HFT & Advanced Concepts

Question 1 (12)

(a) (4)

• Quoted spread = \text{ask} - \text{bid} = 100.02 - 99.98 = \0.04$. (1)
• Half-spread = 0.04/2 = \0.02$. (1)
• Theoretical edge per round-trip $=$ ask $-$ bid = \0.04$/share (buy at 99.98, sell at 100.02). (1)
• Assumption: symmetric fills — the maker gets filled equally on both sides so inventory returns to target; edge only realized if both legs execute. (1)

(b) (4)

• Spread capture: two-sided round-trip earns full \0.04$; with 8,000 shares/side, model as 8,000 round-trips →$8000 \times 0.04 = $320$. (1)
• Rebates: added on both sides = 16{,}000 \times 0.0020 = \32$. (1)
• Adverse selection: 16{,}000 \times 0.015 = \240$. (1)
• Net P&L = 320 + 32 - 240 = \112$/day. (1)

(c) (4)

• Inventory risk: holding a directional position exposes the maker to price moves unrelated to spread income. (1)
• To reduce it the maker skews quotes: shift both bid and ask to encourage trades that flatten inventory. (1)
• When long 50,000 beyond target the maker wants to sell, so it lowers the ask (more aggressive to sell) and lowers the bid (less eager to buy more). (1)
• Net effect: quotes shifted down / offer improved, discouraging further buys and inviting fills that reduce the long. (1)

Question 2 (10)

(a) (5)

• Gross gain if order arrives in time $= \delta - c$. (1)
• If order misses (prob $1-p$), assume it earns $\approx 0$ (quote pulled, no fill). (1)
• Expected profit $E = p(\delta - c) + (1-p)(0) = p(\delta - c)$. (1)
• Profitability condition: $E > 0 \Rightarrow p(\delta - c) > 0$, i.e. require $\delta > c$ and $p > 0$. (1)
• For a minimum required edge $m$: $p(\delta - c) \ge m$. (1)

(b) (5)

• $\delta - c = 0.03 - 0.005 = 0.025$. (2)
• Require $p(0.025) \ge 0.010$. (1)
• $p_{\min} = 0.010 / 0.025 = 0.4$. (2)

Question 3 (14)

(a) (2) $\text{VWAP} = \frac{\sum_{i=1}^{n} P_i Q_i}{\sum_{i=1}^{n} Q_i}$ (1 for numerator being $\sum P_iQ_i$, 1 for denominator $\sum Q_i$.)

(b) (4)

• TWAP slices evenly over time: shares per interval $= Q/k$ (independent of volume). (2)
• VWAP slices in proportion to expected volume: shares in interval $j = Q \cdot \dfrac{V_j}{\sum_{i=1}^{k} V_i}$, where $V_j$ is expected volume in interval $j$. (2)

(c) (5)

def pov_algo(parent_qty, rho, volume_feed):
    remaining = parent_qty
    for interval_volume in volume_feed:
        if remaining <= 0:            # guard: order complete
            break
        target = rho * interval_volume
        child = min(target, remaining)  # guard: don't over-fill
        send_order(child)
        remaining -= child
    return parent_qty - remaining       # total executed

Marks: loop over volume feed (1), target = ρ·volume (1), child = min(target, remaining) over-fill guard (1), remaining<=0 completion guard (1), decrement/coherent structure (1).

(d) (3)

• Numerator: $50.00(300) + 50.10(200) + 49.90(500) = 15000 + 10020 + 24950 = 49970$. (1)
• Denominator: $300+200+500 = 1000$. (1)
• VWAP = 49970/1000 = \49.97$. (1)

Question 4 (10)

(a) (4)

• DMA: client sends orders directly to the exchange matching engine via broker infrastructure, bypassing manual desk handling. (1) Reduces one layer of latency and gives order-type control. (1)
• Colocation: physically placing the firm's servers in the same data center as the exchange's matching engine. (1) Shortens the physical signal path → propagation & handling reduced to microseconds, a decisive edge in racing/latency-sensitive strategies. (1)

(b) (6)

• Cheapest first: Venue X @ 10.00 with $0.003 fee → effective$10.003$/sh; take all 400. (1)
• Venue Y @ 10.01 with $0.001 fee → effective$10.011$/sh; take remaining 600. (1)
• (Compare effective prices: X 10.003 < Y 10.011, so fill X fully then Y — this ordering is optimal.) (1)
• Cost X: 400 \times 10.003 = \4001.20$. (1)
• Cost Y: 600 \times 10.011 = \6006.60$. (1)
• Total = 4001.20 + 6006.60 = \10{,}007.80$. (1)

Question 5 (8)

(a) (4)

• Flash crash: a very rapid, deep, and short-lived price collapse (and often recovery) over seconds/minutes. (2)
• Feedback: as prices fall, HFT market makers widen/withdraw quotes to avoid adverse selection → liquidity vanishes → same sell flow moves price further → stop-loss and momentum algos add sells → self-reinforcing spiral. (2)

(b) (4)

• Band $= 5\%$ of 200 = \10$. (1)
• Upper band = 200 + 10 = \210$. (1)
• Lower band = 200 - 10 = \190$. (1)
• If price hits a band and doesn't move back within band within the window, trading pauses (a brief trading halt / limit state) to let liquidity replenish. (1)

Question 6 (6)

• Central concern: HFT can create unfair speed advantages and fragile/illusory liquidity that vanishes under stress, raising systemic and fairness risks. (2)
• Two mitigations (1 for naming + 1 for justification each, up to 4):
  • Circuit breakers / LULD: halt or band trading to stop cascade feedback and restore orderly pricing. (2)
  • Speed bumps / minimum resting times / order-to-trade ratios / message throttling: neutralize pure latency races and discourage quote-stuffing / spoofing. (2) (Accept alternatives: kill switches, market-maker obligations, tick-size regimes — each requires a justification.)

[
  {"claim":"MM net daily P&L = 112", "code":"spread=8000*0.04; rebate=16000*0.0020; adverse=16000*0.015; result=(spread+rebate-adverse==112)"},
  {"claim":"Latency arb p_min = 0.4", "code":"delta=Rational(3,100); c=Rational(5,1000); m=Rational(10,1000); pmin=m/(delta-c); result=(pmin==Rational(2,5))"},
  {"claim":"VWAP of three fills = 49.97", "code":"num=50.00*300+50.10*200+49.90*500; den=300+200+500; result=(abs(num/den-49.97)<1e-9)"},
  {"claim":"SOR total cost = 10007.80", "code":"x=400*10.003; y=600*10.011; result=(abs((x+y)-10007.80)<1e-6)"},
  {"claim":"LULD bands 210 and 190", "code":"ref=200; band=0.05*ref; result=(ref+band==210 and ref-band==190)"}
]