Effective Spread

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244

export type TradeSide = "buy" | "sell";

function numberValue(name: string, value: unknown, positive = false): number {
  if (typeof value !== "number" || !Number.isFinite(value)) {
    throw new Error(`${name} must be a finite number`);
  }
  if (positive && value <= 0) throw new Error(`${name} must be greater than zero`);
  return value;
}

function sideDirection(side: unknown): number {
  if (side === "buy") return 1;
  if (side === "sell") return -1;
  throw new Error("side must be 'buy' or 'sell'");
}

function quoteValues(bid: unknown, ask: unknown): [number, number, number] {
  const bidValue = numberValue("bid", bid, true);
  const askValue = numberValue("ask", ask, true);
  if (askValue < bidValue) throw new Error("ask must be greater than or equal to bid");
  return [bidValue, askValue, (bidValue + askValue) / 2];
}

export function quotedSpread(bid: unknown, ask: unknown) {
  const [bidValue, askValue, midpoint] = quoteValues(bid, ask);
  const spread = askValue - bidValue;
  const relative = spread / midpoint;
  return {
    model: "quoted-spread",
    bid: bidValue,
    ask: askValue,
    midpoint,
    quoted_spread: spread,
    quoted_spread_relative: relative,
    quoted_spread_bps: relative * 10_000,
    state: spread === 0 ? "locked" : "two-sided",
  };
}

export function effectiveSpread(
  bid: unknown,
  ask: unknown,
  tradePrice: unknown,
  side: TradeSide,
) {
  const quote = quotedSpread(bid, ask);
  const trade = numberValue("trade_price", tradePrice, true);
  const direction = sideDirection(side);
  const effective = 2 * direction * (trade - quote.midpoint);
  const relative = effective / quote.midpoint;
  const quoteSidePrice = direction === 1 ? quote.ask : quote.bid;
  const priceImprovementPerShare = direction * (quoteSidePrice - trade);
  const roundTripSpreadReduction = quote.quoted_spread - effective;
  const tolerance = 1e-12;
  let state = "outside-quote";
  if (effective < -tolerance) state = "through-midpoint-improvement";
  else if (roundTripSpreadReduction > tolerance) state = "inside-quote";
  else if (Math.abs(roundTripSpreadReduction) <= tolerance) state = "at-quote";
  return {
    ...quote,
    model: "effective-spread",
    trade_price: trade,
    side,
    direction,
    effective_spread: effective,
    effective_spread_relative: relative,
    effective_spread_bps: relative * 10_000,
    quote_side_price: quoteSidePrice,
    price_improvement_per_share: priceImprovementPerShare,
    round_trip_spread_reduction: roundTripSpreadReduction,
    state,
  };
}

export function realizedSpread(
  bidAtTrade: unknown,
  askAtTrade: unknown,
  tradePrice: unknown,
  side: TradeSide,
  bidAfter: unknown,
  askAfter: unknown,
  horizonSeconds: unknown = 300,
) {
  const initial = effectiveSpread(bidAtTrade, askAtTrade, tradePrice, side);
  const [futureBid, futureAsk, futureMidpoint] = quoteValues(bidAfter, askAfter);
  const horizon = numberValue("horizon_seconds", horizonSeconds, true);
  const realized = 2 * initial.direction * (initial.trade_price - futureMidpoint);
  const priceImpact = 2 * initial.direction * (futureMidpoint - initial.midpoint);
  const relative = realized / initial.midpoint;
  const impactRelative = priceImpact / initial.midpoint;
  return {
    model: "realized-spread",
    side,
    direction: initial.direction,
    trade_price: initial.trade_price,
    midpoint_at_trade: initial.midpoint,
    midpoint_after: futureMidpoint,
    bid_after: futureBid,
    ask_after: futureAsk,
    horizon_seconds: horizon,
    effective_spread: initial.effective_spread,
    realized_spread: realized,
    realized_spread_relative: relative,
    realized_spread_bps: relative * 10_000,
    price_impact: priceImpact,
    price_impact_relative: impactRelative,
    price_impact_bps: impactRelative * 10_000,
    decomposition_error: initial.effective_spread - realized - priceImpact,
    state: realized < 0 ? "negative-realized" : "nonnegative-realized",
  };
}

function finiteSeries(name: string, values: unknown, minimum: number): number[] {
  if (!Array.isArray(values)) throw new Error(`${name} must be an array`);
  const result = values.map((value, index) => numberValue(`${name}[${index}]`, value, true));
  if (result.length < minimum) throw new Error(`${name} must contain at least ${minimum} observations`);
  return result;
}

function mean(values: number[]): number {
  return values.reduce((total, value) => total + value, 0) / values.length;
}

