Two Processes for Two Prices

Abstract

Postulating additivity of bid and ask prices for claims comonotone with a long or short stock position, two pricing processes are identified from data on bid and ask prices for options. It is observed that there are two separate put call parity relations in place, with the ask price for call less bid prices for put delivering an ask price for the forward stock. Likewise, the ask for puts less the bid for calls identifies the bid for the forward stock. Two processes are introduced to determine bid and ask prices for claims comonotone with a long or short position in the stock. For a claim comonotone with a long position, one uses the so-called increasing process for the ask price and the so-called decreasing process for the bid price, and vice versa for a claim comonotone with a short position. As candidates for the two processes, one may employ any of the traditional one dimensional Markov processes. We illustrate the theory by using a Sato process, a model known to produce a smile conforming fit over strike and maturity. The two processes are observed to have marginals related by first order stochastic dominance. The increasing process dominates the decreasing process in this sense. These two processes are also used to construct upper and lower bounds for bid and ask prices for claims not comonotone with a long or short stock position. The two processes and their properties are illustrated with data on bid and ask prices for options on the exchange traded fund, SPY.