Contracts I Whaley

							   Bid/Ask Spreads:
A Comparative Analysis

Nicolas Bollen, Vanderbilt University
Hans R. Stoll, Vanderbilt University
 Robert E. Whaley, Duke University
Background
   Two types of studies of market maker
    spreads:
       Develop and test theoretical models of spread’s
        determinants.
           Pioneering work by Demsetz (1968).
           Stoll (2003) provides comprehensive review.
Background
   Two types of studies of market maker
    spreads:
       Apply models of spread to compare costs/benefits
        of different trading structures or assess effects of
        intervention.
           Tinic and West (1974) – agent vs dealer-dominated
            markets.
           Bacidore (1997) – decimalization
           Bessembinder (1999) – order-processing rules
Background
   With few exceptions, models of spreads and
    policy examinations have focused on stock
    market.
       Most actively-traded corporate security.
       Long histories of exchange trade and quote data
        available.
Background
   Some studies have focused on stock option market to
    assess effects of multiple listing.
       Neal (1987) – Difference in spreads of AMEX options in
        1985 and 1986.
       Mayhew (2002) – Difference in spreads of CBOE options
        in 1986 to 1997.
       De Fontnouvelle et al (2003) – Reduction in spreads
        during August 1999 when competition among exchanges
        was unleashed.
       Battalio et al (2003) shows reduction in spreads in recent
        years due to competition.
Background
   Studies have not agreed on an appropriate
    structural model.
       Neal (1987) - absolute quoted spread function of:
           Trading volume (--)
           Option price (++)
           ISD times elasticity (++/--)
Background
   Studies have not agreed on an appropriate
    structural model.
       Jameson and Wilhelm (1992) - absolute quoted
        spread function of:
           ISD2 times elasticity2 times stock price (++)
           Gamma (++)
           Vega (++)
           Abs(1-PVX/S) (++)
           Elasticity (++)
Background
   Studies have not agreed on an appropriate
    structural model.
       de Fontnouvelle et al (2003) - absolute effective
        spread function of:
           Trading volume (--)
           Option price (++)
           Delta (++/--)
           Gamma (+/-)
           ISD (++)
           Stock spread (++)
Background
   Studies have not agreed on an appropriate
    structural model.
       Battalio et al (2003) - absolute effective spread
        function of:
           Inverse of option price (--)
           Stock volatility (ln of high/low) (++)
           Ln of market cap (-)
           Trade size (+)
Purpose of research
   Develop and test new model of bid/ask spread
    for stock options.
       Simple and parsimonious structural form.
       Supported empirically.
   Use model to compare and analyze
    differences between stock and stock option
    spreads.
Outline
   Describe option price dynamics
   Introduce concept of inventory-holding
    premium (IHP)
   Apply concept to stock spreads
   Extend model to stock option spreads
   Examine estimation results
   Discuss planned future work
Model development
   Determinants of option value:


              Ot  f  St ,  t , X , r , T 
Model development
   Approximate change in option value through
    time.

                        f      1 2 f      f
     O  Ot t  Ot     S         S 
                                         2
                                               
                        S      2 S 2
                                            
Model development
   Approximate change in option value through
    time.
                     1
    O  delta  S   gamma  S 2  vega  
                     2
Model development
   Can hedge delta, gamma, and vega risks using other
    securities.

                     1
    O  delta  S   gamma  S  vega  
                                  2

                     2


   Total cost of hedge is sum product of number of
    each security bought/sold and its bid/ask spread.
Model development
   Can hedge delta, gamma, and vega risks using
    other securities.
                     1
    O  delta  S   gamma  S 2  vega  
                     2



    E.g., use delta times stock spread as cost of
    delta-hedging option.
Issues
   Motivates use of delta, gamma, and vega in
    regression model.
   Hedging costs on a series-by-series basis
    would be prohibitive.
       Reduced by the fact the market maker hedges at a
        portfolio level.
       Nonetheless, incremental hedging costs are likely
        to be related to risk measures.
Inventory-holding premium
   Perfect hedge is not possible because of trading costs
    in stocks or stock options market because of high
    trading costs.
Inventory-holding premium
   Assume market maker sets bid/ask spread so as to be
    compensated for expected adverse price movements.
       Suppose market is long – risk is that price will fall while
        security is in inventory, i.e.,
                             S  0
   Expected loss conditional on a loss occurring times
    probability of loss occurring is

                E  S | S  0  Pr  S | S  0 
Inventory-holding premium
   This “inventory-holding” premium

           IHP  E  S | S  0  Pr  S  0 

    has value


                                     
                IHP  S  2 N .5 t  1
                                      
Inventory-holding premium
   Note functional form.

