Algorithmic & Quant Trading

Question 1

(a) Spread $s = P_A - \beta P_B = 152 - 1.8(82) = 152 - 147.6 = 4.4$. (2) Z-score $= (s - \mu)/\sigma = (4.4 - 0)/2.5 = 1.76$. (2) Since $|1.76| < 2$, no trade is triggered. (1) (If it were triggered at $z>+2$: spread too high → short A, long B. At $z<-2$: long A, short B.)

(b) Trade $\beta = 1.8$ shares of B for every 1 share of A. (1) Market neutral: the dollar exposure to broad market moves cancels because the position bets only on the spread $s_t$ reverting to its mean, not on the direction of either stock's price. (2)

(c) Null hypothesis $H_0$: the spread series has a unit root (is non-stationary / not cointegrated). (1) Since test stat $-3.9 < -2.9$ (critical value), reject $H_0$ (1) → the spread is stationary/mean-reverting. (1) This is a prerequisite because pairs trading profits only if the spread reliably returns to its mean; without cointegration the spread can drift indefinitely and diverge, causing unbounded losses. (1)

(d) With ADF $= -1.2 > -2.9$, we fail to reject the unit root → the spread is likely non-stationary. (1) Trading it risks the spread drifting away permanently ("divergence"), so mean-reversion entries never revert and losses accumulate. (1)

Question 2

(a) $z = (91 - 100)/4 = -9/4 = -2.25$. (2) Price is 2.25σ below the mean; $|z| > 2$ so a long mean-reversion signal fires (buy the dip expecting reversion up). (2)

(b) $w = -k \cdot z = -0.5 \times (-2.25) = 1.125$. (2) Since $|1.125| > 1$, the cap binds → target weight $= +1.0$ (fully long). (2)

(c) High win rate: reversions occur frequently, so most trades close for small gains near the mean. (1) The asymmetry: occasionally the mean-reversion assumption breaks (regime change, trend, or structural break) and the position is held while the price keeps moving against it, producing a rare large loss that dwarfs many small wins. (2) Mitigation: a stop-loss (e.g. exit if $|z|$ exceeds a wider threshold like 4), or a time-based exit / position cap. (1)

Question 3

(a) Rank descending: X(+45), U(+34), V(+12), Y(+8), W(−5), Z(−18). Top third (2 names) = long X, U; bottom third (2 names) = short Z, W. (2) Equal weight within each leg: +0.5 on X, +0.5 on U; −0.5 on Z, −0.5 on W. (2)

(b) Long leg return = $0.5(+2\%) + 0.5(-1\%) = 1\% - 0.5\% = 0.5\%$. (2) Short leg: we are short, so profit = −(realized return). Short basket realized = $0.5(+3\%) + 0.5(-4\%) = 1.5\% - 2\% = -0.5\%$; short contribution = $-(-0.5\%) = +0.5\%$. (2) Total strategy return $= 0.5\% + 0.5\% = 1.0\%$. (1)

(c) A momentum crash is a sharp reversal in which prior losers rally hard and prior winners fall, so the long-winner/short-loser book suffers heavy losses. (2) Trigger: a rebound/recovery following a severe market decline (high-volatility panic bottoms), when beaten-down losers snap back fastest. (1)

Question 4

(a) Phenomenon: overfitting / curve fitting. (2) Signs: (i) implausibly high in-sample Sharpe (3.1) relative to realistic markets; (ii) large in-sample vs out-of-sample degradation (3.1 → 0.4), the classic generalization gap; also 8 tunable parameters gives high model flexibility to fit noise. (2, any two)

(b) Walk-forward: split data into sequential windows. Optimize parameters on an in-sample window, then apply the fixed parameters to the immediately following out-of-sample window and record performance; roll both windows forward and repeat, stitching the OOS segments into one continuous track record. (3) It is more honest because every reported return comes from data not used to choose the parameters, so it simulates real deployment where you must fix parameters before seeing the future — capturing parameter instability across regimes. (2)

(c) Testing 500 combinations and keeping the maximum is a multiple-comparisons / selection bias: even random strategies produce some high Sharpe by chance, so the best is inflated by luck. (2) Adjustment: the Deflated Sharpe Ratio (or Bonferroni correction / adjust for number of trials). (1)

Question 5

(a) (1 each)

• Alpha/signal: generate predictive forecasts / entry–exit signals.
• Portfolio construction/sizing: convert signals into position sizes respecting diversification and capital.
• Execution: route and fill orders minimizing slippage and market impact.
• Risk: monitor and cap exposures, drawdown, leverage; enforce limits/kill-switch.

(b) (1 each, any three) 62% may be from overfitting 400 features to noise; no accounting for transaction costs/slippage that erode a small edge; possible look-ahead/data leakage in features; non-stationarity — market regime may change; directional accuracy ≠ profitability (misses trade sizing and asymmetric payoffs).

(c) Look-ahead bias: using information not available at decision time (e.g. using the day's close to trade at the open, or restated fundamentals). (1.5) Overfitting: fitting model parameters to random noise in the sample so it fails out-of-sample (e.g. tuning a moving-average length to the exact value that maximized past backtest profit). (1.5)

[
  {"claim":"Q1a spread and z-score", "code":"PA=152; PB=82; beta=Rational(18,10); s=PA-beta*PB; z=(s-0)/Rational(25,10); result = (s==Rational(44,10)) and (float(z)==1.76)"},
  {"claim":"Q2a z-score = -2.25", "code":"z=(91-100)/4; result = (z==-2.25)"},
  {"claim":"Q2b uncapped weight 1.125 then capped to 1", "code":"z=Rational(-225,100); w=-Rational(1,2)*z; capped=1 if abs(w)>1 else w; result = (w==Rational(1125,1000)) and (capped==1)"},
  {"claim":"Q3b strategy monthly return = 1.0%", "code":"long=Rational(1,2)*2 + Rational(1,2)*(-1); short_basket=Rational(1,2)*3 + Rational(1,2)*(-4); short_contrib=-short_basket; total=long+short_contrib; result = (total==1)"}
]