Detailed Examples of the Modifications to Accommodate. any Decimal or Fractional Price Grid

Transcription

1 eailed Examples of he Modificaions o ccommodae any ecimal or Fracional Price Grid The Holden Model on any ecimal or Fracional Price Grid This secion presens he modificaions of he Holden model o accommodae any decimal or fracional price grid. ccouning for decimal price grids adds considerably complexiy o he general formulas. I use a sandard example of a decimal price grid. Specifically, he possible effecive spreads ( he 's) s are 0.0, 0.05, 0.0, 0.5, and.00 and J = 5. To build up o he general formulas, I inroduce several new variables o eep rac of various aribues of a decimal (or fracional) price grid. Le and J + be he oal number of (rade) prices and (no-rade) midpoins, respecively, corresponding o he h spread (,,, J) = K. For prices, here are 00 pennies, 0 nicels, 0 dimes, quarers, and dollar, so 0,, 3, =, and 5 =. For midpoins, here are 00 odd pennies, 0 odd nicels, 0 odd dimes, odd quarers, and odd dollars, so 6 0, 7,, 9 =, and 0 =. To build up his general formula, I need o inroduce a new variable. efine special price incremens for he h spread as price incremens ha can be generaed by he h spread, bu no by any larger spreads. Le and J + be he number of special prices and special midpoins, respecively, corresponding o he h spread (,,, J) number of special price incremens for he h spread (,,, J) incremens of he h spread. = K. Le be he = K which overlap he price Table summarizes he,, and variables for his sandard example.

2 Table,, and for a ecimal Price Grid Corresponding Prices or Spread midpoins? 0.0 Prices Prices 0 =, = Prices 0 3 =, 3 =, 33 = 0.5 Prices 3 = 3, = 3, 3 =, = Prices 5 =, 5 =, 53 =, 5 =, 55 = Midpoins Midpoins = 6, 7 = Midpoins 0 0,, Midpoins 9 =, 9 =, 93, 9 = 0.00 Midpoins 0,, 0,, 0,3, 0,, 0,5 = Firs consider he s for he prices. u of he 00 penny price incremens, 0 are special for he 0.0 spread, because hey can be generaed by a 0.0 spread, bu no by 0.05 spread (or any larger spread). I call hese special incremens off pennies. They are all price incremens where he las digi is,, 3,, 6, 7,, or 9. u of he 0 nicels, are special for he 0.05 spread, because hey can be generaed by a 0.05 spread, bu no by 0.0 spread (or any larger spread). These off nicels are where he las wo digis are 05, 5, 35, 5, 55, 65, 5, or 95. u of he 0 dimes, off dimes are special for he 0.0 spread, namely 0, 0, 30, 0, 60, 70, 0, or 90. u of he quarers, 3 off quarers are special for he 0.5 spread, namely 5, 50, or 75. Hence,, =, 3 =, and = 3. For he larges ( he Jh ) effecive spread, all of is price incremens are special by definiion J = =. J Now consider he J + s for he midpoins. u of 00 odd pennies, 0 are also odd nicels and 0 are special because hey are no odd nicels or above. u of 0 odd nicels, none are odd dimes, bu are odd quarers, so 6 odd nicels are special. ll 0 of he odd

3 dimes are special because hey are no odd quarers or above. ll of he odd special because hey are no odd dollars. The single odd is he highes spread. Firs consider he quarers are dollar is special, by definiion, because i ' s for he prices. ll 0 special price incremens for a 0.0 spread are pennies. ll special price incremens for a 0.05 spread are boh nicels and pennies. ll special price incremens for a 0.0 spread are dimes, nicels, and pennies. ll 3 special price incremens for a 0.5 spread are quarers, nicels, and pennies, bu imporanly one is a dime ( 0.50 ). The single special price incremen for a.00 spread is a dollar, quarer, dime, nicel, and penny. Now consider he ' s for he midpoins. ll 0 of he special off pennies are off pennies. ll 6 of he special off nicels are boh off nicels and off pennies. ll 0 of he special odd dimes are odd dimes, bu none of hem are off nicels or off pennies. ll of he special odd quarers are odd quarers, off nicels, and off pennies, bu none of hem are odd dimes. The single special odd dollar is a odd dollar, bu i is no an odd quarer, odd dime, off nicel, or off penny. y creaing a ree similar o Figure for a decimal price grid, i is sraigh-forward o compue he cluser probabiliies Pr ( C ) cluser Pr ( H h C ) = and condiional probabiliy of a half spread given a paricular price = =. Using he firs par of a ree similar o Figure for a decimal price grid as a emplae, he probabiliies of he rade price clusers are γµ = Pr ( C = ) = =,, K, J. () Similarly, he probabiliies of he no-rade midpoin clusers are =, Pr ( C = J + ) = γ ( µ ) =,, K, J. ()

