1 Detailed Examples of the Modi fications to Accommodate any Decimal or Fractional Price Grid The Holden Model on any Decimal or Fractional Price Grid This section presents the modifications of th e Holden model to accommodate any decimal or fractional price grid. Accounting for decimal price gr ids adds considerably complexity to the general formulas. I use a standard example of a decimal pri ce grid. Specifically, the possible effective spreads . s are $0.01, $0.05, the 's $0.25, and $1.00 and 5 J = $0.10, () j To build up to the general formulas, I introduce sev eral new variables to keep track of various attributes of a decimal (or fractional) price grid. Let A and number of (trade) prices total A be the + j j J jJ th . For prices, K j , spread = 1, 2 , and (no-trade) midpoints, respectively, corresponding to the ) ( there are 100 pennies, 20 nickels, 10 dimes, 4 quarters, and 1 dollar, so 100 A = , 20 A = , = 10, A 3 2 1 1 1 1 = , and dimes 1 A = . For midpoints, there are 100 odd , 4 pennies , 20 odd A , 10 odd nickels 5 4 2 2 2 1 1 4 odd , and 1 odd A . dollars , so = 100 A = , A 20 A = , quarters 10, 1 = , and 4 A = 9 8 10 7 6 2 2 To build up this general formula, I need to introduce a new variable. special price increments for the Define jth spread as price increments that can be generated by jth spread, but not by any larger spreads. Let be the number of special prices and and the B B j + j J jth special midpoints, respectively, corresponding to the be the = 1, 2 , , jJ spread K . Let D ( ) jk the price jth spread overlap which 1, 2 , , jJ = K number of special price increments for the ) ( kth spread. increments of the variables for this standard example. , , and D AB Table 1 summarizes the jk jj

2 Table 1 AB D for a Decimal Price Grid , , and jj jk Corresponding Prices or D A B j j jk j midpoints? Spread = D 80 $0.01 11 Prices 100 80 1 8 8, DD = = $0.05 21 22 Prices 20 8 2 8, == = 8, 8 DDD $0.10 33 32 31 Prices 10 8 3 === = DDDD 3 1, 3, 3, $0.25 44 41 43 42 Prices 4 3 4 ==== = DDDDD 1 1, 1, 1, 1, $1.00 55 52 51 53 54 Prices 1 1 5 = D 80 $0.01 61 80 Midpoints 100 6 = = 16, 16 DD $0.05 72 71 Midpoints 20 16 7 == = DDD 10 0, 0, $0.10 83 81 82 Midpoints 10 10 8 DDDD = === 4, 4, 0, 4 $0.25 93 94 91 92 Midpoints 4 4 9 0, 1 DD DD D ==== 0, 0, 0, = 10,1 10,5 10,4 10,3 10,2 $1.00 Midpoints 1 1 10 First consider the B ’s for the prices. Out of the 100 penn y price increments, 80 are special for j $0.01 spread, because they can be generated by a $0.01 spread (or any $0.05 spread, but not by the last increments “off pennies.” They are all price increments where the larger spread). I call these special digit is 1, 2, 3, 4, 6, 7, 8, or 9 . Out of the 20 nickels, 8 are special for the $0.05 spread, because they $0.05 spread, but not by $0.10 spread (or any larger sp read). These “off nickels” can be generated by a the last two digits are 05, 15, 35, 45, 55, 65, 85, or 95 . Out of the 10 dimes, 8 “off dimes” are where $0.10 are special for the 10, 20, 30, 40, 60, 70, 80, or 90. Out of the 4 quarters, 3 “off spread, namely $0.25 quarters” are special for the spread, namely 25, 50, or 75. Hence, and 80 B = , = 8 B = , B 8, 3 2 1 3 B = . For the largest increments are special by definition effective spread, all of its price the Jth ) ( 4 1 BA == . JJ 1 pennies , 20 are also odd B ’s for the midpoints. Out of 100 odd Now consider the + J j 2 1 1 1 and 80 are special because they are not odd nickels , nickels or above. Out of 20 odd nickels 2 2 2 1 1 1 dimes , but 4 are odd quarters , so 16 odd nickels are special. All 10 of the odd none are odd 2 2 2

