, "gamma": { "exact": "2", "decimal": 2.0 }, "rf": { "exact": "0", "decimal": 0.0 } }, "upper": { "exact": "199/998", "decimal": 0.19939879759519039 }, "x": { "exact": "199/998", "decimal": 0.19939879759519039 }, "cost": { "exact": "3/4990", "decimal": 0.0006012024048096192 }, "cash": { "exact": "4/5", "decimal": 0.8 }, "post_cost_wealth": { "exact": "4987/4990", "decimal": 0.9993987975951903 }, "post_weight_risky": { "exact": "995/4987", "decimal": 0.19951874874674153 }, "post_weight_cash": { "exact": "3992/4987", "decimal": 0.8004812512532584 }, "nominal_expected_net_return": { "exact": "477/199600", "decimal": 0.0023897795591182364 }, "worst_expected_net_return": { "exact": "477/199600", "decimal": 0.0023897795591182364 }, "risk_penalty": { "exact": "39601/99600400", "decimal": 0.000397598804824077 }, "objective": { "exact": "99211/49800200", "decimal": 0.0019921807542941593 }, "no_trade_mu_band": [ { "exact": "1/125", "decimal": 0.008 }, { "exact": "3/250", "decimal": 0.012 } ], "w0_feasible_interior": false } }, "integer": { "default": { "feasible": true, "input": { "mu": { "exact": "1/500", "decimal": 0.002 }, "rho": { "exact": "0", "decimal": 0.0 }, "cash_min": { "exact": "2000", "decimal": 2000.0 }, "cap": 80, "mode": "spread_and_commission", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 37, 38 ], "display_q": 37, "largest_feasible_q": 79, "objective": { "exact": "453/250", "decimal": 1.812 }, "exact_rows_csv": "q,feasible,spread,commission,cost,cash,x,post_cost_wealth,post_weight_risky,expected_gross_pnl,nominal_expected_net_pnl,worst_expected_net_pnl,risk_penalty_dollars,objective_dollars,terminal_mid_wealth,returns_prewealth\n0,True,0,0,0,10000,0,10000,0,0,0,0,0,0,\"[\"\"10000\"\",\"\"10000\"\"]\",\"[\"\"0\"\",\"\"0\"\"]\"\n1,True,1/20,1,21/20,197979/20,1/100,199979/20,2000/199979,1/5,-17/20,-17/20,1/500,-213/250,\"[\"\"199943/20\"\",\"\"200023/20\"\"]\",\"[\"\"-57/200000\"\",\"\"23/200000\"\"]\"\n2,True,1/10,1,11/10,97989/10,1/50,99989/10,2000/99989,2/5,-7/10,-7/10,1/125,-177/250,\"[\"\"99953/10\"\",\"\"100033/10\"\"]\",\"[\"\"-47/100000\"\",\"\"33/100000\"\"]\"\n3,True,3/20,1,23/20,193977/20,3/100,199977/20,2000/66659,3/5,-11/20,-11/20,9/500,-71/125,\"[\"\"199869/20\"\",\"\"200109/20\"\"]\",\"[\"\"-131/200000\"\",\"\"109/200000\"\"]\"\n4,True,1/5,1,6/5,47994/5,1/25,49994/5,1000/24997,4/5,-2/5,-2/5,4/125,-54/125,\"[\"\"49958/5\"\",\"\"50038/5\"\"]\",\"[\"\"-21/25000\"\",\"\"19/25000\"\"]\"\n5,True,1/4,1,5/4,37995/4,1/20,39995/4,400/7999,1,-1/4,-1/4,1/20,-3/10,\"[\"\"39959/4\"\",\"\"40039/4\"\"]\",\"[\"\"-41/40000\"\",\"\"39/40000\"\"]\"\n6,True,3/10,1,13/10,93987/10,3/50,99987/10,2000/33329,6/5,-1/10,-1/10,9/125,-43/250,\"[\"\"99879/10\"\",\"\"100119/10\"\"]\",\"[\"\"-121/100000\"\",\"\"119/100000\"\"]\"\n7,True,7/20,1,27/20,185973/20,7/100,199973/20,14000/199973,7/5,1/20,1/20,49/500,-6/125,\"[\"\"199721/20\"\",\"\"200281/20\"\"]\",\"[\"\"-279/200000\"\",\"\"281/200000\"\"]\"\n8,True,2/5,1,7/5,45993/5,2/25,49993/5,4000/49993,8/5,1/5,1/5,16/125,9/125,\"[\"\"49921/5\"\",\"\"50081/5\"\"]\",\"[\"\"-79/50000\"\",\"\"81/50000\"\"]\"\n9,True,9/20,1,29/20,181971/20,9/100,199971/20,2000/22219,9/5,7/20,7/20,81/500,47/250,\"[\"\"199647/20\"\",\"\"200367/20\"\"]\",\"[\"\"-353/200000\"\",\"\"367/200000\"\"]\"\n10,True,1/2,1,3/2,17997/2,1/10,19997/2,2000/19997,2,1/2,1/2,1/5,3/10,\"[\"\"19961/2\"\",\"\"20041/2\"\"]\",\"[\"\"-39/20000\"\",\"\"41/20000\"\"]\"\n11,True,11/20,1,31/20,177969/20,11/100,199969/20,2000/18179,11/5,13/20,13/20,121/500,51/125,\"[\"\"199573/20\"\",\"\"200453/20\"\"]\",\"[\"\"-427/200000\"\",\"\"453/200000\"\"]\"\n12,True,3/5,1,8/5,43992/5,3/25,49992/5,250/2083,12/5,4/5,4/5,36/125,64/125,\"[\"\"49884/5\"\",\"\"50124/5\"\"]\",\"[\"\"-29/12500\"\",\"\"31/12500\"\"]\"\n13,True,13/20,1,33/20,173967/20,13/100,199967/20,26000/199967,13/5,19/20,19/20,169/500,153/250,\"[\"\"199499/20\"\",\"\"200539/20\"\"]\",\"[\"\"-501/200000\"\",\"\"539/200000\"\"]\"\n14,True,7/10,1,17/10,85983/10,7/50,99983/10