4Why There Is a Spread

A market maker quotes 100.00 bid, 100.02 offered. If nobody knew more than she did, the two cents would pay for her time, her systems and the risk of the inventory she carries between trades. A buyer who has read the news first makes the spread pay for something else: every trade with him loses money, and the spread must recover the loss from everyone else. Three theories of the spread answer three questions: what it costs to process a trade, what it costs to hold what a trade leaves behind, and what it costs to trade with someone who knows more. They lead to different spreads, to different dynamics of quotes, and to different ways of measuring each part (chapter 5).

4.1 Order-processing costs

If processing were the only cost, a competitive market maker would quote a half-spread equal to it, the value would not react to trades, and transaction prices would bounce between bid and ask around a random-walk value. Roll (1984) turned the bounce into an estimate: with a constant effective half-spread $c$ and independent trade signs, the price changes have autocovariance $-c^2$, so $s = 2\sqrt{-\mathrm{Cov}(\Delta p_t,\Delta p_{t-1})}$ (Roll’s estimator, One Quant Book 4, chapter 21). On the simulated hour of firm.tape, the estimate is 0.88 ticks against a quoted spread of 1.08: the simulator’s trade signs are positively autocorrelated (0.29 at lag one, because noise orders come in metaorders), which removes part of the bounce, as it does in real markets (One Quant Book 7, chapter 9).

4.2 Inventory

Garman (1976) modelled a dealer whose inventory is a random walk driven by the arrival of buyers and sellers, and who must set prices to avoid ruin; Stoll (1978) and Ho and Stoll (1981) priced the risk. The simplest version already gives the two features every inventory model shares.

A dealer who is long shades both quotes down: she wants to sell and discourages buying from others. The inventory term is the root of the market-making models of One Quant Book 11 (the Avellaneda–Stoikov reservation price has the same form); here it matters that it produces a spread without any information at all.

4.3 Information: the sequential-trade model

def gm_quotes(delta: float, mu: float, v_low: float = 0.0, v_high: float = 1.0) -> tuple[float, float]:
    dv = v_high - v_low
    ask = v_low + dv * delta * (1 + mu) / (delta * (1 + mu) + (1 - delta) * (1 - mu))
    bid = v_low + dv * delta * (1 - mu) / (delta * (1 - mu) + (1 - delta) * (1 + mu))
    return bid, ask


def gm_update(delta: float, side: int, mu: float) -> float:
    ph = (1 + mu) / 2 if side > 0 else (1 - mu) / 2        # P(side | v_high)
    pl = (1 - mu) / 2 if side > 0 else (1 + mu) / 2        # P(side | v_low)
    return delta * ph / (delta * ph + (1 - delta) * pl)

The spread is entirely adverse selection (One Quant Book 1, chapter 1): the provider breaks even across traders, losing to the informed exactly what she earns from noise. It narrows as trades reveal the value (Figure 4.2): with 30% informed arrivals the median path needs 19 trades for the spread to fall to a tenth of its start, with 10% it needs 148. Informed trading does two things at once: it widens the spread and makes prices learn faster. With a processing cost $c$ per trade added to each quote, the adverse-selection share of the spread is $\mu\Delta v/(\mu\Delta v+2c)$: with $\Delta v=1$ and $c=0.05$, half the spread at $\mu=0.1$ and three quarters at $\mu=0.3$.

4.4 Information: the strategic trader

The informed trader is strategic: he knows his trades move the price and hides in the noise. The linear equilibrium is short to derive.

The slope $\lambda$ is Kyle’s lambda (One Quant Book 7, chapter 9): the price impact of a unit of order flow, the inverse of the market’s depth. More noise makes the market deeper and the insider richer, and leaves the price exactly as informative. On 100 000 simulated draws with $\sqrt{\Sigma_0}=1$ and $\sigma_u=2$, the regression of $p$ on $y$ returns 0.250, the residual variance of $v$ is 0.498 and the insider’s average profit 0.991, against the proposition’s 0.25, 0.5 and 1. In Kyle’s continuous-time version the insider trades gradually, $\lambda$ is constant through time and all of his information is in the price by the end (

; derived in Kyle’s paper).

4.5 Measuring informed trading

Each day’s buys and sells are Poisson with rates set by the day’s hidden state, and the parameters are fitted by maximum likelihood on daily counts (the likelihood mixes three Poisson pairs; its logarithm must be computed in a numerically stable way).

def pin_loglik(theta, buys, sells) -> float:
    a, d, mu, eb, es = theta
    b = np.asarray(buys, float)
    s = np.asarray(sells, float)
    terms = np.stack([np.log(1 - a) + _logpois(b, eb) + _logpois(s, es),
                      np.log(a * d) + _logpois(b, eb) + _logpois(s, es + mu),
                      np.log(a * (1 - d)) + _logpois(b, eb + mu) + _logpois(s, es)])
    return float(logsumexp(terms, axis=0).sum())

The model reads any common swing in buying and selling as information when the swing is not one-sided in its data, and Duarte and Young (2009) found that PIN’s association with returns came from its illiquidity component rather than its information component. Figure 4.3 shows why on simulated data. Days with no information at all, whose uninformed rates move together from day to day (a lognormal factor on both), are fitted with a PIN of 0.085 when the factor’s standard deviation is 0.25, 0.153 at 0.5 and 0.207 at 0.75, against 0.006 without the swings. A market with true PIN 0.107 is estimated at 0.107 without swings and 0.195 with the largest.

4.6 Tutorial: three spreads in code

4.7 Build: the spread models

4.8 Exercises

4.9 Problem: The Specialist’s Two Cents

4.10 Interview questions