and thus the density, or PDF
is,
The cumulative distribution
function for the lognormal with parameters mu and delta is,
where N(* ) is the cumulative
normal density function. The lognormal density is,
No-Arbitrage option pricing models are used to recover implied probability distributions of underlying futures prices. In contrast to standard implied volatility approaches, all the parameters of the ex ante price distribution are simultaneously recovered without using the corresponding underlying asset price. Tests of empirical moments suggest that the Burr III distribution may serve as a useful candidate parameterization of price distributions. To assess the value of moving to more flexible specification of the underlying price distribution, comparisons between the lognormal and the Burr III distributions are performed. The Burr III distribution generally exhibits lower pricing errors and more accurately characterizes ex ante distributions than the lognormal, although both distributions perform well.
Despite its widespread application, the Black-Scholes model suffers from the imposition of a potentially restrictive price diffusion process and resulting end-of-period distribution that is frequently empirically rejected. Rubinstein (1994) also characterizes Black-Scholes type models as being increasingly unreliable and requiring increasingly sophisticated mechanisms for dealing with their known biases. He argues that financial economists should consider more flexible distributional specifications whose parameters could be simultaneously estimated from the complex of option strikes that trade on any given day. Rubinstein (1994) also questions the standard distributional assumptions made and notes that there have only been limited efforts to extract probabilistic information from option prices using more general distributional characterizations.
Under no-arbitrage restrictions, Cox and Ross (1976) derive a general asset pricing theory in which they show that the current price of any asset is equivalent to the stream of expected payoffs discounted to the present at the appropriate rate. Under fairly general conditions, expectations can be made with respect to an artificial distribution, called the risk-neutralized valuation measure. However, except for requiring the prices to be non-negative, economic theory provides little guidance regarding the distributional form of RNVM. In practice, the appropriate representation of price distributions is largely an empirical issue. The lognormal distribution is frequently used because of its relative simplicity and its correspondence to the Black and Black-Scholes option pricing models. However, because of its known problems, other distributions may be useful in characterizing expected prices.
Consistent with the spirit of Rubinstein s comments, this paper employs a straightforward technique for simultaneously recovering all the parameters of an expected RNVM from options data alone, without the use of the underlying asset price. The procedure searches for probabilities which are most consistent with observed option premiums over multiple strike prices. The process is undertaken with two candidate parameterizations of the probability measure, the lognormal and the Burr III, using daily data on options on soybean futures for all contracts that traded over the period 1986-91. Comparisons of the implied distributions are made between alternate characterizations and through time to assess the evolution of information and uncertainty about the underlying asset. Because the estimation makes no use of the underlying asset prices, important comparisons also can be made between the options and the underlying asset market regarding information contained in their respective prices.
The paper is organized as follows. First, a brief discussion of option pricing theory under no-arbitrage conditions is provided. Next, data and methods are discussed including the rationale for the choice of Burr type III distribution as an alternative candidate distribution. Results are then presented including: comparisons of pricing errors under alternate parameterizations, comparisons of the location of the implied distributions to the futures prices, and comparisons of implied volatility to the resulting price variability. Summary remarks and suggestions for future research then follow.
where Vp and Vc are the
prices of put and call options respectively expiring at time T, xp and
xc are the strike prices for puts and calls respectively, YT is the random
price of the underlying asset at expiration, b(T) is the discount factor,
and g(.) is the probability density function of the asset price at maturity.
This approach for option pricing does not require particular assumptions
about the dynamics of price change, or other information about the time
path of prices. The only required condition is that there are no arbitrage
opportunities in the complex of market. In this context, no-arbitrage is
simply the condition that any two portfolios with identical distributions
of future payoffs have identical current prices. Under this approach, the
assumption of a lognormal RNVM results in option pricing formulas for puts
and calls on futures contracts which are equivalent to the familiar option
pricing formulas derived by Black and Scholes and others (Fackler and King,
1990; Garven, 1986).