,14000/99983,14/5,11/10,11/10,49/125,177/250,\"[\"\"99731/10\"\",\"\"100291/10\"\"]\",\"[\"\"-269/100000\"\",\"\"291/100000\"\"]\"\n15,True,3/4,1,7/4,33993/4,3/20,39993/4,2000/13331,3,5/4,5/4,9/20,4/5,\"[\"\"39885/4\"\",\"\"40125/4\"\"]\",\"[\"\"-23/8000\"\",\"\"1/320\"\"]\"\n16,True,4/5,1,9/5,41991/5,4/25,49991/5,8000/49991,16/5,7/5,7/5,64/125,111/125,\"[\"\"49847/5\"\",\"\"50167/5\"\"]\",\"[\"\"-153/50000\"\",\"\"167/50000\"\"]\"\n17,True,17/20,1,37/20,165963/20,17/100,199963/20,34000/199963,17/5,31/20,31/20,289/500,243/250,\"[\"\"199351/20\"\",\"\"200711/20\"\"]\",\"[\"\"-649/200000\"\",\"\"711/200000\"\"]\"\n18,True,9/10,1,19/10,81981/10,9/50,99981/10,2000/11109,18/5,17/10,17/10,81/125,263/250,\"[\"\"99657/10\"\",\"\"100377/10\"\"]\",\"[\"\"-343/100000\"\",\"\"377/100000\"\"]\"\n19,True,19/20,1,39/20,161961/20,19/100,199961/20,38000/199961,19/5,37/20,37/20,361/500,141/125,\"[\"\"199277/20\"\",\"\"200797/20\"\"]\",\"[\"\"-723/200000\"\",\"\"797/200000\"\"]\"\n20,True,1,1,2,7998,1/5,9998,1000/4999,4,2,2,4/5,6/5,\"[\"\"9962\"\",\"\"10042\"\"]\",\"[\"\"-19/5000\"\",\"\"21/5000\"\"]\"\n21,True,21/20,1,41/20,157959/20,21/100,199959/20,14000/66653,21/5,43/20,43/20,441/500,317/250,\"[\"\"199203/20\"\",\"\"200883/20\"\"]\",\"[\"\"-797/200000\"\",\"\"883/200000\"\"]\"\n22,True,11/10,1,21/10,77979/10,11/50,99979/10,2000/9089,22/5,23/10,23/10,121/125,333/250,\"[\"\"99583/10\"\",\"\"100463/10\"\"]\",\"[\"\"-417/100000\"\",\"\"463/100000\"\"]\"\n23,True,23/20,1,43/20,153957/20,23/100,199957/20,46000/199957,23/5,49/20,49/20,529/500,174/125,\"[\"\"199129/20\"\",\"\"200969/20\"\"]\",\"[\"\"-871/200000\"\",\"\"969/200000\"\"]\"\n24,True,6/5,1,11/5,37989/5,6/25,49989/5,4000/16663,24/5,13/5,13/5,144/125,181/125,\"[\"\"49773/5\"\",\"\"50253/5\"\"]\",\"[\"\"-227/50000\"\",\"\"253/50000\"\"]\"\n25,True,5/4,1,9/4,29991/4,1/4,39991/4,10000/39991,5,11/4,11/4,5/4,3/2,\"[\"\"39811/4\"\",\"\"40211/4\"\"]\",\"[\"\"-189/40000\"\",\"\"211/40000\"\"]\"\n26,True,13/10,1,23/10,73977/10,13/50,99977/10,26000/99977,26/5,29/10,29/10,169/125,387/250,\"[\"\"99509/10\"\",\"\"100549/10\"\"]\",\"[\"\"-491/100000\"\",\"\"549/100000\"\"]\"\n27,True,27/20,1,47/20,145953/20,27/100,199953/20,6000/22217,27/5,61/20,61/20,729/500,199/125,\"[\"\"198981/20\"\",\"\"201141/20\"\"]\",\"[\"\"-1019/200000\"\",\"\"1141/200000\"\"]\"\n28,True,7/5,1,12/5,35988/5,7/25,49988/5,3500/12497,28/5,16/5,16/5,196/125,204/125,\"[\"\"49736/5\"\",\"\"50296/5\"\"]\",\"[\"\"-33/6250\"\",\"\"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",\"\"2431/200000\"\"]\"\n58,True,29/10,1,39/10,41961/10,29/50,99961/10,58000/99961,58/5,77/10,77/10,841/125,243/250,\"[\"\"98917/10\"\",\"\"101237/10\"\"]\",\"[\"\"-1083/100000\"\",\"\"1237/100000\"\"]\"\n59,True,59/20,1,79/20,81921/20,59/100,199921/20,118000/199921,59/5,157/20,157/20,3481/500,111/125,\"[\"\"197797/20\"\",\"\"202517/20\"\"]\",\"[\"\"-2203/200000\"\",\"\"2517/200000\"\"]\"\n60,True,3,1,4,3996,3/5,9996,500/833,12,8,8,36/5,4/5,\"[\"\"9888\"\",\"\"10128\"\"]\",\"[\"\"-7/625\"\",\"\"8/625\"\"]\"\n61,True,61/20,1,81/20,77919/20,61/100,199919/20,122000/199919,61/5,163/20,163/20,3721/500,177/250,\"[\"\"197723/20\"\",\"\"202603/20\"\"]\",\"[\"\"-2277/200000\"\",\"\"2603/200000\"\"]\"\n62,True,31/10,1,41/10,37959/10,31/50,99959/10,62000/99959,62/5,83/10,83/10,961/125,153/250,\"[\"\"98843/10\"\",\"\"101323/10\"\"]\",\"[\"\"-1157/100000\"\",\"\"1323/100000\"\"]\"\n63,True,63/20,1,83/20,73917/20,63/100,199917/20,14000/22213,63/5,169/20,169/20,3969/500,64/125,\"[\"\"197649/20\"\",\"\"202689/20\"\"]\",\"[\"\"-2351/200000\"\",\"\"2689/200000\"\"]\"\n64,True,16/5,1,21/5,17979/5,16/25,49979/5,32000/49979,64/5,43/5,43/5,1024/125,51/125,\"[\"\"49403/5\"\",\"\"50683/5\"\"]\",\"[\"\"-597/50000\"\",\"\"683/50000\"\"]\"\n65,True,13/4,1,17/4,13983/4,13/20,39983/4,26000/39983,13,35/4,35/4,169/20,3/10,\"[\"\"39515/4\"\",\"\"40555/4\"\"]\",\"[\"\"-97/8000\"\",\"\"111/8000\"\"]\"\n66,True,33/10,1,43/10,33957/10,33/50,99957/10,2000/3029,66/5,89/10,89/10,1089