Fractional Shares

Effect Upon Options Contracts
By Steven J. Grisafi, PhD.

In the previous two 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. In the second research article, Derivative Dynamics, it was found necessary to make one further assumption added to the body of original assumptions required in the development of the pricing models. We found it necessary to assume that the spatial gradient of the unknown distribution function vanishes at least as fast as the square of the security price as it approaches infinity. Here we wish to expand upon the logic of that assumption.

For there to be any price change to the value of any financial security there must occur a financial transaction. Every financial transaction must include a buyer and a seller. This implies that there exists a bid price and an asking price. The product of the bid price and the asking price of any financial transaction constitutes the square of the security price within our assumption. In the real world of finance no security price can ever reach infinity because there can be no buyer at such a price. So we know that the spatial gradient of the unknown distribution function for the financial derivative price will terminate at some finite value for the price of its underlying security. What is important is that the approach to this finite boundary will be proportional to the square of the security price.

But we must now address the phenomenon of fractional shares. Often it now occurs that the price for one share of stock of highly desirable companies becomes so large that an individual investor cannot purchase shares of this particular stock while still maintaining a diversified portfolio. Consequently, brokerage houses enable their clients to purchase fractional shares of such stock. This differs little from the practise of a brokerage house holding whole shares of stock for an individual investor in the “street name.” So one would think that the practise of offering for sale fractional shares of stock ought not alter the dynamics for the trading of derivative contracts. We can be certain that the practise serves to promote the ability of the highly desirable stocks to reach even higher prices. But other than that it should not alter the approach of the gradient to the ultimate finite share price for a stock. Regardless of whatever accounting rules they may employ, the brokerage house is one party to the transaction with their client serving as the other.