Algorithmic & Quant Trading

Question 1

(a) [4 marks]

• $\beta$ = OLS slope of $p^A$ on $p^B$; it is the number of units of $B$ to short per unit of $A$ long so the combined position is dollar/price neutral in the common factor. (1)
• Correlation measures co-movement of returns/changes but two random walks can be highly correlated yet drift apart without bound. (1)
• Cointegration means a linear combination of two $I(1)$ (unit-root) series is $I(0)$ (stationary) — the spread does not wander off, it reverts. This is the property required for a bounded, tradeable spread. (1)
• The spread $s_t$ must be stationary (mean-reverting, $I(0)$) — constant mean & variance, so deviations are temporary. (1)

(b) [6 marks] Solve the OU SDE. Consider $d(e^{\theta t}s_t)$: $d(e^{\theta t}s_t) = \theta e^{\theta t}s_t\,dt + e^{\theta t}ds_t = \theta e^{\theta t}\mu\,dt + \sigma e^{\theta t}dW_t.$ (2) Integrate $0\to t$: $e^{\theta t}s_t - s_0 = \mu(e^{\theta t}-1) + \sigma\int_0^t e^{\theta u}dW_u.$ Taking expectation kills the Itô integral term (mean 0): (1) $\boxed{\mathbb{E}[s_t\mid s_0] = \mu + (s_0-\mu)e^{-\theta t}.}$ (1) Half-life $H$ = time for deviation $(s_0-\mu)$ to halve: $e^{-\theta H} = \tfrac12 \Rightarrow \theta H = \ln 2 \Rightarrow H = \frac{\ln 2}{\theta}.$ (2)

(c) [4 marks]

• Discrete AR(1): $b = e^{-\theta \Delta t}$ so $\theta = -\ln b$ (with $\Delta t = 1$ day). (1)
• $H = \dfrac{\ln 2}{-\ln b} = \dfrac{\ln 2}{-\ln 0.94}$. (1)
• $-\ln 0.94 = 0.061875$, $\ln 2 = 0.693147$. (1)
• $H = 0.693147/0.061875 \approx 11.2$ trading days. (1)

(d) [10 marks] Rule set (award up to 5):

• Entry: short spread when $z_t \ge +z_{\text{in}}$ (e.g. $+2$): sell $A$, buy $\beta$ of $B$. Long spread when $z_t \le -z_{\text{in}}$. (2)
• Exit: close when $z_t$ reverts toward 0 (e.g. $|z_t|\le z_{\text{out}}=0.5$ or crosses mean). (1)
• Stop-loss: exit if $|z_t| \ge z_{\text{stop}}$ (e.g. $\pm 3.5$) or if cointegration test fails / holding time > cap. (2)

Boundedness of expected holding period (up to 3):

• The spread process $z$ is a positive-recurrent OU/AR(1) with $\theta>0$; a stationary ergodic mean-reverting process visits every neighbourhood of the mean in finite expected time. (1)
• Formally the expected first-passage (hitting) time from level $z_{\text{in}}$ to the exit band is finite because the OU process has a proper stationary distribution and drift $-\theta z$ pushes it toward 0; hence $\mathbb{E}[\tau]<\infty$. (2)

Fixed-$\beta$ danger (up to 2):

• The cointegrating relationship / hedge ratio is not constant — regime changes, corporate actions, changing fundamentals cause $\beta$ to drift, so the "spread" stops being stationary and the position accumulates unbounded loss. Mitigation: rolling/Kalman-filtered $\beta$ and periodic re-testing of cointegration. (2)

Question 2

(a) [4 marks]

• Rank on months $t-12$ to $t-1$, skip month $t$ before forming the portfolio held in $t+1$. (2)
• Justification: the most recent month exhibits short-term reversal (bid-ask bounce, liquidity/microstructure effects, one-month mean reversion); including it contaminates the momentum signal. Skipping isolates the persistent intermediate-horizon momentum. (2)

(b) [4 marks] $\text{SR}_{\text{ann}} = \frac{\bar r}{\sigma}\sqrt{252} = \frac{0.0006}{0.011}\sqrt{252}.$ (2) $= 0.0545455 \times 15.8745 = 0.8659 \approx 0.87.$ (2)

(c) [6 marks]

• Under the null each of $M$ trials has SR $\sim \mathcal N(0,1)$; picking the best is an extreme-value (maximum) selection. (1)
• $\mathbb{E}[\max_M] \approx \sqrt{2\ln M}$. (1)
• $\ln 200 = 5.2983$, $2\ln 200 = 10.5966$, $\sqrt{10.5966} = 3.255$. (2)
• So even with zero true skill, the best of 200 tries has expected Sharpe ≈ 3.26 — an impressive-looking number that is pure selection artefact. The reported best is not evidence of skill. (1) Phenomenon: multiple-testing / data-snooping bias (backtest overfitting, selection bias). (1)

