Pseudocode

This file provides the pseudocode to simplify implementations of the estimator in any programming language.

Input

Vectors of open, high, low, and close prices. The vectors must be sorted in ascending order of the timestamp. The function should also accept the argument sign specifying whether to return signed estimates.

Output

Numeric spread estimate. A value of 0.01 corresponds to a spread of 1%.

Algorithm

# check that the open, high, low, and close prices have the same length
nobs = len(open)
if len(high) != nobs or len(low) != nobs or len(close) != nobs:
	raise error

# return missing if there are less than 3 observations
if nobs < 3:
	return missing

# compute log-prices
o = log(open)
h = log(high)
l = log(low)
c = log(close)
m = (h + l) / 2.

# shift log-prices by one period
h1 = lag(h)
l1 = lag(l)
c1 = lag(c)
m1 = lag(m)

# compute log-returns
r1 = m - o
r2 = o - m1
r3 = m - c1
r4 = c1 - m1
r5 = o - c1

# compute indicator variables
tau = (h != l or l != c1) if h, l, c1 are non-missing else missing
po1 = (tau and o != h) if tau, o, h are non-missing else missing
po2 = (tau and o != l) if tau, o, l are non-missing else missing
pc1 = (tau and c1 != h1) if tau, c1, h1 are non-missing else missing
pc2 = (tau and c1 != l1) if tau, c1, l1 are non-missing else missing

# compute probabilities
pt = mean(tau)
po = mean(po1) + mean(po2)
pc = mean(pc1) + mean(pc2)

# return missing if there are less than two periods with tau=1 
# or po or pc is zero
if sum(tau) < 2 or po == 0 or pc == 0:
  return missing

# compute de-meaned log-returns
d1 = r1 - mean(r1)/pt*tau
d3 = r3 - mean(r3)/pt*tau
d5 = r5 - mean(r5)/pt*tau

# compute input vectors
x1 = -4./po*d1*r2 + -4./pc*d3*r4 
x2 = -4./po*d1*r5 + -4./pc*d5*r4 

# compute expectations
e1 = mean(x1)
e2 = mean(x2)

# compute variances
v1 = mean(x1*x1) - e1*e1
v2 = mean(x2*x2) - e2*e2

# compute square spread by using a (equally) weighted 
# average if the total variance is (not) positive
vt = v1 + v2
s2 = (v2*e1 + v1*e2) / vt if vt > 0 else (e1 + e2) / 2.

# compute signed root
s = sqrt(abs(s2))
if sign and s2 < 0: 
    s = -s

# return the spread
return s

Testing

To test your implementation, import the data available here. The file contains sample OHLC simulated price data to simplify testing. The data have been generated by simulating a price process as described in Ardia, Guidotti, & Kroencke (2024) with 390 trades per day and a 1% probability to observe a trade. The simulation uses a constant spread of 1%. By running the estimation, you should obtain a spread estimate of 0.0101849034905478. If you obtain a different result, you may use the following table to check and debug the intermediate steps.

variable	value
pt	0.9820982098209821
po	1.227922792279228
pc	1.2052205220522052
e1	0.00010702425689560482
e2	0.000101595812797079
v1	2.074215642985551e-06
v2	1.3461279919743572e-06
s2	0.00010373225911177194

To check that your implementation correctly handles missing values, import the data available here. The data have been generated by setting to missing a random subset of the previous data file. By running the estimation, you should obtain a spread estimate of 0.01013284969780197. If you obtain a different result, you may use the following table to check and debug the intermediate steps.

variable	value
pt	0.9822078447230085
po	1.2272254421162134
pc	1.205827632480371
e1	0.00010337780767834583
e2	0.00010219271972776808
v1	2.0045420261850617e-06
v2	1.373839551967266e-06
s2	0.00010267464299824543

Contribute

Have you implemented the estimator in a new programming language? If you want your implementation to be included in this repository, please open a pull request