Knowledge of the RNVM and the discount rate are sufficient to compute option premiums. Conversely, given observed option premiums and a discount rate, a RNVM can be recovered. The only requirements for the latter are an assumption regarding the distributional characterization for the underlying asset prices and an estimation criterion. The most common parameterization of the RNVM is lognormal. However, several authors have pointed out that lognormality is not an accurate description of some price distributions and have suggested the use of alternative candidate distributions, or mixtures of distributions (Hall, Brorsen, and Irwin, 1989; So, 1987). Stable Paretian, Burr XII, Burr III, gamma, Weibull and exponential are among the several alternative candidate distributions suggested in the literature (Fackler, 1990; McCulloch, 1978; Sherrick, Irwin and Forster, 1992). Others have suggested modifications to the data generating process to impose conditional dependence or otherwise admit non- independent price changes (Myers and Hanson, 1993). Nevertheless, the use of lognormality is appealing because of its consistency with the widely used equilibrium option pricing models and because of its relative simplicity. In this study, the performance of the candidate distribution which seems to be most representative of the data is tested against the lognormal which serves as a convenient benchmark.
The futures data were analyzed using moment-ratio diagrams of the observed data as suggested by Rodriguez (1977). Moment-ratio diagrams are useful for preliminary analyses of data and can help identify suitable distributions to characterize sample data. Essentially, these diagrams, drawn in the skewness and kurtosis plane, indicate boundaries and regions of relative moments admitted under different distribution parameterizations. For example, the skewness and kurtosis of a normal distribution are 0 and 3 respectively. Hence, the normal distribution plots as a single point in moment- ratio diagrams. In contrast, Pearson distributions admit a wide array of relative moments, and hence, occupy a region rather than a point in the skewness-kurtosis plane. Plotting empirical moments highlights data characteristics and identifies distributions that are unable to generate similar data. Thus, although the procedures do not prescribe a single acceptable distribution, they do provide some evidence about distributions that are inconsistent with sample moments.
Table II provides results of the examinations of the daily soybean futures price data both in detrended levels and log relative returns for each contract. The moment- ratio diagrams indicate that data in detrended levels fall in the regions with positive skewness and varying kurtosis, but often outside the range admitted by any distribution as a set of independent draws from a single distribution. As a complementary test, normality is rejected using Jarque-Bera's statistic in all cases. Under returns (changes in log prices), normality was rejected for about a third of the 22 contracts, further indicating that the assumption of lognormality of prices may introduce inaccuracies. The implication of these results are that higher moment flexibility is a desirable property when modeling soybean futures price data.
Previous studies, as well as the moment-ratio charts, suggest the use of a distribution which allows a wide range of skewness and kurtosis values. Because much of the data fall in the space covered by Burr type III distribution, it is chosen as the alternate distribution to compare to the lognormal benchmark.
Although Burr distributions have received limited attention in modeling of prices, they have been shown to be useful in describing other economic data with zero support, particularly for loss distributions in the insurance industry. Further, it is important to note that Burr III covers all the space regions in the skewness-kurtosis plane occupied by Pearson types IV, VI, and bell-shaped curves of Pearson type I, gamma, Weibull, normal, lognormal, exponential, and logistic distributions. Additional details about this distribution are provided in Tadikamalla (1980).
The Burr-III cumulative distribution function (CDF) with parameters
alpha, lamda,and tau is:
and thus the density, or PDF
is,
The cumulative distribution
function for the lognormal with parameters mu and delta is,
where N(* ) is the cumulative
normal density function. The lognormal density is,
Using the no-arbitrage pricing model described in (1) and (2), observed option premia are used to recover the parameters of the candidate price distributions by numerically searching for the values of parameters that result in implied option premia closest to the observed option premia. Intuitively, this procedure is similar to that of recovering an implied volatility except that the dimension of the choice variable vector corresponds to the number of parameters of the candidate distribution (two for lognormal distribution and three for the Burr III distribution).
The implied set of distributional parameters under the distributional
assumptions were obtained by minimizing squared error between observed
and model option premia. Specifically, the following equation was solved
for each day's data and for each contract to obtain the parameter vector,
, using the lognormal and Burr type III distributions for g(YT),
This procedure simply minimizes the sum of squared differences between the model premia conditional upon the parameters of the distribution and the observed option premia. Daily samples of "m" puts and "n" calls were used subject to the requirement that (m+n) be greater than the number of parameters of each of the distributions. Equation (7) was solved using data from one day at a time using non- linear least squares optimization methods1. To keep the problem computationally manageable, the process was restricted to no more than 150 trading days of each contract's life, or the entire history for contracts that were active for less than 150 days. Thus, implied distributional parameters were obtained for each day under both Burr III and lognormal distributional assumptions for each of the twenty two contracts. Figures 1 and 2 display a sample of the distributions as implied in option prices for the November 1990 soybeans futures contract at several points in the contract life under the Burr III and lognormal parametizations, respectively.
