The Log-Logistic Option Pricing Model
Muhannad R. Al Najjab
Graduate Student
Lehigh University
200 W Packer Ave Rm 421
Bethlehem PA 18015 USA
mra2@lehigh.edu
+1-454-515-6621
Aurélie Thiele
P.C. Rossin Assistant Professor
Lehigh University
200 W Packer Ave Rm 329
Bethlehem PA 18015 USA
aurelie.thiele@lehigh.edu
+1-610-758-2903
Corresponding author
ABSTRACT
We value European call options based on a Log-Logistic model of the stock prices. We
argue that such a model captures the movements of stock processes more accurately than
the traditional Log-Normal assumption in the Black-Scholes formula. We analyze the
impact of the number of data points used to fit the Logistic distribution, and compare the
option prices obtained in the Log-Logistic and the Log-Normal models through extensive
numerical experiments involving historical data. Our results suggest that European call
options are overpriced in the Log-Normal model. This provides new profit-making
opportunities for investors.
SELECTING HOW MUCH DATA TO KEEP
European call options have been traditionally priced using the Black-Scholes formula,
which assumes that stock returns follow a Log-Normal distribution (see Hull [2000] for an
overview.) The existence of a closed-form solution in that framework explains in large part
its popularity; in this paper we consider an alternative pricing scheme based on the LogLogistic distribution, which we argue predicts stock prices more accurately.
A Log-Normal distribution is entirely characterized by two parameters, μ and σ,
which are estimated from the historical stock prices as follows (see Hull [2000]):
  St  
 

S
t

1



  stdev ln 
  St    2
  

S
  t 1   2
  average ln 
where St is the closing price at the end of time t. Furthermore, the two parameters that
determine a Log-Logistic distribution, μ and s, are estimated by (see Johnson et. al.
[1995]):
  S 
3
s  stdev ln  t   
  S t 1   
  St  
 

S
  t 1  
  average ln 
We obtained daily closing prices from finance.yahoo.com for five years of data, ending
March 14th 2007. Exhibit 1 presents the stocks used for detailed analysis; these stocks
were selected to illustrate a wide range of price movements.
A key issue in estimating the parameters of a distribution is to determine how much
historical data is relevant. If the distribution of a stock price did not vary over time, it
would be best to keep as much data as possible; in practice, however, changes in
ownership, threats of litigations, and the introduction of new products all create nonstationary effects. It then becomes critical to remove obsolete data points before computing
the parameters of the Log-Normal and Log-Logistic distributions. Exhibits 2-4 show the
estimated parameters for PNC Financial Services (PNC), Eastman Kodak (EK) and
Harrah’s Entertainment (HET), respectively, as a function of the number of trading days up
to March 14th 2007. All of our conclusions also hold for the rest of the stocks in the study.
We make the following observations after varying this number from three months to five
years:

σ and s are much less volatile than μ. Specifically, the estimates of σ and s do not
vary significantly if the time horizon between 1 and 4 years.

Smaller time horizon (up to 1 year of trading) induces higher volatility.

The estimates appear to stabilize for time horizons of about 2 years (500 trading
days). This is true in particular for the mean, which as mentioned above is the more
volatile parameter.

Significant volatility occurs for longer time horizons. From a practical standpoint,
five-year-old data points have little value in helping the decision maker understand
current stock prices.
These observations suggest that the parameters are estimated most accurately with about
two years of data.
COMPARING THE DISTRIBUTIONS
In this section, we compare the Log-Normal and Log-Logistic models using (1) a chisquare test and (2) an out-of-sample test.
Chi-Square Test
The Chi-Square test is a statistical tool used to determine the goodness of fit of a specific
distribution for the data set considered (see DeGroot and Schervish [2001].) It compares
the number Ei of expected observations in interval i (i=1..n) to the number of observed
occurrences Oi within each interval and is computed as follows:
n
2 
i 1
E i
 Oi 
Ei
2
Hence, the better the fit, the lower the value of the Chi-Square test statistic. We used the
BestFit module in the DecisionTools Industrial Suite of Palisade, Inc, to compare the fits
of the data with the Log-Logistic and Log-Normal distributions. Based on the results of the
Chi-Square test, the Logistic distribution yielded a better fit for the data points ln(St / St-1)
in eighteen of the twenty cases considered; the only assumption we made was that these
data points all came from the same unknown distribution. Exhibit 5 summarizes the values
of the Chi-Square test for the twenty stocks and the two possible (Logistic or Normal)
distributions for ln(St / St-1). The results motivate the choice of the Log-Logistic model as a
more accurate representation of stock prices than the traditional Log-Normal framework.
Out-of-Sample Test
For the out-of-sample test, we used only a subset of the historical data, consisting of the
closing prices between 12/16/2004 and 12/7/2006. The distribution parameters were
calculated based on that subset and were in turn used to predict the closing prices from
12/8/2006 to 3/14/2007, which were computed as follows:
Lognormal Distribution:
S t  S t 1e
 2

