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中文题目A desk is short a positive residual and books one unit of carry each day equal to that day's expected residual. If X_(t+1) = 3/4 X_t + epsilon_(t+1) with zero-mean shocks and X_0 = 12 bp, what is the total expected carry over the next 3 days?
打开 →A scalar parameter has value w_t=2, gradient g_t=0.5, learning rate eta=0.1, and decoupled weight decay lambda=0.05. What is w_{t+1}?
打开 →Under decoupled weight decay with learning rate eta, decay lambda, parameters w_t, and gradient g_t, derive w_{t+1}.
打开 →A test block has 25 trading days. A signal generated on day t is executed on day t+1 and evaluated on the open-to-close return from day t+1 through day t+4. How many signals inside the block can be scored without the label running past the block end?
打开 →For a GARCH(1,1) with $\omega=0.2$, $\alpha=0.1$, $\beta=0.8$, the one-step-ahead conditional variance is $h_{t+1}=3$. Using the closed form $E_t[h_{t+k}]=\bar h+(\alpha+\beta)^{\,k-1}(h_{t+1}-\bar h)$, compute the 5-step-ahead forecast $E_t[h_{t+5}]$ as a decimal.
打开 →A residual spread follows X_(t+1) = 0.8 X_t + epsilon_(t+1) with zero-mean shocks. In trading days, what is the half-life of mean reversion, i.e. the horizon h at which the expected residual has decayed to half its current value?
打开 →A stationary mean-reverting spread obeys X_(t+1) = 1/2 X_t + epsilon_(t+1), where Var(epsilon_(t+1)) = 4. Starting from the current level, what fraction of the same-horizon random-walk forecast-error variance does the 4-step mean-reverting forecast-error variance represent?
打开 →Let $Z_t$ be a branching process with immigration. Each individual in generation $t$ produces offspring with PGF $\phi(s)$, independently, and the number of immigrants arriving at the next generation has PGF $\psi(s)$, independently of everything else. Express the PGF of $Z_{t+1}
打开 →A desk wants a mean-reverting signal whose shocks lose half their expected size every 5 trading days. If the signal is modeled as AR(1), X_(t+1) = phi X_t + epsilon_(t+1), what value of phi is implied?
打开 →A mean-reverting residual follows X_(t+1) = 1/2 X_t + epsilon_(t+1) with Var(epsilon_(t+1)) = 4. What is the stationary variance of X_t?
打开 →A GARCH(1,1) has $\omega=0.00001$, $\alpha=0.08$, $\beta=0.90$. Today's conditional variance is $h_t=0.0004$ and today's return is $r_t=-0.03$. Compute tomorrow's conditional variance $h_{t+1}$ as a decimal.
打开 →A residual spread follows X_(t+1) = 2/3 X_t + epsilon_(t+1) with E[epsilon_(t+1)] = 0. If today's residual is 9 bp, what is E[X_2 | X_0 = 9 bp]?
打开 →A feature at time t uses a rolling mean computed from t-19 through t+1. Why is that unacceptable even if it is only one extra day?
打开 →A stationary spread obeys X_(t+1) = -0.4 X_t + epsilon_(t+1) with iid zero-mean shocks. What is the lag-1 autocorrelation of X_t, and what does its sign say about period-to-period dynamics?
打开 →In a GARCH(1,1) model with $\omega=\frac{1}{10}$, $\alpha=\frac{1}{5}$, and $\beta=\frac{7}{10}$, suppose the current squared return is $r_t^2=4$ and the current conditional variance is $h_t=2$. Compute $h_{t+1}$.
打开 →For a GARCH(1,1) process with $\omega=1$, $\alpha=\frac{1}{10}$, $\beta=\frac{4}{5}$, suppose you already know the one-step-ahead conditional variance $h_{t+1}=5$. Compute $E_t[h_{t+2}]$ and $E_t[h_{t+3}]$.
打开 →Returns are generated by a stationary AR(1) with autoregressive coefficient 0.5. The Lo-MacKinlay variance ratio at lag 2 is VR(2) = Var(r_t + r_(t+1)) / (2 Var(r_t)). Compute VR(2) and state whether it signals momentum or mean reversion.
打开 →For a GARCH(1,1) process with $\omega=\frac{1}{10}$, $\alpha=\frac{1}{5}$, $\beta=\frac{3}{5}$, suppose you already know the one-step-ahead conditional variance $h_{t+1}=2$. Compute $E_t[h_{t+2}]$ and $E_t[h_{t+3}]$.
打开 →A deterministic recurrence over mod 13 is fully determined by the ordered pair (x_t, x_(t+1)). How many consecutive ordered pairs force a repeated state pair?
打开 →In a GARCH(1,1) model with $\omega=1$, $\alpha=\frac{3}{20}$, and $\beta=\frac{3}{5}$, suppose the current squared return is $r_t^2=4$ and the current conditional variance is $h_t=5$. Compute $h_{t+1}$.
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