3 1 1 1 quarters or above. All 4 of the odd quarters are dimes are special because they are not odd 2 2 2 1 1 is special, by definition, because it dollars dollar . The single odd special because they are not odd 2 2 is the highest spread. $0.01 Ds for the prices. All 80 special price increments for a ' spread are First consider the jk spread are both nickels and pennies. All 8 special $0.05 pennies. All 8 special price increments for a spread are dimes, nickels, and pennies. All 3 special price increments for a $0.10 price increments for a spread are quarters, nickels, and pennies, but importantly one is a dime $0.50 . The single $0.25 () $1.00 spread is a dollar, quarter, dime, nickel, and penny. special price increment for a 1 Ds for the midpoints. All 80 of the special off pennies are off ' Now consider the jk 2 1 1 1 1 pennies nickels nickels . All 16 of the special off pennies . All 10 of the are both off and off 2 2 2 2 1 1 1 1 , but none of them are off pennies dimes . All 4 of dimes are odd nickels or off special odd 2 2 2 2 1 1 1 1 the special odd , off are odd nickels , and off , but none of them quarters pennies quarters 2 2 2 2 1 1 1 1 dollar is a odd , . The single special odd dollar , but it is not an odd dimes quarter are odd 2 2 2 2 1 1 1 odd dime , off nickel , or off penny . 2 2 2 By creating a tree similar to Figure 1 for a decimal price grid, it is straight-forward to compute the cluster probabilities spread given a particular price and conditional probability of a half Pr = Cj () t Pr | == HhC j . Using the first part of a tree similar to Figure 1 for a decimal price grid as a cluster () tkt template, the probabilities of the trade price clusters are j D jk Cj 1, 2, K Pr = == γμ J , j . (A1) () ∑ tk A k k = 1 Similarly, the probabilities of the no-trade midpoint clusters are j D , Jjk + j 1, 2, , . (A2) = K Pr − =+ = 1 CJj J γμ ()() ∑ tk A + Jk k = 1

4 Using the first part of a tree similar to Figure 1 for a decimal price grid as a template, the conditional probability of a half spread given a particular price cluster is D jk μ γ ( ) 2 A k k , (A3) J | 0, K ≠≤ = HhC j k kj j 1, 2, === Pr , , and () tkt Cj = Pr () t and Pr | 0 1, 2, , . (A4) HhC j j J ==== K () tt 0 Similarly, the conditional probability of a half spread given a particular midpoint cluster is HhCJj | 1 1, 2, , Pr (A5) j J ==+== K () 0 tt K Pr | 0 0, , and 1, 2, , ==+= ≠≤ = . (A6) k kj j J HhCJj and () tkt 6) are substituted into th e likelihood function equation To complete the computation, equations (A1) – (A and estimated. To illustrate how a fractional price grid fits into the general formula of the Holden model, 1 1 1 1 1 consider a , $ price grid where the possible effective spreads are $ , . Table 2 $1 $ , $ , and $ 2 4 16 16 8 variables for this fractional price grid. A , D B , and shows the jk j j for a Fractional Price Grid Table 2 A BD , , and jj jk Corresponding Prices or O A B j jk j j Spread midpoints? 1 8 = $ D 11 16 Prices 16 8 1 1 $ = 4, 4 DD = 22 21 8 Prices 8 4 2 1 == $ 2, 2, 2 DDD = 31 32 33 4 Prices 4 2 3 1 1, === $ 1, 1, 1 DDDD = 43 44 41 42 2 Prices 2 1 4 ==== = DDDDD 1 1, 1, 1, 1, $1 55 53 52 51 54 Prices 1 1 5 1 $ 16 D = 61 16 Midpoints 16 16 6 1 8 $ = = DD 0, 72 71 8 Midpoints 8 8 7 1 0, $ == = DDD 4 0, 83 82 81 4 Midpoints 4 4 8 1 === $ = DDDD 2 0, 0, 0, 94 91 92 93 2 Midpoints 2 2 9 1 DD DD D ==== 0, 0, 0, 0, = 10,1 10,2 10,3 10,4 10,5 $1 Midpoints 1 1 10 By inspection of the coefficients in Table 2, it is clear that

5 1 − jk jk −+ D D 1 1 jk jk K 1 , 2, 3, = = . (A7) and jJ jJ ==− 2 2 A A k k and (A3) yield the corresponding fractional Holden Substituting these coefficients into equations (A1) formulas. Also, by inspection of the coefficients D D + , , Jjk + Jjk and 1 for < = . (A8) == 0 for kj kj A A k k elds the corresponding fractional Holden formula. Substituting these coefficients into equation (A2) yi Equations (A4) – (A6) match the remaining fracti onal equations, which completes the demonstration. The Effective Tick Model on any Decimal or Fractional Price Grid This section presents the general formula for th e Effective Tick model, which works on any decimal or fractional price grid. Many price grids exhibit the property that the price increments overlap 100% between adjacent , all wholes are halves, all halves are quarters, all spread levels. For example in a fractional price grid wever, this property does not hold in general. In the quarters are eighths, all eighths are sixteenths, etc. Ho decimal price grid under consideration, all dollars are quarters, all dim es are nickels, and all nickels are pennies, but quarters are different. Two quarters are dimes $0.50, $1.00 and two quarters are not ) ( $0.25, $0.75 . The latter two quarters overlap with nickels (two spread layers down), but not dimes () dimes (one spread layer down). Hence, there is a need for a relatively elaborate way of tracking these overlaps. Let O be the number of price increments for the jth spread overlap which 1, 2 , , jJ = K ) ( jk kth spread and do not overlap the price increments of any spreads between the price increments of the the spread and the kth spread. Similarly, let O jth be the number of overlapping midpoints for the jk + , J kth K not overlap 1, 2 , , jJ = spread which overlap the midpoints of the jth spread and do the ()