    . Z 


2


where Z is standard Normal random variable, and r is the risk free rate of return
Logistic Distribution:
St  St 1e L 
where L is a Logistic random variable calculated using the parameter estimates discussed
above.
We ran the simulation over 100 iterations and compared the projected closing price
on 3/14/2007 with the actual closing price on that day. Exhibit 6 shows our results. To
compare the two models we computed the relative forecast errors in both models,
presented in Exhibit 7. We argue that the Logistic distribution is a superior model because
(i) it outperforms the Normal distribution in eleven out of twenty cases, and (ii) for three of
the cases in which the Normal distribution outperforms the Logistic model, neither
distribution performs very well, i.e., both models fail to predict the closing price
accurately.
COMPARING OPTION VALUATIONS
We now compare the prices of a European call option obtained using the following two
techniques:
1. Closed-form solution using the Black and Scholes formula (see Hull [2000]).
2. Simulation-based solution using the Log-Logistic model, for which no closed-form
solution exists.
We calculated the option value for three different strike prices. The middle strike price was
equal to the closing price rounded off to the nearest $5 on 3/14/2007. The other two option
strike prices were equal to either $5 above or $5 below the middle strike price. Exhibits 810 show the price of the three options plotted against the amount of historical data for
PNC, EK and HET, respectively. These figures, which are representative of the trends for
the other stocks (not shown due to space constraints), indicate that the price obtained in the
Log-Logistic model is lower than the Black-Scholes price for most of the time horizons,
especially when the sample size amounts to about two years (the time horizon we
motivated earlier). The higher volatility in the Log-Logistic model does not affect these
insights. Consequently, these options are overvalued from the perspective of an investor
who agrees that the Log-Logistic model provides a more accurate depiction of reality than
the Log-Normal one. This investor will be able to generate profit opportunities by selling
these options at the market price (determined by the Black-Scholes formula) and taking
advantage of the mispricing.
We put our assertions to the test in the following analysis. We calculated the option
price on 12/14/2006 for both the Log-Normal and the Log-Logistic models, using
historical data from 12/22/2004 to 12/14/2006 and under the assumption that the market
values the option using the Black-Scholes formula and the investor uses the Log-Logistic
model instead. We shorted the option whenever the Black-Scholes price exceeded the price
given in the Log-Logistic model; if the price in the Log-Logistic model exceeded the
Black-Scholes price, we did not take any position, because the Log-Logistic price is rarely