/125,47/250,\"[\"\"98769/10\"\",\"\"101409/10\"\"]\",\"[\"\"-1231/100000\"\",\"\"1409/100000\"\"]\"\n67,True,67/20,1,87/20,65913/20,67/100,199913/20,134000/199913,67/5,181/20,181/20,4489/500,9/125,\"[\"\"197501/20\"\",\"\"202861/20\"\"]\",\"[\"\"-2499/200000\"\",\"\"2861/200000\"\"]\"\n68,True,17/5,1,22/5,15978/5,17/25,49978/5,17000/24989,68/5,46/5,46/5,1156/125,-6/125,\"[\"\"49366/5\"\",\"\"50726/5\"\"]\",\"[\"\"-317/25000\"\",\"\"363/25000\"\"]\"\n69,True,69/20,1,89/20,61911/20,69/100,199911/20,46000/66637,69/5,187/20,187/20,4761/500,-43/250,\"[\"\"197427/20\"\",\"\"202947/20\"\"]\",\"[\"\"-2573/200000\"\",\"\"2947/200000\"\"]\"\n70,True,7/2,1,9/2,5991/2,7/10,19991/2,14000/19991,14,19/2,19/2,49/5,-3/10,\"[\"\"19739/2\"\",\"\"20299/2\"\"]\",\"[\"\"-261/20000\"\",\"\"299/20000\"\"]\"\n71,True,71/20,1,91/20,57909/20,71/100,199909/20,142000/199909,71/5,193/20,193/20,5041/500,-54/125,\"[\"\"197353/20\"\",\"\"203033/20\"\"]\",\"[\"\"-2647/200000\"\",\"\"3033/200000\"\"]\"\n72,True,18/5,1,23/5,13977/5,18/25,49977/5,4000/5553,72/5,49/5,49/5,1296/125,-71/125,\"[\"\"49329/5\"\",\"\"50769/5\"\"]\",\"[\"\"-671/50000\"\",\"\"769/50000\"\"]\"\n73,True,73/20,1,93/20,53907/20,73/100,199907/20,146000/199907,73/5,199/20,199/20,5329/500,-177/250,\"[\"\"197279/20\"\",\"\"203119/20\"\"]\",\"[\"\"-2721/200000\"\",\"\"3119/200000\"\"]\"\n74,True,37/10,1,47/10,25953/10,37/50,99953/10,74000/99953,74/5,101/10,101/10,1369/125,-213/250,\"[\"\"98621/10\"\",\"\"101581/10\"\"]\",\"[\"\"-1379/100000\"\",\"\"1581/100000\"\"]\"\n75,True,15/4,1,19/4,9981/4,3/4,39981/4,10000/13327,15,41/4,41/4,45/4,-1,\"[\"\"39441/4\"\",\"\"40641/4\"\"]\",\"[\"\"-559/40000\"\",\"\"641/40000\"\"]\"\n76,True,19/5,1,24/5,11976/5,19/25,49976/5,4750/6247,76/5,52/5,52/5,1444/125,-144/125,\"[\"\"49292/5\"\",\"\"50812/5\"\"]\",\"[\"\"-177/12500\"\",\"\"203/12500\"\"]\"\n77,True,77/20,1,97/20,45903/20,77/100,199903/20,14000/18173,77/5,211/20,211/20,5929/500,-327/250,\"[\"\"197131/20\"\",\"\"203291/20\"\"]\",\"[\"\"-2869/200000\"\",\"\"3291/200000\"\"]\"\n78,True,39/10,1,49/10,21951/10,39/50,99951/10,26000/33317,78/5,107/10,107/10,1521/125,-367/250,\"[\"\"98547/10\"\",\"\"101667/10\"\"]\",\"[\"\"-1453/100000\"\",\"\"1667/100000\"\"]\"\n79,True,79/20,1,99/20,41901/20,79/100,199901/20,158000/199901,79/5,217/20,217/20,6241/500,-204/125,\"[\"\"197057/20\"\",\"\"203377/20\"\"]\",\"[\"\"-2943/200000\"\",\"\"3377/200000\"\"]\"\n80,False,4,1,5,1995,4/5,9995,1600/1999,16,11,11,64/5,-9/5,\"[\"\"9851\"\",\"\"10171\"\"]\",\"[\"\"-149/10000\"\",\"\"171/10000\"\"]\"\n", "rows_scope": "All q=0..80, including infeasible candidates", "format": "CSV with header; scalar rationals are exact. Two-state arrays list exact lower/upper values." }, "no_cost": { "feasible": true, "input": { "mu": { "exact": "1/500", "decimal": 0.002 }, "rho": { "exact": "0", "decimal": 0.0 }, "cash_min": { "exact": "2000", "decimal": 2000.0 }, "cap": 80, "mode": "no_cost", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 50 ], "display_q": 50, "largest_feasible_q": 80, "objective": { "exact": "5", "decimal": 5.0 }, "exact_rows_csv": "q,feasible,spread,commission,cost,cash,x,post_cost_wealth,post_weight_risky,expected_gross_pnl,nominal_expected_net_pnl,worst_expected_net_pnl,risk_penalty_dollars,objective_dollars,terminal_mid_wealth,returns_prewealth\n0,True,0,0,0,10000,0,10000,0,0,0,0,0,0,\"[\"\"10000\"\",\"\"10000\"\"]\",\"[\"\"0\"\",\"\"0\"\"]\"\n50,True,0,0,0,5000,1/2,10000,1/2,10,10,10,5,5,\"[\"\"9910\"\",\"\"10110\"\"]\",\"[\"\"-9/1000\"\",\"\"11/1000\"\"]\"\n80,True,0,0,0,2000,4/5,10000,4/5,16,16,16,64/5,16/5,\"[\"\"9856\"\",\"\"10176\"\"]\",\"[\"\"-9/625\"\",\"\"11/625\"\"]\"\n", "rows_scope": "q=0, all tied optima, display q and largest feasible q", "format": "CSV with header; scalar rationals are exact. Two-state arrays list exact lower/upper values." }, "spread_only": { "feasible": true, "input": { "mu": { "exact": "1/500", "decimal": 0.002 }, "rho": { "exact": "0", "decimal": 0.0 }, "cash_min": { "exact": "2000", "decimal": 2000.0 }, "cap": 80, "mode": "spread_only", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 37, 38 ], "display_q": 37, "largest_feasible_q": 79, "objective": { "exact": "703/250", "decimal": 2.812 }, "exact_rows_csv": "q,feasible,spread,commission,cost,cash,x,post_cost_wealth,post_weight_risky,expected_gross_pnl,nominal_expected_net_pnl,worst_expected_net_pnl,risk_penalty_dollars,objective_dollars,terminal_mid_wealth,returns_prewealth\n0,True,0,0,0,10000,0,10000,0,0,0,0,0,0,\"[\"\"10000\"\",\"\"10000\"\"]\",\"[\"\"0\"\",\"\"0\"\"]\"\n37,True,37/20,0,37/20,125963/20,37/100,199963/20,74000/199963,37/5,111/20,111/20,1369/500,703/250,\"[\"\"198631/20\"\",\"\"201591/20\"\"]\",\"[\"\"-1369/200000\"\",\"\"1591/200000\"\"]\"\n38,True,19/10,0,19/10,61981/10,19/50,99981/10,38000/99981,38/5,57/10,57/10,361/125,703/250,\"[\"\"99297/10\"\",\"\"100817/10\"\"]\",\"[\"\"-703/100000\"\",\"\"817/100000\"\"]\"\n79,True,79/20,0,79/20,41921/20,79/100,199921/20,158000/199921,79/5,237/20,237/20,6241/500,-79/125,\"[\"\"197077/20\"\",\"\"203397/20\"\"]\",\"[\"\"-2923/200000\"\",\"\"3397/200000\"\"]\"\n", "rows_scope": "q=0, all tied optima, display q and largest feasible q", "format": "CSV with header; scalar rationals are exact. Two-state arrays list exact lower/upper values." }, "low_mu": { "feasible": true, "input": { "mu": { "exact": "3/2500", "decimal": 0.0012 }, "rho": { "exact": "0", "decimal": 0.0 }, "cash_min": { "exact": "2000", "decimal": 2000.0 }, "cap": 80, "mode": "spread_and_commission", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 0 ], "display_q": 0, "largest_feasible_q": 79, "objective": { "exact": "0", "decimal": 0.0 }, "exact_rows_csv": "q,feasible,spread,commission,cost,cash,x,post_cost_wealth,post_weight_risky,expected_gross_pnl,nominal_expected_net_pnl,worst_expected_net_pnl,risk_penalty_dollars,objective_dollars,terminal_mid_wealth,returns_prewealth\n0,True,0,0,0,10000,0,10000,0,0,0,0,0,0,\"[\"\"10000\"\",\"\"10000\"\"]\",\"[\"\"0\"\",\"\"0\"\"]\"\n79,True,79/20,1,99/20,41901/20,79/100,199901/20,158000/199901,237/25,453/100,453/100,6241/500,-994/125,\"[\"\"984653/100\"\",\"\"1016253/100\"\"]\",\"[\"\"-15347/1000000\"\",\"\"16253/1000000\"\"]\"\n", "rows_scope": "q=0, all tied optima, display q and largest feasible q", "format": "CSV with header; scalar rationals are exact. Two-state arrays list exact lower/upper values." }, "box0006": { "feasible": true, "input": { "mu": { "exact": "1/500", "decimal": 0.002 }, "rho": { "exact": "3/5000", "decimal": 0.0006 }, "cash_min": { "exact": "2000", "decimal": 2000.0 }, "cap": 80, "mode": "spread_and_commission", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 22, 23 ], "display_q": 22, "largest_feasible_q": 79, "objective": { "exact": "3/250", "decimal": 0.012 }, "exact_rows_csv": "q,feasible,spread,commission,cost,cash,x,post_cost_wealth,post_weight_risky,expected_gross_pnl,nominal_expected_net_pnl,worst_expected_net_pnl,risk_penalty_dollars,objective_dollars,terminal_mid_wealth,returns_prewealth\n0,True,0,0,0,10000,0,10000,0,0,0,0,0,0,\"[\"\"10000\"\",\"\"10000\"\"]\",\"[\"\"0\"\",\"\"0\"\"]\"\n22,True,11/10,1,21/10,77979/10,11/50,99979/10,2000/9089,22/5,23/10,49/50,121/125,3/250,\"[\"\"99583/10\"\",\"\"100463/10\"\"]\",\"[\"\"-417/100000\"\",\"\"463/100000\"\"]\"\n23,True,23/20,1,43/20,153957/20,23/100,199957/20,46000/199957,23/5,49/20,107/100,529/500,3/250,\"[\"\"199129/20\"\",\"\"200969/20\"\"]\",\"[\"\"-871/200000\"\",\"\"969/200000\"\"]\"\n79,True,79/20,1,99/20,41901/20,79/100,199901/20,158000/199901,79/5,217/20,611/100,6241/500,-1593/250,\"[\"\"197057/20\"\",\"\"203377/20\"\"]\",\"[\"\"-2943/200000\"\",\"\"3377/200000\"\"]\"\n", "rows_scope": "q=0, all tied optima, display q and largest feasible q", "format": "CSV with header; scalar rationals are exact. Two-state arrays list exact lower/upper values." }, "cap10": { "feasible": true, "input": { "mu": { "exact": "1/500", "decimal": 0.002 }, "rho": { "exact": "0", "decimal": 