HFT & Advanced Concepts

Question 1

(a) Speed in fibre $v = c/n = 3.0\times10^8 / 1.47 = 2.0408\times10^8$ m/s. Fibre latency $t_f = \dfrac{1.18\times10^6}{2.0408\times10^8} = 5.782\times10^{-3}$ s $= 5782\,\mu s$. (2) Microwave (vacuum) $v \approx c = 3.0\times10^8$: $t_{mw} = \dfrac{1.18\times10^6}{3.0\times10^8} = 3.933\times10^{-3}$ s $= 3933\,\mu s$. (2) Advantage $= 5782 - 3933 = 1849\,\mu s \approx 1.85$ ms. (2) Why: microwave travels ~straight line at near-c; fibre both slows light (n) and is straight here for the comparison.

(b) Fibre path $= 1.15 \times 1180 = 1357$ km. $t_f' = \dfrac{1.357\times10^6}{2.0408\times10^8} = 6.650\times10^{-3}$ s $= 6650\,\mu s$. (2) Microwave advantage $= 6650 - 3933 = 2717\,\mu s \approx 2.72$ ms. (2) Why: physical routing overhead compounds the refractive slowdown.

(c) Both receive signal simultaneously, so propagation cancels; only last-mile differs.

• Colocated: $30\,\mu s$.
• DMA: $250\,\mu s$. Colocated trader wins by $250 - 30 = 220\,\mu s$. (3) Two components each optimises: (1) propagation latency (network distance — solved by microwave/colo proximity) and (2) processing/last-mile latency (order gateway + local switching — solved by colocation inside the matching-engine datacentre). (2)

(d) Speed bumps add a fixed, deterministic delay to incoming orders so that a marginally faster trader cannot pick off stale quotes (latency arbitrage against slow market makers). (2) Rationale: level playing field, protect resting liquidity, reduce "picking off." (1) A fixed speed bump applied equally does not neutralise the (c)-type race, because it adds the same 350 µs to both traders — the 220 µs relative advantage persists; the winner is unchanged. (2) (Speed bumps only help when asymmetric — e.g., applied to aggressive orders but not to the market maker's cancels, as in IEX's design protecting the pegged midpoint.)

Question 2

(a) Total $= 600{,}000$; 6 intervals. TWAP = equal: $600000/6 = 100{,}000$ shares per interval. (2) VWAP proportional to $V_i$. $\sum V_i = 200+400+800+600+300+200 = 2500$k. Fraction $\times 600000$:

• $i1: 200/2500 \times 600000 = 48{,}000$
• $i2: 96{,}000$
• $i3: 192{,}000$
• $i4: 144{,}000$
• $i5: 72{,}000$
• $i6: 48{,}000$ (3) (sum = 600,000 ✓)

(b) TWAP avg price $= \frac{1}{6}\sum P_i = \frac{50.0+50.2+50.5+50.3+50.1+50.0}{6} = \frac{301.1}{6} = 50.1833$. (2) VWAP avg price $= \frac{\sum w_i P_i}{\sum w_i}$ with $w_i \propto V_i$: $\sum V_i P_i = 200(50.0)+400(50.2)+800(50.5)+600(50.3)+300(50.1)+200(50.0)$ $= 10000 + 20080 + 40400 + 30180 + 15030 + 10000 = 125{,}690$k. $\text{VWAP}_{mkt} = 125690/2500 = 50.276$. (2) VWAP algo achieves exactly $50.276$; it tracks the benchmark exactly because it weights each interval's fill in the same proportion as market volume, so the achieved price = volume-weighted benchmark by construction. TWAP ($50.1833$) differs because equal weighting under-buys in high-volume/high-price intervals. (2)

(c) POV: child $= \rho \times V_i^{realised}$, $\rho = 0.15$, realised $= 1.2 V_i$. So child $= 0.15 \times 1.2 \times V_i = 0.18 V_i$ (in k shares):

• $i1: 0.18\times200 = 36$k
• $i2: 72$k
• $i3: 144$k
• $i4: 108$k
• $i5: 54$k
• $i6: 36$k Total $= 0.18 \times 2500 = 450$k shares. (3) POV executes only 450,000, leaving 150,000 residual — it does NOT complete. (1) Handling: POV is volume-driven, not deadline-driven — if volume is insufficient it under-fills. Options: (i) leave order incomplete (participation constraint honoured), or (ii) increase $\rho$ / switch to aggressive completion near close. Risk: forcing completion raises participation → higher market impact and signalling; leaving residual → timing/opportunity risk (unfilled at target). (2)