Summary statistics of the option pricing errors are reported in III. Although daily results are available, the findings are reported over roughly monthly intervals for brevity. The entries in Table III are calculated as follows. For each contract, the pricing errors between observed and implied option prices across all strikes are computed under both distributions. The average option pricing error is then computed as the unweighted average of the mean absolute difference between the option premia and the implied option premia over all contracts. Then, for each interval, the mean and variance of these average errors is calculated. The number of contracts for which each distribution registered lower average pricing errors is reported in the table. For some cases, the total number of contracts is less than 22 where insufficient trading activity existed.
The magnitudes of the daily errors for both distributions are small, averaging only 0.14 and 0.16 cents under Burr III and lognormal assumptions, respectively. As expected, the magnitude of the pricing error decreases as time to maturity decreases. The Burr III has lower average pricing errors closer to maturity, the period of most heavy trading. The lognormal distribution performs slightly better for intervals of 120- 150 days to maturity. While the Burr displays more cases with a lower average pricing errors, the economic significance of the improvement needs to be assessed to justify the complication of moving to a more flexible distribution. On the other hand, if the user finds the movement to a more flexible distribution to be fairly low cost, there are many cases which result in smaller average pricing errors.
These results permit at least two interpretations. If the current futures price is thought to be an unbiased estimate of a future price, the option-implied means are unbiased as well; or, if the current futures price is somehow otherwise related to a future price, the options markets likewise contain similar information. To illustrate graphically, Figure 3 depicts the close relationship between the futures prices and the daily implied means for the November 1990 contract.
For each day, the implied variance for the remaining interval until expiration is computed for both distributions and compared to the actual variance of resulting futures prices for the same interval. Absolute differences are calculated between the lognormal estimate and the realized variance and between the Burr III and the realized variance. The distribution with the smaller absolute difference is then recorded as the closer candidate for that interval. The results across all contracts are tabulated in Table V over the same intervals presented earlier. Burr III substantially outperforms the lognormal in tracking variance of the underlying price distribution across different time intervals. Figure 4 graphically depicts these results demonstrating that the more flexible Burr III quite often outperforms the lognormal at depicting ex ante price variability. Note that although the Burr III is "better" than the lognormal in a vast majority of cases, the results do not imply that either are necessarily "good" indicators of resulting price behavior. Nevertheless, the relative improvement by the Burr III identifies the potential value in using more flexible distributions when higher ordered moments are of interest.
A series of tests are performed to compare the two functional forms including: evaluation of average pricing errors; comparisons of implied mean prices with futures prices; and assessment of the predictability of resulting futures price variability. In the first two cases, there is at least evidence of marginal improvement in moving to the more flexible specification. In depicting ex ante price variability, the Burr III substantially outperforms the lognormal. These benefits come at the cost of the loss of some convenience in estimation and the loss of familiarity with more traditional models.
Implications of the results depend on the context of the use of the information. For example, if an estimate of the implied futures prices were needed during a day with a futures price limit-move, but during which options continue to trade, either distribution will likely give very accurate results in recovering an implied futures price. However, for higher moment estimates, more flexible distributions are just that -- more flexible and are less likely to have induced inaccuracies in estimating higher moments. Thus, on that same limit-move day, a more flexible distribution will probably provide a more accurate estimate of the remaining volatility for the life of the contract.
The merits of moving toward more flexible distributional specifications appear greatest under conditions that are likely to involve more complicated information. In the present context, the greatest likelihood of detecting additional economic value of more flexible specifications is under conditions with large amounts of information impacting the market-implied distributions. These conditions exist during heavily traded intervals close to maturity when much probabilistic information is summarized in option prices.
These procedures and the implied distributions provide a rich background for additional work. Future tests should consider the use of trading strategies based on differences in implied distributions and the data. For example, the data indicate that the options-based estimates of variance are often too low far from maturity. Hence, a strategy of buying option straddles far from maturity may prove profitable. Further research also should consider other distributional candidates and procedures for matching data characteristics to distributions for use in option pricing models.
