A State Contingent Claim
Approach To Asset Valuation
Kate Barraclough
Overview
• Aim is to empirically test the application of
Arrow-Debreu state preference theory
• State preference approach is applied to pricing
both to stocks and options
• Model values under state pricing are compared
to other asset pricing models
• State preference approach is found to provide
an overall improvement on the other models for
both stocks and options
Background
• Basic form of any asset pricing equation:
Pit  Et  M t 1 X it 1 
i, t
where Pt is the price at time t, Et is a conditional
expectations operator, Xt+1 is the asset’s payoff and
Mt+1 is the stochastic discount factor
• The stochastic discount factor represents
investors’ marginal rates of substitution between
consumption in the current period, and
consumption in time t, state s.
State Contingent Claims
• Stochastic discount factor may be characterised
by the set of state contingent claim prices
• A state contingent claim will have a positive
payoff in state s and zero elsewhere
• Ross (1976) - state contingent claims will be
implicit in the price of traded securities and in a
complete market investors may form a portfolio
with a positive payoff in state s and zero
elsewhere
State Preference Approach
• State price is the price today of one unit of
consumption at time t, state s
 tsU (Cts )
ts 
U (C0 )
• The price of a risky asset may be determined as
the payoff in time t state s multiplied by the state
price and summed over all possible S states:
Pt j   s d sj
s
j
State Price Computation
• Breeden and Litzenberger (1978) – state
contingent claim may be modelled as the
second derivative of a call option price
• Construct a butterfly
spread with a unit payoff
– buy one call with strike ΔM
M-ΔM, one call with strike
M+ΔM, and sell two calls
with strike M
M-ΔM
M
M+ΔM
State Price Computation
• If the value of the underlying asset is M next period,
then the payoff on the option portfolio will be ΔM and
zero otherwise.
• Normalising for a unit payoff:
1
M C ( M  M , T )  2C ( M , T )  C ( M  M , T )

M
M
• Taking the limit as ΔM tends to zero then the price of a
portfolio paying one unit will be given by:
C ( M  M , T )  2C ( M , T )  C ( M  M , T )
M 
M 2
P  lim
• Evaluating the portfolio at X=M provides
 2C ( X , T )
P
X 2
X M
State Price Computation
• Black-Scholes option pricing formula provides a
closed form solution:
 2C
X 2

X  MT
e  rT
M T T
where
n( d 2 ) 
1  d2 / 2
e
, d2 
2
n  d 2  X  M T  
ln  ( M 0  PVD) / M T    r  1/ 2 2  T
 T
• But cannot compute in continuous increments
for each strike
Delta Securities
• Delta security – unit payoff if the price of the
underlying asset is greater than or equal to
some level Y:
G Y   

Y
e rT n(d 2 )
dX e  rT N  d 2  X  Y  
X T
• State price - cost of a security with a unit payoff
if the level of the underlying asset is between
some levels is between some levels Y1 and Y2:
 Yi , Yi 1   e  rT  N  d 2  X  Yi    N  d 2  X  Yi 1  
Advantages
• Incorporate market microstructure features
– Minimum tick size in the underlying instrument
– Price limits, circuit breakers etc
• Limit range of distribution with empirical
maximum and minimum returns
– Not price extreme observations