4 Using he firs par of a ree similar o Figure for a decimal price grid as a emplae, he condiional probabiliy of a half spread given a paricular price cluser is µ ( ) γ Pr ( H = h C = ) = 0,, and =,, K, J, (3) Pr and ( ) 0 ( C = ) Pr H = h C = =,, K, J. () Similarly, he condiional probabiliy of a half spread given a paricular midpoin cluser is ( ) Pr H = h C = J + = =,, K, J (5) and ( ) 0 Pr H = h C = J + 0,, and =,, K, J. (6) To complee he compuaion, equaions () (6) are subsiued ino he lielihood funcion equaion and esimaed. To illusrae how a fracional price grid fis ino he general formula of he Holden model, consider a 6 price grid where he possible effecive spreads are 6,,,, and. Table shows he,, and variables for his fracional price grid. Table,, and for a Fracional Price Grid Corresponding Prices or Spread midpoins? 6 Prices 6 = Prices =, = 3 Prices 3 =, 3 =, 33 = Prices =, =, 3 =, = 5 Prices 5 =, 5 =, 53 =, 5 =, 55 = 6 6 Midpoins = 6 7 Midpoins 7, 7 = Midpoins,, 3 = 9 Midpoins 9, 9, 93, 9 = 0 Midpoins 0,, 0,, 0,3, 0,, 0,5 = y inspecion of he coefficiens in Table, i is clear ha

5 + = =,3, K, J and = = J. (7) Subsiuing hese coefficiens ino equaions () and (3) yield he corresponding fracional Holden formulas. lso, by inspecion of he coefficiens, = for = and, for <. () Subsiuing hese coefficiens ino equaion () yields he corresponding fracional Holden formula. Equaions () (6) mach he remaining fracional equaions, which complees he demonsraion. The Effecive Tic Model on any ecimal or Fracional Price Grid This secion presens he general formula for he Effecive Tic model, which wors on any decimal or fracional price grid. Many price grids exhibi he propery ha he price incremens overlap 00% beween adacen spread levels. For example in a fracional price grid, all wholes are halves, all halves are quarers, all quarers are eighhs, all eighhs are sixeenhs, ec. However, his propery does no hold in general. In he decimal price grid under consideraion, all dollars are quarers, all dimes are nicels, and all nicels are pennies, bu quarers are differen. Two quarers are dimes ( 0.50,.00 ) and wo quarers are no dimes ( 0.5, 0.75 ). The laer wo quarers overlap wih nicels (wo spread layers down), bu no dimes (one spread layer down). Hence, here is a need for a relaively elaborae way of racing hese overlaps. Le be he number of price incremens for he h spread (,,, J) = K which overlap he price incremens of he h spread and do no overlap he price incremens of any spreads beween he h spread and he h spread. Similarly, le J +, be he number of overlapping midpoins for he h spread (,,, J) = K which overlap he midpoins of he h spread and do no overlap he

6 midpoins of any spreads beween he h spread and he h spread. and are disinc in wo ways. Firs, is he number of oal price incremens which overlap vs. in he number of special price incremens which overlap. Second, he coun excludes hose incremens which overlap he price incremens of any spreads beween he h spread and he h spread vs. he coun includes hose incremens. Table 3 summarizes he,, and variables for a sandard example of a decimal price grid. Table 3,, and for a ecimal Price Grid Corresponding Prices or Spread midpoins? 0.0 Prices Prices Prices 0 3, Prices 3, =, 3 = 5.00 Prices 5, 5, 53, 5 = Midpoins Midpoins Midpoins 0 0, Midpoins 9, 9 =, Midpoins 0,, 0,, 0,3, 0, Now consider he ' s for he prices. ll 0 nicels are also pennies and all 0 dimes are also nicels. f he quarers, are dimes and are nicels, bu no dimes. The single dollar is also a quarer. Finally consider, ' s for he midpoins. ll 0 of he odd nicels overlap wih odd pennies. None of he odd dimes overlap wih odd nicels or odd quarers overlap wih odd dimes, bu all of hem overlap wih odd pennies. None of he odd nicels. The odd dollar doesn overlap wih anyhing below.

7 Given all of he infrasrucure variables in Table 3, he general formula of he Effecive Tic model can be saed. The general formula for he unconsrained probabiliy of he h spread is U F J + + F J + = J + =, F F + F F =,3,, J = = K. (9) The res of he effecive ic compuaion is he same as he fracion grid case. To illusrae how a fracional price grid fis ino he general formula of he Effecive Tic model, consider a 6 price grid where he possible effecive spreads are 6,,,, and. Table shows he,, and variables for his fracional price grid. Table,, and for a Fracional Price Grid Corresponding Spread Prices or midpoins? 6 Prices 6 Prices = 3 Prices 3, 3 = Prices,, 3 = 5 Prices 5, 5, 53, 5 = 6 6 Midpoins Midpoins 7 Midpoins, 9 Midpoins 9, 9, 93 0 Midpoins 0,, 0,, 0,3, 0, Consider he coefficiens of equaion (9). From Table, i is clear ha = for =,, K, J, J =, and = for =,, K, J. (0) J

8 In fracional price grids, he price incremens exhibi 00% overlap beween adacen spread levels (i.e., all wholes are halves, all halves are quarers, all quarers are eighhs, ec.). This implies ha = for =- and for all < and for,3,, = K J. () In fracional price grids, he midpoin clusers exhibi 0% overlap beween adacen spread levels (i.e., no odd midpoins are odd midpoins, no odd midpoins are odd midpoins, no odd midpoins are odd 6 midpoins, ec.). This implies ha, for all and for =,3, K, J. () Subsiuing he fracional price grid coefficiens in equaions (0), (), and () ino he general Effecive Tic formula (9) yields he fracional Effecive Tic formula.