much greater than the Black-Scholes price but is much more volatile, as shown in Exhibits
8-10. We then observed the price at maturity (3/14/2007) and calculated our payoff. The
results are summarized in Exhibits 11 and 12. We observe that we made a profit every
time we assumed the short position. This suggests that the Log-Logistic model describes
stock price movements more accurately than the Log-Normal model and offers valuable
profit-making opportunities by exploiting market mispricings.
CONCLUSIONS
The Black-Scholes model became popular at a time where computing power did not allow
for extensive simulations of asset prices. The analysis presented in this paper, however,
suggests that the Log-Logistic model describes stock prices more accurately and should be
preferred. We have also shown that the parameters characterizing the Log-Normal and the
Log-Logistic distributions vary greatly depending on the length of the time horizon
considered, and motivated keeping two years worth of data. Finally, we illustrated how an
investor can take advantage of market mispricings using the Log-Logistic model.
REFERENCES
DeGroot, Morris and Mark Schervish [2001] Probability and Statistics (3rd Edition),
Addison Wesley.
Hull, John C. [2000] Options, Futures, and Other Derivatives (4th Edition), Prentice Hall.
Johnson, Norman, Samuel Kotz and Narayanaswamy Balakhrishnan [1995] Continuous
Univariate Distributions, vol.2, Wiley-Interscience.
EXHIBIT 1: Company list
Company
Ticker Symbol
3M Company
Air Products & Chemicals
BMC Software
Eastman Kodak
Fannie Mae
Harrah's Entertainment
Johnson & Johnson
Kroger Co.
Lockheed Martin Corp.
Merck & Co.
Merrill Lynch
New York Times
PepsiCo Inc.
PNC Financial Services
QLogic Corp.
RadioShack Corp
Texas Instruments
Unum Group
MMM
APD
BMC
EK
FNM
HET
JNJ
KR
LMT
MRK
MER
NYT
PEP
PNC
QLGC
RSH
TXN
UNM
SLE
ORCL
Sara Lee Corp.
Oracle Corp.
EXHIBIT 2: Estimated parameters for PNC as a function of time horizon
Lognormal Distribution
Sigma
Mu
0.001
0.016
0.0008
0.014
0.0006
0.012
0.01
0.0004
0.008
0.0002
0.006
0
0
200
400
600
800
1000
1200
1400
0.004
-0.0002
0.002
-0.0004
0
0
-0.0006
200
400
600
800
1000
1200
1400
800
1000
1200
1400
Logistic Distribution
Mu
S
0.001
0.009
0.0008
0.008
0.007
0.0006
0.006
0.0004
0.005
0.0002
0.004
0.003
0
0
-0.0002
200
400
600
800
1000
1200
1400
0.002
0.001
-0.0004
0
-0.0006
0
200
400
600
EXHIBIT 3: Estimated parameters for EK as a function of time horizon
Lognormal Distribution
Sigma
Mu
0.025
0.002
0.0015
0.02
0.001
0.0005
0.015
0
-0.0005
0
200
400
600
800
1000
1200
1400
0.01
-0.001
-0.0015
0.005
-0.002
-0.0025
0
0
-0.003
200
400
600
800
1000
1200
1400
800
1000
1200
1400
Logistic Distribution
S
Mu
0.012
0.0015
0.001
0.01
0.0005
0.008
0
-0.0005
0
200
400
600
800
1000
1200
1400
0.006
-0.001
0.004
-0.0015
-0.002
-0.0025
-0.003
0.002
0
0
200
400
600
EXHIBIT 4: Estimated parameters for HET as a function of time
horizon
Lognormal Distribution
Sigma
Mu
0.02
0.003