– Reduce computational burden
S&P 500 Index Options
• First test of the paper is to apply the state
preference approach to pricing S&P 500 index
options
• Compare model values to the Black-Scholes
option pricing formula and Stutzer’s canonical
valuation approach
State Preference Approach
• Historical maximum and minimum T-day returns
are determined from the empirical distribution of
returns on the S&P 500 index
• Possible future index levels are calculated in
increments of 5c
• The state price corresponding to each index
level are determined by the delta security
method
State Preference Approach
• The option price is calculated as the sum of the
expected payoff at each index level multiplied by
the respective state price
• Call option prices will be given by:
C   max  M 0  PVD   Rh  X , 0 h
h
  max  M Th  X , 0h
h
• And put options:
P   max  X   M 0  PVD   Rh , 0h
h
  max  X  M Th , 0h
h
Canonical Valuation
• Stutzer (1996) – determine risk-neutral
probabilities from the empirical distribution of
returns on the underlying instrument
• Solving the unconstrained minimisation problem:
 *  arg min  exp   Rh / r T  1

h
provides the probability distribution:
ˆ h* 
exp  *  Rh / r T  
 exp   R
*
h
h
/ r  
T
,
h  1, 2,
, H T
Canonical Valuation
• Option prices are determined with reference to
historical index returns
• Calculated as the sum of the expected payoff at
each index level multiplied by the respective
risk-neutral probability:
C
h
P
h
max  M t  PVD  Rh  X , 0 
r
ˆ h*
T
max  X   M t  PVD  Rh , 0 
rT
ˆ h*
Options Data
• S&P 500 index options
• Weekly observations January 1990 through
December 1993
• Two proxies for expected stock market volatility:
– CBOE Market Volatility Index (VIX)
– 40 day historical volatility
• Daily observations on the S&P 500 index used
to determine historical distribution of index
returns
Results – Table 1
A: Aggregate Results
Model
State Price
Black-Scholes
Stutzer
B: Call Options
Moneyness
Total
All Options
1.599
1.712
6.376
Call Options
1.475
1.590
6.053
Put Options
1.731
1.843
6.719
Model
State Price
Black-Scholes
Stutzer
Total
1.475
1.590
6.053
T ≤ 10
0.359
0.374
0.482
11 ≤ T ≤ 20
0.672
0.676
1.672
21 ≤ T ≤ 30
0.935
0.987
2.928
31 ≤ T ≤ 40
1.289
1.414
3.509
41 ≤ T ≤ 50
1.897
2.066
5.902
51 ≤ T ≤ 60
2.088
2.251
8.353
61 ≤ T ≤ 70
1.778
1.853
10.954
71 ≤ T ≤ 80
2.171
2.333
11.963
In-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
1.063
1.068
2.994
0.264
0.291
0.259
0.483
0.504
0.719
0.760
0.843
0.834
0.831
0.618
1.608
0.620
0.749
3.417
1.587
1.663
4.601
1.396
1.493
6.511
1.729
1.640
7.645
2.501
2.540
5.786
2.969
2.868
6.660
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
1.453
1.567
7.778
0.425
0.432
0.639
0.798
0.791
2.305
1.053
1.084
4.268
1.459
1.657
4.800
2.290
2.472
7.634
1.919
2.046
10.627
1.565
1.616
13.746
1.910
2.076
14.463
2.100
2.275
16.024
1.694
2.054
16.157
Out-of-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
3.393
3.973
4.526
0.157
0.197
0.031
0.364
0.398
0.142
0.656
0.753
0.462
1.584
2.257
1.071
3.607
3.814
2.005
4.826
5.508
3.956
4.133
4.294
6.943
4.832
5.608
8.438
6.133
6.529
12.732
4.551
6.805
11.152
Model
State Price
Black-Scholes
Stutzer
Total
1.731
1.843
6.719
T ≤ 10
