# Microsoft Word Examples.doc

## Transcript

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

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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