Phase 18 Exam

Integrated Quant Capstones - Phase Exam

20 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches

Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.

Q1 Medium

You halve \(h\) in a central-difference derivative and the error drops by a factor of ~4. This confirms:

First-order accuracy

\(O(h^2)\) second-order accuracy

Round-off error

A bug

Q2 Easy

In simulating \(W\) on a grid with step \(\Delta t\), each increment should be drawn with standard deviation:

\(\Delta t\)

\(\sqrt{\Delta t}\)

\(1\)

\((\Delta t)^2\)

Q3 Medium

Applying Itô’s lemma to \(\log S\) under \(dS=\mu S\,dt+\sigma S\,dW\) gives a drift of:

\(\mu\)

\(\mu-\tfrac12\sigma^2\)

\(r\)

\(\mu+\tfrac12\sigma^2\)

Q4 Medium

A flat implied-volatility curve across strikes would be consistent with:

Real option markets

Exact constant-\(\sigma\) Black–Scholes (which markets violate)

GARCH

A volatility smile

Q5 Medium

Why is ordinary shuffled k-fold cross-validation invalid for a trading signal?

It is too slow

It trains on future data to predict the past, leaking look-ahead information

It uses too little data

It cannot compute Sharpe

Q6 Easy

The central trade-off in optimal execution is between:

Commission and taxes

Market impact (from trading fast) and timing/volatility risk (from trading slow)

Long and short positions

Equities and bonds

Q7 Easy

A market maker’s gross compensation comes from:

Predicting direction

Earning the bid–ask spread by providing liquidity on both sides

Paying the spread

Holding large inventory

Q8 Medium

Which deliverable would a skeptical reviewer weight most heavily when judging a quant research project?

A high in-sample Sharpe

Reproducible, leakage-free, net-of-cost out-of-sample results with stated uncertainty

An elegant model

A large number of features

Q9 Medium

For an ill-conditioned least-squares problem, why prefer QR over the normal equations?

QR is always faster

Forming \(A^\top A\) squares the condition number, amplifying error; QR avoids that

Normal equations are wrong

QR needs less memory

Q10 Medium

The quadratic variation of Brownian motion on \([0,T]\) converges to:

\(0\)

\(T\)

\(\sqrt T\)

\(T^2\)

Q11 Medium

For pricing a European option by Monte Carlo, the simulated terminal should use drift:

\(\mu\) (real-world)

\(r\) (risk-free, risk-neutral)

\(\mu-r\)

zero

Q12 Medium

In GARCH(1,1), persistence \(\alpha+\beta\) close to 1 means:

Volatility is constant

Shocks to variance decay slowly, so turbulence lingers

The model is misspecified

Returns are independent

Q13 Medium

A strategy has a great gross Sharpe but a flat net Sharpe once costs are included. The correct conclusion is:

Report the gross Sharpe; costs are a detail

The edge does not survive realistic costs, so it is not tradeable as-is

Increase leverage

Lower the cost assumption until it works

Q14 Medium

In the Almgren–Chriss objective, raising the risk-aversion \(\lambda\) causes you to:

Trade more slowly

Trade faster / front-load to shed risky inventory sooner

Ignore impact

Trade at random

Q15 Medium

When a market maker accumulates a long inventory, the Avellaneda–Stoikov reservation price:

Rises above the mid

Falls below the mid, making the ask more attractive so inventory is sold down

Stays at the mid

Becomes undefined

Q16 Hard

If your properly-validated study finds no reliable edge net of costs out-of-sample, the correct action is:

Keep tuning until it looks profitable

Report the null result clearly - it is a complete and valuable answer

Drop the transaction costs

Report the best of the 50 variants you tried

Q17 Easy

After computing a Cholesky factor \(L\) of \(\Sigma\), the validation you must run is:

Check \(L\) is symmetric

Check \(LL^\top=\Sigma\) to tolerance

Check \(L\) is orthogonal

No check needed

Q18 Medium

You estimate \(\E[g(W_T)]=0.500\) from \(N=10^4\) paths with sample SD 0.8. A correct report is:

Exactly 0.500

0.500 with no uncertainty

0.500 ± \(1.96\cdot0.8/100=0.0157\)

0.500 ± 0.8

Q19 Hard

‘Risk-neutral pricing sets the drift to \(r\)’ means:

The stock is forecast to grow at \(r\)

For pricing by no-arbitrage/replication we take expectations under a measure where the drift is \(r\); it is not a real-world forecast

Investors are indifferent to risk in reality

The formula is only approximate

Q20 Medium

You estimate annualized vol as 0.18 from one year of daily data (\(n\approx252\)). The approximate standard error is:

~0.008 (i.e. \(0.18/\sqrt{2\cdot252}\)

~0.18

~0.09

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