0.292
0.311
0.511
11 ≤ T ≤ 20
0.665
0.710
1.659
21 ≤ T ≤ 30
1.055
1.129
3.719
31 ≤ T ≤ 40
1.572
1.666
3.955
41 ≤ T ≤ 50
2.149
2.254
5.733
51 ≤ T ≤ 60
2.657
2.861
9.954
61 ≤ T ≤ 70
2.446
2.598
11.328
71 ≤ T ≤ 80
2.586
2.710
13.023
In-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
2.701
2.927
4.439
0.321
0.328
0.342
0.722
0.778
0.757
1.127
1.242
1.449
1.842
1.991
1.502
3.386
3.593
2.690
4.692
5.116
5.166
4.291
4.672
7.930
4.696
4.983
10.496
5.953
6.455
15.462
6.174
6.997
18.545
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
1.700
1.803
8.383
0.325
0.351
0.654
0.798
0.853
2.315
1.296
1.381
5.197
1.826
1.925
5.472
2.180
2.281
7.558
2.480
2.656
12.600
2.377
2.514
14.119
2.256
2.364
15.534
3.268
3.427
21.602
2.307
2.533
20.564
Out-of-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
0.576
0.568
3.357
0.058
0.057
0.065
0.127
0.125
0.352
0.177
0.175
1.150
0.254
0.249
1.794
0.582
0.578
3.005
0.626
0.643
5.597
0.949
0.933
6.888
1.141
1.116
6.366
1.610
1.570
7.744
2.033
1.969
7.881
C: Put Options
Moneyness
Total
81 ≤ T ≤ 90 91 ≤ T ≤ 100
2.556
2.382
2.715
2.781
13.058
12.636
81 ≤ T ≤ 90 91 ≤ T ≤ 100
3.593
3.122
3.795
3.421
18.304
17.420
Results – Table 2
A: Aggregate Results
Model
State Price
Black-Scholes
Stutzer
B: Call Options
Moneyness
Total
All Options
1.165
1.262
5.464
Call Options
1.074
1.175
5.126
Put Options
1.262
1.354
5.823
Model
State Price
Black-Scholes
Stutzer
Total
1.074
1.175
5.126
T ≤ 10
0.209
0.215
0.314
11 ≤ T ≤ 20
0.453
0.452
1.297
21 ≤ T ≤ 30
0.640
0.673
2.430
31 ≤ T ≤ 40
0.962
1.086
2.892
41 ≤ T ≤ 50
1.513
1.660
4.938
51 ≤ T ≤ 60
1.552
1.697
7.087
61 ≤ T ≤ 70
1.268
1.325
9.367
71 ≤ T ≤ 80
1.578
1.729
10.266
In-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
0.511
0.508
2.006
0.069
0.079
0.068
0.178
0.183
0.322
0.283
0.323
0.370
0.395
0.227
0.919
0.223
0.301
2.171
0.788
0.837
3.116
0.643
0.711
4.621
0.892
0.845
5.598
1.407
1.449
4.110
1.756
1.696
4.805
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
1.123
1.222
6.815
0.301
0.305
0.481
0.615
0.608
1.918
0.834
0.859
3.710
1.163
1.342
4.142
1.906
2.078
6.708
1.485
1.600
9.327
1.167
1.203
12.129
1.418
1.556
12.737
1.599
1.744
14.234
1.191
1.483
14.268
Out-of-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
3.037
3.587
4.162
0.131
0.169
0.019
0.323
0.357
0.115
0.583
0.675
0.409
1.393
2.034
0.932
3.270
3.469
1.785
4.375
5.030
3.598
3.651
3.804
6.402
4.274
5.005
7.812
5.529
5.904
11.868
4.055
6.167
10.359
Model
State Price
Black-Scholes
Stutzer
Total
1.262
1.354
5.823
T ≤ 10
0.175
0.190
0.363
11 ≤ T ≤ 20
0.442
0.476
1.321
21 ≤ T ≤ 30
0.736
0.792
3.169
31 ≤ T ≤ 40
1.117
1.191
3.308
41 ≤ T ≤ 50
1.602
1.687
4.907
51 ≤ T ≤ 60
1.984
2.161
8.662
61 ≤ T ≤ 70
1.792
1.919
9.882
71 ≤ T ≤ 80
1.870
1.972
11.404
In-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
1.830
2.012
3.330
0.129
0.133
0.134
0.339
0.375
0.331
0.591
0.662
0.783
1.146
1.254
0.862
2.338
2.495
1.729
3.338
3.698
3.681
3.005
3.320
5.995
3.267
3.501
8.326
4.279
4.705
12.666
4.488
5.176
15.530
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
1.279
1.365
7.407
0.218
0.238
0.504
0.574
0.618
1.927
0.962
1.032
4.554
1.358
1.442
4.699
1.684
1.771
6.618
1.905
2.060
11.178
1.770
1.886
12.512
1.652
1.741
