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5784Where The Inventory Limit BindsA maker keeps adding to a long position only while the per-lot edge it still captures, 0.30, exceeds the marginal inventory risk it takes on, modeled as gamma*sigma 2*q with gamma = 0.02 and sigma 2 = 0.25. Beyond what position size q does the marginal inventory risk exceed the edge, defining the maker's effective long-side inventory limit?金融与交易中等数值题未尝试免费5787How Far To Skew Given InventoryHolding 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 sells. By what additional multiplicative factor (1 + s) must you scale the base 0.40 skew, and what is s?金融与交易中等数值题未尝试免费5791Width From Volatility DoublingA desk sets half-spread proportional to expected holding-period volatility: h = c*sigma*sqrt(T). It currently quotes h=0.05 at sigma=0.02 per unit-sqrt-time and T=1. If realized volatility doubles to sigma=0.04 and expected holding time rises to T=4, what half-spread should it quote (same c)?金融与交易简单数值题未尝试面试订阅5794Inventory Inflates The Sell-Side WidthA maker is long inventory q=200 units. It widens the side that would add to its position using h buy = base + gamma*sigma2*q and tightens nothing else, with base=0.02, gamma=0.0001, sigma2=2 (variance). What half-spread does it quote on the buy side (where filling makes it longer)?金融与交易中等数值题未尝试面试订阅5796Minimum Width To Clear FeesA venue charges a 0.003 taker-removal fee that the maker effectively pays when it must cross to flatten, and the maker pays a 0.002 clearing fee per fill but receives a 0.0015 maker rebate. Expected adverse-selection loss is 0.004 per fill. What minimum half-spread leaves non-negative expected PnL per fill?金融与交易中等数值题未尝试面试订阅5797Width Sensitivity To Informed FractionOf the flow that fills the maker, a fraction f is informed and costs 0.06 in adverse move; the rest (1-f) is uninformed and costs 0. The maker quotes the break-even half-spread h(f) = f*0.06. By how much must it widen if the informed fraction rises from f=0.20 to f=0.35?金融与交易中等数值题未尝试面试订阅5799Width For A Latency-Limited QuoteA maker cannot cancel instantly: from a price signal to its cancel landing there is a fixed exposure window of 50 milliseconds during which its quote is stale. Per-second volatility is 0.4 price units (so variance scales linearly with time). The maker sets its half-spread equal to one standard deviation of the price move over the exposure window. What half-spread does it quote?金融与交易中等数值题未尝试面试订阅5803Minimum Width For A Profit TargetA maker wants to earn at least 5.0 in total net edge over a session in which it expects exactly 250 fills. Each fill earns (h - loss) with adverse-selection loss 0.008, and the maker assumes the fill count is independent of the half-spread over the relevant range. What is the minimum half-spread that meets the target?金融与交易中等数值题未尝试面试订阅5809EV Comparison Of Two Quote PricesQuote A sits one tick inside: fill probability 0.40, net edge 0.015 per share. Quote B sits one tick wider: fill probability 0.25, net edge 0.028 per share. Both quote 100 shares. Which quote has higher expected PnL, and what is that higher value?金融与交易中等数值题未尝试免费5811Edge Required To Cover Inventory RiskEach fill of 200 shares leaves you holding inventory that carries an expected risk charge of 0.50 until you can hedge. The fill earns rebate of 0.001 per share. What minimum gross half-spread per share must you capture so the fill at least breaks even after the rebate and the inventory charge?金融与交易中等数值题未尝试免费5819Edge Per Unit Of RiskA quote earns an expected net edge of 0.018 per fill, and the PnL of each fill has a standard deviation of 0.12. What is the edge-per-unit-of-risk (expected edge divided by standard deviation) for this quote? Give the answer to three decimals.金融与交易简单数值题未尝试免费5829In a market the per-trade adverse-selection cost to a dealerIn a market the per-trade adverse-selection cost to a dealer is alpha*delta, where alpha is the informed fraction and delta is the value gap. Noise traders are willing to pay at most a half-spread of 0.30 before they stop trading entirely. The value gap is delta = 1.0. Above what informed fraction alpha does the market break down (no spread can both cover adverse selection and retain noise traders)?金融与交易困难数值题未尝试面试订阅5833A dealer quotes a two-sided market and earns the half-spreadA dealer quotes a two-sided market and earns the half-spread s on every fill. A fraction alpha = 0.15 of fills are informed and cost the dealer the full value gap delta = 0.8 (the dealer is on the wrong side for the entire move). The remaining fills are noise and the dealer keeps s. The dealer wants expected profit per trade of at least 0.02. What is the minimum half-spread s the dealer must charge?金融与交易中等数值题未尝试面试订阅5835Two venues route flow to your quotes. Venue A flow is 5% infTwo venues route flow to your quotes. Venue A flow is 5% informed; Venue B flow is 40% informed. Both have the same value gap delta = 0.5, and informed flow costs you delta per fill while you earn half-spread s = 0.10 on all fills. You can only post on one venue. Compute expected profit per fill on each venue, and explain in one line which venue you choose and the adverse-selection reason.金融与交易中等数值题未尝试面试订阅5837Roll Implied Spread From AutocovarianceUnder Roll's model, transaction price changes have a first-order serial autocovariance of -0.0009 (in price-squared units). Estimate the implied effective spread.金融与交易中等数值题未尝试面试订阅5869Hedged P&L From The Variance GapApproximate the total P&L of a long delta-hedged option by 0.5·gamma·S 2·(sigma real 2 - sigma impl 2)·T. With gamma 0.05, S = 50, realized vol 0.30, implied vol 0.22, and T = 20/252 years, what is the approximate P&L, to four decimals?数理金融困难数值题未尝试面试订阅5871Vega P&L From A Vol RepricingYour position has total vega of 20 dollars per one full volatility point (i.e. per 1.00 change in sigma). Overnight the implied vol used to mark the book rises from 22% to 27% with spot unchanged. Ignoring all other Greeks, what is the mark-to-market vega P&L in dollars, to two decimals?数理金融简单数值题未尝试免费6025Five-Step Forecast via the Mean-Reversion FormulaFor a GARCH(1,1) with \omega=0.2, =0.1, =0.8, the one-step-ahead conditional variance is h t+1 =3. Using the closed form E t[h t+k ]= h+( + ) \,k-1 (h t+1 - h), compute the 5-step-ahead forecast E t[h t+5 ] as a decimal.统计困难derivation未尝试面试订阅6027When GARCH(1,1) Becomes EWMARiskMetrics EWMA updates variance as h t=(1- )r t-1 2+ h t-1 . State the constraints on (\omega, , ) that make GARCH(1,1) coincide exactly with EWMA, and give the implied when =0.06. Report as a decimal.统计中等数值题未尝试面试订阅6028Fat Tails: Unconditional Kurtosis of GARCH ReturnsLet r t= h t \,z t with z t\sim N(0,1) i.i.d. and GARCH(1,1) variance. The unconditional kurtosis (when finite) is K=\dfrac 3\,[1-( + ) 2] 1-( + ) 2-2 2 . For =0.1, =0.85, compute K and state whether returns are leptokurtic. Give K as a decimal.统计困难derivation未尝试面试订阅