0.0 }, "cash_min": { "exact": "2000", "decimal": 2000.0 }, "cap": 10, "mode": "spread_and_commission", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 10 ], "display_q": 10, "largest_feasible_q": 10, "objective": { "exact": "3/10", "decimal": 0.3 }, "exact_rows_csv": "q,feasible,spread,commission,cost,cash,x,post_cost_wealth,post_weight_risky,expected_gross_pnl,nominal_expected_net_pnl,worst_expected_net_pnl,risk_penalty_dollars,objective_dollars,terminal_mid_wealth,returns_prewealth\n0,True,0,0,0,10000,0,10000,0,0,0,0,0,0,\"[\"\"10000\"\",\"\"10000\"\"]\",\"[\"\"0\"\",\"\"0\"\"]\"\n10,True,1/2,1,3/2,17997/2,1/10,19997/2,2000/19997,2,1/2,1/2,1/5,3/10,\"[\"\"19961/2\"\",\"\"20041/2\"\"]\",\"[\"\"-39/20000\"\",\"\"41/20000\"\"]\"\n", "rows_scope": "q=0, all tied optima, display q and largest feasible q", "format": "CSV with header; scalar rationals are exact. Two-state arrays list exact lower/upper values." }, "cash9000": { "feasible": true, "input": { "mu": { "exact": "1/500", "decimal": 0.002 }, "rho": { "exact": "0", "decimal": 0.0 }, "cash_min": { "exact": "9000", "decimal": 9000.0 }, "cap": 80, "mode": "spread_and_commission", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 9 ], "display_q": 9, "largest_feasible_q": 9, "objective": { "exact": "47/250", "decimal": 0.188 }, "exact_rows_csv": "q,feasible,spread,commission,cost,cash,x,post_cost_wealth,post_weight_risky,expected_gross_pnl,nominal_expected_net_pnl,worst_expected_net_pnl,risk_penalty_dollars,objective_dollars,terminal_mid_wealth,returns_prewealth\n0,True,0,0,0,10000,0,10000,0,0,0,0,0,0,\"[\"\"10000\"\",\"\"10000\"\"]\",\"[\"\"0\"\",\"\"0\"\"]\"\n9,True,9/20,1,29/20,181971/20,9/100,199971/20,2000/22219,9/5,7/20,7/20,81/500,47/250,\"[\"\"199647/20\"\",\"\"200367/20\"\"]\",\"[\"\"-353/200000\"\",\"\"367/200000\"\"]\"\n", "rows_scope": "q=0, all tied optima, display q and largest feasible q", "format": "CSV with header; scalar rationals are exact. Two-state arrays list exact lower/upper values." } }, "official_commission_examples": [ { "q": 100, "price": { "exact": "25", "decimal": 25.0 }, "commission": { "exact": "1", "decimal": 1.0 } }, { "q": 1000, "price": { "exact": "25", "decimal": 25.0 }, "commission": { "exact": "5", "decimal": 5.0 } }, { "q": 1000, "price": { "exact": "1/4", "decimal": 0.25 }, "commission": { "exact": "5/2", "decimal": 2.5 } }, { "q": 10, "price": { "exact": "1/5", "decimal": 0.2 }, "commission": { "exact": "1/50", "decimal": 0.02 } } ], "complete_exact_results": { "uri": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/results.json", "locator": "cost.integer..rows", "purpose": "Read the selected scenario here when reproducing every candidate with exact fractions. The packet includes all default candidates and exact worked rows for all taught scenarios; full attachment is unchanged." } } }, "entry_id": "zh-qt23", "content_version": "2026-09-21-QT-DE-review-v2", "audience": "高年级本科至研究生；默认微积分、线性代数与基本概率", "learning_task": "由精确费用预算推导不交易带，再用整股佣金枚举检验连续近似的边界。" } ``` ## Supplied entry 预测给出的是关于结果的判断，决策还要回答“现在允许买多少、花掉哪些现金、愿意承受什么风险”。本篇先完整求解一个比例费用模型，再换成整股委托和实际最低佣金规则。两层使用同一种预算思路，却不是同一个优化问题。读完后，你应能独立核对现金、推导不交易带，并解释为什么最优数量可能不唯一。 ## 1. 决策变量先有分母，成本才有位置 先取一期现金收益 $r_f=0$，调仓前财富为 1，原风险资产金额为 $w_0$。令 $x$ 表示调仓后风险资产的中价金额，仍除以调仓前财富；交易成本为 $C(x)$，从期初现金扣除。无新增资金时 $$ c=1-x-C(x),\qquad x+c+C(x)=1. $$ $c$ 是留下的现金金额，而非在交易后财富下重新归一化的权重。期初中价标记的剩余财富是 $1-C$，因此真正的交易后权重是 $x/(1-C)$ 与 $c/(1-C)$，两者才合计为 1。把 $x+c=1$ 和另一处“费用已扣掉”同时写入账本，会让同一笔现金出现两次。[^cvx] 若风险资产一期简单收益为 $R$，期末财富是 $x(1+R)+c$，所以以原财富为分母的组合净回报为 $R_p=xR-C$。若另取确定现金收益 $r_f$，相同推导给 $R_p=x(R-r_f)+r_f-(1+r_f)C$；费用在期初支付，会同时失去相应现金利息。本篇固定 $r_f=0$，不把这项时点差异藏进近似。 下面的收益均值、波动、价差和风险参数都是教学设定，不是从 BusEq 历史均值直接宣称得到的预测。只有后面注明的佣金组件来自真实公开规则。 ## 2. 比例成本为什么产生不交易带？ 本实验取 $0\le w_0,u\le1$、$0\le\kappa<1$、$c_{\min}\ge0$，并要求 $\gamma>0,\sigma>0$；可能没有可行解，届时先报告预算不成立。取 $E[R]=\mu$、$\operatorname{Var}(R)=\sigma^2$，成本为 $C(x)=\kappa|x-w_0|$。明确选择一个均值—方差准则：最大化净回报期望减去 $\gamma/2$ 倍方差。这是本例的决策目标，不宣称代表所有人的效用。