13.827
2.554
2.690
19.452
1.758
1.935
18.567
Out-of-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
0.452
0.445
3.051
0.038
0.037
0.047
0.091
0.090
0.295
0.130
0.128
1.029
0.184
0.180
1.600
0.447
0.444
2.729
0.475
0.492
5.096
0.744
0.730
6.324
0.910
0.889
5.814
1.312
1.275
7.077
1.681
1.623
7.187
C: Put Options
Moneyness
Total
81 ≤ T ≤ 90 91 ≤ T ≤ 100
1.892
1.653
2.030
2.012
11.378
10.866
81 ≤ T ≤ 90 91 ≤ T ≤ 100
2.737
2.358
2.909
2.601
16.233
15.469
Results – Table 3
A: Aggregate Results
Model
State Price
Black-Scholes
Stutzer
B: Call Options
Moneyness
Total
All Options
0.428
0.455
1.546
Call Options
0.321
0.336
1.492
Put Options
0.543
0.582
1.605
Model
State Price
Black-Scholes
Stutzer
Total
0.321
0.336
1.492
T ≤ 10
0.131
0.103
0.224
11 ≤ T ≤ 20
0.264
0.239
0.724
21 ≤ T ≤ 30
0.233
0.215
1.053
31 ≤ T ≤ 40
0.451
0.528
1.117
41 ≤ T ≤ 50
0.637
0.643
1.609
51 ≤ T ≤ 60
0.383
0.389
1.882
61 ≤ T ≤ 70
0.323
0.311
2.542
71 ≤ T ≤ 80
0.390
0.440
2.536
In-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
-0.440
-0.441
0.754
-0.048
-0.081
0.013
-0.148
-0.180
0.229
-0.379
-0.413
0.289
-0.374
-0.298
0.479
-0.246
-0.244
0.968
-0.662
-0.678
1.067
-0.676
-0.712
1.773
-0.751
-0.710
1.734
-1.023
-1.025
1.513
-1.154
-1.091
1.642
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
0.568
0.580
1.879
0.243
0.217
0.366
0.485
0.462
1.038
0.525
0.511
1.518
0.753
0.817
1.502
0.941
0.946
2.000
0.632
0.637
2.345
0.563
0.558
2.962
0.629
0.668
2.925
0.525
0.555
3.220
0.264
0.371
3.081
Out-of-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
1.457
1.559
1.362
0.264
0.295
0.007
0.380
0.395
0.139
0.589
0.623
0.416
1.016
1.179
0.597
1.575
1.604
1.026
1.851
1.937
1.146
1.736
1.755
2.223
1.925
2.056
2.319
2.175
2.231
3.213
1.783
2.159
2.535
Model
State Price
Black-Scholes
Stutzer
Total
0.543
0.582
1.605
T ≤ 10
0.102
0.127
0.251
11 ≤ T ≤ 20
0.274
0.303
0.741
21 ≤ T ≤ 30
0.480
0.513
1.346
31 ≤ T ≤ 40
0.651
0.687
1.184
41 ≤ T ≤ 50
0.689
0.724
1.490
51 ≤ T ≤ 60
0.806
0.854
2.196
61 ≤ T ≤ 70
0.648
0.692
2.707
71 ≤ T ≤ 80
0.734
0.775
2.663
In-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
0.904
0.966
1.062
0.014
0.045
0.042
0.241
0.286
0.228
0.499
0.550
0.528
0.780
0.836
0.522
1.154
1.211
0.856
1.479
1.556
1.429
1.456
1.538
2.024
1.611
1.680
2.299
1.810
1.903
3.156
1.922
2.078
3.393
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
0.691
0.729
1.924
0.183
0.209
0.362
0.434
0.465
1.014
0.713
0.749
1.765
0.874
0.910
1.552
0.859
0.894
1.867
0.912
0.957
2.588
0.914
0.957
3.087
0.840
0.879
3.029
0.908
0.954
3.960
0.642
0.729
3.498
Out-of-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
-0.497
-0.488
1.097
-0.162
-0.161
0.058
-0.258
-0.255
0.354
-0.312
-0.308
0.799
-0.354
-0.347
0.750
-0.446
-0.436
0.936
-0.539
-0.520
1.626
-0.807
-0.797
2.302
-0.830
-0.817
1.689
-0.984
-0.969
2.035
-1.176
-1.155
2.153
C: Put Options
Moneyness
Total
81 ≤ T ≤ 90 91 ≤ T ≤ 100
0.264
-0.039
0.288
0.081
2.773
2.568
81 ≤ T ≤ 90 91 ≤ T ≤ 100
0.824
0.545
0.875
0.634
3.512
3.189
Results – Table 4
A: Aggregate Results
Model
State Price
Black-Scholes
Stutzer
B: Call Options
Moneyness
Total
All Options
3.151
3.199
6.376
Call Options
3.545
3.681
6.053
Put Options
2.732
2.686
6.719
Model