0.018
0.0025
0.016
0.014
0.002
0.012
0.0015
0.01
0.008
0.001
0.006
0.004
0.0005
0.002
0
0
0
200
400
600
800
1000
1200
1400
0
200
400
600
800
1000
1200
1400
800
1000
1200
1400
Logistic Distribution
Mu
S
0.012
0.003
0.0025
0.01
0.002
0.008
0.0015
0.006
0.001
0.004
0.0005
0.002
0
0
0
200
400
600
800
1000
1200
1400
0
200
400
600
EXHIBIT 5: Results of the Chi-Square test
Ticker Symbol
APD
BMC
EK
FNM
HET
JNJ
KR
LMT
MER
MMM
MRK
NYT
ORCL
PEP
PNC
QLGC
RSH
SLE
TXN
UNM
Logistic
Normal
Logistic Score/
Normal Score
18.67
14.36
20.87
28.62
55.81
19.38
15.24
14.98
21.05
21.93
10.93
20.61
26.42
20.26
24.83
21.58
33.63
28.26
25.98
44.1
35.92
43.49
41.82
50.62
95.85
31.26
28
20.78
24.22
66.1
36.89
67.51
40.06
13.57
34.51
70.42
101.9
63.55
20.87
120.1
0.520
0.330
0.499
0.565
0.582
0.620
0.544
0.721
0.869
0.332
0.296
0.305
0.660
1.493
0.720
0.306
0.330
0.445
1.245
0.367
EXHIBIT 6: Simulation results
Ticker
Symbol
APD
BMC
EK
FNM
HET
JNJ
KR
LMT
MER
MMM
MRK
NYT
ORCL
PNC
PEP
QLGC
RSH
SLE
TXN
UNM
Lognormal
Actual Close
72.83
30.93
22.6
53.6
83.91
60.71
26.57
98.14
79.22
75.8
43.27
24.46
16.88
69.73
62.69
17.01
25.18
16.47
31.67
21.84
Average close
73.086
35.073
25.874
58.834
82.465
66.480
24.159
97.245
94.837
79.702
46.287
22.283
18.205
74.659
65.023
22.485
16.174
16.339
30.817
21.147
Standard
Deviation
6.971
5.252
3.865
7.474
10.770
4.351
2.568
7.910
8.809
7.126
5.525
2.229
2.284
5.967
3.834
3.405
2.817
1.516
4.424
2.728
Logistic
Average close
73.100
35.052
25.857
58.853
82.380
66.476
24.153
97.249
94.784
79.696
46.284
22.293
18.185
74.679
65.016
22.516
16.162
16.332
30.812
21.167
Standard
Deviation
7.079
5.098
3.714
7.564
10.062
4.311
2.510
7.966
8.330
7.121
5.455
2.344
2.093
6.209
3.671
3.637
2.752
1.459
4.371
2.908
EXHIBIT 7: Relative errors
Ticker
Symbol
APD
MER
MMM
TXN
QLGC
RSH
FNM
PNC
ORCL
JNJ
KR
NYT
PEP
UNM
HET
SLE
LMT
MRK
BMC
EK
Percentage Error
Normal
Logistic
0.352%
0.371%
13.394%
13.328%
14.485%
14.411%
9.765%
9.801%
1.722%
1.823%
9.505%
9.498%
9.074%
9.098%
0.911%
0.908%
19.713%
19.646%
5.147%
5.140%
6.973%
6.966%
8.899%
8.859%
7.848%
7.734%
7.069%
7.097%
3.722%
3.710%
32.186%
32.370%
35.767%
35.812%
0.796%
0.838%
2.692%
2.709%
3.171%
3.081%
EXHIBIT 8: Option Prices for PNC
$65 Option
10
9
8
7
6
BS Price
5
Logistic Price
4
3
2
1
0
0
200
400
600
800
1000
1200
1400
$70 Option
6
5
4
BS Price
3
Logistic Price
2
1
0
0
200
400
600
800
1000
1200
1400
$75 Option
3
2.5
2
BS Price
1.5
Logistic Price
1
0.5
0
0
200
400
600
800
1000
1200
1400
EXHIBIT 9: Option Prices for EK
$20 Option
5
4.5
4
3.5
3
BS Price
2.5
Logistic Price
2
1.5
1
0.5
0
0
200
400
600
800
1000
1200
1400
$25 Option
1.4
1.2
1
0.8
BS Price
0.6
Logistic Price
0.4
0.2
0
-0.2
0
200
400
600
800
1000
1200
1400
$30 Option
0.25
0.2
0.15
BS Price
0.1
Logistic Price
0.05
0
0
-0.05
200
400
600
800
1000
1200
1400
EXHIBIT 10: Option Prices for HET
$80 Option
25
20
15
BS Price
Logistic Price
10
5
0
0
200
400
600
800
1000
1200
1400
$85 Option
16
14
12
10
BS Price
8
Logistic Price