                                
                 IHP  S  2 N .5 t  1
                                         
   IHP is nonlinear function of:
         share price (S)
         return volatility ()
         market maker’s holding period (t)
   Entering variables separately obfuscates their role.
Simulation of IHP
   Assume:
       Stock price is $27.50 a share
       Volatility rate ranges from 0% to 100%.
       Number of minutes between offsetting trades ranges from 0
        to 20.
Simulation of IHP
   IHP as a function of time between trades and
    volatility.
                                                              0.2


                                                              0.15

                                                                     Inventory-holding
                                                              0.1
                                                                         premium

                        20                                   0.05
                              15
                                   10
            Minutes between                                   0
                                        5
                trades                                     100%
                                            0
                                                     50%
                                                0%
                                                           Volatility
Model specification
            SSPRDi   0  1IHPi   2 InvTVi   i

   Expect:
       Intercept to be minimum tick size.
       Coefficient on IHP to be positive.
       Coefficient on InvTV to be fixed costs.
Data
   CBOE stock options listed on 16 NYSE
    stocks during February 2001.
       Most active option classes (>50,000 contracts
        traded during month).
       Both options and underlying stocks trade in
        decimals.
Stock spreads
    Descriptive statistics
                                                            Percentiles
               Variable        Mean       5%        25%        50%        75%       95%
    Spread measures
               EWQS           0.0573    0.0307     0.0435     0.0510      0.0623   0.1048
               VWES           0.0516    0.0252     0.0359     0.0461      0.0578   0.0947

    Determinants of spread
               S                47.08    17.74      28.00     45.68     52.99     104.92
               TV            8,265,146 2,451,470 3,959,100 6,055,050 10,048,300 18,859,990
               Inv TV        0.0001879 0.0000530 0.0000995 0.0001652 0.0002526 0.0004079
               S              0.6050    0.2578    0.4320    0.5436    0.8173     1.0672
               Sqrt(t)         0.3961    0.3034    0.3484    0.3880    0.4502     0.4902
               IHP S          0.01258   0.00667   0.00885   0.01100   0.01383    0.02703
Stock spreads
    Correlation structure

           EWQS     VWES       S       TV      Inv TV    S      Sqrt(t)
VWES       0.921
S          0.511    0.366
TV         0.210    0.241    -0.264
Inv TV     -0.073   -0.120    0.319   -0.656
S         0.168    0.227    -0.453    0.467   -0.529
Sqrt(t)    -0.145   -0.133   -0.030   -0.537    0.687   -0.456
IHP S      0.709    0.625     0.674   -0.072    0.007    0.206   -0.111
Stock spreads
   Regression results
                    Number of
                                         2
                   observations       R        a 0/t(a 0)   a 1/t(a 1)   a 2/t(a 2)
      A. Equal-weighted quoted spread
                       304          0.5012      0.0218       2.8288
                                                 9.62         17.48

                       304           0.5057     0.0249       2.8310      -16.8037
                                                 8.98         17.57        -1.94

      B. Volume-weighted effective spread
                       304            0.3887    0.0163       2.8045
                                                 5.79         13.92

                       304           0.4021     0.0219       2.8084      -29.9176
                                                 6.38         14.09        -2.79
Option spreads
   Descriptive statistics
         A. Number of contracts traded
         Calls                               Moneyness categories
                           0.875        0.625      0.375         0.125       0 to         All
              Days to       to 1        0.875      0.625         0.375      0.125     moneyness
            expiration     DITM         ITM        ATM           OTM       DOTM      categories
               n <=7      26,662       15,593     30,911        23,704     32,298      129,168
             7<n <=30     15,108       64,343     174,415      163,303     51,270      468,439
            30<n <=90      6,046       31,755     154,423      131,231     11,908      335,363
           90<n <=270      4,063       26,247     136,606       80,137      5,198      252,251
              270<n        6,683       66,852     94,544        31,949      3,528      203,556
              All days    58,562       204,790    590,899      430,324     104,202    1,388,777