State Price
Black-Scholes
Stutzer
Total
3.545
3.681
6.053
T ≤ 10
0.337
0.374
0.482
11 ≤ T ≤ 20
0.621
0.681
1.672
21 ≤ T ≤ 30
1.372
1.481
2.928
31 ≤ T ≤ 40
2.134
2.242
3.509
41 ≤ T ≤ 50
2.759
2.899
5.902
51 ≤ T ≤ 60
5.058
5.250
8.353
61 ≤ T ≤ 70
5.800
6.011
10.954
71 ≤ T ≤ 80
7.151
7.357
11.963
In-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
3.367
3.550
2.994
0.280
0.310
0.259
0.640
0.706
0.719
1.242
1.386
0.834
2.099
2.250
1.608
2.655
2.856
3.417
5.637
5.936
4.601
5.220
5.516
6.511
7.320
7.614
7.645
8.560
8.915
5.786
9.716
10.046
6.660
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
3.990
4.120
7.778
0.382
0.423
0.639
0.650
0.711
2.305
1.566
1.666
4.268
2.437
2.541
4.800
3.133
3.263
7.634
5.364
5.536
10.627
6.941
7.146
13.746
8.172
8.373
14.463
8.446
8.648
16.024
10.859
11.046
16.157
Out-of-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
0.576
0.578
4.526
0.034
0.034
0.031
0.030
0.031
0.142
0.018
0.018
0.462
0.338
0.345
1.071
0.347
0.348
2.005
1.115
1.121
3.956
0.643
0.652
6.943
0.759
0.763
8.438
0.708
0.720
12.732
0.762
0.717
11.152
Model
State Price
Black-Scholes
Stutzer
Total
2.732
2.686
6.719
T ≤ 10
0.323
0.311
0.511
11 ≤ T ≤ 20
0.684
0.670
1.659
21 ≤ T ≤ 30
0.826
0.823
3.719
31 ≤ T ≤ 40
1.881
1.844
3.955
41 ≤ T ≤ 50
3.152
3.093
5.733
51 ≤ T ≤ 60
3.639
3.583
9.954
61 ≤ T ≤ 70
4.516
4.432
11.328
71 ≤ T ≤ 80
6.612
6.500
13.023
In-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
0.638
0.636
4.439
0.317
0.314
0.342
0.599
0.610
0.757
0.520
0.568
1.449
0.377
0.373
1.502
0.697
0.689
2.690
1.010
1.018
5.166
0.470
0.429
7.930
1.056
1.006
10.496
0.845
0.854
15.462
1.039
1.005
18.545
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
3.299
3.230
8.383
0.363
0.345
0.654
0.798
0.773
2.315
0.979
0.960
5.197
2.463
2.406
5.472
3.851
3.763
7.558
4.246
4.160
12.600
5.843
5.720
14.119
8.005
7.853
15.534
5.862
5.747
21.602
9.740
9.552
20.564
Out-of-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
3.316
3.303
3.357
0.121
0.121
0.065
0.372
0.371
0.352
0.633
0.631
1.150
1.840
1.831
1.794
3.627
3.611
3.005
4.714
4.696
5.597
4.619
4.602
6.888
8.342
8.310
6.366
7.409
7.376
7.744
9.023
8.986
7.881
C: Put Options
Moneyness
Total
81 ≤ T ≤ 90 91 ≤ T ≤ 100
7.801
9.501
8.027
9.711
13.058
12.636
81 ≤ T ≤ 90 91 ≤ T ≤ 100
5.031
7.623
4.954
7.502
18.304
17.420
Results – Table 5
A: Aggregate Results
Model
State Price
Black-Scholes
Stutzer
B: Call Options
Moneyness
Total
All Options
2.532
2.570
5.464
Call Options
2.748
2.860
5.126
Put Options
2.303
2.262
5.823
Model
State Price
Black-Scholes
Stutzer
Total
2.748
2.860
5.126
T ≤ 10
0.173
0.195
0.314
11 ≤ T ≤ 20
0.368
0.410
1.297
21 ≤ T ≤ 30
0.919
1.000
2.430
31 ≤ T ≤ 40
1.570
1.655
2.892
41 ≤ T ≤ 50
2.046
2.159
4.938
51 ≤ T ≤ 60
3.963
4.125
7.087
61 ≤ T ≤ 70
4.602
4.783
9.367
71 ≤ T ≤ 80
5.718
5.897
10.266
In-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
2.290
2.428
2.006
0.078
0.090
0.068
0.284
0.323
0.322
0.597
0.686
0.370
1.293
1.396
0.919
1.620
1.763
2.171
3.962
4.199
3.116
3.560
3.797
4.621
5.311
5.554
5.598
6.280
6.579
4.110
7.214
7.495
4.805
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
3.253
3.365
6.815
0.239
0.267
0.481
0.438
0.485
1.918
1.174
1.258
3.710
1.908
1.997
4.142
