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1955Large Buy-Side Imbalance Skew 10A quoting engine values upside fill opportunities much more than downside ones, so the optimal skew should be meaningfully positive. A market maker chooses a skew x in (-1,1) to maximize G(x) = 9 ln(1+x) + 3 ln(1-x). What skew is optimal?数学困难derivation未尝试免费1956Carry Drag Versus Quadratic Inventory 11A carry term explodes as the state approaches -1, so the desk cannot simply push x downward. Minimize H(x) = 4 x 2 + 9/(1+x) over x > -1.数学简单数值题未尝试免费1957Funding Buffer Versus Risk Penalty 12A funding buffer term falls with x while the quadratic penalty rises, creating an interior optimum. Minimize H(x) = 3 x 2 + 24/(1+x) over x > -1.数学简单数值题未尝试免费1958Verify the Designed Minimizer in a Quadratic-Reciprocal Objective 13Let r>0 and a>0. Show that x=r uniquely minimizes H(x)=a x 2 + 2 a r(1+r) 2 /(1+x) over x>-1.数学中等derivation未尝试免费1959Large Cushion Interior Minimum 14The reciprocal term is strong enough that the optimum sits well away from the boundary. Minimize H(x) = 1 x 2 + 36/(1+x) over x > -1.数学困难derivation未尝试免费1960Derive the Optimizer for an Exponential Asymmetry Objective 15For A>0 and B>0, derive the unique minimizer of J(x)=A e x + B e -2x .数学困难derivation未尝试免费1961Exponential Imbalance Optimizer 16A smooth response score is J(x)=e x + 4 e -2x . What x minimizes J?数学简单数值题未尝试免费1962Balanced Exponential Penalties 17A desk minimizes J(x)=6 e x + 3 e -2x . What x is optimal?数学中等derivation未尝试免费1963How the Exponential Optimum Moves With the Back-End Penalty 18For J(x)=A e x + B e -2x , suppose A stays fixed and B is quadrupled. How does the optimizer x* move?数学中等derivation未尝试免费1964Negative Exponential Tilt 19Minimize J(x)=16 e x + 4 e -2x . What x minimizes J?数学困难derivation未尝试免费1965Derive the Optimizer for a Linear-Softplus Objective 20For 0<m<n, derive the unique maximizer of K(x)=m x - n ln(1+e x).数学困难derivation未尝试免费1966Balanced Softplus Tradeoff 21The linear reward and saturation penalty balance exactly at a central point. The desk maximizes K(x) = 2 x - 4 ln(1+e x). What x is optimal?数学简单数值题未尝试免费1967Positive Softplus Tilt 22The linear reward is strong relative to the saturation penalty, so the optimum should be positive. The desk maximizes K(x) = 3 x - 4 ln(1+e x). What x is optimal?数学简单derivation未尝试免费1968Negative Softplus Tilt 23The saturation penalty dominates until x turns sufficiently negative. The desk maximizes K(x) = 1 x - 3 ln(1+e x). What x is optimal?数学中等derivation未尝试免费1969Which Side of Zero Is Optimal in the Linear-Softplus Problem 24For K(x)=m x - n ln(1+e x) with 0<m<n/2, is the unique optimizer positive, zero, or negative?数学中等derivation未尝试免费1970Maximum Value at the Interior Utility-Impact Optimum 25For F(x)=4 ln(1+x)-x 2 on x>-1, what x maximizes F and what is the maximum value?数学困难derivation未尝试免费1972Derive the Two-Book Hedge Formula 2Derive the minimizer of a x 2 + b y 2 subject to u x + v y = c for positive a,b.数学简单derivation未尝试免费1974Weighted Exposure Hedge Split 4The first hedge line loads twice as much on the required exposure as the second, but also has its own quadratic penalty. Minimize L(x,y) = 2x 2 + 3y 2 subject to 2x + 1y = 12.数学中等数值题未尝试免费1977Derive the Three-Book Total-Size Allocation 7Derive the minimizer of a x 2 + b y 2 + c z 2 subject to x+y+z=N for positive a,b,c.数学中等derivation未尝试免费1978Why the Three-Book Solution Is Inverse-Coefficient Weighted 8In min a x 2 + b y 2 + c z 2 subject to x+y+z=N, why do larger quadratic coefficients receive smaller allocations?数学中等derivation未尝试免费