         Puts                                Moneyness categories
                            0 to        -0.125      -0.375       -0.625    -0.875        All
             Days to       -0.125     to -0.375   to -0.625    to -0.875    to -1    moneyness
            expiration    DOTM           OTM         ATM          ITM      DITM      categories
              n <=7       57,853       30,764      20,569         7,610    10,450     127,246
            7<n <=30      29,916       151,733     113,862      67,952     19,951     383,414
           30<n <=90      13,478       102,096     85,311       12,627      5,992     219,504
           90<n <=270      6,694       101,729     49,030       13,311       173      170,937
             270<n         1,706       66,631      21,899         1,953      466      92,655
             All days     109,647      452,953     290,671      103,453    37,032     993,756
Option spreads
   Descriptive statistics
         B. Percent of total volume
         Calls                              Moneyness categories
                              0.875    0.625      0.375         0.125       0 to          All
              Days to          to 1    0.875      0.625         0.375      0.125     moneyness
            expiration       DITM       ITM       ATM           OTM        DOTM     categories
               n <=7         1.92%     1.12%      2.23%         1.71%      2.33%        9.30%
             7<n <=30        1.09%     4.63%     12.56%        11.76%      3.69%       33.73%
            30<n <=90        0.44%     2.29%     11.12%         9.45%      0.86%       24.15%
           90<n <=270        0.29%     1.89%      9.84%         5.77%      0.37%       18.16%
              270<n          0.48%     4.81%      6.81%         2.30%      0.25%       14.66%
              All days       4.22%    14.75%     42.55%        30.99%      7.50%      100.00%


         Puts                                Moneyness categories
                             0 to      -0.125      -0.375       -0.625     -0.875       All
             Days to       -0.125     to -0.375   to -0.625    to -0.875    to -1   moneyness
            expiration     DOTM         OTM         ATM           ITM      DITM     categories
              n <=7        5.82%        3.10%       2.07%        0.77%     1.05%     12.80%
            7<n <=30       3.01%       15.27%      11.46%        6.84%     2.01%     38.58%
           30<n <=90       1.36%       10.27%       8.58%        1.27%     0.60%     22.09%
           90<n <=270      0.67%       10.24%       4.93%        1.34%     0.02%     17.20%
             270<n         0.17%        6.70%       2.20%        0.20%     0.05%      9.32%
             All days      11.03%      45.58%      29.25%       10.41%     3.73%     100.00%
Option spreads
   Equal-weighted quoted spreads
                                            Option bid price (O )
    A. Calls             All O    O <2     2<O <=5      5<=O <10    10<=O <20   O >=20
    No. of obs.          9,611    3,654     2,549         2,022       1,028      358

    Option spread
    Mean                 0.3255   0.1623   0.2967       0.4115       0.6279     0.8423
    Median               0.2875   0.1520   0.2953       0.4000       0.6000     0.8000

    Underlying stock spread
    Mean                0.0608    0.0546   0.0579       0.0625       0.0756     0.0940
    Median              0.0515    0.0481   0.0508       0.0534       0.0651     0.0955

    Option relative to stock
    Mean                   5.35    2.97      5.12         6.58        8.31       8.96
    Median                 5.58    3.16      5.81         7.49        9.22       8.38
Option spreads
   Volume-weighted effective spreads
                                            Option bid price (O )
    A. Calls             All O    O <2     2<O <=5      5<=O <10    10<=O <20   O >=20
    No. of obs.          9,611    3,654     2,549         2,022       1,028      358

    Option spread
    Mean                 0.2005   0.1011   0.1841       0.2597       0.3768     0.4930
    Median               0.1512   0.0959   0.1783       0.2461       0.3717     0.5000

    Underlying stock spread
    Mean                0.0550    0.0488   0.0523       0.0569       0.0701     0.0826
    Median              0.0463    0.0416   0.0457       0.0491       0.0597     0.0851

