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中文题目You start short q0 = -150 shares; lot size is 50. Under a symmetric quote, bid-fill probability is 0.30 and ask-fill probability is 0.30. Under a bid-skew policy designed to buy back, bid-fill probability rises to 0.50 while ask-fill probability falls to 0.20. Compute the expecte
打开 →Use a second-order Taylor approximation around 0 to estimate (1+4x)^(-1) * (1+-2x)^(1/2) at x=1/25.
打开 →You are long 100 lots. Skewing the ask down attracts an expected sell of 60 lots this period, each lot offloaded at +0.08 of edge versus your reservation price. The 40 lots that remain carry an expected holding cost of 0.15 per lot. What is the expected PnL of the skew policy thi
打开 →If downside skew gets steeper, what usually happens to the sensitivity of fixed-strike vol to spot moves under sticky-moneyness?
打开 →After a round-1 buy fill leaves you long 1 unit, you choose a round-2 quote skew s (in cents) that shifts both quotes down to encourage a sell. Selling probability is 0.3 + 0.1*s and expected edge per sell is 0.05 - 0.01*s, for s between 0 and 5. Round-2 expected edge is (selling
打开 →Long inventory, you skew quotes to sell. The ask fills with probability 0.45 earning net edge 0.020 per share; the bid fills with probability 0.10 but is toxic, earning net edge -0.030 per share. Each side quotes 100 shares. Assuming at most one side fills, what is the expected P
打开 →Without skew, expected bid-fill probability is 0.32 and ask-fill probability is 0.18. If you increase ask-side skew by s, bid-fill probability becomes 0.32 - 0.02*s and ask-fill probability becomes 0.18 + 0.015*s. What is the smallest nonnegative s that makes expected inventory c
打开 →Let $X_1, X_2, \ldots, X_n$ be i.i.d.\ $\operatorname{Bernoulli}(p)$ with $p = 0.01$ and $n = 10{,}000$. Define $S_n = \sum_{i=1}^{n} X_i$. **(a)** Using the CLT, approximate $P(S_n \le 80)$. **(b)** The Berry-Esseen theorem states that $\sup_x |P(Z_n \le x) - \Phi(x)| \le \fra
打开 →Holding inventory q = 40, your base reservation shift below mid is lambda*q with lambda = 0.01, i.e. 0.40. You believe an adverse downward drift of 0.60 will hit before you can offload, and you want your effective quote center to drop by at least the full 0.60 to keep encouraging
打开 →On one quote axis, the maker gets more value from aggressive bids than from aggressive offers. A market maker chooses a skew x in (-1,1) to maximize G(x) = 5 ln(1+x) + 3 ln(1-x). What skew is optimal?
打开 →A 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?
打开 →Order-flow asymmetry is present but not extreme, so the optimal skew should stay close to flat. A market maker chooses a skew x in (-1,1) to maximize G(x) = 7 ln(1+x) + 5 ln(1-x). What skew is optimal?
打开 →If a market maker maximizes G(x)=a ln(1+x)+a ln(1-x) on x in (-1,1), what skew is optimal?
打开 →Why do market makers treat quote width and quote skew as different levers?
打开 →For a>0 and b>0, derive the unique maximizer of G(x)=a ln(1+x)+b ln(1-x) on x in (-1,1).
打开 →At launch there were twice as many regional funds as global funds. Today, 18 regional funds and 18 global funds remain live. If the regional-fund survival rate was 45%, what was the global-fund survival rate?
打开 →Use a second-order Taylor approximation around 0 to estimate exp(1x)/(1+-1x) at x=1/25.
打开 →Why can a strategy with severe crash risk still look reassuring in ordinary validation windows?
打开 →Why should the optimal query tree usually isolate high-probability states earlier than low-probability states?
打开 →With a typical downside skew, what happens to fixed-strike implied vol under sticky-moneyness after a larger selloff?
打开 →Why do negative jumps create downside skew even when the diffusion part is symmetric?
打开 →Why can stochastic volatility explain a persistent equity-index downside skew more naturally than a constant-volatility Black-Scholes model?
打开 →A maker centers quotes on its reservation price r = 100.0 (already shifted below the 100.4 fair mid by a long inventory). It quotes a total spread of 0.20 but, to attract sells, places the ask only 0.06 above r and the bid the remaining width below r. What are the bid and ask pri
打开 →Let $U_1, \ldots, U_n$ be i.i.d.\ $\mathrm{Uniform}(0,1)$ and $S_n = \sum_{i=1}^n U_i$. The Berry-Esseen theorem states $$\sup_x \left|P\!\left(\frac{S_n - n/2}{\sigma\sqrt{n}} \le x\right) - \Phi(x)\right| \le \frac{C\,\rho}{\sigma^3 \sqrt{n}},$$ where $\sigma^2 = \mathrm{Var}(U
打开 →If the smile has negative slope in log-moneyness, what happens to the fixed-strike implied vol shift under sticky-moneyness when the spot rally becomes larger?
打开 →Let $X_1, \ldots, X_n$ be i.i.d.\ $\operatorname{Exp}(\lambda)$ with $\lambda = 4$ (so $E[X_i] = 1/4$, $\operatorname{Var}(X_i) = 1/16$). Define $T_n = \sqrt{\bar{X}_n}$. **(a)** Using the delta method, find the asymptotic distribution of $\sqrt{n}\,(T_n - \sqrt{\mu})$ where $\m
打开 →If the smile becomes flatter in log-moneyness, what happens to the difference between sticky-strike and sticky-moneyness for small spot moves?
打开 →A desk uses the simplified risk-neutral drift relation mu_Q = r - lambda*kappa for a jump-diffusion. If r = 3.00%, lambda = 1.2, and mu_Q = 0.60%, what jump compensator kappa is implied?
打开 →In a simplified jump-diffusion, mu_Q = r - lambda*kappa. If r = 2.50%, kappa = 1.60%, and mu_Q = 0.50%, what jump intensity lambda is implied?
打开 →A risk-neutral jump-diffusion uses mu_Q = r - lambda*kappa. If r = 4.00%, lambda = 0.8, and kappa = 1.00%, what is mu_Q?
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