export function rollSpread(prices: unknown) {
  const priceValues = finiteSeries("prices", prices, 4);
  const changes = priceValues.slice(1).map((value, index) => value - priceValues[index]);
  const lagged = changes.slice(0, -1);
  const current = changes.slice(1);
  if (current.length < 2) throw new Error("prices must create at least two covariance pairs");
  const laggedMean = mean(lagged);
  const currentMean = mean(current);
  const covariance = lagged.reduce(
    (total, value, index) => total + (value - laggedMean) * (current[index] - currentMean),
    0,
  ) / (current.length - 1);
  const meanPrice = mean(priceValues);
  const covarianceTolerance = 1e-15;
  const spread = covariance < -covarianceTolerance ? 2 * Math.sqrt(-covariance) : null;
  const relative = spread === null ? null : spread / meanPrice;
  return {
    model: "roll-spread",
    observation_count: priceValues.length,
    change_count: changes.length,
    covariance_pair_count: current.length,
    lag1_price_change_covariance: covariance,
    covariance_tolerance: covarianceTolerance,
    mean_price: meanPrice,
    spread_estimate: spread,
    spread_estimate_relative: relative,
    spread_estimate_bps: relative === null ? null : relative * 10_000,
    state: spread === null ? "not-identified" : "estimated",
  };
}

export function amihudIlliquidity(
  closes: unknown,
  dollarVolumes: unknown,
  scale: unknown = 1_000_000,
) {
  const closeValues = finiteSeries("closes", closes, 2);
  const volumeValues = finiteSeries("dollar_volumes", dollarVolumes, 2);
  if (closeValues.length !== volumeValues.length) {
    throw new Error("closes and dollar_volumes must have equal length");
  }
  const scaleValue = numberValue("scale", scale, true);
  const returns = closeValues.slice(1).map((value, index) => value / closeValues[index] - 1);
  const ratios = returns.map((value, index) => Math.abs(value) / volumeValues[index + 1]);
  const raw = mean(ratios);
  return {
    model: "amihud-illiquidity",
    observation_count: closeValues.length,
    return_count: returns.length,
    returns,
    daily_illiquidity: ratios,
    illiquidity_per_currency_unit: raw,
    scale: scaleValue,
    illiquidity_per_scaled_volume: raw * scaleValue,
    state: "estimated",
  };
}

export function corwinSchultzSpread(
  highDay1: unknown,
  lowDay1: unknown,
  highDay2: unknown,
  lowDay2: unknown,
  clipNegative = true,
) {
  const h1 = numberValue("high_day_1", highDay1, true);
  const l1 = numberValue("low_day_1", lowDay1, true);
  const h2 = numberValue("high_day_2", highDay2, true);
  const l2 = numberValue("low_day_2", lowDay2, true);
  if (h1 < l1 || h2 < l2) throw new Error("each daily high must be greater than or equal to its low");
  const beta = Math.log(h1 / l1) ** 2 + Math.log(h2 / l2) ** 2;
  const gamma = Math.log(Math.max(h1, h2) / Math.min(l1, l2)) ** 2;
  const denominator = 3 - 2 * Math.sqrt(2);
  const alphaRaw = (
    (Math.sqrt(2 * beta) - Math.sqrt(beta)) / denominator
    - Math.sqrt(gamma / denominator)
  );
  const alpha = clipNegative ? Math.max(0, alphaRaw) : alphaRaw;
  const expAlpha = Math.exp(alpha);
  const spread = 2 * (expAlpha - 1) / (1 + expAlpha);
  return {
    model: "corwin-schultz-spread",
    beta,
    gamma,
    alpha_raw: alphaRaw,
    alpha_used: alpha,
    spread_estimate_relative: spread,
    spread_estimate_bps: spread * 10_000,
    clip_negative: clipNegative,
    state: clipNegative && alphaRaw < 0 ? "clipped-to-zero" : "estimated",
  };
}

export function calculate(topicId: string, inputs: Record<string, any>) {
  if (topicId === "D11-F02-A01") return quotedSpread(inputs.bid, inputs.ask);
  if (topicId === "D11-F02-A02") return effectiveSpread(inputs.bid, inputs.ask, inputs.trade_price, inputs.side);
  if (topicId === "D11-F02-A03") return realizedSpread(
    inputs.bid_at_trade,
    inputs.ask_at_trade,
    inputs.trade_price,
    inputs.side,
    inputs.bid_after,
    inputs.ask_after,
    inputs.horizon_seconds ?? 300,
  );
  if (topicId === "D11-F02-A04") return rollSpread(inputs.prices);
  if (topicId === "D11-F02-A05") return amihudIlliquidity(
    inputs.closes,
    inputs.dollar_volumes,
    inputs.scale ?? 1_000_000,
  );
  if (topicId === "D11-F02-A06") return corwinSchultzSpread(
    inputs.high_day_1,
    inputs.low_day_1,
    inputs.high_day_2,
    inputs.low_day_2,
    inputs.clip_negative ?? true,
  );
  throw new Error(`unsupported topic_id: ${topicId}`);
}