(d) [8 marks]

• Deflated Sharpe ratio (DSR): adjusts the observed Sharpe by (i) the number of trials $M$, (ii) skew/kurtosis of returns and (iii) sample length, giving the probability the true Sharpe exceeds 0 after accounting for multiple testing — it "deflates" the naive SR toward the selection-corrected benchmark. (2)
• WFA / splits: parameters are tuned only on training/validation windows; the test window is never seen during optimisation, so reported performance is genuinely out-of-sample and reflects the model's ability to generalise, not memorise. Rolling windows also detect non-stationarity. (2)

Pseudocode (up to 4):

data sorted by time; define K folds by time
oos_results = []
for k in 1..K:
    train = data[fold_start(k) : fold_train_end(k)]      # in-sample
    test  = data[fold_train_end(k) : fold_test_end(k)]   # unseen, later in time
    best_params = None; best_score = -inf
    for params in param_grid:
        score = backtest(strategy, train, params)         # optimise on TRAIN only
        if score > best_score:
            best_score, best_params = score, params
    oos = backtest(strategy, test, best_params)           # evaluate on TEST only
    oos_results.append(oos)
report( aggregate(oos_results) )   # report ONLY out-of-sample metrics

Marks: correct time-ordered no-shuffle split (1), optimise on train only (1), evaluate fixed params on future test (1), report only OOS (1).

Question 3

(a) [6 marks] — 1 mark per block correctly named + role (max 6):

[Market Data Feed] -> [Data Cleaning/Storage] -> [Signal/Alpha Model]
        -> [Portfolio Construction/Sizing] -> [Risk Management]
        -> [Execution/Order Router] -> [Broker/Exchange]
                          ^                          |
                          |__________ [Monitoring / P&L / Logging] <--

• Data feed: ingest real-time & historical prices/fundamentals.
• Cleaning/storage: handle missing data, adjustments, survivorship-free DB.
• Signal/alpha model: generate trading signals from strategy logic.
• Portfolio construction/sizing: convert signals to target positions, apply capital allocation.
• Risk management: limits, exposure, stop-losses, drawdown control.
• Execution/router: slice & send orders, minimise slippage/impact.
• Monitoring/logging: track live vs expected performance, alerts.

(b) [5 marks] — any three, 1 mark reason + shared mark for mitigation (max 5):

• Curse of dimensionality / overfitting: 300 features vs 500 samples → model memorises noise. Mitigation: feature selection / regularisation / fewer features, more data.
• In-sample evaluation: 82% is on training data; no honest estimate of generalisation. Mitigation: purged/embargoed cross-validation or walk-forward OOS test.
• Serial correlation / non-IID data: standard CV leaks temporal info; financial data is non-stationary with autocorrelation. Mitigation: time-series CV, purge & embargo.
• Low signal-to-noise: market returns are ~random; 82% accuracy on direction is implausibly high → data leakage suspected. Mitigation: audit features for look-ahead. (Award 5 for three sound reason+mitigation pairs.)

(c) [3 marks]

• Look-ahead bias: using data not available at decision time (e.g. same-day close to trade at that close, restated fundamentals) inflates accuracy because the model effectively "sees the future." (1.5)
• Survivorship bias: dataset contains only firms that survived; delisted/bankrupt names removed, so backtest overstates returns and understates risk — the model is trained on a rosier universe than reality. (1.5)
• Both operate through the data, so even bug-free model code produces optimistic, non-reproducible live results.

[
  {"claim":"OU half-life for AR(1) b=0.94 is ~11.2 days","code":"from sympy import log, N; b=0.94; H=log(2)/(-log(b)); result = abs(N(H)-11.2)<0.1"},
  {"claim":"Annualised Sharpe = 0.0006/0.011*sqrt(252) ~ 0.866","code":"from sympy import sqrt, N; SR=(0.0006/0.011)*sqrt(252); result = abs(N(SR)-0.866)<0.01"},
  {"claim":"Expected max Sharpe over 200 trials ~ 3.26","code":"from sympy import log, sqrt, N; e=sqrt(2*log(200)); result = abs(N(e)-3.255)<0.01"},
  {"claim":"Half-life formula: e^(-theta*H)=1/2 gives H=ln2/theta","code":"from sympy import symbols, Eq, solve, log, exp; theta,H=symbols('theta H',positive=True); sol=solve(Eq(exp(-theta*H),0.5),H)[0]; result = sol.equals(log(2)/theta)"}
]