[^cvx] $$ J(x)=\mu x-\tfrac12\gamma\sigma^2x^2-\kappa|x-w_0|, $$ 约束为 $0\le x\le u$，以及 $1-x-\kappa|x-w_0|\ge c_{\min}$。上限 $u$ 限风险金额，现金约束又会进一步缩小可行范围；它们不是一回事。 令 $a=\gamma\sigma^2>0$。目标是连续严格凹函数，可行集若非空且有界，就是一个闭区间，因此存在唯一最优解。这里的一维证明不需要先讲整套 KKT。凸目标的支撑不等式，或等价地凹目标的切线界，保证符合可行方向条件的解是全局最优。[^optimality] 在 $w_0$ 两侧，普通导数分别为 $$ J'(x)= \begin{cases} \mu-ax+\kappa,&xw_0. \end{cases} $$ 若 $w_0$ 是可行区间内点，不交易恰好要求左导数非负、右导数非正，于是 $$ |\mu-aw_0|\le\kappa. $$ 若 $\mu-aw_0>\kappa$，向右买入，先求 $x=(\mu-\kappa)/a$；若 $\mu-aw_0<-\kappa$，向左卖出，先求 $x=(\mu+\kappa)/a$。再将该无约束候选截到可行区间。严格凹性使这一局部导数判断足够：越过最优点后，导数不可能再升回正值。 这个带的经济含义是：小幅改变持仓带来的边际好处，还不足以支付买卖方向上的边际费用。不是所有预测变动都值得下单。但若原持仓本身违反新现金要求，就不能拿内点不等式为“不交易”辩护，只能沿可行方向重新选择。 ## 3. 把默认解与资金账一起算出来 固定 $w_0=0.5,\mu=0.015,\sigma=0.1,\gamma=2,\kappa=0.002,u=0.8,c_{\min}=0.2$。$a=0.02$，不交易的均值带为 $[0.008,0.012]$。默认均值在带上方，所以 $$ x^*=\frac{0.015-0.002}{0.02}=0.65. $$ 此处候选满足所有约束。费用 $0.002(0.65-0.5)=0.0003$，现金为 $0.3497$，成本后的财富是 $0.9997$。真正的风险权重为 $0.65/0.9997=500/769\approx0.650195$，不是恰好 65%。 | 默认连续解：均以调仓前财富为基数 | 数值 | |---|---:| | 风险资产金额 $x$ | 0.65 | | 现金 $c$ | 0.3497 | | 费用 $C$ | 0.0003 | | 名义均值下净回报期望 $\mu x-C$ | 0.00945 | | 风险罚项 $\gamma\sigma^2x^2/2$ | 0.004225 | | 目标 $J$ | 0.005225 | 目标 0.005225 不是期望净回报，更不是某一期会赚到的回报。三者必须分开。默认可行上界实际为 $\min(0.8,(1+\kappa w_0-c_{\min})/(1+\kappa))=267/334\approx0.799401$，所以仅看 $u=0.8$ 会遗漏费用占用。[^calculation] 把风险金额上限收紧到 $u=0.6$，最优点被截到 0.6；不是重新发现了一个不同的市场均值。如果把最低现金改为 0.8，原持仓 0.5 已不可行。卖出侧现金为 $1-\kappa w_0-(1-\kappa)x$，因此 $x\le(1-\kappa w_0-0.8)/(1-\kappa)\approx0.199399$，必须大幅减仓。 ## 4. 预测不确定性怎样进入目标？ 现在不改名义 $\mu$，而是指定一组可能均值 $\mu+\delta$，其中 $|\delta|\le\rho$。对 long-only 的 $x\ge0$，最坏均值是 $\mu-\rho$。多资产盒集的计算同样直接： $$ \inf_{|\delta_i|\le\rho_i}(\mu+\delta)^Tx =\mu^Tx-\sum_i\rho_i|x_i|. $$ 证明是逐项选择 $\delta_i=-\rho_i\operatorname{sign}(x_i)$；各坐标约束互不耦合，所以可以同时达到这个下界。它来自明确指定的集合，不是把所有未知量统一打折。Boyd 等人的 2024 稿件把这类均值区间放进组合构建，并分开讨论约束、交易成本预测和非凸实施。[^robust] 本例取 $\rho=0.004$，最坏均值成为 0.011，落入不交易带，所以 $x^*=0.5$。名义净回报期望仍按 0.015 算，是 0.0075；最坏均值下是 0.0055；风险罚项 0.0025；稳健目标为 0.003。不同 $\rho$ 改变的是所优化的问题，不能比较两次目标的大小，就宣称哪次预测更准确。 若要把 $\rho$ 称为统计置信半径，必须另外给出估计对象、数据和推断条件。本例没有这样做，也没有把论文中对未来成交量的预测换成事后已知成交量。现金预算仍按这里的精确期初账执行，不沿用忽略费用的权重近似。 ## 5. 真正下整股委托时，连续模型还剩下多少？ 换一个独立但同类的教学账户：财富 $W=10000$ 美元、原持仓零，假设中价 100、买价 100.05，$q$ 为 0—80 的整数，最低现金 2000，现金收益为零。假设一期股票收益为 $\mu\pm\sigma$，各概率 $1/2$，其中 $\mu=0.002,\sigma=0.02,\gamma=10$。这些是研究概率下的构造情景，不是报价或定价测度。 作为真实规则组件，2026-09-21 所核 IBKR Pro 美国 Fixed 普通整股单笔委托的佣金写成：下单时 $$ C_{comm}(q,p)=\min\{0.01p|q|,\;\max(1,0.005|q|)\}, $$ 不下单则为零。这里 $p$ 为成交价，规则包括每股费、最低费与成交金额上限。100 股×25 美元收 1 美元；1000 股×25 收 5；1000 股×0.25 收 2.50；10 股×0.20 时上限只有 0.02，即按 0.02 而不是 1 收取。其他监管、交易所等费用、税、冲击及退出交易没有包含，不能称作全部账单。[^commission] 本例还计入假设价差 $0.05q$。令 $C_q=0.05q+C_{comm}(q,100.05)$，于是 $$ \begin{aligned} \text{现金}&=W-100q-C_q,\\ x&=100q/W,\\ J_{\$}(q)&=100q\mu-C_q-\frac{W\gamma}{2}(x\sigma)^2. \end{aligned} $$ $J_{\$}$ 是原财富乘以均值—方差准则的美元表示。整数和最低佣金使它不再是前面那个连续凸优化问题；这里只有 81 个候选，逐个枚举即可，无须用求导假装获得唯一整股解。 | 默认整股例 | $q=37$ | $q=38$ | |---|---:|---:| | 假设价差（美元） | 1.85 | 1.90 | | 佣金组件（美元） | 1.00 | 1.00 | | 剩余现金（美元） | 6297.15 | 6197.10 | | 名义净损益期望（美元） | 4.55 | 4.70 | | 风险罚项（美元） | 2.738 | 2.888 | | 目标（美元） | 1.812 | 1.812 | 两个股数并列最优。显示 37 只是预先约定的同分取较小股数，不把另一个最优解藏起来。在选中的 $q=37$ 下，两个期末中价标记财富分别为 9930.55 和 10078.55，按原财富计算回报是 −0.6945% 和 0.7855%；平均净损益 4.55。期末尚未发出退出委托，所以不称为往返交易净利。[^calculation] | 改一项条件 | 全部最优股数 | 最优目标（美元） | |---|---|---:| | 无费用对照：按中价交易 | 50 | 5 | | 只计假设价差 | 37、38 | 2.812 | | 价差与佣金组件 | 37、38 | 1.812 | | 名义 $\mu=0.0012$ | 0 | 0 | | 指定 $\rho=0.0006$ | 22、23 | 0.012 | | 教学可买数量上限为 10 | 10 | 0.300 | | 最低现金为 9000 | 9 | 0.188 | 稳健行的数字是稳健目标，不是实际利润。默认 $q=80$ 的现金只有 1995，现金低于最低要求，即使它没有违反股数上限；最大可行数量是 79，现金为 2095.05。[^calculation]