6 midpoints of any spreads between the jth spread. spread and the D and kth O are distinct in two jk jk in the number of total price increments which overlap vs. is the number of special ways. First, D O jk jk price increments which overlap. Second, the count those increments which overlap the excludes O jk jth spread and the spread vs. the count includes kth price increments of any spreads between the D jk those increments. Table 3 summarizes the , , and AB O variables for a standard example of a jk jj decimal price grid. Table 3 O for a Decimal Price Grid , , and AB jk jj Corresponding Prices or B A O j jk j j midpoints? Spread $0.01 Prices 100 80 1 20 O = $0.05 21 Prices 20 8 2 OO = 10 0, = $0.10 32 31 Prices 10 8 3 == = 0, OOO 2 2, $0.25 43 41 42 Prices 4 3 4 === 0, 0, 0, 1 OOOO = $1.00 53 54 51 52 Prices 1 1 5 $0.01 Midpoints 100 80 6 20 O = $0.05 71 Midpoints 20 16 7 = = OO 0, 0 $0.10 82 81 Midpoints 10 10 8 == = 0, OOO 0 4, $0.25 93 92 91 Midpoints 4 4 9 === = OO OO 0 0, 0, 0, 10,4 10,3 10,1 10,2 $1.00 10 Midpoints 1 1 ' Os for the prices. All 20 nickels are also pennies and all 10 dimes are also Now consider the jk nickels. Of the 4 quarters, 2 are dimes and 2 are nickel s, but not dimes. The single dollar is also a quarter. 1 Finally consider nickels for the midpoints. All 20 of the odd ' Os overlap with odd , Jjk + 2 1 1 1 1 . None of the odd pennies dimes overlap with odd pennies . None of the odd nickels or odd 2 2 2 2 1 1 1 overlap with odd dimes nickels . The odd , but all 4 of them overlap with odd quarters 2 2 2 1 dollar doesn’t overlap with anything below. 2

7 Given all of the infrastructure variables in Tabl e 3, the general formula of the Effective Tick a for the unconstrained probability of the model can be stated. The general formul spread is jth A A 1 J + 1 FF j += 1 + 11 J BB J + 11 U . (A9) = j −− 11 jj O AOA ++ , jjkJj Jjk FFF FjJ , 2, 3, K = −+− ∑∑ ++ jkJj Jk B BBB == 11 kk ++ jkJj Jk The rest of the effective tick computation is the same as the fraction grid case. To illustrate how a fractional price grid fits into the general formula of the Effective Tick model, 1 1 1 1 1 consider a , $ $1 $ , price grid where the possible effective spreads are $ . Table 4 , $ , and $ 2 4 16 16 8 , A , and O variables for this fractional price grid. B shows the jk j j Table 4 AB O for a Fractional Price Grid , , and jj jk Corresponding O B A j j jk j Spread Prices or midpoints? 1 $ 16 Prices 16 8 1 1 = $ O 8 21 8 Prices 8 4 2 1 OO = 0, $ = 4 32 31 4 Prices 4 2 3 1 $ == = OOO 0, 0, 2 43 42 41 2 Prices 2 1 4 Prices 1 1 5 $1 0, 0, 0, 1 OOOO = === 53 54 51 52 1 $ 16 Midpoints 16 16 6 1 = $ O 0 71 8 Midpoints 8 8 7 1 OO = $ = 0 0, 82 81 4 Midpoints 4 4 8 1 $ == = OOO 0, 0, 0 93 91 92 2 Midpoints 2 2 9 === OO O O = 0 0, 0, 0, 10,4 10,3 10,1 10,2 $1 Midpoints 1 1 10 Consider the coefficients of equation (A9). From Table 4, it is clear that A A A Jj j + J 1, 2, , 1 j J =− K , 1 K j . (A10) , = , and 1 = 1, 2 , J for for = 2 = B B B + j Jj J

8 In fractional price grids, the price increments exhib it 100% overlap between adjacent spread levels (i.e., all wholes are halves, all halves are quarters, all quarters are eighths, etc.). This implies that O O jk jk K = and 0 for all 1 = J . (A11) j , 1 for k=j-1 <− and for 2, 3, = kj B B k k it 0% overlap between adjacent spread levels (i.e., no In fractional price grids, the midpoint clusters exhib 1 1 1 1 1 odd odd midpoints, no odd midpoints are midpoints are odd midpoints, no odd 8 4 2 8 4 1 midpoints are odd midpoints, etc.). This implies that 16 O , Jjk + kj J = K , 2, 3, for all and for = 0 . (A12) B + Jk (A10 ), (A11), and ( A12) into the general Substituting the fractional price grid coefficients in equations Effective Tick formula (A9) yields the fractional Effective Tick formula.

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