Algorithmic & Quant Trading

Question 1 (10 marks)

(a) Components in data-flow order (6 marks — 1 per correct component/role, max 6):

1. Market data handler — ingests live/historical price & reference data, normalises it.
2. Signal / alpha model — transforms data into predictive signals (strategy logic).
3. Portfolio / position sizing — converts signals into target positions (risk-weighting, capital allocation).
4. Risk management — enforces limits (max position, drawdown, exposure) before orders.
5. Execution / order management (OMS) — routes orders, handles fills, slippage minimisation.
6. Monitoring / logging / backtester — records state, PnL, reconciliation, alerts.

Marks: 1 per component correctly named and described, in roughly correct order.

(b) Separation of execution from signal (4 marks):

• Modularity/testability: signal logic can be tested independently of broker connectivity (1).
• Prevents a broker/API bug or latency spike from corrupting signal state (1).
• Failure mode 1: a stuck fill or partial execution should not cause the signal engine to re-fire or double-count positions (1).
• Failure mode 2: execution outages must be handled (retry/cancel) without the alpha model wrongly assuming a position exists (1).

Question 2 (14 marks)

(a) Spread derivation (4 marks):

• Cointegration assumes a linear combination of $p_A$ and $p_B$ is stationary (1).
• Run OLS regression $p_{A,t} = \alpha + \beta\, p_{B,t} + \varepsilon_t$ (1).
• Hedge ratio $\beta$ = slope; it is the number of units of $B$ per unit of $A$ (1).
• Spread $s_t = p_{A,t} - \beta\, p_{B,t}$ (residual $\varepsilon_t$ up to intercept) (1).

(b) Correlation vs cointegration (3 marks):

• Correlation measures co-movement of returns/changes, short-term, can drift apart (1).
• Cointegration means a linear combination of the levels is stationary — the spread reverts (1).
• Mean-reversion trade needs the spread to return to its mean → cointegration guarantees a long-run equilibrium; correlation alone does not (1).

(c) Entry levels (4 marks):

• Entry spread = $\mu \pm z\sigma = 0 \pm 2(0.04) = \pm 0.08$ (2).
• Upper: $s = +0.08$ → spread rich → short A, long B (revert down) (1).
• Lower: $s = -0.08$ → spread cheap → long A, short B (revert up) (1).

(d) Test (3 marks):

• Augmented Dickey–Fuller (ADF) test on the spread (or Engle–Granger / Johansen) (2).
• Null hypothesis $H_0$: the series has a unit root (is non-stationary) (1).

Question 3 (10 marks)

(a) AR(1)/OU form (4 marks):

• $x_t = \phi\, x_{t-1} + \varepsilon_t$ (mean-centred), or with mean: $x_t = \mu + \phi(x_{t-1}-\mu)+\varepsilon_t$ (2).
• The coefficient $\phi$ (or $\theta = 1-\phi$) controls reversion speed; $|\phi|<1$ required for stationarity/reversion (2).

(b) Half-life (6 marks):

• Formula: $t_{1/2} = \dfrac{\ln(0.5)}{\ln(\phi)}$ (or $-\ln 2 / \ln\phi$) (2).
• With $\phi = 0.6$: $\ln 0.6 = -0.5108$, $\ln 0.5 = -0.6931$ (2).
• $t_{1/2} = -0.6931 / -0.5108 = 1.357$ periods $\approx 1.36$ (2).

Question 4 (10 marks)

(a) Pseudocode (6 marks):

def cross_sectional_momentum(prices):      # prices: DataFrame [date x ticker]
    # lookback: 12 months, skip most recent 1 month (12-1)
    mom = prices.shift(21) / prices.shift(252) - 1   # ~12m ret ending 1m ago
    ranks = mom.rank(axis=1, pct=True)               # cross-sectional rank
    longs  = ranks >= 0.9                             # top decile
    shorts = ranks <= 0.1                             # bottom decile
    weights = longs.astype(float) - shorts.astype(float)
    weights = weights.div(weights.abs().sum(axis=1), axis=0)  # normalise
    return weights

Marks: inputs stated (1); 12-1 lookback (1); cross-sectional ranking (1); top/bottom decile selection (1); long−short weighting (1); normalisation/output (1).

(b) Why skip recent month (4 marks):

• Short-term (1-month) returns exhibit reversal, not continuation (2).
• Including the last month contaminates the momentum signal / adds microstructure & liquidity noise, weakening returns (2).

Question 5 (10 marks)

(a) Overfitting (4 marks):

• Definition: tuning parameters so the model fits noise/idiosyncrasies of historical data, giving great backtest but poor live performance (1).
• Warning signs (1 each, max 3): too many parameters vs data; unrealistically smooth/high Sharpe; performance collapses out-of-sample; sensitivity to small parameter changes; strategy relies on a few outlier trades.

(b) Walk-forward (6 marks):

• Split data into consecutive segments (1).
• In-sample (train) window: optimise/fit parameters (1).
• Out-of-sample (test) window immediately following: apply frozen params, record performance (1).
• Roll forward the window (anchored or rolling) by the test length and repeat (1).
• Concatenate the out-of-sample results only — never re-use test data for fitting (1).
• Aggregate OOS metrics (Sharpe, drawdown, return) across all folds to judge robustness (1).

Question 6 (6 marks)

• Low signal-to-noise / non-stationarity (3): financial data is noisy with a low, unstable signal; unlike images, the "labels" (future returns) are near-random, so ML easily fits noise and the data-generating process shifts (regime change) so training distribution ≠ live distribution.
• Feedback / adversarial & small effective sample (3): markets adapt — an edge disappears once traded (non-stationary target), and independent samples are few (autocorrelated, limited history), so cross-validation assumptions break and models overfit; look-ahead/leakage is easy and costly.

Award 3 marks per distinct, correct data/regime-related reason.

[
  {"claim":"Pairs entry levels at z=2, sigma=0.04 are +/-0.08","code":"sigma=Rational(4,100); z=2; upper=z*sigma; lower=-z*sigma; result=(upper==Rational(8,100) and lower==Rational(-8,100))"},
  {"claim":"Half-life for phi=0.6 is approx 1.357 periods","code":"import sympy as sp; hl=sp.log(sp.Rational(1,2))/sp.log(sp.Rational(6,10)); result=abs(float(hl)-1.3569)<0.001"},
  {"claim":"Momentum lookback offsets: 252 trading days ~12m, 21 days ~1m skip","code":"result=(252-21==231)"},
  {"claim":"phi=0.6 satisfies stationarity condition |phi|<1","code":"phi=sp.Rational(6,10); result=abs(phi)<1"}
]