先在连续模式指出预计会生效的约束，再改变均值或费用；随后在整数模式同时看所有候选、并列最优、现金和期望净损益。曲线最高的位置必须属于可行集，且目标轴不是利润轴。 ## 6. 独立重建与解析 解释题。 连续默认解 $x=0.65$，为什么风险权重不是恰好 65%？为什么目标 0.005225 不是净回报期望？ 解析。 $x$ 以调仓前财富为分母，费用后财富为 0.9997；重新归一化后风险权重为 $0.65/0.9997$。净回报期望是 $0.015\times0.65-0.0003=0.00945$，再减风险罚项 0.004225 才得到目标。风险罚项没有付给交易所，也不从现金中再扣一次。 迁移题一。 连续模型改成 $\mu=0.011$，其余默认，是否交易？若另把最低现金改成 0.8，能否继续使用相同答案？ 解析。 原默认可行域下 $|0.011-0.02\times0.5|=0.001\le0.002$，原持仓为可行内点，所以不交易。最低现金 0.8 时原持仓不可行，内点条件的前提失效；卖出侧可行上界约为 0.199399；整个可行区间内 $J'(x)=0.011-0.02x+0.002>0$，所以最优点就是该上界，不能继续维持 0.5。 迁移题二。 重算 $q=37$ 的现金、平均净损益和目标；再判断 $q=80$ 是否可行。 解析。 现金 $10000-3700-1.85-1=6297.15$；名义毛损益期望 $3700\times0.002=7.4$，净期望 $7.4-2.85=4.55$；罚项 $10000\times10\times0.37^2\times0.02^2/2=2.738$；目标 $4.55-2.738=1.812$。$q=80$ 的现金为 $10000-8000-4-1=1995<2000$，不可行。反过来，37 和 38 的目标相同并不表示两者的风险或期望损益相同。 迁移题三。 多资产盒集中有一项 $x_i<0$，为什么最坏均值是 $\mu_i+\rho_i$ 而非 $\mu_i-\rho_i$？ 解析。 负头寸乘以更高收益，贡献更差，因此对手在该坐标选最大均值。最坏贡献统一写作 $\mu_i x_i-\rho_i|x_i|$。换成耦合的不确定性集合后不能再逐坐标独立取端点，必须重做极小化。 至此，预测、验证与决策的分工才闭合：预测量需要合适的数据与评价，决策把它当作有不确定性的输入，在实际资金边界内求解。求得最优只说明模型内比较已完成，不能替代对输入和制度的核实。 [^cvx]: Boyd、Vandenberghe，[Convex Optimization](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf)，2004，Markowitz portfolio optimization 子节，印刷 pp.155–156。本篇使用明确的期初费用预算，并固定 $r_f=0$。 [^optimality]: 同书 §4.2.2–4.2.3，pp.138–140，可行凸集上的最优性与完整方向证明；本篇的一维不交易带由左右导数直接证明，不依赖未展开的一般 KKT。 [^robust]: Boyd 等，[Markowitz Portfolio Construction at Seventy](https://web.stanford.edu/~boyd/papers/pdf/markowitz.pdf)，所读稿为 2024-01-05，§2.1–2.4、§3.4–3.5、§4.1–4.3；盒集最坏均值见 p.23 式(16)。论文发表与稿件日期分开，本篇不采用其绩效主张。 [^commission]: IBKR，[美国股票 Fixed 佣金表](https://www.interactivebrokers.com/en/pricing/commissions-stocks.php?re=amer)，2026-09-21 取得的 US table、Fixed Examples、最低/最高费用及相关说明。适用对象为普通整股单笔委托的佣金组件，非全部成本。 [^calculation]: [输入与身份](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/shared_inputs.json)、[全部精确分数及枚举结果](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/data/results.json) cost 组、[复算程序](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/compute/reproduce.py)。教学收益、价差和容量与佣金事实分开。 ## Experiment inputs and static equivalents ```json [ { "id": "EXP-QT23-COST-01", "title": "成本、约束与稳健决策：可复算实验", "anchor": "qt23-integer", "description": "由精确费用预算推导不交易带，再用整股佣金枚举检验连续近似的边界。", "inputs": { "identity": "QT-DE-adopted-inputs-20260921-v1", "manifest": "https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/shared_inputs.json" }, "outputs": { "default_continuous": { "feasible": true, "input": { "mu": { "exact": "3/200", "decimal": 0.015 }, "rho": { "exact": "0", "decimal": 0.0 }, "cap": { "exact": "4/5", "decimal": 0.8 }, "cash_min": { "exact": "1/5", "decimal": 0.2 }, "kappa": { "exact": "1/500", "decimal": 0.002 }, "w0": { "exact": "1/2", "decimal": 0.5 }, "sigma": { "exact": "1/10", "decimal": 0.1 }, "gamma": { "exact": "2", "decimal": 2.0 }, "rf": { "exact": "0", "decimal": 0.0 } }, "upper": { "exact": "267/334", "decimal": 0.7994011976047904 }, "x": { "exact": "13/20", "decimal": 0.65 }, "cost": { "exact": "3/10000", "decimal": 0.0003 }, "cash": { "exact": "3497/10000", "decimal": 0.3497 }, "post_cost_wealth": { "exact": "9997/10000", "decimal": 0.9997 }, "post_weight_risky": { "exact": "500/769", "decimal": 0.6501950585175552 }, "post_weight_cash": { "exact": "269/769", "decimal": 0.34980494148244473 }, "nominal_expected_net_return": { "exact": "189/20000", "decimal": 0.00945 }, "worst_expected_net_return": { "exact": "189/20000", "decimal": 0.00945 }, "risk_penalty": { "exact": "169/40000", "decimal": 0.004225 }, "objective": { "exact": "209/40000", "decimal": 0.005225 }, "no_trade_mu_band": [ { "exact": "1/125", "decimal": 0.008 }, { "exact": "3/250", "decimal": 0.012 } ], "w0_feasible_interior": true }, "default_integer": { "feasible": true, "input": { "mu": { "exact": "1/500", "decimal": 0.002 }, "rho": { "exact": "0", "decimal": 0.0 }, "cash_min": { "exact": "2000", "decimal": 2000.0 }, "cap": 80, "mode": "spread_and_commission", "wealth": { "exact": "10000", "decimal": 10000.0 }, "mid": { "exact": "100", "decimal": 100.0 }, "ask": { "exact": "2001/20", "decimal": 100.05 }, "sigma": { "exact": "1/50", "decimal": 0.02 }, "gamma": { "exact": "10", "decimal": 10.0 }, "rf": 0 }, "optimal_q_all_ties": [ 37, 38 ], "display_q": 37, "largest_feasible_q": 79, "objective": { "exact": "453/250", "decimal": 1.812 }, "all_candidate_rows": "agent_packet.supplied_inputs.cost_results_for_teaching.integer.default.exact_rows_csv", "exact_worked_optima": [ { "q": 37, "feasible": true, "spread": { "exact": "37/20", "decimal": 1.85 }, "commission": { "exact": "1", "decimal": 1.0 }, "cost": { "exact": "57/20", "decimal": 2.85 }, "cash": { "exact": "125943/20", "decimal": 6297.15 }, "x": { "exact": "37/100", "decimal": 0.37 }, "post_cost_wealth": { "exact": "199943/20", "decimal": 9997.15 }, "post_weight_risky": { "exact": "74000/199943", "decimal": 0.3701054800618176 }, "expected_gross_pnl": { "exact": "37/5", "decimal": 7.4 }, "nominal_expected_net_pnl": { "exact": "91/20", "decimal": 4.55 }, "worst_expected_net_pnl": { "exact": "91/20", "decimal": 4.55 }, "risk_penalty_dollars": { "exact": "1369/500", "decimal": 2.738 }, "objective_dollars": { "exact": "453/250", "decimal": 1.812 }, "terminal_mid_wealth": [ { "exact": "198611/20", "decimal": 9930.55 }, { "exact": "201571/20", "decimal": 10078.55 } ], "returns_prewealth": [ { "exact": "-1389/200000", "decimal": -0.006945 }, { "exact": "1571/200000", "decimal": 0.007855 } ] }, { "q": 38, "feasible": true, "spread": { "exact": "19/10", "decimal": 1.9 }, "commission": { "exact": "1", "decimal": 1.0 }, "cost": { "exact": "29/10", "decimal": 2.9 }, "cash": { "exact": "61971/10", "decimal": 6197.1 }, "x": { "exact": "19/50", "decimal": 0.38 }, "post_cost_wealth": { "exact": "99971/10", "decimal": 9997.1 }, "post_weight_risky": { "exact": "38000/99971", "decimal": 0.3801102319672705 }, "expected_gross_pnl": { "exact": "38/5", "decimal": 7.6 }, "nominal_expected_net_pnl": { "exact": "47/10", "decimal": 4.7 }, "worst_expected_net_pnl": { "exact": "47/10", "decimal": 4.7 }, "risk_penalty_dollars": { "exact": "361/125", "decimal": 2.888 }, "objective_dollars": { "exact": "453/250", "decimal": 1.812 }, "terminal_mid_wealth": [ { "exact": "99287/10", "decimal": 9928.7 }, { "exact": "100807/10", "decimal": 10080.7 } ], "returns_prewealth": [ { "exact": "-713/100000", "decimal": -0.00713 }, { "exact": "807/100000", "decimal": 0.00807 } ] } ] } } } ] ``` ## Sources - [QT-D/E adopted teaching experiments](https://ou-liu-red-sugar.github.io/notebook/labs/qt-de/compute/reproduce.py): 本轮实际沙盒复算，预测遵守冻结算法及数据；选择沿PCG64调用顺序并核strict/inclusive，成本用Fraction。数学演算与假设模型不等于真实投资有效性。 - [Convex Optimization](https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf): 已读可行凸集最优性完整证明和组合预算子节。本站不交易带由一维严格凹直接推出，不引用未读强对偶或KKT证明。 - [US stocks IBKR Pro Fixed commission component](https://www.interactivebrokers.com/en/pricing/commissions-stocks.php?re=amer): 普通整股单笔委托每股0.005、最低1、最高成交金额1%，上限低于最低时按上限。仅佣金组件；其余收费与零股/改单/退出未纳入。 - [Markowitz Portfolio Construction at Seventy](https://web.stanford.edu/~boyd/papers/pdf/markowitz.pdf): 规划轮完整单元；本轮回核p.16/18/23原页。成本、约束、均值盒集、非凸实施；本站采用精确期初现金账而非稿中忽略/预测费用的近似。未采用§5回测绩效。 ## Content relations ```json [ { "from": "zh-qt23", "relation": "part_of", "to": "quant-estimation", "reason": "主要 topic 归属" }, { "from": "zh-qt23", "relation": "requires", "to": "zh-qt04", "required_competence": "期望、方差与量纲", "reason": "本篇使用该能力，不要求整门随机过程课程。" }, { "from": "zh-qt23", "relation": "requires", "to": "zh-qt19", "required_competence": "估计量与指定不确定集的区别", "reason": "本篇使用该能力，不要求整门随机过程课程。" }, { "from": "qt23-integer", "relation": "illustrated_by", "to": "EXP-QT23-COST-01", "reason": "完整输入、默认及静态等价支持本篇独立任务。" }, { "from": "zh-qt23", "relation": "supported_by", "to": "QTDE-CVX", "locator": "§4.2.2–4.2.3 pp.138–140；Markowitz子节pp.155–156", "scope": "凸可行集最优性、预算与费用子单元", "reason": "核严格凹直接证明，不补造KKT依赖" }, { "from": "zh-qt23", "relation": "supported_by", "to": "QTDE-MARKOWITZ", "locator": "§2.1–2.4 pp.13–18；§3.4–3.5 pp.21–24；§4.1–4.3 pp.24–28", "scope": "交易约束、成本预测、盒集及非凸实施完整采用单元", "reason": "分清论文近似与本站精确预算；区间半径不是已估置信界" }, { "from": "zh-qt23", "relation": "supported_by", "to": "QTDE-IBKR", "locator": "US Fixed table、Fixed Examples、minimum/maximum以及相关适用脚注完整单元", "scope": "普通整股单笔佣金组件", "reason": "确认规则日期、上限优先与费用排除" } ] ``` ## Related entries ## Optional reading path 理解模型并亲手算: step 9/9 从期初资金账推导不交易带，再枚举有最低佣金的整股可行集。