(d) Pseudo-code: (3 for correct logic, 2 for part iv commentary)

def pov_execute(total_qty, rho, intervals):
    remaining = total_qty
    for i, iv in enumerate(intervals):
        realised_vol = read_realised_volume(iv)      # (i)
        cap = int(rho * realised_vol)                # (ii)
        if i == len(intervals) - 1:                  # (iv) final interval
            child = remaining                        # force completion
        else:
            child = min(cap, remaining)
        send_order(child, iv.price)
        remaining -= child                           # (iii) track remaining
        if remaining <= 0:
            break
    return remaining

Commentary on (iv): forcing child = remaining in the last interval can demand a quantity far exceeding $\rho \times V$, so the order becomes a large fraction of available liquidity — market impact spikes (walks the book), and the forced aggression reveals urgency, worsening execution price. This is the fundamental impact-vs-completion trade-off. (2)

Question 3

(a) Gross spread revenue: half-spread $= 0.02$ captured per round trip. Round-trip volume 8,000,000 shares. Interpretation: capturing half the 4¢ spread ($0.02$) per share of round-trip flow. Gross = 8{,}000{,}000 \times 0.02 = \160{,}000$. **(2)** Adverse selection$= 0.35 \times 160000 = $56{,}000$. **(2)** Rebate credit$= 0.0002 \times 8{,}000{,}000 = $1{,}600. Net P&L = 160000 - 56000 + 1600 = $105{,}600$. (2)

(b) Market order size $Q$ consumes book levels. With depth $d$ per level and spacing $\Delta$, filling $Q$ needs $k = \lceil Q/d \rceil$ levels. Prices paid: $p_0, p_0+\Delta, \dots, p_0+(k-1)\Delta$ (buy; symmetric for sell walking down). Average fill price for a sell walking down from $p_0$: $\bar p = \frac{1}{Q}\sum_{j=0}^{k-1} d\,(p_0 - j\Delta) = p_0 - \frac{\Delta}{Q}\cdot d\cdot\frac{k(k-1)}{2}.$ With $k \approx Q/d$: impact $\approx \frac{\Delta}{Q}\cdot d \cdot \frac{(Q/d)^2}{2} = \frac{\Delta Q}{2d}$. (4) As $d \to 0$ (liquidity evaporates in a flash crash), impact $\frac{\Delta Q}{2d} \to \infty$ — price impact is unbounded, i.e., the order walks arbitrarily far down the book (this is how the 2010 Flash Crash saw prices hit stub quotes near $0.01). (2)

(c) Level 1 trigger = 40.00 \times (1 - 0.07) = 40.00 \times 0.93 = \37.20$. (2) Argument: LULD bands (single-stock, e.g. ±5–10% over a rolling 5-min window) prevent individual runaway prints and stub quotes — directly addresses the 2010 mechanism where individual stocks printed at absurd prices while the index moved modestly. Market-wide breakers halt the whole market on aggregate index decline. (2)

• LULD failure mode: fragmented liquidity / momentum can push a stock repeatedly to band edges causing many short halts ("limit state" cycling), harming price discovery; also inconsistent bands across venues.
• Market-wide breaker failure mode: too coarse — a single-stock or ETF cascade (as in 2010) may not breach the aggregate index threshold, so the market-wide breaker never fires while individual names collapse. (2)

[
 {"claim":"Fibre one-way latency for 1180km at n=1.47 is ~5782 microseconds","code":"c=Rational(3,1)*10**8; n=Rational(147,100); v=c/n; t=Rational(118,100)*10**6 / v; result = abs(float(t)*1e6 - 5782) < 2"},
 {"claim":"Microwave advantage with 15pct longer fibre is ~2717 microseconds","code":"c=3.0e8; n=1.47; v=c/n; d=1.15*1.18e6; tf=d/v; tmw=1.18e6/c; adv=(tf-tmw)*1e6; result = abs(adv-2717) < 3"},
 {"claim":"VWAP achieved price equals market VWAP 50.276","code":"V=[200,400,800,600,300,200]; P=[50.0,50.2,50.5,50.3,50.1,50.0]; vwap=sum(v*p for v,p in zip(V,P))/sum(V); result = abs(vwap-50.276) < 1e-6"},
 {"claim":"TWAP average price is 50.18333","code":"P=[50.0,50.2,50.5,50.3,50.1,50.0]; twap=sum(P)/len(P); result = abs(twap-50.183333)<1e-4"},
 {"claim":"POV total executed at 0.18 factor is 450000 shares, residual 150000","code":"V=[200,400,800,600,300,200]; ex=sum(0.18*v for v in V)*1000; result = abs(ex-450000)<1e-6 and abs((600000-ex)-150000)<1e-6"},
 {"claim":"Market maker net daily P&L is 105600 dollars","code":"gross=8000000*0.02; adv=0.35*gross; rebate=0.0002*8000000; net=gross-adv+rebate; result = abs(net-105600)<1e-6"},
 {"claim":"Level 1 circuit breaker trigger price on 40.00 is 37.20","code":"trig=40.00*(1-0.07); result = abs(trig-37.20)<1e-9"}
]