2.494
2.606
6.708
4.393
4.545
9.327
5.788
5.971
12.129
6.780
6.961
12.737
7.039
7.223
14.234
9.189
9.362
14.268
Out-of-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
0.456
0.458
4.162
0.022
0.022
0.019
0.020
0.020
0.115
0.009
0.009
0.409
0.278
0.285
0.932
0.258
0.259
1.785
0.945
0.950
3.598
0.496
0.504
6.402
0.562
0.565
7.812
0.524
0.533
11.868
0.558
0.522
10.359
Model
State Price
Black-Scholes
Stutzer
Total
2.303
2.262
5.823
T ≤ 10
0.214
0.205
0.363
11 ≤ T ≤ 20
0.490
0.478
1.321
21 ≤ T ≤ 30
0.637
0.628
3.169
31 ≤ T ≤ 40
1.540
1.510
3.308
41 ≤ T ≤ 50
2.661
2.611
4.907
51 ≤ T ≤ 60
3.071
3.020
8.662
61 ≤ T ≤ 70
3.892
3.820
9.882
71 ≤ T ≤ 80
5.727
5.630
11.404
In-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
0.311
0.307
3.330
0.123
0.122
0.134
0.243
0.250
0.331
0.246
0.265
0.783
0.148
0.148
0.862
0.348
0.342
1.729
0.552
0.552
3.681
0.215
0.188
5.995
0.592
0.558
8.326
0.411
0.420
12.666
0.518
0.489
15.530
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
2.809
2.748
7.407
0.270
0.256
0.504
0.619
0.599
1.927
0.788
0.769
4.554
2.047
1.999
4.699
3.285
3.210
6.618
3.616
3.540
11.178
5.042
4.935
12.512
6.933
6.798
13.827
4.992
4.890
19.452
8.554
8.385
18.567
Out-of-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
2.990
2.977
3.051
0.092
0.092
0.047
0.309
0.308
0.295
0.539
0.537
1.029
1.622
1.613
1.600
3.231
3.216
2.729
4.254
4.236
5.096
4.139
4.123
6.324
7.645
7.615
5.814
6.740
6.708
7.077
8.231
8.195
7.187
C: Put Options
Moneyness
Total
81 ≤ T ≤ 90 91 ≤ T ≤ 100
6.272
7.710
6.471
7.896
11.378
10.866
81 ≤ T ≤ 90 91 ≤ T ≤ 100
4.282
6.671
4.213
6.562
16.233
15.469
Results – Table 6
A: Aggregate Results
Model
State Price
Black-Scholes
Stutzer
B: Call Options
Moneyness
Total
All Options
-1.022
-1.029
1.546
Call Options
-1.129
-1.168
1.492
Put Options
-0.908
-0.881
1.605
Model
State Price
Black-Scholes
Stutzer
Total
-1.129
-1.168
1.492
T ≤ 10
-0.182
-0.218
0.224
11 ≤ T ≤ 20
-0.354
-0.391
0.724
21 ≤ T ≤ 30
-0.685
-0.729
1.053
31 ≤ T ≤ 40
-0.856
-0.892
1.117
41 ≤ T ≤ 50
-1.054
-1.096
1.609
51 ≤ T ≤ 60
-1.505
-1.549
1.882
61 ≤ T ≤ 70
-1.790
-1.834
2.542
71 ≤ T ≤ 80
-2.024
-2.061
2.536
In-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
-1.054
-1.110
0.754
-0.062
-0.099
0.013
-0.249
-0.292
0.229
-0.581
-0.643
0.289
-0.857
-0.913
0.479
-1.014
-1.078
0.968
-1.663
-1.733
1.067
-1.724
-1.793
1.773
-2.079
-2.138
1.734
-2.368
-2.430
1.513
-2.590
-2.646
1.642
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
-1.270
-1.306
1.879
-0.261
-0.297
0.366
-0.432
-0.468
1.038
-0.803
-0.841
1.518
-0.969
-1.001
1.502
-1.202
-1.239
2.000
-1.608
-1.646
2.345
-2.048
-2.086
2.962
-2.263
-2.296
2.925
-2.270
-2.302
3.220
-2.665
-2.691
3.081
Out-of-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
-0.262
-0.264
1.362
-0.107
-0.108
0.007
-0.063
-0.065
0.139
-0.008
-0.010
0.416
-0.143
-0.141
0.597
-0.094
-0.099
1.026
-0.295
-0.302
1.146
-0.438
-0.443
2.223
-0.487
-0.486
2.319
-0.433
-0.437
3.213
-0.437
-0.419
2.535
Model
State Price
Black-Scholes
Stutzer
Total
-0.908
-0.881
1.605
T ≤ 10
-0.225
-0.201
0.251
11 ≤ T ≤ 20
-0.374
-0.348
0.741
21 ≤ T ≤ 30
-0.442
-0.419
1.346
31 ≤ T ≤ 40
-0.752
-0.725
1.184
41 ≤ T ≤ 50
-1.116
-1.087
1.490
51 ≤ T ≤ 60
-1.144