    Option relative to stock
    Mean                   3.65    2.07      3.52         4.56        5.38       5.97
    Median                 3.27    2.31      3.90         5.01        6.23       5.88
Model specification
    OSPRDi   0  1 IHPdelta,i   2 IHPgamma,i   3 IHPvega,i
                4 InvTVi   5 d C/P,i   6 d$3,i   i


   Separate IHP’s for each source of risk. E.g.,


                                         
         IHPvega  vega   S  2 N .5 V T  1
                                                      
Model specification
    OSPRDi   0  1 IHPdelta,i   2 IHPgamma,i   3 IHPvega,i
                4 InvTVi   5 d C/P,i   6 d$3,i   i
   Expect:
       Intercept to be minimum tick size.
       Coefficients on IHP’s to be positive.
       Coefficient on InvTV to be fixed costs.
       Coefficient on $3 dummy to be five cents.
Model specification
 OSPRDi   0  1 IHPdelta,i   2 IHPgamma,i   3 IHPvega,i
             4 InvTVi   5 d C/P,i   6 d$3,i   i

Benchmark model:



  OSPRDi   0  1delta i   2 gamma i   3 vega i
               4 InvTVi   5 d C/P,i   6 d$3,i   i
Model specification
     Regression results - EWQS

               R2     a 0/t(a 0)   a 1/t(a 1)   a 2/t(a 2)   a 3/t(a 3)   a 4/t(a 4)   a 5/t(a 5)   a 6/t(a 6)
Benchmark    0.6420    0.0671       0.3788      -1.4856       0.0073      -20.0678      0.0704       0.0692
                        21.86       65.55        -37.73       59.22         -5.51       33.20        22.20

IHP          0.6749    0.1289       0.0294       0.0001      -0.0100       6.0976       0.0598       0.1599
                        65.31       88.47         2.85        -10.19        1.76        30.62        74.63
Model specification
     Regression results - VWES


               R2     a 0/t(a 0)   a 1/t(a 1)   a 2/t(a 2)   a 3/t(a 3)   a 4/t(a 4)   a 5/t(a 5)   a 6/t(a 6)
Benchmark    0.2775    0.0385       0.2488      -0.9179       0.0043      -15.0115      0.0404       0.0388
                        9.44        32.37        -17.52       26.08         -3.10       14.31         9.36

IHP          0.2804    0.0840       0.0163       0.0001      -0.0027       0.9657       0.0304       0.1028
                        30.56       35.26         3.05        -1.96         0.20        11.15        34.44
 Model specification
       Results appear to be influenced by other factors.
             Use deFontnouvelle (2003) et al dummies for maximum
              spread categories.
OSPRD   0  1 IHPdelta   2 IHPgamma   3 IHPvega
                4 InvTV   5 dCP   6 d 25   7 d510  8 d1020   9 d 20   

              R2     a 0/t(a 0)   a 1/t(a 1)   a 2/t(a 2) a 3/t(a 3) a 4/t(a 4)   a 5/t(a 5)   a 6/t(a 6)   a 7/t(a 7)   a 8/t(a 8)   a 9/t(a 9)
 EWQS       0.8177    0.1438       0.0090      0.000025 0.006173 13.972            0.0227       0.1118       0.2090       0.3995       0.5688
                      91.84         28.35         1.76      8.17       5.37        15.11        62.46        100.33       138.15       108.94

 VWES       0.3447    0.0940       0.0032      0.000062 0.008036        5.875      0.0067       0.0704       0.1379       0.2468       0.3754
                      33.83         5.61          2.49    5.99           1.27       2.52        22.16         37.29        48.09        40.50
Summary
   Project is incomplete.
       Develop simple, parsimonious model for option
        spread.
           Permits understanding why previous regression
            models performed well, even though their
            specifications varied.
           Works better than competing models, but does not
            explain why option price effect.
Summary
   Next steps.
       Examine more recent periods.
           Given increased competition, price effect may have
            become smaller.
           Also, need to develop a more accurate proxy for
            length of market maker’s expected holding period.
       Once problems are overcome, estimate spread
        model across stock and stock spreads
        simultaneously.
           Isolate differences in cost components.