25 interleaved questions spanning multiple phases. Cumulative review is one of the most powerful retention tools - it forces retrieval of older material right when it is starting to fade.
Q1 MediumThe second-order condition for convexity of twice-differentiable \(f\) is:
\(\nabla f(x)=0\)
\(\nabla^2 f(x)\succeq0\) everywhere
\(\nabla^2 f(x)\preceq0\)
\(f(x)\ge0\)
Q2 MediumThe empirical square-root law of market impact states that the cost of a metaorder scales approximately as:
Linearly in \(Q\)
\(\propto\sigma\sqrt{Q/V}\), sublinearly in size
Independently of size
\(\propto Q^2\)
Q3 MediumA market buy order for size larger than the best-ask depth will execute at an average price that is:
Equal to the best ask
Equal to the mid
Worse than the best ask, because it walks up through deeper levels
Better than the best ask due to rebates
Q4 EasyTo price an \(n\)-period claim by backward induction you:
Average terminal payoffs under \(\Prob\)
Roll one-period risk-neutral values back node by node, discounting by \(R\) each step
Discount the maximum payoff
Use \(q^n\) only
Q5 MediumAn AR(2) process is best identified by:
An ACF that cuts off after lag 2
A PACF that cuts off after lag 2
Both ACF and PACF cutting off after lag 2
Neither function showing structure
Q6 MediumThe inverse-transform method sets \(X=F^{-1}(U)\) with \(U\sim\mathrm{Unif}(0,1)\). Why does \(X\) have CDF \(F\)?
Because \(F\) is always linear
Because \(\Prob(F^{-1}(U)\le x)=\Prob(U\le F(x))=F(x)\) by monotonicity
Because \(U\) is normal
Because \(F^{-1}\) is random
Q7 MediumWhich condition guarantees strong duality for a convex problem?
The objective is bounded
Slater's condition (a strictly feasible point exists)
All constraints are equalities
The Hessian is diagonal
Q8 EasyA backtest that trades using each day’s closing price but assumes execution at that same close exhibits:
Survivorship bias
Look-ahead bias
Data snooping
No bias
Q9 EasyBox–Muller consumes and produces how many values?
One uniform, one normal
Two uniforms, two independent normals
One uniform, two normals
Two normals, one uniform
Q10 HardYou test 1000 zero-alpha strategies over 10 years and report the best Sharpe. Its expected value is roughly:
0
About 1.2, entirely from luck
Exactly the true Sharpe of the best strategy
Negative
Q11 MediumAs the penalty \(\lambda\) increases from 0, the bias and variance of the estimator:
Both increase
Bias decreases, variance increases
Bias increases, variance decreases
Both decrease
Q12 MediumUnder the risk-neutral measure \(\mathbb{Q}\), the discounted stock price \(\tilde S_t=S_t/R^t\) is:
A submartingale with positive drift
A martingale: \(\E_{\mathbb{Q}}[\tilde S_{t+1}\mid\mathcal F_t]=\tilde S_t\)
Deterministic
A martingale only under \(\Prob\)
Q13 HardIn an incomplete market, a non-replicable claim has:
No arbitrage-free price
Exactly one arbitrage-free price
A range of arbitrage-free prices, one per EMM
A negative price
Q14 MediumBagging improves a single deep tree primarily by:
Reducing bias
Reducing variance by averaging many high-variance trees
Making the tree deeper
Removing the need for cross-validation
Q15 MediumTemporary market impact differs from permanent impact in that temporary impact:
Persists forever and depends on total size
Decays after you stop trading and depends on your trading rate
Is caused only by exchange fees
Is always larger than permanent impact
Q16 HardWhich Greek estimator best handles a discontinuous (digital) payoff?
Pathwise method
Likelihood-ratio method
Forward finite difference with fresh random numbers
None can
Q17 MediumWith a risk-free asset, two-fund separation says every mean–variance investor holds:
A different risky portfolio each
The risk-free asset plus the same tangency portfolio, scaled by risk appetite
Only the risk-free asset
Only the highest-return asset
Q18 MediumTrading a signal that is computed from today's closing price, executed at today's close, exhibits:
Survivorship bias
Look-ahead bias
Selection bias
No bias
Q19 MediumWhich statement about the model’s assumptions is correct?
Volatility is observed to be constant across strikes
A volatility smile shows the constant-\(\sigma\) assumption fails
The formula accounts for jumps
Trading is assumed costly
Q20 EasyImplied volatility is best described as:
The market’s forecast of realized volatility
The \(\sigma\) that makes the Black–Scholes price equal the market price
The historical standard deviation of returns
A number that is the same for all strikes
Q21 MediumWhy does logistic regression require iterative optimization?
The likelihood is non-convex
The score equation \(X^\top(y-p)=0\) is nonlinear in \(\beta\) because \(p=\sigma(X\beta)\)
There are too many parameters
The data must be standardized first
Q22 MediumWeak (covariance) stationarity requires:
The full joint distribution to be shift-invariant
Constant mean and variance, and autocovariance depending only on the lag
Independent observations
A zero mean
Q23 HardGlosten–Milgrom show that a bid–ask spread arises even for a costless, risk-neutral market maker because:
Exchanges mandate a minimum spread
Inventory must be financed
A buy is more likely to come from an informed trader, so \(\E[V\mid buy]\gt \E[V\mid sell]\)
Volatility is always positive
Q24 MediumIn the Kalman filter, the predict step changes the state covariance by:
Shrinking it by the measurement
Inflating it by the process noise \(Q\)
Setting it to zero
Leaving it unchanged
Q25 MediumA trader is long a delta-hedged straddle. Over the next month realized volatility comes in far below the implied vol paid. The position:
Profits, because it is delta-hedged
Loses, because gamma P&L cannot offset the theta paid
Is unaffected by realized vol
Profits only if the stock rises