-1.115
2.196
61 ≤ T ≤ 70
-1.476
-1.450
2.707
71 ≤ T ≤ 80
-1.845
-1.814
2.663
In-the-Money
(0.05 ≤ X/S-1 ≤ 0.1)
State Price
Black-Scholes
Stutzer
-0.046
-0.008
1.062
-0.022
0.011
0.042
0.024
0.070
0.228
0.110
0.145
0.528
-0.016
0.014
0.522
-0.064
-0.023
0.856
-0.025
0.014
1.429
-0.218
-0.187
2.024
-0.292
-0.250
2.299
0.040
0.079
3.156
-0.216
-0.162
3.393
At-the-Money
(-0.05 ≤ X/S-1 ≤ 0.05)
State Price
Black-Scholes
Stutzer
-1.094
-1.064
1.924
-0.298
-0.273
0.362
-0.479
-0.453
1.014
-0.560
-0.534
1.765
-0.942
-0.911
1.552
-1.339
-1.307
1.867
-1.362
-1.331
2.588
-1.749
-1.716
3.087
-2.184
-2.150
3.029
-1.720
-1.685
3.960
-2.453
-2.418
3.498
Out-of-the-Money
(-0.1 ≤ X/S-1 ≤ -0.05)
State Price
Black-Scholes
Stutzer
-1.330
-1.327
1.097
-0.229
-0.229
0.058
-0.451
-0.450
0.354
-0.622
-0.621
0.799
-1.082
-1.079
0.750
-1.585
-1.581
0.936
-1.766
-1.762
1.626
-1.888
-1.884
2.302
-2.526
-2.521
1.689
-2.383
-2.377
2.035
-2.720
-2.713
2.153
C: Put Options
Moneyness
Total
81 ≤ T ≤ 90 91 ≤ T ≤ 100
-2.135
-2.422
-2.173
-2.453
2.773
2.568
81 ≤ T ≤ 90 91 ≤ T ≤ 100
-1.446
-2.005
-1.414
-1.971
3.512
3.189
Results – Figure 1
Black Scholes - VIX
Black-Scholes - Historical Volatility
10
10
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
Moneyness
0
0.05
0.1
0.05
0.1
Moneyness
State Preference - VIX
State Preference - Historical Volatility
10
10
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
-0.1
-0.05
0
0.05
0.1
Moneyness
Stutzer
8
6
4
2
0
-2
-4
-6
-8
-10
-0.05
0
Moneyness
-0.05
0
Moneyness
10
-0.1
-0.1
0.05
0.1
Results – Figure 2
Black Scholes - VIX
Black-Scholes - Historical Volatility
10
10
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
-0.1
-0.05
0
0.05
0.1
-0.1
-0.05
Moneyness
0
0.05
0.1
0.05
0.1
Moneyness
State Preference - VIX
State Preference - Historical Volatility
10
10
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
-0.1
-0.05
0
0.05
0.1
Moneyness
Stutzer
8
6
4
2
0
-2
-4
-6
-8
-10
-0.05
0
Moneyness
-0.05
0
Moneyness
10
-0.1
-0.1
0.05
0.1
Results – Figure 3
0.2
Canonical Risk-Neutral Probabilities
State Price (VIX) Probabilities
0.18
State Price (σ) Probabilities
0.16
0.14
Probability
0.12
0.1
0.08
0.06
0.04
0.02
0
290
311
331
352
372
392
Index level
413
433
454
474
Results – Figure 4
0.2
Canonical Risk-Neutral Probabilities
State Price (σ) Probabilities
0.18
State Price (VIX) Probabilities
0.16
0.14
Probability
0.12
0.1
0.08
0.06
0.04
0.02
0
280
308
335
362
389
416
Index level
444
471
498
525
Results – Figure 5
0.2
Canonical Risk-Neutral Probabilities
State Price (σ) Probabilities
0.18
State Price (VIX) Probabilities
0.16
0.14
Probability
0.12
0.1
0.08
0.06
0.04
0.02
0
289
316
343
371
398
425
Index level
453
480
507
534
Stock Valuation
• The second test is to apply the state preference
approach to stock valuation
• Comparison is made between the state
preference approach and Ohlson’s (1995)
residual income model
• Stutzer’s canonical valuation approach is also
applied
State Preference Approach
• Linear projection to determine stock movements
in reference to the market:
Rt j     Rtm   t
j
• Payoffs are given by:
dt j  P0 j exp  Rt j   P0j exp  j   j Rtm 
• Providing the valuation expression:
Pt j   s d sj   s  P0 j exp  j   j Rsm  
S
