Standard & Poors Index 500
By Steven J. Grisafi, PhD.

In two previous posts of research articles, Derivatives Pricing and Derivative Dynamics, the Black-Scholes-Merton derivatives pricing models were reformulated to eliminate the unknown initial condition of the boundary value problem. A new Option Contracts analysis is has been developed and will be continued daily each business day that the U.S. Department of Treasury reports the four week treasury bill interest rate. This interest rate is used as the risk free interest rate in the analysis of the Black-Scholes-Merton derivatives pricing model. In our development of the pricing model we showed how to make use of only quantities known for certain at time zero when the model is applied. Proper usage of the model requires particular attention to the use of consistent time units for the two free parameters: the risk free interest rate and the variance of prices for the underlying financial security upon which the option contract is based. This means that one must choose an interest rate with the same time span upon which the price variance of the underlying security is measured. Choosing the four week treasury bill interest rate compels the analysis to evaluate the price variance over the same four week time period recorded for the interest rate. Usually this means that one cannot rely upon parameter values reported in news media but must evaluate those parameters one’s self. We choose the four week treasury bill interest rate, and not a longer interest rate time period, such as a customary annualized interest rate, because of the need to keep our time periods short due the inconstancy of both interest rates and price variance.

To show the utility of the modified pricing model we select the Standard & Poors Index 500 as the underlying security upon which one may purchase an option contract. So now one may ask: Of what utility is the modified pricing model? The pricing model serves the very useful purpose of indicating just how much one should pay for an option contract. Whether one is purchasing a option to sell the underlying security at a premium to the current price, or to buy the underlying security at a discount to the current price, the pricing model shows in normalized fashion what the value of the contract is worth both at the moment the contract is purchased and as time progresses.

The results of analysis are presented in tabular form within three tables. The first table, the Four Week Rolling S&P500 Statistics table, presents the variance, standard deviation, and mean for the S&P500 index each day at the end of a four week rolling time period. The fifth column in the table shows the normalized variance, that means the variance displayed as a Gaussian statistic. The sixth column of the table shows the time constant θ which measures the first order exponential decay of the initial value of the option contract. That value is based upon time measured in four week intervals. The seventh column shows the value of θ measured in one week intervals. We choose not to break apart the time interval into days because of the ambiguity as to whether one should choose a five day business week or a seven day calendar week. Although business may not be conducted on weekends and on holidays, an option contract still remains in force on those days as well.

The second, Normalized 10% Premium Contract Values, and third, Normalized 10% Discount Contract Values, tables present the value of the contract at time zero when the contract is enacted and one week afterward for both a 10% premium contract and a 10% discount contract, respectively. The values of the contracts are normalized to the value of the underlying security when the contract is placed. Examining all three tables one can see that the decay of value for a contract depends significantly upon the price variance of the underlying security. When the variance is large, the time constant θ is large which causes a more significant decay in the value of the contract after one week. Let the option contract buyer beware. These three tables ought be constructed whenever an investor seeks to gamble with options.