题目5764 · 金融与交易
Convexity Contribution To Return
A bond has modified duration 7 and convexity 90. For a yield increase of 150 basis points, what is the convexity adjustment alone (the second-order term) as a percentage of price?
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中文题目题目2012 · 数学
Convexity of a Tighter Utilization Penalty 17
The utilization cap is tighter, but the same barrier argument applies. Show that r(x) = -ln(1-3x) + 2x^2 is convex on x < 0.333333.
打开 →题目2005 · 数学
Convexity of a Two-Asset Quadratic With a Balance-Sheet Term 10
Show that H(w_1,w_2)=2w_1^2+5w_2^2+3(w_1+w_2)^2 is convex.
打开 →题目2019 · 数学
Convexity of Logistic Loss 24
Show that ell(z)=ln(1+e^{-z}) is convex on R.
打开 →题目2424 · 机器学习
Convexity of the Log-Cosh Loss 4
Show that ell(r)=ln cosh(r) is convex in the residual r.
打开 →题目5131 · 金融与交易
Duration-Convexity Approximation 1
A bond has current price 102, modified duration 4.3, and convexity 18. Using the duration-convexity approximation, what price do you estimate after a yield change of 0.01?
打开 →题目5132 · 金融与交易
Duration-Convexity Approximation 2
A bond has current price 98.5, modified duration 3.1, and convexity 11. Using the duration-convexity approximation, what price do you estimate after a yield change of -0.015?
打开 →题目5133 · 金融与交易
Duration-Convexity Approximation 3
A bond has current price 105.2, modified duration 5.5, and convexity 25. Using the duration-convexity approximation, what price do you estimate after a yield change of 0.02?
打开 →题目5134 · 金融与交易
Duration-Convexity Approximation 4
A bond has current price 99, modified duration 2.8, and convexity 9. Using the duration-convexity approximation, what price do you estimate after a yield change of -0.01?
打开 →题目5138 · 金融与交易
Why Convexity Helps
Why is positive convexity generally good for a bond holder when rates move a lot in either direction?
打开 →题目2011 · 数学
Joint Convexity of Size-Time Execution Cost 16
The cost couples size and trading time through a perspective form. Show that P(x,t)=x^2/t + 3 t is convex on the domain t>0.
打开 →题目2002 · 数学
Shifted Log-Sum-Exp Convexity 7
One stress term slopes down and the other slopes up, but the smooth envelope remains convex. Show that g(x) = ln(exp(-1x) + exp(3x + 0)) is convex on R.
打开 →题目1997 · 数学
Strict Convexity of a Barrier-Regularized Cost 2
The desk wants a direct curvature argument, not a vague appeal to 'it looks bowl-shaped'. Show that f(q) = 3 q^2 + 1/(1-q) is strictly convex on q<1.
打开 →题目3361 · 数学
Why Convexity Makes KKT So Powerful
Why do KKT conditions become sufficient, not just necessary, in many convex optimization problems?
打开 →题目2016 · 数学
Why Convexity Matters Operationally 21
Why do trading desks care whether an execution-cost model is convex rather than merely smooth?
打开 →题目2438 · 机器学习
Why Convexity Makes Averaging Predictions Safe
Why does convexity of a loss function support the intuition that averaging similar predictors often cannot hurt too much?
打开 →题目1418 · 脑筋急转弯
Duration-Convexity Approximation 10
Use dP/P ~= -D*dy + 0.5*C*dy^2. If D=6, C=80, and dy=0.01, what percentage price move is implied?
打开 →题目4725 · 数理金融
Butterfly Repair Or Ceiling 10
For equally spaced strikes 100, 110, 120, calls trade at 11.5, 8.4, and 5.8. How much could the middle call price rise before butterfly convexity is first violated?
打开 →题目4721 · 数理金融
Butterfly Repair Or Ceiling 6
For equally spaced strikes 90, 100, 110, call prices at the wings are C(90)=18.2 and C(110)=8.1. What is the largest arbitrage-free value the middle call C(100) can take under butterfly convexity?
打开 →题目4722 · 数理金融
Butterfly Repair Or Ceiling 7
For equally spaced strikes 80, 100, 120, call prices at the wings are C(80)=26 and C(120)=7. What is the largest arbitrage-free value the middle call C(100) can take under butterfly convexity?
打开 →题目4724 · 数理金融
Butterfly Repair Or Ceiling 9
For equally spaced strikes 85, 95, 105, calls trade at 20, 14.8, and 10.5. How much could the middle call price rise before butterfly convexity is first violated?
打开 →题目2003 · 数学
Execution Cost With Stronger Capacity Wall 8
The blow-up term is steeper, but the same convexity logic should still go through. Show that f(q) = 2 q^2 + 4/(1-q) is strictly convex on q<1.
打开 →题目2015 · 数学
Inventory Cost Near a Hard Capacity Limit 20
The PM wants a formal convexity check before using the function in an optimizer. Show that f(q) = 5 q^2 + 2/(1-q) is strictly convex on q<1.
打开 →题目2007 · 数学
Three-Sleeve Convex Balance-Sheet Penalty 12
The third sleeve is much more expensive on its own, but the aggregate term still preserves convexity. Prove that F(w_1,w_2,w_3) = 1w_1^2 + 4w_2^2 + 9w_3^2 + 2(w_1+w_2+w_3)^2 is convex.
打开 →课程利率风险与信用 · 固定收益
凸性与关键利率风险
周三午盘，CPI 数据偏软，长端 CGB（中国国债）一口气下行 30bp。你持有 10 年期 CGB，早盘只用久期算出预期盈利 800 万元；收盘实际记账 850 万元。多出来的 50 万元不是算错，而是凸性（convexity）——久期线性近似系统性遗漏的二阶项。跨周期看，大幅利率变动时无论方向忽略凸性都会偏离实际损益，且决定哪些 CFFEX 上的交易结构...
打开 →模块1.3.2 · 金融与量化投资 · 固定收益
利率风险与信用
fixed-income · duration · dv01 · modified-duration · macaulay-duration · rates-risk · convexity · key-rate-dv01
打开 →课程利率风险与信用 · 固定收益
久期与 DV01：利率的一阶敏感性
周二早上九点半，CFFEX 开盘前。某保险资管（私募 之外的机构主力）固收交易台调出今日风险日报，组合 DV01 行写着 ¥1,240,000——中债利率曲线平移 1bp，账户当日盈亏约 124 万元。这是风险总监开盘前唯一会盯的数字。本课把这行数字拆开：从麦考利久期（Macaulay duration）出发，过到修正久期（modified duration...
打开 →题目1519 · 数学
Adjusted Gross Multiplier 4
Use a second-order Taylor approximation around 0 to estimate (1+3x)^(2) * (1+1x)^(-1) at x=1/40.
打开 →题目2008 · 数学
Barrier Cost With Strong Quadratic Curvature 13
Both the quadratic inventory term and the capacity wall are active sources of curvature. Show that f(q) = 4 q^2 + 3/(1-q) is strictly convex on q<1.
打开 →题目4723 · 数理金融
Butterfly Repair Or Ceiling 8
For equally spaced strikes 95, 100, 105, calls trade at 14, 12.5, and 10. By how much must the middle call price be reduced to remove the butterfly arbitrage?
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