The Log-Logistic Option Pricing Model
Muhannad R. Al Najjab Aurélie Thiele
Graduate Student P.C. Rossin Assistant Professor Lehigh University Lehigh University 200 W Packer Ave Rm 421 200 W Packer Ave Rm 329 Bethlehem PA 18015 USA Bethlehem PA 18015 USA firstname.lastname@example.org email@example.com
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  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 Log-Logistic 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 ):
where S is the closing price at the end of time t. Furthermore, the two parameters that t
determine a Log-Logistic distribution, μ and s, are estimated by (see Johnson et. al. ):
We obtained daily closing prices from finance.yahoo.com for five years of data, ending
thMarch 14 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 non-stationary 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
thto March 14 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
： 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 chi-square test and (2) an out-of-sample 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 .) It compares the number E of expected observations in interval i (i=1..n) to the number of observed i
occurrences O within each interval and is computed as follows: i
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(S / S) tt-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(S / S). The results motivate the choice of the Log-Logistic model as a tt-1
more accurate representation of stock prices than the traditional Log-Normal framework.
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:
where Z is standard Normal random variable, and r is the risk free rate of return Logistic Distribution:
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 ).
2. Simulation-based solution using the Log-Logistic model, for which no closed-form
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 8-
10 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.
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
rdDeGroot, Morris and Mark Schervish  Probability and Statistics (3 Edition),
thHull, John C.  Options, Futures, and Other Derivatives (4 Edition), Prentice Hall.
Johnson, Norman, Samuel Kotz and Narayanaswamy Balakhrishnan  Continuous
Univariate Distributions, vol.2, Wiley-Interscience.
EXHIBIT 1: Company list
Company Ticker Symbol3M CompanyMMM
Air Products & Chemicals APD BMC SoftwareBMC
Eastman Kodak EK
Harrah's EntertainmentHETJohnson & JohnsonJNJ
Kroger Co. KR
Lockheed Martin Corp. LMTMerck & Co. MRK
Merrill Lynch MER
New York TimesNYT
PepsiCo Inc. PEP
PNC Financial Services PNCQLogic Corp. QLGC
RadioShack Corp RSH
Texas Instruments TXN
Unum Group UNM
Sara Lee Corp. SLEOracle Corp. ORCL
EXHIBIT 2: Estimated parameters for PNC as a function of time horizon