S
s 1
s 1
Canonical Valuation
• Determine risk-neutral probabilities from the
historical distribution of index returns as
described previously
• Apply to the payoff function from the previous
slide to provide the valuation expression:
Pt j   h  P0j exp  j   j Rhm 
h
Residual Income Model
• The residual income model specifies a
relationship between market value, book value,
and contemporaneous and future earnings
• Based on the dividend discount model:

Pt j  
i 1
Et  Dt ji 
1  r 
i
• And a clean surplus relation:
bt j  bt j1  xtj1  dt j1
Residual Income Model
• Combining the dividend discount model and the
clean surplus relation:

Pt j  bt j  
i 1
Et  xtji  rbt ji 1 
1  re 
i
• Test empirically using cross sectional regression
estimates:
Pt  0  1bt   2 xt   t
Pt  0  1bt  2 xt  3 ft   t
Stock Data
• Sample covers all companies in the
COMPUSTAT database from 1993 through 2004
• Linear projection – based on monthly
observations over the previous 5 years
• Stock and index returns are from the CRSP
database
• Consensus earnings from I/B/E/S proxy for the
market’s expectation of future earnings
Results – Table 7
A. Mean Squared Errors
Model
State Price
Canonical Valuation
Residual Income 1
Residual Income 2
Total
2.102
2.592
332.687
332.517
1993
0.753
0.950
340.438
340.039
1994
0.537
0.519
185.748
183.789
1995
1.426
1.297
233.540
234.285
1996
0.899
1.278
199.373
195.846
1997
1.439
1.132
314.823
294.867
1998
4.931
5.063
361.292
381.421
1999
3.286
3.966
563.076
562.904
2000
3.610
7.351
574.774
589.513
2001
1.805
2.284
265.559
266.237
2002
1.394
0.462
279.508
273.716
2003
1.994
2.866
283.105
276.658
1993
0.977
0.965
-0.259
0.329
1994
1.989
2.198
-0.297
0.225
1995
-4.137
-4.151
-0.415
0.142
1996
-1.665
-1.711
-0.316
0.523
1997
-4.587
-4.354
-0.548
0.334
1998
2.440
2.600
-0.529
0.213
1999
1.304
1.313
-0.125
0.650
2000
2.254
2.240
0.494
1.182
2001
0.396
-0.889
-0.268
0.479
2002
3.582
3.586
-0.759
-0.209
2003
-4.267
-4.456
-0.329
0.512
B. Mean Outside Errors
Model
State Price
Canonical Valuation
Residual Income 1
Residual Income 2
Total
-0.310
0.401
-0.119
-0.228
Results – Figure 6
Canonical Valuation Model
State Preference Model
500
500
400
400
300
300
200
200
100
100
0
0
-100
-100
-200
-200
-300
-300
-400
-400
-500
-500
Residual Incom e Model 1
Residual Incom e Model 2
500
500
400
400
300
300
200
200
100
100
0
0
-100
-100
-200
-200
-300
-300
-400
-400
-500
-500
Summary
• S&P 500 index options:
– state preference approach provides an overall
improvement compared to Black-Scholes and
canonical valuation
• Stocks
– state preference approach provides a significant
improvement on the residual income model for stock
valuation
Future Directions
• Extend canonical valuation approach
– Additional constraint of previous day’s call option price
– similarity to using previous day’s implied volatilities
for Black-Scholes
– Include market microstructure considerations - 5c/10c
price increments rather than empirical movement
• Implications for investor risk preferences
– VIX index vs realised volatility
Future Directions
• Compare results for stock valuation over
different maturities
– 1 week
– 1 month
– Quarterly
• May be improvements available for residual
income model