6
4
2
0
0
200
400
600
800
1000
1200
1400
$90 Option
12
10
8
BS Price
6
Logistic Price
4
2
0
0
200
400
600
800
1000
1200
1400
EXHIBIT 11: Strategy testing (stocks 1 to 10)
Underlying Stock
QLogic Corp.
Air Products & Chemicals
BMC Software
Eastman Kodak
Fannie Mae
Harrah's Entertainment
Johnson & Johnson
Kroger Co.
Lockheed Martin Corp.
Merrill Lynch
Strike
15
20
25
65
70
75
25
30
35
20
25
30
55
60
65
75
80
85
60
65
70
20
25
30
85
90
95
85
90
95
Logistic Price
12/14/2006
8.3250719
3.4541144
0.7187793
9.8078228
5.4713305
2.3489725
10.4804100
5.4744978
1.7176227
6.5474872
2.3820630
0.5521874
5.9524498
2.9866335
1.1229813
8.9700918
5.7557946
3.3127546
7.1111816
2.9650053
0.6092022
5.5805262
1.1953106
0.0356172
11.3255301
6.6579942
3.2841304
11.5236764
7.4013063
4.4714900
BS Price
12/14/2006
2.8184743
0.3361454
0.0121160
10.7962780
6.5311279
3.2539296
7.0764457
3.0115787
0.7937614
3.6526040
0.7655331
0.0693375
3.1563588
1.2533153
0.4021411
12.6694216
8.7219594
5.5240497
3.7099726
0.9141198
0.0996975
7.4503612
2.8592697
0.3558963
16.8984848
12.2374449
7.9459124
1.9616352
0.7132573
0.2100570
Action
0
0
0
-1
-1
-1
0
0
0
0
0
0
0
0
0
-1
-1
-1
0
0
0
-1
-1
-1
-1
-1
-1
0
0
0
Spot Price
3/14/2007
17.01
17.01
17.01
72.83
72.83
72.83
30.93
30.93
30.93
22.6
22.6
22.6
53.6
53.6
53.6
83.91
83.91
83.91
60.71
60.71
60.71
26.57
26.57
26.57
98.14
98.14
98.14
79.22
79.22
79.22
Result
0.00
0.00
0.00
2.97
3.70
3.25
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3.76
4.81
5.52
0.00
0.00
0.00
0.88
1.29
0.36
3.76
4.10
4.81
0.00
0.00
0.00
EXHIBIT 12: Strategy testing (stocks 11 to 20)
Underlying Stock
3m Company
Merck & Co.
New York Times
Oracle Corp.
PNC Financial Services
PepsiCo Inc.
RadioShack Corp
Sara Lee Corp.
Texas Instruments
Unum Group
Strike
75
80
85
40
45
50
20
25
30
15
20
25
70
75
80
55
60
65
10
15
20
10
15
20
25
30
35
15
20
25
Logistic Price
12/14/2006
4.8666305
2.0000190
0.4805810
6.3122052
2.3698689
0.4155344
2.6001378
0.2032920
0.0000000
3.9268263
0.3733097
0.0000000
6.5957473
3.0201317
1.0689812
9.3481717
4.3612469
1.1111948
6.8931624
2.1665014
0.2096187
6.7349227
1.6054216
0.0245763
6.5742976
2.5564113
0.5795964
5.8871064
1.4017844
0.1078731
BS Price
12/14/2006
5.0824368
2.2757005
0.7910428
5.3653328
2.1024171
0.5559488
5.3450918
1.2742422
0.0640870
2.6027715
0.1458905
0.0008011
3.8038325
1.3363989
0.3202226
10.1109032
5.4116316
1.6900654
15.6200252
10.8403362
6.1350485
6.9100252
2.1677026
0.0329141
7.7981124
3.5591784
1.0289760
7.5002767
2.9116368
0.3989121
Action
-1
-1
-1
0
0
-1
-1
-1
-1
0
0
-1
0
0
0
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
Spot Price
3/14/2007
75.8
75.8
75.8
43.27
43.27
43.27
24.46
24.46
24.46
16.88
16.88
16.88
69.73
69.73
69.73
62.69
62.69
62.69
25.18
25.18
25.18
16.47
16.47
16.47
31.67
31.67
31.67
21.84
21.84
21.84
Result
4.28
2.28
0.79
0.00
0.00
0.56
0.89
1.27
0.06
0.00
0.00
0.00
0.00
0.00
0.00
2.42
2.72
1.69
0.44
0.66
0.96
0.44
0.70
0.03
1.13
1.89
1.03
0.66
1.07
0.40